theoretical manual
TRANSCRIPT
RReeccuurrDDyynn// SSoollvveerr
TThheeoorreettiiccaall MMaannuuaall
Copyright © 2012 FunctionBay, Inc. All rights reserved
User and training documentation from FunctionBay, Inc. is subjected to the copyright laws of the
Republic of Korea and other countries and is provided under a license agreement that restricts
copying, disclosure, and use of such documentation. FunctionBay, Inc. hereby grants to the licensed
user the right to make copies in printed form of this documentation if provided on software media,
but only for internal/personal use and in accordance with the license agreement under which the
applicable software is licensed. Any copy made shall include the FunctionBay, Inc. copyright notice
and any other proprietary notice provided by FunctionBay, Inc. This documentation may not be
disclosed, transferred, modified, or reduced to any form, including electronic media, or transmitted or
made publicly available by any means without the prior written consent of FunctionBay, Inc. and no
authorization is granted to make copies for such purpose.
Information described herein is furnished for general information only, is subjected to change
without notice, and should not be construed as a warranty or commitment by FunctionBay, Inc.
FunctionBay, Inc. assumes no responsibility or liability for any errors or inaccuracies that may
appear in this document.
The software described in this document is provided under written license agreement, contains
valuable trade secrets and proprietary information, and is protected by the copyright laws of the
Republic of Korea and other countries.
UNAUTHORIZED USE OF SOFTWARE OR ITS DOCUMENTATION CAN RESULT IN CIVIL
DAMAGES AND CRIMINAL PROSECUTION.
Edition Note
This theoretical manual documents the theoretical background of the RecurDyn /
Solver.
RecurDyn/Solver THEORETICAL MANUAL
Registered Trademarks of FunctionBay, Inc. or Subsidiary
RecurDyn is a registered trademark of FunctionBay, Inc.
RecurDyn/Professional, RecurDyn/ Modeler, RecurDyn/Solver, RecurDyn/ProcessNet,
RecurDyn/AutoDesign, RecurDyn/CoLink, RecurDyn/FFlex, RecurDyn/RFlex,
RecurDyn/Linear, RecurDyn/CATIA Read & Write, RecurDyn/EHD, RecurDyn/ECFD_EHD,
RecurDyn/Control, RecurDyn/Hydraulic, RecurDyn/MeshInterface, RecurDyn/Bearing,
RecurDyn/Driver, RecurDyn/Spring, RecurDyn/Tire, RecurDyn/Belt ,RecurDyn/Chain,
RecurDyn/Crank, RecurDyn/Engine, RecurDyn/Gear, RecurDyn/HAT, RecurDyn/MTT2D,
RecurDyn/MTT3D, RecurDyn/Piston ,RecurDyn/Track_HM, RecurDyn/Track_LM,
RecurDyn/TimingChain, RecurDyn/Valve are trademarks of FunctionBay, Inc.
Third-Party Trademarks
Windows and Windows NT are registered trademarks of Microsoft Corporation.
ProENGINEER and ProMECHANICA are registered trademarks of PTC Corp. Unigraphics and I-
DEAS are registered trademark of UGS Corp. SolidWorks is a registered trademark of SolidWorks
Corp. AutoCAD is a registered trademark of Autodesk, Inc.
CADAM and CATIA are registered trademark of Dassault Systems. FLEXlm is a registered
trademark of GLOBEtrotter Software, Inc. All other brand or product names are trademarks or
registered trademarks of their respective holders.
Revision History
First printed, April 2001
1st Revision, January 2002
2nd
Revision, July 2002
3rd
Revision, August 2002
4th
Revision, September 2003
5th
Revision, September 2005
6th
Revision, February 2009
7th
Revision, April 2011
TABLE OF CONTENTS
I. EQUATION OF MOTION ………………………………………… 1
1. EQUATION OF MOTION…………………………………………………… 2
1.1 GENERALIZED RECURSIVE FORMULATION……………………………… 3
1.2 DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION………………………………………………………………
29
1.3 LINEARIZED EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS WITH
CLOSE LOOPS……………………………………………………………………
42
1.4 STATIC EQUILIBRIUM ANALYSIS OF MULTI PHYSICS SYSTEM………………...………………...………………...………………
61
2. CONTACT………………………………………………………….….….… 69
2.1 AN EFFICIENT CONTACT SEARCH ALGORITHM FOR GENERAL
MULTIBODY SYSTEM DYNAMICS …………………………………………
70
2.2 AN EFFICIENT AND ROBUST CONTACT ALGORITHM FOR A COMPLIANT CONTACT FORCE MODEL BETWEEN BODIES OF COMPLEX GEOMETRY ………………………………………………………………
88 2.3 A STUDY ON THE STICK AND SLIP ALGORITHM IN CONTACT
PROBLEMS OF MULTIBODY SYSTEM DYNAMICS………………………………………………………………
118 II. IMDD…………………………………………………………
1. MFBD………………………………………………………..…..…..…..… 133
1.1 FFLEX………………………………………………………………… 134
132
RecurDyn / Solver THEORETICAL MANUAL
1.1.1 RELATIVE NODAL METHOD FOR LARGE DEFORMATION PROBLEM………………………………………………………
135
1.1.2 MULTI FLEXIBLE BODY DYNAMICS USING INCREMENTAL
FINITE ELEMENT FORMULATION………………………………
154
1.2 RFLEX……………………………………………………………… 173
1.2.1 FLEXIBLE MULTIBODY DYNAMICS USING A VIRTUAL BODY AND
JOINT…………………………………………………………….
174 1.2.2 GENERALIZED RECURSIVE FORMULATION FOR FLEXIBLE
MULTIBODY DYNAMICS…………………………………………
198 1.2.3 RELATIVE NODAL METHOD FOR LARGE DEFORMATION
PROBLEM………………………………………………………
223 2. OPTIMIZATION………………………………………………………..….
239
2.1 DFSS AND ROBUST OPTIMIZATION OF A PAPER FEEDING
MECHANISM…………………………………………………………. 240
2.2 THE ROBUST DESIGN OPTIMIZATION OF HIGH MOBILITY TRACKED VEHICLE SUSPENSION SYSTEM……………………………
253
2.3 EFFICIENT DESIGN OPTIMIZATION TOOL FOR INTERDISCIPLINARY ANALYSIS SYSTEM ………………………………………. …….. ……
2.4 EFFICIENT OPTIMIZATION METHOD FOR NOISY RESPONSES OF MECHANICAL SYSTEMS………………………………………. ……..
2.5 ROBUST DESIGN OPTIMIZATION OF THE MCPHERSON SUSPENSION SYSTEMWITH CONSIDERATION OF A BUSH COMPLIANCE UNCERTAINTY………………………………………. …….. …….. ..
268
288
301
3. MECHATONICS………………………………………………………..….
3.1 A CASE STUDY OF MECHATRONIC SYSTEM SIMULATION: FORKLIFT
ELECTRONIC CONTROL……………………………………………….
3.2 THE INTER-DISCIPLINARY SIMULATION ENVIRONMENT INCLUDING
THE FIRMWARE AND THE MECHANICAL SYSTEM…………………….
III. APPLICATION……………………………………….............
1. TRACK VEHICLE…………………………………………………………
1.1 DYNAMIC ANALYSIS OF HIGH-MOBILITY TRACKED
VEHICLES……………………………………….……………………
1.2 DYNAMIC TRACK TENSION OF HIGH MOBILITY TRACKED
VEHICLES……………………………………….……………………
1.3 EFFICIENT CONTACT AND NONLINEAR DYNAMIC MODELING FOR
TRACKED VEHICLES……………………………………….…………
2. CHAIN………………………………………………………………………
2.1 NONLINEAR DYNAMIC MODELING OF SILENT CHAIN
DRIVE……………………………………….……………………
2.2 THE RESERCH OF MULTIPLE AXES CHAIN COUPLER METHOD FOR
AUTOMOTIVE ENGINE SYSTEM………………………………….…
2.2 SYSTEMETIC ENVIRONMENT CONSTRUCTION FOR EFFICIENT TIMING
CHAIN ANALYSIS OF MOTORCYCLE’S ENGINE………………………
3. BELT………………………………………………………………………
3.1 HYDRAULIC AUTO TENSIONER (HAT) FORBELT DRIVE SYSTEM…..
310
311
325
335
336
337
369
393
410
411
428
446
461
462
RecurDyn / Solver THEORETICAL MANUAL
4. GEAR………………………………………………………………………
4.1 DYNAMIC ANALYSIS OF CONTACTING SPUR GEAR PAIR FOR FAST SYSTEM SIMULATION…………………………………………………
5. BEARING……………………………………………………………………
5.1 DYNAMIC ANALYSIS OF CONTACTING SPUR GEAR PAIR FOR FAST
SYSTEM SIMULATION…………………………………………………
5.2 NUMERICAL MODELING AND ANALYSIS OF JOURNAL BEARING WITH COUPLED ELASTOHYDRODYNAMIC LUBRICATION AND FLEXIBLE MUTIBODY DYNAMICS………………………………………………
6. MEDIA TRANSPORT SYSTEM………………………………….……...
6.1 DYNAMIC ANALYSIS AND CONTACT MODELING FOR TWO
DIMENSIONAL MEDIA TRANSPORT SYSTEM…………………………
481
482
501
502
517
530
531
Theoretical Manual for Equation of Motion
FunctionBay, Inc.
1. Equation of Motion
3
1.1
GENERALIZED RECURSIVE FORMULATION
1.1.1. INTRODUCTION
In Ref. 1, the equations of motion for the constrained mechanical systems
were derived with respect to Cartesian coordinates. Then the equations were
transformed into the corresponding ones that employ the relative coordinates by
using the velocity transformation method. Since the virtual displacement and
acceleration of the entire system were simultaneously substituted into the
variational form of the equations of motion, the resulting equations of motion
were compact. In spite of the compactness, they are not computationally efficient
since the recursive nature of the relative kinematics was not exploited.
In Ref. 2, Hooker proposed a recursive formulation for the dynamic analysis
of a satellite which has a tree topology. It was shown that the computational
cost of the formulation increases only linearly with respect to the number of
bodies. In Ref. 3, Featherstone proposed a recursive formulation to calculate the
acceleration of robot arms using screw notation. These ideas were extended by
using the variational vector calculus for constrained mechanical systems in Ref.
4.
Constrained mechanical systems are represented by differential equations of
motion and algebraic constraint equations, which are often called differential
algebraic equations (DAE). Several DAE solution methods using the BDF have
been proposed in Refs. 5-7. In particular, the parameterization method treated
the DAE as an ordinary differential equation (ODE) on the kinematic constraint
manifolds of the system. The stability and convergence of the method were
proved in Ref. 8. The present research employs this method, due to its
mathematical soundness.
In Ref. 9, a recursive formulation was presented to obtain the Jacobian in the
linearization of the equations of motion. Recursive formulas for each term in
the equations of motion were directly derived, using the state vector notation.
4
Similar approach was taken in Ref. 8 to implement the implicit BDF
integration with the relative coordinates. Since the recursive formulas were
derived term by term, the resulting equations and algorithm became much
complicated.
To avoid the complication involved in Ref. 8, the equations of motion are
derived in a compact matrix form by using the velocity transformation method in
the present study. Computational structure of the equations of motion in the joint
space is carefully examined to classify all computational operations that can be
done in a recursive way into several categories. The generalized recursive
formula for each category of the computational operations is then developed and
applied whenever such a category is encountered. Many common factors, which
are not easily observed when they are derived term by term, can be observed
among terms in the equations of motion. Furthermore, the matrix form of the
equations makes it easy to debug and understand the program while
computational efficiency is achieved by the recursive computational operation. A
library of the generalized recursive formulas is developed to implement a
dynamic analysis algorithm using the backward difference formula (BDF) and
the relative generalized coordinate.
Section 2 introduces relative coordinate kinematics. Generalization of velocity
and force recursive formulas is treated in Sections 3 and 4, respectively. Also,
computational equivalence between the recursive method and velocity
transformation method for a mechanical system is shown in Section 3. Section 5
presents a graph representation of mechanical systems. Section 6 presents the
equations of motion and a solution method for the DAE. A library of the
generalized recursive formulas are developed and applied in Sections 7 and 8.
Numerical examples are given in Section 9. Conclusions are drawn in Section 10.
1.1.2. RELATIVE COORDINATE KINEMATICS
1.1.2.1. COORDINATE SYSTEMS
Orientation of a body in Fig. 2.1 is given as
5
hgfA
333231
232221
131211
aaa
aaa
aaa
(2-1)
where f , g , and h are unit vectors along the x , y , and z axes,
respectively. The zyx frame is the body reference frame and the
ZYX frame is the inertial reference frame.
Z
X
Y
rp
r
p
x
y
z
s
o
Fig. 2.1 Coordinate systems and a rigid body
Velocities and virtual displacements of point O in the ZYX frame are
defined as
wr
(2-a)
r (2-b)
Their corresponding quantities in the zyx frame are defined as
6
wArA
wr
YT
T (3-a)
T
T
ArAr
(3-b)
1.1.2.2. RELATIVE KINEMATICS FOR A PAIR OF CONTIGUOUS BODIES
A pair of contiguous bodies is shown in Fig. 2.2. Body 1)(i is assumed
to be an inboard body of body i and the position of point iO is
1)i(i1)i(i1)i(i1)(ii sdsrr (2-4)
The angular velocity of body i in its local reference frame, using Eq. 2-3a and
defining i
T
1)(i1)i(i AAA , is
1)i(i1)i(i
T
1)i(i1)-(i
T
1)i(ii qHAwAw (2-5)
where H is determined by the axis of rotation.
zi
X
Z
Y
ri-1 r i
s(i-1)i s i(i-1)
yi-1
xi-1
zi-1
zi-1
x i-1
x i
yi
x i
z i
y i
yi-1
d(i-1)i
o ioi-1
Fig. 2.2 Kinematic relationship between two adjacent rigid bodies
7
Differentiation of Eq. 2-4, using Eq. 2-3a, yields
1)i(i
'
1)i(i1)(i
'
i
'
1)i(ii
'
1)(i
'
1)i(i1)(i
'
1)(i
'
1)i(i1)(i
'
1)(i1)(i
'
ii
1)i(i)(
~~
~
qdA
sAdA
sArArA
q
(2-6)
where symbols with tildes denote skew symmetric matrices comprised of their
vector elements that implement the vector product operation (Ref. 1) and 1)i(iq
denotes the relative coordinate vector. Substituting '
iω of Eq. 2-5 and
multiplying both sides of Eq. 2-6 by T
iA yields
1)i(i1)i(i1)i(iT
1)i(i1)i(i1)i(i1)i(iT
1)(i1)i(iT
1)-i(i1)i(i1)i(i1)i(i1)i(iT
1)(i1)i(iT
i
)~)((
)~~~(
1)i(i
qHAsAdA
AsAdsA
rAr
q
(2-7)
where iii~ AA is used. Combining Eqs. 2-5 and 2-7 yields the recursive
velocity equation for a pair of contiguous bodies.
1)i(i1)i2(i1)(i1)i1(ii qBYBY (2-8)
where
I0AsAdsI
A00A
B )~~~( 1)i(iT
1)i(i1)i(i1)i(i1)i(i
1)i(iT
1)i(iT
1)i1(i
1)i(i
1)i(i1)i(iT
1)i(i1)i(i1)i(i
1)i(iT
1)i(iT
1)i2(i
~)(1)i(i
HHAsAd
A00A
B q (2-9)
It is important to note that matrices 1)i1(iB and 1)i2(iB are functions of only
relative coordinates of the joint between bodies 1)(i and i . As a
consequence, further differentiation of the matrices 1)i1(iB and 1)i2(iB in
Eq.2-9 with respect to other than 1)i(iq yields zero. This property plays a key
8
role in simplifying recursive formulas in Section 7.
Similarly, the recursive virtual displacement relationship is obtained as follows.
1)i(i1)i2(i1)(i1)i1(iiδ δqBδZBZ (2-10)
1.1.3. GENERALIZATION OF THE VELOCITY RECURSIVE FORMULA
1
2
n-1
n
0
Fig. 3.1 A serial chain mechanism
Before proceeding to generalize the recursive velocity formula, the
computational equivalence between the recursive method and the velocity
transformation method is demonstrated using the mechanical system shown in
Fig. 3.1. The Cartesian velocity mY is obtained by replacing i by m in
Eq. 2-8.
1)m(m1)m2(m1)(m1)m1(mm qBYBY (3-1)
Substitutions of Eq. 2.8 for 1)-(mY , 2)-(mY , . . . , and 0Y yield
9
1)m-(m1)m2-(m
1-m
1j
1)j-(j1)j2-(j
j-m
1k
1)1-mm)(-kk(
m
1k
01)1-mm)(-kk(m
qB
qBB
YBY
(3-2)
Thus, the Cartesian velocity Y for all bodies is obtained as
qBY (3-3)
where B is the collection of coefficients of 1)i(iq and
T
1nc
TT
2
T
1
T
0 nY,,Y,Y,YY (3-4)
T
1nr
T
)1(
T
12
T
01
T
0 nnq,,q,q,Yq (3-5)
where nc and nr denote the numbers of the Cartesian and relative coordinates,
respectively.
The Cartesian velocity ncRY , with a given nrRq , can be evaluated either
by using Eq. 3-3 or by using Eq. 2-8 with recursive numerical substitution of iY .
Since both formulas give an identical result, and recursive numerical substitution
is proven to be more efficient in Ref. 4, matrix multiplication qB with a given
q will be evaluated by using Eq. 2-8.
Since q in Eq. 3-3 is an arbitrary vector in nrR , Eqs. 2-8 and 3-3, which are
computationally equivalent, are actually valid for any vector nrRx such that
xBX (3-6)
and
1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX (3-7)
where ncRX is the resulting vector of multiplication of B and x . As a
10
result, transformation of nrRx into ncRBx is calculated by recursively
applying Eq. 3.7 to achieve computational efficiency.
1.1.4. GENERALIZATION OF THE FORCE RECURSIVE FORMULA
It is often necessary to transform a vector G in ncR into a new vector
GBg T in nrR . Such a transformation can be found in generalized force
computation in the joint space with a known force in the Cartesian space. The
virtual work done by a Cartesian force ncRQ is
QZW Τδδ (4-1)
where Zδ must be kinematically admissible for all joints in Fig. 3.1.
Substitution of qBZ δδ into Eq. 4-1 yields
*TTT δδδ QqQBqW (4-2)
where QBQ T* . Equation 4-2 can be written in a summation form as
1-n
0i
*
)!i(i
T
1)i(iδδ QqW (4-3)
On the other hand, the symbolic substitution of the recursive virtual
displacement relationship Eq. 2-10 into Eq. 4-1, along the chain in Fig. 3.1
starting from the body n toward inboard bodies, yields
1-n
0i
1i1i
T
1)2i(i
T
1)i(iδδ SQBqW (4-4)
where
11
2i2i2)11)(i(iT
1i
0
SQBS0S
(4-5)
Equating the right sides of Eqs. 4-3 and 4-4, the following recursive formula for *Q is obtained:
0 ...., 1,-ni,1i1i)21i(iT
1)i(i* SQBQ (4-6)
where 1iS is defined in Eq. 4-5.
Since Q is an arbitrary vector in ncR , Eqs. 4-5 and 4-6 are valid for any
vector G in ncR . As a result, the matrix multiplication of GBT is
evaluated to achieve computational efficiency by
0 ...., 1,i1i1i)11i(iT
i
n
1i1i
T
1)2i(i1)i(i
nSQBS0S
SGBg (4-7)
where g is the result of GBT .
1.1.5. GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS
In the previous section, a serial chain mechanism is considered to derive
recursive formulas for Bx and GBT where x is a vector in nrR and G
in ncR . In general, a mechanical system may have various topological
structures. To cope with the various topological structures, an automatic
preprocessing is required for a general purposed program, which employs a
relative coordinate formulation. The preprocessing identifies the topological
structure of a constrained mechanical system to achieve computational efficiency.
A graph theory was used to represent bodies and joints for mechanical systems
12
in Refs. 1 and 4. A node and an edge in a graph represented a body and a joint,
respectively. The preprocessing based on the graph theory yielded the path and
distance matrices that are provided to automatically decide execution sequences
for a general purposed program. As an example, a governor mechanism and its
graph representation are shown in Figs. 5.1 and 5.2.
4 3
2
87
6
5
1
U1
R2 R3
U2
S1 S2
T2
R1
T1
: Cut joint
Fig. 5.1 Governor mechanism
7
6
4
5
3
8
2
1
R1
T1
R3
U2
R2
U1
T2
Cut JointCut Joint
Fig. 5.2 Graph representation of the governor mechanism
13
1.1.6. EQUATIONS OF MOTION AND DAE SOLUTION METHOD
The variational form of the Newton-Euler equations of motion for a
constrained mechanism is
0)QλΦYMZ ΤΖ
T ( (6-1)
where Z must be kinematically admissible for all joints except cut joints [1].
In the equation, Φ and λ , respectively, denote the cut joint constraint and the
corresponding Lagrange multiplier. The mass matrix M and the force vector
Q are defined as
nbd21 ,,,diag MMMM (6-2)
i
i
i J00Im
M (6-3)
T
nbd
T
3
T
2
T
1 ,,,, QQQQQ (6-4)
ωJωnrωmf
Q ~
~
i
(6-5)
where nbd denotes the number of bodies, I denotes the identity matrix, J denotes the moment of inertia, f denotes the external force, and n denotes
the external torque. Substituting the virtual displacement relationship into Eq.
6-1 yields
0)QλΦYMBq ΤΖ
T (T (6-6)
Since q is arbitrary, the following equations of motion are obtained:
0)QλΦYMBF ΤΖ
T ( (6-7)
The equations of motion, the constraint equations, vq , and av
constitute the following differential algebraic equations[8]:
14
0
avvqavqΦ
vqΦqΦ
)λa,vq(F
t,,,
t,,
t,
t,,,
(6-8)
Application of 'tangent space method' in Ref. 7 to Eq. 6-8 yields the following
nonlinear system that must be solved at each time step:
0
βavUβvqU
avqΦvqΦ
qΦ)λa,vq(F
pH
)β(
)β(
t,,,
t,,
t,
t,,,
)(
2n0n
T
0
1n0n
T
0
nnn
nnn
nn
nnnnn
n
n
(6-9)
where TT
n
T
n
T
n
T
nn λ,a,v,qp , 0β , 1β , and 2β are determined by the
coefficients of the BDF, and 0U is an ncut)-(nrnr such that the augmented
square matrix
qΦU T
0 is nonsingular.
Applying the Newton's method to solve the nonlinear system in Eq. 6-9 yields
HΔpHp (6-10)
1,2,3,...i,i
n
1i
n Δppp (6-11)
where
15
0UU000UU0ΦΦΦ00ΦΦ000ΦFFFF
H
T0
avq
vq
q
λavq
p
T
00
T
00
T
0
β
β
(6-12)
Since F and Φ are highly nonlinear functions of q , v , a , and λ , care
must be taken in deriving the non-zero expressions in pH , so that they can be
efficiently evaluated.
1.1.7. GENERALIZED RECURSIVE FORMULAS
Inspection of the residual H and Jacobian matrix pH shows that types of
necessary recursive formulas are classified into Bx , GBT , xB , qBx ,
qGBT , qxB , and vxB , where nrRx into ncRG are arbitrary constant
vectors, and q are relative coordinates. Formulas Bx and GBT were
derived in Sections 3 and 4, and the formulas for the rest will be derived in this
section. All recursive formulas are tabulated in Appendix A. Note that the
recursive formulas are quite simple. This simplicity is achieved by exploiting
the relative kinematics in the local reference frame instead of the global
reference frame.
To derive the formulas systematically, bodies in a graph are divided into four
disjoint sets (associated with a generalized coordinate kq ) as follows:
}coordinate dgeneralize its as havingjoint theofbody outborad{k kqqI
kk of bodies outboard all qIqII
body inboard and base theincluding
, ofbody inboard theandbody base ebetween th bodies all k
k
qIqIII
kkkk ofset ary complement the qIIIqIIqIqIV
16
For example, the body sets associated with 24q (relative coordinate between
bodies 2 and 4) for the graph shown in Fig. 5.2 are obtained as follows:
4Body 24 qI
7 and 6 Bodies 24 qII
2 and 1 Bodies 24 qIII
8 and 5 3, Bodies 24 qIV
1.1.7.1. RECURSIVE FORMULA FOR xBX
Recursive formula for ncRxB is easily obtained by differentiating Eq. 3-7.
(7-1)
This recursive formula can be applied to compute the Cartesian acceleration Y
with known relative velocity and acceleration.
1.1.7.2. RECURSIVE FORMULA FOR RM BOLD qq BxX )(
To obtain the recursive formula for qBx)( , Eq. 3-7 is partially differentiated
with respect to nr ..., 1, k,k q .
1)i(i1)2(i1-i1)i1-(i1i1)i1(ii kkk)()()()( xBXBXBX qqqqk
(7-2)
Since matrices 1)i1(iB and 1)i2(iB depend only on the relative coordinates for
joint 1)i-(i , their partial derivatives with respect to generalized coordinates
other than 1)i1(iq are zero. In other words, the partial derivatives are zero if kq
does not belong to set kqI . Therefore if body i is an element of set kqII ,
Eq. 7-2 becomes
kk)()( 1i1)i1(ii qq XBX (7-3)
1)i(i1)i2(i1i1)i1(i1i1)i1(ii XBXBXBX
17
If body i belongs to set kk qIVqIII , iX is not affected by kq . As a
result, Eq. 7-3 is further simplified as follows
0X q k
)( i (7-4)
If body i is an element of set kqI , body 1)-(i is naturally its inboard body
and it belongs to set kqIII . Using Eq. 7-4, Eq. 7-2 becomes
1)i(i1)2(i1i1)i1(ii kkk)()()( xBXBX qqq (7-5)
This recursive formula can be applied to compute the partial derivative of the
Cartesian velocity with respect to relative coordinates qY . For example, if
24k qq in Fig. 5.2, 24qY is shown in Fig. 7.1.
1.1.7.3. RECURSIVE FORMULA FOR qT
q G)(Bg
Recursive formula for qTG)(B is obtained by using the recursive formula in
Eq. 4-7. By replacing i by 1)-(i , Eq. 4-7 can be rewritten as
)(
)(
ii
T
1)i1-(i1-i
ii
T
1)i2-(i1)i-(i
SGBS
SGBg
(7-6)
Taking partial derivative of Eq. 7-6 with respect to kq yields
18
(Y1)q24
= 0
(B671)q24 = 0,
(B672)q24 = 0
(B461)q24 = 0,
(B462)q24 = 0
(B241)q24 ,
(B242)q24
(B121)q24 = 0,
(B122)q24 = 0
(B231)q24 = 0,
(B232)q24 = 0
(B351)q24 = 0,
(B352)q24 = 0
(B281)q24 = 0,
(B282)q24 = 0
(Y2)q24
= 0
(Y4)q24
=(B241)q24Y2
+(B242)q24q24
(Y3)q24
= 0
(Y8)q24
= 0
(Y5)q24
= 0
.
(Y6)q24
=
B461 (Y4)q24
(Y7)q24
=
B671 (Y6)q24
Fig. 7.1 Computation Sequence for
24qY
kkk
kkk
)()()()()(
)()()()()(
ii
T
1)i1(iii
T
1)i1(i1i
ii
T
1)i2(iii
T
1)i2(i1)i(i
qqq
qqq
SGBSGBS
SGBSGBg
(7-7)
Since ncRG is a constant vector, 0Gq k
. If . kkki qIVqIIIqII ,
1)i1(iB and 1)i2(iB are not functions of kq . Therefore their partial derivatives
with respect to kq are zero. As a result, Eq. 7-7 can be simplified to
kk
kk
)()()(
)()()(
i
T
1)i1(i1i
i
T
1)i2(i1)i(i
SBS
SBg
(7-8)
Since 0S q k
)( i for the tree end bodies, 0S q k
)( 1-i by the second equation
19
of Eq. 7-8 for kki qIVqII . Thus, for kki qIVqII , Eq. 7-8
becomes
0g q k)( 1)i(i (7-9)
If ki qI , body 1)(i belongs to set kqII , and 0S q k
)( i . Thus, Eq. 7-7
becomes
)()()(
)()()(
i
T
1)i1(i1i
i
T
1)i2(i1)i(i
kk
kk
SBS
SBg
(7-10)
For example, if 24k qq in Fig. 5.2, GB q24
T is shown in Fig. 7.2.
1
(g67)q24 = 0
(S6)q24 = 0
2
8
4
6
7
3
5
(g46)q24 = 0
(S4)q24 = 0(g35)q24 = 0
(S3)q24 = 0
(g23)q24 = 0
(S2)q24 = 0
(g28)q24 = 0
(S2)q24 = 0
(g24)q24 = (B242)q24 S4
(S2)q24 =(B241)q24 S4
(g12)q24 = (B122)( S2) q24
(S12)q24 =(B121)(S2) q24
Fig. 7.2 Computation Sequence for
2424
T
qq gGB
20
1.1.7.4. RECURSIVE FORMULAS FOR qq x)B(X AND vv xBX )(
To obtain the recursive formula for qx)B( and vx)B( , Eq. 7-1 is partially
differentiated with respect to kq and kv for nr ..., 1, k .
1)i(i1)i2(i1i1)i1(i1i1)i1(i
1i1)i1(i1i1)i1(ii
)()()(
)()()(
xBXBXB
XBXBX
qqq
qqq
kkk
kkk
(7-11)
1)i(i1)i2(i1i1)i2(i
1i1)i1(i1i1)i1(ii
)()(
)()()(
xBXB
XBXBX
vv
vvv
kk
kkk
(7-12)
The recursive formulas for qx)B( and vx)B( are obtained as in Appendix A
by following the similar steps taken in the previous sections.
1.1.8. APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS
A library of the generalized recursive formulas is developed in Section 7. This
section shows how the library can be utilized to compute the terms in H and
pH in Eqs. 6.9 and 6.12. Inspection of H and pH reveals that the residual
F and partial derivatives of qF , vF , aF , qΦ , qΦ , and qΦ need to be
computed. Only F and qF are presented in this section and the rest are
omitted for simplicity of the presentation.
1.8.1 COMPUTATION OF THE RESIDUAL F
The generalized force Q , λΦZT and the Cartesian acceleration Y need to be
computed to obtain F shown in Eq. 6-7. The term Y is obtained by applying
the recursive formula in Eq. 7.1. The recursive formula GBT with
)( T QλΦYMG Z in Eq 4.7 can be applied to evaluate F in nrR since
G is a vector in ncR .
21
1.8.2 COMPUTATION OF THE JACOBIAN qF In Eq. 6-7, differentiation of matrix B with respect to vector q results in a
three dimensional array. To avoid the complexity, Eq. 6-7 is differentiated with
respect to a typical generalized coordinate kq . Thus,
nr.....,2,1,k,)(
)(
k
kk
TT
TT
qZ
Zqq
QλΦYMB
QλΦYMBF
(8-1)
Since the term )( T QλΦZ can be easily expressed in terms of the Cartesian
coordinates, k
)( T
qZ QλΦ is obtained by applying the chain rule, as
k
TT )()(k
BQλΦQλΦ ZZqZ (8-2)
where BqZ
is used and kB denotes the kth column of the matrix B . The
resulting equation for kqF becomes
nr.....,2,1,k),)((
)(
k
TT
TT
k
kk
BQλΦYMB
QλΦYMBF
ZZq
Zqq
(8-3)
The first term in Eq. 8-3 can be obtained by applying the recursive formula for
GB qk
T , with )( T QλΦYMG Z , as explained in section 7.3. Collection of
k
T )( BQλΦ ZZ , for all k, constitutes BQλΦ ZZ )( T , which is equivalent to
TTTT ))(( ZZ QλΦB . Matrix TT )( ZZ QλΦ consists of nc columns which are
vectors in ncR . Therefore, the application of GBT , where G is each column
of matrix TT )( ZZ QλΦ , yields the numerical result of
TTT )( ZZ QλΦB . Finally,
the second term in Eq. 8-3 is also obtained by applying GBT , where
))(( k
T
kBQλΦYMG ZZq .
22
1.1.9. NUMERICAL EXAMPLES
1.1.9.1. A GOVERNOR MECHANISM
The mechanism shown in Fig. 9.1 consists of seven bodies, a spring-damper,
five revolute joints, and a translational joint. The material properties and spring
and damping constants of the system are shown in Table 9-1. The mechanism
has redundant constraints that are removed by the Gaussian elimination with full
pivoting. Consequently, it has only 2 degrees of freedom.
Dynamic analysis is carried out for 2 seconds with error tolerance of 5103
for the system. The Z acceleration of body 4 is drawn in Fig. 9.2. The result
obtained by the other commercial program and that obtained by the proposed
method are almost identical. The average step size, the numbers of residual
function evaluations and CPU time on SGI R3000 are shown in Table 9-2. The
CPU time spent by the other commercial program is about 6 times larger than
that by the proposed method. Note that the number of function evaluations of the
proposed method is smaller than that of the other commercial program.
4 3
2
5
6
1
R4
R2 R3
R5
S1 S2
R1
T17
YX
Z
0.16
0.5
0.2
0.109
45
Fig. 9.1 A governor mechanism
23
Table 9-1 Inertia properties of the governor mechanism and spring and damping
constants
Mass xI yI zI xyI yzI
zxI
Body 1 (Ground) not necessary
Body 2 200.0 25.0 50.0 25.0 0.0 0.0 0.0
Body 3 1.0 0.1 0.1 0.1 0.1 0.1 0.1
Body 4 1.0 0.1 0.1 0.1 0.1 0.1 0.1
Body 5 1.0 0.15 0.125 0.15 0.0 0.0 0.0
Body 6 0.1 0.1 0.1 0.1 0.0 0.0 0.0
Body 7 0.1 0.1 0.1 0.1 0.0 0.0 0.0
Spring constant 1000
Damping constant 30
Table 9-2 Integration output information
Program TOL Average step size No. fevals CPU time (sec)
Other 5103 2101.1 748 41
Proposed 5103 2102.1 441 7
Fig. 9.2 Z acceleration of Body 4
— PROPOSED
… OTHER
24
1.1.9.2. A MULTI-WHEELED VEHICLE
A vehicle example shown in Fig. 9.3 is chosen to show the practicality of the
proposed method. The vehicle runs over a bump whose radius is 0.3048(m).
The system consists of a chassis and twelve road wheels and arms. The
material properties and spring and damping constants are shown in Table 9-3.
The road wheel and arm are considered as a single body. As a result, the
system has 18 degrees of freedom.
Fig. 9.3 A multi-wheeled vehicle
Figure 9.4 shows the vertical acceleration of the chassis. It is shown that the
proposed method and the other commercial program yield almost identical
results. The average stepsize, number of residual function evaluations, and
CPU time on SGI R3000 are shown in Table 9-4. It can be shown that the
proposed method performs much smaller number of residual function
evaluations with larger step sizes and the CPU time by the proposed method is
much shorter than that by the other commercial program. Since there is no closed
chain in the system, the governing equations of motion are formulated as an
ODE problem by the proposed method. On the other hand, the equations of
motion by the other commercial program are formulated as an DAE problem.
The DAE problem is generally more difficult to solve than the ODE problem.
This general argument is supported by the numbers of function evaluation and
average stepsize.
25
Table 9-3 Inertial properties of the vehicle mechanism and spring and damping constants
Mass xI yI zI xyI yzI zxI
Body 1
(Ground)
not necessary
Body 2 40773.
36
231800
.0
60840
.0
251700
.0
-
863.6
234.
5
-
496.3 Body 3
~ Body 14 340.27 32.86 20.76 26.85 0.0 0.0 0.0
Spring
constant 200000
Damping
constant 40000
Table 9-4 Integration output information
Fig. 9.4 Vertical acceleration of the chassis
Program TOL. Average step size No. fevals CPU time (sec)
Other 410 3104 1359 330
Proposed 410 3106.6 1167 69
— OTHER
… PROPOSED
26
1.1.10. CONCLUSIONS
The recursive formulas are generalized in this research. The velocity
transformation method is employed to transform the equations of motion from
the Cartesian to the joint spaces. Computational structure of the equations of
motion is examined to classify all necessary computational operations into
several categories. The generalized recursive formula for each category is then
applied whenever such a category of computation is encountered. Since the
velocity transformation method yields the equations of motion in a compact form
and computational efficiency is achieved by the generalized recursive formulas,
the proposed method is not only easy to implement but also efficient. A dynamic
analysis algorithm using the backward difference formula (BDF) and the relative
generalized coordinate is implemented using the library of generalized recursive
formulas developed in this research. Numerical studies showed that obtained
solutions were numerically stable and computation time was reduced by an order
of magnitude compared to a well-known commercial program.
REFERENCES
1. J. Wittenburg, Dynamics of Systems of Rigid Bodies, B. G. Teubner,
Stuttgart, 1977.
2. Hooker, W., and Margulies, G., The Dynamical Attitude Equtation for an
n-body Satellite, Journal of the Astrnautical Science, Vol. 12, pp. 123-128,
1965.
3. R. Featherstone, The Calculation of Robot Dynamics Using Articulated-Body
Inertias, Int. J. Roboics Res., Vol 2 : 13-30, 1983.
4. D. S. Bae and Edward J. Haug, A Recursive Formulation for Constrained
Mechanical System Dynamics: Part II. Closed Loop Systems, Mech. Struct.
and Machines, Vol. 15, No. 4, pp. 481-506
5. Potra, F. A. and Petzold, L. R., ODAE Methods for the Numerical Solution
of Euler-Lagrange Equations. Applied Nume. Math., Vol. 10, pp. 397-413,
1992
6. Potra, F. A. and Rheinboldt, W. C., 1989, On the Numerical Solution of
Euler-Lagrange Equations, NATO Advanced Research Workshop on Real-
Time Integration Methods for Mechanical System Simulation, Snowbird,
Utah, U. S. A..
7. Jeng Yen, Edward J. Haug, and Florian A. Potra, 1990, Numerical Method
27
for Constrained Equations of Motion in Mechanical Systems
Dynamics,Technical Report R-92, Center for Simulation and Design
Optimization, Department of Mechanical Engineering, and Department
of Mathematics, The University of Iowa, Iowa City, Iowa.
8. Ming-Gong Lee and Edward J. Haug, 1992, Stability and Convergence for
Difference Approximations of Differential-Algebraic Equations of
Mechanical System Dynamics, Technical Report R-157, Center for
Simulation and Design Optimization, Department of Mechanical
Engineering, and Department of Mathematics, The University of Iowa,
Iowa City, Iowa.
9. Lin, T. C. and Yae, K. H., 1990, Recursive Linearization of Multibody
Dynamics and Application to Control Design, Technical Report R-75,
Center for Simulation and Design Optimization, Department of
Mechanical Engineering, and Department of Mathematics, The University
of Iowa, Iowa City, Iowa.
APPENDIX A : RECURSIVE FORMULAS
Recursive
formulas )(i kqI )(i kqII
qq BxX )( 1)i(i1)2(i
1i1)i1(ii
k
kk
)(
)()(
xB
XBX
q
kk 1i1)i1(ii qq )(XB)(X
qT
q G)(Bg
i
T
iii
i
T
iiii
k
k
( SBS
SBg
)()
)()(
1)1(1
2)1()1(
k
k
0S
0g
q
q
k
k
)(
)(
1i
1)i(i
qq x)B(X
1)i(i1)i2(i
1i1)i1(i
1i1)i1(ii
k
k
kk
)(
)(
)()(
xB
XB
XBX
q
q
k
kk
)(
)()(
1i1)i1(i
1i1)i1(ii
q
XB
XBX
vv xBX )( 1)i(i1)i2(i
1i1)i1(ii
k
kk
)(
)()(
xB
XBX
v
vv
k
kk
)(
)()(
1i1)i2(i
1i1)i1(ii
v
vv
XB
XBX
Recursive
formulas )(i kqIII )(i kqIV
qq BxX )( 0X q k
)( i 0X q k
)( i
28
qq GBg )( T
kk
kk
)()(
)()(
i
T
1)i1(i1i
i
T
1)i2(i1)i(i
SBS
SBg
0S
0g
q
q
k
k
)(
)(
1i
1)i(i
qq xBX )( 0X q k
)( i 0X q
k)( i
vv xBX )( 0X v k
)( i 0X v
k)( i
Recursive
formulas )(i kqI or )(i kqII or )(i kqIII or )(i kqIV
BxX 1)i(i1)i2(i1i1)i1(ii xBXBX
GBg T
)(
0
)(
1i1iT
1)1i(ii
n
1i1iT
2)1i(i)1i(i
SGBS
SSGBg
xBX i)1i(2i)1i(1i1i)1i(1i1i)1i(i xBXBXBX
29
1.2
DECOUPLING SOLUTION METHOD FOR
IMPLICIT NUMERICAL INTEGRATION
1.2. 1. INTRODUCTION
The dynamic behavior of a constrained mechanical system is often represented
by differential algebraic equations (DAEs)[1]. Solutions of DAEs are generally
more difficult to obtain than those of ordinary differential equations (ODEs)[2].
To solve DAEs, a direct discretization method was proposed by Gear[3]. Since
the solution obtained by Gear does not satisfy the velocity level constraints,
consistent initial conditions cannot be obtained. It was found that the
inconsistency often resulted in a poor local error estimation[4]. A series of
stabilization methods[5-7] which employ either Lagrange multipliers or
constraint violation penalty terms were followed.
Recently several solution methods[8], projecting the differential equations on
the inflated constraint manifolds, have appeared. Two kinds of solution process
are available. In the first solution process, the numerical integration is carried
out first and the integrated variables are corrected so that the position level
constraints, the velocity level constraints, and the acceleration level constraints
are satisfied. Since the correction is made sequentially level-by-level, the size
of system equations to be solved remains small. However, the integration
stepsize can be excessively small for highly nonlinear or stiff problems due to a
narrow stability region of the explicit method. In order to overcome this
difficulty, the second solution process is developed. In the second solution
process, the numerical integration formula, kinematic constraints and their
derivatives, and equations of motion are solved simultaneously. Therefore, the
size of the system equations to be solved becomes larger although the problem of
excessive small step size is resolved. In addition to the problem of large size of
the matrix equation, the condition of the matrix becomes poor as the stepsize
gets smaller for discontinuous systems. The poor condition of the matrix often
30
results in large error in the solution of the matrix equation.
In this paper, a decoupling solution method for the implicit numerical
integration method is proposed. This method is free from the problems of the
poor matrix condition and the excessively small step size as well as the large
matrix size.
In section 2, overdetermined DAEs for constrained mechanical systems are
given. A decoupling solution method is given in section 3. In section 4, the
numerical algorithm is provided. The numerical examples are given in section
5 to demonstrate the efficiency of the proposed method. Conclusions are drawn
in section 6.
1.2.2. IMPLICIT NUMERICAL INTEGRATION FOR DIFFERENTIAL
ALGEBRAIC EQUATIONS
The equations of motion for a constrained mechanical system can be
implicitly described as
0qv (2.1.a)
0λ)a,v,F(q, (2.1.b)
0(q) (2.1.c)
where q is the generalized coordinate vector in Euclidean space nR , and λ
is the Lagrange multiplier vector for constraints in mR , represents the
position level constraint vector in mR , and its Jacobian is expressed
nm
q R that is assumed to have full row-rank. Successive differentiations
of Eq. 2.1.c yield velocity and acceleration level constraints,
0υvv)(q, q (2.2.a)
0γaa)v,(q, q (2.2.b)
Equations 2.1 and 2.2 comprise a system of overdetermined differential algebraic
31
equations (ODAE). An algorithm based on backward differentiation formula
(BDF) to solve the ODAE is given in Ref. 1 as follows:
0
δqvU
δvaU
Φ(q)
υvΦ
γaΦ
λ)a,v,F(q,
RU
RU
Φ
Φ
Φ
F(x)
H(x)
T
2
T
1
q
q
2
T
2
1
T
1
2
0
1
0
0
0
b
h
b
h
b
h
b
h
(2.3)
where
k
1i
1ni
0
1 bb
1vδ ,
k
1i
1ni
0
2 bb
1qδ , k is the order of integration and
ib are the BDF coefficients. Here, ]q,v,a,[λxTTTT and the columns of
)2,1i()mn(n
i RU constitute bases for the parameter space of the position
and velocity level constraints. iU are chosen so that
T
q
U
Φ
i
has an inverse.
Therefore, the parameter space spanned by the columns of iU and the subspace
spanned by the columns of T
qΦ constitute the entire space nR .
The number of equations and the number of unknowns in Eq. 2.3 are the same,
so Eq. 2.3 can be solved. Newton's numerical method can be applied to obtain
the solution x .
iii
HxH x (2.4.a) ii1i
xxx (2.4.b)
LU-decomposition of the matrix i
xH not only increases the computation time
but also produces an ill-conditioned matrix as h approaches zero [4]. In order to
32
eliminate these problems, Eq. 2.4.a will be divided into several pieces to obtain
q , v , a and λ separately in the next section.
1.2.3. A DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL
INTEGRATION
Equation 2.4.a can be rewritten in detail as follows:
0xFΔλFΔaFΔvFΔqF λavq )( (3.1.a)
0xΦΔqΦΔvΦΔaΦ qvq )( (3.1.b)
0xΦΔqΦq )( (3.1.c)
0xΦΔqΦΔvΦ qq )( (3.1.d)
0xRΔvΔaU ))(( 1
T
1 hh (3.1.e)
0xRΔqΔvU ))(( 22 hhT (3.1.f)
where 0b
hh . Equation 3.1.e can be rewritten in an equivalent inflated form
by choosing 1U such that 0T
q
1
a
T
1 ΦFU . as follows [7]:
0τΦFUxRΔvΔa
1
1
1)( T
qa
T
i hhh (3.2)
where aF is a mass matrix and is generally nonsingular. The aF can be
singular if a parametric formulation is employed. If aF is singular, Eqs. 3.1
must be solved simultaneously to obtain q , v , a and λ . The vector
m
1 Rτ is a new unknown variable. The a is thus obtained from Eq. 3.2 in
terms of v as
1
T1
a1 )(h
1τΦFxRΔvΔa q
(3.3)
33
Substituting Eq. 3.3 into Eq. 3.1.a yields
)()()(h
1 xRFxFτΔλΦΔvF
FΔqF a1
T
q
a
vq
(3.4)
Equation 3.1.f can be rewritten in an equivalent inflated form by choosing 2U
such that 0ΦF
FUT
q
a
v
1
T
2h
as follows:
0τΦF
FxRΔqΔvT
q
a
v2
2
1
)(h
hhh (3.5)
where h
av
FF is assumed to be a nonsingular matrix and m
2 Rτ is a new
unknown variable. The solution process for the case of a singular matrix will be
explained later in this section. Equation 3.5 can be solved for v in terms of
q as follows:
2
1
2h
)(h
1τΦ
FFxRΔqΔv
T
q
a
v
(3.6)
Substituting Eq. 3.6 into Eq. 3.4 and multiplying both sides of Eq. 3.4 by 2h
yields
3
2h RβΦΔqKT
q
* (3.7)
where
avq
*FFFK hh 2
(3.8.a)
21 ττΔλβ (3.8.b)
)()h(h)(h)(h 21
22
3 xRFFxRFxFR ava (3.8.c)
Equations 3.7 and 3.1.c are combined to obtain
34
)(h0
3
2 xΦ
R
β
Δq
Φ
ΦK
q
T
q
*
(3.9)
Equation 3.9 is then solved for q and β . Note that β is scaled by 2h to
avoid the ill-conditioned coefficient matrix in Eq. 3.9, even as h approaches to
zero.
Multiplying both sides of Eq. 3.5 by h
av
FF yields
)(hh
1
h22 xRΔq
FFτΦΔv
FF a
v
T
q
a
v
(3.10)
Equations 3.10 and 3.1.d are combined to obtain
ΔqΦxΦ
xRΔqF
F
τ
Δv
Φ
ΦF
F
q
a
v
q
T
q
a
v
)(
)(hh
1
0h
2
2
(3.11)
where q has been obtained from Eq. 3.9. Equation 3.11 is solved for the v
and 2τ . Multiplying both sides of Eq. 3.3 by aF yields
vxRFτΦΔaF aah
1)(11
T
q (3.12)
Equations 3.12 and 3.1.b are combined to obtain
35
qΦΔvΦxΦ
vxRF
τ
Δa
Φ
ΦF
qv
a
q
T
qa
)(
h
1)(
0
1
1
(3.13)
Equation 3.13 is solved for a and 1τ . Once β , 1τ and 2τ are obtained, the
Δλ is evaluated from Eq. 3.8.b, as follows:
21 ττβΔλ (3.14)
Since aF is a mass matrix and vF is a tangent damping matrix, va FF h is
generally not ill conditioned. If an ill-conditioned case is encountered, Eqs. 3.1
must be solved simultaneously to obtain q , v , a and λ . However, the
aF and va FF h are rarely singular, so q , v , a and λ are obtained by
using Eqs. 3.9, 3.11, 3.13, and 3.14 for most of practical problems.
1.2.4. NUMERICAL ALGORITHM
The DASSAL subroutine [4] is employed to integrate the system variables.
Computational flow for the proposed DAE solution method is given in Fig.
1.(Page 2-7)
1.2.5. NUMERICAL EXAMPLES
1.2.5.1 QUICK-RETURN MECHANISM
The quick-return mechanism as shown in Fig. 5.1 is mounted on a body
translating with respect to the ground. The system consists of 6 bodies, 2
translational joints, and 5 revolute joints. The system has two degrees of freedom
if the redundant constraints are eliminated.
Dynamic analyses were performed for 1 sec with error tolerances of 10-4
and
10-6
by using the program developed in this paper and the other commercial
program
36
Read initial conditions
Compute initial Accelerations and Lagrange multipliers from Eqs. 2.1.b and 2.2.b
t = t + h
Predict q v a, , , and
t > tout ?
YN
End
Y
FF
va
h '
N
Compute in Eq. 3.9 q and
Compute in Eq. 3.11v and 2
Compute
in Eqs. 3.1
q, v, a, and
Update q v a, , , and
Convergence?N
Compute in Eq. 3.13a and 1
Compute in Eq. 3.14
Faor is singular ?
Y
Fig. 1 Flowchart for the proposed DAE solution method
37
Front view Side view
Body1
Body2
Body3
Body5
Body6
Body4
Fig. 5.1 A quick-return mechanism
, which employs the implicit numerical integration with the BDF. The results are
shown in Fig. 5.2. and the integration information is shown in Table 5.1.
Fig. 5.2 Results of the quick-return mechanism
Table 5.1 Integration information for the quick return mechanism
—OTHER
…PROPOSED
38
Method Error
Tolerance
No.
Steps
No.
Function
Evaluation
No.
Jacobian
Evaluation
No.
Newton
Iteration
Failure
No.
Integration
Failure
CPU
Time
Proposed 1.0d-4 293 342 180 15 0 16 sec
Other
commercia
l program
1.0d-4 115 554 NA NA NA 34 sec
Proposed 1.0d-6 315 722 336 22 0 22 sec
Other
commercia
l program
1.0d-6 Failed to integrate.
(NA means Not Available)
Note that the proposed method converged successfully for the small error
tolerance (10-6
) while the other commercial program did not. We think that the
reason is the ill-conditioned Jacobian matrix of them.
1.2.5.2 AIR COMPRESSOR
This system was modeled as four bodies, two revolute joints, two translational
joints, and 2 ball joints as shown in Fig. 5.3. The system has 1 degree of
freedom if the redundant constraints are eliminated. Dynamic analyses were
carried out for 1.0 sec with initial angular velocity. The proposed method and the
other commercial program yielded identical results, as shown in Fig. 5.4. The
system is conservative and the total energy should be constant. Figure 5.5
shows the total energy change during the integration. It is shown that the total
energy obtained from the present program is numerically more stable than that
obtained from the other commercial program. Thus, the other commercial
program failed to integrate (while the proposed method did not) as the error
tolerance became small. The integration information is also given in Table 5.2.
50 rad/sec
Fig. 5.3 An air compressor mechanism
39
Table 5.2 Integration information for the air compressor mechanism
Method Error
Tolerance
No.
Steps
No.
Function
Evaluation
No.
Jacobian
Evaluation
No.
Newton
Iteration
Failure
No.
Integration
Failure
CPU
Time
Proposed 1.0d-4 349 707 351 0 0 16 sec
Other 1.0d-4 295 1185 NA NA NA 31 sec
Proposed 1.0d-6 529 1067 531 0 0 20 sec
Other 1.0d-6 Failed to integrate.
Fig. 5.4 Results for the air compressor
PROPOSED OTHER
— OTHER
… PROPOSED
40
Fig. 5.5 Total energy comparison for the air compressor
1.2.6. CONCLUSIONS
A decoupling solution method for the implicit numerical integration is
proposed in this paper. The size of the Jacobian matrix is significantly reduced
by decoupling the iteration equations. The ill-conditioning problem of the
implicit numerical integration is resolved in this method. Numerical study
showed that the proposed method yields numerically more stable solution than
the commercial program with smaller number of function evaluation.
41
REFERENCES
1. J. Yen, Constrained Equations of Motion in Multibody Dynamics as ODE's on Manifolds,
SIAM J. Numer. Anal., vol. 30 , pp. 553-568, (1993).
2. P. L. stedt and L. R.. Petzold, Numerical Solution of Nonlinear Differential Equations
with Algebraic Constraints I: Convergence Results for Backward Differentiation
Formulas, Math. Comp., vol. 46, pp. 491-516, (1986).
3. C. W. Gear, The Simultaneous Numerical Solution of Differential Algebraic Equations,
IEEE Trans. Circuit Theory, vol. 18, pp. 89-95, (1971).
4. K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value
Problems in Differential-Algebraic Equations, SIAM Press, (1995).
5. J. Baumgarte, Stabilization of Constraints and Integrals of Motion in Dynamical
Systems, Comput. Methods Appl. Mech. Engrg., vol. 1, pp. 1-16, (1972).
6. Javier Garcia de Jalon and Eduardo Bayo, Kinematic and Dynamic Simulation of
Multibody Systems, Springer-Verlag, (1993).
7. F. A. Potra, Implementation of Linear Multistep Methods for Solving Constrained
Equations of Motion, SIAM J. Numer. Anal., vol. 30, pp. 74-789, (1993).
8. Ming-Gong Lee and Edward J. Haug, Stability and Convergence for Difference
Approximations of Differential-Algebraic Equations of Mechanical System Dynamics,
Technical Report R-157, August, (1992).
42
1.3
LINEARIZED EQUATIONS OF MOTION FOR
MULTIBODY SYSTEMS WITH CLOSED LOOPS
1.3.1. INTRODUCTION
Linearization is an important tool in understanding the system behavior of a
nonlinear system at a certain state. As an example, the eigenvalues of the
linearized equations of motion are very useful information in developing control
logics. Linearization of an unconstrained system is relatively easier than that of
the constrained systems due to the algebraic constraint equations and
corresponding Lagrange multipliers. This research proposes a linearization
method for the constrained mechanical systems and compares the results with
those obtained from other methods.
Sohoni [1] presented an approach for automatically generating a linearized
dynamical model, which is derived from the nonlinear equations of motion. The
Lagrange multiplier term was kept constant in the linearized equations of motion.
The velocity and acceleration level constraints have not been considered in the
resulting linearized equations of motion. Neuman symbolically generated the
dynamic robot model by Lagrange-Euler formulation and linearized the dynamic
model about a nominal trajectory [2]. Balafoutis presented a computational
method for recursive evaluation of linearized dynamic robot model about a
nominal trajectory [3]. The formulation was applied to the robot systems, which
are unconstrained systems. This formulation was generalized by Gontier [4] for
general unconstrained mechanical systems. Similar formulations have been
developed by the variational approach in Refs. [5,6]. A recursive formulation
using the relative coordinates was proposed by Bae in Ref. [7]. The equations of
motion were derived in a compact matrix form by using the velocity
transformation method. The actual computation was carried out by using the
recursive formulas developed for each joints. Realtime simulation of a vehicle
42
system has been carried out by the recursive method in Ref. [8]. The Jacobian
matrix was updated once in a while during time marching of the numerical
integration. The recursive method was extended to the flexible body dynamics of
constrained mechanical systems in Ref. [9]. A virtual body concept was
employed to relieve the implementation burden of the flexible body dynamics
coding. A compliant track link model was developed for tracked vehicles in Ref.
[10]. A minimum set of the equations of motion was obtained by the recursive
method. Concept of the configuration design variable with the recursive
formulation was introduced in Ref. [11].
The equations of motion for multibody systems are highly nonlinear with
respect to the relative positions, velocities, and accelerations. The equations of
motion are perturbed to obtain the linearized equations of motion. Since the
equations of motion are highly nonlinear, their perturbation involves with many
arithmetic operations for a multibody system consisting of many bodies and
joints. In case of open loop systems which do not have any constraints, the
equations of motion result in the ordinary differential equations whose partial
derivatives with respect to the relative coordinates, velocities, and accelerations
has been obtained by several different methods in Refs. [2,3,4]. In case of closed
loop systems which have constraints, these method cannot be used directly any
more due to the constraints and corresponding Lagrange multipliers.
One of the intuitive methods to handle the constraints is to directly express the
equations of motion only in terms of the independent relative positions,
velocities, and accelerations. In order to achieve this goal, the relative
coordinates must be divided into the independent and dependent coordinates and
the dependent coordinates, velocities, and accelerations must be directly
expressed in terms of independent ones. However, the independent and
dependent coordinates, velocities, and accelerations are tightly and nonlinearly
coupled by the position, velocity, and acceleration level constraints and the
equations of motion are implicit function of the coordinates, velocities, and
accelerations. As a result, it is very difficult to directly express the dependent
coordinates, velocities, and accelerations in terms of independent ones and
consequently to express the equations of motion only in terms of the independent
coordinates, velocities, and accelerations.
The null space of the constraint Jacobian is first pre-multiplied to the
42
equations of motion to eliminate the Lagrange multiplier and the equations of
motion are reduced down to a minimum set of ordinary differential equations.
The resulting differential equations are still functions of all relative coordinates,
velocities, and accelerations. Since the coordinates, velocities, and accelerations
are tightly coupled by the position, velocity, and acceleration level constraints,
direct substitution of the relationships among these variables yields very
complicated equations to be implemented. As a consequence, the reduced
equations of motion are perturbed with respect to the variations of all coordinates,
velocities, and accelerations, which are coupled by the constraints. The position,
velocity and acceleration level constraints are also perturbed to obtain the
relationships between the variations of all relative coordinates, velocities, and
accelerations and variations of the independent ones. The perturbed constraint
equations are then simultaneously solved for variations of all coordinates,
velocities, and accelerations only in terms of the variations of the independent
coordinates, velocities, and accelerations. Finally, the relationships between the
variations of all coordinates, velocities, accelerations and these of the
independent ones are substituted into the variational equations of motion to
obtain the linearized equations of motion only in terms of the independent
coordinate, velocity, and acceleration variations.
The proposed method is implemented in the commercial program RecurDyn.
Vibration analyses of a four bar mechanism and a vehicle system are carried out
to demonstrate the validity of the proposed method.
51
1.3.2. RELATIVE COORDINATE KINEMATICS
Figure 1 Coordinate systems and a rigid body
Figure 1 shows the coordinate system fixed on a body i . In the figure, the
iii zyx frame is the body reference frame and the ZYX frame is the
inertial reference frame. Point O is the origin of ZYX , point iO is the
origin of iii zyx , and ir is the position vector of iO from O . The if , ig ,
and ih are unit vectors along the x , iy , and iz axes, respectively.
Orientation matrix of the body is given as
iiii hgfA (1)
Velocities and virtual displacements of point iO in the ZYX frame are
defined as (see Refs. [4-5])
i
i
iω
rY
(2)
i
i
δπ
δrδZ (3)
Their corresponding quantities in the iii zyx frame are defined as
X
Z
Y
ir
iy
ix
iz
iO
Oif
ig
ih
44
i
T
i
i
T
i
i
i
iωA
rA
ω
rY
(4)
i
T
i
i
T
i
i
i
πA
rA
πδ
rδδZ
(5)
Figure 2 Kinematic relationships between two adjacent rigid bodies
A pair of contiguous bodies is shown in Figure 2. Body 1i is assumed to
be an inboard body of body i and the position of point iO is
1)i(i1)i(i1)i(i1)(ii sdsrr (6)
By using Eq. (5), the angular virtual displacement of body i in its local
reference frame is
1 ) i(i1)i(i
T
1)i(i1)-(i
T
1)i(ii δδδ qHAπAπ (7)
where ii )1( H is determined by the axis of rotation and 1)i(iA is defined as
i
T
1)(i1)i(i AAA (8)
Taking variation of Eq. (6) yields
X
Z
Y
iy
ix
iz
)1( iis
ii )1( d
)1(y i
)1(x i
)1(z i
iO)1( iO
ir)1( ir
ii )1( s
45
1)i(i1)i(i1)i(iT
1)i(i1)i(i1)i(i1)i(iT
1)(i1)i(iT
1)-i(i1)i(i1)i(i1)i(i1)i(iT
1)(i1)i(iT
i
δ)~)((
δ)~~~(
δδ
1)i(i
qHAsAdA
πAsAdsA
rAr
q
(9)
where symbols with tildes denote skew symmetric matrices comprised of their
vector elements that implement the vector product operation and 1)i(iq denotes
the relative coordinate vector.
Combining Eqs. (7) and (9) yields the recursive virtual displacement equation
for a pair of contiguous bodies
1)i(i1)i2(i1)(i1)i1(iiδ δqBZδBZ (10)
where
I0
AsAdsI
A0
0AB
)~~~( 1)i(iT
1)i(i1)i(i1)i(i1)i(i1)i1(i
1)i(i
1)i(i
T
T
(11)
1)i(i
1)i(i1)i(iT
1)i(i1)i(i1)i(i
1)i(i
1)i(i
1)i2(i
~)(1)i(i
H
HAsAd
A0
0AB
q
T
T
(12)
It is important to note that matrices 1)i1(iB and 1)i2(iB are functions of only
relative coordinates of the joint between bodies 1)(i and i . As a
consequence, further differentiation of the matrices 1)i1(iB and 1)i2(iB in Eqs.
(11) and (12) with respect to other than 1)i(iq yields zero. The virtual
displacement relationship between the absolute and relative coordinates for the
whole system can be obtained by repetitive application of Eq. (10) as
qBZ (13)
where B is the velocity transformation matrix with relationship between
Cartesian and relative coordinates. The relationship between Cartesian velocity
46
Y and relative velocity q can be derived in the same manner.
qBY (14)
1.3.3. EQUATIONS OF MOTION
The variational form of the Newton-Euler equations of motion for a
constrained multibody system is
0QλΦYMZ Ζ )(δ TT (15)
where M and Q are the mass matrix and general force vector in Cartesian
space, respectively. Zδ must be kinematically admissible for all joints except
cut joints [12]. In the equation, Φ and λ , respectively, denote the constraint
equations and the corresponding Lagrange multiplier in mR in which m is the
number of the constraint equations. Substituting the virtual displacement
relationship and acceleration relationships qBqBY into Eq. (15) yields (see
Ref. [5])
n*T* R F0QλΦqMF q (16)
where n is the number of generalized coordinates and the mass matrix *M
and force vector *Q are defined as
BMBMT*
(17)
)(*qBMQBQ
T (18)
A recursive method has been proposed to compute Eqs. (17) and (18) in Ref.
[7].
47
1.3.4. ELIMINATION OF LAGRANGE MULTIPLIERS AND
LINEARIZATION OF THE EQUATIONS OF MOTION
The relative coordinates q can be partitioned into dependent coordinates Dq
and independent coordinates Iq such that the sub-Jacobian DqΦ is well-
conditioned. Variational form of the cut constraint equations can be written as
0qΦqΦΦ qq ID δδδID
(19)
The Dδq can be obtained from Eq. (19) as
I
1
D δδID
qΦΦq qq
(20)
By using the relationship in Eq. (20), I
1
D δδID
qΦΦq qq
is represented as
Iδδ qNq (21)
where
I
ΦΦN qq ID
1
(22)
Direct calculation of TT
qΦN shows that N is the null space of qΦ as
0Φ
ΦIΦΦΦN
q
q
qqq
T
T
T1-TTT
I
D
DI)( (23)
As a result, pre-multiplication of Eq. (16) by TN gives
0QNqMNF *T*T* (24)
where Lagrange multiplier λ term was eliminated since N is the null space of
48
qΦ . However, the equations of motion *F are dependent on not only the
dependent variables Dq , Dq and Dq but also independent variables Iq , Iq
and Iq . Taking variation of Eq. (24) yields
0qFqFqFF qqq **** (25)
Equation (25) can be rewritten in a matrix form as
0
q
q
q
FFF qqq
***
(26)
Variations of position, velocity and acceleration level constraints are
0qΦqΦqΦ
0qΦqΦ
0qΦ
qqq
q
(27)
Appending the trivial identity relationships for the variations of independent
coordinates, velocities and accelerations to Eq. (19) yields
I
I
I
q
q
q
I
000
000
000
q
q
q
I
ΦΦΦ
0ΦΦ
00Φ
qqq
q
(28)
Equation (28) is solved for the Tqqq and substituted into the
linearized equations of motion in Eq. (26) to yield the following linearized
equations of motion only in terms of the variations of independent coordinates,
velocities and accelerations:
49
0
q
q
q
I
000
000
000
I
ΦΦΦ
0ΦΦ
00Φ
FFFqqq
q
qqq
I
I
I
1
***
(29)
Direct comparison of Eq. (29) and the following linearized equations of
motion yields the M
, C
and K
matrices:
0qKqCqMFq
III
* δˆδˆδˆδ * (30)
1.3.5. NUMERICAL EXAMPLES
1.3.5.1 . FOURBAR MECHANISM WITH A SPRING
Figure 3 shows a four bar mechanism with a spring. The system consists of
four revolute joints and one spring and their material properties are defined in
Table 1. As a result, three generalized coordinates, 1 , 2 and 3 are defined
for the first three revolute joints and the remaining one revolute joint is defined
as a cut joint. The constraint equations are introduced from the cut joint.
50
Figure 3 A four-bar mechanism with a spring
Table 1 Material property of bodies and a spring
Mass (kg) Inertia Moment
(kg*mm^2)
Body
Link A 7.707 161760.83 Link B 3.946 53005.79 Link C 7.707 161760.83
Spring Stiffness (N/mm) Damping (N*sec/mm)
10.0 0.0
Dynamic analysis of the mechanism is performed to obtain the time domain
response. FFT of the time response is performed to extract dominant frequency
domain response. Figures 4 and 5 show the time and frequency responses,
respectively.
The proposed linearization method is applied for the system. The dominant
frequency and corresponding mode shape are shown in Figure 6 and Table 2.
The frequency obtained from the proposed method and that obtained from FFT
analysis of the time domain responses are shown to be very close, which
validates the proposed method.
2
1 3
Cut joint 400
500
Link
A
Link
B
Link
C
51
Figure 4 Angle of link C in time domain
Figure 5 Response in frequency domain
52
F
Figure 6 Mode shape of fourbar mechanism
Table 2 Undamped natural frequency and mode shape from the proposed method
Undamped Natural
Frequency (Hz) Mode
5.040164E+00 1 2 3
5.773503E-01 -5.773503E-01 5.773503E-01
1.3.5.2. CANTILEVER BEAM DRIVEN BY A MOTION
The system characteristics of a rotating cantilever beam differ from those of
beam in a static state, because the stiffness of the beam is changed by a
centrifugal force due to the rotational motion. (see Ref. [13]). A cantilever beam
rotating with the angular velocity ω is shown in Figure 7.
Figure 7 A rotating cantilever beam
53
Length of the beam is 6.8 m, density of the material is 14705.88 kg/m3,
Young's modulus of the material is 7.0×108 N/m
2. Area of the cross section is
0.002 m2, the moment of inertia 4.0×10
-7 m
4. The beam is divided into 21
lumped mass and 20 beam elements. Figure 8 shows the lowest three natural
frequencies of the rotating beam. As the angular speed increases, the bending
natural frequencies are shown to be increased.
Figure 8 The relationship between angular velocity and natural frequencies
1.3.5.3 A SPRING SYSTEM WITH 2 D.O.F.
A spring model shown in Fig. 9 is a system with two D.O.F, and the system
has two masses, joints and spring elements. Their material properties, spring and
damping coefficients are shown in Table 3.
Figure 9 A spring model
54
Table 3 Material properties, spring and damping coefficients
Mass1 5 Kg
Mass2 3 Kg
Length of m1 300 mm
Spring coefficient (k1) 10 N/mm
Spring coefficient (k2) 20 N/mm
If the rotational angle is small, sin and the equation of motion of this
system can be derived as:
0
ykk
l
kl
kl
kl
ym
I
22
22
2
1
2
2
2
240
0
(31)
From Table 3, Eq. (31) can be replaced as:
0200003000
30001350
30
015.0
yy
(32)
The characteristic equation of this spring system is derived from Eq (32).
03200003000
300015.01350
(33)
Also, the analytic natural frequencies can be computed as:
)Hz(76.17fsec)/rad(6.11112455
)Hz(019.9fsec)/rad(66.563211
22
11
(34)
Finally, the eigenvalues of this spring system is validated shown in Table 4.
55
Table 4 Eigenvalues of spring model
Mode
number
Undamped Natural Frequency (Hz)
RecurDyn/Eigenvalue Analytic solution
1 9.01862E+00 9.019
2 1.77621E+01 17.76
1.3.5.4 . A CANTILEVER BEAM
Two cantilever beam models shown in Figs. 10 and 11 have a fixed-free end
condition and ten lumped masses. One is modeled by using ten beam force
elements and the other is modeled by using one flexible body of RecurDyn. The
flexible beam model is originally generated in ANSYS. The material properties
and geometry conditions of the beam are shown in Table 4.
Figure 10 Beam model using RecurDyn/Beam element
Figure 11 Beam model using RecurDyn/Flexible body element
56
Table 5 The material properties and geometry conditions of beam
Length 0.4 m
Mass 3.9888 Kg
Young’s
modulus 9101 N/m
2
Inertia of area 810215.1 m4
Area 0.0018 m2
In Ref. [14], the analytic natural frequencies of these beams are computed as:
4
2
1 875.1AL
EI
, 4
2
2 694.4AL
EI
, 4
2
3 855.7AL
EI
(35)
By replacing Eq. (35) with Table 5, the natural frequencies can be computed as:
8601.32537.24875.14
2
1 nfAL
EI
1929.240085.152694.44
2
2 nfAL
EI
7477.676714.425855.74
2
3 nfAL
EI
Finally, the eigenvalues of this beam model is validated shown in Table 6.
57
Table 6 Eigenvalues of cantilever beam model
Mode
number
Undamped Natural Frequency (Hz)
Beam element Flexible Body Analytic solution
1 3.84002E+00 3.84259E+00 3.8426
2 2.37455E+01 2.38154E+01 23.8154
3 6.55744E+01 6.60152E+01 66.0152
4 1.26483E+02 1.28016E+02 128.016
5 2.05481E+02
6 2.65264E+02
In addition, RecurDyn can show the mode shapes of the beam model through 3D
animation, as shown in Figs. 12 and 13.
(a) 1st mode shape (b) 2nd mode shape
(c) 3rd mode shape
Figure 12 The mode shapes of model using RecurDyn /Beam element
58
(1) 1st mode shape (2) 2nd mode shape
(3) 3rd mode shape
Figure 13 The mode shapes of model using RecurDyn/Flexible body
1.3. 6. CONCLUSIONS
In this paper, a linearization method for constrained multibody systems is
proposed for the non-linear equations of motion employing the relative
coordinates. Null space of the constraint Jacobian is pre-multiplied to the
equations of motion to eliminate the Lagrange multipliers and to reduce the
number of equations. The set of differential equations are perturbed in terms of
all relative positions, velocities and accelerations. The position, velocity and
acceleration level constraints are perturbed to express the variations of all
relative positions, velocities and accelerations in terms of the variations of
independent positions, velocities and accelerations, which are substituted into the
perturbed equations of motion. The equations of motion perturbed with respect
to the q , q and q finally become the corresponding equations perturbed with
respect to the Iq , Iq and Iq . Eigenvalues and eigenvectors are then computed
from the equations of motion perturbed with respect to the Iq , Iq and Iq . The
proposed method is implemented in a commercial program RecurDyn.
Numerical results obtained from the proposed method are in good agreement
with the results reported in the literature and obtained by other methods.
REFERENCES
1. Sohoni VN, Whitesel J. Automatic Linearization of Constrained Dynamical
Models. ASME. Journal of Mechanism, Transmission, and Automation in Design,
59
Vol. 108, pp 300-304, 1986.
2. Neuman CP, Murray JJ. Linearization and Sensitivity Functions of Dynamic
Robot Models. IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-
14, No.6, pp.805-818, 1984.
3. Balafoutis CA, Misra P, Patel RV. ecursive Evaluation of Linearized Dynamic
Robot Models. IEEE Journal of Robotics and Automation, Vol.RA-2, No.3,
pp.146-155, 1986.
4. Gontier C, Li Y. Lagrangian Formulation and Linearization of Multibody System
Equations. Computers & Structures, Vol.57. No.2, pp.317~331, 1995.
5. Bae DS, Haug EJ. A Recursive Formulation for Constrained Mechanical System
Dynamics: Part I, Open Loop Systems. Mech. Struct. and Machines, Vol. 15, No.
3, pp.359-382, 1987.
6. Bae DS, Haug EJ. A Recursive Formulation for Constrained Mechanical System
Dynamics: Part II, Closed Loop Systems. Mech. Struct. and Machines, Vol. 15,
No. 4, pp. 481-506, 1987.
7. Bae DS, Han JM, Yoo HH. A Generalized Recursive Formulation for
Constrained Mechanical System Dynamics. Mech. Struct. & Mach., Vol. 27(3),
pp. 293-315, 1999.
8. Bae DS, Lee JK, Cho HJ, Yae H. An Explicit Integration Method for Realtime
Simulation of Multibody Vehicle Models. Computer Methods in Applied
Mechanics and Engineering, Vol. 187, pp. 337-350, 2000.
9. Bae DS, Han JM, Choi JH, Yang SM. A Generalized Recursive Formulation for
Constrained Flexible Multibody Dynamics. International Journal for Numerical
Methods in Engineering, Vol. 50, pp. 1841-1859, 2001.
10. Ryu HS, Bae DS, Choi JH, Shabana AA. A Compliant Track Link Model for
High-speed, High-mobility Tracked Vehicles. International Journal for
Numerical Methods in Engineering, Vol. 48, pp. 1481-1502, 2000.
60
11. Kim HW, Bae DS, Choi KK. Configuration Design Sensitivity Analysis of
Dynamics for Constrained Mechanical Systems. Computer Methods in Applied
Mechanics and Engineering, Vol. 190, pp. 5271-5282, 2001.
12. Wittenburg J. Dynamics of Systems of Rigid Bodies. B. G. Teubner Stuttgart,
1977.
13. Southwell R, Gough F. The Free Transverse Vibration of Airscrew Blades.
British A.R.C. Reports and Memoranda No. 766, 1921.
14. L. Meirovitch, “ Analytical Methods in Vibrations”, MACMILLAN, 1967.
61
1.4
STATIC EQUILIBRIUM ANALYSIS OF
MULTI PHYSICS SYSTEM
1.4.1. INTRODUCTION
The desire of describing the real world makes the integration of multi body
dynamics (MBD) and finite element analysis (FEA). These virtual systems are used
to figure out what happens in the real world.
Unlike the static analysis, the dynamic analysis of mechanical systems requires the
pre-analysis for finding static equilibrium. For example: the real-world vehicle is
statically stable under gravity condition. However it is impossible that the practical
engineers measure the equilibrium positions of whole chassis components. Thus,
the CAE engineers can not define the hard positions of each chassis components in
their multi-body dynamic models. Thus, they require the pre-analysis of finding
static equilibrium.
Suppose that a mechanical system is composed of only mass, spring and dampers
without rigid contacts. Then, typical numerical optimization method or typical
nonlinear
equation solvers can find the equilibrium position. Now, suppose that the MBD
systems include many rigid contact conditions among bodies. In other words, this
represents that many inequality constraints are included in numerical optimization
and nonlinear equation problems. Unlike the conventional inequality design
constraints in design optimization, these contact constraints can make the disjoint
space in the solution space. The conventional Newton-Raphson method cannot
solve these problems. Thus, dynamic analysis approaches is widely used to find the
static equilibrium even though it takes much computational time. However,
recently the CAE models become more complicated and it requires much
computational time. Even, the CAE engineers should determine the design
improvements from the virtual systems as fast as they could. Hence, at now, the
pre-analysis for finding the static equilibrium becomes a key process for multi-
body dynamics.
In this study, an augmented Newton-Raphson method will be presented for the pre-
analysis. Chapter 1.4.2. explains the equation of motion of multi flexible body
dynamics (MFBD). In chapter 1.4.3., the basic concept of an augmented Newton-
Raphson method is explained. Then, several numerical case studies will be
62
presented in chapter 1.4.4. Finally, chapter 1.4.5. will summarize the proposed
study.
1.4.2. THE EQUATION OF MOTION FOR MFBD BASED ON
RECURSIVE FORMULATION
From the global coordinate system, let’s define the translational and angular
velocities of the body coordinate system as Then, their corresponding
quantities.
with respect to the body coordinate system are defined as
where is the combined velocity of the translation and rotation. Y
The recursive velocity and virtual relationship for a pair of contiguous bodies are
obtained in [1] as
where denotes the relative coordinate vector. It is important to note that
matrices and are only functions of the . Similarly, the
recursive virtual displacement relationship is obtained as 1)
If the recursive formula in Eq. (2) is respectively applied to all joints, the
following relationship between the Cartesian and relative generalized velocities can
be obtained:
where is the collection of coefficients of the . Also, and are
composed of
and
X
Z
Y
ir
iy
ix
iz
iO
Oif
ig
ih
63
respectively. In these equations, the subscripts nc and nr denote the number of the
Cartesian and relative coordinates, respectively. Since in Eq. (4) is an arbitrary
vector in , two equations of (2) and (4), which are computationally equivalent,
are actually valid for any vector such that
and
In this equation, is the resulting vector of
multiplication of and . As a result, the transformation of into
is actually calculated by recursively applying Eq. (8) to achieve
computational efficiency in this research. Inversely, it is often necessary to
transform a vector in into a new vector in . Such a
transformation can be found in the generalized force computation in the joint space
with a known force in the Cartesian space. The virtual work done by a Cartesian
force is obtained as
where should be kinematically admissible for all joints in a system.
Substitution of into Eq. (9) yields
where .
The equations of motion for constrained multi-body dynamic systems can be
obtained as
where the is the Lagrange multiplier vector for cut joints [2] in and Φ
64
represents the position level constraint vector in . Also, and are the
mass matrix and force vector in the Cartesian space including the contact forces,
respectively.
Similarly, the equation of motion of finite element system can be obtained as
In this study, we call the combined equations of (11) and (12) as the equation of
motion of multi flexible body dynamics (MFBD).
1.4.3. AUGMENTED NEWTON-RAHPSON METHOD
In the equilibrium state, the velocity and acceleration of the body should be zero-
valued. Thus, the equilibrium equations of (11) and (12) can be simplified as
Now, in order to solve this nonlinear equation, the Newton-Raphson method is
generally used. First, nonlinear equation of (13) is linearized at the position
vector and rearranged as
where . Then, obtained by solving the linear equation of
(14). Second, the position vector is updated as
The conventional Newton-Raphson method repeats those two steps until satisfying
the convergence criteria. However, this method does not guarantee the convergence
for the following two cases:
When stiffness matrix becomes singular due to no ground stationary forces
of MBD system.
When contact forces are encountered by only small change of position
vector .
In this study, in order to prevent the singular of equation (14), trust-region concept
[3] is introduced as
65
The multiplier 0>ν will be operated when is nearly singular. Also, when the
position vectors are updated, the following line search scheme is employed for
guaranteeing the convergence.
where is determined by solving
We call the new process of the equations (16) through (18) as Augmented Newton-
Raphson method.
Now, let’s consider the convergence of equilibrium. It is noted that equation (13) is
a necessary condition of equilibrium. In other words, it cannot guarantee the
equilibrium. Thus, we introduce the total potential energy term into the
descent function of (18). Then, the descent function is augmented as
From the viewpoint of numerical optimization, this descent function of (19) can be
a local or global optimization problem. Hence, in this study, when
, a global line search algorithm is employed. Otherwise, a local
line search algorithm is done. The former is based on Lipschitzian concept [4] and
the latter is done on variable-order polynomial approximations [5].
1.4.4. NUMERICAL RESULTS
The multi physics simulation program RecurDyn is used to examine the numerical
results.
1.4.4.1. Simple Pendulum
The simple pendulum model and the result are shown as Fig. 1. The total potential
energy term drive the pendulum downward.
The steps are listed as table 1. The Augmented Newton-Raphson is the proposed
and Robust Newton-Raphson [6] is the previous. NJ and NR is the number of
evaluation of jacobian and the residual respectively.
66
(a) Initial Position (b) Final Position
Figure 1. THE SIMPLE PENDULUM MODLEING & RESULT
Table 1. THE STEP OF THE SIMPLE PENDULUM
proposed previous
NJ 3 102
NR 81 454
1.4.4.2. Simple Pendulum with the Contact
The simple contact model and the result are shown in Fig. 2. The sphere and the
cylinder contact in the model. The stationary force is generated in contact and the
Augmented Newton-Raphson method finds the solution of static equilibrium. The
steps are listed in table 2.
(a) Initial Position (b) Final Position
Figure 2. THE SIMPLE CONTACT MODLEING & RESULT
67
Table 2. THE STEP OF THE SIMPLE CONTACT
proposed previous
NJ 2 238
NR 42 1071
1.4.4.3. Paper Sheet
The paper model is shown in fig. 3. The paper element is finite shell element of
MFBD in the RecurDyn. The model is composed 100 shell elements and is the
square paper of 100mm. The left side is fixed and the gravity is applied. The steps
are listed in table 3
(a) Initial Position (b) Final Position
Figure 3. THE PAPER MODLEING & RESULT
Table 3. THE STEP OF THE PAPER MODEL
proposed previous
NJ 57 587
NR 2754 8318
68
1.4.5. Conclusion In this study, static equilibrium of Augmented Newton-Raphson method is
proposed. Augmented Newton-Raphson method is applied to MBD and MFBD for
3 models as chapter 4. The efficiency is better than previous one.
REFERENCES [1] Bae, D. S., Han, J. M., and Yoo., H. H., 1999, “A Generalized Recursive
Formulation for Constrained Mechanical System Dynamics”, “Mech. Struct. &
Mach.”, Vol. 27(3), pp. 293-315
[2] Wittenburg, J., 1977, "Dynamics of Systems of Rigid Bodies", B. G. Teubner,
Stuttgart
[3] Conn, A.R., Gould, N.I.M. and Toint, P.L., 2000, Trust-Region Methods, Siam,
Philadelpia.
[4] Jones, C.D., Perttunen, C.D. and Stuckman, B.E., 1993, “Lipschitzian
optimization without the Lipschitzian Constant”, Journal of Optimization Theory
and Application, Vol. 79, No.1, pp.157-181.
[5] Kim M.-S. and Choi, D.-H., 1995, “Development of an Efficient Line Search
method by using the Sequential polynomial Approximation”, KSME(in Korean),
Vol. 19, No.2, pp.433-442.
[6] Functionbay, “RecurDyn version 6.4 Solver Theoretical Manual”, 2007.
2. Contact
70
2.1
AN EFFICIENT CONTACT SEARCH ALGORITHM
FOR GENERAL MULTIBODY SYSTEM DYNAMICS
2.1.1. INTRODUCTION
This paper presents a contact analysis algorithm employing the relative
coordinate system for the multibody system dynamics. Multiple-contact higher
pairs are widely used in mechanical systems such as walking machines, feeding
systems, driving chains, and tracks of off-road vehicles. Common design
problems due to the multiple contacts among bodies are undercutting, jamming,
backlash, and body interference.
The configuration space representation of a higher pair was proposed by
Lozano-Perez [1] for robot motion planning. Sacks extended the configuration
space concept in [2] for efficient detection of contact pairs. The relative position
and orientation of a pair were mapped into the configuration space. The degrees
of freedom of a pair became the dimension of the configuration space, which is
divided into free space and contact space in the preprocessing stage of a dynamic
analysis and is tabulated into a database. Run time query is made to decide
whether a pair is currently in contact or not. When a higher pair has many
degrees of freedom, formation of the configuration space and processing effort
for a run time query may become extensive.
Wang presented an interference analysis method in [3]. Relative coordinates
were defined for a contact pair and a kinematic closed loop including the contact
pair was formed. Constraint equations arising from closed loops are solved for
the relative coordinates including the ones for the contact pair. The canonical
Hamiltonian formulation is used to derive a minimal set of dynamic equations of
motion.
Mirtich proposed a contact detection algorithm consisting of narrow and broad
phases in [4]. Candidate features are selected in the broad phase and contact
71
inspection is carried out in the narrow phase only among the candidate features.
Haug presented a formulation for domains of mobility that characterizes
kinematic boundaries of multiple contact pairs in [5]. A surface-surface contact
joint was developed by Nelson in [6]. Piecewise dynamic analysis method for a
contact problem was employed in [7, 8]. Dynamic analysis is halted when a
contact pair is detected to be in contact and is resumed with new velocities that
are calculated from the momentum balance equations. One of drawbacks of this
method is that too frequent halting and resuming of the numerical integration
may occur when a contact pair toggles between contact and not contact status.
Zhong summarized many contact search algorithms in the area of the finite
element analysis in [9]. All geometric variables necessary to detect a contact
were expressed in the absolute Cartesian coordinate system. The penalty and
Lagrange multiplier methods were proposed. The compliant contact model that
is based on the Herzian law was used in [10]. Since the contact force is large and
varied significantly, the differential equations of motion for this method are
generally stiff.
A recursive formulation using the relative coordinates was proposed by Bae in
Ref. [11]. The equations of motion were derived in a compact matrix form by
using the velocity transformation method. The actual computation was carried
out by using the recursive formulas developed for each joints. Realtime
simulation of a vehicle system is carried out by the recursive method in Ref. [12].
The Jacobian matrix was updated once in while during time marching of the
numerical integration. The recursive method was extended to the flexible body
dynamics of constrained mechanical systems in Ref. [13]. A virtual body
concept was employed to relieve the implementation burden of the flexible body
dynamics coding. A compliant track link model was developed for tracked
vehicles in Ref. [14]. A minimum set of the equations of motion was obtained by
the recursive method. Concept of the configuration design variable with the
recursive formulation was introduced in Ref. [15]. The recursive method is
applied to efficiently detect a contact in this research.
This paper presents a hybrid contact detection algorithm of the configuration
space method and bounding box method in conjunction with the compliant
contact model. Two bodies of a contact pair are logically considered as a defense
72
body on which the contact reference frame is defined and as a hitting body that
moves relative to the defense body, respectively. Contour of the defense body is
approximated by many triangular patches which are projected on axes of the
contact reference frame. Bounding box inside which contains base surface is
divided into several blocks each of which is indexed on axis of the contact
reference frame. Contact inspection for a contact pair is processed in the sequence
of broad and narrow phases. Relative position vector of the hitting body to the
defense body is projected on the axes of the contact reference frame and select
candidate features that may come in contact shortly in the broad inspection phase,
which greatly reduces the searching effort. It is not needed any database to be
built prior to an analysis. Since the searching algorithm is coupled with stepping
algorithm of the numerical integration, a strategy for deciding an integration
stepsize is proposed. A numerical example is presented to demonstrate the
validity of the proposed method.
2.1.2. KINEMATIC NOTATIONS OF A CONTACT PAIR
Consider a contact pair shown in Fig. 1. Two bodies of the contact pair will be
referred as a hitting body and a defense body for convenience in the following
discussions, respectively. The contours of the hitting and defense bodies will be
referred as the hitting and target boundaries, respectively.
Figure 1 Kinematic notations of a contact pair
73
Figure 2 Contact reference frame and generalized coordinate
The ZYX coordinate system is the inertial reference frame and the
zyx primed coordinate systems are the body reference frames. The
orientation and position of the body reference frame is denoted by A and r ,
respectively.
Double primed coordinate systems are the node reference frame of the hitting
body and the surface and contact reference frames of the defense body,
respectively. All geometric variables of the defense body are measured on the
surface reference frame. The contact reference frame for the contact pair is
defined on the left corner of the bounding box of the defense body, as shown in
Fig. 2. The relative position and orientation of the hitting body to the defense
body are defined as the generalized coordinates, which are denoted by chd and
chA as shown in Fig. 2. Therefore the generalized coordinates are directly used
to detect a contact for the pair.
2.1.3. DIVISION OF THE CONTACT DOMAIN
A surface-to-surface contact problem can be replaced by multiple sphere-to-
surface contact problems. Therefore, the sphere-to-surface contact problem will
be discussed in this research.
74
Contour of a smoothly shaped body has been represented by the 3D
NURBS(Non-Uniform Rational B-Spline)[16] in many commercial CAD
programs. Since it is computationally extensive to find intersection lines or
points between two surfaces, the defense surface is approximated by triangular
patches and the boundary of the hitting body is represented by a set of spheres,
as shown in Fig. 3. The numbers of patches and spheres must be decided by the
degree of accuracy required.
Figure 3 Approximated defense and hitting surfaces
The bounding box of the defense surface in space can be divided into many
blocks each of which has a list of patches lying inside or on the block to
efficiently process a contact detection, as shown in Fig. 4. Since the block
locations are tabulated with respect to the contact reference frame attached to the
defense body, they are constant. As a result, the locations do not needed to be
calculated at every time steps, which significantly reduces computation time
associated with the contact search.
75
Figure 4 Relationship patch and block: The patch, p belongs in the block, b
2.1.4. PRE-SEARCH
Every pairs of the boundary nodes of the hitting body and the patches on the
defense body must be examined to detect a contact between two bodies, which is
computationally extensive. In order to save the extensive computation, each node
of the hitting body searches to find blocks of the contact domain to which it
belongs in the pre-search stage, as shown in Fig. 5.
The relative position and orientation of the hitting body reference frame with
respect to the contact reference frame shown in Fig. 2 can be directly available
from the generalized coordinate chd and chA . Therefore, the relative nodal
position of the hitting body with respect to the contact reference frame is
obtained as
nchchcn sAdd (1)
where ns is the nodal position with respect to the hitting body reference
frame. Direct comparison of the cnd with this of the block locations of the
defense body yields the state of a contact.
76
Figure 5 Node and blocks in pre-search stage
If a pair of a node and a block is in contact, post-search step will be proceeded.
The bounding box of the defense body is divided into many blocks. Each block
has a list of patches lying within or on the block boundary. Therefore, the post-
search step will be carried out only for the patches belonging to the blocks that
have found to be in contact in pre-search step, as shown in Fig. 5.
2.1.5. POST-SEARCH AND COMPLIANCE CONTACT FORCE
The candidate patches on the defense surface have been selected for the post
search step in the pre-search step. For the candidate patches, it is necessary to
compute the amount of penetration to generate the contact forces, as shown in
Fig. 6.
Figure 6 Node and patch in post-search stage
78
The relative position pnd of a node with respect to the patch reference frame
is obtained as follows.
1pcnpn sdd (2)
where
The vector pnd is projected into the patch reference frame as
pn
T
ppn dCd (3)
where pC is the orientation matrix of the patch reference frame with respect
to the contact reference frame.
The first step in the post search is to check whether the node is in contact with
the patch or not by inspecting pnd . In case of non-contact, the rest of procedures
must be skipped. Otherwise, the penetration of the node into the patch is
calculated by
pn
T
p-rδ dn (4)
where δ is always positive. The pn is a normal vector of a patch and a
constant vector with respect to the patch reference frame.
Thus, the contact normal force is obtained by
3
21 m
mm
n δδδ
δckδf
(5)
where k and c are the spring and damping coefficients which are
determined by an experimental method, respectively and the δ is time
differentiation of δ . The exponents 1m and 2m generates a non-linear contact
force and the exponent 3m yields an indentation damping effect. When the
penetration is very small, the contact force may be negative due to a negative
damping force, which is not realistic. This situation can be overcome by using
78
the indentation
damping exponent greater than one.
The friction force is obtained by
nf fμf (6)
where μ is the friction coefficient and its sign and magnitude can be
determined from the relative velocity of the pair on contact position.
2.1.6. KINEMATICS AND EQUATION OF MOTION FOR THE
RECURSIVE FORMULAS
A contact search algorithm is proposed in the previous sections. The proposed
method makes use of the relative position and orientation matrix for a contact
pair. This section presents the relative coordinate kinematics for a contact pair as
well as for joints connecting two bodies.
Translational and angular velocities of the zyx frame in the ZYX
frame are respectively defined as
w
r (7)
Their corresponding quantities in the zyx frame are defined as
wA
rA
w
rY
T
T (8)
where Y is the combined velocity of the translation and rotation. The
recursive velocity and virtual relationship for a pair of contiguous bodies are
obtained in [17] as
1)i(i1)i2(i1)(i1)i1(ii qBYBY (9)
80
where 1)i(iq denotes the relative coordinate vector. It is important to note that
matrices 1)i1(iB and 1)i2(iB are only functions of the 1)i(iq . Similarly, the recursive
virtual displacement relationship is obtained as follows
1)i(i1)i2(i1)(i1)i1(iiδ δqBδZBZ (10)
If the recursive formula in Eq. (9) is respectively applied to all joints, the
following relationship between the Cartesian and relative generalized velocities
can be obtained:
qBY (11)
where B is the collection of coefficients of the 1)i(iq and
T1nc
TT
2
T
1
T
0 nY,,Y,Y,YY (12)
T1nr
T
)1(
T
12
T
01
T
0 nnq,,q,q,Yq (13)
where nc and nr denote the number of the Cartesian and relative coordinates,
respectively. Since q in Eq. (11) is an arbitrary vector in nrR , Eqs. (9) and (11),
which are computationally equivalent, are actually valid for any vector nrRx
such that
xBX (14)
and
1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX (15)
where ncRX is the resulting vector of multiplication of B and x . As a
result, transformation of nrRx into nc
RBx is actually calculated by
recursively applying Eq. (15) to achieve computational efficiency in this
research.
Inversely, it is often necessary to transform a vector G in ncR into a new
81
vector GBgT in nr
R . Such a transformation can be found in the generalized
force computation in the joint space with a known force in the Cartesian space.
The virtual work done by a Cartesian force ncRQ is obtained as follows.
QZWΤδδ (16)
where Zδ must be kinematically admissible for all joints in a system.
Substitution of qBZ δδ into Eq. (16) yields
*TTT δδδ QqQBqW (17)
where QBQT* .
The equations of motion for constrained systems have been obtained as
follows.
0)QλΦYMBFΤ
Ζ
T ( (18)
where the λ is the Lagrange multiplier vector for cut joints [18] in mR
and Φ represents the position level constraint vector in mR . The M and Q
are the mass matrix and force vector in the Cartesian space including the contact
forces, respectively.
The equations of motion and the position level constraint can be implicitly
rewritten by introducing vq as
0λa,vqF ),,( (19)
0qΦ )( (20)
Successive differentiations of the position level constraint yield
0υvΦvqΦ q ),( (21)
0γvΦvvqΦ q ),,( (22)
82
Equation (19) and all levels of constraints comprise the over determined
differential algebraic system (ODAS). An algorithm for the backward
differentiation formula (BDF) to solve the ODAS is given in [19] as follows.
0
βvvU
βvqU
vvqΦ
vqΦ
qΦ
)λv,vq(F
xH
)β(
)β(
,,
,
,,
)(
20
T
0
10
T
0
(23)
where TTTTTλ,v,v,qx , 0β ,
1β and 2β are determined by the
coefficients of the implicit integrators and 0U is an m)(nrnr matrix such
that the augmented square matrix
qΦ
UT
0 is nonsingular.
The number of equations and the number of unknowns in Eq. (23) are the
same, and so Eq. (23) can be solved for nx . Newton Raphson method can be
applied to obtain the solution nx .
HΔxHx (24)
1,2,3,...i,i1i Δxxx (25)
0
0UU0
00UU
0ΦΦΦ
00ΦΦ
000Φ
FFFF
H
T
0
avq
vq
q
qqqq
x
T
00
T
00
T
0
β
β
(26)
Recursive formulas for xH and H in Eq. (24) are derived to evaluate them
efficiently.
83
2.1.7. NUMERICAL INTEGRATION STRATEGY
The sufficient condition for a successful numerical integration step is to
satisfy both accuracy and stability of the state variables for a system without
contact. Satisfaction of the accuracy and stability is not sufficient for a system
with a contact. Suppose a bullet collides with an object. If the object is thin, the
bullet passes through the object without noticing it. If the object is thick and a
moderately large step size satisfies both the accuracy and stability, the bullet
penetrates too deep
at the first step of a contact. Large and sudden contact force due to the large
penetration generally introduces a large numerical error in the state variables.
The large numerical error often causes the integration step to fail. Therefore, the
contact condition must be considered in deciding an integration step. In order to
make a system transition from a non-contact status to a contact status smooth as
much as possible, time of contact must be predicted accurately. However, the
computationally extensive search algorithm must be triggered to predict the
exact time of a contact even though two bodies of a contact pair are located in a
distance. Easy and practical solution to this problem is to use the method of
backtracking.
Figure 7 Buffer radius of a node
This paper adopted the concept of buffer radius shown in Fig. 7. In post-search
stage, if no nodes with radius in the hitting body is contacted with the candidate
85
lines in the defense body and some nodes with buffer radius are contacted, the
integrating step will be decreased.
2.1.8. NUMERICAL EXAMPLE
The proposed algorithm is implemented in the commercial program RecurDyn.
A paper-feeding problem of a copying machine is solved to demonstrate the
efficiency and validity of the proposed method.
Figure 8 Copying machine
The system has 255 degree of freedom and consists of five roller pairs and one
paper shown in Fig. 8. Each roller pair is modeled by using two driving rollers,
two idlers, two driving bars, two idler bars, six joints and one nip spring. The
paper is modeled by using 40-segmented bodies and 28 plate force elements. The
segmented paper bodies and the roller pairs are contacted and it is modeled by
using 160 sphere to surface contacts.
The paper goes through a path while contacting the roller pairs. The angular
velocity of each driving roller reaches 10 rad/sec during one second. The
tangential velocities of a driving roller and a leading segment body of the paper
are shown in Fig. 9. The static and dynamic friction coefficient is 0.5 and 0.3,
respectively.
85
The analysis was performed on an IBM compatible computer (PIII-933Mhz)
and took about 260 sec. per 1 sec. for simulation. A copying machine is solved to
demonstrate the effectiveness of the proposed algorithm
Figure 9 Tangential velocities of a driving roller and a leading segment body of paper
on a contact point
2.1.9. CONCLUSIONS
This research proposes an efficient implementation algorithm for contact
mechanisms. The contact domain is divided into many blocks each of which
contains the list of patches inside it. The search process consists of pre-search
and post search steps. In the pre-search step, the bounding box technique is
employed to find approximate contact state. Once the contact is detected in the
pre-search step, the detailed contact condition is further examined in the post-
search step. The compliance contact model is used to generate the contact force
which is applied to the hitting and defense bodies. The relative coordinate
formulation is used to generate the equations of motion. The local
parameterization method is used to solve the differential algebraic equations.
The integration stepsize is automatically reduced when a contact is expected
soon. The proposed algorithm is implemented in the commercial program
RecurDyn and a copying machine example is successfully solved.
86
REFERENCES
1. Lozano-Perez, T., "Spatial Planning: A Configuration Space Approach", IEEE
Transactions on Computers, Vol. C-32, IEEE Press, 1983.
2. Sacks, E. and Joskowicz, L., "Dynamical Simulation of Planar Systems with Changing
Contacts Using Configuration Spaces", "Journal of Mechanical Design", Vol. 120, pp.
181~187, 1998.
3. Wang, D., Conti, C. and Beale, D., "Interference Impact Analysis of Multibody Systems",
"Journal of Mechanical Design", Vol. 121, pp. 121-135, 1999.
4. Mirtich, B. V., "Impulse-based Dynamic Simulation of Rigid Body Systems", Ph. D thesis,
University of California, Berkeley, 1996.
5. Haug, E. J., Wu, S. C. and Yang, S. M., "Dynamic mechanical systems with Coulomb
friction, stiction, impact and constraint addition-deletion, I: Theory", "Mech. Mach.
Theory", Vol. 21(5), pp. 407-416, 1986.
6. Nelson, D. D. and Cohen, E., "User Interaction with CAD Models with Nonholonomic
Parametric Surface Constraints", Proceedings of the ASME Dynamic Systems and
Control Division, DSC-Vol. 64, pp. 235-242, 1998.
7. Wang, D., Conti, C., Dehombreux, P. and Verlinden, O., "A Computer-aided
Simulation Approach for Mechanisms with Time-Varying Topology", "Computers and
Structures", Vol. 64, pp. 519-530, 1997.
8. Wang, D., "A Computer-aided Kinematics and Dynamics of Multibody Systems with
Contact Joints", Ph. D Thesis, Mons Polytechnic University Belgium, 1996.
9. Zhong, Z. Z., "Finite Element Procedures for Contact-Impact Problems", Oxford
University Press, 1993
10. Lankarani, H. M., "Canonical Impulse-Momentum Equations for Impact Analysis of
Multibody System", ASME, "Journal of Mechanical Design", Vol. 180, pp. 180-186,
1992
11. Bae, D. S., Han, J. M., and Yoo., H. H., “A Generalized Recursive Formulation for
87
Constrained Mechanical System Dynamics”, “Mech. Struct. & Mach.”, Vol. 27(3), pp.
293-315, 1999.
12. Bae, D. S., Lee, J. K., Cho, H. J., and Yae, H., “An Explicit Integration Method for
Realtime Simulation of Multibody Vehicle Models”, “Computer Methods in Applied
Mechanics and Engineering”, Vol. 187, pp. 337-350, 2000.
13. Bae, D. S., Han, J. M., Choi, J. H., and Yang, S. M., “A Generalized Recursive
Formulation for Constrained Flexible Multibody Dynamics”, “International Journal for
Numerical Methods in Engineering”, Vol. 50, pp. 1841-1859, 2001.
14. Ryu, H. S., Bae, D. S., Choi, J. H., and Shabana, A. A., “A Compliant Track Link Model
for High-speed, High-mobility Tracked Vehicles”, “International Journal for Numerical
Methods in Engineering”, Vol. 48, pp. 1481-1502, 2000.
15. Kim, H. W., Bae, D. S., and Choi, K. K., “Configuration Design Sensitivity Analysis of
Dynamics for Constrained Mechanical Systems”, “Computer Methods in Applied
Mechanics and Engineering”, Vol. 190, pp. 5271-5282, 2001.
16. Farin, G., "Curves and Surfaces for Computer-aided Geometic Design", Academic
Press, 1997.
17. Angeles, J., "Fundamentals of Robotic Mechanical Systems", Springer, 1997.
18. Wittenburg, J., "Dynamics of Systems of Rigid Bodies", B. G. Teubner, Stuttgart, 1977.
19. Yen, J., Haug, E. J. and Potra, F. A., "Numerical Method for Constrained Equations
of Motion in Mechanical Systems Dynamics", Technical Report R-92, Center for
Simulation and Design Optimization, Department of Mechanical Engineering, and
Department of Mathematics, University of Iowa, Iowa City, Iowa, 1990.
88
2.2.
AN EFFICIENT AND ROBUST CONTACT
ALGORITHM FOR A COMPLIANT CONTACT
FORCE MODEL BETWEEN BODIES OF
COMPLEX GEOMETRY
2.2.1. INTRODUCTION
Recently the contact problem for multibody dynamics has been an issue not only
for engineering problems but also for video game engines. Contact analysis
between highly complex geometries which cannot be represented by simple
surfaces, such as spheres and cylinders, is especially challenging. It is even
challenging to perform contact analyses between less complex surfaces that can be
represented with smooth splines.
In general, contemporary multibody dynamics problems for both the analysis of
realistic mechanical systems and virtual reality environments like those found in
video games are composed of a large number of rigid bodies, constraints, and
external forces. In a game engine, generally, accuracy is not a main concern
because fast and reliable visualization is more important than accuracy. However,
in engineering problems, accuracy is as important as solving speed because the
main objective is to find a realistic solution for a given problem.
There are a number of significant issues in the modeling of contact. One issue is
the enforcement of non-penetration between contacting bodies. Contact can be
modeled using algebraic constraints that strictly prevent the interpenetration of
bodies. Or it can be modeled with a compliant contact force model that allows
slight interpenetration of the undeformed body surfaces. Contact models based on
compliant forces have the ability to easily approximate the deformation that would
occur in the real material in the region of contact [1]. For that reason, in order to
get a realistic and continuous contact force during the entire period of contact, this
paper uses a one-dimensional compliant contact force model based on a penalty
method for each contact region [2-8].
Another significant issue in modeling contact is the representation of the contact
surfaces of the bodies. Various methods for representing surfaces exist, but some
are more appropriate for modeling contact than others. Simple shapes can be
modeled with analytical equations defining their surfaces. Others can be very
closely modeled using smooth splines. But in many of these cases, no good contact
algorithm exists yet that is fast and robust when the shape of the geometry is very
89
irregular and complex. Also, even though a smooth spline approach is more
accurate and efficient than a triangle approach for representing the shape, the
development of a fast and robust algorithm for finding contact locations for
complex geometries using smooth splines is more a challenging and difficult issue
than using triangles. But the triangle approach can be more suitable than the
smooth spline approach for various complex contact problems because it is simpler
and easier to handle. Therefore, before developing a general contact algorithm
using smooth splines, in this study, triangles are used to represent the complex
surfaces.
In recent contact analyses of multibody dynamics, the most important and
difficult things are to perform the collision detection (pre search) and to query the
collision response (detailed search) for the geometrical information such as a
penetration depth or a contact reference frame (a contact point and normal and
tangent directions). A fast and robust algorithm for finding the penetration depth is
essential when a compliant contact force model is used. Recently, these kinds of
contact search algorithms have been studied widely by using triangles for the
surface representation method in computer graphics, robotics, and computational
geometry literature [9-13]. Bounding volume hierarchies such as axis-aligned
bounding box (AABB) trees [14] or oriented bounding box (OBB) trees [15,16]
have been widely used to accelerate the performance of collision detection
algorithms.
But until now, there still exist severe bottlenecks or problems in pre and detailed
search algorithms when a large number of triangles are used to represent complex
(convex or non-convex) contact surfaces. For example, Teschner et al. [19] or Cho
et al. [20] used a spatial hashing concept as a spatial partitioning method in order to
improve the performance of pre search in the contact analysis between complex
rigid or deformable bodies, but they did not develop an efficient detailed search
algorithm. They just used a node-to-surface contact concept to calculate the
penetration depth as the detailed contact search algorithm. As a result, even though
they used a fast pre search algorithm, the solving time can increase rapidly as the
number of triangles and nodes increases for more accurate surface representation,
because the penetration depth should be calculated for all combinations of expected
primitives such as nodes or triangles. Furthermore, there are a number of
shortcomings to spatial partitioning methods when they are used for collision
detection. Their speed and memory efficiency depends highly on an appropriate
choice of the grid size. However, determining the appropriate size is not easy.
Furthermore, such methods only find potentially intersecting triangle pairs rather
than actually intersecting pairs. Some detailed search algorithms require finding
intersecting triangle pairs, in which case an algorithm for testing the potential pairs
for intersection is required. On the other hand, as a similar example, Hippmann [21]
used a collision detection algorithm based on an AABB tree to enhance the
90
performance of collision detection. He proposed a method to find sets for master
and slave active triangles, which include intersected and inner triangles, in order to
solve multiple-contact-region or multiply-bordered-contact cases. He also proposed
a triangle-to-triangle collision concept as a method for generating the contact force.
However, because the triangle-to-triangle collision concept is used for all
combinations between master and slalve active triangles, even though an efficient
pre search algorithm is used, the solving time can increase rapidly as the number of
active triangles increases.
Also, because of the geometrical complexity, a fast and general penetration depth
calculation algorithm for complex geometries is still a significant challenge in
contact analyses [22-24]. Until now, contact algorithms have been resolved clearly
only in the case of convex geometries. For non-convex models, no algorithms for
finding penetration depth have yet been proposed that are fast, general, and
computationally simple. In order to calculate the penetration depth between
complex geometries in realtime, many researchers have developed penetration
depth calculation algorithms that rely on the speed of graphic hardware [25-27].
But, because those algorithms have been mainly developed and used for the
physically-based animation such as the kind found in game engines, solving speed
is more important than solution accuracy which is important in engineering
problems. Therefore, in this paper, an efficient, robust, and computationally simple
contact algorithm between bodies of complex geometry, which is called the triangle
soup average plane contact (TSAPC) algorithm, is proposed. The TSAPC
algorithm does not use graphic hardware and is illustrated by being compared with
an analytic method for penetration depth calculation.
In section 2, the kinematic notation conventions and the surface representation
method are explained. In section 3, the bounding box tree and the overlap test are
used to enhance the contact pre search performance and the connectivity
information is used to separate multiple contact regions into each individual contact
region. From the results of pre search, in section 4, the efficient and robust detailed
search algorithm for the penetration depth and contact reference frame is proposed.
In section 5, the modified compliant contact force model to generate the contact
force is explained. The solution accuracy and performance are discussed with
numerical examples in section 6 and the conclusion is presented in section 7.
2.2.2. KINEMATIC NOTATION CONVENTIONS AND
SURFACE REPRESENTATION
2.2.2.1. KINEMATIC NOTATION CONVENTIONS
This investigation examines contact between two bodies. The two bodies can be
91
labeled arbitrarily; in this paper, one is called the base body and the other is called
the action body. The TSAPC algorithm is invariant with respect to the choice of
base and action body except for numerical differences. To define contact in 3D
space, a base contact surface on the base body and a contact reference frame in
global space are considered, as shown in Figure 1. The X-Y-Z coordinate system is
the inertial or global reference frame. The x -y -z primed coordinate systems are
body reference frames. The x -y -z double-primed coordinate systems are surface
reference frames. The x -y -zc c c
coordinate systems are contact reference frames.
The subscripts i and c indicate that the values are for base and contact
information, respectively. iA and cA are orientation matrices with respect to
X-Y-Z . ir and cr are position vectors from the origin of X-Y-Z to the origin of
x -y -zi i i and x -y -z
c c c, respectively. iC is the orientation matrix of x -y -zi i i
with
respect to x -y -zi i i . is is a position vector from the origin of x -y -zi i i
to the origin
of x -y -zi i i . id and ics are position vectors from the origin of X-Y-Z to the
origin of x -y -zi i i and from the origin of x -y -zi i i
to the origin of x -y -zc c c
,
respectively. Similarly, kinematic notation conventions for an action contact
surface can be represented using subscript j .
Fig. 1 Kinematic notation conventions of a base contact surface and a contact reference
frame.
2.2.2.2. SURFACE REPRESENTATION METHOD
Generally, there are many approaches to represent surfaces. One method is to
represent the surfaces using analytical functions. The analytic surface approach is
very efficient and exact in the case of simple surfaces such as spheres or cylinders.
But it is difficult to apply to complicated surfaces such as those found in general
Ci
Ai
si
ri
di = ri + Ai si’
sic
Contact Point
Ac
X
Z
Y
xi’
yi’
zi’
xi’’yi’’
zi’’
yc
xc
zc
Inertial Ref. Frame
rc
Body Reference Frame
Contact Reference Frame
Surface Reference Frame
Ci
Ai
si
ri
di = ri + Ai si’
sic
Contact Point
Ac
X
Z
Y
xi’
yi’
zi’
xi’’yi’’
zi’’
yc
xc
zc
Inertial Ref. Frame
rc
Body Reference Frame
Contact Reference Frame
Surface Reference Frame
92
CAD models. CAD models generally use rational and non-rational polynomial
splines, which are capable of representing a wide range of complex, smooth
surfaces that have complex curvature.
Another method for representing surfaces is to use triangles. The triangle
approach is not exact in describing simple, curved primitive geometry that can be
exactly described by analytical functions, and it is not as exact as splines for
complex, smooth surfaces. But because analytical functions can only describe
simple surfaces, they have limited usefulness in contact algorithms for engineering
problems. Furthermore, contact algorithms for smooth splines are very complex.
On the other hand, algorithms for finding contact with surfaces represented by
triangles are relatively simple and can analyse contact with arbitrarily complex
geometry. In this paper, the triangular representation is adopted to represent the
surfaces.
To define the contact, two rigid bodies are needed, each with its own surface.
Figure 2 shows an example of a cam-valve contact problem between cam body and
valve body; the cam body is chosen as the base body. Figure 3 shows the surface
representation with triangles for the cam-valve contact example model. Generally,
in order to get accurate contact results for the complicated surfaces, a large number
of triangles should be used.
Fig. 2 A cam-valve contact problem example. The cam body is chosen as the base body
and the valve as the action body.
X
Y
Z
YBase Body(i)
Action Body(j)
Spring
Revolute Joint & Motion
Translational Joint
X
Y
Z
YBase Body(i)
Action Body(j)
Spring
Revolute Joint & Motion
Translational Joint
93
Fig. 3 Surface representation using triangles for the base and action surfaces in the cam-
valve contact problem example.
Generally, the geometry of bodies used in engineering problems are modeled
using CAD software, which uses splines to represent the surfaces. In order to
generate triangles from the CAD models, the spline surfaces must be faceted or
meshed [28]. After faceting or meshing the contact surface with triangles, bounding
box trees for the base and action surfaces should be built [14-16]. When building
the tree, triangle and node data is needed. The triangle data includes three node ids,
and the node data includes its position vector expressed in the surface reference
frame x -y -z . When using bounding box trees with rigid bodies, the node
positions do not change during simulation, because they are expressed in the
surface reference frame. Therefore, there is no need to update the bounding box
tree at each time step. This makes the bounding box tree approach very efficient for
rigid body contact problems.
In this study, the triangle and node data must satisfy the following conditions in
order to use triangle connectivity information, which stores neighbor triangle ids
sharing one of edges of a triangle, in the pre search algorithm:
(a) Each node must be unique.
(b) The number of edges connected to every node must be greater than or equal to 2.
(c) Every triangle edge must not belong to more than 2 triangles.
(d) The area of each triangle must be greater than zero.
2.2.2.3. AVAILABLE SURFACE
Figure 4 shows the concept of an available surface. An available surface is a
surface which can create an enclosed contact volume. This algorithm is designed to
allow non-closed surfaces (i.e., surfaces that do not completely enclose a volume)
to be used as contact surfaces. However, the algorithm is designed to only allow for
contact to exist between the surfaces if the two surfaces overlap in such a way as to
enclose a volume. If the surfaces intersect in 3D space but do not enclose a volume
between them, then the TSAPC algorithm will not be able to detect contact
appropriately in this intersecting region. Therefore, the understanding of the
94
available surface concept is essential because the detailed search algorithm in
section 4 is developed from the assumption that the base and action surfaces satisfy
the definition of available surfaces. As shown in Figure 4(a), the available base and
action surfaces make an enclosed contact volume in 3D space. On the other hand, if
the base and action surfaces do not make an enclosed contact volume at any
simulation time as shown in Figure 4(b), those surfaces cannot be used as available
surfaces in the detailed search algorithm. From the geometrical definition, solid,
completely enclosed surfaces can always be used as an available surface.
(a) Available surfaces.
(b) Not available surfaces.
Fig. 4 The concept of an available surface. (a) Available surfaces, (b) Not available
surfaces.
2.2.3. PRE SEARCH
As discussed in the previous section, in order to represent the surface accurately
for complex geometry, in general, a large number of triangles should be used. But,
in general, at any given time, the number of penetrating triangles is expected to be
very small compared to the total number of triangles. If we check all triangles of
the base and action surfaces to calculate the penetration depth and contact reference
frame in detailed search algorithm, the high computation effort would lead to
unacceptable simulation performance. Therefore we need to efficiently find all of
Available Surfaces
Enclosed Contact Volume
Surface Normals
Available
Available Surfaces
Enclosed Contact Volume
Surface Normals
Available
Not Available Surfaces
Not Enclosed Volume
Surface Normals
Not Available
Not Available Surfaces
Not Enclosed Volume
Surface Normals
Not Available
95
the intersecting or penetrating triangles which should be used in the detailed search
algorithm. The intersecting triangles are found in pairs that are composed of one
triangle on the base body and one on the action body. This is the objective of the
pre search.
In this paper, the pre search is composed of two parts. The first part is to find
intersecting triangle pairs by using the recursive overlap tests between base and
action bounding boxes along each bounding box tree. To achieve this, this study
used an AABB tree as a intersection detection algorithm. But other algorithms such
as the OBB tree can also be used instead of an AABB tree. These methodologies
are well studied in references [14-18]. Therefore, using this kind of methodology,
all intersecting triangle pairs can be found efficiently at each time step. In this study,
the collection of all intersecting triangle pairs is called the “global triangle pair set”,
as shown in Figure 5.
The pre search algorithm also contains a second component. The second
component is responsible for identifying separate regions of contact. In general,
when two complex bodies are in contact, it is possible that there are multiple,
separate contact regions. Each region is identified by a single enclosed contact
volume. If this is the case, then the global triangle pair set will contain intersecting
pairs of triangles for all of the contact regions. However, the detailed search
requires that each contact region be individually identified. Therefore, the pre
search must also separate the intersecting triangle pairs in the global triangle pair
set into local sets for each individual contact region, which are called sub local
triangle pair sets. The algorithm that separates the triangle pairs for different
contact regions can be implemented by using the triangle connectivity information.
The algorithm steps for the pre search can be summarized as follows:
(a) Find all intersecting triangle pairs (the global triangle pair set, ( )gp gpnS )
between the base and action surfaces as shown in Figure 5 by using the
bounding box trees and recursive overlap tests. Here, gpn is the total
number of triangle pairs. ( )gp gpnS is a set of id pairs, in which each pair
contains a base triangle id and an action triangle id, as follows:
( ) ( ( ) ( )),gp i jk k kS G G , 1 gpk n
Here, G is a global set of intersecting triangle ids, and subscripts i and
j mean base and action, respectively. The subscript gp means global
pairs, and k is the index for a set or array.
(b) Separate the global triangle pair set into the sub local triangle pair sets for
each contact region by using the triangle connectivity information. As
described in Section 2.2, during the contact surface representation process,
96
neighbor triangle ids sharing an edge with the current triangle are found. As a
result, for each contact surface of the base and action bodies, triangles can be
separated into sub local triangle sets. And then, each base and action sub
local triangle set is defined as a contact region if it has intersected triangles.
As a result, each local triangle pair set ( ( )rnlp lpnS ) makes the -thrn contact
region. Here, superscript rn is the contact region index and lpn is the total
number of intersecting triangle pairs in the contact region rn . Therefore
( )rnlp lpnS can be expressed as follows:
( ) ( ( ) ( )),r r rn n nlp i jk k kS L L , 1 rn
lpk n , 1 r mrn n
Here, L is a local set of intersecting triangle ids and mrn is the total
number of contact regions. The subscript lp means local pairs, and k is the
index for a set or array.
(c) Each contact region ( rn ), call the detailed search algorithm to find the
penetration depth and contact reference frame, which includes the contact
point and the normal and tangent (or friction) directions.
Fig. 5 Multiple contact regions.
2.2.4. DETAILED SEARCH
The detailed search algorithm finds the penetration depth and the contact
reference frame for each individual contact region. It finds only one penetration
depth and contact reference frame for each contact region, which allows the
Action Surf.
Base Surf.
Surface Normals
Intersecting Triangle Pairs
Global Triangle Pair Set
Local Triangle Pair Set 1 ( Contact Region 1 )
Local Triangle Pair Set 2( Contact Region 2 )
Action Surf.
Base Surf.
Surface Normals
Intersecting Triangle Pairs
Global Triangle Pair Set
Local Triangle Pair Set 1 ( Contact Region 1 )
Local Triangle Pair Set 2( Contact Region 2 )
97
TSAPC algorithm to be computed quickly. This information is generated by finding
a plane that closely passes through the points of intersection between the base and
action surfaces. Because the surfaces in contact must satisfy the available surface
property, the intersection of the two surfaces forms a polygon in space. This
polygon is not guaranteed to be planar. Therefore, a single plane is found by
minimizing the error between the polygon and the plane. This plane is then used to
define the penetration depth and the contact reference frame.
From the pre search, the local triangle pair sets ( )r rn nlp lpnS are found for every
contact region rn . Each local triangle pair set identifies one contact region. In each
contact region, one plane, one contact point, and one contact reference frame are
defined. The intersection points of the triangles of each triangle pair in the local
triangle pair set ( )lp lpnS are used to find these properties.
Fig. 6 The concept of the detailed search.
The algorithm steps for the detailed search can be summarized as follows:
(a) Find intersecting points ( )ip ipnS between the triangles of the base and
action surfaces, as shown in Figure 6. ipS is a set of position vectors and
ipn is the total number of intersecting points. ipS can be expressed as
follows:
( ) { ( ), ( ), ( )}ip k k k kS x y z , 1 ipk n , 3ipn
Here, x , y , and z are coordinate components in the inertial reference
frame X-Y-Z . Here, ipS is calculated in two stages. In the first stage, the
intersection points are found between the base triangle plane and the three
edges of the triangle on the action body. In the second stage, the
Action Surf.
Base Surf.
Contact Plane
un
pcj
pci
Surface Normals
Intersecting Points ( Intersecting Points ( SSip ip ))
pc
Action Surf.
Base Surf.
Contact Plane
un
pcj
pci
Surface Normals
Intersecting Points ( Intersecting Points ( SSip ip ))
pc
98
intersection points are found between the three edges of the triangle on the
base body and the action triangle plane. Therefore an algorithm for finding
an intersecting point between triangle and line is needed, as shown in
Figure 7.
In Figure 7, x -y -z is the triangle reference frame and it is defined with
respect to the surface reference frame x -y -z . The x and z axes are
parallel with 12d and the normal direction of triangle plane, respectively.
tA is the orientation matrix of x -y -z with respect to x -y -z . 1p is
a position vector expressed in x -y -z from the origin of x -y -z to the
origin of x -y -z . 1p , 2
p , and 3p are the node positions of the
triangle expressed in x -y -z . 1n and 2
n are the start and end
positions of triangle edges expressed in x -y -z , respectively. All triple-
primed vectors are defined in x -y -z .
Fig. 7 Schematic diagram to find the intersecting point between a triangle and a
line.
One of the necessary conditions for a point of intersection between the
line segment and the triangle to exist is that the two points of the line
segment must be on opposite sides of the plane containing the triangle.
This can be tested for by multiplying the z components of the endpoints
expressed in the reference frame of the triangle:
1 2 0z z
n n
Here, the subscript z means the z component of a vector. If Equation
TrianglePlane
Edge of Triangle
Intersecting Point
x’’’
z’’’
y’’’
n1’’’
n2’’’
p1’’’ p2’’’
p3’’’
d12’’’
d13’’’
d1ip’’’
d23’’’
d2ip’’’
dn12’’’
At
pip’’’
x’’
z’’
y’’
p1’’
C
TrianglePlane
Edge of Triangle
Intersecting Point
x’’’
z’’’
y’’’
n1’’’
n2’’’
p1’’’ p2’’’
p3’’’
d12’’’
d13’’’
d1ip’’’
d23’’’
d2ip’’’
dn12’’’
At
pip’’’
x’’
z’’
y’’
p1’’
C
(1)
99
(1) is satisfied, the point ipp on the triangle plane can be found, though
this point is not guaranteed to be inside the triangle. Therefore 1ipd can
be calculated as follows:
1 1 2 1 1 12( )ip nt t d n n n n d ,
1 12/ nz zt n d
Here, the end point of 1ipd is the intersecting point ip
p and it is defined
with respect to the triangle reference frame x -y -z . The equations
above do not indicate whether the intersecting point is inside or outside
of the triangle. Therefore, 1ipd must satisfy the following conditions to
be an intersecting point:
12 1( ) 0ip z d d and 1 13( ) 0ip z
d d ,
12 2( ) 0ip z d d and 2 23( ) 0ip z
d d
If 1ipd and 2ip
d satisfy Equation (2) and (3), calculated intersecting point
can be added to ipS .
(b) Calculate the approximated contact point cp as follows:
1
1( )
ipn
c ipkip
kn
p S
Calculate the contact normal direction nu . As shown in Figure 8, a plane
equation which passes through cp and closely passes through intersecting
points is calculated in this section. A plane equation can be expressed as
follows:
( ) ( ) ( ) 0c c ca x x b y y c z z
where coefficients a , b , and c compose a plane normal vector { , , }a b c
which is an unit vector nu . If 0a is assumed, the plane equation can be
expressed as follows,
(2)
(3)
(4)
(c)
100
1 2( ) ( ) ( ) ( ) ( ) ( ) 0c c c c c c
b cx x y y z z x x y y z z
a a
Fig. 8 Schematic diagram to calculate the unit normal vector nu .
If all intersecting points ( )ip ipnS are substituted in the Equation (4), the
following matrix equation can be obtained:
Kζ f
Here, K , ζ , and f are defined as follows:
1 1
2 2
( 2)ip
ip ip
c c
c c
n
n c n c
y y z z
y y z z
y y z z
K ,
1
(2 1)2
ζ ,
1
2
( 1)ip
ip
c
c
n
n c
x x
x x
x x
f (6)
Now, we define the residual ε , which means the distances between the
intersection points and the plane, as
ε Kζ f (7)
The function can be defined as
1 1
( ) ( ) ( )2 2
T Tζ ε ε Kζ f Kζ f (8)
un = { a , b , c }
pc = { xc , yc , zc }
Plane Eq : a ( x - xc ) + b ( y - yc ) + c ( z - zc ) = 0
Intersecting Points
un = { a , b , c }
pc = { xc , yc , zc }
Plane Eq : a ( x - xc ) + b ( y - yc ) + c ( z - zc ) = 0
Intersecting Points
(5)
101
and we can find the plane equation coefficients ζ which minimize the function
through the equations
[ ] 0T
K Kζ fζ
, (9)
1
[ ]T T
ζ K K K f
If TK K is not a singular matrix, nu can be calculated from the
definition of ζ . On the other hand, if TK K is a singular matrix, then
Equations (4)~(10) should be re-solved with an assumption of 0b or
0c . On very rare occasions, TK K will still be singular even when
assuming 0b or 0c . For example, this situation can be occured
when contact point and intersection points compose an almost straight
line. In these cases the detailed search algorithm presented in this paper
cannot be used. A different method must be used to find a penetration
depth and a contact reference frame, such as the node-to-surface contact
algorithm, as shown in Figure 9. In the node-to-surface contact algorithm,
all nodes of rnjL are compared to all triangles of rn
iL . For more detailed
information about the node-to-surface contact, refer to references
[12,19,20].
Fig. 9 The schematic algorithm concept of the node-to-surface contact.
On the other hand, noticing that the vector ζ is not continuous during
a simulation can be important to understand the results of the TSAPC
algorithm. During simulation, because intersection points can be added
Triangle Nodes ( Not Penetrating )
Triangle Surface
Triangle Node ( Penetrating )
Surface Normal
Penetration Depth
Contact Force
Contact Point
Triangle Nodes ( Not Penetrating )
Triangle Surface
Triangle Node ( Penetrating )
Surface Normal
Penetration Depth
Contact Force
Contact Point
(10)
102
or removed suddenly from the current intersection points set, the
vector ζ and all following results can be discontinuous. But, this kind
of discontinuity can be reduced by using a finer mesh.
(d) Find base contact point cip and action contact point cjp along
nu , which is passing through cp as shown in Figure 6. To find the
triangle ids which include cip or cjp , an overlap test or ray tracing
between bounding boxes and the line nu is used [10,12]. Then, cip
and cjp are calculated from the algorithm in section 4(a).
But, for the some special contact regions such as a ring shaped
volume between a torus and a sphere geometry as shown in Figure 10,
either cip or cjp cannot be found. As a result, in such cases, the
proposed detailed search algorithm cannot be used. If cip or cjp
cannot be found, an alternative detailed search algorithm such as the
node-to-surface contact algorithm should be used.
Fig. 10 The example of ring shaped contact region.
(e) Update cp , which should be re-calculated as the center position
between cip and cjp , as follows:
1( )
2c ci cj p p p
(f) Calculate the penetration depth , which is a distance between cip
and cjp , as follows:
cj ci p p
(g) Calculate the friction force direction f
u with relative velocity at
Ring shaped contact regionRing shaped contact region
103
cp between base and action bodies as shown in Figure 11. And then
determine the contact reference frame with nu and f
u .
Fig. 11 Relative velocity and contact reference frame.
The relative velocity cd at cp can be determined as follows:
( )c j j jc i i ic j j j jc i i i ic
d
dtd r A s r A s r A s r A s (11)
Here, is the angular velocity of the body defined with respect to
the body reference frame x -y -z and tilde (~) means the skew matrix
of the vector. ics and jcs are the position vectors of the contact point
expressed in the base and action body reference frame, respectively.
Now, f
u can be determined by the following equation:
( )f n c nu u d u (12)
If cd is zero, f
u is selected as an arbitrary unit vector which is
orthogonal to nu . Finally, the contact reference frame ( cr , cA ) can be
expressed as follows:
Ai
rj
Contact Plane
pc
X
Z
Y
xi’
yi’
zi’
ycxczc
Inertial Ref. Frame
Ajxj’
yj’
zj’
ri
sjc’sic’
Ac
Ai
rj
Contact Plane
pc
X
Z
Y
xi’
yi’
zi’
ycxczc
Inertial Ref. Frame
Ajxj’
yj’
zj’
ri
sjc’sic’
Ac
104
c cr p ,
c c c c f n f n A x y z u u u u
(j) Generate the contact force with the compliant contact force model
described in next section and apply the contact force to the base and
action bodies.
2.2.5. COMPLIANT CONTACT FORCE MODEL
In the previous section, the penetration depth and contact reference frame are
calculated for each individual contact region. In this section, a modified contact
force model based on a compliant contact force model [2-7, 20] is explained.
In the modified compliant contact force model, the contact normal force can be
separated into the spring force nsf and damping force
ndf as follows:
sm
nsf K ,
d
im m
ndf D
where K and D are the spring and damping coefficients, respectively, which are
determined through an experimental method. sm ,
dm , and
im are the spring,
damping, and indentation exponents, respectively. and are the penetration
depth and the time derivative of the penetration depth, respectively. can be
calculated from the relative velocity and contact normal direction as follow:
c nd u
If the relative velocity is large when the bodies are separating, ns ndf f can have a
negative value. This negative contact normal force is not realistic and can cause a
significant error. Therefore, in order to avoid the negative contact normal force and
obtain the realistic hysteresis loop for the energy dissipation during the contact, the
minimum contact normal force and the rebound normal force coefficient are
introduced. The rebound normal force coefficient controls the rebound damping
force when bodies are in the restitution phase, as shown in Figure 12. And the
minimum contact normal force can be expressed with the rebound normal force
coefficient and contact spring force as follows:
105
minn c nsf R f
where cR is the rebound normal force coefficient, which is a value between 0 and
1. Therefore, the contact normal force nf can be obtained by
minMax , n ns nd nf f f f
Fig. 12 The hysteresis loop and the rebound normal force coefficient.
Also, the friction force is obtained by
f nf f
where is the friction coefficient. And its sign and magnitude can be
determined from the relative velocity v between the base and action bodies of the
contact point as follows:
havsin( , , , , ),
havsin( , , , , ),
s s s s s
s s d d s
for v
for v
v v v v
v v v v
Here, sv and dv are static and dynamic threshold velocities, respectively. s
and d are static and dynamic friction coefficients, repectively. The function
“ havsin ” [29, 30] is defined by
fn
Penetration Depth
fns + fnd
fns
fnmin = Rc fns
+ damping( loading )
- damping (unloading) : restitution phase
+ -
fn
Penetration Depth
fns + fnd
fns
fnmin = Rc fns
+ damping( loading )
- damping (unloading) : restitution phase
+ -
106
0 0 1 1 0 0
0 1 1 0 00 1
1 0
1 1
havsin( , , , , ) ,
sin , 2 2 2
,
x x y x y y for x x
y y y y x xfor x x x
x x
y for x x
2.2.6. NUMERICAL EXAMPLES
Two numerical examples are presented in order to illustrate the accuracy and
performance of the TSAPC algorithm. The first example is the cam-valve contact
problem introduced in section 2.2, and the second is an example in which there are
multiple contact regions between two bodies with a large number of triangles. In
order to demonstrate the accuracy and performance of the TSAPC algorithm, it is
compared with the spatial partitioning method for the pre search and a node-to-
surface contact algorithm for the detailed search [19, 20]. Also, these solutions are
compared with analytical methods such as 2D contact or sphere-to-sphere contact
algorithm in order to calculate the penetration depth and contact reference frame
which includes the contact point and contact directions. In the analytical methods,
the modified compliant contact force model mentioned in this study is used to
generate the contact force.
In order to solve the equations of motion, a recursive formulation using relative
coordinates and an implicit integration method are used [20, 31]. As a variable
step-size implicit integration method, the generalized-α method [32] is used. The
error tolerance of 1.0E-3 is used to solve the Newton-Raphson equation for the
equations of motion. The analysis was performed on a computer using an AMD
Athlon(tm) 64 X2 Dual Core Processor 4600+ 2.42 GHz with 2GB RAM.
2.6.6.1 CAM-VALVE PROBLEM
As a first demonstration model, the cam-valve contact problem is used. In order to
illustrate the accuracy of the contact force, which contributes to the motion of
bodies, the results of a 2D contact method are also presented. Because this 2D
contact algorithm uses the equation of a circle to represent the valve geometry and
3rd
order cubic-spline equations for the cam curve profile, the results of 2D contact
can be treated as an analytic solution in current cam-valve contact problem. In 2D
contact algorithm, similar with the case of Figure 6, the penetration depth is
defined as the length of a distance vector ( cj cip p ) between base contact point
107
( cip ) and action contact point ( cjp ) which are lying on the both spline curves. At
the same time, the distance vector must be orthogonal to the tangential directions of
the spline curves at the base and action points, respectively. Also, two curve normal
direction vectors at the base and action contact points should be opposite. This
concept is similar with a method for finding minimum distance between two curves
[33].
Table 1 and Table 2 show the simulation and contact parameters, respectively,
used in cam-valve contact problem. Also, the analysis cases for comparison and
simulation results are summarized in Table 3. Table 3 presents the number of
triangles and nodes on the available contact surfaces, which are shown in Figure 13.
Table 1 The simulation parameters of the cam-valve contact problem.
Simulation parameters Values
Spring stiffness coefficient 5.0 N/mm
Spring damping coefficient 0.1 N∙s/mm
Initial spring compression
length 40.0 mm
Motion of cam revolute joint 40πt rad.
(1200 rpm)
Simulation end time 0.1 s
Simulation steps for output
(out
N ) 720
Table 2 The contact parameters of the cam-valve contact problem.
Contact parameters
2D Contact,
Node-To-
Surface
TSAPC
Spring coefficient (K ) 1000.0 N/mm 1000.0 N/mm
Damping coefficient (D ) 1.0 N∙s/mm 1.0 N∙s/mm
Spring exponent (sm ) 2.0 2.0
Damping exponent (dm ) 1.0 1.0
Indentation exponent (im ) 2.0 2.0
Rebound normal force
coefficient (cR ) Not Used 0.25
Dynamic friction coefficient
( d ) 0.2 0.2
108
Static friction coefficient ( s ) 0.3 0.3
Dynamic threshold velocity (dv ) 0.1 mm/s 0.1 mm/s
Static threshold velocity (sv ) 0.01 mm/s 0.01 mm/s
Table 3 The analysis cases and simulation results of the cam-valve contact
problem.
2D
Conta
ct
TSAPC Node-To-Surface
Case
1
Case
2
Case
3
Case
1
Case
2
Case
3
Cam
No. of
Triangles - 124 190 250 124 190 250
No. of Nodes - 124 190 250 124 190 250
Valv
e
No. of
Triangles - 1012 1200 1404 1012 1200 1404
No. of Nodes - 530 626 730 530 626 730
Number of
Evaluations 4627 2536 2592 2524 2590 2656 2631
Average Error (avgE ) - 2.025 1.244 1.020 3.107 4.104 3.365
CPU Time (s) 0.828 0.937 1.078 1.110 8.094 11.71
9
16.14
1
Fig. 13 An available surface representation used in analysis cases on Table 3.
Figure 14 shows the magnitude of the normal and friction contact forces. As
shown in Figure 14(a), the results of 2D contact for the magnitude of contact force
are smooth and used in the comparison with TSAPC and node-to-surface contact
algorithms. Figure 14(b) shows the contact force magnitude for TSAPC. The
Action available surface
Base availablesurface
No triangles. Not needed in contact test.Action
available surface
Base availablesurface
No triangles. Not needed in contact test.
109
results for the contact force magnitude become less noisy and more accurate as the
number of triangles used to represent each surface increases. As mentioned in
Section 4(c), the results such as a contact force magnitude can be non-smooth
during simulation because intersection points are added or removed discontinuouly
in the intersection points set ipS . The results of TSAPC can have numerical errors
due to the triangular facetting of contact surfaces. On the other hand, as shown in
Figure 14(c), node-to-surface contact shows more noise than TSAPC. Also, in the
node-to-surface contact, the contact force does not become smooth even when the
number of triangles is increased.
In order to compare the accuracy and solving speed related to the number of
triangles, average errors and CPU time are plotted in Figure 15. Here, the average
error for the contact force magnitude is defined from the results of 2D contact as
follows:
2
, ,
21 ,
100 (%)
1 outDN
mag i mag i
avg Diout mag avg
F FE
N F
where avgE is the average error for the contact force magnitude and
outN is the
simulation steps for output. ,mag iF is the i-th contact force magnitude, which is
defined from a force vector summed by i-th normal and friction contact forces. The
superscript 2D means the results of 2D contact and subscript avg means the
average value during the simulation. Also, in Table 3, “Number of Evaluations”
means the total number of evaluations of the equations of motion including contact
forces during simulation.
As shown in Figure 15(a), the average errors of TSAPC are less than those of
node-to-surface contact. The average error of TSAPC decreases as the number of
triangles increases. Also, the solving speed of TSAPC is about 10 times faster than
node-to-surface contact. The slope for the CPU time of TSAPC, as shown in
Figure 15(b), is much less than the slope of node-to-surface contact, which implies
that TSAPC will be more efficient as the number of triangles increases.
110
(a) 2D contact.
(b) TSAPC. (c) Node-to-surface contact.
Fig. 14 The comparison results for the magnitude of contact force. (a) 2D contact, (b)
TSAPC, (c) Node-to-surface contact.
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
Mag
nit
ud
e o
f co
nta
ct f
orc
e (N
)
2D Contact
Avg. Error = 0.000 %
CPU Time = 0.828 s
Normal Force
Friction Force
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
Mag
nit
ud
e o
f co
nta
ct f
orc
e (N
)
2D Contact
Avg. Error = 0.000 %
CPU Time = 0.828 s
Normal Force
Friction Force
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
Mag
nit
ud
e o
f co
nta
ct f
orc
e (N
)
TSAPC - Case 3
Cam Tri. No = 250
Valve Tri. No = 1404
Avg. Error = 1.020 %
CPU Time = 1.110 s
TSAPC - Case 2
Cam Tri. No = 190
Valve Tri. No = 1200
Avg. Error = 1.244 %
CPU Time = 1.078 s
TSAPC - Case 1
Cam Tri. No = 124
Valve Tri. No = 1012
Avg. Error = 2.025 %
CPU Time = 0.937 s
Normal Force
Friction Force
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
Mag
nit
ud
e o
f co
nta
ct f
orc
e (N
)
TSAPC - Case 3
Cam Tri. No = 250
Valve Tri. No = 1404
Avg. Error = 1.020 %
CPU Time = 1.110 s
TSAPC - Case 2
Cam Tri. No = 190
Valve Tri. No = 1200
Avg. Error = 1.244 %
CPU Time = 1.078 s
TSAPC - Case 1
Cam Tri. No = 124
Valve Tri. No = 1012
Avg. Error = 2.025 %
CPU Time = 0.937 s
Normal Force
Friction Force
Mag
nit
ud
e o
f co
nta
ct f
orc
e (N
)
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
Node-To-Surface - Case 2
Cam Tri. No = 190
Valve NodeNo = 626
Avg. Error = 4.104 %
CPU Time = 11.719 s
Node-To-Surface - Case 1
Cam Tri. No = 124
Valve NodeNo = 530
Avg. Error = 3.107 %
CPU Time = 8.094 s
Node-To-Surface - Case 1
Cam Tri. No = 250
Valve NodeNo = 730
Avg. Error = 3.365 %
CPU Time = 16.141 s
Normal Force
Friction Force
Mag
nit
ud
e o
f co
nta
ct f
orc
e (N
)
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
100
200
300
400
500
600
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
Node-To-Surface - Case 2
Cam Tri. No = 190
Valve NodeNo = 626
Avg. Error = 4.104 %
CPU Time = 11.719 s
Node-To-Surface - Case 1
Cam Tri. No = 124
Valve NodeNo = 530
Avg. Error = 3.107 %
CPU Time = 8.094 s
Node-To-Surface - Case 1
Cam Tri. No = 250
Valve NodeNo = 730
Avg. Error = 3.365 %
CPU Time = 16.141 s
Normal Force
Friction Force
111
(a) Average error (Eavg). (b) CPU time (s).
Fig. 15 The comparison results for the average error and CPU time according to the
number of triangles. (a) Average error (Eavg), (b) CPU time (s).
2.6.6.2 MULTIPLE CONTACT REGIONS
In order to illustrate the TSAPC algorithm for the case of multiple contact
regions with a large number of triangles, a contact example which can be solved
with an analytical sphere-to-sphere contact algorithm is introduced, as shown in
Figure 16. The sphere-to-sphere contact algorithm is an analytic algorithm to
calculate the contact information and it is well defined in the reference [29]. In
this example, the base body is a single sphere and the action body geometry is
composed of 4 spheres. Even though the action geometry includes 4 sphere
geometries, the action geometry is treated as one general geometry in the new
and node-to-surface contact algorithms. In the case of the analytical sphere-to-
sphere contact algorithm, 4 sphere-to-sphere contact elements are used. Also,
Figure 16(b) shows the triangles used in the analysis of TSAPC. In this case, the
base surface is composed of 7568 triangles and 3786 nodes, and action surface is
composed of 78028 triangles and 39014 nodes. In the case of node-to-surface
contact, the triangles of TSAPC cannot be used because it was too slow.
Therefore, this study uses 2808 triangles for base geometry and 4932 nodes for
action geometry. Table 4 and Table 5 show the analysis parameters and contact
parameters, respectively. And Table 6 shows the summary for the analysis cases
and simulation results. In this example, the time step size is reduced before the
collision is detected in order to prevent an unrealistically deep penetration during
the first time step in which collision starts.
2.025 1.2441.02
3.107
4.104
3.365
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Case 1 Case 2 Case 3
TSAPC
Node-To-Surface
Av
erag
eE
rro
r (%
)
Number of Triangles
2.025 1.2441.02
3.107
4.104
3.365
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Case 1 Case 2 Case 3
TSAPC
Node-To-Surface
Av
erag
eE
rro
r (%
)
Number of Triangles
1.111.0780.937
16.141
11.719
8.094
0
2
4
6
8
10
12
14
16
18
Case 1 Case 2 Case 3
TSAPC
Node-To-Surface
CP
U T
ime
(s)
Number of Triangles
1.111.0780.937
16.141
11.719
8.094
0
2
4
6
8
10
12
14
16
18
Case 1 Case 2 Case 3
TSAPC
Node-To-Surface
CP
U T
ime
(s)
Number of Triangles
112
(a) Contact model description.
(b) Surface representation used in the case of TSAPC.
Fig. 16 Multiple contact region problem. (a) Model description, (b) Surface
representation.
Table 4 The analysis parameters of the multiple contact region problem.
Simulation parameters Values
Radius of sphere 100 mm
Material density 7.85E-06
Base Body
Inertial Ref. Frame (0,0,0)
Action Body
X
Y
Z
Base Body
Inertial Ref. Frame (0,0,0)
Action Body
X
Y
ZX
Y
Z
X
Y
500 mm 500 mm500 mm
Spring Free Length = 1000 mm K=100.0 N/mm, C=1.0 Nsec/mmFixed at ground.
Spring 1 Spring 2
X
Y
X
Y
500 mm 500 mm500 mm
Spring Free Length = 1000 mm K=100.0 N/mm, C=1.0 Nsec/mmFixed at ground.
Spring 1 Spring 2
Action Geometry, No. of Patches = 78024.
Base Geometry, No. of Patches = 7568.
Action Geometry, No. of Patches = 78024.
Base Geometry, No. of Patches = 7568.
113
kg/mm^3
Spring stiffness coefficient 1000 N/mm
Spring damping coefficient 1 N∙s/mm
Spring free length 1000 mm
Gravity Not Used
Simulation end time 1.0 s
Simulation steps for output
(out
N ) 2000
Table 5 The contact parameters of the multiple contact region problem.
Contact parameters
Sphere-To-
Sphere, Node-
To-Surface
TSAPC
Spring coefficient (K ) 1.0E+05 N/mm 1.0E+05
N/mm
Damping coefficient (D ) 10.0 N∙s/mm 10.0 N∙s/mm
Spring exponent (sm ) 2.0 2.0
Damping exponent (dm ) 1.0 1.0
Indentation exponent (im ) 0.0 0.0
Rebound normal force
coefficient (cR ) Not Used 0.25
Table 6 The analysis cases and simulation results of multiple contact region problem.
Sphere
- To-
Sphere
TSAPC
Node-
To-
Surface
Base
No. of
Triangles - 7568 2808
No. of Nodes - 3786 1406
Actio
n
No. of
Triangles - 78028 9864
No. of Nodes - 39014 4932
Number of
Evaluations 2428 5581 3979
CPU Time (s) 0.3910 19.61 3233.0
Figure 17 shows the contact force vectors at all contact regions at the end of
114
the simulation time. In order to compare the results of the various methods, the
center position of the X-coordinate for the action body is plotted in Figure 18. In
Figure 18, the results of TSAPC and node-to-surface contact are very close with
the results of sphere-to-sphere contact. The average errors of contact positions
for both cases are less than 0.5%. But, although the number of triangles used in
TSAPC is much larger than node-to-surface contact, the CPU time of TSAPC is
much faster (164 times) than the node-to-surface contact case. Also, even though
the solving speed of TSAPC is much slower than sphere-to-sphere contact, the
TSAPC algorithm is designed as a general purpose algorithm to solve the contact
problem between complex rigid geometries. Therefore, if the shape of contact
geometries is not simple, the analytic approach such as sphere-to-sphere contact
cannot be used. In those cases, the TSAPC algorithm is very efficient.
Fig. 17 Display for contact force vectors at all contact regions (time = 1.0 sec) using
the TSAPC algorithm.
Fig. 18 The center position of X-coordinate for the action body.
4 contact points. ( at t = 1.0 s )
4 contact points. ( at t = 1.0 s )
0
25
50
75
100
125
150
175
200
225
250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Sphere-To-Sphere
TSAPC
Node-To-Surface
Time (sec)
Po
siti
on
of
X (
mm
)
0
25
50
75
100
125
150
175
200
225
250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Sphere-To-Sphere
TSAPC
Node-To-Surface
Time (sec)
Po
siti
on
of
X (
mm
)
115
2.2.7. CONCLUSIONS
This paper presents an efficient and robust contact search algorithm for a
compliant contact force model between bodies of complex geometry in multibody
dynamics. The proposed contact algorithm contains two parts, the pre search and
the detailed search. In the pre search algorithm, the bounding box tree and
recursive overlap tests are used to find the global triangle pair set between two
objects. And the connectivity information between triangles is used to separate the
global triangle pair set into the sub local triangle pair sets to handle multiple
contact region problems. Then, in the detailed search algorithm, the penetration
depth and contact reference frame for every contact region are calculated by using
the proposed efficient and robust TSAPC algorithm.
In order to illustrate the TSAPC algorithm, a cam-valve contact problem and a
multiple contact region problem are simulated. The simulation results of the
TSAPC algorithm show a good agreement with analytic-based solutions such as 2D
contact or sphere-to-sphere contact. And the solving speed of the TSAPC algorithm
is superior to the node-to-surface contact. Furthermore, even though the number of
triangles is increased in order to represent the contact surface accurately, the
solving time is increased slowly.
ACKNOWLEDGEMENTS
This research is supported by the KyungHee University research program.
REFERENCES
1. Johnson KL. Contact mechanics. Cambridge University Press; 1985.
2. Hertz H. On the contact of elastic solids. Journal für die reine und
angewandte Mathematik 1882;92:156-71.
3. Hunt KH, Crossley FRE. Coefficient of restitution interpreted as damping
in vibroimpact. ASME Journal of Applied Mechanics 1975;42:440-5.
4. Khulief YA, Shabana AA. A continuous force model for the impact
analysis of flexible multibody systems. Mechanism and Machine Theory
1987;22(3):213-24.
5. Lankarani HM, Nikravesh PE. A contact force model with hysteresis
damping for impact analysis of multibody systems. ASME Journal of
Mechanical Design 1990;112:369-76.
6. Lankarani HM, Nikravesh PE. Cannonical impulse-momentum equations
116
for impact analysis of multibody system, ASME Journal of Mechanical
Design 1992;180:180-6.
7. Lankarani HM, Nikravesh PE. Continuous contact force models for impact
analysis in multibody systems, Journal of Nonlinear Dynamics
1994;5:193-207.
8. Sharf I., Zhang Y. A contact force solution for non-colliding contact
dynamics simulation, Multibody system dynamics 2006;16:263-90.
9. van den Bergen G. Collision Detection in Interactive 3D Environments.
Morgan Kaufmann; 2003.
10. Arvo J, Kirk D. A survey of ray tracing acceleration techniques. In: An
Introduction to Ray Tracing. 1989. p. 201-62.
11. Lin MC, Manocha D. Collision and proximity queries. Handbook of
Discrete and Computational Geometry, 2003.
12. Ericson C. Real-Time Collision Detection. Morgan Kaufmann; 2004.
13. Teschner M, Kimmerle S, Heidelberger B, Zachmann G, Raghupathi L,
Fuhrmann A, Cani MP, Faure F, Magnenat-Thalmann N, Strasser W,
Volino P. Collision detection for deformable objects. Computer Graphics
Forum 2005;24(1):61-81.
14. van den Bergen G. Efficient collision detection of complex deformable
models using AABB trees. Journal of Graphics Tools 1997;2(4):1-14.
15. Gottschalk S. Separating axis theorem. Technical Report TR96-024, Dept.
of Computer Science, UNC Chapel Hill, 1996.
16. Gottschalk S, Lin MC, Manocha D. OBBTree: A hierarchical structure for
rapid interference detection, In: Proc. of ACM Siggraph’96. 1996. p. 171-
80.
17. Gottschalk S, Lin MC, Manocha D. RAPID,
http://www.cs.unc.edu/~geom/OBB/OBBT.html, 1997.
18. Terdiman P. OPCODE, http://www.codercorner.com/Opcode.htm, 2003.
19. Teschner M, Heidelberger B, Müller M, Pomeranets D, Gross M.
Optimized spatial hashing for collision detection of deformable objects. In:
Proc. of Vision, Modeling, and Visualization. 2003. p. 47-54.
20. Cho HJ, Bae DS, Ryu HS, Choi JH. An efficient contact search algorithm
using the relative coordinate system for multibody system dynamics, In:
Proc. of the 1st Asian Conference on Multibody Dynamics. 2002. p. 520-
527.
21. Hippmann G, An algorithm for compliant contact between complexly
shaped bodies, Multibody system dynamics 2004;12:345-62.
22. Zhang L, Kim YJ, Varadhan G, Manocha D. Generalized penetration depth
computation. Computer-Aided Design 2007;39:625-38.
23. Dobkin D, Hershberger J, Kirkpatrick D, Suri S. Computing the
intersection-depth of polyhedra, Algorithmica 1993;9:518-33.
117
24. Heidelberger B, Teschner M, Keiser R, Müller M, Gross M. Consistent
penetration depth estimation for deformable collision response. In: Proc. of
Vision, Modeling, and Visualization. 2004. p. 339-46.
25. Faure F, Barbier S, Allard J, Falipou F. Image-based collision detection and
response between arbitrary volume objects, In: ACM
Siggraph/Eurographics Symposium on Computer Animation. 2008.
26. Govindaraju NK, Kabul I, Lin MC, Manocha D. Fast continuous collision
detection among deformable models using graphics processors, Computers
and Graphics 2007;31(1):5-14.
27. Sud A, Otaduy MA, Manocha D. Difi: Fast 3d distance field computation
using graphics hardware, Computer Graphics Forum 2004;23(3):557-66.
28. Carey GF. Computational Grids: Generations, Adaptation, and Solution
Strategies. Taylor & Francis; 1997.
29. RecurDynTM
Help Library, FunctionBay, Inc.,
http://www.functionbay.co.kr/, 2008.
30. MD Adams Manual, MSC Software, http://www.mscsoftware.com/, 2008.
31. Bae DS, Han JM, Yoo HH. A Generalized recursive formulation for
constrained mechanical system dynamics, Mechanics of Structures and
Machines 1999;27(3):293-315.
32. Chung J, Hulbert GM. A Time Integration Algorithm for Structural
Dynamics With Improved Numerical Dissipation: The Generalized-α
Method, Journal of Applied Mechanics 1993;60;371-75.
33. Mortenson ME, Geometric Modeling Third Edition. Industrial Press Inc.;
2006.
118
2.3.
A STUDY ON THE STICK AND SLIP
ALGORITHM IN CONTACT PROBLEMS
OF MULTIBODY SYSTEM DYNAMICS
2.3.1. INTRODUCTION
Friction can be represented by two states such as stick and slip. A belt system like
Fig. 1 is a well-known model to describe the states of friction (McMillan 1997).
Figure 1. BELT MODEL TO DESCRIBE THE FRICTION PHENOMENON
A box in the initial state is stuck on the belt. Therefore, in the initial stage, the box
is in a stick state. As a result, the box moves with the belt because the box is stuck
on the belt. The stick state is preserved until the spring force is reached up to the
same level of a static friction force which can be called a break-away force. At the
moment, the slip state is started. The box begins to move to the opposite side of the
belt moving direction. And the slip state is kept until the spring force is equal to the
dynamic friction force. These two states of friction phenomenon generates a non-
linearity of a system (McMillan 1997). But, the conventional friction model is only
the function of relative velocity. As a result, the conventional friction force model
always doesn’t have the stick state because the relative velocity must be a non-zero
value to generate the friction force. Therefore a stick-slip friction force model
should be considered in order to solve this problem. A stick-slip friction force
Stick state Slip state
Belt
Spring
Driven roller
Box
Contact point between box and belt
Belt contact model for friction phenomenon
x
Driving roller
Stick state Slip state
Belt
Spring
Driven roller
Box
Contact point between box and belt
Belt contact model for friction phenomenon
x
Driving roller
120
model, in which the friction force works like as a spring force, was proposed by
Dahl (1968). And also, the stick-slip friction models have been developed by many
researchers (Canudas-de-Wit et al. 1995, Olsson et al. 1998).
In this paper, a stick-slip and a conventional friction force model is introduced. And
then, the two friction force models are applied in the contact algorithm in multi-
body dynamics (MBD) system. And the two friction models are compared with
some numerical examples.
2.3.2. MBD FORMULATION
The MBD formulation used in this study is described well in Bae el al. (2001) and
Choi (2009). In this section, the brief formulations for MBD are introduced.
Velocities and virtual displacements of the origin of body reference frame
x y z with respect to the global reference frame X Y Z , respectively, defined
as
r
ω (1)
and
r
π (2)
Their corresponding quantities with respect to the body reference frame x y z
are, respectively, defined as
T
T
r A rY
ω A ω (3)
and
T
T
r A rZ
π A π (4)
where A is the orientation matrix of the x y z frame with respect to the
X Y Z frame.
The recursive velocity equations for a pair of contiguous bodies is obtained as
1 2
j ij i ij ij Y B Y B q (5)
120
where Y is the combined velocity of the translation and rotation as defined in Eq.
(3) and 1
ijB and 2
ijB are defined as follows:
1
T T
ij ij ij ij ji ij
ij T
ij
A 0 I s d A s AB
0 A 0 I (6)
and
2
ij
TTij ij ji ij ijij
ij T
ij
q
I d A s A HA 0B
0 A 0 I
(7)
It is important to note that matrices 1
ijB and 2
ijB are only functions of ijq .
Similarly, the recursive virtual displacement relationship is obtained as follows:
1 2
j ij i ij ij Z B Z B q (8)
If the recursive formula in Eq. (5) is respectively applied to all joints along the
spanning tree, the following relationship between the Cartesian and relative
generalized velocities can be obtained:
Y Bq
(9)
where B is the collection of coefficients of the ijq and
0 1 2, , , ,T
T T T T
nnc
Y Y Y Y Y (10)
and
0 01 12 ( 1), , , ,T
T T T T
n nnr
q Y q q q (11)
where nc and nr denote the number of the Cartesian and relative generalized
coordinates, respectively. The Cartesian velocity ncY R with a given nrq R can
be evaluated either by using Eq. (9) obtained from symbolic substitutions or by
using Eq. (5) with recursive numerical substitution of iY .
It is often necessary to transform a vector G in ncR into a new vector Tg B G in
nrR . Such a transformation can be found in the generalized force computation in the
joint space with a known force in the Cartesian space. The virtual work done by a
Cartesian force ncQ R is obtained as follows:
121
T W Z Q (12)
where Z must be kinematically admissible for all joints in a system. Substitution
of Z B q into Eq. (12) yields
*T T T W q B Q q Q (13)
where * TQ B Q .
The equations of motion for a constrained mechanical system (García de Jalón et al.
1986) in the joint space (Wittenburg 1977) have been obtained by using the
velocity transformation method as follows:
( )T T Z
F B MY Φ λ Q 0 (14)
where Φ and λ , respectively, denote the cut joint constraint and the
corresponding Lagrange multiplier. M is a mass matrix and Q is a force vector
including the external forces in the Cartesian space.
2.3.3. CONTACT FORCE MODEL
Contact force can be classified by two parts. First is the contact normal force and
the other is the contact friction force. In this section, the contact force models are
introduced. The Fig. 2 shows a coordinate system and contact parameters to
compute the contact normal and friction forces. In the figure, is a penetration
depth which is used for computing the contact normal force. And refA is a contact
reference frame. In the contact reference frame refA , the normal direction is
defined as y -axis. And then, a relative velocity is calculated. The x -axis of refA
is defined as the same direction with a relative velocity v which is a value
projected on x z plane and measured on the contact point s .
122
Figure 2. THE COORDINATE SYSTEM OF THE CONTACT FORCE MODEL
2.3.3.1 CONTACT NOTMAL FORCE
The compliance contact force model is used in this study. The contact normal force
is defined as the function of penetration depth and its velocity as follows:
m n
nf k c (15)
where k , c , m and n are a spring coefficient, damping coefficient, stiffness
exponent and indentation exponent, respectively.
2.3.3.2 COVENTIONAL FRICTION FORCE MODEL
The convectional general friction model is shown in Fig. 3. Where s ,
d , sv
and dv are a static friction coefficient, dynamic friction coefficient, static threshold
velocity, and dynamic threshold velocity, respectively.
Also, the convectional friction force model can be simplified as the static
coefficient and the dynamic friction coefficient is the same ( t s d ) and the
dynamic threshold velocity and the static threshold velocity is the same (t s dv v v ).
The simplified friction force model, which is used in this study, is shown in Fig. 4.
The friction force is the function of the contact normal force and the relative
velocity as Eq. (16). The relative velocity v is defined on the contact point
between two contacted bodies as shown in Fig. 2.
sgn( ) ( )f nf v v f (16)
sAs ref
yx
y
x
Contact point
Base body
Action body
br
ar
refA
bA
aA
v
sArs bb
Ground.Inertia frame
x
y
yx
sAs ref
yx
y
x
Contact point
Base body
Action body
br
ar
refA
bA
aA
v
sArs bb
Ground.Inertia frame
x
y
yx
(16)
123
Where ff , v , and
nf are the contact friction force, relative velocity, friction
coefficient and contact normal force, respectively. The friction coefficient is
calculated as defined in Table 1.
Figure 3. THE CONVENTIONAL FRICTION FORCE MODEL
Figure 4. THE SIMPLIFIED CONVENTIONAL FRICTION FORCE MODEL
Table 1. FRICTION COEFFICIENT FUNCTION FOR CONTACT FRICTION
FORCE MODEL
State Slip Stick
s
d
sv dv
Friction coefficient
( )
Relative velocity
on contact pointsvdv
s
d
)sgn(v
Conventional friction force model
Slip state
s
d
sv dv
Friction coefficient
( )
Relative velocity
on contact pointsvdv
s
d
)sgn(v
Conventional friction force model
Slip state
)( dst
tv
Friction coefficient
( )
Relative velocity
on contact pointtv
t
)sgn(v
Simplified Conventional friction force model
Slip state
)( dst
tv
Friction coefficient
( )
Relative velocity
on contact pointtv
t
)sgn(v
Simplified Conventional friction force model
Slip state
124
v tv v 0 tv v
( )v t ( , , , , )t t t tstep v v v
Where t and
tv are the static friction coefficient and the threshold velocity,
respectively. The step function is defined in RecurDyn Manual (2010) or Choi
(2009).
2.3.4. STICK-SLIP FRICTION FORCE MODEL
Figure 5. DEFINITION OF THE STICTION DEFORMATION
In this section, we introduce a new friction force model which is called a stick-slip
algorithm. A big problem of conventional friction force model is that the stick state
doesn’t exist because the relative velocity must be a non-zero value to generate the
friction force. To solve this problem, the new stick-slip algorithm is proposed. The
key concept is that the stick-slip friction force model has a displacement function in
order to generate the friction force during the stick state. The displacement value is
defined as the stiction deformation . The stiction deformation is shown in Fig. 5.
2.3.4.1 STICK-SLIP FRICTION MODEL
The stick-slip friction force model has two parts which are the stiction and sliding
forces as follows:
sgn( )stiction sliding
f f f total nf f f v f (17)
where v and total are the relative velocity on the contact point and the total
friction coefficient of the contact friction force. The total friction coefficient is
calculated as follows:
Stiction deformation
Force
Friction force
Stiction deformation
Force
Friction force
125
( 0)total f n nf f if f (18)
The sliding and stiction forces are calculated as Eq. (19) and (20).
sgn( )(1 ) ( , )stiction
f nf v f (19)
sgn( ) ( )sliding
f v nf v v f (20)
Where ,
v , and are the friction coefficient for stiction force, friction
coefficient for sliding force, weighting value of stiction friction force and static
deformation, respectively. These parameters can be calculated as defined in Table 2.
In the table, max ,
tv and t are the maximum static deformation, threshold
velocity and threshold friction coefficient, respectively.
The value controls that the friction force becomes the static friction force t nf
when the stiction deformation is equal or greater than max . In the case of a slip
state, the stiction friction force becomes zero. Therefore, if the stiction deformation
is greater than max , the friction coefficient can be defined as shown in Fig. 6. If
the stiction deformation has a value, then the friction force must be a non-zero
value even though the relative velocity reaches to zero.
Table 2. PARAMETERS FOR STICK-SLIP FRICTION FORCE MODEL
State Slip Stick
v tv v tv v
1.0 ( , , 1.0, ,1.0)t tstep v v v
0.0 max max( , , , , )t tstep
v t ( , , , , )t t t tstep v v v
ff sliding
ff sliding stiction
f ff f
126
Figure 6. EXAMPLE OF THE STICK-SLIP FRICTION FORCE MODEL
2.3.4.2. STICTION DEFORMATION
In order to generate the friction force using the stick-slip friction force model, we
need an important parameter which is the stiction deformation . The stiction
deformation is calculated from the position vector of the contact point. We already
defined that x -axis of the contact reference frame is the direction of the relative
velocity v . Eq. (21) shows how to define the stiction deformation.
*
* * *
0
0
x x
T
x y
T
x y
s s
s s s
s s s
(21)
Where xs and *
xs are the x -axis value of the current contact point and the
reference x -axis value of a previous contact point, respectively. And the reference
contact point should be updated when the first collision is detected.
2.3.5. NUMERICAL EXAMPLES
Two numerical example models are shown in this section. In order to build and
tv
Friction coefficient
( )
Relative velocity
on contact pointtv
t
)sgn(v
Stick-slip friction force model
( Assumption : )
Stick state Slip stateSlip state
t
max
tv
Friction coefficient
( )
Relative velocity
on contact pointtv
t
)sgn(v
Stick-slip friction force model
( Assumption : )
Stick state Slip stateSlip state
t
max
127
test the models, this study uses the commercial software RecurDynTM
/MTT2D
(RecurDyn 2010). The MTT2D is developed to simulate a media transport system
in 2-dimensional. The sheet bodies of MTT2D consists of the finite number of rigid
bodies connected by the revolute joint and the rotational spring and damping force
elements.
2.3.5.1 BELT CONTACT MODEL
Fig. 7 shows a similar model with the belt contact model. There are 3 sheet
elements, an ideal spring force element and a roller. The roller is rotated and the
rotational velocity is (rad/sec). The all sheet bodies are constrained by the
translation joints.
Figure 7. EXAMPLE OF THE BELT CONTACT MODEL
The simulation results are shown in Fig 8. Fig. 8 shows a displacement along the
x -axis of a sheet body. In the case of conventional friction force model, the sheet
body is stop after about 1 sec. This means only slip state is occurred. In the case of
stick-slip friction force model, the sheet bodies move to backward and forward
continuously. It means that the stick and slip state occurs.
Figure 8. DISPLACEMENT RESULTS OF THE SHEET BODY
Driving Roller
Ideal spring
The sheet body (3 segment with translation joint)
Driving Roller
Ideal spring
The sheet body (3 segment with translation joint)
Stick Slip
Stick-slip friction force model
Conventional friction force model
Stick Slip
Stick-slip friction force model
Conventional friction force model
128
Figure 9. STATE PLANE OF THE SHEET BODY IN THE CASE OF STICK-SLIP
FRICTION FORCE MODEL
Figure 10. FRICTION FORCE VERSUS RELATIVE VELOCITY FOR
SHEET-ROLLER CONTACT
The example model uses an ideal spring which doesn’t have damping. Therefore
the system will vibrate continuously. The closed loop diagram for a state plane and
the friction force versus the relative velocity is shown in Fig. 9 and Fig. 10.
2.3.5.2 THE SLOPED ROLLER GUIDES MODEL
The other simulation model is the sloped roller guide model. The system has 10
rollers and a sheet body which consists of 50 rigid bodies as shown in Fig. 11. All
the rollers are fixed in space. And the sheet is on the rollers with no constraints.
The slope is 15 degree. If the static friction coefficient is more than tan(15 ) , the
sheet should be stay on the rollers. The all static friction coefficients ( t ) are used
as 1.0. Therefore the sheet body must be stuck in the first position because the
a
bc
d
e
f
a b c d e f c d e …
a
bc
d
e
f
a b c d e f c d e …
a
b
c
d
e
a b c d e c) d e c … a
b
c
d
e
a b c d e c) d e c …
129
friction coefficient is greater than tan(15 ) .
Fig. 12 shows only the stick-slip friction force model stuck in the slope. The
conventional friction force model is sliding with a steady velocity but the results of
stick-slip friction force model shows more realistic results.
The solving speed is compared in Table 3. And the model parameters are shown in
Table 4. The solving speed of stick-slip friction force model is faster than the
conventional friction force model. Also, the number of jacobian and residual
calculation calls is lower than the conventional friction force model. In the case of
the stick-slip friction force, the numerical integration step size can be used with a
large value. But in the case of conventional friction model, the numerical step size
keeps a small value because of the direction changes of the friction force.
Figure 11. THE SLOPED ROLLER GUIDES MODEL
Figure 12. SIMULATION RESULTS OF THE SLOPED ROLLER GUIDES
MODEL
Slope 15˚
Sheet body
10 Rollers (Don’t rotate)
Static friction coefficient : 1.0Slope 15˚
Sheet body
10 Rollers (Don’t rotate)
Static friction coefficient : 1.0
Stick-slip friction force model
Conventional friction force model
Stick-slip friction force model
Conventional friction force model
130
Figure 13. FRICTION FORCE RESULTS OF THE SLOPED ROLLER GUIDES MODEL
Table 3. SOLVER PERFORMANCE
- Conventional Stick-slip
Solving
Time 24 sec 10 sec
Number of
Jacobian
Calls
5465 883
Number of
Residual
Calls
1832 2649
Table 4. MODEL PARAMETERS
Parameters Value
End time (sec) 5
Spring coefficient k 1.2
Damping coefficient c 0.012
Stiffness exponent m 1.3
Indentation exponent n 2
Threshold velocity tv
(mm/sec) 10
Static friction coeff. t 1.0
Stick-slip friction force model (Conversion value : 0.00011)
Conventional friction force model
Note )
Contact normal force ( fn ) = 0.00039 N
ff = fn * tan(15degree) = 0.00011
Stick-slip friction force model (Conversion value : 0.00011)
Conventional friction force model
Note )
Contact normal force ( fn ) = 0.00039 N
ff = fn * tan(15degree) = 0.00011
131
Max. stiction deformation
max
(mm, Only stick-slip friction
force)
1
2.3.6. CONCLUSIONS
In this paper we introduced the MBD and the contact force model including the
normal and friction force. The friction force is classified into two parts, one is the
conventional friction model and the other is the stick-slip friction force model. If a
MBD system includes a stiction phenomenon, the stick-slip friction force model is
recommended in order to get more realistic and better numerical performance.
A difference between two friction force models is weather the force model
includes the stiction deformation to describe the stick state. In the stick-slip friction
model, the stiction deformation generates a friction force even though the relative
velocity reaches to zero.
REFERENCES
1. Bae D. S., Han J. M., Choi J. H., and Yang S. M., A Generalized Recursive
Formulation for Constrained Flexible Multibody Dynamics, International
Journal for Numerical Methods in Engineering, Vol. 50, pp.1841-1859,
2001.
2. Canudas-de-Wit, C., Olsson, H., Astrom, K. J., and Lischinsky., P., A new-
model for control of system with friction, IEEE Transactions on Automatic
Control, Vol. 40, No. 3, pp.419-425, 1995.
3. Choi, J., A Study on the Analysis of Rigid and Flexible Body Dynamics
with Contact, PhD Dissertation, Seoul National University, Seoul (2009).
4. Dahl P., A solid friction model, Aerospace Corporation, El Segundo, CA,
Technical Report TOR-0158(3107-18)-1, 1968.
5. García de Jalón D. J., Unda J., and Avello A., Natural coordinates for the
computer analysis of multibody systems, Computer Methods in Applied
Mechanics and Engineering, Vol. 56, pp.309-327, 1986.
6. McMillan, A. J., A non-linear friction model for self-excited vibrations,
Journal of sound and vibration, Vol. 205, pp.323-335, 1997.
7. Olsson, H., Astrom, K. J., Canudas de Wit, C., Gafvert, M., Lischinky, P.,
Friction models and friction compensation, European J. Control, Vol. 4,
No. 3, pp.176-195, 1998.
8. RecurDynTM
Manual, www.functionbay.co.kr, FunctionBay, Inc. , 2010.
9. Wittenburg J., Dynamics of Systems of Rigid Bodies, B. G. Teubner,
Stuttgart, 1977.
Theoretical Manual for IMDD
FunctionBay, Inc.
1. MFBD
1.1 FFlex
135
1.1.1
RELATIVE NODAL METHOD FOR LARGE
DEFORMATION PROBLEM
1.1.1.1. INTRODUCTION
Geometrically nonlinear analyses[1-4] have been investigated by many
researchers. Their equations of equilibrium are based on either the total
Lagrangian formulation or the updated Lagrangian formulation. Since all
displacements are referred to the initial configuration in the total Lagrangian
formulation, the resulting equations of equilibrium are relatively simple.
However, if a structure undergoes a large displacement, some difficulties may be
encountered due to the nonlinearity associated with rotation. All displacements
are referred to the last calculated configuration in the updated Lagrangian
formulation and the rotational nonlinearity is relieved if the load increment is
small. The same difficulties as the total Lagrangian formulation can be
encountered in the case of a large load increment.
Avello[5] referred kinematic variables relative to the initial configuration and
he expressed the strains in a moving frame. Therefore, the strains were invariant
for finite rigid body deformations. Shabana[6-8] presented an absolute nodal
coordinate formulation for flexible multibody dynamics. All finite elements were
reformulated. Shimizu[9] considered the rotary inertia effects. This method is
based on the absolute nodal coordinate formulation.
Moving reference frame approaches were proposed by some researchers in
Refs. 10-14. A moving reference frame is introduced to represent a finite rigid
body motion. Deformation at a point of a flexible body was super-imposed on
the rigid body motion.
136
1.1.1.2. RELATIVE DEFORMATION KINEMATICS
(1) GRAPH THEORETIC REPRESENTATION OF A STRUCTURE
This paper proposes to use the relative nodal displacements in formulating the
equations of equilibrium. Since the absolute nodal deformations are obtained by
accumulating the relative deformations along a path, element connectivity
information must be identified prior to generating the equations of equilibrium
for a general system. Therefore, a topology analysis must be carried out for a
structural system discretized into many finite elements.
Figure 1(a) A cantilever beam with five nodes
Figure 1(b) Graphic theoretic representation for the cantilever beam
The discretized systems can be represented by a graph. A node and an element
are represented by a node and an edge in the corresponding graph, respectively.
As an example, the graph theoretic representation for the system in Fig. 1(a) is
shown in Fig. 1(b). If a structure possesses a loop in its graph theoretic
representation, it is called as a closed loop system. Otherwise, it is called as an
open loop system.
A spanning tree denotes a graph which does not have a closed loop. A node
which does not have a child node is called as a terminal node. A node which
does not have a parent node is called as a base node. The terminal node and the
0 1 2 3 4
Forward path sequence
Backward path sequence
0 1 2 3 4
137
base node for the system in Fig. 1(b) are nodes 4 and 0, respectively. Two
computational sequences must be defined in the proposed relative displacement
formulation. One is the forward path sequence which traverses a graph from the
base node towards the terminal nodes. The other is the backward path sequence
which is the reverse of the forward path sequence. Two sequences for the graph in
Fig. 1(a) are shown in Fig. 1(b).
(2) KINEMATIC DEFINITIONS
Consider a system consisting of two beam finite elements as shown in Fig. 2(a)
and (b). Nodes 1i and i are assumed to be inboard nodes of nodes i and
1i in a graph, as shown in Fig. 2(b), respectively. ZYX is the inertial
reference frame and kkk zyx ),( jik is the nodal reference frame attached
to a node k , and kr is a position vector of the node k . iiiiii )1()1()1( zyx is
the reference frame attached to a node i and the first subscript 1i denotes
the inboard node number of the second subscript i . The orientation of
iiiiii )1()1()1( zyx coincides with that of )1()1()1( zyx iii in the undeformed
state. The absolute nodal displacements measured in the ZYX frame have
been solved for in the conventional finite element analysis methods(see Refs. 1-
4). In contrast to conventional methods, the relative nodal displacements
measured in its inboard nodal reference frame are solved in this paper.
Figure 2(a) Two finite beam elements
ix
iz
iy
1x i
1z i
1y i
i-1
i
X
Z
Y
ir1ir
ii )1(x
ii )1(z
ii )1(y
i+1
138
Figure 2(b) Graphic theoretic representation for the beam elements
The generalized coordinates for the relative nodal position and orientation
displacements of a node are denoted by '
)1( iiu and '
)1( ii , respectively. The
nodal position and orientation of node i in the ZYX frame can be
expressed in terms of these of node 1i and the relative nodal displacements as
follows:
'
)1(
'
0)1()1()1( iiiiiii usArr (1)
iiiiiiii )1(
'
)1()1(1 )( CDAA (2)
where
T
iiiiiiii
'
3)1(
'
2)1(
'
1)1(
'
)1( (3)
In Eqs.(1) and (2), kA ),1( iik denotes the transformation matrix for nodal
reference frame k , ii )1( C denotes the constant transformation matrix from
iii zyx to iiiiii )1()1()1( zyx , '
0)1( iis denotes the location vector of node i
measured in )1()1()1( zyx iii in the undeformed state, and '
)1( iiu denotes the
deformation vector of node i relative to the nodal frame 1i . ii )1( D is the
transformation matrix due to a rotational displacement of iiiiii )1()1()1( zyx
relative to the nodal frame 1i and can be expressed by the 1-2-3 Euler angle
as
)()()( '
3)1(3
'
2)1(2
'
1)1(1)1( iiiiiiii DDDD (4)
Taking a variation of Eq. (1) yields
Forward path sequence
Backward path sequence
ii-1 i+1
139
'
)1()1(
'
)1(
'
)1(
'
0)1()1(
'
)1()1(
' ~~ii
T
iiiiiii
T
iii
T
iii uAusArAr (5)
where a symbol with tilde denotes a skew symmetric matrix which consists of its
vector elements, and ii )1( A is defined as
i
T
iii AAA )1()1( (6)
The virtual rotation relationship between nodes i and 1i is given as
'
)1()1()1(
'
)1()1(
'
iiii
T
iii
T
iii HAA (7)
where
)cos()cos()sin(0
)cos()sin()cos(0
)sin(01
'
2)1(
'
1)1(
'
1)1(
'
2)1(
'
1)1(
'
1)1(
'
2)1(
)1(
iiiiii
iiiiii
ii
ii
H (8)
Combining Eqs.(5) and (7) yields the following recursive virtual displacement
equation for a pair of contiguous elements.:
iiiiiiii )1(2)1()1(1)1( qBZBZ (9)
where
),1(,' iikTT
k
T
kk rZ (10)
TT
ii
T
iiii
'
)1(
'
)1()1( uq (11)
I0
usI
A0
0AB
)~~( '
)1(
'
0)1(
)1(
)1(
1)1(
iiii
T
ii
T
ii
ii (12)
iiT
ii
T
ii
ii
)1()1(
)1(
2)1( H
I
A0
0AB (13)
It is important to note that matrices 1)1( iiB and 2)1( iiB are only functions of the
relative displacement ii )1( q between nodes 1i and i .
The virtual displacement relationship between the absolute and relative nodal
coordinates for the whole system can be obtained by repetitive application of Eq.
140
(9) along a chain in a graph. As an example, the virtual displacement relationship
for the Cartesian and relative coordinate systems in Fig. 1 is as follows.
qBZ (14)
where
TTTTT
4321 ZZZZZ (15)
TTTTT
34231201 qqqqq (16)
342232341122231341012121231341
232122231012121231
122012121
012
BBBBBBBBBB
0BBBBBB
00BBB
000B
B (17)
1.1.1.3. GOVERNING EQUATIONS OF EQUILIBRIUM
(1) STRAIN ENERGY
The strain energy in a finite element having multiple nodes is affected only by
the relative displacements of nodes relative to the inboard nodal frame of the
element and is free from its rigid body motion. As a result, the variational form
of the strain energy for a system can be obtained in a summation form as
n
k
T
kkkk
T
kkW1
)1()1()1( KqqqKq (18)
where q must be kinematically admissible for all constraints. Since the
stiffness matrix is generated in the nodal reference frame, the strain energy due
to a rigid body motion of a node does not appear in Eq. (18). The element
stiffness matrix kk )1( K is contributed from linear and nonlinear terms as (see
Ref. 3)
nL
kk
L
kkkk )1()1()1( KKK (19)
where
141
kkl
kk
L
kk
T
kk
L
kk dx)1(
0
*
)1()1(
*
)1()1(K (20)
)1(
0
*
)1()1()1(
*
)1()1( )(iil
kkkk
nL
kk
T
kk
nL
kk dxqK (21)
In Eqs. (19) - (21), L
kk )1( K denotes a linear stiffness matrix, nL
kk )1( K denotes a
nonlinear stiffness matrix, and kkl )1( denotes the undeformed length of the
element between the nodes 1k and k . Note that the significance of nL
kk )1( K
depends on the magnitude of kk )1( q . nL
kk )1( K becomes negligible when the
magnitude of kk )1( q is small, which is true when the element size is small. It is
very difficult analytically to prove the significance of nL
kk )1( K . As a consequence,
the significance of nL
kk )1( K has been demonstrated through a numerical example
in § 5.
(2) EXTERNAL FORCE
The virtual work done by both nodal forces Q described in the absolute
nodal coordinate system and R described in the relative nodal coordinate
system is obtained as follows:
RqQZ TTW (22)
where Z must be admissible for the kinematic relationship between Z and
q . Substitution of qBZ into Eq. (22) yields
*QqRQBq TTTW (23)
where
RQBQ T* (24)
142
(3) CONSTRAINT
Figure 3 A closed loop system
A nodal displacement is measured relative to its inboard nodal frame in the
proposed method. The relative nodal displacement can be defined only in
structures having a tree topology. Therefore, if a structural system has a closed
loop, it must be opened to form the tree topology. The cut joint method (see Ref.
12) is employed to treat the closed loops. A node in a closed loop is removed and
the corresponding cut constraint equations are introduced to compensate for the
removed node. As an example, Fig. 3 shows a closed loop system. The graphical
representation of the system is presented in Fig. 4.
Figure 4 Graphic representation of the system of the closed loop system in Fig. 3
cut
0
23
4
5
1
0
1 5
2 4
3
cut
143
Figure 5 Tree structure corresponding to the system in Fig. 3
A cut has been made at node 5 to form the tree structure shown in Fig. 5. The cut
constraint can be formulated from the geometric compatibility relationships.
From Eqs. (1) and (2), the position and orientation matrix of node 5 is obtained
along the forward path sequence as
*
5
'
)1(
'
0)1(
5
1
)1(5 rusAr
kkkk
k
k (25)
5
1
*
5)1()1(05
k
kkkk ACDAA (26)
where *
5r and *
5A are given by the boundary conditions at node 5. Since Eq.
(26) comprises of nine dependent equations, only three are independent. The
three independent constraint equations can be extracted by imposing
perpendicularity between the axes of reference frames. As a result, the six
independent constraint equations are given as
*
5251
*
5253
*
5153
*
55
aa
aa
aa
rr
Φ
T
T
T
(27)
where
5352515 aaaA (28)
*
53
*
52
*
51
*
5 aaaA (29)
In Eqs. (28) and (29), i5a and *
5ia )3,2,1( i denote the i -th column vector
of 5A and *
5A , respectively.
0 2 31 4 5
144
(4) EQUATIONS OF EQUILIBRIUM
For a closed loop system, relative deformation q is not independent, and q
must satisfy the constraint Eq. (27). Taking variation of Eq. (27) yields
0qΦΦ q (30)
The Lagrange multiplier theorem (see Refs. 12 and 15) can be applied to obtain
the following equations of equilibrium for a constrained system:
0ΦQKqq q TT * (31)
where the q is arbitrary. Since q is arbitrary, its coefficient must be zero,
which yields
0QΦKqqF q *, T (32)
Since the number of equations is less than that of unknown variables in Eq. (32),
the unknown variables cannot be determined. Thus, constraint equations given in
Eq. (27) are supplemented to find the solution of q and . Deformations q
can be obtained by solving Eqs. (27) and (32) simultaneously. Since the , q ,
and *Q in the equations are the nonlinear function of q , q can be solved by
using Newton-Raphson method as
Φ
Fq
0Φ
ΦF
q
T
(33)
where
qqq QΦKF
* T (34)
By solving Eq. (33), the improved solution of q for the next iteration can be
obtained as follows:
qqq (35)
145
By using Eqs. (33) and (35), the iteration continues until the solution variance
remains within a specified allowable error tolerance. Before solving Eq. (33), it
is necessary to calculate qF . However, the calculation of qF is numerically
difficult and tedious.
In order to save computing time in solving Eq. (33), some numerical
approximation techniques may be applied. As an example, the coefficient matrix
of Eq. (33) may remain near constant if the variation of q is small, which is the
case when the lengths of finite elements are small. In such case, the coefficient
matrix of Eq. (33) can be hold during Newton-Raphson iterations, which
significantly reduces the computation time. However, the approximation
technique may not converge for a system whose q is large. To overcome this
numerical difficulty, a combined incremental and iterative method (see Ref. 16)
can be used.
1.1.1.4. NUMERICAL ALGORITHM
Kinematics of the relative nodal displacements and the equations of
equilibrium are presented in the section 3. This section explains how the
equations are implemented to obtain the relative and absolute nodal
displacements of a structure. The numerical algorithm for closed loop systems is
as follows:
1) Perform the graph theoretic preprocessing to determine computational path
sequences.
2) Form a stiffness matrix K .
3) Compute , q , and *Q for k
q in the backward path sequence.
4) Solve the Eq. (37) to obtain q and .
5) If F and q remains within the specified allowable error tolerance, then
go to step 6. Otherwise, improve the solution using Eq. (35). Go to step 3.
6) Compute the Cartesian deformations in the forward path sequence by using
146
Eqs. (1) and (2).
1.1.1.5 NUMERICAL EXAMPLES
Static analysis of a cantilever beam subjected to end moment M , as shown in
Fig. 6 is carried out.
Figure 6 A cantilever beam subjected to end moment
M
X
Y
][0.1
][0.1
][0.12
0.0
]/[100.3
4
2
27
mI
mA
mL
mNE
L
-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13-2
0
2
4
6
8
10
12
14
16
Undeformed
Proposed
ANSYS: nonlinear
ANSYS: linear
Y [
m]
X [m]
147
Figure 7 Deformed shape of the beam
In the figure, E , , L , A , and I denotes Young's modulus, Poisson ratio,
the length of the beam, the cross sectional area of the beam, and the second area
moment of the cross section, respectively. M =6.545106 [N·m] is applied at
the end node. Fig. 7 shows the deformed shapes of the beam by the proposed
method by the proposed method and a commercial program ANSYS. In the
figure, Proposed, ANSYS: nonlinear, and ANSYS: linear denote numerical
results by the proposed method, a commercial program ANSYS using nonlinear
analysis, and ANSYS using linear analysis, respectively. It shows that the
numerical results obtained by the proposed method and ANSYS(nonlinear
analysis) are almost identical, but the numerical results by ANSYS(linear
analysis) shows large difference with the remaining two numerical results.
0 2 4 6 8 10
-10
-8
-6
-4
-2
0
ue: Proposed
ue: ANSYS
ue [m
]
The number of elements
Figure 8 Convergence of axial deformation at the end node vs. the number of elements
Fig. 8 shows the convergence of the axial deformation eu at the end node.
When fewer elements are used for static analysis, the numerical results of
148
ANSYS are more accurate than those of the proposed method, but eu obtained
by the proposed method converges rapidly as the number of elements is
increased. In the figure, the numerical results with more than 6 elements by the
two methods are almost identical. From the analysis results, it is known that the
effect of the nonlinear stiffness matrix is diminished rapidly as the number of
elements is increased.
Figure 9 A closed loop system subjected to concentrated force and Moment
F
Y
M
X
][002.0
][01.0
][14.14
0.0
]/[100.3
4
2
27
mI
mA
mL
mNE
4
4
L
P
149
Figure 10 Comparison of deformed shapes of the closed system
Fig. 9 shows a closed loop system subjected to a concentrated force F and
moment M at a point P . When F =[3×104 -1×10
4]T
[N] and M =0.0
[N·m] are applied at the point P the deformed shapes of the system are shown
in Fig. 10. It shows that the numerical results obtained by the proposed method
with 20 elements and a commercial program ANSYS are almost identical.
-1 0 1 2 3 4 5 6 7 8
0
1
2
3
4(F = [3.0E+4, -1.0E+04 ]
T [N], M= 0.0E+0[Nm])
Undeformed shape
Deformed shape: 20 elements
Deformed shape: ANSYS
Y [m
]
X [m]
150
Figure 11 Undeformed and deformed shapes of the closed loop system
Figure 12 Deformed shapes of the closed loop system at each load step
When F =[3×104 -3×10
4]T
and M =3.0×104[N·m] are applied at the point
P , the deformed shape of the system is shown in Fig. 11. While the numerical
solution by the proposed method converges after the 7-th iteration, that by the
commercial program ANSYS does not converge. Fig. 12 shows the deformed
shape of the system at each load step.
-1 0 1 2 3 4 5 6 7 8-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0(F=[3.0E+4, -3.0E+4]
T[N], M=3.0E+4[Nm])
Y[m
]
X[m]
Undeformed shape
Deformed shape: 20 elements
0 1 2 3 4 5 6 7 8
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
(F=[3.0E+4, -3.0E+4]T[N], M=3.0E+4[Nm])
Y[m
]
X[m]
0.33F, 0.33M
0.67F, 0.67M
1.00F, 1.00M
151
1.1.1.6 CONCLUSIONS
A geometric nonlinear formulation for structures undergoing large
deformations is investigated in this research. Nodal displacements in the
proposed method are referred to its adjacent nodal reference frame. Since the
nodal displacements are measured relative to its inboard nodal frame, quantity of
the nodal displacements is still small for a structure undergoing large
deformations if the element sizes are small. Relative coordinate kinematics is
developed to define relative position and orientation of the nodal displacements.
As a consequence, many element formulations developed under small
deformation assumptions are reusable for structures undergoing large
deformations, which makes it easy to develop a computer program. A structural
system is represented by a graph to systematically develop the governing
equations of equilibrium for general systems. Closed loops are opened to form a
tree topology by cutting nodes. Two computational sequences are defined for a
graph. One is the forward path sequence that is used to recover the Cartesian
nodal deformations from relative nodal displacements and traverses a graph from
the base node towards the terminal nodes. The other is the backward path
sequence that is used to recover the nodal forces in the relative coordinate
system from the known nodal forces in the absolute coordinate system and
traverses from the terminal nodes toward the base node. A solution algorithm is
developed to implement the proposed method. Static analyses are performed for
structures undergoing large deformations. The proposed method can solve the
problem which cannot be solved by the commercial program ANSYS.
152
REFERENCES
(1) El Damatty, A. A., Korol, R. M and Mirza, F. A., "Large Displacement Extension of
Consistent Shell Element for Static and Dynamic Analysis," Computers & Structures, Vol.
62, No. 6, (1997), p. 943-960.
(2) Mayo, J and Domínquez, J., "A Finite Element Geometrically Nonlinear Dynamic
Formulation of Flexible Multibody Systems using a New Displacements
Representation," J. Vibration and Acoustics, Vol. 119, (1997), p.573-580.
(3) Dhatt, G and Touzot, G., The Finite Element Method Displayed, John Wiley & Sons,
(1984).
(4) Bathe, K. J., Finite Element Procedures, Prentice-Hall, (1996).
(5) Avello, A. J., Jolón, G. D. and Bayo, E., "Dynamics of Flexible Multibody Systems
using Cartesian Co-ordinates and Large Displacement Theory," Int. J. Numer. Methods
Eng., Vol. 32, No. 8, (1991), p.1543-1564.
(6) Shabana, A. A, "An Absolute Nodal Co-ordinate Formulation for the Large Rotation
and Deformation Analysis of Flexible Bodies," Technical Report MBS 96-1-UIC,
Department of Mechanical Engineering, University of Illionois at Chicago, (1996).
(7) Shabana, A. A. and Christensen, A., "Three Dimensional Absolute nodal coordinate
formulation: Plate Problem," Int. J. Nuner. Methods Eng., Vol. 40, No. 15, (1997),
p.2275-2790.
(8) Shabana, A. A., Dynamics of Multibody Systems, 2nd edition, Cambridge University
Press, (1998).
(9) Takahashi, Y. and Shimizu, N, “Study on Elastic Forces of the Absolute Nodal
Coordinate Formulation for Deformable Beams,” Proceedings of the ASME Design
Engineering Technical Conferences, (1999).
(10) Featherstone, R., "The Calculation of Robot Dynamics Using Articulated-Body
Inertias, " Int. J. Roboics Res., Vol. 2, (1983), p. 13-30.
(11) Bae, D. S. and Haug, E. J., "A Recursive Formulation for Constrained Mechanical
System Dynamics: Part I. Open Loop Systems," Mech. Struct. and Machines, Vol. 15, No.
153
3, (1987), p. 359-382.
(12) Bae, D. S. and Haug, E. J., "A Recursive Formulation for Constrained Mechanical
System Dynamics: Part II. Closed Loop Systems," Mech. Struct. and Machines, Vol. 15,
No. 4, (1987), p. 481-506.
(13) Lin, T. C. and Yae, K. H., Recursive Linearization of Multibody Dynamics and
Application to Control Design, Technical Report R-75, Center for Simulation and
Design Optimization, Department of Mechanical Engineering, and Department of
Mathematics, The University of Iowa, Iowa City, Iowa, (1990).
(14) Yoo, H., Ryan, R. and Scott, R., "Dynamics of Flexible Beams undergoing Overall
Motion," J. Sound and Vibration, Vol. 181, No. 2, (1995), p.261-278.
(15) Haug, E. J., Computer-Aided Kinematics and Dynamics of Mechanical Systems:
Volume I. Basic Methods, Allyn and Bacon, (1989).
(16) Crisfield, M. A., Non-Linear Finite Element Analysis of Solids and Structures, Wiley,
(1997).
154
1.1.2
MULTI FLEXIBLE BODY DYNAMICS
USING INCREMENTAL FINITE
ELEMENT FORMULATION
1.1.2.1 INTRODUCTION
As an advent of high performance computer and the development of proper
numerical algorithms, the analytic approach for the engineering design has been
shifted to the computer aided design processes. From the 70ths, the computational
methods have been extensively used in statics to perform the local stress analyses
in the part level design and in dynamics to understand the multibody dynamics in
the system level design. From mid 80ths, structural dynamics, large deformation,
and nonlinear FEA had become main research topics in structural community. On
the other hand, flexible body dynamics, linear small deformation, modal synthesis
technique, and co-simulation interface had become main research topics in
dynamics community. But, recently, both communities have recognized that they
need a same platform which can integrate large deformation finite element
formulations with flexible multibody system algorithms. The goal of both
communities is to develop computer simulation programs for the analysis of
physical or engineering models such as a Multi Flexible Body Dynamics (MFBD),
which is also called as a flexible multibody dynamics system in the literature. In
general, a MFBD system consists of many rigid and flexible bodies connected by
forces, joints and contacts. And a flexible body may experience large deformation,
plasticity, and fracture. But the solution of these new multiphysics and multiscale
problems requires the development of a new generation of computer codes that
integrate large deformation finite element and multibody system algorithms. The
needs for the new generation of computer codes are well explained by Shabana
et al. [1].
Until now, most existing general-purpose multibody system computer codes are
designed to solve rigid body systems and small deformation problems. This
general-purpose computer codes are based on the floating frame of reference
formulation. Even though this formulation is widely used in the flexible multibody
system dynamics for solving small deformation problems, it is not acceptable for
solving large deformation problems [2,3]. As a result, it cannot be used in the
analysis of many physical or engineering applications. In contrast, existing large
154
deformation finite element algorithms and computer codes are not designed for
multibody systems. For this reason, it becomes necessary to successfully integrate
large deformation finite element formulations and multibody system algorithms in
order to develop a new generation of computer algorithms which can handle
MFBD problems.
In general, system components of multibody systems undergo finite relative
displacements and they are connected by mechanical joints that impose restrictions
on their motion. In particular, the finite rotations introduce geometric nonlinearities
and mechanical joints introduce differential algebraic constraint equations [1,4].
Because of these kinds of characteristics of multibody sytems, recently, the
computer implementation approaches for the multibody system formulations are
mainly based on the nonincremental solution procedures such as absolute nodal
coordinate formulations [5-10].
But, most existing finite element algorithms are based on a corotational
formulation and it is very efficient for large displacements but small strains
problems. The co-rotational method was originally introduced by Wempner [11]
and Belytschko et al. [12] and has much in common with the natural approach of
Argyris et al. [13]. And this approach has generated an increased amount of interest
in the last decade [14-18]. In this study, we use the incremental finite element
formulation using corotational procedure in order to propose a new integration
method for the MFBD problems.
The rigid body formulation using a recursive formulation is introduecd in Section
2 and the incremental formulation with corotational procedure are explained in
Section 3. In Section 4, in order to formulate the kinematic relations between rigid
or flexible bodies, the virtual body and flexible body joint are proposed. In Section
5, the whole system equations for the MFBD system are explained. The numerical
example for the double pendulum MFBD problem is simulated and discussed in
Section 6. Summary and conclusions are presented in Section 7.
1.1.2.2. RIGID BODY FORMULATION
The coordinate systems for two contiguous rigid bodies in 3D space are shown in
Fig. (1). Two rigid bodies are connected by a joint, and an external force F is
acting on the rigid body j . The X-Y-Z frame is the inertial or global reference
frame and the x -y -z is the body reference frame with respect to the X-Y-Z
frame. The subscript i means the inboard body of body j in the spanning tree of
a recursive formulation [19]. And, in this section, the subscript j can be replaced
with the subscript ( 1)i .
156
Figure 1: Two contiguous rigid bodies.
Velocities and virtual displacements of the origin of body reference frame x -y -z
with respect to the global reference frame X-Y-Z , respectively, defined as
r
ω
and
r
π
Their corresponding quantities with respect to the body reference frame x -y -z
are, respectively, defined as T
T
r A rY
ω A ω
and
T
T
r A rZ
π A π
where A is the orientation matrix of the x -y -z frame with respect to the X-Y-Z
frame. Also, as shown in Fig. (1), two contiguous bodies are connected by a joint
between the i1 i1 i1x -y -z and j1 j1 j1x -y -z frames.
ri
X
Z
Y
xi1’’
yi1’’zi1’’
Inertial Ref. Frame
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Rigid Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
rj
F
di1(j1)
Joint
si(i1)
sj(j1)
ri
X
Z
Y
X
Z
Y
xi1’’
yi1’’zi1’’
Inertial Ref. Frame
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Rigid Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
rj
F
di1(j1)
Joint
si(i1)
sj(j1)
(1)
(2)
(3)
(4)
157
The origin of the j j jx -y -z frame can be expressed as
( 1) 1( 1) ( 1)j i i i i j j j r r s d s
The angular velocity in the body reference frame is obtained as
T T
j ij i ij ij ij ω A ω A H q
where H is determined by the axis of rotation and T
ij i jA A A . Differencitaion
of Eq. (5) with j j jr A r , ij i ij
d A d , ij i ijs A s , and ji j ji
s A s yields
ij
j j i i i ij i i iijj i j ji j i ij ij
qA r A r A s ω A d ω A s ω A d q
where symbols with tildes denote skew symmetric matrices associated with their
vector elements and ijq denotes a relative coordinate vector. Substituting jω of
Eq. (6) and multiplying both sides of Eq. (7) by T
jA yield
T T T T T
ijj ij i ij iij iiijj ij ji ij i ij ij ij ji ij ij ij
qr A r A s d A s A ω A d A s A H q
for which j j jA A ω is used. Combining Eqs. (6) and (8) yields the following
velocity recursive equations for a pair of contiguous bodies.
1 2
j ij i ij ij Y B Y B q
where Y is the combined velocity of the translation and rotation as defined in Eq.
(3) and 1
ijB and 2
ijB are defined as follows:
TT
1
T
iij iiijj ij ji ijij
ij
ij
I s d A s AA 0B
0 A 0 I
and
(5)
(6)
(7)
(8)
(9)
(10)
158
TT
2
Tij
ij ij ji ij ijij
ij
ij
qI d A s A HA 0
B0 A 0 I
It is important to note that matrices 1
ijB and 2
ijB are only functions of the ijq .
Similarly, the recursive virtual displacement relationship is obtained as follows:
1 2
j ij i ij ij Z B Z B q
If the recursive formula in Eq. (9) is respectively applied to all joints along the
spanning tree, the following relationship between the Cartesian and relative
generalized velocities can be obtained:
Y Bq
where B is the collection of coefficients of the ijq and
T
T T T T
0 1 2 nc? 1, , , , n
Y Y Y Y Y
and
T
T T T T
0 01 12 ( 1) nr? 1, , , , n n
q Y q q q
where nc and nr denote the number of the Cartesian and relative generalized
coordinates, respectively. The Cartesian velocity ncY R with a given nrq R
can be evaluated either by using Eq. (13) obtained from symbolic substitutions or
by using Eq. (9) with recursive numerical substitution of jY . Since both formulas
give an identical result and recursive numeric substitution is proven to be more
efficient, matrix multiplication Bq with a given q will be actually evaluated by
using Eq. (9). Since q in Eq. (13) is an arbitrary vector in nrR , Eqs. (9) and (13),
which are computationally equivalent, are actually valid for any vector nrx R
such that
X Bx
and
(11)
(12)
(13)
(14)
(15)
(16)
159
1 2
j ij i ij ij X B X B x
where ncX R is the resulting vector of multiplication of B and x . As a result,
transformation of nrx R into ncBx R is actually calculated by recursively
applying Eq. (17) to achieve computational efficiency in this research.
Inversely, it is often necessary to transform a vector G in ncR into a new vector
Tg B G in nrR . Such a transformation can be found in the generalized force
computation in the joint space with a known force in the Cartesian space. The
virtual work done by a Cartesian force ncQ R is obtained as follows:
Τ W Z Q
where Z must be kinematically admissible for all joints in a system. Substitution
of Z B q into Eq. (18) yields
T Τ T * W q B Q q Q
where * TQ B Q .
The equations of motion for a constrained mechanical system [21] in the joint
space[22] have been obtained by using the velocity transformation method as
follows:
( T Τ
ΖF B MY Φ λ Q ) 0
where Φ and λ , respectively, denote the cut joint constraint and the
corresponding Lagrange multiplier. M is a mass matrix and Q is a force vector
including the external forces in the Cartesian space.
1.1.2.3. INCREMENTAL FORMULATION WITH
COROTATIONAL PROCEDURE
The incremental formulation has been widely and successfully used in the
nonlinear finite element analysis of large rotation structural problems. The idea of
this approach is to decompose the motion of the element into rigid body and pure
deformation parts, through the use of a local element frame which continuously
translates and rotates with element. The schematic diagram for the incremantal
formulation with corotational procedure is shown in Fig. (2). In this procedure,
(17)
(18)
(19)
(20)
160
which is independent of the element formulation, any rigid body motion
contribution is eliminated from the global displacement field in order to determine
the pure deformation. The contribution of the rigid body rotations of the element is
eliminated by using a local element reference frame (or a convected coordinate
system) that moves with the element. The element equations are first defined in the
element coordinate system and then transformed in order to define element
equations in the global inertial frame. These equations are solved for the
displacement increments that are then used to update the global displacement field
of the element. The incremental finite element formulation using corotational
procedure [11-18] has been used in several general purpose finite element structural
programs, and has been used in the analysis of many large rotation and deformation
problems. In this study, the incremental finite element formulation using
corotational procedure is used to represent the finite element equations in the
general purpose multi flexible body dynamics solver.
In this approach, in general, the nonlinear kinematics of the finite element is
defined in terms of a large reference motion and a small deformation. Therefore, in
order to accurately represent the element equations in the current configuration
from the previous configuration, the displacement increments should be small at
each time step. This assumption implies that in one time step there is no large
variation in the deformation within each element and the large reference motion.
Consequently, the most important parameter that governs this procedure is the
integration time step which can accurately represent the current configuration from
the previous configuration [1,2].
Figure 2: The schematic diagram for the incremantal formulation with corotational
procedure.
In this study, for the incremental formualtion uisng the corotational procedure, the
X
Z
Y
Inertial Ref. Frame
Original or Previous Configuration (i)
Deformed or Current Configuration (i+1)
x
x
y
y
ri, θi
∆ri+1 , ∆θi+1Ai
Ai+1
ri+1, θi+1 Rigid Body Motion
Deformation
X
Z
Y
X
Z
Y
Inertial Ref. Frame
Original or Previous Configuration (i)
Deformed or Current Configuration (i+1)
x
x
y
y
ri, θi
∆ri+1 , ∆θi+1Ai
Ai+1
ri+1, θi+1 Rigid Body Motion
Deformation
161
generalized coordinated for the finite elements are defined as,
1
1
1
ie
i
i
rq
θ
where, the 1iθ is defined from Eq. (22) to Eq. (24).
1 1i i i A A A
and
1
cos cos cos sin sin cos sin cos cos sin sin sin
cos sin 0 cos 0 sin 1 0 0
sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
y z z x y x z x z y x
z z y y
z z x x
y y x x
i
A
cos sin cos cos sin sin sin cos sin sin cos sin
sin cos sin cos cos
z
y z x z x y z x y z z x
y y x x y
where, if x ,
y , and z are infinitesimal, the matrix
1iA can be
approximated as
T
1 1
1 sin sin 1
sin 1 sin 1
sin sin 1 1
z y z y
i i i z x z x
y x y x
A A A
Here, we define 1iθ as follows:
1
x
i y
z
θ
1.1.2.4. KINEMATIC RELATIONS BETWEEN RIGID OR
FLEXIBLE BODIES
(1) Virtual rigid bodies
The coordinate systems for two adjacent rigid and flexible bodies in 3D space are
shown in Fig. (3). Two bodies are connected by a joint and an external force F is
(21)
(22)
(23)
(24)
(25)
162
acting on the flexible nodal body. The XYZ frame is the inertial reference frame,
and the i i ix y z and j j jx y z frame are the body reference frame of the rigid body i
and flexible nodal body j with respect to XYZ frame, respectively.
Figure 3: The coordinate systems for two adjacent rigid and flexible bodies.
Kinematic admissibility conditions among the reference frames can be divided
into two types. One is the admissibility conditions between the two joint frames
and the other is the admissibility conditions among the frames within a flexible
body. These two types of conditions have been mixed in formulating the kinematic
joint constraints and generalized forces. As a result, every joint and force module in
a flexible multibody code has been developed separately for rigid and flexible
bodies. This takes a long time for computer implementation. In particular, flexible
body programming requires much more effort than rigid body programming does
due to the complexity of generalized coordinates for flexible bodies. Therefore, in
order to minimize the programming effort, the concept of the virtual body is
introduced in this section. At every joint and force reference frame, a virtual rigid
body, whose mass and moment of inertia is zero, is introduced. As an example, in
the case of Fig. (3), two rigid virtual bodies are introduced as shown in Fig. (4).
Through this virtual rigid body concept, the flexible nodal bodies have no joints
or applied forces. The flexible bodies are subjected to only the kinematic
admissibility conditions among its body frame and the virtual body frames.
Therefore, in the general purpose program, the joint and force modules are
developed only for rigid bodies as described in Section (2) and one flexible body
joint is to be added in the joint module. The kinematic admissibility conditions for
the flexible body joint are formulated in the Section (3).
ri
X
Z
Y
xi1’’
yi1’’zi1’’
Inertial Ref. Frame
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Flexible Nodal Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
rj
F
di1(j1)
Joint
ri
X
Z
Y
X
Z
Y
xi1’’
yi1’’zi1’’
Inertial Ref. Frame
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Flexible Nodal Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
rj
F
di1(j1)
Joint
163
Figure 4: The concept of virtual body.
(2) Joint constraints between two rigid bodies
A joint has been represented by imposing the parallel or orthogonal conditions on
vectors attached to two adjacent rigid bodies. A library of such conditions for rigid
bodies has been well developed and has become the basics in building various
joints [3,24]. The conditions are formulated by using geometric vectors that are
defined within or between two joint reference frames. A joint reference frame, in
general, does not coincide with the body reference frame. But, in the case of virtual
rigid body, the body reference frame is used as a joint reference frame in this study.
As a result, the kinematic admissibility conditions for a joint connecting a virtual
rigid body is simplified and the number of non-zero entries of the constraint
Jacobian is reduced.
(3) Flexible body joint constraints between a flexible body and a virtual body
As shown in Fig. (5), the origin and orientation matrix of the body reference frame
are jr and jA , respectively. Similarly, the origin and orientation matrix of the
virtual rigid body are 1jr and 1jA , respectively.
xi1’’
yi1’’
zi1’’
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Flexible Nodal Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
F
Virtual Rigid Body j1
Virtual Rigid Body j2
di1(j1)
xi1’’
yi1’’
zi1’’
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Flexible Nodal Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
F
Virtual Rigid Body j1
Virtual Rigid Body j2
di1(j1)
164
Figure 5: Flexible body joint.
For the joint constraint equations between flexible nodal body and virtual rigid
body, in this study, the constraints for all translational and rotational degree of
freedom are fixed as Eq. (26).
1
1
1( ) ( )
j jj j
T
j j j
r rdΦ 0
θ dA θ A C A I
where, jC is the orientation matrix of 1jA with respect to jA in the original
configuration. And dA is the relative orientation matrix induced by the rotation. If
the Euler angle(1-2-3) is employed, dA is expressed as follows:
cos sin 0 cos 0 sin 1 0 0
sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
cos cos cos sin sin cos sin cos cos sin sin sin
cos sin cos cos sin
z z y y
z z x x
y y x x
y z z x y x z x z y x z
y z x z
dA I
sin sin cos sin sin cos sin
sin cos sin cos cos
x y z x y z z x
y y x x y
I
where, if x ,
y , and z are infinitesimal, the matrix dA can be approximated as
1
0 sin sin 0
( ) sin 0 sin 0
sin sin 0 0
z y z y
T
j j j z x z x
y x y x
dA A C A I
Aj
xj’
yj’
zj’
Flexible Nodal Body j
Aj1 xj1’’
yj1’’zj1’’
Virtual Rigid Body j1
X
Z
Y
Inertial Ref. Frame
rj
rj1dj(j1)
Aj
xj’
yj’
zj’
Flexible Nodal Body j
Aj1 xj1’’
yj1’’zj1’’
Virtual Rigid Body j1
X
Z
Y
X
Z
Y
Inertial Ref. Frame
rj
rj1dj(j1)
(26)
(27)
(28)
165
Here, we define θ as follows:
1( ) ( )
x
T
j j j y
z
θ dA θ A C A I
1.1.2.5. SYSTEM MATRIX FOR EQUATION OF MOTION
The equation of motion for the rigid body can be expanded from the Eq. (20) as
follows:
T T 0r T rr rr er er r z zF B MY Φ λ Φ λ Q
where, the superscript r means the quantity for the rigid body. The superscripts
rr means the quantities between rigid bodies and the superscript er means the
quantities between a flexible nodal body and a rigid body. Also, the constraints
equations between rigid bodies is expressed as the funtion of rigid body
generalized coordintates rq as follows:
rrrrrqΦΦ
Similary, we can derive the equations of motion for the flexible body as follows:
T 0e
e e e er er e q
F M q Φ λ Q
where, the superscript e means the quantities for the flexible nodal body and eq
is the generalized coordinate for the flexible nodal bodies. The superscripts ee
means the quantities between flexible nodal bodies and the superscript er means
the quantities between a flexible nodal body and a rigid body. Here, the force eQ
between flexible nodal bodies can include the element and gravity forces as follows:
e element gravity Q Q Q
Also, from Eq. (26) for the flexible body joint constraints between a flexible nodal
body and a virtual body, we can express the erΦ as follows:
(29)
(30)
(31)
(32)
(33)
166
1
T
1
,
e rj j
er e r
j j j
r r
Φ q q 0θ A C A I
Finally, we can make the whole system matrix for the MFBD problems as Eq. (35)
and we can solve the Eq. (35) using the sparse matrix solver for the incremental
quantities.
T
T T T T
e
r
e r
e eer
e ree e
r r rr rrr er
e rrr rr
rrer er
er er
q
z z
q
q q
F F0 Φ
q q q F
F F q FB Φ B Φq q λ Φ
0 Φ 0 0 λ Φ
Φ Φ 0 0
1.1.2.6. NUMERICAL RESULTS
As a numerical model, a rigid-rigid (Model A) and a rigid-flexible (Model B)
double pendulum models are used as shown in Fig. (6) and Fig. (7), respectively.
The model parameters and material properties for both models are shown in Table 1
and only difference between two models is the use of flexible shell element instead
of the box rigid geometry.
Figure 6: A rigid-rigid double pendulum model (Model A).
50 mm 100 mm
Rigid Body ( Cylinder )
Rigid Body ( Box )
Revolute Joints
A0
y
z
x
x
Gravity = 9806.65 mm/s2
50 mm 100 mm
Rigid Body ( Cylinder )
Rigid Body ( Box )
Revolute Joints
A0
y
z
x
x
Gravity = 9806.65 mm/s2
(34)
(35)
167
Figure 7: A rigid-flexible double pendulum model (Model B).
Table 1. The model parameters and material properties.
Model A ( rigid-rigid ) Model B ( rigid-flexible )
Rigid Body :
Cylinder
Rigid Body : Box Rigid
Body :
Cylinder
Flexible
Body : Shell
Elements
Radius (mm) 1.0 - 1.0 -
Thickness,
Depth (mm)
- 0.1, 2.0 - 0.1, 2.0
Density
(kg/mm3)
7.85e-06 7.0e-07 7.85e-06 7.0e-07
Young’s
Modulus
(N/mm2)
- - - 10.0
Poisson’s Ratio - - - 0.2
Damping Ratio - - - 0.01
50 mm 100 mm
Gravity = 9806.65 mm/s2Revolute Joints
Rigid Body ( Cylinder )
Flexible Body ( Shell Elements )
B1 B2 B3 B4 B5 B6B0
x
x
y
z
50 mm 100 mm
Gravity = 9806.65 mm/s2Revolute Joints
Rigid Body ( Cylinder )
Flexible Body ( Shell Elements )
B1 B2 B3 B4 B5 B6B0
x
x
y
z
168
Figure 8: Position trajectory results for the Model A and Model B.
As the simulation results, the position trajectory results are displayed at the same
time for the both model in Fig. (8). The rigid body motion of cylinder-shaped rigid
body are almost same between two models, and the flexible shell elements undergo
large rotation, large displacement, and large deformation. For more detailed
comparison for the rigid body motion, the displacement results at the center
positions (A0 and B0) for the cylinder-shaped rigid bodies are plotted in Fig. (9). In
Fig. (9), the position results for the x and y coordinates position show a good
agreement between Model A and B. And Fig. (10) shows the x and y coordinate
position results for the given node positions of flexible body for Model B.
T = 0.0 s
T = 0.04 s
T = 0.08 s
T = 0.12 s
T = 0.16 s
T = 0.20 s
Model A
Model B
T = 0.0 s
T = 0.04 s
T = 0.08 s
T = 0.12 s
T = 0.16 s
T = 0.20 sT = 0.0 s
T = 0.04 s
T = 0.08 s
T = 0.12 s
T = 0.16 s
T = 0.20 s
Model A
Model B
169
Figure 9: The results of positions for Model A and B at the center positions A0 and B0.
Figure 10: The results of positions for the flexible body of Model B at the given nodes B1,
B2, B3, B4, B5, and B6.
1.1.2.7 CONCLUSIONS
In order to solve the multi flexible body dynamics (MFBD) problems, this study
Position X
Position Y
Position X
Position Y
B1
B2
B3
B4
B5
B6
Position X
Position Y
B1
B2
B5
B4
B3
B6
B1
B2
B3
B4
B5
B6
Position X
Position Y
B1
B2
B5
B4
B3
B6
170
uses an incremental finite element formulation using a corotational procedure in
which the nodal coordinates are referred to the last calculated configuration. Also,
in order to easily and efficiently implement the general purpose program which can
integrate the large deformation finite element and multibody dynamics, the virtual
body and the flexible body joint are introduced. The system equations for the
MFBD problems are introduced and the numercal example for the very flexible
double pendulum problem is simulated and compared with the rigid body model.
As a result, the use of the incremental finite element formulation using corotational
procedure for the MFBD problems is acceptable for the general-purpose MFBD
program. And the use of virtual bodies and flexible body joint constraints is also
acceptable.
REFERENCES
1. A. Shabana, O. A. Bauchau, and G. M. Hulbert, Integration of Large
Deformation Finite Element and Multibody System Algorithms, Journal of
Computational and Nonlinear Dynamics, 2, 351-359, 2007.
2. A. Shabana, Computational Continuum Mechanics, Cambridge University
Press, 2008.
3. A. Shabana, Dynamics of Multibody Systems, 3rd Edition, Cambridge
University Press, 2005.
4. E. J. Haug, Computer Aided Kinematics and Dynamics of Mechanical
Systems, Volume I: Basic Methods, Allyn and Bacon Series in Engineering,
1989.
5. M. Campanelli, M. Berzeri, and A. A. Shabana, Performance of the
Incremental and Non-Incremental Finite Element Formulations in Flexible
Multibody Problems, Transactions of ASME, Journal of Mechanical
Design, 122, 498-507, 2000.
A. A. Shabana and A. P. Christensen, Three-Dimensional Absolute Nodal
Co-ordinate Formulation: Plate Problem, International Journal for
Numerical Methods in Engineering, 40, 2775-2790, 1997.
6. D. García-Vallejo, J. L. Escalona, J. Mayo, and J. Domínguez, Describing
Rigid-Flexible Multibody Systems Using Absolute Coordinates, Nonlinear
Dynamics, 34, 75-94, 2003.
7. H. Sugiyama, J. L. Escalona, and A. A. Shabana, Formulation of Three-
Dimensional Joint Constraints Using the Absolute Nodal Coordinates,
Nonlinear Dynamics, 31, 167-195, 2003.
8. W. S. Yoo, S. J. Park, O. N. Dmitrochenko, and D. Y. Pogorelov,
Verification of Absolute Nodal Coordinate Formulation in Flexible
Multibody Dynamics via Physical Experiments of Large Deformation
171
Problems, Transactions of ASME, Journal of Computational and Nonlinear
Dynamics, 1, 81-93, 2006.
9. D. García-Vallejo, J. Mayo, J. L. Escalona, and J. Domínguez, Three-
Dimensional Formulation of Rigid-Flexible Multibody Systems with
Flexible Beam Elements, Multibody System Dynamics, 20, 1-28, 2008.
10. G. Wempner, Finite Elements, Finite Rotations and Small Strains of
Flexible Shells, International Journal of Solids and Structures, 5, 117-153,
1969.
11. T. Belytschko and B. J. Hsieh, Non-Linear Transient Finite Element
Analysis with Convected Co-ordinates, International Journal for
Numerical Methods in Engineering, 7, 255-271, 1973.
12. J. H. Argyris, H. Balmer, J. S. Doltsinis, P. C. Dunne, M. Haase, M.
Kleiber, G. A. Malejannakis, H. P. Mlejnek, M. Müller, and D. W. Scharpf,
Finite Element Method - The Natural Approach, Computer Methods in
Applied Mechanics and Engineering, 17, 1-106, 1979.
A. Eriksson and C. Pacoste, Element Formulation and Numerical
Techniques for Stability Problems in Shells, Computer Methods in
Applied Mechanics and Engineering, 191, 3775-3810, 2002.
B. C. Rankin and F. A. Brogan, An Element Independent Corotational
Procedure for the Treatment of Large Rotations, ASME Journal of
Pressure Vessel Technology, 108, 165-174, 1986.
13. M. A. Crisfield and G. F. Moita, A Unified Co-rotational Framework for
Solids, Shells and Beams, International Journal of Solids and Structures,
33, 2969-2992, 1996.
14. A. Felippa and B. Haugen, A Unified Formulation of Small-Strain
Corotational Finite Elements: I. Theory, Computer Methods in Applied
Mechanics and Engineering, 194, 2285-2335, 2005.
15. J. M. Battini and C. Pacoste, On the Choice of the Linear Element for
Corotational Triangular Shells, Computer Methods in Applied Mechanics
and Engineering, 195, 6362-6377, 2006.
16. S. Bae, J. M. Han, J. H. Choi, and S. M. Yang, A Generalized Recursive
Formulation for Constrained Flexible Multibody Dynamics, International
Journal for Numerical Methods in Engineering, 50, 1841-1859, 2001.
17. S. Bae, J. M. Han, and J. H. Choi, An Implementation Method for
Constrained Flexible Multibody Dynamics Using a Virtual Body and Joint,
Multibody System Dynamics, 4, 297-315, 2000.
18. J. García de Jalón, J. Unda, A. Avello, Natural coordinates for the computer
analysis of multibody systems, Computer Methods in Applied Mechanics
and Engineering, 56, 309-327, 1986.
19. J. Wittenburg, Dynamics of Systems of Rigid Bodies, B. G. Teubner,
172
Stuttgart, 1977.
20. J. Angeles. Fundamental of Robotic Mechanical Systems. Springer, 1997.
21. RecurDynTM
Help Library, FunctionBay, Inc., http://ww
1.2 RFlex
174
1.2.1
FLEXIBLE MULTIBODY DYNAMICS USING A
VIRTUAL BODY AND JOINT
1.2.1.1. INTRODUCTION
A rigid body in space is described by the position and orientation generalized
coordinates with respect to the inertial reference frame. Contrast to
implementation of a rigid body dynamic analysis program, it is generally
complicated to implement a flexible body dynamic formulation and to expand it
for a general purpose program, regardless of whatever formulation has been
chosen. This is because the flexible body dynamic formulations handle
additional generalized coordinates to these of the rigid body dynamics. One of
the most tedious works involved with the implementation of the flexible body
dynamics is to build a set of joint and force modules. Whenever a new force or
joint module is developed for the rigid body dynamics, the corresponding
module for the flexible body dynamics has to be formulated and programmed
again. In order to avoid such a repetitive process, this investigation proposes a
concept of virtual body and joint.
Shabana [1] presented a coordinate reduction method for multibody systems
with flexible components. The local deformation of a flexible component was
expressed in terms of the nodal coordinates and was then spanned by a set of
mode shapes obtained from a normal mode analysis. Yoo and Haug [2] spanned
the deformation by a set of static correction modes obtained by applying a unit
force or unit displacement at a node where a large magnitude of force is expected
during the dynamic analysis. Mani [3] used Ritz vectors in spanning the local
deformation and the Ritz vectors were generated by spatially distributing the
inertial and joint constraint forces on a flexible body. Gartia de Jalon et al [4]
presented a fully Cartesian coordinate formulation for rigid multibody dynamics.
This formalism was extended to the flexible body dynamics by Vukasovic et al
175
[5]. Nonlinearity associated with an orientational transformation matrix was
relieved by defining all necessary vectors for the equations of motion and
constraints as the generalized coordinates.
Several formulations have been recently developed for flexible body systems
that undergo large deformation. Simo [6] had formulated the equations of motion
for a flexible beam, based on the inertial reference frame. Since displacement of
a point on the beam was directly measured from the inertial reference frame, the
inertia terms become linear and uncoupled, while the strain energy related terms
become nonlinear. Yoo and Ryan [7] proposed a mixed formulation of inertial
and floating reference frames for a rotating beam. Axial deformation was
measured from a deformed state of the rotating beam, while other deformations
were measured from an undeformed state. Shabana [8,9] presented a non-
incremental absolute coordinate formulation in which the global location
coordinates and slopes were defined as the generalized coordinates. Since the
finite rotation coordinates were not used as the generalized coordinates, the
difficulties associated with the finite rotation were resolved.
Contrast to implementation of a rigid body dynamic analysis program, it is
generally complicated to implement a flexible body dynamic formulation and to
expand it for a general purpose program, regardless of whatever formulation has
been chosen. This is because the flexible body dynamic formulations handle
additional generalized coordinates to these of the rigid body dynamics. One of
the most tedious works involved with the implementation of the flexible body
dynamics is to build a set of joint and force modules. Whenever a new force or
joint module is developed for the rigid body dynamics, the corresponding
module for the flexible body dynamics has to be formulated and programmed
again. In order to avoid such a repetitive process, this investigation proposes a
concept of virtual body and joint. The kinematics of virtual body and joint is
presented in Section 2. The equations of motion for a flexible body system are
presented in Section 3. Computer implementation and its impact on a sparse
oriented algorithm are explained in Section 4. Two flexible body systems are
dynamically analyzed by using the proposed method to show its validity in
section 5. Conclusions are drawn in Section 6.
176
1.2.1.2. KINEMATICS OF TWO CONTIGUOUS FLEXIBLE BODIES
(1) COORDINATE SYSTEMS AND VIRTUAL BODIES
Figure 1 Two adjacent flexible bodies
Two flexible bodies connected by a joint and their reference frames are shown
in Fig. 1. The iii ZYX ,, frame is the body reference frame of flexible body i
and the ZYX ,, frame is the inertial reference frame. Suppose there exists a
joint between the iii ZYX 111 ,, and jjj ZYX 111 ,, frames, and a force applied at the
origin of the iii ZYX 222 ,, frame. Kinematic admissibility conditions among the
reference frames can be divided into two categories. One is the admissibility
conditions between the two joint frames and the other is the admissibility
conditions among the frames within a flexible body. These two types of
conditions have been mixed in formulating the kinematic joint constraints and
generalized forces in the previous works. As a result, every joint and force
modules in a flexible multibody code, such as ADAMS [10] and DAMS [11],
has been developed separately for rigid and flexible bodies. This would take long
time for computer implementation and prone to coding errors. Especially,
flexible body programming requires much more effort than rigid body
programming does due to complexity associated with flexibility generalized
177
coordinates. In order to minimize the programming effort, a concept of
the virtual body is introduced in this section. At every joint and force
reference frames, a virtual rigid body, whose mass and moment of inertia are
zero, is introduced.
Figure 2 Two adjacent flexible bodies and three virtual bodies
As an example, three rigid virtual bodies are introduced for two adjacent
deformable bodies as shown in Fig. 2. This makes the flexible body has no joint
or applied force and is subjected to only the kinematic admissibility conditions
among its body frame and the virtual body frames. Therefore, the joint and force
modules are developed only for rigid bodies and one flexible body joint is to be
added in the joint module. The kinematic admissibility conditions for the flexible
body joint are formulated in the following subsections.
(2) JOINT CONSTRAINTS BETWEEN TWO RIGID BODIES
A joint has been represented by imposing condition of parallelism or
orthogonality on vectors attached to two adjacent rigid bodies. A library of such
condition for rigid bodies has been well developed and becomes the primitives in
building various joints [10, 11]. The conditions are formulated by using
178
geometric vectors that are defined within or between two joint reference frames.
A joint reference frame does not generally coincide with the body reference
frame. The body reference frame for a virtual body also serves as a joint
reference frame in the proposed method. Therefore, the kinematic admissibility
conditions for a joint connecting a virtual body is simplified and the number of
non-zero entries of the constraint Jacobian is reduced.
(3) FLEXIBLE BODY JOINT CONSTRAINT BETWEEN A FLEXIBLE BODY AND A
RIGID VIRTUAL BODY
Figure 3 Flexible body joint constraint between a flexible and a virtual body
Origin of the body reference frame for the virtual body in Fig. 3 can be
expressed as follows:
i
f
iii
iiii
uuAR
uARr
0
1
(1)
where i
0u and i
fu are the undeformed location vector and deformation vector of
a point on the body with respect to a body reference frame and iA is the
orientation matrix of body reference frame. The deformation vector i
fu at the
179
nodal position can be spanned by linear combination of a set of mode shapes [12]
as i
f
i
R
i
f pu (2)
where i
R is a modal matrix whose columns consist of the translational mode
shapes and i
fp is a modal coordinate vector.
Orientation of the virtual body 1i is obtained as follows:
1,1 iii
f
iiAAAA (3)
where i
fA is the relative orientation matrix induced by the rotational
deformation and 1, iiA is the orientation matrix between the reference frames of
the flexible body i and virtual body 1i in an undeformed state. If the Bryant
angle (1-2-3) [13] is employed, the i
fA is expressed as follows:
i
y
i
x
i
z
i
y
i
x
i
z
i
x
i
z
i
y
i
x
i
z
i
x
i
y
i
x
i
z
i
y
i
x
i
z
i
x
i
z
i
y
i
x
i
z
i
x
i
y
i
z
i
y
i
z
i
y
i
f
coscossinsincossinsincossincossinsin
cossinsinsinsincoscoscossinsinsincos
sinsincoscoscos
A (4)
If Ti
z
i
y
i
x
i ε is infinitesimal, the matrix i
fA can be approximated as
1
1
1
i
x
i
y
i
x
i
z
i
y
i
z
i
f
A (5)
The rotational deformation vector iε can be represented by linear
combination of rotational mode shapes of body i as
i
f
iipε (6)
180
where i
is a modal matrix whose columns are composed of rotational mode
shapes and i
fp is the vector of modal coordinate.
Finally, kinematic constraints between two body frames of the flexible and
virtual bodies can be obtained from Eqs. (1) and (3) as follows:
0uuARrC i
f
iiiii
R 0
1 (7)
0
hAAfhAAf
hAAghAAg
gAAfgAAf
C
1,1
1,1
1,1
iii
f
TiTiT
iii
f
TiTiT
iii
f
TiTiT
i
(8)
where
100
010
001
hgf (9)
Orthogonality conditions would have been used in deriving the orientational
constraints. However, the i
C in Eq. (8) is employed in this research for simple
implementation. Eqs. (7) and (8) yields algebraic constraint equations that
describe the flexible joint between flexible body i and virtual body 1i .
Taking variation of Eqs. (7) and (8) yields
0qC
CqC
q
q
q
i
i
Ri
flex
i (10)
where
TTiTiTi
f
TiTii 11 πrpπRq (11)
and the constraint Jacobian matrix flex
i
qC is obtained as
181
1
1
1
i
h
Tif
h
Ti
h
T
i
h
Tif
h
Ti
h
T
i
g
Tif
g
Ti
g
T
i
i
R
iii
R
Bf0BfBf0
Bg0BgBg0
Bf0BfBf0
C
0IABIC
q
q
(12)
where
hgk
skew
skew
skew
skew
iTii
k
iif
k
iiTii
k
iii
,,
)(
)(
)(
)(
11
1,
1
kAAB
kAB
AkAAB
uAB
(13)
and the vectors )( iskew u , )( 1kA
iskew , )( 1,kA
iiskew , and )(kskew are the
skew symmetric matrices of vectors, iu , kA
1i , kA1, ii , and k , respectively.
In order to obtain the acceleration level constraint, one can differentiate Eqs. (7)
and (8) twice with respect to time to yield
4321
4321
4321
2
2
2
)(2)(
h
T
h
T
h
T
h
T
h
T
h
T
h
T
h
T
g
T
g
T
g
T
g
T
i
f
i
R
iii
flex
i
c
ii
flex
ii
flex
i
skewskew
HfHfHfHf
HgHgHgHg
HfHfHfHf
ωpAωωuA
QqqCqCq
(14)
where the ω is the angular velocity with respect to the body reference frame
and the generalized velocity vector q is
TTiTiTi
f
TiTii 11 ωrpωRq (15)
and
182
hgk
skew
skewskew
skewskewskew
skewskew
i
f
iiii
f
i
f
i
k
iiiTi
k
iiTii
k
iTiii
k
,,
)(
)()(
)()()(
)()(
1,4
1113
112
11
pkAApH
kωωAAH
ωkAAωH
kAAωωH
(16)
1.2.1.3. EQUATIONS OF MOTION
Even though the proposed method is applicable to a general system consisting
of many flexible bodies, a slider crank mechanism with one flexible body in Fig.
4(a) is used to clearly show the impact of the proposed method on the equations
of motion. An equivalent virtual system, modeled by using the rigid virtual
bodies proposed in this investigation, is shown in Fig. 4(b). The augmented
equations of motion for the system is obtained by using the general form of
equations of motion as [11]
c
sve
T
Q
QQQ
λ
q
0C
CM
q
q
(16)
where M is the mass matrix of the system. The vector q consists of
translational acceleration for rigid and flexible bodies, angular acceleration, and
modal acceleration for the flexible body.
183
(a) Two rigid bodies and one flexible body
(continue)
(b) Two rigid bodies, one flexible body and two virtual bodies
Figure 4 Slider crank mechanism with one flexible body
The λ is the vector of Lagrange multipliers and sQ , vQ and cQ are the
strain energy terms, velocity induced forces and externally forces. The vector
cQ absorbs terms that are quadratic in the velocities, defined clearly by Shabana
[11].
FLEXIBLE BODYFLEXIBLE BODY
RIGID BODYRIGID BODY
CR
AN
K
CR
AN
K
SLIDERSLIDER
COUPLER
COUPLER
12
3
Y
Z
X
P1P1
FLEXIBLE BODYFLEXIBLE BODY
RIGID BODYRIGID BODY
VIRTUAL BODYVIRTUAL BODY
CR
AN
K
CR
AN
K
COUPLER
COUPLER
SLIDERSLIDER
VIRTUAL BODYVIRTUAL BODY1
2
4
5
3
184
(1) COEFFICIENT MATRIX OF CONVENTIONAL AUGMENTED FORMULATION
The mass matrix for the system in Fig. 4(a) is
3
2
1
r
r
f
M0
M
0M
M (17)
where fM and rM are the mass matrix for flexible body and for a rigid body,
respectively. They can be represented as follows :
)3,2(,
66
)6()6(
1
k
symmetric
k
k
rrk
r
nfnfffffr
r
rr
f
m0
0mM
mmm
mm
m
M
(18)
where nf is the number of modal coordinates. The constraint Jacobian matrix
C)( qC of the slider crank mechanism with flexible crank is
int
30
int
23
,
12
,
01
)(
jo
jo
cflex
cflex
c
q
q
q
q
q
C
C
C
C
C (19)
where, cflex,qC is the constraint Jacobian matrix of the flexible joint obtained
by the conventional method[11].
185
(2) COEFFICIENT MATRIX OF THE PROPOSED AUGMENTED FORMULATION
The mass matrix for the system in Fig. 4(b) is
5
4
3
2
1
r
r
v
f
v
M
M0
M
0M
M
M (20)
where the mass matrix for virtual body, vM , the mass matrix for flexible body,
fM , and the mass matrix for rigid body, rM are
3,1,66 kv 0M
)6()6(
2
nfnfffffr
r
rr
f
symmetric
mmm
mm
m
M
(21)
)5,4(,
66
kk
k
rrk
r
m0
0mM
The proposed constraint Jacobian matrix pqC of the slider crank
mechanism with flexible crank is
int
50
int
45
int
34
,
23
,
12
int
01
jo
jo
jo
pflex
pflex
jo
p
q
q
q
q
q
q
q
C
C
C
C
C
C
C (22)
186
where pflex,qC is the constraint Jacobian matrix of the flexible body joint
obtained by the proposed method. As shown in Eq. (22), the constraint Jacobian
matrix can be clearly divided into flexible and rigid body joint modules by
introducing rigid virtual bodies.
(3) NON-SINGULARITY OF AUGMENTED MASS MATRIX
If the constraint Jacobian matrix qC has a full row rank, the coefficient
matrix of Eq. (16) is non-singular, which can be proved by showing that the
following equations have only trivial solutions under the same assumption.
0yCyM q 31 N
T
N (23)
0yCq 3V
T (24)
0yCyC qq 21 VN (25)
where NM is the mass matrix of non-virtual body, NqC and
VqC are the
constraint Jacobian matrix of non-virtual and virtual bodies, respectively. After
pre-multiplying Eq. (23) by T
1y and Eq. (24) by T
2y , their summation of Eq.
(26) can be simplified by using Eq. (25).
0yMyyCyCyyMy qq 1121311 N
T
VN
T
N
T (26)
Now, we can see 0y 1 from Eq. (26). Then, Eqs. (23) and (24) reduces to
0yCq 3
T (27)
Since the qC has full row rank, 3y must be zero. Substituting 0y 1 into Eq.
(25) yields
0yCq 2V (28)
Since rank of VqC is the same as the size of 2y , 2y must be zero. Since
187
0y 1 , 0y 2 , 0y 3 are only solutions of Eqs. (23), (24), and (25), their
coefficient matrix is non-singular.
1.2.1.4. COMPUTER IMPLEMENTATION AND DISCUSSIONS
In the above sections, the equation of motion and the kinematic constraints for
flexible body explained by using the virtual body concept. This section describes
the computational procedure for those equations developed in section 3.2 and 3.3.
(1) NUMERICAL ALGORITHM
A general purpose program for the dynamic analysis of mechanical systems
can be implemented in many different ways, depending on the DAE solution
method employed. The generalized coordinate partitioning method [12] is
employed in this investigation and the proposed program structure is shown in
Fig. 5. Note that there exist joint and force modules only for rigid bodies, and
one flexible body joint is added in the joint library. Those modules can handle
any system consisting of rigid bodies as well as flexible bodies.
Figure 5 A program structure for proposed flexible multibody dynamics
188
(2) COMPARISON OF DIFFERENT IMPLEMENTATION METHODS
The joint and force modules must be expanded whenever a user group of the
flexible body dynamics code demands a special type of joint or force element.
Since the proposed implementation method for a flexible body dynamics code
reuses all joints and force modules for the rigid body, only necessary modules to
be added to a rigid body dynamics code are the flexible body joint and the
equations of motion for a flexible body. As a result, the proposed method is not
only easy to implement but also to maintain, because the proposed method
eliminates the additional programming effort for the flexible body modules when
an expansion of the joint or force library is required.
However, there are some computational overheads, because extra bodies and
joints must be introduced to a flexible body system if the proposed method is
employed. It is very difficult to analyze the computational overheads for general
rigid and flexible multibody systems, because various models and flexible body
dynamics theories may end up with various situations. In order to simplify the
presentation, the slider crank mechanism in section 3 is reconsidered in this
section.
Numerical experiments with the Cartesian coordinate formulation [12] showed
that more than 70% of the total computation time is consumed in the Gaussian
elimination of matrices arising from various equations. Direct Gaussian
elimination of Eq. (16) would require a number of arithmetic operations
proportional to approximately cube of the matrix size. However, the number of
arithmetic operations for a sparse solver such as the Harwell Library [14] is
increased only linearly to the number of non-zero entries if the structure of the
non-zero entries is exploited. A sparse solver reduces the number of operations
by minimizing the number of fill-ins and performing the Gaussian elimination
only on the non-zero entries and fill-ins. Therefore, it is important to add the new
non-zero entries so that overall non-zero structure of the resulting matrix is not
disturbed and is well suited for minimization of the fill-ins. The structures of the
non-zeros are shown in Eqs. (20) and (21), respectively. No non-zero entry in the
mass matrix of the proposed method is added, because the mass and moment of
inertia of the virtual body are zero. Total numbers of non-zero entries of Eq. (16)
189
are shown in Table 1. Note that redundant constraints are eliminated and
coincidence of the virtual body and joint reference frames is utilized in reducing
the number of non-zeros. Since the new non-zero entries in Eq. (16) are scattered
around the existing ones, the overall structure of the non-zeros is not disturbed
and a similar reordering sequence in sparse Gaussian elimination to the original
reordering sequence in a sparse linear solver can be used. As a result, expected
computation time increment with the proposed method would be about 50% for
the slider crank mechanism, when a sparse solver is employed.
Table 1 Number of non-zero entries for the slider-crank mechanism
Implementation Methods No. of non-zero entries
Conventional 122+10×nmode
Propose 188+12×nmode
* nmode: the number of mode shapes
The number of non-zeros for a most frequently used joints such as, revolute
joint, spherical joint and translational joint, are also given in Table 2. It can be
easily shown that the percent ratio of the computation time would become
smaller if the number of flexible bodies in a system is small, which is true in
many cases. However, the computation time may be increased significantly for a
flexible body system which has many joints and force elements, because the
number of virtual bodies in such a system is large.
Table 2 Number of non-zero entries for the slider-crank mechanism
Joint Increment of non-zero
entries
Revolute joint (33 + nomde) ×nvirtualr
Spherical joint (33 + 3×nomde)×nvirtuals
Translational
joint (33 + nomde) ×nvirtualt
190
* nvirtualr : the number of virtual bodies which are connected with revolute
joint
* nvirtuals : the number of virtual bodies which are connected with spherical
joint
* nvirtualt : the number of virtual bodies which are connected with
translational joint
Another way of implementing the virtual body concept is to mix the proposed
implementation method with the conventional one. The conventional method
may be used to implement the frequently used joint and force modules such as
the revolute and translational joints and an applied force at a point. Meanwhile,
the proposed method may be used to implement the less frequently used joint
and force modules such as an universal joint or a planar joint. This
implementation method will improve both the computational overhead as well as
the coding convenience. This mixed formulation can be very effective if a set of
basic joint and force modules have already been developed and more modules
for the flexible bodies need to be added.
1.2.1.5. NUMERICAL RESULTS
Dynamic analysis of a flexible slider crank mechanism and a flexible
pendulum mechanism is presented in order to validate the results from the
proposed method. The examples are solved by using both the proposed method
and the nonlinear approach developed by Simo [6].
(1) FLEXIBLE SLIDER CRANK MECHANISM
The system consists of two rigid bodies and one flexible body, as shown in
Fig. 4. Length, cross sectional area, and area moment of inertia of the elastic
crank are 0.4 m, 0.0018m2, and 1.215 10
-4m
4, respectively. The crank is
modeled by using 10 two-dimensional elastic beam elements of equal lengths.
191
The material mass density of the beam is 5540.0 kg/m3 and its Young's modulus
is 1.0109 N/m
2. Vibration analysis of the crank is carried out with fixed-free
boundary condition and the resulting mode shapes are shown in Fig. 6.
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
X(M)
MAG
NIT
UD
E..
1st mode
2nd mode
3rd mode
4th mode
Figure 6 Mode shapes of the crank
Four mode shapes are selected to span the deformation of the crank. As a result,
the system has 5 degrees of freedom. Dynamic analysis using the generalized
coordinate partitioning method is performed for 5 sec under the constant
acceleration condition of the joint between the ground and the body 1. The
acceleration, displacement, and relative deformation of the pin joint connecting
the crank and the coupler both from the proposed method and the nonlinear
approach [6] are shown in Figs. 7, 8, and 9, respectively. Note that since the
results from both models are almost identical as shown in these figures, the
proposed implementation methods using rigid virtual body can be validated.
192
-30
-20
-10
0
10
20
30
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
TIME (SEC)
Y (
M/S
EC
^2).
..NONLINEAR
PROPOSED
Figure 7 Y Acceleration of P1
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
TIME (SEC)
Y (
M)
NONLINEAR
PROPOSED
Figure 8 Y Displacement of P1
193
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
TIME (SEC)
Y (
M)
NONLINEAR
PROPOSED
Figure 9 Deformation of P1
(2) FLEXIBLE PENDULUM MECHANISM
Figure 10 Simple flexible pendulum model
The pendulum body shown in Fig. 10 is modeled with 10 beam elements
having a length of 0.4m, a cross sectional area of 0.0018m2, and a mass of
3.9888kg. Dynamic analysis is performed for 1 sec under the free falling
condition. Mode shapes of the pendulum are obtained by ANSYS[15] with the
simply supported-free(pin-free) boundary condition. Mode Shapes of the
pendulum are shown in Fig. 11. The acceleration and relative transverse
deformation of the tip point both from the proposed method and the nonlinear
BEAMBEAM
(FLEXIBLE BODY)(FLEXIBLE BODY)
RIGID BODYRIGID BODY
REVOLUTEREVOLUTE
JOINT JOINT
GRAVITYGRAVITY
X
Y
194
approach [6] are shown in Figs. 12, and 13, respectively. It is clear from these
results that the proposed method and nonlinear approach are in good agreement,
accordingly.
Figure 11 Mode Shapes of the pendulum
-30
-20
-10
0
10
20
30
40
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
TIME (SEC)
Y (
M/S
EC
^2).
..
NONLINEAR
PROPOSED
Figure 12 Y Acceleration of beam tip
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
MAG
X(M)
1s
t
m
o…
195
-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
TIME (SEC)
Y (
M)
NONLINEAR
PROPOSED
Figure 13 Deformation of beam tip
1.2.1.6. SUMMARY AND CONCLUSIONS
An implementation method is proposed for general purpose rigid and flexible
multibody dynamics with the Cartesian coordinate formulation. A concept of the
virtual body and joint is introduced to make a flexible body free from all
kinematic admissibility conditions except these from the virtual-flexible body
joint. This eliminates extra programming efforts for the flexible body whenever a
joint or force module is added to a general purpose dynamic analysis program.
The computational overhead of the proposed method is turned out to be
moderate if a sparse solver is employed, while implementation convenience is
dramatically improved. A flexible slider crank mechanism and a simple
pendulum are analyzed and the results are validated against these from a
nonlinear approach.
196
REFERENCES
1. A. A. Shabana, "Substructure Synthesis Methods for Dynamic Analysis of Multibody
Systems", Computers & Structures, Vol. 20. No. 4, pp 737-744, 1985
2. W. S. Yoo, and E. J. Haug, "Dynamics of Flexible Mechanical Systems Using Vibration
and Static Correction Modes", Journal of Mechanisms, and Transmissions, and
Automation in Design, 1985
3. H. T. Wu, and N. K. Mani, "Modeling of Flexible Bodies for Multibody Dynamic Systems
Using Ritz Vectors", Journal of Mechanical Design, Vol. 116, pp. 437-444, 1994.
4. J. Garcia de Jalon, J. Unda, and A. Avello, "Natural Coordinates for the Computer
Analysis of Three-Dimensional Multibody Systems", Computer Methods in Applied
Mechanics and Engineering, Vol. 56, pp. 309-327, 1985.
5. N. Vukasovic, J. T. Celigueta, J. Garcia de Jalon, and E. Bayo, "Flexible Multibody
Dynamics Based on a Fully cartesian System of Support Coordinates", Journal of
Mechanical Design, Vol. 115, pp. 294-299, 1993.
6. J. C. Simo, and L. Vu-Quoc, "On the Dynamics of Flexible Beams Under Large Overall
Motions-The Plane Case: Part I", Journal of Applied Mechanics, Vol. 53, pp. 849-854,
1986.
7. H. H. Yoo, R. R. Rion, and R. A. Scott, "Dynamics of Flexible Beams Undergoing
Overall Motions", Journal of Sound and Vibration, Vol. 181, pp. 261-278, 1994
8. A. A. Shabana, A. P. Christensen, "Three Dimensional Absolute Nodal Coordinate
Formulation : Plate Problem", International Journal for Numerical Methods in
Engineering, Vol. 40, pp. 2775-2790, 1997
9. A. A. Shabana, H. A. Hussien, and J. L. Escalona, "Application of the Absolute Nodal
Coordinate Formulation to Large Rotation and Large Deformation Problems", Journal
of Mechanical Design, Vol. 120, pp. 188-195, 1998
10. ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann
Arbor, MI 48105.
11. A. A. Shabana, Dynamics of Multibody Systems, John & Wiley, New York, 1989.
197
12. R. A. Wehage and E. J. Haug, "Generalized Coordinate Partitioning for Dimension
Reduction in Analysis of Constrained Dynamic Systems", Journal of Mechanical Design,
Vol. 104, pp. 247-255, 1982.
13. P. E. Nicravesh, Computer-Aided Analysis of Mechanical systems, Prentice-Hall, 1988
14. I. S. Duff, A. M. Erisman, and R. K. Reid, Direct Methods for Sparse Matrices,
Clarendon Press, Oxford, 1986
15. ANSYS Reference Manual, ANSYS, Inc., Southpointe 275 Technology Drive,
Canonsburg, PA 15317.
198
1.2.2
GENERALIZED RECURSIVE FORMULATION
FOR FLEXIBLE MULTIBODY DYNAMICS
1.2.2.1. INTRODUCTION
The equations of motion for the general constrained mechanical systems were
derived in terms of the relative coordinates by Wittenburg [1]. The velocity
transformation method with the graph theory was employed to transform the
equations of motion in the Cartesian coordinate space to the joint space
systematically. Hooker [2] proposed a recursive formulation for the dynamic
analysis of a satellite which has a tree topology. It was shown that the
computational complexity of the formulation increases only linearly to the
number of bodies. Fetherstone [3] used the recursive formulation to perform the
inverse dynamic analysis of manipulators. Bae and Haug [4] further developed
the formulation for constrained mechanical systems by using the variational
vector calculus. The recursive formulation was applied to linearize the equations
of motion [5]. Recursive formula for each term in the equations of motion was
directly derived, using the state vector notation. Similar approach was taken in
Ref. 6 to implement the implicit BDF integration with the relative coordinates.
Since the recursive formulas were derived term by term, the resulting equations
and algorithm became much complicated. To avoid the complication, the
equations of motion were derived in a compact matrix form by using the velocity
transformation method in Ref [7]. The generalized recursive formula for each
category of the computational operations was developed and applied whenever
such a category was encountered. This research applies the generalized recursive
formulas for the multibody flexible dynamics.
Shabana [8] presented a coordinate reduction method for multibody systems
with flexible components. The local deformation of a flexible component was
expressed in terms of the nodal coordinates and was then spanned by a set of
199
mode shapes obtained from a mode analysis. A fully Cartesian
coordinateformulation for rigid multibody dynamics by Jalon [9] was extended to
the flexible body dynamics by Vukasovic and Celigueta [10]. Nonlinearity
associated with an orientational transformation matrix was relieved by defining
all necessary vectors for the equations of motion and constraints as the
generalized coordinates. Variational equations of motion for flexible multibody
systems were derived in Ref. 11. The variational approach was applied to extend
the rigid body recursive formulation to flexible multibody systems. An extended
kinematic graph concept was employed to develop a new recursive formulation
for the dynamic analysis of flexible multibody systems by Lai and Haug [12].
Cardona and Geradin [13] dealt with substructuring for dynamic analysis of
flexible multibody systems. The joint coordinates and the finite element method
were employed for the flexible body dynamics by Nikravesh [14]. Pereira [15]
presented a systematic method for deriving the minimum number of equations of
motion for spatial flexible multibody systems.
Contrast to implementation of a rigid body dynamic analysis program, it is
generally complicated to implement a flexible body dynamic formulation and to
expand it for a general purpose program, regardless of whatever formulation has
been chosen. This is because the flexible body dynamic formulations handle
additional generalized coordinates to these of the rigid body dynamics. One of
the most tedious works involved with the implementation of the flexible body
dynamics is to build a set of joint and force modules. Whenever a new force or
joint module is developed for the rigid body dynamics, the corresponding
module for the flexible body dynamics has to be formulated and programmed
again. In order to avoid such a repetitive process, this investigation proposes a
concept of virtual body and joint. The relative coordinate kinematics and the
virtual body concept are presented in section 2. A graph representation of
flexible multibody systems is presented in section 3. The forward recursive
formula and backward recursive formula respectively are treated in sections 4
and 5. A solution method of the equations of motion for a flexible body system is
presented in section 6. Flexible slider crank mechanism is dynamically analyzed
by using the proposed method to show its validity in section 7. Conclusions are
drawn in section 8.
200
1.2.2.2. RELATIVE COORDINATE KINEMATICS OF TWO
CONTIGUOUS FLEXIBLE BODIES
(1) COORDINATE SYSTEMS AND VIRTUAL BODIES
Figure 1 Two adjacent flexible bodies
The ZYX frame is the inertial reference frame and the zyx frame
is the body reference frame in Fig. 1. Velocities and virtual displacements of
point O in the ZYX frame are respectively defined as
ω
r (1)
and
ω
r
δ
δ (2)
Their corresponding quantities in the zyx frame are respectively defined as
ωA
rA
ω
rY
T
T (3)
201
and
πA
rA
π
rZ
T
T
(4)
where A is the orientation matrix of the zyx frame with respect to the
ZYX frame. Two flexible bodies connected by a joint and their reference
frames are shown in Fig. 1.
Suppose there exists a joint between the 111 iii zyx and 111 jjj zyx
frames, and a force applied at the origin of the 222 jjj zyx frame. Kinematic
admissibility conditions among the reference frames can be divided into two
categories. One is the admissibility conditions between the two joint frames and
the other is the admissibility conditions among the frames within a flexible body.
These two types of conditions have been mixed in formulating the kinematic
joint constraints and generalized forces in the previous works. As a result, every
joint and force modules in a flexible multibody code, such as ADAMS [16] and
DAMS [17], have been developed separately for rigid and flexible bodies. This
would take long time for computer implementation and prone to coding errors.
Especially, flexible body programming requires much more effort than rigid
body programming does due to complexity associated with flexibility
generalized coordinates and the strain energy.
202
Figure 2 Two adjacent flexible bodies and three virtual bodies
In order to minimize the programming effort, a concept of the virtual body is
introduced in this section. At every joint and force reference frames, a virtual
rigid body, whose mass and moment of inertia are zero, is introduced. The
virtual body and the original flexible body are then connected by a virtual joint.
As an example, three virtual rigid bodies are introduced for two adjacent
deformable bodies as shown in Fig. 2. Note that the flexible bodies have no joint
or applied force except the virtual joints which are represented by the kinematic
admissibility conditions among the flexible body frame and the virtual body
frames. Therefore, the joint and force modules are developed only for rigid
bodies and one flexible body joint of the virtual joints to be added in the joint
module. The recursive kinematic relationships representing the admissibility
conditions of the flexible body joint are formulated in the following subsections.
(2) RELATIVE KINEMATICS FOR A FLEXIBLE BODY JOINT
Figure 3 Flexible body joint between a flexible body and a virtual body
A virtual body is always connected to the original flexible body by a flexible
body joint. Origin of the virtual body reference frame in Fig. 3 can be expressed
as follows:
203
)( )1()1(011 iiiiiii usArr (5)
where ii )1(0 s and ii )1(
u are the undeformed location vector and deformation
vector of the origin of the virtual body with respect to the flexible body reference
frame, 1iA is the orientation matrix of the flexible body reference frame. The
deformation vector ii )1( u can be spanned by linear combination of a set of
mode shapes [8] as f
ii
R
iii )1(1)1( qΦu (6)
where R
i 1Φ is a modal matrix whose columns consist of the translational mode
shapes and superscript f in f
ii )1( q denotes the modal coordinate vector. Subscripts
i and 1i denote the generalized coordinate between the i and 1i body
reference frames.
The angular velocity in the local reference frame is obtained as follows
f
iii
T
iii
T
iii )1(1)1(1)1( qΦAωAω (7)
where i
T
i
T
ii AAA )1()1( is used. Differentiating Equation (5) and multiplying by
T
iA yields
f
iii
T
iiiii
T
iii
T
iii )1(1)1(1)1()1(1)1(~
qΦAωsArAr (8)
where iiiiii )1()1(0)1(
~ uss , symbol with tilde denotes skew symmetric matrix
which consists of their vector elements, and iii ωAA ~ wide tilde iω are used.
Combining Equations (7) and (8) yields the following recursive velocity
equation for a flexible body joint.
f
ii
f
iii
f
iii )1(2)1(11)1( qBYBY (9)
where
204
1)1(
1)1(
2)1(
)1(
)1()1()1(
1)1(
~
i
T
ii
R
i
T
iif
ii
T
ii
ii
T
ii
T
iif
ii
ΦA
ΦAB
A0
sAAB
(10)
It is important to note that matrices f
ii 1)1( B and f
ii 2)1( B are function of only
modal coordinates of the flexible body i-1. As a result, further differentiation of
the matrices f
ii 1)1( B and f
ii 2)1( B in Equation (9) with respect to other than bf
f
ii )1( q yields null. This property will play a key role in simplifying recursive
formulas in sections 4 and 5.
Equation (9) defines the kinematic relationships between an inboard flexible
body and an outboard rigid body. The kinematic relationships between an inboard
rigid body and an outboard flexible body can be derived similarly. Similarly, the
recursive virtual displacement relationship between a flexible body and a virtual
body is obtained as follows
f
ii
f
iii
f
iii )1(2)1(11)1( qBYBY (11)
where
1
11)1(
2)1(
)1(
)1()1()1(
1)1(
~
~
i
R
iiiir
ii
T
ii
T
iiii
T
iir
ii
Φ
ΦΦsB
A0
AsAB
(12)
(3) RELATIVE KINEMATICS FOR A RIGID BODY JOINT
The recursive velocity relationship for a rigid body joint connecting two rigid
bodies can be derived by following the similar steps as in Equations (5)-(9) as
r
ii
r
iii
r
iii )1(2)1(11)1( qBYBY (13)
205
where superscript r denotes the generalized co-ordinate from a rigid body joint
and
ii
T
ii
ii
T
iiiiiiii
T
iir
ii
T
ii
T
iiiiiiiiii
T
ii
T
iir
ii
ii
)1()1(
)1()1()1()1()1()1(
2)1(
)1(
)1()1()1()1()1()1()1(
1)1(
)~)~
((
)~~~(
)1(
HA
HAsAdAB
A0
AsAdsAAB
q
(14)
where ii )1( H is determined by the axis of rotation. Node that the B matrices
are function of only r
ii )1( q .
(4) GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS
The graph theory was used to automatically preprocess mechanical systems
having various topological structures in References [1, 4]. A node and an edge in
a graph have represented a body and a joint, respectively. The preprocessing,
based on the graph theory, yields the path and distance matrices that are provided
to automatically decide computational sequences. Two computational sequences
are required in a general purpose program. One is the forward path sequence
starting from the base body and moving towards the terminal bodies. The other is
the backward path sequence starting from the terminal bodies and moving
towards the base body.
Figure 4 Flexible slider crank mechanism
206
Figure 5 Graph representation and computational sequence
In order to derive systematically the recursive formulas, bodies in a graph are
divided into four disjoint sets (associated with a generalized coordinate kq ) as
follows :
)( kqI ={adjacent outboard body of the joint having kq as its generalized co-
ordinate}
)( kqII ={all outboard bodies of )( kqI , excluding all bodies in )( kqI }
)( kqIII ={all bodies between the base body and the inboard body of )( kqI ,
including the base and inboard bodies and excluding all bodies in )( kqI }
)( kqIV ={ the complementary set of )()()( kkk qqq IIIIII }
As an example, the graph theoretic representation and computational path
sequences of the system in Fig. 4 are shown in Fig. 5. The four disjoint sets for
207
the system in Fig. 5, if kq belongs to the joint between bodies 3 and 4, are
)( 34qI ={body 4}, )( 34qII ={bodies 5, 6, and 7}, )( 34qIII ={bodies 1, 2, and 3},
)( 34qIV ={bodies 8, 9, 10, 11, and 12}
1.2.2.3. FORWARD RECURSIVE FORMULAS
(1) GENERALIZATION OF THE VELOCITY RECURSIVE FORMULA
Generalization of the velocity recursive formula can be achieved by
computational equivalence between the recursive method and the velocity
transformation method. The velocity Y for all bodies in a system can be
obtained by repetitive symbolic substitutions of the recursive formula in
Equations (9), (11) and (13), depending on the type of a joint, along the forward
path sequence of a graph and by appending the trivial equation of ffqq as
follows :
qBq
q
I0
BB
q
YY
f
rzfzr
f (15)
where rq and f
q are the relative and modal co-ordinates vectors for a system,
respectively. The dimension of Y , rq and f
q are, respectively, assumed to
be nc , nr , and nf . The velocity nfncR Y with a given nfnrR q can be
evaluated either by using Equation (15) obtained from symbolic substitutions or
by using (9), (11) and (13) with recursive numeric substitution of iY 's. Since
both formulas give an identical result and recursive numeric substitution is
proven to be more efficient [4], matrix multiplication qB with a given q will
be actually evaluated by using Equations (9), (11) and (13). Since q in
Equation (15) is an arbitrary vector in nfnrR , Equations (9), (11), (13) and (15)
which are computationally equivalent, are actually valid for any vector
208
nfnrR x such that
Bxx
XX
(16)
and
BXBXX (17)
where nfncR X is the resulting vector of multiplication of B and x and B
matrices depend on a joint type. As a result, transformation of nfncR Y into nfncR Bx is actually calculated by recursively applying Equation (17) to
achieve computational efficiency in this research.
(2) RECURSIVE FORMULA FOR qq )(BxX
Equation (17) is partially differentiated with respect to kq for
)(,...,1 nfnrk to obtain the recursive formula for q)(Bx as follows.
iiqiiqiiiiqiiqi kkkk )1(2)1(11)1(11)1( )()()()( XBXBXBX (18)
Since B matrices depend only on the generalized coordinates for joint
ii )1( , their partial derivatives with respect to generalized coordinates other than
ii )1( q become null. In other words, the partial derivatives become null if kq
does not belong to set )( kqI . If body i is an element of set )( kqII , Equation
(18) becomes
kk qiiiqi )()( 11)1( XBX (19)
209
Figure 6 Computation sequence of
iiqY
If body i belongs to set )()( kk qq IVIII , iX is not affected by kq . As a
result, Equation (18) is further simplified as follows.
0X kqi )( (20)
210
There are two recursive formulas in the case of body )( kqi I . If body i is an
element of set )( kqI , body 1i is naturally its inboard body and belongs to set
)( kqIII . Equation (18) becomes
iiqiiiqiiqi kkk )1(2)1(11)1( )()()( XBXBX (21)
If bodies 1i and i are elements of set )( kqI , the recursive formula in
Equation (18) is expressed as follows:
iiqiiqiiiiqiiqi kkkk )1(2)1()1(1)1(11)1( )()()()( XBXBXBX (22)
As an example, the recursive formula in Equation (19)-(22) can be applied to
compute 34qY for the system in Fig. 4, as shown in Fig. 6. Note that since the
recursive formulas for Bx and q)(Bx can be obtained similarly, they are
omitted.
1.2.2.4. BACKWARD RECURSIVE FORMULAS
(1) GENERALIZATION OF THE FORCE RECURSIVE FORMULA
A generalized recursive formula for transformation of nfnrR x into a new
vector BxX in nfncR is derived in section 4. Inversely, it is often necessary
to transform a vector G in nfncR into a new vector GBgT in nfnrR . Such
a transformation can be found in the generalized force computation in the joint
space with a known force in the Cartesian space. The virtual work done by nfncR Q is obtained as follows.
f
cfTTT
Q
QqZQZW (23)
where Z must be kinematically admissible for all joints in a system and cQ
211
and fQ are the Cartesian and modal forces, respectively. Substitution of virtual
displacement relationship into Equation (23) yield
**)( ffTrrTfcT
zr
fTcT
zr
rT QqQqQQBqQBqW (24)
where cT
zr
rQBQ * and fcT
zf
fQQBQ * . Equation (24) can be written in a
summation form as
fjtsii
f
ii
fT
ii
rjtsii
r
ii
rT
ii
)1(
*
)1()1(
)1(
*
)1()1( QqQqW (25)
where rjts and fjts respectively denote all rigid body joints and all flexible body
joints.
On the other hand, the symbolic substitution of the recursive virtual
displacement relationship into Equation (23) along the chain (starting from the
terminal bodies toward inboard bodies) and the reorganization of the equation
about the virtual relative displacement and modal displacement yield
fjtsii ql
i
c
i
fT
ii
f
ii
fT
ii
rjtsii ql
i
c
i
rT
ii
rT
ii
ii
ii
)1( )(
112)1()1()1(
)1( )(
112)1()1(
)1(
)1(
I
I
SQBQq
SQBqW
(26)
where
)(
221
1
)2)(1(
)2)(1()(
body terminala is 1 if
ii
ii
ql
i
c
i
T
i
i i
I
SQBS
0S
(27)
The recursive formula for bf *fQ and
*rQ is obtained by equating Equations
(25) and (26) as follows:
212
)(
112)1()1(
*
)1(
)1( iiql
i
c
i
T
iiiiii
I
)S(QBQQ (28)
where 0Q )1(ii for a rigid body joint and for a flexible body joint connecting
an inboard flexible body and an outboard virtual body, and f
iiii )1()1( QQ for a
flexible body joint connecting an inboard virtual body and an outboard flexible
body, and 1iS is defined in Equation (27).
Since Q in Equation (23) is an arbitrary vector in nfncR , Equations (23) and
(28)are valid for any vector G in nfncR . As a result, the matrix multiplication
of GBT is actually evaluated to achieve computational efficiency in this
research by
)(
112)1()1()1(
)1( iiqIl
i
c
i
T
iiiiii )S(GBGg (29)
where g is the result of GBT and )1( iiG is defined as )1( iiQ in Equation (28)
and
body terminala is 1 if1 ii 0S (30)
)(
221)2)(1(1
)2)(1(
)(iiql
i
c
i
T
iii
I
SQBS (31)
Recursive formula in Equation (29) must be applied for all joints in the
backward path sequence to obtain GBgT where G is a constant vector in
nfncR .
(2) RECURSIVE FORMULA FOR kk q
T
q )( GBg
The Recursive formula for kq
T )( GB is obtained by replacing i by 1i in
Equation (29) and 1i by 1i in Equation (31) and taking partial derivative
with respect to kq yield
213
)(
2)1(
)(
2)1()1()1(
)1(
)1(
)()()(
ii
k
ii
kkk
ql
qi
c
i
T
ii
ql
i
c
iq
T
iiqiiqii
I
I
)S(GB
)S(GBGg
(32)
)(
1)1(
)(
1)1()1(
)1()1(
)()(ii
k
ii
kk
ql
qi
c
i
T
ii
ql
i
c
iq
T
iiqi
II
)S(GB)S(GBS (33)
Since nfncR G is a constant vector, 0G kq . If
)()()( kkk qqqi IVIIIII , B matrices are not functions of kq . Therefore,
their partial derivatives with respect to kq become null. As a result, Equations
(32) and (33) can be simplified as follows.
)(
2)1()1(
)1(
)()(ii
kk
ql
qi
T
iiqii
I
SBg (34)
)(
)1(1
)1(
)()(ii
kk
qIl
qi
T
iiqi SBS (35)
Since 0S kqi )( for the terminal bodies, 0S
kqi )( for )()( kk qqi IVII .
Thus, for )()( kk qqi IVII , Equation (34) becomes
0g kqii )( )1( (36)
There are two recursive formulas in the case of body )( kqi I . If body
)( kqi I and body 1i belongs to set )( kqII , and 0S kqi )( . Thus, Equation
(32) and (33) become
)(
2)1()1(
)1(
)()(ii
kk
ql
i
c
iq
T
iiqii
I
)S(GBg (37)
)(
1)1()1(
)1(
)()(ii
kk
ql
i
c
iq
T
iiqi
I
)S(GBS (38)
where iS must be saved when GBT is computed. This recursive formula can
be applied to compute q
T )( GB . As an example, 3 43 4
)( q
T
q GBg for the system
in Fig. 4 is obtained, as shown in Fig. 7 for the case of 34qqk . Note that the
214
components of 3 4qg are either zero or simple to compute.
Figure 7 Computation sequence of
iiii qq )()( BGg .
1.2.2.5. THE GOVERNING EQUATIONS OF SOLUTION
(1) IMPLICIT INTEGRATION OF THE EQUATIONS OF MOTION
The dynamic equations of motion for a constrained mechanical system in the
215
joint space have been obtained in Reference [1] by the velocity transformation
method as follows.
)QλΦY(MBFZ
TT (41)
where Φ and λ , respectively, denote the cut joint constraint and the
corresponding Lagrange multiplier. The M is a mass matrix and Q is a force
vector including external forces, strain energy terms, and velocity induced forces.
The equations of motion, the constraint equations, vq , and av
constitute the differential algebraic equations(DAE). Application of 'tangent
space method' in Reference [19] to the DAE yields the following nonlinear
system of equations
0
),z,v,(qΦ
)v,(qΦ
),Φ(q
),λ,a,v,F(q
)βa(vU
)βv(qU
)H(p
nnnn
nnn
nn
nnnnn
nn
T
nn
T
n
t
t
t
t
,
200
100
(42)
where TT
n
T
n
T
n
T
n
T
n λavqp ,,, , 0 , 1β , and 2β are determined by the coefficients
of the BDF. The 0U must be chosen such that the augmented square matrix
qΦ
UT
0 is nonsingular. Applying Newton's method to solve the nonlinear system
in Equation (42) yields
Hp)H(p n (43)
,...3,2,1,)()1( ii
n
i
n ppp (44)
where
216
0ΦΦΦ
00ΦΦ
000Φ
FFFF
0UU0
00UU
)H(p
avq
vq
q
λavq
TT
TT
n
000
000
(45)
Since qF and qΦ are highly nonlinear functions of av,q, and λ , some
cautions must be taken in deriving the non-zero expressions in matrix pH so
that they can be efficiently evaluated.
(2) APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS
A set of the generalized recursive formulas has been developed in the sections
3 and 4. This section shows how these formulas can be utilized to efficiently
compute the qF in pH in Equation (45). Inspection of pH reveals that partial
derivatives of qF , vF , aF , qΦ , qΦ and qΦ are needed to be computed.
Only the qF is presented in this section and the rest can be derived similarly.
In Equation (41), differentiation of matrix B with respect to vector q
results in a three dimensional matrix. To avoid the notational complexity for the
three dimensional matrix, Equation (41) is differentiated with respect to each
generalized coordinate kq one by one. Thus,
nfnrkkk
kk
q
T
q
T
TT
,...,3,2,1),()( )QλΦY(MB
)QλΦY(MBF
Z
Z
(46)
Since the term )QλΦZ
T( can be easily expressed in terms of the Cartesian
coordinates, kq
T)QλΦ
Z( is obtained by applying the chain rule as follows.
nfnrkk
T
q
T
k ,...,3,2,1,(( B)QλΦ)QλΦ ZZZ
(47)
217
where Bq/Z is used and kB denotes the k th column of the matrix B .
The first term in Equation (46) can be obtained by applying the recursive
formula for GBT
qk with )QλΦY(MG
Z T . Collection of )QλΦ
ZT( for all
k constitutes )QλΦZ
T( , which is equivalent to )QλΦZ
T( . Matrix TT
ZZ)QλΦ (
consists of nc+nf column vectors in nfncR . Therefore, the application of GBT
qk,
where G is each column of matrix TT
ZZ)QλΦ ( , yields the numerical result of
)QλΦZ
T( . Finally, the second term in Equation(46) is also obtained by applying
GBT
qk, where )QλΦY(MG
Z T and
kqY is recursively obtained.
1.2.2.6. NUMERICAL RESULTS
Dynamic analysis of a flexible slider crank mechanism is presented in order to
validate the results from the proposed method. The example problem is solved
by using both the proposed method and the nonlinear approach developed by
Simo [19].
The system consists of two rigid bodies and one flexible body, as shown in
Fig. 4. Length, cross-sectional area, and area moment of inertia of the elastic
crank are 0.4 m, 0.0018 m2, and 1.3510
-7 m
4, respectively. The crank is
modeled by using 10 two-dimensional elastic beam elements of equal length.
The material mass density of the beam is 5540.0 kg/m3 and its Young's modulus
is 1.0 times 109 N/m
2. Vibration analysis of the crank is carried out with fixed-
free boundary condition and the resulting mode shapes are shown in Fig. 8, 9.
Four mode shapes are selected to span the deformation of the crank. As a result,
the system has 5 degrees of freedom.
Dynamic analysis is performed for 5 sec under the constant acceleration
condition of the joint between the ground and the body 1. The acceleration,
displacement, and relative deformation of the pin joint connecting the crank and
the coupler both from the proposed method and the nonlinear approach[19] are
shown in Figs.10,11 and 12, respectively. Note that since the results from both
models are almost identical as shown in these figures, the proposed
218
implementation methods using rigid virtual body can be validated.
Figure 8 Mode shapes of the crank
Figure 9 Mode shapes of coupler
219
Figure 10 Y acceleration of 1P
Figure 11 Relative deformation of 1P
220
Figure 12 The strain energy of crank
1.2.2.7. CONCLUSION
This research extends the generalized recursive formulas for the rigid
multibody dynamics to the flexible body dynamics using the backward
difference formula(BDF) and the relative generalized coordinate. When a new
force or joint module is added to the general purpose program in the relative
coordinate formulations, the modules for the rigid bodies are not reusable for the
flexible bodies. In order to relieve the implementation burden, a virtual rigid
body is introduced at every joint and force reference frames and a virtual flexible
body joint is introduced between two body reference frames of the virtual and
original bodies. The notationally compact velocity transformation method is used
to derive the equations of motion in the joint space. The terms in the equations of
motion which are related to the transformation matrix are classified into several
categories each of which recursive formula is developed. Whenever one category
is encountered, the corresponding recursive formula is invoked. Since
computation time in a relative coordinate formulation is approximately
proportional to the number of the relative coordinates, computational overhead
due to the additional virtual bodies and joints is minor. Meanwhile,
implementation convenience is dramatically improved.
221
REFERENCE
1. J. Wittenburg, Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart, 1977.
2.W. Hooker, and G. Margulies, "The Dynamical Attitude Equtation for an n-body
Satellite", it Journal of the Astrnautical Science, Vol. 12, pp. 123-128, 1965.
3. R. Featherstone, "The Calculation of Robot Dynamics Using Articulated-Body
Inertias",it Int. J. Roboics Res., Vol 2, pp. 13-30, 1983.
4. D. S. Bae and E. J. Haug, "A Recursive Formulation for Constrained Mechanical
System Dynamics: Part II. Closed Loop Systems", it Mech. Struct. and Machines, Vol. 15,
No. 4, pp. 481-506.
5. T. C. Lin, and K. H. Yae, "Recursive Linearization of Multibody Dynamics and
Application to Control Design", it Technical Report R-75, Center for Simulation and
Design Optimization, Department of Mechanical Engineering, and Department of
Mathematics, The University of Iowa, Iowa City, Iowa, 1990.
6. Ming-Gong Lee and E. J. Haug, "Stability and Convergence for Difference
Approximations of Differential-Algebraic Equations of Mechanical System Dynamics", it
Technical Report R-157, Center for Simulation and Design Optimization, Department of
Mechanical Engineering, and Department of Mathematics, The University of Iowa, Iowa
City, Iowa, 1992.
7. D. S. Bae, J. M. Han, H. H. Yoo, and E. J. Haug. "A Generalized Recursive Formulation
for Constrained Mechanical Systems", it Mech. Struct. and Machines, To appear.
8. A. A. Shabana, "Substructure Synthesis Methods for Dynamic Analysis of Multibody
Systems", it Computers & Structures, Vol. 20. No. 4, pp 737-744, 1985.
9. J. Garcia de Jalon, J. Unda, and A. Avello, "Natural Coordinates for the Computer
Analysis of Three-Dimensional Multibody Systems", it Computer Methods in Applied
Mechanics and Engineering, Vol. 56, pp. 309-327, 1985.
10. N. Vukasovic, J. T. Celigueta, J. Garcia de Jalon, and E. Bayo, "Flexible Multibody
Dynamics Based on a Fully cartesian System of Support Coordinates", it Journal of
Mechanical Design, Vol. 115, pp. 294-299, 1993.
222
11. S. S. Kim and E. J. Haug, "A Recursive Formulation for Flexible Multibody
dynamics:Part I, Open loop systems", it Comp. Methods Appl. Mech.Eng, Vol. 71,
pp.293-314, 1988.
12. H. J. Lai, E. J. Haug, S. S. Kim, and D. S. Bae. "A Decoupled Flexible-Relative
Coordinate Recursive Approach for Flexible Multibody Dynamics", it International
Journal for Numerical Methods in Engineering, Vol. 32, pp.1669-1689, 1991.
13. A. Cardona and M. Geradin, "Modelling of Superelements in Mechanism Analysis", it
International Journal for Numerical Methods in Engineering, Vol. 32, pp.1565-1593,
1991.
14. P. E. Nikravesh and A. C. Ambrosio, "Systematic Construction of Equations of Motion
for Rigid-Flexible Multibody Systems Containing Open and Closed Kinematic Loops", it
International Journal for Numerical Methods in Engineering, Vol. 32, pp.1749-1766,
1991.
15. M. S. Pereira and P. L. Proenca, "Dynamic Analysis of Spatial Flexible Multibody
Systems Using Joint Coordinates", it International Journal for Numerical Methods in
Engineering, Vol. 32, pp.1799-1812, 1991.
16. ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann
Arbor, MI 48105.
17. A. A. Shabana, Dynamics of Multibody Systems, John & Wiley, New York, 1989.
18. Jeng Yen, E. J. Haug, and F. A. Potra, "Numerical Method for Constrained Equations
of Motion in Mechanical Systems Dynamics", it Technical Report R-92, Center for
Simulation and Design Optimization, Department of Mechanical Engineering, and
Department of Mathematics, The University of Iowa, Iowa City, Iowa2 1990.
19. J. C. Simo, and L. Vu-Quoc, "On the Dynamics of Flexible Beams Under Large Overall
Motions-The Plane Case: Part I", it Journal of Applied Mechanics, Vol. 53, pp. 849-854,
1986.
223
1.2.3.
CONTACT MODELING TECHNIQUE OF A
FLEXIBLE PISTON ANDCYLINDER USING
MODAL SYNTHESIS METHOD
1.2.3.1. INTRODUCTION
Since a modal synthesis method is very cheaper than a nodal synthesis method
for a solving time, the modal synthesis method has been widely used in CAE. But
the method has a disadvantage in representing a large deformed or local deformed
system. To cover these problems, a static or dynamics correction modes have been
additionally used on normal modes and Craig and Bampton [1] generalize theseto
huge structural problems. This technique can be the most efficient analysis method
in a special purposed mechanical system. The generalized recursive formulation
and virtual body technique for a constrained flexible multibody dynamics system [2,
3] is used to represent the dynamics behavior of the flexible piston and cylinder.
Shabana [4] presented a coordinate reduction method for multibody systems with
flexible components. The local deformation of a flexible component was expressed
in terms of the nodal coordinates and was then spanned by a set of mode shapes
obtained from a mode analysis. Yoo and Haug [5] spanned the deformation by a set
of static correction modes obtained by applying a unit force or unit displacement at
a node where a large magnitude of force is expected during the dynamic analysis.
Mani [6] used Ritz vectors in spanning the local deformation and the Ritz vectors
were generated by spatially distributing the inertial and joint constraint forces on a
flexible body. Gartia de Jalon et al [7] presented a fully Cartesian coordinate
formulation for rigid multibody dynamics. This formalism was extended to the
flexible body dynamics by Vukasovic et al [8]. Nonlinearity associated with an
orientation matrix was relieved by defining all necessary vectors for the equations
of motion and constraints as the generalized coordinates. Several formulations have
been recently developed for flexible body systems that undergo large deformation.
Simo [9] had formulated the equations of motion for a flexible beam, based on the
inertial reference frame. Since displacement of a point on the beam was directly
measured from the inertial reference frame, the inertia terms become linear and
uncoupled, while the strain energy related terms become nonlinear. Yoo and Ryan
[10] proposed a mixed formulation of inertial and floating reference frames for a
224
rotating beam. Axial deformation was measured from a deformed state of the
rotating beam, while other deformations were measured from an undeformed state.
Shabana [11, 12] presented a non-incremental absolute coordinate formulation in
which the global location coordinates and slopes were defined as the generalized
coordinates. Since the finite rotation coordinates were not used as the generalized
coordinates, the difficulties associated with the finite rotation were resolved. Bae [2,
3] presented a recursive formulation which is both notationally compact and
computationally efficient for the constrained flexible body dynamics and a concept
of virtual body for multibody flexible dynamics to relieve implementation
complexity. Cho and Bae [13] present an efficient method to search a contact point
of a multibody system. Sohn and Kim [14] developed a numerical modeling
technique for the flexible piston and cylinder contact. Horiuchi [15] suggested the
computer modeling techniques for the dynamics analysis of total crank system.
In this investigation, a numerical modeling method and dynamic analysis of the
contact model between the flexible piston and cylinder is presented by using modal
synthesis method. First normal modes of piston and cylinder under a boundary
condition are computed, and then static correction modes due to a contact force
applied at contacted nodes are added to the normal modes. The nodal positions of
the original geometry are newly calculated from the updated geometry data with an
interpolation method. And then the updated nodes are used to compute a precise
contact force. Finally a crank system with the flexible bodies is modeled as a
numerical example. The proposed methods have good agreement with results of a
nodal synthesis technique and proved that it is very efficient design method.
1.2.3.2 NUMERICAL MODEL OF A CRNAK SYSTEM
The engine systems mainly consist of crank and valve systems, and so on.
Figure 1 shows an example of a crank system.
Figure 1. An example of a crank system
The piston is connected with connecting rod through a piston pin. In the
cranksystem, twokinds of the piston, free-free normal modes are recommended
225
rather than fixed-fixed normal modes to prevent a pretty higher frequency. Figure 3
shows first, second, and third eigenvalues and eigenvectors of piston.
(1) Finite Element Model of Piston
A piston transmits an explosion force to crank system and the translational
motion of the piston is translated to rotational motion of the crankshaft. Figure 2
shows us a piston model composed by finite element. The model has 8643 nodes
and 4830 elements that is tetrahedral type.
Figure2. A piston model
Figure3. Normal modes of piston
The piston is connected with connecting rod through a piston pin. In the crank
system, two kinds of forces are always acting on the piston. The one is from
contact and the other is from explosion. In the modal synthesis method, static
correction modes are needed to span a local deformation. In the piston, free-free
normal modes are recommended rather than fixed-fixed normal modes to prevent a
pretty higher frequency. Figure 3 shows first, second, and third eigenvalues and
eigenvectors of piston.
(2) Finite Element Model of Cylinder
The engine system holds a crank system and cylinder that is a part of crank
system restrict a piston motion. Figure 4 shows a cylinder model and its wall
composed by finite element. The model has 40440 nodes and 21663 elements that
226
is tetrahedral type.
Figure4. A cylinder model
Figure5. Normal modes of cylinder
Actually, cylinder is covered with cylinder head that is related with valve system.
But this paper is only considering a cylinder that is most closely related with piston
motion. In the cylinder wall, contact forces are continuously occurred while piston
is keep moving. So the cylinder model using modal synthesis method must
containstatic correction modes to span a local deformation of wall. In the cylinder,
fixed-fixed normal modes are used. So all nodes laid on cylinder wall are fixed to
generate normal modes. Figure 5 shows first, second, and fourth eigenvalues and
eigenvectors of cylinder.
1.2.3.3 CONTACT MODELING
Static Correction Modes: In generally, unit displacement is imposed in a node
while others are fixed to generate static correction mode. And the final modes are
obtained from synthesizing normal modes and static correction modes. But cylinder
model that consist of solid element is not a suitable to use a unit displacement
227
control. It has a possibility to generate extremely local deformation. To overcome
such a problem, distributed load should be applied to get static correction modes. It
is good for solid element to get a continuous deformation result because it is not a
point load. Figure 6 shows one of the synthesized mode shapes with a distributed
load.
Figure 6. Synthesized mode shape
(1) Interpolation of Nodal Positions
Cylinder wall has a different shapes depend on environment condition. For
example, the temperature of cylinder, the wear of wall or assembling condition can
be one of the reasons. And it does not have effect to system property such as
Eigenvalue because the quantity of the changing is too small but have a huge effect
to contact condition. So it is very useful that the direct using new geometric
information without repeated Eigenvalue analysis. Figure 7 shows us a concept of
surface interpolation and node projection.
Figure 7. Projection to a surface
According to the above figure, point data can make a spline surface and original
228
nodes position can be projected to the surface. The projected point is used as a new
initial position and the updated data are used to calculate contact force. And the
mode shapes that represent system properties and span a deformation are used
original one.
(2) Contact Force
Contact force between cylinder and piston is calculated from the penetration.
Figure 8 shows the schematic diagram of contact force analysis used in the piston
and cylinder body.
Figure 8. Penetration between two bodies
When the contact occurs, the cylinder penetrates into the piston. The contact force
can be generated with penetration, its time derivative, and compliance
characteristics of two bodies. Thus, the contact normal force is obtained by
32
1 mm
m
n δδδ
δckδf
where k and c are the spring and damping coefficients which are determined byan
experimental method, respectively and the δ is time differentiation of δ . The
exponents 1m and 2m generates a non-linear contact force and the exponent 3m
yields an indentation damping effect. In general steel material, 1m equal 1.0 and
2m exist between 2.2 and 2.5 and these values are related with contact shape. When
the penetration is very small, the contact force may be negative due to a negative
damping force, which is not realistic. This situation can be overcome by using the
indentation exponent greater than one.
The friction force is obtained by
(1)
229
nf fμf
where μ is the friction coefficient and its sign and magnitude can be determined
from the relative velocity of the pair on contact position. The equation of friction is
based on coulomb friction and μ isdeterminedbyexperimental method.
The force is distributed to the neighboring nodes and project to modal space.
dj
nmode
j
m fΨF
1
where df is a distributed nodal force and jΨ is a j
th mode shape of a node on which
force is applied. The nmode means the number of used modes
1.2.3.4 NUMERICAL SOLUTION
Figure 9. Two adjacent flexible bodies
(2)
(2)
(3)
230
Figure 10. Adjacent flexible bodies and virtual bodies
This paper uses general DAE solution technique to solve a dynamic piston and
cylinder contact problem. The kinematics and equation of motion of the rigid and
flexible bodies are presented as relative coordinates and recursive formulations.
Two flexible bodies connected by a joint and their reference frames are shown in
Fig. 9. Flexible body programming especially requires much more effort than rigid
body programming does due to the complexity associated with deformation. In
order to minimize the programming effort, a concept of the virtual body has been
used in [3]. At every joint and force reference frames, a virtual rigid body, whose
mass and moment of inertia are zero, is introduced as shown inFigure10. A virtual
joint is connected the virtual body and the flexible body. Therefore, the joint and
forces can be developed only for rigid bodies and one virtual joint is to be added in
the joint module. The recursive kinematic relationships representing the
admissibility conditions of the flexible body joint are formulated in the following
subsections.
(1) Kinematic Definitions and Recursive Formulation
The ZYX frame is the inertial reference frame and the zyx
frame is the body reference frame in Fig. 9. Velocities and virtual displacements of
point O in the ZYX frame are respectively defined as
w
r
Their corresponding quantities in the zyx frame are respectively defined as
(4)
231
wA
rA
w
rY
T
T
where A is the orientation matrix of the zyx frame with respect to the
ZYX frame. Y is the combined velocity of the translation and rotation. The
recursive velocity formulas for a rigid body [18] and a flexible body [2] are
obtained as
1)i(i1)i2(i1)(i1)i1(ii qBYBY
where ii )1( q denotes the relative velocity vector, respectively. It is important to
note that matrices 1)i1(iB and
1)i2(iB are only functions of the1)i(iq . Similarly,
the recursive virtual displacement relationship of the rigid and flexible bodies is
obtained as follows
1)i(i1)i2(i1)(i1)i1(iiδ δqBδZBZ
If the recursive formula in Eq. (6) is respectively applied to all joints, the following
relationship between the Cartesian and relative generalized velocities can be
obtained:
qBY
where B is the collection of coefficients of the 1)i(iq and
T1nc
TT
2
T
1
T
0 nY,,Y,Y,YY
T1nr
T
)1(
T
12
T
01
T
0 nnq,,q,q,Yq
wherenc and nr denote the number of the Cartesian and relative coordinates,
respectively. Since q in Eq. (8) is an arbitrary vector in nrR , Eqs. (6) and (8),
which are computationally equivalent, are actually valid for any vector nr
Rx
such that
(5)
(6)
(7)
(8)
(9)
(10)
(11)
232
xBX
and
1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX
wherenc
RX is the resulting vector of multiplication of B and x . As a result,
transformation of nr
Rx into ncRBx is actually calculated by recursively
applying Eq. (12) to achieve computational efficiency in this research.
Inversely, it is often necessary to transform a vector G in ncR into a new
vector GBgT in nr
R . Such a transformation can be found in the generalized
force computation in the joint space with a known force in the Cartesian space. The
virtual work done by a Cartesian force nc
RQ is obtained as follows.
QZW Τδδ
where Zδ must be kinematically admissible for all joints in a system. Substitution
of qBZ δδ into Eq. (13) yields
*TTT δδδ QqQBqW
where QBQ T* .
(2) Equation of Motion and Numerical Solution
The equations of motion for a constrained mechanical system in the joint space
[16] have been obtained by using the velocity transformation method as follows.
0QΦYMBF Z TT
whereΦand λ respectively denote the cut joint constraint and the corresponding
Lagrange multiplier. M is a mass matrix and Q is a force vector including external
forces, strain energy terms, and velocity induced forces.The equations of motion,
the constraint equations, vq , and av constitute the differential algebraic
equations(DAE). Application of tangent space method in Ref. [17] to the DAE
yields the following nonlinear system of equations:
(12)
(13)
(14)
(15)
233
)t,,,(
)t,,(
)t,(
)t,,,,(
)(
)(
)(
nnnn
nnn
nn
nnnnn
2n0n
T
o
1n0n
T
o
n
avqΦ
vqΦ
qΦ
λavqF
βavU
βvqU
pH
where TT
n
T
n
T
n
T
n
T
n ,,, λavqp , 0 1β , and 2β are determined by the coefficients ofthe
BDF. The 0U must be chosen such that the augmented squarematrix.
qΦ
UT
0
isnonsingular. Applying Newton's method to solve the nonlinear system in Eq. (16)
yields
HΔppH )( n
,...23,1,i
n
1i
n iΔppp
where
0ΦΦΦ
00ΦΦ
000Φ
FFFF
0UU0
00UU
H
aaa
v
q
λavq
p
B
T
00
T
0
T
00
T
0
Since the F and Φare highly nonlinear functions of q , v , a , and λ some
precautions must be taken in deriving the non-zero expressions in matrixpH so
that they can be efficiently evaluated.
1.2.3.5 NUMERICAL EXAMPLES
In this paper, one crank system based on 4-stroke cycle is used for numerical
example. Figure 11 shows us a crank system.
(16)
(17)
(18)
(19)
234
Figure 11. A crank system
Young’s modulus and Poisson’s ratio of the cylinder and piston are 2.0e9 (N/mm2)
and 0.28for each other and the model consists of fully solid element. The modes of
cylinder are made from 20 normal modes and 30 static correction modes and the
modes of piston are made from 20 normal modes and 30 static correction modes.
The other components are constrained by joint and contact force is applied to the
piston and cylinder wall. Equivalent explosion force is acting on the piston top part
as a 2206.1834 (N/mm2). The system analyzed for 0.03 sec. and rotational velocity
of crankshaft at the end of the time almost reached at 5000 RPM. In this system,
the piston is moved along to the x-axis due to contact force with the cylinder wall.
And the maximum contact force is occurred in y direction due to the rotational
motion of the crankshaft and deformations of the flexible bodies can be maximized.
To validate a reliance of a result, displacements of the piston in two models are
compared as shown in Figure 12. One is a rigid body model in which the piston and
cylinder are modeled as rigid bodies. The other is a flexible body model where they
are modeled as flexible bodies. As shown in the figure, the y displacement of the
flexible piston is compared with that of a rigid body. The first stage of analysis, the
results are looks like almost same because of low speed of piston but finally the
flexible body deformation due to contact force made a difference between rigid and
flexible components.
235
Figure 12. The comparison of positions of piston
Figure 13. Effect of updated position
In real engine system, cylinder wall radius can be bigger than unused state
because of the wear due to endless contact of piston. The wear makes a restricted
piston motion to be more largely in the y direction and such motion makes more
big contact impact and accelerates the wear. The effect of the wear can get by
renewing the nodal positions. The nodes that lay on cylinder wall are moved to the
outer radius direction as 0.05mm. The original radius of cylinder wall is 26.75mm
and modified radius is 26.8mm. Figure 13 shows us an effect of the position
change. As shown in the figure, the piston motion in y direction is greater than that
of original state although the changing of radius is very small.
236
1.2.3.6 CONCLUSION
The contact modeling technique and dynamics analysis of piston and cylinder
system are presented by using modal synthesis method. The normal modes of the
piston are computed under free-free boundary condition. The normal modes of the
cylinder are calculated under a boundary condition that some nodes are fixed.
Static correction modes of the piston and cylinder are computed under static loads
and added to the normal modes. The nodal positions of the original geometry are
newly calculated from the updated geometry data with an interpolation method.
And then the updated nodes are used to compute a precise contact force. A crank
system is modeled as a numerical example and this proposed method has proved
that it is very efficient design method.
REFERENCES
[1] Craig, R. R., and Bampton, M. C. C., 1965, “Coupling of substructures for
dynamics analyses.” AIAA Journal, Vol. 3, No. 4, pp. 678-685.
[2] Bae, D. S.,Han, J. M., Choi, J. H. and Yang, S. M., 2001, "A Generalized
Recursive Formulation for Constrained Flexible Multibody Dynamics”,
International Journal for Numerical Methods in Engineering, Vol. 50,No. 6,
pp.1841-1859.
[3] Bae, D. S.,Han, J. M. and Choi, J. H., 2000, "An Implementation Methods for
Constrained Flexible Multibody Dynamics using a Virtual Body and Joint”, The
Journal of Multibody System Dynamics, Vol. 4, pp. 297-315.
[4] Shabana, A. A., 1985, "Substructure Synthesis Methods for Dynamic Analysis
of Multibody Systems", Computers Structures, Vol. 20. No. 4, pp 737-744.
[5] Yoo, W. S. and Haug, E. J., 1985, "Dynamics of Flexible Mechanical Systems
Using Vibration and Static Correction Modes", Journal of Mechanisms, and
Transmissions, and Automation in Design.
[6] Wu, H. T. and Mani, N. K., 1994, "Modeling of Flexible Bodies for Multibody
Dynamic Systems Using Ritz Vectors",Journal of Mechanical Design, Vol. 116,
pp. 437-444.
237
[7] Garcia de Jalon, J., Unda, J. and Avello, A., 1985, "Natural Coordinates for the
Computer Analysis of Three-Dimensional Multibody Systems", Computer
Methods in Applied Mechanics and Engineering, Vol. 56, pp. 309-327.
[8] Vukasovic, N., Celigueta, J. T., Garcia de Jalon,J. and Bayo, E., 1993, "Flexible
Multibody Dynamics Based on a Fully cartesian System of Support Coordinates",
Journal of Mechanical Design, Vol. 115, pp. 294-299.
[9] Simo,J. C. and Vu-Quoc, L., 1986, "On the Dynamics of Flexible Beams Under
Large Overall Motions-The Plane Case: Part I", Journal of Applied Mechanics,
Vol. 53, pp. 849-854.
[10] Yoo, H. H., Rion, R. R. and Scott, R. A., 1994, "Dynamics of Flexible Beams
Undergoing Overall Motions", Journal of Sound and Vibration, Vol. 181, pp. 261-
278.
[11] Shabana, A. A. and Christensen, A. P., 1997, "Three Dimensional Absolute
Nodal Coordinate Formulation: Plate Problem", International Journal for
Numerical Methods in Engineering, Vol. 40, pp. 2775-2790.
[12] Shabana, A. A., Hussien,H. A. and Escalona, J. L., 1998, "Application of the
Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation
Problems", Journal of Mechanical Design}, Vol. 120, pp. 188-195.
[13] Cho, H. J., Bae, D. S.,Ryu, H. S. and J. H. Choi, 2002,
“An Efficient Contact SearchAlgorithm using the Relative Coordinate System for
Multibody System Dynamics”, Proceedings of ACMD’02, The First Asian
Conference on Multibody Dynamics 2002, July 31-August 2, Iwaki, Fukushima,
Japan.
[14] Sohn, S. H. and Kim, W. K., 2004, “Crank development specification”,
Technical report, FunctionBay Inc.
[15] Horiuchi, S., 2004, “The meeting of Crank development”, Yamaha Motors Co.
LTD., Japan.
[16] Wittenburg, J., 1977, “Dynamics of Systems of Rigid Bodies”, B. G. Teubner,
Stuttgart.
[17] Jeng Yen, 1993, “Constrained Equations of Motion in Multibody Dynamics as
ODE’s on manifolds”, SIAM Journal of Numerical Analysis, Vol. 30, pp. 553-568.
238
[18] Angeles, J., "Fundamentals of Robotic Mechanical Systems", Springer, 1997.
2. Optimization
240
2.1.
DFFS AND ROBUST OPTIMIZATION OF A
PAPER FEEDING MECHANISM
2.1.1. INTRODUCTION
Recently the media transport systems, such as printers, copiers, fax, ATMs,
cameras, film develop machines, etc., have been widely used and being developed
rapidly. In this field, it is an important key technology to determine kinematic
mechanisms of parts dimensions, and materials, etc for the media machine
developers. To shorten the develop time, reduce the cost, and improve the machine
performance, most of early works has done to develop the computer simulation for
analyzing the paper feeding and separation process. Among them,
RecurDyn/MTT2D and RecurDyn/MTT3D are widely used in these areas.
Although the analysis process has well developed in the media transport systems,
however, its’ design optimization is very difficult, because the analytical design
sensitivity process is very difficult in the multi-body dynamics (Kim and Heo, 2003;
Kim and Choi, 2001; Kim and Choi, 1998). What is worse, a lower-pass filter are
frequently used to signify the dynamic responses. In this case, analytical approach
for design sensitivity is impossible.
This study introduces a meta-model based design optimization for dynamic
response optimization, which can avoid the design sensitivity analysis and
overcome the numerical noise. Especially, Design For Six Sigma (DFSS) and
robust optimization is easily implemented by using the gradient information of
meta-models. Chapter 2 reviews the early works for modeling the flexible paper
and analyzing the media transport systems. Chapter 3 presents the proposed
optimization strategy. Chapter 4 explains the DFSS optimization of a paper feeding
mechanism. Finally, Chapter 5 presents the conclusion of this study. 2.1.2. REVIEW ON THE PAPER MODELINGS
Since a thin plate model with an orthotropic material property suggested to
model a paper behavior (Trope, 1981), many numerical techniques have tried to
analysis contact normal and frictional forces between paper and guide with respect
to paper velocity and attack angle. Among them, beam elements are widely used to
represent a flexible paper (Stole and Benson, 1992; Stole and Benson, 1993). A
240
separation mechanism of paper while developing numerical analysis models for a
copy machine has been investigated (Stack, 1993; Stack, 1999). The local static
mechanics of electrometric nip system for media transport system has been
introduced. The nonlinear finite element method and experimental measurement
techniques are used to investigate the large deformable rollers. Several unique
phenomena, such as skewing sheet, etc., of nip feeding system are well described
(Diehl, 1995). The stiffness of a coated paper has been computed (Koji, 1999). The
computer modeling techniques is introduced for the design and analysis of film
feeding mechanisms. The primitive dynamic analysis of two-dimensional film-
feeding models is presented by using commercial computer program (Ashida,
2000). Computational modeling techniques and a computer simulation tools for
two- and three- dimensional film feeding mechanisms have been developed (Cho
and Choi, 2003; Cho and Choi, 2005). The modeling techniques such as a
mathematical representation of a flexible paper, guide and rollers and contact
algorithm have been implemented on a commercial dynamics analysis program of
RecurDyn/MTT2D and RecurDyn/MTT3D. An experimental way is investigated to
estimate a slip between paper and roller in a simple paper feeding mechanism while
validating the results of experiment and simulation. RecurDyn/MTT2D have been
used to simulate (Ryu and Choi, 2004).
2.1.3 META-MODEL BASED OPTIMIZATION 2.1.3.1 SIMULTANEOUS KRIGING MODEL
Meta-Models such as RSM, Kriging and Radial Basis Function (RBF) are
increasingly used to approximate expensive responses in engineering fields. RSM
was introduced in the classical DOE, which used a polynomial type regression
model. Hence, it required the rotatable characteristics for sampling points such as
CCD and SCD. However, Kriging (Farhang-Mehr and Azarm, 2005) and RBF
(Wang and Liu, 2002) are Bayesian models. Hence, they used a space filled
sampling points such as Latin hypercube or descriptive designs (Kim, 2006).
Kriging models can be defined as a combination of a regression model plus a
departure term:
y z Xβ x ,
where y is the approximate model, Xβ is a polynomial type regression model,
(1)
241
and z x is a Gaussian random process with 20,N . If the regression model
Xβ globally approximates the design space, the departure term z x represents
the localized deviations so that the Kriging model interpolates the sn sampled
points. In my experience, the regression model plays an important role in design
optimization especially for insufficient sampling points. The covariance matrix of
z x is given by
2 ,i j i jCov z z R x x R x x ,
where R is the correlation matrix and ,i jR x x is the correlation function
between any two of the sn sampled points. Hence, R is a s sn n symmetric
matrix with ones in the diagonal term. There are many correlation functions
,i jR x x . Among them, the Gaussian type is widely used
2
1
, expk
l l
i j l i j
l
R
x x x x ,
where l are the unknown correlation parameters to fit model. The estimates,
y x of the response y x at the untried values of x are given by
1Ty x X x β r x R y X x β .
The correlation vector between x and the sampled points 1 2, ,...,snx x x is given
by:
1 2, , , ,..., ,s
TT
nR R R
r x x x x x x x
In the estimates, the unknown coefficients of regression model is determined as
1
1 1T T
β X R X X R y .
Also, in order to determine the unknown correlation parameters l , the estimate of
(2)
(3)
(4)
(5)
(6)
242
the variance 2 (not the variance in the observed data) is introduced. Hence, the
correction parameters l is determined by solving
1
0min det
sn
θ
R θ θ )
While any values for θ create an interpolation model, the best kriging model is
found by solving the k-dimensional unconstrained optimization problems described
in the above.
From the viewpoint of numerical optimization, equation (7) can be non-smooth
because the correlation matrix R θ is frequently singular during optimization
process. Hence, some special techniques are required to avoid the singular
phenomena and non-linearity of it. Hence, we use a singular value decomposition
(SVD) and normalization and scaling techniques. Also, multi-objective formulation
is introduced in equation (7) to solve the multiple kriging models simultaneously.
This approach uses only one correlation matrix R θ even for multiple kriging
models (Kim, 2006).
2.1.3.2 DFSS & ROBUST DESIGN FORMULATION
Lets’ consider the general optimization formulation. Fundamentally, all the
functions are composed of meta-models.
Minimize 0 0f k x x x
Subject to 0, 1,2,...,ih i l x
0, 1, 2 , . . . ,j j jg k j m x x
x
where , 0,1,...,j j m x are the standard deviation of f and jg that evaluated
from meta-models. The value of and ik are alpha weight and robust index,
respectively. If 0 and 0 1k , the design objective is a minimization of the
variance of f x . If 6jk is defined, the inequality constraints become DFSS
constraints. If multiple objectives are given, the objective function of equation (8)
is replaced by a preference function as
(7)
(8)
(9)
(10)
(11)
243
Minimize , 1,2,...k objP k N x
In order to represent our preference function, let’s consider following two
objectives.
1minx
x and 2maxx
x
There are many preference functions in multi-objective optimization strategy.
Among them, we uses following two types.
1 2
1 2
1 2
1 1G G
P w w
x xx
1 2
1 2
1 2
max 1 , 1G G
P w w
x xx
where, the values of iw are the user defined weighting coefficients and the
relaxation factors and are automatically determined. Also, the ideal solution G
if is internally determined.
2.1.3.3 NUMERICAL OPTIMIZATION PROCESS
The approximate optimization problems, based on meta-modes, are sequentially
solved with augmented Lagrange multiplier method (ALM) (Kim and Choi, 1998).
In order to avoid the convergence difficulty for the insufficient sampling points, the
initial design is selected from the best points in the given DOE and move limit is
automatically adjusted. Also, the polynomial types are automatically switched to
the degree of convergence of optimization process.
In the first iteration, the sequential approximate optimization (SAO) process
requires the sampling points. We provide a discrete Latin hypercube design,
incomplete small composite design-I (Kim and Heo, 2003), incomplete small
composite design-II, generalized small composite design and other classical DOE
methods such as CCD and BBD etc (Kim, 2006).
In the subsequent iteration of SAO, a new optimal design is given. Next, exact
analysis is done for this point. Then, this new information is added to the design
database. Hence, meta-model is newly developed and repeat these processes until
(12)
(13)
(14)
(15)
244
the convergence criterion is satisfied for their tolerance values.
Figure 1. A paper feeding system model in Recurdyn/MTT2D
2.1.4. DFSS OF THE PAPER FEEDING SYSTEM
2.1.4.1 SYSTEM ANALYSIS
Figure 1 shows a simple model of a paper feeding mechanism in the
Recurdyn/MTT2D, which includes a printing part, feeding part and output part. In
the mechanism, the upper part roller feeds the paper to the lower part roller. When
the end part of feeding paper passes the sensor, the lower part roller rotates in the
reverse direction. Then, the paper is feed to the upper part roller. During this
reverse feeding process, the printing part is operated. The printing quality is fully
depends on the slip of the feeding paper. Larger nip force can reduce the slip
amount but manufacturing cost is increased. Hence, the slip amounts should be
minimize within the allowable nip force limitation. The flexible paper is composed
of rigid bars, revolute joints and rotational springs and dampers. Thus, the dynamic
responses have numerical noise due to non-smoothness between bars. Hence, we
use a lower-pass filter is needed to signify them.
2.1.4.2 RANDOM DESIGN VARIABLE SELECTION
Figure 2 shows the 8 random design variables. The 1st through 3
rd design
variables are Nip spring stiffness, damping coefficient and the initial pre-load in the
lower part roller pair. The 4th
through 6th
design variables are the same
characteristics in the upper roller pair. Also, in order to control the feeding direction
in the lower part roller pair, the installation position and angles of moving roller are
245
selected as the 7th
and 8th
design variables. The lower and upper limits on the
design variables are listed in Table 1 side by side. All the design variables are
regarded as random variables. The nip spring data has 5% COV, the roller
position has 0.1( )mm deviation and the roller angle has 0.1(deg.) deviation.
Figure 2 Random Design variables
Table 1. Lower and upper bounds on design variables
Lower
bound
Upper bound
DV1 6.00E-4 1.50E-3
DV2 6.00E-5 1.50E-4
DV3 2.00E-3 8.00E-2
DV4 6.00E-4 1.50E-3
DV5 6.00E-5 1.50E-4
DV6 2.00E-3 8.00E-2
DV7 -3.00E+0 3.00E+0
DV8 -1.00E+0 1.00E+0
2.1.4.3 DESIGN FORMULATION
Now, in order to enhance the printed quality of printer, the slip between the
lower part roller pairs and the feeding paper should be minimized. Also, the nip
force should be less than 0.025(N) within 6-sigma variance. Hence, the
performance indexes are selected as the slip amounts and nip force during the
reverse rotation of the lower part roller system.
Minimize the average of slips and the sum of slips
246
subject to max max6 0.025( )nip nipF F N
in the design variable limits.
In the practical implementation, RecurDyn/AutoDesign (Kim, 2006) uses a lower-
pass filter to remove the numerical noise.
2.1.4.4. OPRIMIZATION RESULTS
From the viewpoint of meta-modeling constructions, the 8 design variable
problem is quite large design. Hence, a discrete Latin-hypercube design is
employed for the initial sampling in RecurDyn/AutoDesign. First, the 16 sampling
points are selected in the design range given in Table 1 and the current design is
added. Hence, total 17 sampling points is used. The performance index for the 17
sampling points are listed in Table 2. It is noted that the nip force values do not
include 6-sigma values. Nevertheless, most of trials is greater than 0.025 (N).
Table 2 Performance index for the initial design
Trials Average of Slip Sum of Slip Nip
Force
1 2.04E+00 4.10E+02 5.39E-02
2 7.26E-01 1.45E+02 7.33E-02
3 4.56E+00 8.98E+02 3.26E-02
4 2.80E+00 5.60E+02 3.75E-02
5 1.33E+00 2.66E+02 6.55E-02
6 1.20E+00 2.42E+02 1.68E-02
7 1.09E+00 2.17E+02 2.19E-02
8 3.60E+00 7.23E+02 2.69E-02
9 9.64E-01 1.94E+02 7.15E-02
10 1.72E+00 3.45E+02 5.74E-02
11 1.32E+00 2.59E+02 5.33E-02
12 1.17E+01 2.16E+03 1.19E-02
13 9.52E-01 1.89E+02 6.17E-02
14 5.04E+00 9.73E+02 6.47E-03
15 2.47E+00 4.96E+02 4.28E-02
16 5.98E-01 1.20E+02 6.84E-02
17 2.03E+00 4.08E+02 4.76E-02
In this study, simultaneous Kriging models combined with pure quadratic
247
polynomials are employed for meta-models. RecurDyn/AutoDesign requires 6
iterations. The iteration summary is listed in Table 3. The final design successfully
satisfies the nip force constraints and reduces the slip amounts.
Table 3 Performance index from SAO
SAO Average of Slip Sum of Slip Nip
Force
1 4.727 945.4 2.135E-
02
2 1.216 242.1 1.899E-
02
3 1.166 234.4 2.070E-
02
4 1.096 220.4 1.927E-
02
5 1.091 218.2 2.034E-
02
6 1.088 217.6 2.019E-
02
The convergence criteria are selected as the relative change of objectives between
consecutive iterations and the maximum violation of constraints. Their
convergence tolerances are selected as 0.01, respectively. RedurDyn/AutoDesign
provides the approximate value of standard deviation for the performance index.
Then, the DFSS constraint violation is checked as
: 6 limitviolation .
The approximate standard deviation for nip force is given as 0.0008362. Thus, the
violation values is evaluated as 0.02019-6*0.0008362, which is less than the limit
within its’ convergence tolerance. Figure 3 shows the convergence history of SAO.
249
Also, in order to validate the DFSS results, we check the constraint violation by
using the sampled variance. To do this, 10 latin-hypercube sampling points are
sampled in the neighborhood of the final design. The sampled range is random
variable deviation. The sampled standard deviation is evaluated from the final
design and additional 10 values. The sample standard deviation is obtained as
0.00050226. This is less than the approximate standard deviation. It represents that
the proposed design satisfies the 6-sigma constraints by using only 23 evaluations.
For the initial and final designs, the nip forces are shown in Figure 4 and 5. The
final design is much less than the initial. These comparisons show the role of a
lower-pass filter. Figure 6 shows the additional 10 analysis results for DFSS
validation.
Finally, the final design values are (8.00E-4, 1.19E-4, 1.59E-4, 1.24E-3, 6.00E-5,
8.00E-2, 1.0, 2.165). Also, the initial design values are (1.00E-3, 1.00E-4, 5.00E-2,
1.00E-3, 1.00E-4, 5.00E-2, 0.0, 0.0). In this comparison, it is noted that the
installation position and angles are changed.
Figure 4 Nip force for the initial design
0 1 2 3 4 5 6
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.065
0.070
0.075
0.080
0.085
0.090
0.095
0.100
Perf
orm
ace Indexes
SAO Iteration
Average of Slip(*1.0E+2)
Sum of Slip (*1.0E+4)
Nip Force
Figure 3 Convergence history of SAO
250
Figure 5 Nip force for the final design
Figure 6 Nip force for the additional 10 analysis used in DFSS validation
2.1.5 CONCLUDING REMARKS
This study introduces the meta-model based design strategy for dynamic
response optimization. This can avoid the difficulty of design sensitivity analysis,
especially when a lower-pass filter is employed. Also, it shows that DFSS can be
easily implemented by using the approximate variance from meta-model. In the
numerical test, it successfully solved for 6-sigma design of the paper feeding
mechanism only for 23 analyses including the initial samplings. Finally, the DFSS
optimization results are validated by the sampled variance.
ACKNOWLEDGMENT
Some of researchers of this research got financial support of the second stage of
BK21 program and the Basic Research Fund of the Agency for Defense and
Development Grant No. ADD-04-05-02.
Nip
Forc
es
251
REFERENCES
1. Ashida, T., The Meeting Material of The Japan Society for Precision
Engineering, Japan, 2000.
2. Cho, H. J., Bae, D. S., Choi, J. H. and Suzuki, T., Dynamic Analysis and
Contact Modeling for Two Dimensional Media Transport System,
Proceedings of DETC’03, DETC2003/MECH-48338, ASME 2003 Design
Engineering Technical Conferences and Computers and Information in
Engineering Conference, Chicago, Illinois, USA, September 2-6, 2003.
3. Cho, H. J., Bae, D. S., Choi, J. H., Lee, S. G. and Rhim, S. S., Simulation
and Experimental Methods for Media Transport System: Part I, Three-
Dimensional Sheet Modeling Using Relative Coordinate, Journal of
Mechanical Science and Technology, Vol. 19, No. 1, pp. 305~311, 2005.
4. Diehl, T., Two dimensional and three dimensional analyses of nonlinear
nip mechanics with hyperelastic material formulation, Ph. D. Thesis,
University of Rochester, Rochester, NewYork, 1995.
5. Farhang-Mehr, A. and Azarm, S., Bayesian meta-modeling of engineering
design simulations: A sequential approach with adaptation to irregularities
in the response behaviour, Int. J. Numer. Meth. Engng., Vol. 62, pp.
2104~2126, 2005.
6. Kim, M.-S. and Heo, S.-J., Conservative quadratic RSM combined with
incomplete small composite design and conservative least squares fitting,
KSME International Journal, Vol. 17, No. 5, pp. 698~702, 2003
7. Kim, M-S and Choi, D.-H. An efficient dynamic response optimization
using the design sensitivities approximated within the estimate confidence
radius, KSME International Journal, Vol. 15, No. 8, pp. 1143~1155, 2001.
8. Kim M.-S and Choi, D.-H., Min-max dynamic response optimization of
mechanical systems using approximate augmented Lagrangian, Int. J.
Numer. Meth. Engng., Vol 43, pp. 549~564, 1998.
9. Kim, M.-S., RecurDyn/AutoDesign: Meta-Model based design optimizer,
Theoretical Manual, FunctionBay, 2006.
10. Koji, O., Toshiharu, E. and Fumihiko, O., Evaluation and Control of
Coated paper Stiffness, Proceedings of Tappi advanced coating
fundamentals symposium, Tappi press, Atlanta, USA, pp. 121-132, 1999.
11. Ryu, J. K., Song, I. H., Lee, S. G., Rhim, S. S. and Choi, J. H., Simulation
and Experimental Methods for Media Transport System: Part II, Effect of
Normal Force on Slippage of Paper, Proceedings of ACMD’04, The Second
Asian Conference on Multibody Dynamics 2004, Olympic Parktel, Seoul,
Korea August 1-4, 2004.
12. Stack, K. D., A Study of Friction Feed Paper Separation, Transaction of the
252
ASME-B-Journal of Engineering for Industry, Vol. 115, pp. 236-241, 1993.
13. Stack, K. D, Stole, J. and Benson, R. C., A review of computer simulation
models for sheet, Advances in information storage systems, Vol. 10, pp.
173-184, 1999.
14. Stolte, J. and Benson, R.C., Dynamic Deflection of Paper Emerging from a
Channel, ASME Journal of Vibration and Acoustics, Vol. 114, pp. 187-193,
1992.
15. Stolte, J. and Benson, R.C., And Extending Dynamic Elastica: Impact With
a Surface, ASME Journal of Vibration and Acoustics, Vol. 115, pp. 308-313,
1993.
16. Thorpe, J.L., Paper as an orthotropic thin plate, Tappi, Vol. 64, No.3, pp.
119-121, 1981.
17. Wang, J.G. and Liu, G.R., A point interpolation meshless method based on
radial basis functions, Int. J. Numer. Meth. Engng., Vol. 54, pp. 1623~1648,
2002
253
2.2.
THE ROBUST DESIGN OPTIMIZATION
OF HIGH MOBILITY TRACKED
VEHICLE SUSPENSION SYSTEM
2.2.1. INTRODUCTION
Up to now, most of researches for tracked vehicle systems has been the
suspension systems modeling and the track system modeling in the multi-body
system dynamic[1-6]. However, recently, a design study is required for a tracked
vehicle system in order to improve the ride characteristics, which can be realized
by using the well-developed CAE software for tracked vehicle system.
Although the analysis process has well developed for the tracked vehicle systems,
however, its’ design optimization is very difficult because the analytical design
sensitivity process is very difficult in the multi-body dynamics[7-9]. What is worse,
a lower-pass filter are frequently used to signify the dynamic responses. In this case,
analytical approach for design sensitivity is impossible.
This study introduces a meta-model based design optimization for dynamic
response optimization, which can avoid the design sensitivity analysis and
overcome the numerical noise. Especially, robust optimization is easily
implemented by using the gradient information of meta-models. Section 2 reviews
the early works for modeling the track system and analyzing the tracked vehicle
systems. Section 3 presents the proposed optimization strategy. Section 4 explains
the robust optimization of a tracked vehicle system. Finally, section 5 presents the
conclusion of this study.
2.2.2 REVIEW OF TRACKED VEHICLE MODELING AND
ANALYSIS
In early 1980’s several dynamic modeling techniques for track systems have
been developed in universities, and research institutes and companies. McCullough
and Haug[1] designed a super element that represented spatial dynamics of high
mobility tracked vehicle suspension systems. Their track was modeled as an
internal force element that acted in ground, wheels and the chassis of the vehicle.
Also, track tension was computed from a relaxed catenary relationship. Nakanish
254
and Shabana[2] introduced a contact search approach for planar rigid body track
system. This was extended for spatial dynamic analysis by Choi, Lee and
Shabana[3-3]. In these approaches, track link was modeled by rigid body and
connected by one-degree-of-freedom pin joint and bushing force element. In order
to overcome the numerical difficulty of Choi et.al., Ryu et.al.[5] extended the track
system modeling techniques for the high mobility military tracked vehicle in the
context of G-Alpha numerical integrator. Recently, Ryu et.al[6] proposed the
nonlinear dynamic modeling for a tracked vehicle and validated it’s performance
by comparing them with test results.
2.2.2.1 Multibody Tracked Vehicle Model
The tracked vehicle model used in this investigation is a military purpose high
speed tank system which has sophisticated suspension system to damp out impacts
from hostile ground. The suspension units of the vehicle include Hydro-pneumatic
Suspension Units and torsion bar systems that are modeled as force elements
whose compliance characteristics are obtained from analytical and empirical
methods. The extracted stiffness and damping characteristics from test machines
are converted into spline curves and implemented directly to express the nonlinear
characteristics of the Hydropneumatic Suspension Units. The torsion bars are
mounted on the middle stations for this vehicle model. Since the torsion bar has a
linear stiffness force only, a simple torsional spring model is used in this
investigation to represent the stiffness of the torsional bars. The hydraulic passive
tension adjustor is installed on the idler to maintain a proper track tension of the
tracked vehicle model. The equivalent spring-damper force model from analytical
method of incompressible fluid is employed for passive tensioning system.
In general this type of vehicle can be divided four subsystems for overall motion
analysis of vehicle dynamics. These subsystems are two track subsystem with
suspension units, main body subsystem with power pack, and turret subsystem with
main gun. The each right and left track subsystems is composed of rubber bushed
track link, double sprockets with single retainer, seven road wheels and arms, and
three upper rollers. The sprockets, road arms, road wheels, upper rollers
and turrets are mounted on main body by revolute joints which allow single
degrees of freedom. Total 38 revolute joints are used for the vehicle modeling and
generate 190 nonlinear algebraic constraint equations. Two busing force elements
to connect each track links and total 304 bushing forces elements for both track
systems are used in this investigation. The modeled vehicle has 191 rigid bodies
and 956 degrees of freedom. The threedimensional model, which is shown in Fig. 1,
represents the third generation of a military vehicle weighing approximately 50
tons and can be driven at a speed
higher than 60 km/hr.
255
(1) SUSPENSION UNIT
The suspension unit includes a Hydro-pneumatic Suspension Unit (HSU), and
torsion bar that are modeled as force elements whose compliance characteristics are
evaluated using analytical and empirical methods. The HSU systems are mounted
on front and rear stations to damp out pitching motion and to decrease the vehicle
speed when the vehicle is running over large obstacles.
Figure 1. Computer graphics of high speed tracked vehicle model
(a) Hydro-pneumatic suspension unit
256
(b) Torsion bar systems
Figure 2. Schematic diagram of spring damper suspension units
The spring torque of the HSU systems can be written as
1PALTHSU
where P is the gas pressure, A is the area of piston, and 1L is the distance shown
in Fig. 2. The pressure P in the gas chamber of HSU system with respect to
rotation angle of a road arm is defined as
22/ LLllPP isii
where iP , il and iL2 are the initial pressure and distances when the road arm is in
its initial configuration, γ is a constant which is equal to 1.4, and 2l is the distance
shown in Fig. 2. The distance sl can be adjusted by charging or discharging oil into
the oil chamber. The torsion bars are mounted on the middle stations for this
vehicle model. A simple torsional spring model is used in this investigation to
represent the stiffness of the torsional bars. The stiffness coefficient of the torsion
bar spring is approximately 5×104 N-m/rad.
Figure 2 shows the schematic diagram of the HSU and the torsion bar systems.
2.2.2.2 Equations of Motion
In this investigation, the relative generalized coordinates are employed in order to
reduce the number of equations of motion and to avoid the difficulty associated
with the solution of differential and algebraic equations. Since the track chains
(1)
(2)
257
interact with the chassis components through contact forces and adjacent track
links are connected by compliant force elements, each track chain link in the track
chain has six degrees of freedom which are represented by three translational
coordinates and three Euler angles. Recursive kinematic equations of tracked
vehicles were presented by [6] and the equations of motion of the chassis are given
as follows :
)( r
i
Tr
i
T qBMQBqMBB
where r
iq and B are relative independent coordinates, velocity transformation
matrix, and M is the mass matrix, and Q is the generalized external and internal
force vector of the chassis subsystem, respectively. Since there is no kinematic
coupling between the chassis subsystem and track subsystem, the equations of
motion of the track subsystem can be written simply as
ttt QqM
where tM ,
tq and tQ denote the mass matrix, the generalized coordinate and
force vectors for the track subsystem, respectively. Consequently, the accelerations
of the chassis and the track links can be obtained by solving Eqs. (3) and (4). The
G-Alpha integration algorithm is used to find the accelerations of Eqs. (3) and (4).
[5]
2.2.2.3. Experimental Measurements
The experimental measurement has been performed for two objectives in this
investigation. One is to construct the reliable virtual tracked vehicle simulation
model for the sake of efficient development of the vehicle system without having
super-expensive real prototypes at early design stage. It can reduce the failure cases
significantly when the physical prototypes are constructed. Second is to collect the
data base for simulation and correlation. The positions, velocities, accelerations,
forces are measured with respect to time. The real time data acquisition methods
are used for chassis system measurements by storing analog and digital signals
from the sensors. The measured data stored in memory are down loaded into
portable computer for post-processing. The tracked vehicle is tested on the various
heavy vehicle proving grounds with different speed and conditions. The pitch angle
of the center of the chassis when the vehicle runs on profile IV cross country
course with constant velocity 15km/h are shown in Fig. 3. A mechanical gimbal
type of gyro sensor is used to measure the pitch angle with 65Hz cut off frequency.
Figure 4 shows FFT result of the vertical acceleration. The track vehicle runs on
(3)
(4)
258
flat ground with 10 km/h forward velocity. The result shows exact match of track
polygon excitation between experimental and numerical correlation. [6]
Figure 3. Pitch angle of the center of chassis on profile IV cross country course
Figure 4. FFT of the vertical acceleration of the center of the chassis
2.2.3. META-MODEL BASED OPTIMIZATION
2.2.3.1 SIMULATANEOUS KRIGING MODEL SIMULATANEOUS
Meta-Models such as RSM, Kriging and Radial Basis Function (RBF) are
increasingly used to approximate expensive responses in engineering fields. RSM
was introduced in the classical DOE, which used a polynomial type regression
model. Hence, it required the rotatable characteristics for sampling points such as
CCD and SCD. However, Kriging[7] and radial basis function[8] are Bayesian
259
models. Hence, they used a space filled sampling points such as Latin hypercube or
descriptive designs[9].
Kriging models can be defined as a combination of a regression model plus a
departure term:
y z Xβ x ,
where y is the approximate model, Xβ is a polynomial type regression model,
and z x is a Gaussian random process with 2
0,N . If the regression model Xβ
globally approximates the design space, the departure term z x represents the
localized deviations so that the Kriging model interpolates the s
n sampled points.
In my experience, the regression model plays an important role in design
optimization especially for insufficient sampling points. The covariance matrix of
z x is given by
2,
i j i jCov z z Rx x R x x ,
where R is the correlation matrix and ,i j
R x x is the correlation function
between any two of the sn sampled points. Hence, R is a s s
n n symmetric
matrix with ones in the diagonal term. There are many correlation functions
,i j
R x x . Among them, the Gaussian type is widely used
2
1
, expk
l l
i j l i j
l
R
x x x x ,
where l
are the unknown correlation parameters to fit model. The estimates,
y x of the response y x at the untried values of x are given by
1Ty
x X x β r x R y X x β .
The correlation vector between x and the sampled points 1 2, ,...,
snx x x is given
by:
1 2, , , ,..., ,
s
TT
nR R R r x x x x x x x
(5)
(6)
(7)
(8)
(9)
260
In the estimates, the unknown coefficients of regression model is determined as
1
1 1T T
β X R X X R y .
Also, in order to determine the unknown correlation parameters l
, the estimate of
the variance 2 (not the variance in the observed data) is introduced. Hence, the
correction parameters l
is determined by solving
1
0
min detsn
θ
R θ θ
While any values for θ create an interpolation model, the best kriging model is
found by solving the k-dimensional unconstrained optimization problems described
in the above.
From the viewpoint of numerical optimization, equation (11) can be non-smooth
because the correlation matrix R θ is frequently singular during optimization
process. Hence, some special techniques are required to avoid the singular
phenomena and non-linearity of it. Hence, we use a singular value decomposition
(SVD) and normalization and scaling techniques. Also, multi-objective formulation
is introduced in equation (11) to solve the multiple kriging models simultaneously.
This approach uses only one correlation matrix R θ even for constructing
multiple kriging models[9].
2.2.3.2 ROBUST OPTIMIZATION FORMULATION
Lets’ consider the general optimization formulation for robust design.
Fundamentally, all the functions are composed of meta-models.
Minimize 0 0f k x x x
subject to 0, 1, 2,...,i
h i l x
0, 1, 2,...,j j j
g k j m x x
x
where , 0,1,...,j
j m x are the standard deviation of f and j
g that evaluated
from meta-models. The value of and i
k are alpha weight and robust index,
respectively. If 0 and 0
1k , the design objective is a minimization of the
(10)
(11)
(12)
(13)
(14)
(15)
261
variance of f x . If 6j
k is defined, the inequality constraints become DFSS
constraints. If multiple objectives are given, the objective function of equation (12)
is replaced by a preference function as
Minimize , 1, 2,...k obj
P k N x
In order to represent our preference function, let’s consider following two
objectives.
1
minx
x and 2
maxx
x
There are many preference functions in multi-objective optimization strategy.
Among them, we uses the following two types.
1 2
1 2
1 2
1 1G G
P w w
x xx
1 2
1 2
1 2
max 1 , 1G G
P w w
x xx
where, the values of i
w are the user defined weighting coefficients and the
relaxation factors and are automatically determined. Also, the ideal solution G
if is internally determined.
2.2.3.3 NUMERICAL OPTIMIZATION PROCESS
The approximate optimization problems, based on meta-modes, are sequentially
solved with the augmented Lagrange multiplier method[10]. In order to avoid the
convergence difficulty for the insufficient sampling points, the initial design is
selected from the best points in the given DOE and move limit is automatically
adjusted. Also, the polynomial types are automatically switched to the degree of
convergence of optimization process.
In the first iteration, the sequential approximate optimization (SAO) process
requires the sampling points. We provide a discrete Latin hypercube design,
incomplete small composite design-I (ISCD-I), incomplete small composite design-
II (ISCD-II), generalized small composite design (GSCD) and other classical DOE
methods such as orthogonal arrays, CCD and BBD etc.[9,11].
(16)
(17)
(18)
(19)
262
In the subsequent iteration of SAO, a new optimal design is given. Next, exact
analysis is done for this point. Then, this new information is added to the design
database. Hence, meta-model is newly developed and repeat these processes until
the convergence criterion is satisfied for their tolerance values.
Figure 5. A tracked vehicle model in RecurDyn/Track HM
2.2.4. ROBUST OPTIMIZATION OF A TRACKED VEHICLE
SYSTEM
2.2.4.1 SYSTEM ANALYSIS
Figure 5 shows a high mobility tracked vehicle model of military tank in the
RecurDyn/Track HM, which consists of a chassis and two track systems. The
chassis includes a chassis, sprockets, road wheel, road arm, Idler & tensioner and
the suspension system. The one-side suspension system includes three hydro-
pneumatic suspension uints(HSU) and three torsion bars. The HSU is installed in
the 1st, 2
nd and 6
th road wheels. And the Torsion Bar is installed in the 3
rd, 4
th and 5
th
Road Wheel. Then, this high mobility tracked vehicle goes through the 10 inch
25.4cm bump as a velocity of 40 km h .
This tracked vehicle model consists of total 189 bodies; 37 bodies for the chassis
such as sprocket, road wheel, road arm, etc., 76 track link bodies for each track
subsystem, 36 revolute joints and 152 bushing elements. Therefore, it has 954
degree of freedom.
263
2.2.4.2 RANDOM DESIGN VARIABLE SELCTION
Figure 6 Random Design variables
Figure 6 shows the 11 random design variables. The 1st through 3rd design
variables are tensioner stiffness, damping coefficient and the tensioner free length
in the idler & tensioner part. The 4th through 7th design variables are 1st, 2nd, 6th
HSU stiffness scale factor in the HSU part. The 8th and 9th design variables are
torsion bar stiffness, damping coefficient. The 10th and 11th design variables are
3rd & 5th torsion bar free angle and 4th torsion bar free angle. All the design
variables are regarded as random variable. All of design variables have 1%
coefficient of variance (COV), which represents that the design variables are the
mean values and their deviations are 1% of them. As design variables are changed,
the magnitudes of their deviations will be simultaneously changed with their COV.
2.2.4.3 DESIGN FORMULATION
Now, in order to enhance the comfortable to ride and vehicle performance of high
mobility tracked vehicle, when the vehicle goes through the 25.4 cm hemi-cylinder
type bump as a velocity of 40 km h , the magnitude of the maximum vertical
acceleration ,CG
z tb and it’s standard deviation
0,
max ,CG
t T
t
z b should be
simultaneously minimized, while satisfy the wheel travels of three torsion bars
3 4 5, , and front road wheel 6 within 1 ranges. Hence, all performance
indexes are selected as the maximum value when the vehicle goes through the
bump.
Minimize
0, 0,
max , max ,CG CG
t T t T
t t
b z b z b
264
subject to
0, 0,
,max max , , 3, 4,..., 6a
it T t T
i it t i
b b
and
, 1, 2,...,11L U
k k k kb b b k ,
where the deviation of each random design variable is based on 1 % coefficient of
variance (COV). Hence they are evaluated as 0.01k k
b in the design process. In
the practical implementation, a lower-pass filter is used to remove the numerical
noise in the time dependant responses. Hence, all the performance indexes, used in
the above formulation, are evaluated from the filtered results.
2.2.4.4 OPTIMIZATION RESULTS
In this study, simultaneous Kriging models combined with pure quadratic
polynomials are employed for constructing meta-models. First, the meta-models
are constructed from only 12 points that have the current design plus 11 sampling
points are selected from discrete Latin hypercube method. SAO process requires
only 14 iterations until satisfying the convergence tolerances. The convergence
criteria are selected as the relative change of objectives between consecutive
iterations and the maximum violation of constraints. Their convergence tolerances
are selected as 0.05 and 0.01, respectively.
In order to validate the inequality constraints including robust concept, we check
the constraint violation by using the sampled variance. To do this, 12 points are
sampled from a Latin hypercube method in the neighborhood of the final design
( *b ). The sampled range is the same random variable deviation ( * *
0.01 b b ). The
sampled standard deviation is evaluated from the final design and additional 12
values. Table 1 lists the approximate and the sampled standard deviations for the
wheel travel constraints. The approximate standard deviation values are evaluated
at *b by the Taylor series from meta-models.
Table 1. Comparisons of the approximate and the sampled standard deviations
Approximate
Values
Sampled
Values s
3
0.01546 0.02183
265
4 0.01951 0.06536
5
0.01587 0.01873
6
0.01684 0.03302
Although the approximate standard deviation is used during the sequential
approximate optimization process, when the final convergence checking, this
approach used s in place of . If this final checking is not satisfied, this
approach optionally restarts the optimization process with these additional
sampling points that are used to evaluate the sampled variance. However, this
optimization result can successfully satisfy the final convergence check even
though the approximate values are slightly different from the sampled ones.
Figure 7 compares the acceleration of mass center for the initial and the final
designs. Figure 8 shows the wheel travel results for the final design.
Figure 7. Comparison of the acceleration of mass center for the initial and the final designs
2.2.5. CONCLUDING REMARKS
This study introduces the meta-model based design strategy for dynamic response
optimization. This can avoid the difficulty of design sensitivity analysis, especially
when a lower-pass filter is employed. Also, it shows that the robust design concept
can be easily implemented by using the approximate variance from meta-model. In
the numerical test, it successfully solved for the tracked vehicle system only for 26
analyses including the initial samplings. Finally, the robust optimization results are
validated by the sampled variance.
266
Figure 8. Comparison of wheel travels 6 between the initial and the final designs
ACKNOWLEDGEMENT
Some of researchers of this research got financial support of the second stage of
BK21 program.
REFERENCES
1. McCullough, M.C. and Haug, E.J., Dynamics of High Mobility Tracked
Vehicles, ASME, Journal of Mechanisms Transmissions, and Automation
in Design, Vol. 108, pp. 189-196, 1996
2. Nakanishi,T. and Shabana,A.A., Contact forces in the nonlinear dynamic
analysis of tracked vehicles, International Journal for Numerical Method in
Engineering, Vol 37, pp. 1251-1275, 1994
3. Choi,J.H. Lee, H.C. and Shabana, A.A., Spatial dynamics of multibody
tracked vehicles: Spatial Equation of Motion, International Journal of
vehicle mechanics and mobility, Vehicle System Dynamics, Vol. 29, pp.
27-49, 1998
4. Lee, H.C., Choi, J.H. and Shabana, A.A., Spatial dynamics of multibody
tracked vehicles: Contact Forces and Simulation Results, International
Journal of Vehicle Mechanics and Mobility, Vehicle System Dynamics, Vol.
29, pp. 113-137, 1998.
5. Ryu, H.S., Bae, D.S., Choi, J.H. and Shabana, A.A., A Compliant Track
Model for High Speed, High Mobility Tracked Vehicle, International
267
Journal for Numerical Method in Engineering, Vol. 48, pp. 1481-1502,
2000.
6. Ryu, H.S., Choi, J.H. and Bae, D.S., Dynamic Modeling and Experiment
of Military Tracked Vehicle, Paper No. 2006-01-0929, SP-2040, SAE,
2006.
7. Farhang-Mehr, A. and Azarm, S., Bayesian meta-modeling of engineering
design simulations: A sequential approach with adaptation to irregularities
in the response behaviour, Int. J. Numer. Meth. Engng., Vol. 62, pp.
2104~2126, 2005.
8. Wang, J.G. and Liu, G.R., A point interpolation meshless method based on
radial basis functions, Int. J. Numer. Meth. Engng., Vol. 54, pp. 1623~1648,
2002
9. Kim, M.-S., RecurDyn/AutoDesign: Meta-Model based design optimizer,
Theoretical Manual, FunctionBay, 2006.
10. Kim M.-S and Choi, D.-H., Min-max dynamic response optimization of
mechanical systems using approximate augmented Lagrangian, Int. J.
Numer. Meth. Engng., Vol 43, pp. 549~564, 1998.
11. Kim, M.-S. and Heo, S.-J., Conservative quadratic RSM combined with
incomplete small composite design and conservative least squares fitting,
KSME International Journal, Vol. 17, No. 5, pp. 698~702, 2003
268
2.3.
EFFICIENT DESIGN OPTIMIZATION
TOOL FOR INTERDISCIPLINARY
ANALYSIS SYSTEM
2.3.1. INTRODUCTION
In the CAE field, engineers want to analyze more realistic model. Then they
require the interdisciplinary system analysis, like as the multi body dynamic system
include flexible bodies or control system. The CAE softwares are going to be
developed by this kind of requests of engineers. Some kind of software satisfies
these users‟ request. But now, the engineers have more requests for design. That is
that they want to validate their design.
Industrial design requires the validation of multi-body dynamics, non-linear FE
analysis, and test etc. However, the conventional DSA(Design Sensitivity Analysis)
based approaches cannot be used in these disciplinary. Multi-body dynamics and
non-linear FE analysis is very complicated. Thus, their design sensitivity analysis
process can be very tedious and difficult. This represents that analytical design
sensitivity is impossible in these areas. Then the meta-model based optimization is
required in the industrial design process. However, this approach requires a
difficult integration of DOE, meta-modeling methods and numerical optimization
process. And more and more, end users of optimization tools require an easy-of-use
and powerful design environment.
From these kinds of requests, the efficient and easy design optimization tool for
interdisciplinary analysis system, AutoDesign is developed in the context of inter-
disciplinary analysis software RecurDyn. In this paper, an easy-of-use and
powerful user interface and application example of AutoDesign are explained.
Chapter 2 explains the basic requirements for good design tools from the viewpoint
of meta-model based optimization. Also, it explains the basic guidelines for good
application of optimization tools. Chapters 3 through 6 explain the basic functions
of AutoDesign. Chapter 3 explains the procedure of design optimization
formulation. Chapter 4 describes the effect analysis and design variables screening.
Chapter 5 explains the meta-model based optimization. It explains the deterministic
optimization, the robust design optimization and the 6-sigma optimization. Chapter
6 shows the case studies of AutoDesign. Finally, Chapter 7 summarizes this paper.
269
2.3.2 APPLICATIONS OF DESIGN OPTIMIZATION
2.3.2.1 BASIC REQUIREMENTS FOR GOOD DESIGN OPTIMIZATION TOOLS
Figure 1 shows three major steps for the meta-model based optimization such as
DOE, meta-model construction and numerical optimization. However, it is noted
that these three techniques are independently developed for different purpose. Thus,
their integration is very difficult. In this study, we think that good design
optimization tools are simple and easy but robust. Thus, from the viewpoint of
meta-model based optimization, we think that following three items are basic
requirements for good design tools:
Figure 1.THREE STEPS FOR META-MODEL BASED SAO
-DOE step should minimize the number of experiment or analyses. To do this,
various DOE methods are considered and developed. But engineer is confused for
selecting DOE method in optimization process. Then easy and automatic DOE
method selection environment is needed.
-Meta-modeling step should overcome the lack of fit due to insufficient data. Even
though nearly duplicated design points are encountered during SAO, it should
overcome the singular.
-Optimization process should give the feasible optimum, even though the given
meta-models are highly nonlinear. Also, it does not require mathematical decision
270
such as design variable scaling, constraint normalization, multi-objective
formulation, optimization algorithm selection etc. These serious selections make
most of CAE engineers who are not familiar to optimization theory to be perplexed.
2.2.2.2. GUIDELINES FOR DESIGN OPTIMIZATION APPLICATION
Let‟s consider the procedure for good application of design optimization tools.
Following guidelines are summarized from the experience of practical design
applications and educations.
Step 0: Validate the CAE models from test results. If the analysis results of CAE
do not meet those of test, one should tune the analysis parameters by their
experience or optimization tools. Otherwise, go to Step 1.
Step 1: If ones fully understand the physical concepts for the systems for design,
then go to Step 2. Otherwise, ones select many variables or parameters as design
Figure 2. THE DESIGN PARAMETER LIST
variables as possible as. Also, they select many analysis responses as the
performance indexes as possible as. Then, perform the effect analysis. Effect
analysis gives them the correlation among design variables and performance
indexes. Also, it provides the design variable screening function, which can
automatically reduce the number of design variables. Then, go to Step 3.
Step 2: Ones select the major variables or parameters as design variables. Also,
271
they select the major analysis responses as the performance indexes
Step 3: For the selected performance indexes, one select the objectives to be
minimized or maximized and define the constraints to be less than or greater than.
Also, one selects the meta-model types.
Step 4: After defining the convergence tolerances and maximum iteration, ones
push the run button.
2.3.3. DESIGN OPTIMIZATION FORMULATION
In general process of optimization, the first step is to define optimization problem,
such as to define design variables and objective functions. In the dynamic field,
mass of rigid body, stiffness of spring-damper, etc can be defined as the design
variable. And the gain value of control algorithm or young‟s modulus of flexible
body, etc can be, too. For defining object function, it is needed to define some kind
of analysis result value as analysis responses, like as velocity of body, force of
spring-damper, stress of node of flexible body, etc. In optimization process,
analysis responses are selected and redefined as objective functions.
2.2.3.1 DESIGN VARIABLE DEFINITION
In design study and design optimization processes, design variables are
automatically varied and applied to
Figure 3. DEFINE DESIGN PARAMETER
272
Figure 4. THE ANALYSIS RESPONSE LIST
Figure 5. DEFINE ANALYSIS RESPLNSE
make models. In developed user interface of optimization tool, design parameters
are needed to define design variables. The checked design parameter like as Figure
2 is applied as design variable in design study and optimization process.
Design parameters are defined using parametric values in Figure 3, which are
linked with the values of analyzed model such as mass of body, stiffness of force
element, gain value of control algorithm, etc. Especially shape relationship of
nodes of flexible body can be easily defined as design variable using the developed
tool and RecurDyn functions.
2.2.3.2. Objective Function Definition
For defining objective functions, analysis responses are first defined using
function expressions, which define some values of analysis results like as Figure 5.
Max, Min, Average, Min/Max ABS, RMS, End values can be selected
273
Figure 6. THE DESIGN VARIABLES
Figure 7. THE EFFECT ANALYSIS PAGE
as option for analysis response treatment.
The checked analysis responses like as Figure 4 is applied as performance indices
in design studyand optimization process. In optimization problem, performance
index is defined as objective function or constraint condition.
274
2.3.4. EFFECT ANALYSIS AND SCREEENING Sometime, the optimization process can be useless because the selected design
variable cannot affect to the selected performance index. Then engineer wants to
know sensitivity of design variable before optimization process. From this kind of
request, a design study tool is developed.
In the design study tool of the Figure 6, the trial number of design study process is
automatically calculated from the number and level of design variables and
selected DOE(Design Of Experiment) method. The supported DOE method options
are six types:
Extended Plackett-Burman design
Full factorial design
Three-level orthogonal design
Level balanced descriptive design
Two-level orthogonal design
Bose‟s orthogonal design
Figure 8. THE EFFECTIVE VALUE PLOT
275
Figure 9. THE SCREENING VARIABLES
In the effect analysis page of the design study tool in Figure 7, engineer can easily
draw the relationship plot like as Figure 8 between design variables and
performance indices, and they can consider and decide the design variable include
on the optimization process.
In the screening variable page of the design study tool in Figure 9, engineer can
easily turn on and off design variables and update model.
2.3.5. META-MODEL BASED OPTIMIZATION
2.3.5.1. META-MODEL CONSTRUCTION
In the DOE & Meta modeling methods page in the Figure 12, engineer can easily
define DOE methods and meta modeling methods from just selecting options. If
users want to use previous analysis results, user can import data from the „Get from
simulation history‟ option. From
276
Figure 10. THE DESIGN VARIABLE PAGE
Figure 11. THE PERFORMANCE INDEX PAGE
Figure 12. THE DOE & META MODELING METHODS PAGE
277
selected DOE method, the number of trial is automatically calculated by the
selected DOE method. And optimization process is automatically executed by
using the defined meta model and polynomial functions.
AutoDesign provides four-types meta-model techniques.
-Simultaneous Kriging
- Radial Basis Function (Gaussian)
- Radial Bssis Function (Multi-quadratics)
- Conservative Response Surface Model
Unlike the conventional Kriging, the simultaneous Kriging requires only one time
solving for multiple performance. Also, it provides special DOE methods for meta-
modeling.
For small scaled design:
-Generalized small composite design
-Central composite design
- Box and Behaken Design
For large scaled design:
-Discrete Latin-hypercube design
-Incomplete small composite design-1
Incomplete small composite design-2
For the detailed information for those DOE and meta-model methods, one may
refer to the theoretical manual for AutoDesign.
2.5.5.2. DETERMINISTIC OPTIMIZATION
To define optimization problem, first step is to define design variables and
objective functions. And the next is to define DOE method and parameters for
meta-model method. After optimization process, engineer need some tool for
understand optimization process. The each page of deterministic optimization tool
is considered in the purpose.
In the design variable page in the Figure 10, engineer can define the type of design
variables. Default type of design variable is „Variable‟. When some design
variables are satisfied design goal but need more optimization process, engineer
can change the type of the design variable as „Constant‟ and execute another
optimization process.
In the performance index page in the Figure 11, engineer can define objective
functions or constraint conditions of optimization problem. For defining objective
278
function, user can easily define the function from just selecting the option of „Goal‟
as „MIN‟ or „MAX‟. For defining constraint conditions, users just select the
„Constraint‟ type, „Goal‟ type, and input the Limit value.
In the result sheet page in the Figure 13 engineer can check the convergence history,
the values of performance indices and violations
2.5.5.3. ROBUST DESIGN AND DFSS(DESIGN FOR SIX SIGMA) OPTIMIZATION
For robust design, statistic information about design variables is need for defining
optimization problem. From that purpose, the dialogues are upgraded from
deterministic optimization tool like as Figure 14, 15. Engineer can just select and
input the statistic option and information in dialogue.
The definition page for DOE and meta modeling methods is same as the page of
the deterministic optimization tool. The DFSS/Robust Design Control option is
added on the optimization control page of the deterministic optimization. The result
sheet page in Figure 16 is upgraded, too.
Figure 13. THE RESULT SHEET PAGE OF DETERMINISTIC OPT
279
.
Figure 14. THE DESIGN VARIABLE PAGE
Figure 15. THE PERFORMANCE INDEX PAGE
Figure 16. THE RESULT SHEET PAGE OF ROBUST DESIGN AND DFSS
280
OPTIMIZATION
Figure 17. . THE CATAPULT MECHANISM
281
Figure 18. . CONVERGENCE HISTORY FOR CATAPULT DESIGN
2.3.6. CASE STUDIES
The applied examples show the efficient and easy to use user interface. First
example is the catapult mechanism. It shows the function of the design study. The
second example is the landing gear mechanism. The control mechanism is included
in that system. The third is paper-feeding mechanism that includes a flexible body
and control mechanism.
2.3.6.1 Catapult Mechanism
Catapult Mechanism throws the ball to the target. It has contact element between
ball and race surface. The
282
Figure 19. . THE LANDING GEAR MECHNISM
Figure 20. . THE CONTROL ALGORITHIM OF LANDING GEAR MECHNISM
objective is the ball pass through the target center. Two design variables are
defined. The first is the angle of the link body that is linked between race body and
283
ground. The second is the spring mount position. Performance indices are position
deviations.
Figure 18 shows convergence history. Even though the analysis results are
nonlinear, AutoDesign gives a good optimization results.
2.3.6.2. Landing Gear Mechanism
Application example landing gear mechanism has a PID control mechanism. That
control algorithm is defined in CoLink, which is the tool of RecurDyn. The design
variables are the gain values of the control algorithm. Objective of optimization is
the gear is arrived to bay faster and safety.
Figure 21 shows the distance between tire center and target position. We try to
solve this problem three times. SAO_3 gives best result, which include the equality
constraint that the distance at the end time = 0.
284
2.3.6.3 PAPER-FEEDING MECHANISM
Paper has the flexible body characteristic. Then that is modeled as flexible element
and analyzed. Two PID controllers are applied on the rollers to control the paper
speed and paper skew angle. This problem is an interdisciplinary system problem.
Paper is orthotropic finite element, and roller mechanism is multi-body dynamics.
The design variables are 6 gains and weight values in the angle control algorithm
and the objective is to minimize the skew angle while maintaining a constant paper
speed.
The Figure 21 compares the initial and the final design. Optimal gains successfully
control the paper feeding system.
2.3.7. CONCLUDING REMARKS
This paper presented the efficient design optimization tool for interdisciplinary
analysis system. Each process of design optimization for the tool is clearly
explained. The developed tool solves three design problems including multi-
physics design. The optimization results show that the proposed tool is efficient
and useful.
Figure 22. . THE PAPER-FEEDING MECHANIM
285
Figure 23. . THE CONTROL ALGORITHIM OF PAPER-FEEDING MECHANIM
Figure 24. THE OPTIMIZATION RESULT
286
REFERENCES
1. Addelman, S. and Kempthorne, O. 1961, “Some Main Effects Plans and
Orthogonal Arrays of Strength Two”, Ann. Math. Statist., Vol. 32, pp.
1167~1176.
2. Barthelemy, J.-F., “Function Approximation”,(eds Kamat, M.P, Structural
Optimization: Status and Promise), Progress in Astronautics and
Aeronautics, AIAA,Vol 150, 1992, pp. 51-66
3. Box, C.E.P. and Hunter, W.G., 1957, “Multi-factor Experimental Designs
for Exploring Response Surfaces”,
4. Figure 23. . THE CONTROL ALGORITHIM OF PAPER-FEEDING
MECHANIM
5. Figure 24. THE OPTIMIZATION RESULT
6. Annals of Mathematical Statistics, Vol. 28, pp. 195~241.
7. Draper, N.R., 1985, “Small Composite Designs”, Technometrics, Vol. 27,
No. 2, pp. 173~180.
8. Draper, N.R. and Lin, D.K., 1990, “Small Response Surface Design”,
Technometrics, Vol. 32, No. 2, pp.
9. Annals of Mathematical Statistics, Vol. 28, pp. 195~241.
10. Draper, N.R., 1985, “Small Composite Designs”, Technometrics, Vol. 27,
No. 2, pp. 173~180.
11. Draper, N.R. and Lin, D.K., 1990, “Small Response Surface Design”,
Technometrics, Vol. 32, No. 2, pp.
12. 187~194.
13. Eduardo, S., 1997, “Descriptive Sampling: An Improvement over Latin
Hypercube Sampling”, Proceeding of the 1997 Winter Simulation
Conference, Andradottir, S., Healy, K.J., Withers, D.H. and Nelson B.L.
(eds.)
14. John, P.W.M., Statistical Design and Analysis of Experiments, 1998,
SIAM, Philadelphia
15. Haftka, R.T., Scott, E.P. and Cruz, J.R., “Optimization and Experiments: A
Survey”, Applied Mechanics Review, Vol. 51, No. 7, 1998, pp. 435-448.
16. Kim M.-S, RD/AutoDesign-Part1:Theoretical Manual, FunctionBay, Inc.
2007
17. Kim,M.-S and Heo S.-J., 2003, “Conservative Quadratic RSM combined
with Incomplete Small Composite Design and Conservative Least Squares
Fitting”, KSME International Journal, Vol. 17, No. 5, pp. 698~707.
18. Kim,M.-S., Choi,D.-H. and Hwang, Y. “Composite Nonsmooth
Optimization Using Approximate Generalized Gradient Vectors”, Journal
of Optimization Theory and Applications, Vol. 112, pp. 145-165, January
2002.
287
19. Kim,M.-S., Choi, J.H., Lee, S.G., Song, I.H., Yoon, J.S. and Baek, S.H.
“Robust Design Optimization of a Tracked Vehicle System” ASME 2007
International Design Engineering Technical Conferences & Computers
and Information in Engineering Conference, DETC2007-34713,
September 4-7, 2007.
20. Matheron G, Principles of geostatistics, Economic Geology 1963;
58:1246-1266.
21. Mckay, M.D., Conover, W.J., and Beckman, R.J., “A Comparison of Three
Methods for Selecting Values of Input Variables in the Analysis of Output
From a Computer Code”, Technometrics, Vol. 21, pp. 239-245.
22. Michael, S., 1987, “Large Sample Properties of Simulations using Latin
Hypercube Sampling”, Technometrics, Vol. 29, No.2, pp.143 ~ 151
23. Myers, R.H., and Montgomery, D.C. Response Surface Methodology, 1995,
Wiley & Sons
24. Sacks J, Susannah SB, Welch WJ. Design for computer experiments.
Technometrics 1989; 31:41-47
25. Sacks, J., Welcj, W.J., Mitchell, T.J. and Wynn, H.P., “Design and
Analysis of Computer Experiments”, Statistical Science, Vol. 4, No.4, 1989,
pp. 409-435.
26. Saliby, E., “Descriptive Sampling: An Improvement over Latin Hypercube
Sampling”, Proceedings of the 1997 Winter Simulation Conference (eds. S.
Andradottir, K.J. Healy, D.H. Withers, and B.L. Nelson), pp. 230-233.
27. Simpson, T.W., Peplinski, J.D., Koch, P.N. and Allen, J.K., “Metamodels
for Computer-Based Engineering Design: Survey and Recommendations”,
Engineering with Computers, Vol. 17, 2001, pp. 129-150.
28. Stein, M., “Large Sample Properties of Simulations Using Latin
Hypercube Sampling”, Technometrics, Vol. 29, No.2, 1987, pp. 143-151.
29. Westlake,W.J., 1965, “Composite Design based on Irregular Fractions of
Factorials”, Biometrics, Vol. 21, pp. 324~336.
288
2.4.
EFFICIENT OPTIMIZATION METHOD
FOR NOISY RESPONSES OF
MECHANICAL SYSTEMS
2.4.1. INTRODUCTION
As non-linear analyses and experiments are frequently encountered in modern
engineering optimizations, sequential approximate optimizations (SAOs) combined
with meta models such as response surface model (RSM), radial basis function, and
Kriging have gained in popularity [1–3]. Thus, in constructing the meta model, it is
important to achieve an acceptable level of accuracy while attempting to minimize
the computational effort, i.e. the number of system analyses or experiments.
Although increasing the number of
experimental points could improve the accuracy of the approximate model, many
studies have concentrated on reducing the number of experimental points that
represent expensive analyses and experiments [4–12].
Among them, the small composite design is one of the minimum designs for
constructing the second-order response surfaces [9, 10]. From the viewpoint of
filling the design space, the Latin hypercube design is widely used for constructing
Kriging [13, 14].
However, the SAO, combined with meta models, requires many analyses and
experiments. The reason is that the SAO remains an iterative process until some
convergence criteria are satisfied. Thus, sequentially updated meta models are
required during SAO iterations.
In practical design, the noisy analysis results prevent the SAO from obtaining a
feasible design, even though many sampling points are employed for constructing
meta models. To improve the feasibility of RSMs during SAO, this study proposes
the conservative
response surface models (CRSMs) and incomplete small composite designs
(ISCDs).
Section 2 presents the basic idea of the proposed ISCDs. Section 3 explains the
numerical procedure for the CRSM. Section 4 presents the computational
procedure of the SAO combined with the CRSM and the ISCDs. In section 5, the
proposed approach is numerically
289
examined by solving the well-known gear reducer design problem and the tracked
vehicle suspension system. The former is a typical test problem for numerical
optimization codes and the latter is a very noisy problem. Finally, section 6
summarizes this study.
2.4.2 INCOMPLETE SMALL COMPOSITE DESIGN
In the original SCD, it is difficult to select the minimum columns of a Plackett–
Burman design for the cube portion to avoid singularity or near singularity while
removing one of each set of duplicates if duplicate runs exist, because it is also an
optimization problem.
Therefore, Draper [9, 10] proposed only three designs assessed on the three cases,
such as k = 5, 7, and 9. ISCD fundamentally uses 2k axial runs plus centre runs to
represent curvatures of the system, and allows for efficient estimation of the pure
quadratic terms. However, for constructing the cube portion, although the Plackett–
Burman design is used, only the
minimum number of runs is directly used, which are available from the Plackett–
Burman design for the k factor. For a more detailed description, the minimum
number of runs to be performed in order to assess the factors under study is listed
inTable 1. In the previously
discussed various composite designs, the total number of points in cube and star
excluding centre points is summarized in Table 2. Figure 1 shows the proposed
ISCD for k = 3. Now, this ISCD is called ISCD-I. Then a more simplified ISCD-II
is proposed, which removes
2k axial points in the ISCD-I. It is composed of one centre point plus the
Plackett–Burman design for the given k value. Figure 2 shows the ISCD-II for k =
3.
This study suggests that the proposed ISCDs can be used only at the first iteration
in the SAO process.
After that the SAO gives an approximate optimum.
290
Table 1 Number of runs assessed on the number of factors in the Plackett–Burman design
The symbol ‘–’ denotes that the design is not provided by the developers.
Table 2 Total experimental points centre points in some SCDs
The symbol ‘–’ denotes that the design is not provided by the developers.
Fig. 1 The ISCD-I for k = 3, q = 1, and n0 = 1
291
During the second iteration, the exact function values are evaluated at this
approximate optimum. Then the approximate models are updated using the
information at the pre sampled ISCDs and this new point. The detailed numerical
procedure is explained in section 4.
2.4.3. CONSERVATIVE RESPONSE SURFACE MODEL
2.5.3.1. LEAST-SQUARE FITTING USING SINGLE VALUE DECOMPOSITION
To simplify the explanation of the construction of RSMs using ISCDs, the
following matrix notation is as considered
where y a vector of N observations, X a matrix of known constant, vector of
parameters to be estimated, and the vector of random errors. In estimating
the unknown constants, , by least squares, a set minimizes the
sum of the squares of the residuals from equation (1). This can be
simplified in the matrix form as
This is called the normal equation. The proposed ISCDs can give a rank deficient X
matrix because its number of points is . Hence, this study suggests a singular-
value decomposition (SVD) method [15] to solve the normal equation (equation
(2)).
292
2.4.3.2. CONSERVATIVE LEAST-SQUARE FITTING METHOD
By solving equation (2), the approximate observations are evaluated as .
Then the violated sets are selected, for the
overestimated approximate function and
for the underestimated approximate function , respectively. Hence, the
formulation of conservative least-square fitting finds to minimize
while satisfying
where the subscript a denotes that their components are included in the violated set
So or Su. Using the Wolfe dual method [16], equations (3) and (4) can be
transformed as
subject to
where is the Lagrange multiplier vector. Using equation (6) to eliminate
from the dual objective function of equation (5), the simplified problem as
max
where . In equation (7) the
constant terms are neglected. The optimum dual variables can be
obtained from equation (7). Hence, substituting into equation (6) gives the
explicit solution of the unknown coefficients for conservative fitting as
In these evaluations, is directly used from equation (2) and A−1 is
computed by SVD in the same way that evaluates . All these steps are
iteratively solved until the conservative condition ofequation (4) is satisfied.
293
2.4.3. CONSERVATIVE RSM APPLICATION IN THE DESIGN
OPTIMIZATION
Figure 3 shows graphical comparison of the conventional and CRSMs. The scatter
squares represent the sample responses. The conventional RSM pass through the
scatter squares, which can minimize the sum of squares of errors. The conservative
RSMs give an envelope curves for their purposes. When these approximated
responses are employed for design optimization, it is recommended that the
overestimated RSM for the less than type inequality constraints and the
underestimated RSM is done for the greater than type inequality constraints.
Suppose that the conventional RSMs are used for inequality constraints, it gives a
serious problem in the convergence of the SAO process. In other words, it cannot
guarantee the feasible region because the conventional RSMs pass through the
sample responses shown in Fig. 3. Even though real responses are violated for the
constraint limit at some design points, the conventional RSM may estimate them in
the feasible region. Hence it requires resampling points and the modified RSMs
many times until it satisfies the convergence tolerances. Practically, these can
retard the convergence of the sequential optimization process or it leads to the
failure of the convergence of the noisy problem. Thus, this study proposes the
CRSMs, which can reduce the number of analyses and improve the convergence of
the noisy problem.
Fig. 3 Comparison of conservative RSM and conventional RSM
294
2.4.4. SAO PROCEDURE
To use the CRSM and ISCDs in the SAO, the following computational procedure
is described.
Step 0: Given the design range and the convergence tolerances and , set
t = 0 and the design set whose number of sampling points is one centre point
plus the points in cube and star listed in Table 2 (ISCDs).
Step 1: Evaluate function values of objective and inequality constraint
functions , for the sampling points in D and storethem into
the sets and , respectively.
Step 2: Construct the approximate functions and
using the CRSM described in section 3.
Step 3: Solve the following approximate optimization problems: minimize
subject to and i for
be the approximate optimum.
Step 4: Evaluate the exact function values at the approximate optimum .
If the convergence and
satisfied for consecutive iterations, then the optimization is terminated. Otherwise,
see step 5.
Step 5: Update the design set Dt and function value sets and by including
and its corresponding function values. Then, return to step .
In step 2, the optimization process optionally uses linear, hybrid linear, and
quadratic polynomials to construct CRSM. In step 3, the approximate optimization
problem can be solved using any constrained optimization algorithms [16]. This
study uses the augmented Lagrange multiplier method [17].
295
2.4.5. NUMERICAL STUDIES
In this section, the numerical performance of the proposed CRSM with ISCDs is
evaluated. First, to show the accuracy of the proposed optimization approach, the
gear reducer design problem is solved [18, 19]. This is widely used to validate the
numerical optimization algorithm. Then, the conservativeness of CRSM is
evaluated by solving the tracked vehicle suspension design problem, i.e. the noisy
problem.
2.4.5.1. GEAR REDUCER DESIGN
The design of the gear reducer, shown in Fig. 4, is considered with the face width
(x1), module of teeth (x2), number of teeth of pinion (x3), length of shaft 1 between
bearings (x4), length of shaft 2 between bearings (x5), diameter of shaft 1 (x6) and
diameter of shaft 2 (x7) as design variables. The constraints include limitations on
the bending stress of gear teeth, contact stress, transverse deflections of shafts 1
and 2 due to transmitted force, and stresses in shafts 1 and
2 [14–20]. From references, the optimum solution was x∗ = {3.5, 0.7, 17.0, 7.3, 7.3,
3.35, 5.29}T, and its objective value was 2987.3, but the maximum violation of
constraints was 0.0574. This design was not feasible. Table 3 lists the side
constraints and the initial and final designs side-by-side. The proposed method
starts with ISCD-II. Sixteen SAO iterations are required until converged. Hence the
proposed method uses only 25 analyses. It is noted that the maximum value of
constraints is _ 0.0. The final design is feasible.
Fig. 4 Gear reducer system
296
Table 3 The optimization results of a gear reducer design
2.4.5.2. TRACKED VEHICLE SUSPENSION DESIGN
Figure 5(a) shows a tracked vehicle suspension system, which is designed to
minimize the extreme acceleration of the mass centre when the vehicle run over a
bump (0.36 m) as shown in Fig. 5(b) for a given speed (40 km/h). The tracked
vehicle model is composed of a hull, two sprockets, six wheels, and a track. The
suspension systems of the wheels have four hydraulic suspension unit (HSU)
systems and two torsion bars. Nine design variables are divided into three groups:
(a)the charging pressures for the HSU systems of the 1st, 2nd, 5th, and 6th wheels
are four design variables of x1, x2, x3, and x4; (b) the static track tension is the 5th
design variable, x5; and (c) the length of a gas chamber, the preload on Bellevile
springs, the diameter of orifices, and the choking flow rate for HSU systems are
design variables of x6, x7, x8, and x9, respectively. The motion of the vehicle is
constrained so that the maximum acceleration of mass centre, wheel travels, and
static wheel loads for the six wheels are within given limits. Moreover, the stiffness
of torsion bars for the Fig. 5 Tracked vehicle system (a) tracked vehicle model
(b) single bump 3rd and 4th wheels is within given limits. A commercial multi
body dynamics software, RecurDyn, is used to solve the dynamic analyses. Table 4
lists the initial and final designs side-by-side. The saturated design for the nine
variables requires 55 points for constructing a quadratic RSM. However, the
proposed ISCD-I uses only the 31 points, such as the 19 points for the linear and
pure quadratic terms and the 12 points from the Placket–Burman design for the
linear and two-factor interaction terms. Then, only three points are sequentially
added as the approximate optimization progresses. Thus, a total 34 analyses are
required to solve this design problem.
Figure 6 shows the convergence histories of the SAO results using two RSM. The
CRSMs denotes that proposed in section 4, and the RSM, is a conventional RSM.
In CRSM, overestimation is employed for approximating the object and constraint
functions. It is noted that RSM is oscillated during optimization processes and
297
finally failed convergence. These oscillations occur whenever the conventional
RSMs are used in dynamic response optimization.
Fig. 5 Tracked vehicle system (a) tracked vehicle model / (b) single bump
Fig. 6 Convergence histories of two SAOs combined with CRSM and RSM
298
Table 4 The optimization results of a tracked vehicle system design
2.4.6. CONCLUSION
In order to overcome the oscillation phenomena and the convergence difficulty in
the sequential approximate optimization combined with a meta-model, the authors
proposed the conservative response surface models (CRSMs). Also, in order to
reduce the number of runs, the incomplete small composite designs(ISCDs) are
proposed. The proposed CRSM combined with ISCDs used SVD to overcome the
rank-deficiency of their normal equations. It was then developed from
the duality optimization theory. Moreover, unlike the original SCD that one could
not determine a unique design assessed on the number of variables, the proposed
ISCDs gave a unique and economic design table, although it might induce the rank
deficiency in the normal equation for quadratic RSM.
To validate the numerical performance of the proposed methods, one typical
design problem is solved. The proposed method used only 25 analyses to give a
good feasible design. The tracked vehicle suspension system design is thereby
solved. The proposed CRSM showed a good converged result, even though the
analysis results are very noisy, and by which the method in this article requires
only 34 analyses for the 9-design variable problem.
ACKNOWLEDGEMENT
This work was finally supported by the second stage of BK 21 programme.
REFERENCES
1. Simpson, T. W., Peplinski, J. D., Koch, P. N., and Allen, J. K.
Metamodels for computer-based engineering design: survey and
299
recommendations. Eng. Comput., 2001, 17, 129–150.
2. Wang, J. G. and Liu, G. R. A point interpolation meshless method based
on radial basis functions. Int. J. Numer. Meth. Eng., 2003, 54, 1623–1648.
3. Kim, M.-S.,Cho, H., Lee, S.-G., Choi, J.,and Bae,D. DFSS and robust
optimization tool for multibody system with random variables. J. Syst. Des.
Dyn., JSME, 2007, 1(3), 583–592.
4. Haftka, R. T., Scott, E. P., and Cruz, J. R. Optimization and experiments:
a survey. Appl.Mech. Rev., 1998, 51(7), 435–448.
5. Box, G. E. P. and Wilson, K. B. On the experimental attainment of
optimum conditions. J. R. Stat. Soc. Ser.B, 1951, 13, 1–45.
6. Box,G. E. P. andHunter,W. G.Multi-factor experimental designs for
exploring response surfaces. Ann.Math. Stat.,1957, 28, 195–241.
7. Hartley, H. O. Smallest composite designs for quadratic response surfaces.
Biometrics, 1959, 15, 611–624.
8. Westlake, W. J. Composite designs based on irregular fractions of
factorials. Biometrics, 1965, 21, 324–336.
9. Draper, N. R. Small composite designs. Technometrics, 1985, 27(2), 173–
180.
10. Draper, N. R. and Lin, D. K. Small response-surface designs.
Technometrics, 1990, 32(2), 187–194.
11. Hedayat, A. S., Sloane, N. J. A., and Stufken, J. Orthogonal array:
theory and applications. 1999, (Springer, NewYork).
12. Santner, T. J., Williams, B. J., and Notz,W. I. The design and analysis of
computer experiments, 2003, (Springer, NewYork).
13. Stein, M. Large sample properties of simulations using latin hypercube
sampling. Technometrics, 1987, 29(2),143–151.
14. Genzi, L. and Azarm, S. Maximum accumulative error sampling strategy
for approximation of deterministic
15. engineering simulations. In the 11th AIAA/ISSMO Multidisciplinary
Analysis and Optimization Conference, Portsmouth, VA, 6–8 September
2006, AIAA 2006-7051.
16. Björck, Å . Numerical methods for least squares problems. 1996, (SIAM,
philadelphia).
17. Fletcher,R. Practical method of optimization, 1987 (John Wiley & Sons,
Chichester).
18. Kim, M.-S. and Choi,D.-H.Min–max dynamic response optimization of
mechanical systems using approximate augmented Lagrangian. Int.
J.NumerMethodsEng., 1998, 43, 549–564.
19. Golinski, J. An adaptive optimization system applied to machine
synthesis.Mech.Mach. Synth., 1973, 8, 419–436.
20. Li, H. L. and Papalambros, P. A production system for use of global
300
optimization knowledge. ASME J. Mech. Transm. Automa. Des., 1985, 107,
277–284.
21. Plackett, R. L. and Burman, J. P. The design of optimum multi-factorial
experiments. Biometrika, 1946, 33,3-5-325.
APPENDIX
Notation
sampling data for the tth optimization process
random error vector for regression model
objective function values according to the set
inequality constraint function values according to the set
number of parameters
number of inequality constraints
number of centre points in the repeated experiments
number of defining relations So violated set for overestimated fitting
violated set for underestimated fitting
iteration number for the optimization process
the ith component in the design variable set x
design variables for the tth optimization process
design matrix for the regression model
observations for the regression model
observations in the violated sets,So or Su
regression model obtained from least-square fitting
overestimated regression model
underestimated regression model
distance along the parameter axis
coefficients for the regression model
coefficient for conservative regression model
design range for optimization process
convergence tolerance for the optimization
process
Lagrange multiplier vector
301
2.5.
ROBUST DESIGN OPTIMIZATION OF
THE MCPHERSON SUSPENSION SYSTEM
WITH CONSIDERATION OF A BUSH
COMPLIANCE UNCERTAINTY
2.5.1. INTRODUCTION
Bush uncertainty and mechanical flexibility influence the movement error of
suspension systems. As such, it is difficult to predict the performance of a
suspension system and to quantify the performance indices. Thus, the mechanism
of the suspension system is designed by trial and error based on the designer’s
experiences and intuition. This requires much time and effort. The performance
index of a suspension system is a function of the maximum and minimum values
over the parameter interval. Thus, it is impossible to apply directly a well
developed optimization algorithm based on gradient information. As such, a special
technique is needed to process dynamic response optimization problems [1] or the
design must be reformulated without those deviations [2, 3]. These challenges have
impeded the study of optimal design in this area compared with structural
optimization.Chun et al. [2] recently studied optimal designs for suspension
systems based on reliability analyses. They performed a mechanical analysis and an
analytical sensitivity analysis of a suspension design, taking into consideration
tolerances and grafting a reliability analysis that applied the mean-value first order
method with tolerance optimization. Chun et al. used an in-house code for
kinematic analysis to evaluate the kinematic to lerances. Thus, their approach
cannot be used in general multi-body dynamic codes such as ADAMS and
RecurDyn. Choi et al. [3] performed a reliability optimization with the single-loop
single-variable method by using results of a deterministic optimization as initial
values of reliability-based optimization using the finite difference design sensitivity.
Choi et al. used the sum of errors as the performance index to avoid a design
sensitivity analysis for the forms of the deviations. Although they solved the robust
design problem, the solution required nearly 1700 analyses for 15 design variables
and four random constants. Additionally, their design results were fully dependent
on the deterministic optimization result. To reduce the number of analyses needed
for robust design optimization for the deviation of kinematical behaviours of the
302
suspension system, taking into consideration the uncertainty in the bush
compliances, this study employs the metamodel based optimization technique. The
design variables are the joint positions of the system. The random constant is the
bush stiffness, which is the noise factor uncertainty. The design goal is to
minimize simultaneously the deviations in and variances of the toe and camber
angles during wheel movement.
In this study, a metamodel technique is used to represent the approximation
functions for those deviations. In other words, the deviations are functions of the
design variables and random constants. Thus, the variances of the performance
indices can easily be evaluated from the metamodel. Then, a sequential
approximate optimization (SAO) technique is used to solve the robust design
optimization problem for the suspension system, which automatically updates the
metamodel and solves the design problem until it satisfies the design criteria.
Section 2 describes the suspension system model and the performance indices
that evaluate the kinematic performance. Section 3 describes the variations in the
performance indices that are caused by changing the bush compliance. Section 4
explains the sequential optimization technique and the metamodel technique for the
robust design optimization.
Section 5 analyses and describes the results of the robust design optimization.
Finally, section 6 summarizes and discusses the contents of this study.
2.5.2 KINEMATIC ANALYSIS MODEL OF THE
MACPHERSON SUSPENSION SYSTEM
Suspension systems can be classified into several groups according to the
mechanical jointing pattern, the type of springs used, the independence of the left
and right wheels, etc. In this study, a MacPherson type suspension system, which is
sensitive to the kinematic performance, is used. Figure 1 shows a dynamic model
of a MacPherson suspension system.
303
Fig. 1 The MacPherson suspension system
Major components include the strut, tie rod, knuckle, and lower control arm.
Connections between individual components are spherical, revolute, and universal
joints, as well as compliance elements such as springs, dampers, and bushings. A
commercial program, RecurDyn, is employed for modeling and analyzing the
suspension system. The design purpose of this study is to determine the positions
of the joints. The rigid body model ignores elastic deformations of the components
except for compliance elements. The optimal design is a balance of the kinematics
and compliance characteristics of the suspension system. Both vehicle dynamic and
kinematic characteristics should be considered to assess the performance of the
suspension system. In general, the dynamic characteristics of the system include
the mass of the tire and wheel and the force elements, such as springs, dampers,
and bushings, acting on the system. The kinematic characteristics of the system
include the positions of the fixed points of the suspension system. This study
performs kinematic optimization of the suspension system. For that purpose, the
suspension performance indices used are the deviations in the camber angle, toe
angle, and wheel centre recession during the wheel stroke. The camber angle is
defined as positive when the top of the wheel moves to the outside. The camber
angle alters the handling qualities of a suspension system; in particular, a negative
camber angle improves grip when cornering.
This effect occurs because a negative camber angle places the tire at a more
304
optimal angle to the road, transmitting the forces through the vertical plane of the
tire, rather than through a shear force across it [4]. The toe angle is defined as toe-
in if the wheel goes inside at the front section of the vehicle when it is viewed from
the top. The toe angle change plays an important role in determining the apparent
transient oversteer or understeer. The front-to-rear change in the wheel centre is the
error of the wheel centre in the longitudinal direction of the vehicle against the
wheel stroke. Large errors will have a negative effect on the behaviour of the
chassis when braking or accelerating.
2.5.3. VARIATION RESULTS DUE TO UNCERTAINTY IN
THE BUSH STIFFNESS
To evaluate the effect on the bush compliance uncertainty for the suspension
system, the bush stiffness is set as a random constant. The random constants are the
translational and rotational stiffnesses of the internal bushes of the lower control
arm and the top mount bush of the strut. There are18 random constants with
uncertainty. The uncertainty deviation has been set to +8 per cent of the nominal
stiffness (the translational stiffness is in newtons per millimetre; the rotational
stiffness is in newton millimetres per degree) [3]. The simulation condition for the
wheel stroke is a bump of 60.0mm and a rebound of 260.0mm. The kinematic
analyses are performed for 36 Latin hypercube samples within the deviation ranges
of 18 random constants. Figure 2 shows the camber angle variations for the 37
analyses including the base model. Figure 3 compares the camber angle
deviations for the base model and 36 samples. The camber angle deviation pi(y) is
defined as a function of the random
305
Fig. 2 Camber angle variations for 37 analyses. The magnified portion of the curve shows
how the curves for the initial analysis (base model) and the analyses for samples #01 to
#36 coincide
Fig. 3 Camber angle deviations for the base model and 36 samples (ID, identification)
constant according to
306
where is the wheel travel and the subscript indicates the sequence of
analyses. The magnitudes of equation (1) are represented as bars in Fig. 3. The
sample variance of those camber angle deviations is evaluated as
where is the deviation when the bush stiffness is the nominal value and is
the deviation when the bush stiffness is the sample value. From the deviations
shown in Fig. 3, the camber angle deviation for the base model is , and its
sample variance is , which indicates that the base model is quite a
robust design for uncertainty in the bush stiffness.
2.5.4. METAMODEL-BASED SEQUENTIAL
APPROXIMATE OPTIMIZATION
To avoid the analytical design sensitivity analysis, this study uses approximate
models in the numerical optimization process. The generic name of an approximate
model is a metamodel. Common among metamodels, this study employs a radial
basis function (RBF). Then, it uses the RBF during the optimization process. When
an optimum is found, the process performs an exact analysis for the optimum and
checks the convergence criteria in the outer loop. When the criteria are not satisfied,
the metamodel is automatically updated by adding only one analysis result to the
sample results. This
design process will be called a metamodel-based SAO.
2.5.4.1. RADIAL BASIS FUNCTION
RBFs are a class of functions used for interpolation purposes [9–11]. Their values
depend only on the radius between the generic point and the centre of the particular
function. The RBF method constructs the approximation function ~zðxÞ to pass
through all sample points using an RBF and a polynomial basis function
according to
307
where is the weighting coefficient for and is the coefficient for
. An RBF has the general form
where is the distance between the interpolating point and the i th sample
point . In general, multi quadratics and a Gaussian spline exp
are widely used in the RBF. To guarantee a unique approximation, the constraints
usually imposed on the polynomial term are
Hence, the RBF method can be constructed by solving the matrix equations
As the distance is a scalar value, the matrix is symmetric. Hence, the unique
solution is guaranteed if the inverse of the matrix exists.
2.5.4.2. ROBUST DESIGN OPTIMIZATION FORMULATION
Fundamentally, all the functions are assumed as the metamodels according to
where are the standard deviations of evaluated from the metamodels.
308
and are the alpha weight and the robust index respectively. If and
, the design objective is a minimization of the variance of . In equation
(10), is the deviation representing the uncertainty and V is the design range.
The variables X include the design variables and random constants. If multiple
objectives are given, the objective function of equation (7) is replaced by a
preference function as
where is called a preference function. The preference function is an equivalent
function that transforms the vector-type objective into a scalar type objective. To
represent the preference function, two objectives are considered and these are given
by
There are many preference functions in a multi objective optimization strategy.
Here a weighted function and a weighted min–max function are used
simultaneously according to
where the values of are the user-defined weighting coefficients and the
relaxation factors and are automatically determined. Also, the ideal
solution i is internally determined by using the analysis results.
2.5.4.3. SEQUENTIAL APPROXIMATE OPTIMIZATION
Figure 4 shows the metamodel-based SAO process. When the optimization
problem is defined, the design variables, including the random constants, should be
selected and the design formulation defined on the basis of objectives and
constraints. Then, the initial sample method is automatically selected on the basis
of the number of design variables and random constants. When the random
constants are given, a Latin hypercube design of 3*k ; numbers [12] is
recommended, where k is the total number of design variables and random
constants.
309
For this design problem, the current design and 108 sample points are used for
the initial design-of experiment
points because the total number of variables is 36 (18 design variables and 18
random constants). The robust design optimization formulation for the suspension
system will be explained in section 5. When the basic information is given to the
SAO process, it solves the user-defined design optimization problem by using
numerical optimization algorithms [12]. For constrained optimization problems,
an augmented Lagrange multiplier method is employed. A quasi-Newton algorithm
and a conjugate gradient algorithm are automatically selected on the basis of the
number of design variables. To overcome divergence due to the lack of sample
points, a proper move limit strategy is automatically introduced. Also, the
polynomial model of equation (5) is automatically switched to the degree of
nonlinearity of responses. When the numerical optimization algorithm converges in
the inner loop, the convergence is verified through actual analysis
Fig. 4 Schematic process of the metamodel-based SAO (DOE, design of
experiments)
results in the outer loop. At the moment, convergence conditions are not satisfied.
Then, this new analysis result is added to the sample results, a new metamodel is
generated, and the procedures above are repeated. The polynomial type and move
limit strategy are automatically selected on the basis of the degree of non-linearity
310
and the magnitude of approximation error.
2.5.5.5. ROBUST DESIGN OPTIMIZATION OF THE
MACPHERSON SUSPENSION SYSTEM
2.5.5.1. DESIGN VARIABLES AND RANDOM CONSTANTS
The design variables are the six positions of the joints illustrated in Fig. 5. Their
upper and lower bounds are set to from the base model in Table 1.
The random variables are the stiffnesses of the bushings with per cent
uncertainty in Table 2.
2.5.5.2. QUANTIFICATION OF DESIGN PERFORMANCE INDICES
The design performance index for the kinematic behaviours of the MacPherson
suspension system are the maximum deviations in the camber angle, toe angle, and
front-to-rear change in the wheel centre when the wheel stroke is given, as
discussed in section 2. The behaviors of these performance indices are functions of
the wheel travel parameter, as shown in Fig. 6. The quantification of design
performance indices is required for the design optimization process. Figure 6
represents the change in the toe angle over the wheel travel.
Table 1 Design variables of the MacPherson suspension system
The design performance index is defined as the maximum deviation in the transient
responses over the wheel travel. Suppose that the wheel travel is defined as the
parameter t, the maximum deviations in the toe angle, the camber angle, and the
wheel centre recession can be expressed as
311
and
respectively. The parameter t moves in the range -60mm<t<60mm. The variable x
is the design variable vector, and y is the random constant vector.
2.2.5.3. ROBUST DESIGN OPTIMIZATION FORMULATION
To improve the kinematic performances of the McPherson suspension system,
the toe angle change as a function of the suspension travel is tuned to optimize the
vehicle’s transient handling response, and the camber angle change is tuned to
optimize the grip and to limit handling [13]. Accordingly, the design objective for
this study is to minimize the maximum deviations of the camber angle and
the toe angle as well as their sample standard deviations and
simultaneously.
As explained in section 3, the base model is a robust design. Thus, three inequality
constraints for the camber angle, toe angle, and front-to-rear error of the wheel
centre are added. The camber and toe angles are set to 10 per cent margins
compared with the results of the base model because their variances are included in
the objective functions. The wheel centre error is set to be reduced by 220 per cent
compared with the base model because its variance is not included in the objective
functions. This design formulation is mathematically represented as
312
Table 2 Random constants of the MacPherson suspension system
Fig. 6 Quantitative representation of the performance index
The upper and lower bounds on the design variables x are set to of the
joint positions of base model, in consideration of the package. Also, the uncertainty
in the random variables y is set to per cent of the bush stiffness of the base
model [3]. In the above formulation, the superscript 0 denotes the results of the
base model.
2.5.5.4. ROBUST DESIGN OPTIMIZATION RESULTS
To generate metamodels of 18 design variables and 18 random constants, the
initial analyses were performed with 109 sample points. Then, 77 analyses were
313
sequentially performed until all design criteria are satisfied. Figure 7 shows the
convergence history of the robust design optimization problem described in section
5.3. Based on the convergence history, small oscillations on SAOs 1 to 34 were
observed. This phenomenon occurs when the optimal solutions, given by the
numerical optimizer, are obtained from premature metamodels. In other words, this
phenomenon may be a transient process that improves the accuracy of the
metamodels by sequentially adding candidate optimum positions.
Similarly, the violations on SAOs 70 to 75 may be a transient process that
improves the stability of variance estimation based on the metamodels. The robust
design optimization process converged on the 77th iteration. The final design
satisfied the constraint that the relative change in objective values between
consecutive iterations is less than 0.01, and all inequality constraints are less than
0.0001. Thus, the total number of analyses is 186 including 109 analyses for the
initial metamodel and 77 analyses for SAOs. The three performance indices
between the base model and an optimum design are compared in Figs 8 to 10. In
the figures, the final design represents the 77th design from the above SAO. Figure
8 compares the toe angle changes. Two designs give different behaviours for the
toe angle. The final design can reduce the deviation by 68 per cent compared with
the base design. Figure 9 compares the camber angle changes. Like the toe angle, it
can be seen that the maximum deviation in the camber angle for the optimal design
has been reduced by 52 per cent compared with the base design. Figure 10
compares the wheel centre changes. It can be seen that the maximum deviation in
the wheel centre change for the optimal design has been reduced by 51 per cent
compared with the base design.
To verify the results of the robust design optimization, the sample variances for
the base and the final designs are compared. First, hypercube
experimental points were selected within the deviation of the random constants of
the base design and an exact analysis was performed to calculate the sample
variances for individual performance indices.
For the final design, the same 36 hypercube experimental points were also selected
and an exact analysis was performed to calculate the sample variances for the
performance indices. Both comparisons use the same values for the random
constants but different values of the design variables. Table 3 lists the sample
standard deviations.
Compared with those of the base design, the final
314
Fig. 7 Convergence history of the metamodel-based SAO
Fig. 8 Comparison of the toe angle changes for the optimal and base designs
315
Fig. 9 Comparison of the camber angle changes for the optimal and base designs
design can reduce the sample standard deviations in the toe angle and camber angle
changes by 37 per cent and 85 per cent respectively. Even though the variance of
the wheel centre errors was not included in the design objectives, the final design
reduces this error by 38 per cent compared with that of the base
design. Figures 11 and 12 show the 37 sample analysis results used for evaluating
the sample variances of the toe and camber angles. Sample #01 denotes the result
for the base model. The results of the analyses appear to be the same because their
variances are small. Finally, Table 4 compares the design variable changes of the
base design and the final design. It can be seen that the front and rear positions of
the internal joint of the lower control arm are changed by 29.8mm and 2 11.5mm
respectively along the longitudinal direction of vehicle. These values determine the
pitch pole of the suspension system. Also, the lengths of the tie rod and the lower
control arm are lengthened by 0.14mm and 44.5mm respectively.
These values influence the toe and camber angles.
2.5.6. CONCLUSION
This study explained a robust design optimization that maximized the kinematic
performance of a
316
Fig. 10 Comparison of the wheel centre recession for the optimal and base designs
Table 3 Comparison of the sample standard deviations
MacPherson suspension system. The quantity processes for kinematic performance
of the suspension system as well as the basic concept and computational
procedures of the metamodel-based SAO were described. To verify the proposed
design concept, a robust design optimization problem for a MacPherson suspension
system, which had 18 design variables (joint positions) and 18 random constants
(bush stiffnesses), was solved. The proposed design process solved the problem
using only 186 analyses, including 109 analyses for the initial metamodel and
77 analyses for SAOs. The final design can reduce the maximum deviations in the
toe angle and the camber angle by 68 per cent and 52 per cent respectively
compared with the base model,. Even though the base design is robust for the
uncertainty in the bush stiffness, the final design reduced the sample standard
deviations in the toe angle and the camber angle by 85 per cent and 37 per cent
respectively compared with the base design.
317
Fig. 11 Toe angle responses for validation. Similarly to Fig. 2, the curves for
sample #01 (base model) and for samples #02 to #36 coincide
Fig. 12 Camber angle responses for validation. Similarly to Fig. 2, the curves for
sample #01(base model) and for samples #02 to #36 coincide
318
Table 4 Optimization results of the MacPherson suspension system
REFERENCES
1. Kim, M.-S. and Choi, D.-H. Direct treatment of a max-value cost function
in parametric optimization. Int. J. Numer. Meth. Engng, 2001, 50, 169–180.
2. Chun, H. H., Kwon, S. J., and Tak, T. Reliability based design
optimization of automotive suspension systems. Int. J. Automot. Technol.,
2007, 8(6), 713–722.
3. Choi, B.-L., Choi, J.-H., and Choi, D.-H. Reliability-based design
optimization of an automotive suspension system for enhancing kinematics
and compliance characteristics. Int. J. Automot. Technol.,2004, 6(3), 235–
242.
4. Jazar, R. N. Vehicle dynamics: theory and application,2008 (Springer,
New York).
5. Kim, M.-S. and Heo, S.-J. Conservative quadratic RSM combined with
incomplete small composite design and conservative least squares fitting.
KSME Int. J., 2003, 17(5), 698–707.
6. Kim, M.-S., Cho, H.-J., Lee, S.-G., Choi, J., and< Bae, D.-S. DFSS and
robust design optimization tool for multibody system with random
variables. J.System Des. Dynamics, 2007, 1(3), 583–592.
7. Kim, M.-S., Kim, C.-W., Kim, J.-H., and Choi, J. H. Efficient optimization
method for noisy responses of mechanical systems. Proc. IMechE, Part C:
J. Mechanical Engineering Science, 2008, 222(12), 2433–2439. DOI:
10.1243/09544062JMES1093.
8. Kim, M.-S. User’s guide for autodesign, RecurDyn <V7R2, 2008
(FunctionBay Inc., Seoul).
9. Hussain, M. F., Barton, R. R., and Joshi, S. B. Meta modeling: radial basis
functions, versus polynomials.Eur. J. Opl Res., 2002, 138, 142–154.
10. Wang, J. G. and Liu, G. R. A point interpolation meshless method based
on radial basis functions Int. J. Numer. Meth. Engng, 2002, 54, 1623–1648.
11. Jin, R., Chen, W., and Simpson, T. W. Comparative studies of
metamodeling techniques under multiple modeling criteria. Struct.
Multidisciplinary Optimization, 2001, 23, 1–13.
319
12. Kim, M.-S. Integrated numerical optimization library: E-INOPL, 2009
(Institute of Design Optimization Inc., Seongnam-si, Gyeonggi-do).
13. Jung, H. K. Vehicle dynamics analysis and chassis design using the
functional suspension model. Doctoral Thesis, Kookmin University, Seoul,
Republic of Korea, 2005.
APPENDIX
Notation
radial basis function
number of design variables
robust index
deviation when the bush stiffness is the ith sample
deviation when the bush stiffness is a nominal value
preference function
distance between the interpolation point x and the ith sampling point
S(y) sample variance
weighting coefficient for (x)
construction approximation function of the radial basis function method
polynomial basis function
alpha weight
coefficient for (x)
deviation representing the uncertainty
standard deviation
design range
3. Mechatronics
311
3.1.
A CASE STUDY OF MECHATRONIC
SYSTEM SIMULATION : FORKLIFT
ELECTRIC CONTROL
3.1.1. INTRODUCTION
In general a mechanical system designer and an electrical system designer use
their own independent design models. Electrical system designers want to use S/W
tools that are useful to build the control system and the electrical system, rather
mechanical system designers prefer to use S/W tools that are useful to build the
realistic nonlinear dynamic system.
Mostly canned packages, Simulink, Simplorer, PSIM, and/or Spice are used to
simulate the electrical system because they are very useful tools to simulate the
motor driving system in the motor control and the electric power filed, and
multibody dynamic analysis tools, such as RecurDyn, ADAMS or etc., are used to
simulate the mechanical system, which can describe the realistic nonlinear dynamic
model.
According to the aim of system design becomes more precise and exact results,
and the simulation environment has improved significantly, the system designers
would like to simulate the whole system including realistic mechanical system,
control system, electrical system, and hydraulic system. In this investigation, the
simulation environment is introduced how to simulate the total system which has
the electrical system, the control system, and mechanical system simultaneously.
As a study example, the dynamic simulation of electric forklift system powered
by PMSM drive is employed to obtain a contemporary simulation environment.
The electric forklift system employed in this study has two parts. One is
mechanical parts that consist of chassis with lifting mechanism, suspension, drive
train, and tire, and the other is electrical parts that the motor drive system consists
of PMSM, PWM, Inverter, and Controller.
The paper is organized as follows. The integrated modeling framework for whole
simulation is described in Section II. The integrated solver framework for total
simulation is described in Section III. A mathematical model of the electrical
forklift system is described in Section IV. Finally we show and discuss the result of
dynamic simulations of the electric forklift system forklift driven PMSM drive in
Section V.
312
3.1.2. INTEGRATED MODELING FRAMEWORK
Since one of modeling method used in RecurDyn software is based on CAD
modeling, it is very useful to build multi-body model like chassis and suspension of
the forklift. But it is very difficult to build PMSM drive system or other control
elements. In order to solve the difficulties, block modeling based CoLink tool is
developed and integrated into RecurDyn environment. Because CoLink is based on
block link modeling method, it is easy to build the control and electric system such
as PMSM drive system or more. However for the complete modeling for
simulation of mechatronic system, strong general purpose script capability is also
necessary. In this investigation, complete C based ChScript [6] is also integrated in
to the environment. Thus integration of RecurDyn/CoLink/ChScript environment
supports complete three modeling methods and firstly used in this study. And it is
very helpful to simulate the mixed whole system. Figure 1 shows three integrated
modeling techniques in order to simulate forklift electric control system.
Figure 1. Three integrated modeling techniques in RecurDyn
3.1.3 INTEGRATED SOLVER FRAMEWORK
This section describes the integrated solver framework so that RecurDyn, CoLin
k and Ch Script are executed by an integrated solver.
3.1.3.1 EQUATION OF MOTION OF MULTI-BODY MODEL
Equation of motion of multi-body is established as follows.
CAD
(Solid Modeler)
SCRIPTBLOCK
0)QλΦYMBFΤ
Ζ
T ( (1)
313
Where, the λ is the Lagrange multiplier vector for cut joints, and Φ
represents the position level constraint vector in mR . The M and Q
are the mass matrix and force vector in the cartesian space including the contact fo
rces, respectively.
The equations of motion and the position level constraint can be implicitly
rewritten by introducing vq as
and
Successive differentiations of the position level constraint yield
and
Equation (2) and all levels of constraints comprise the over determined differential
algebraic system (ODAS). An algorithm for the backward differentiation formula
(BDF) solves the ODAS of
0
βvvU
βvqU
vvqΦ
vqΦ
qΦ
)λv,vq(F
xH
)β(
)β(
,,
,
,,
)(
20
T
0
10
T
0
In equation (6), TTTTTλ,v,v,qx ,
0β , 1β and
2β are determined by the
coefficients of the implicit integrators and 0U is an m)(nrnr matrix such that
the augmented square matrix TT
0 qΦU is nonsingular.
0qΦ )(
0υvΦvqΦ q ),(
0γvΦvvqΦ q ),,(
0λa,vqF ),,( (2)
(3)
(4)
(5)
(6)
314
3.1.3.2 EQUATION OF MOTION IN BLOCK MODEL
Transfer function block shown in Figure 2 can be expressed as the ordinary
differential equation listed in (7).
Figure 2. Transfer function block
And also close loop is modeled as following Figure 4. It can express following the
algebraic equation of (8). This example is easy to get the solution which is z = 0.5.
However, more complicated model must be computed by numerical method.
zz 1
Figure 4. Closed loop block diagram
The equation of motion of block model generally is a equation including ordinary
differential equations and algebraic equations of (9),
where
3.1.3.3 INTEGRATED EQUATION OF MOTION
Integrated solver deals with the multi-body model and block model
simultaneously. Integrated equation of motion is including the multi-body model
and the block model equation as
ubyay
0z)R(y,z
z)F(y,y
)(cG
TTz,yc
(7)
(8)
(9)
315
0G(c)
H(x)c)I(x,
,
TTTTTλ,v,v,qx
and
TTz,yc
Newton-Raphson method can be applied to obtain the solution x and c.
IΔc)(ΔI jac ,x
c
G
x
G
c
H
x
H
I jac
A different thing between using the integrated solver and co-simulation method
whether deals with considering both cH / and xG / or only one in each
package.
3.1.4. FORKLIFT MODEL
The driving machine of the electric forklift vehicle shown in Figure 3 is PMSM
(Permanent Magnetic Synchronous Motor). To use PMSM need the controller and
PWM inverter. The controller of PMSM is PI controller and the PWM inverter is
Space Vector PWM inverter.
(10)
(11)
(12)
(13)
(14)
316
Figure 3. Electric forklift system
3.1.4.1. ELECTRIC MODEL
A Permanent Magnet Synchronous motor (PMSM) has a wound stator, a permanent
magnet rotor assembly and internal or external devices to sense rotor position.
317
(a) Motor Drive System
(b) Multi-body System
Figure 4. Integrated model of Forklift system
The sensing devices provide logic signals for electronically switching the stator
windings in the proper sequence to maintain rotation of the magnet assembly. The
combination of an inner permanent magnet rotor and outer windings offers the
advantages of low rotor inertia, efficient heat dissipation, and reduction of the
motor size. Two configurations of permanent magnet brushless motor are usually
considered: the trapezoidal type and the sinusoidal type. Depending on how the
stator is wounded, the back-electromagnetic force will have a different shape (the
BEMF is induced in the stator by the motion of the rotor). The trapezoidal BEMF
motor called DC brushless motor (BLDC) and the sinusoidal BEMF motor called
PMSM. This paper introduces the implementation of a control for sinusoidal
PMSM motor. The sinusoidal voltage waveform applied to this motor is created by
using the Space Vector PMW inverter technique. The Field Oriented Control
algorithm is used for control of torque and rotation speed of PMSM.
The mathematical equation of the motor driving system consists of three equation
blocks, first is PMSM block, second is controller block, and third is SVPWM
inverter. Figure 4 shows the integrated simulation model of forklift system
constructed by RecurDyn/CoLink/Ch Script.
318
(1) PMSM Block
The PMSM block implements 3-phases permanent magnet excited synchronous
motor. This model assumes that the flux generated by the permanent magnets is
sinusoidal, which implies that the electromotive forces are also sinusoidal. The
voltage balance equation is as
qqmdd
dd iLpRidt
diLv ,
ddmmmq
q
qq iLppRidt
diLv
and
])([5.1 qdqdqme iiLLipT
Table 1 lists the variables in equations (15) through (17).
Table 1. Variables of the voltage valance equation
Variables Desription
dV d-axis voltage
qV q-axis voltage
di d-axis current
qi q-axis current
dL d-axis inductance
qL q-axis inductance
R Resistance of stator windings
m Angular velocity of the rotor
m Flux induced by the permanent magnets
p Number of pole pairs
eT Developed electrical torque
LT Load torque
J The inertia of rotor
B Viscous friction coefficient
Angular position of rotor
(15)
(16)
(17)
319
(2) Control Blocks
In order to drive PMSM motor, the speed controller, computation logic for
reference current, the current controller , and the computation logic for 3 phase AC
voltage are needed.
The speed controller is anti-windup controller as
where, E(s) is the difference between the reference velocity and the rotor velocity,
Kp, Ki, Ka are gains, I*(s) is the reference current. The block modeling method is
used to simulate the speed controller like Figure 5 in CoLink.
Figure 5. Speed controller in CoLink
The computation logic is needed for transformation from the output of speed
controller to reference d-axis current and q-axis current.
Script method is very easy to build this process. In order to run script method, we
use Ch script engine that is an embeddable C/C++ interpreter for cross-platform
scripting.
The current controller is anti-windup controller as
where, E(s) is the difference between the reference current and the motor current,
Kpd, Kid, Kad are d-axis gains, Vd*(s) is the reference d-axis voltage.
))()(()(1
)()( ** sIsIKasEs
KisEKpsI
))()(()(1
)()(
*
*
sVsVKadsEs
Kid
sEKpdsV
dd
d
(18)
(19)
320
Figure 6. Computation logic for reference current by Ch Script
The block modeling method is used to simulate the current controller in CoLink as
shown in figure 7.
Figure 7. Current controller
The current controller makes the d-axis and q-axis voltages from the current of the
motor. The d-axis and q-axis voltages are converted 3-phases voltages to drive
PMSM with SVPWM block.
321
(3) SVPWM Inverter Block
Among various modulation techniques for a inverter, space vector pulse width
modulation (SVPWM) is an attractive candidate due to the following merits. It
directly uses the control variable given by the control system and identifies each
switching vector as a point in complex space. It is suitable for digital signal
processor (DSP) implementation. It can optimize switching sequences.
A numerical algorithm of SVPWM is below. Input of SVPWM is the reference
value of 3-phases voltage. Output of SVPWM is real value of 3-phase voltage to
drive PMSM.
- For the reference voltage:
- For a numerical algorithm:
In equations of (20) and (21), refabcV _
, outabcV _
, and dcV are reference, output voltage,
and input voltage respectively. The SVPWM Inverter is modeled by Ch Script as
shown in Figure 8.
)min()max( __
__ *
refabcrefabcoffset
refabcrefabc
VVK
VV
2/)(
)(
0)2/(
*
*
*
__
___
_
dcrefabcoutabc
refabcrefabcoutabc
refabcdc
VVsignV
else
VVsignV
VVif
(20)
(21)
322
Figure 8. SVPWM Inverter by Ch Script
(4) Ch Script
Ch script is used for the computation logic for SVPWM Inverter and the d-q axis
reference current. Many scripting languages have claimed that their syntax
resembles C or C++, but they are not C or C++. Their coding style and syntax are
different from C/C++. Ch is an embeddable interpreter that provides a superset of C
with salient extensions. It parses and executes C code directly without intermediate
code or byte code. It does not distinguish interpreted code from compiled C/C++
code. Ch is the most complete C interpreter and C virtual machine in existence. Ch
is embeddable in other application programs and hardware. Figure 9 shows Ch
versus other program languages clearly.
323
Figure 9. Ch versus other languages
3.1.4.2. INPUTS AND OUTPUTS OF MULTI-BODY MODEL
PMSM makes the electrical torque that drive the wheel of Forklift and the angle
and rotation speed of the rotor are transferred to PMSM system.
3.1.5 CONCLUSION
In order to simulate the integrated system of the multi-body system and the motor
drive system, the integrated sysyem simulation methods and environment are
developed in this investigation. It is applied to the electric forklift system driven by
PMSM drive.
The developed integrated modeling system helps the electrical desinger to build
the electrical system using the block/script modeling method, and helps the
mechancial desinger to build the mechanical system using the CAD modeling
method. The developed integrated solver helps one integrator to simlate the whole
system.
The electrical forklift system is a multi-body system driven by PMSM drive and a
good example to be applied by the development tool. In this paper, the mathmatical
model of PMSM drive system is described.
The proposed method can give excellent convinient and efficiency for the
mechatronic system designer since at early concept design stage the simulation
result could give detail information which can achive with real hardware.
REFERENCES
1. H. S. Mol, S. H. Kim and Y. H. Cho, “Torque ripple reduction of PMSM ca
used by position sensor error for EPS application,” Electronics Letters, 24th
vol.43, no. 11. 2007, pp.646~647.
2. Functionbay, "RecurDyn version 6.4 Solver Theoretical Manual", 2007.
324
3. Bae, D. S. and Haug, E. J., 1987, "A Recursive Formulation for
Constrained Mechanical System Dynamics: Part I. Open Loop Systems,"
Mech. Struct. and Machines, Vol. 15, No. 3, pp. 359-382.
4. Bae, D. S. and Haug, E. J., 1987, "A Recursive Formulation for
Constrained Mechanical System Dynamics: Part II. Closed Loop Systems,"
Mech. Struct. and Machines, Vol. 15, No. 4, pp. 481-506.
5. Haug, E. J., 1989, Computer-Aided Kinematics and Dynamics of
Mechanical Systems: Volume I. Basic Methods, Allyn and Bacon.
6. ChScript (www.softintegration.com)
325
3.2
THE INTER-DISCIPLINARY SIMULATION
ENVIRONMENT INCLUDING THE
FIRMWARE AND THE MECHANICAL
SYSTEM
3.2.1. INTRODUCTION
Recently, the importance of the embedded software is increasing more and more.
And the importance of the mechatronics which is the integrated system of the
mechanics and electronics are increasing too. As a simple example, the automobile,
the most traditional mechanical system, the percentage of the electronic equipments
is about 20~30%. In the case of hybrid car, the percentage is about 40~50%. For
instance, the various equipments of the car such as a steering system, an engine
control, a body control, a lane departure warning, consist of lots of electronics
equipments to control them
In addition, most of the mechanical products such as robot, printer, camera, now
can be called as mechatronics products which are controlled by firmware, not just
the mechanical products.
As the importance of the mechatronics products are increasing, the importance of
the firmware to control the product is also increasing. Also the products are getting
more and more complex, the firmware is also getting complicated and bigger.
Therefore, the number of the bugs of the firmware is growing. Since the bug of the
firmware can cause lots of troubles, cost and waste of time, it is essential to
discover and fix the bugs in the early stage of the development period by through
validation and debugging.
But in the case of mechatronics products, since the firmware is used to control
the mechanical system, in order to validate and test the firmware under
development, the mechanical system should be ready in advance and the firmware
under development needs to be downloaded to the system. It means that the
firmware should be developed to the level which can be compiled and executed to
control the mechanical system to test if it can control the system as intended. It is a
kind of dilemma. In the view point of the mechanical system engineer, the
mechanical system should be developed to the level which can be operated when it
is controlled by the firmware. But this kind of level is achieved in the late stage of
326
the development. So, even if any bug or any conflict of the specification or
interface is discovered, it is not easy to change the specification or fix it.
Furthermore, the test and debugging of the firmware with the real hardware is
very difficult and needs lots of efforts and time. Therefore, the better development
environment which can help the validation and debugging in the early stage of the
development period is required. If such an environment can be used, it can save lots
of time and cost.
In this paper, an inter-disciplinary simulation environment including the firmware
and the mechanical system is suggested which uses the virtual models of the
firmware and the mechanical model. By using the virtual models, validation and
debugging of the system can be done in the early stage without the real firmware
and hardware. And it is much easier to change the specification when the virtual
models are used than the real models are used.
The suggested environment in this paper uses two simulation tools. For the
firmware, ZIPC, the CASE tool which is based on the state transition matrix (STM)
is used. For the mechanical system, RecurDyn, the CAE tool which can simulate
the virtual multi-body model. To validate the effectiveness of the suggested
environment, we developed a forklift robot with Lego Mindstorms® and we
validated and debugged the firmware to control the robot by using the environment.
In section II, the specification of the Mindstorms® robot is explained. In section III,
it is explained how to make the specification of the firmware using the state
transition matrix on ZIPC and the state transition matrix can be simulated without
the real source code. In section IV, the process to validate and debug the inter-
disciplinary system is explained. This process doesn’t need the real source code and
the real robot. In this stage, the ZIPC model developed in section II and the
RecurDyn model of the robot are used. The environment treated in section IV is the
inter-disciplinary simulation environment which this paper suggests. In section V,
the real firmware is generated from the state transition matrix and the
Mindstorms® robot is assembled according to the virtual robot model tested in
section IV. And they will be tested and the result is compared with the result of
section IV.
3.2.2. SPECIFICATION OF THE MINDSTORMS® ROBOT
Mindstorms® is the programmable robotics kit consists of Lego blocks, motors,
sensors, etc. This paper used Mindstorms® to build a robot for the validation of the
suggested development environment.
Figure 1 is the Mindstorms® robot which was used for this paper. It is a simple
forklift robot which has the track to drive the robot and it has a lift to lift up an
object at the front part of the robot. When the touch sensor is triggered, the robot
327
moves forward. And when the ultrasonic sensor is triggers an object ahead, robot
lift it up. After lifting up, it moves backward and if the sound sensor is triggered, it
put down the object.
Figure 1. Real Mindstorms model
3.2.3. ZIPC AND STATE TRANSITION MATRIX
ZIP is the CASE tool which is based on the state transition matrix. In this paper,
the firmware to control the robot was developed by using ZIPC.
3.2.3.1. STATE TRANSITION MATRIX
State transition matrix has lots of advantages compared to the other methods for
making a specification of the firmware. It can prevent the missed out cases or
exceptional cases.
Figure 2 compares the state transition diagram and the state transition matrix
which describes the same specification. The blanks in the state transition matrix on
the right are the missed cases which were not described in the state transition
diagram on the left. State transition diagram shows the missed cases visually and
prevent them from the stage of making specification. ZIPC is the tool which is
based on state transition matrix and it has other lots of useful functions to make a
specification and generate the C source code for the firmware
328
Figure 2. Comparison between STD and STM
Figure 3 shows the state transition matrix for the robot explained in section II.
Figure 3. STM for the firmware of the robot
3.2.3.2. SIMULATION OF STATE TRANSITION MATRIX
ZIPC provides a simulation of the state transition matrix itself. So it is possible to
validate if the specification works correctly as it is intended and if it has any bug or
exception. Since it is possible to validate the specification without the source code
in the early stage of the development, it helps to find out the problems of the
specification early and to save the time and cost for the debugging and
modification.
Figure 4 shows the simulation of the state transition matrix using ZIPC. It is
possible to issue an event and check if the state is transited correctly. And it is also
possible to create a log file of the simulation and reuse it and show the code
coverage of the test. Since the debugging capabilities such as break point are
available, it is very useful to validate the specification
329
Figure 4. Simulation of STM on ZIPC
3.2.4. INTEGRATED ENVIRONMENT FOR ZIPC-RECURDYN
After the specification is completed and validated, it should be validated with the
mechanical system if the specification works correctly when it is used to control it.
Traditional way to do this is writing and compiling a source code for the firmware
and downloading it to the prototype of the hardware for the validation. Usually the
firmware doesn’t work well, and the firmware or the hardware should be modified.
After the modification, the same procedure such as downloading and testing is
repeated, and it wastes lots of time until the development is completed.
But by using the integrated environment which this paper suggests, it is possible
to validate the firmware without the source code and the real hardware. Since it is
easier to debug and modify with the virtual models, it is possible to reduce the
repeated and cumbersome process.
3.2.4.1. ROBOT MODELING USING RECUDYN
Virtual multi-body model of the robot including the track was built by RecurDyn
330
Figure 5. MBD MODEL OF MINDSTORMS IN RECURDYN
3.2.4.2. MOTOR MODELING USING COLINK
Since the track is driven by motor, the motor drive was built by CoLink which
supports the block modeling method.
Figure 6. CoLink Model for motor and interface with ZIPC
3.2.4.3. COSIMULATION OF ZIPC, RECURDYN AND COLINK
There are 3 virtual models, 1) ZIPC model, 2) RecurDyn model, 3) CoLink model,
so that they needs to be cosimulated as an integrated environment.
Firstly, the following integrated equation of motion (1)~(3) was used for the
331
integration of RecurDyn model and CoLink model, which integrated the equation
of motion of the multi-body model and the equation of motion of the CoLink block
model (Yun et al. 2008).
0G(c)
H(x)c)I(x,
,
TTTTTλ,v,v,qx
and
TTz,yc .
For the integration of CoLink and ZIPC, interface block of CoLink and VIP
technique of ZIPC were used. Since ZIPC model doesn’t need to solve the equation
of motion, there was no need of integration of the equations
Figure 7 shows the integrated environment of 3 virtual models and how they are
connected. By using this integrated environment, it is possible to validate and test if
the state transition matrix designed in section II can control the virtual robot model
correctly.
Since all this process can be conducted on PC, it is very convenient to test and it
is possible to watch all the values which the developer is interested in. When
debugging is needed, it is possible to reproduce the same test condition. And the
models can be modified easily.
Figure 7. Simulation using STM in ZIPC
3.2.5. FIRMWARE-MINDSTORMS® ROBOT
After the state transition matrix and the Mindstorms® model are validated, the next
step is to make the real firmware and Mindstorms® robot.
(1)
(2)
(3)
332
Since ZIPC can generate the c source code directly from the state transition matrix,
it is very easy to making the real firmware. Figure 8 shows the source codes
generated by ZIPC.
Figure 8. C source code generated by ZIPC
After compiling the source code, it is downloaded to Mindstorms® robot of figure
1.
3.2.5.1. VALIDATION OF THE REAL MINDSTORM® ROBOT
The real firmware was generated from ZIPC model, and the Mindstorms® robot
was assembled same to RecurDyn model. After the firmware is downloaded to the
robot, we tested it under the same condition which was conducted in section IV.
Since the firmware and robot model was validated on the virtual environment,
there was no big problem. The real robot moved as intended in the same way of the
virtual model.
But since the virtual model used a simplified signal instead of the real sensor, the
sensitivity of the sensors affected a little to the simulation result. To minimize this
kind of difference, it is required to use the more realistic virtual sensor model
instead of the simplified signal.
Except the problems related to the sensor, we can say that the virtual simulation
environment is very effective to reduce the repeated procedure. Since the firmware
is validated enough under the virtual environment, it is possible to develop the
stable firmware before the real hardware is ready.
3.2.6. CONCLUSION
This paper suggests the inter-disciplinary simulation environment including the
firmware and the mechanical system using CASE and CAE tools which can be
333
used to validate and debug the firmware and mechanical system from the early
stage of the development period.
This paper used a robot using Lego Mindstorms® for validation of the
environment. Firstly, the state transition matrix for the firmware was made. The
state transition matrix could prevent the missed out cases and the exceptional cases
in the early stage. And the simulation capability of ZIPC helped to validtae the state
transition matrix without source code. Secondly, after making a virtual multi-body
model of the robot using RecurDyn, the firmware was validated, tested and
debugged on the integrated environment of ZIPC, RecurDyn and CoLink without
any source code or real hardware.
After the firmware is validated enough, c source code was generated from the
state transition matrix, and real robot model was assembled based on the RecurDyn
model. And it was verified that the real firmware and the real robot model worked
as intended without big problem.
Even though this paper used a simple robot kit, Mindstorms® , the development
process of this paper is basically similar to the development process of the most of
the mechatronics products. So the inter-disciplinary simulation environment which
this paper suggests can be used for the development of the real mechatronics
products and it can save lots of time and cost. And since it provides better
debugging environement, it also can help to develop the better quality products.
In the next sduty, we are planning to use this environment for the development
of the real product rather than Mindstorms® robot, and verify how effective it is.
REFERENCES
1. D.J. Yun, Hyungsoo Mok, K. H. Cho, and J. H. Choi, 2008, “Dynamic
simulations for Electric Forklift System driven by PMSM drive Using
RecurDyn and CoLink”, 4th Asian Conference on Multibody Dynamics
2008, pp. 7-11.
2. H. S. Mol, S. H. Kim and Y. H. Cho, “Torque ripple reduction of PMSM
caused by position sensor error for EPS application,” Electronics Letters,
24th vol.43, no. 11. 2007, pp.646~647.
3. Functionbay, "RecurDyn version 6.4 Solver Theoretical Manual", 2007.
4. Haug, E. J., 1989, Computer-Aided Kinematics and Dynamics of
Mechanical Systems: Volume I. Basic Methods, Allyn and Bacon.
5. Bae, D. S. and Haug, E. J., 1987, "A Recursive Formulation for
Constrained Mechanical System Dynamics: Part I. Open Loop Systems,"
Mech. Struct. and Machines, Vol. 15, No. 3, pp. 359-382.
6. Bae, D. S. and Haug, E. J., 1987, "A Recursive Formulation for
334
Constrained Mechanical System Dynamics: Part II. Closed Loop Systems,"
Mech. Struct. and Machines, Vol. 15, No. 4, pp. 481-506.
Theoretical Manual for Application
FunctionBay, Inc.
1. Track Vehicle
337
1.1
DYNAMIC ANALYSIS OF HIGH-MOBILITY
TRACKED VEHICLES
1.1.1. INTRODUCTION
High-speed, high-mobility tracked vehicles are subjected to impulsive
dynamic loads resulting from the interaction of the track chains with the vehicle
components and the ground. These dynamic loads can have an adverse effect on
the vehicle performance and can cause high stress levels that limit the
operational life of the vehicle components. For this reason, high speed, high
mobility tracked vehicles have sophisticated suspension systems, a more
elaborate and detailed design of the links of the track chains, and improved
vibration characteristics that allow the vehicle to perform efficiently in hostile
operating environments.
Galaitsis [1] demonstrated that the predicted dynamic track tension and
suspension loads in a high speed tracked vehicle developed by an analytical
method are useful in evaluating the dynamic characteristics of the tracked
vehicle components. The predicted track tension was compared with the
measured data from a military tracked vehicle. Bando et al [2] developed a
planar computer model for rubber tracked bulldozers. Steel and fiber molded
continuous rubber track is discretized into several rigid bodies connected by
compliant force elements. Characteristics of track damage, vibration, and noise
are investigated using the simulation results. Nakanishi and Shabana [3]
developed a two-dimensional contact force model for planar analysis of
multibody tracked vehicle systems. The stiffness and damping coefficients in
this contact force model were determined based on experimental observations of
the overall vibration characteristics of the tracked vehicle. The nonlinear
equations of motion of the vehicle were obtained using the Lagrangian approach
and the algebraic constraint equations that describe the joints and specified
338
motion trajectories are adjoined to the system equations of motion using the
technique of Lagrange multipliers. The generalized contact forces associated
with the system generalized co-ordinates were obtained using the virtual work.
Choi [4] presented a large-scaled multibody dynamic model of construction
tracked vehicle in which the track is assumed to consist of track links connected
by single degree of freedom pin joints. In this detailed three-dimensional
dynamic model, each track link, sprocket, roller, and idler is considered as a
rigid body that has a relative rotational degree of freedom. Scholar and Perkins
[5] developed an efficient alternative model of the track chains considering
longitudinal vibrations. The track is assumed to consist of a finite number of
segments and each is modeled as a continuous uniform elastic rod attached to the
vehicle wheels. Each segment consists of several track links which are
collectively lumped as a single body so that overall chain stretching effects are
accounted for.
A detailed three-dimensional tracked vehicle model as the one developed by
Choi [4] may have hundreds or thousands differential and algebraic equations.
These equations are highly non-linear and can only be solved using matrix,
numerical and computer methods. In addition to this dimensionality problem,
tracked vehicles are characterized by impulsive forces due to the contacts
between the track links and the vehicle components as well as the ground. The
impulsive contact forces cause serious numerical problems when the vehicle
equations of motion are integrated numerically. The degree of difficulty may
significantly increase if compliant elements, instead of the ideal pin joints, are
used to model the connection between the links of the track chains, as it is the
case in this investigation. The compliant elements must have very high stiffness
coefficients in order to maintain the link connectivity. These stiffness
coefficients, which are determined experimentally in this investigation, introduce
high frequency oscillatory components to the solution, thereby forcing the
numerical integration routine to take a very small time step size. It is, therefore,
important to adopt a numerical scheme that can be efficiently used in modeling
this type of vehicle. Newmark [6] presented an absolutely stable second-order
numerical integrator in the area of structural dynamics. The Newmark
integrator was modified by Wilson [13] so that highly oscillatory state variables
339
are numerically damped out. The numerical damping algorithms are extended
and generalized in implicit and explicit forms with a constant step size by Chung
[7,8]. The algorithm developed by Chung is employed in this investigation due
to its easy implementation and large stability region.
The objective of this investigation is to develop a computational procedure for
the nonlinear dynamics of high-speed, high-mobility tracked vehicles. The
model developed in this investigation differs from the low-speed tracked vehicle
model previously developed by Choi [4] in two important aspects summarized as
follows:
(1) The high speed tracked vehicle considered in this investigation has a
sophisticated suspension system that consists of road arms and wheels instead
of the simple roller type suspension system previously developed by Choi.
(2) In the model previously developed by Choi [4], the links of the track
chains are connected by pin joints that have one degree of freedom. In the model
developed in this investigation, compliant force elements are used to model the
connectivity between the links of the track chains. The characteristics of these
compliant elements are determined experimentally as discussed in Section 5.
The application of the numerical integration scheme developed by Chung [7,8]
to tracked vehicle dynamics is investigated in this paper using different
simulations scenarios that include accelerated motion, high speed motion,
braking and turning motion.
1.1.2. HIGH-SPEED, HIGH-MOBILITY TRACKED VEHICLES
In this section, the high-speed, high-mobility tracked vehicle model used in
this investigation is described. The three-dimensional model, which is shown in
Fig. 1, represents the third generation of a military vehicle weighing
approximately 50 tons and can be driven at a speed higher than 60 km/h.
340
Figure 1. High mobility tracked vehicle.
The vehicle consists of a chassis and two track subsystems. The chassis
subsystem includes a chassis, sprockets, support rollers, idlers, road arms, road
wheels and the suspension units. The sprockets, support rollers, and road arms
are connected to the chassis by revolute joints. The suspension unit includes a
Hydro-pneumatic Suspension Unit (HSU)[17], and torsion bar that are modeled
as force elements whose compliance characteristics are evaluated using
analytical and empirical methods. The HSU systems are mounted on front and
rear stations to damp out pitching motion and to decrease the vehicle speed when
the vehicle is running over large obstacles. The spring torque of the HSU
systems can be written as
1HSU PALT (1)
where P is the gas pressure, A is the area of piston, and 1L is the distance
shown in Fig. 2. The pressure P in the gas chamber of HSU system with
respect to rotation angle of a road arm is defined as
)( 22 LLl
lPP
is
ii (2)
where iP , il , and iL2 are the initial pressure and distances when the road arm
is in its initial configuration, is a constant which is equal to 1.4, and 2L is
the distance shown in Fig. 2.
341
Figure 2. Schematic diagram of spring-damper suspension units: hydro pneumatic
suspension unit and torsion bar systems.
The distance sl can be adjusted by charging or discharging oil into the oil
chamber. The torsion bars are mounted on the middle stations for this vehicle
model. A simple torsional spring model is used in this investigation to represent
the stiffness of the torsional bars. The stiffness coefficient of the torsion bar
spring is approximately 4105 Nm/rad. Figure 2 shows the schematic diagram
of the HSU and the torsion bar systems. Figure 3 shows the spring characteristics
which are employed in this investigation.
Figure 3. Spring characteristics of suspension unit
Each track subsystem is modeled as a series of bodies connected by rubber
bushings around the link pins which are inserted into a shoe plate with some
radial pressure in order to reduce the non-linear effect of the rubber. When the
342
vehicle runs over rough surfaces, the track chains are subjected to extremely
high impulsive contact forces as the result of their interaction with the vehicle
components such as road wheels, idlers, and sprocket teeth, as well as the ground.
The rubber bushings and double pins tend to reduce the high impulsive contact
forces by providing cushion and reducing the relative angle between the track
links. About 10 percent of the vehicle weight is given as the pre-tension for the
track to prevent frequent separations of the track when the vehicle runs at a high
speed. About 14 degrees of a pre-torsion is also provided in order to reduce the
fluctuation of the torque in the rubber bushing when the track links contact the
sprocket and idler.
The vehicle, which presents a high-mobility military tracked vehicle, consists
of one hundred eighty nine bodies; body 1 is the chassis, bodies 2 and 3 are the
right and left driving sprockets, bodies 4 - 17 are the right and left arms, bodies
18 - 31 are the right and left wheels, bodies 32 - 37 are the right and left support
rollers, and bodies 38 - 113 and 114 - 189 are the right and left track links,
respectively. The sprockets, rollers and arms are connected with chassis by 22
revolute joints, and wheels are connected with arms by 14 revolute joints, each
of which has one degree of freedom. This vehicle model has 152 bushing
elements between track links, and 954 degrees of freedom.
1.1.3. KINEMATIC RELATIONSHIPS AND EQUATIONS OF MOTION
In this investigation, the relative generalized co-ordinates are employed in
order to reduce the number of equations of motion and to avoid the difficulty
associated with the solution of differential and algebraic equations. Since the
track chains interact with the chassis components through contact forces and
since adjacent track links are connected by compliant force elements, each link
in the track chain has six degrees of freedom which are represented by three
translational co-ordinates and three Euler angles [9]. Recursive kinematic
equations of tracked vehicles were presented by Choi [4,16], who showed that
the relationship between the absolute Cartesian velocities of the chassis
components can be expressed in terms of the independent joint velocities as
343
r
iqBq (3)
where r
iq , B , and q are relative independent co-ordinates, velocity
transformation matrix, and Cartesian velocities of the chassis subsystem,
respectively. The equations of motion of the chassis that employs the velocity
transformation defined in the preceding equations are given as follows:
)qBMQBqMBB r
i
Tr
i
T ( (4)
where M is the mass matrix, and Q is the generalized external force vector of
the chassis subsystem. Since there is no kinematic coupling between the chassis
subsystem and the track subsystems, the equations of motion of the chassis
subsystem can be obtained using the preceding equation as follows:
C
i
r
i
C
i QqM (5)
where MBBMTC
i , )( r
i
TC
i qBMQBQ .
For the track subsystems, the equations of motion can be written as
ttt
QqM (5)
where tM , t
q and tQ denote the mass matrix; and the generalized
coordinate and force vectors for the track subsystem, respectively. Consequently,
the accelerations of the chassis and the track links can obtained by solving
Equations (5) and (6).
1.1.4. A COMPLIANT TRACK MODEL
Two models can be used to connect the track links of the high-mobility
tracked vehicle chains. These two models are shown in Fig. 4. In the first model,
shown in Fig. 4(a), a single pin is used to connect two links of the chain. In the
second model, shown in Fig. 4(b), two pins are used to connect the track links. In
344
both models, rubber bushings are inserted between the pins and the track links,
and as a consequence, the relative rotations between the pins and the links are
relatively small. In this section, the force models used for the single pin and
double pin connections are described.
(a) Single pin track links (b) Double pin track links
Figure 4. Track links of high mobility tracked vehicle
1.1.4.1. SINGLE PIN CONNECTION
Figure 5. Single pin connection
Figure 5 shows the details of the link, pin and bushings connection of a single
pin track link. In this investigation, a continuous force model is used to define
345
the pin joint connections. This force model is a non-linear function of the co-
ordinates of the two links. In order to define the generalized compliant bushing
forces, several coordinate systems are introduced. Two centroidal body
coordinate systems i
b
i
b
i
b ZYX and j
b
j
b
j
b ZYX for the track links i and j ,
respectively; a joint coordinate system iii ZYX whose origin is assumed to be
located at the geometric center of the circular groove containing the pin and the
bushing; and a pin coordinate system jjj ZYX whose origin is rigidly attached
to the center of the pin. Note that because of the bushing effect, the origins of the
joint and pin coordinate systems do not always coincide. The displacement of the
pin coordinate system jjj ZYX with respect to the joint coordinate system
iii ZYX is a function of the bushing stiffness. Also note that the location and
orientation of the joint coordinate system iii ZYX can be determined as a
function of the generalized co-ordinates of link i . For simplicity, it is assumed
in this investigation that the location and orientation of the pin coordinate system
can be defined in terms of the co-ordinates of link j . The deviation
T
zyxR ],,[ δ shown in Figure 5 can be used to determine the generalized
forces acting on the two links i and j as the result of the bushing effect. The
bushing force and torque applied to the frame j are given as follows:
δ
δ
0
0C
δ
δ
K
K
Q
Q
RRRR
j
j
R
C0
0
where RR ,, CKK and C are the 3 3 diagonal matrices that contain the
stiffness and damping coefficients of the bushing, and j
RQ is translational force
vector and R is the vector of translational deformations of the frame j relative
to the frame i . Similarly, j
Q is the rotational force vector and δ is the
vector of relative rotational deformations of the frame j relative to the frame i .
The force and torque applied to the frame i are assumed to be equal in
magnitude and opposite in direction to the force and torque acting on frame j .
Once these forces are determined, the generalized bushing forces associated with
346
the generalized co-ordinates of the track links i and j can be determined.
1.1.4.2. DOUBLE PIN CONNECTION
Figure6. Double pin connection
In the double pin assembly, shown in Figure 6, two adjacent track links are
connected with a connector element using two pins and rubber bushings. The
mass and mass moment of inertia of the connector element are relatively small as
compared to those of the track links. Therefore, the dynamic effects of connector
element are modeled in this investigation. This approach has the advantage of
reducing the number of degrees of freedom of the system. The double pin
assembly can be modeled by considering one radial, one axial, and three
rotational springs. The radial spring provides the restoring force due to the
combined translational deformation of the two rubber bushings along the radial
direction of the connector as shown in Fig. 6. The axial spring restricts the
translational motion of the two links along the lateral direction as shown in Fig.
6. The rotational springs are used to model the relative rotational deformation
between the two track links. The length l of the radial spring is assumed to be
the distance between the origins of the coordinate systems iii ZYX and
347
jjj ZYX shown in Fig. 6. This distance is defined as
ijij2ddl
T
(8)
The magnitude of the force produced by the radial spring is
lCllKF r0rr (9)
where rK is the spring stiffness coefficient, and rC is the damping coefficient,
and i is obtained by differentiating Equation (8) with respect to time. Similarly,
the restoring force due to the translational spring along the iZ axis is
i,j
zRz
i,j
zRzz CKF (10)
where i,jz is translational deformation of the jjj ZYX frame with respect to
the iii ZYX frame along the Z axis, RzK and RzC are the stiffness and
damping coefficients.
The first two components of the bushing restoring torque as the result of the
relative rotation of link i with respect to link j are given by
i,j
xxi,j
xxx CKT (11)
i,jyy
i,jyyy CKT
(12)
where i,jx and i,j
y are relative rotational deformations of the jjj ZYX frame
about x-axis and y-axis with respect to the iii ZYX frame, respectively, xK ,
yK , xC , and yC are stiffness and damping coefficients. The restoring bushing
torque about the jZ axis due to the rotation of link j with respect to link i
ib,j
zzib,j
zzz CKT (13)
where ib,jz is relative angle between the link i and the connector element and
348
can be obtained by defining the components of the ijd in the i coordinate
system of link i as
iji dAdT
ijz
ijy
ijx
ij
d
d
d
(14)
It follows that
ijx
ijy
ib,iz d/d-1tan (15)
where iA is the transformation matrix that defines the orientation of link i
with respect to the global frame. Note that since the inertia of the connector
element is neglected, the resultant force acting on this element must be equal to
zero. Using the spring forces defined in this section, the generalized bushing
forces acting on the track links can be systematically defined.
1.1.5. MEASUREMENT OF TRACK COMPLIANCE CHARACTERISTICS
In order to determine the stiffness and damping coefficients of the contact
force models used in this investigation, an experimental study is conducted to
examine the road wheel and track link contact as well as the interaction between
the track links. Since the experimental results are to be used in the dynamic
simulation of the multibody tracked vehicle, the dynamics of the contact is also
considered in the measurement process.
While a viscous damping force is proportional to the velocity, in many cases,
analytic expressions for the damping forces are not directly available. It is,
however, possible to obtain an equivalent viscous damping coefficient by
equating energy expressions before and after a contact. In this investigation, the
effective stiffness and damping coefficient are obtained by employing the
hysteresis loop method [10]. The effective stiffness and damping coefficient of
single degree of freedom system are given as follows [10]:
cosx
FmK
0
02
effeff (16)
349
s i n
0
0
x
FCe f f (17)
where eff
m , 0
F , 0
x , , and are the effective mass, the magnitude of
applied force, the magnitude of displacement, the natural frequency, and the
phase angle of displacement, respectively.
In these experimental studies, forces are applied to the center of the road
wheel which is in contact with a track link fixed to a rigid frame. A LVDT
sensor is attached between the center of the wheel and a track link fixed base to
measure the relative displacement. For a static test, the actuator force is
increased gradually up to 10 ton with 2mm/min velocity. For a dynamic test, the
actuator force is excited harmonically up to 35 Hz. Frequencies higher than 35
Hz are not considered in the measurement because of noise and system
resonance. The relationship between the effective stiffness, damping coefficient,
and frequencies is given by Park et al [11]. It can be shown that the effective
stiffness increases up to a frequency of 10 Hz and does not significantly change
after this frequency. On the other hand, the effective damping coefficient
decreases as the frequency increases.
A LVDT sensor is attached between two adjacent track links to measure the
relative displacement. For a static test, the actuator force is increased gradually
up to 10 ton with 2mm/min velocity. Figure 7 shows the resulting load-
displacement relationship. For a dynamic test, a harmonic actuator force with a
frequency up to 50 Hz is used. Figure 8 shows the hysteresis loop when the load
frequency is 10 Hz with 5 ton pre-static applied force. It can be observed that the
effective stiffness increases up to 12 Hz and does not significantly change after
this frequency. The effective damping coefficient, on the other hand, decreases
as the frequency increases.
350
Figure 7. Load-displacement relationship for Radial static measurement
(Pre-static load: 5ton, Forcing freq.: 10HZ)
Figure 8. Hysteresis loop for radial dynamic test
A connector end is welded to the fixed track shoe plate and the other end of
the connector is attached to a load cell, which is connected to an actuator
cylinder by revolute joint. Fourteen degrees of the pre-set angle is given in the
351
pin. A torque of 500 ton-mm is applied along the directions of rotation. The
static torque versus the rotational angle, the effective torsional stiffness versus
frequency, and the effective damping coefficient versus frequency are plotted by
Park et al [11]. The experimental results showed that the effective torsional
stiffness is less sensitive to the loading frequency and the effective damping
coefficient decreases to a small value when the frequency exceeds 20 Hz.
In this investigation, for the sake of simplicity, the stiffness and damping
coefficients used in the force models are determined using empirical methods
based on the results of the static test only. A spline curve fitting is used to obtain
the compliant characteristics between measurements.
1.1.6. CONTACT FORCES
In this section, the methods used for developing the contact force models used
in this investigation are briefly discussed. The scenarios of the contacts between
the track links and the road wheels, rollers, sprockets, and the ground are
explained. A more detailed discussion on the formulation of the contact forces is
presented by Choi, et al, [4, 12], and Nakanishi and Shabana [3].
(a) inner surface contact (b) edge contact
Figure 9. Track link and wheel interactions
1.1.6.1. INTERACTION BETWEEN TRACK AND ROAD WHEEL, IDLER, AND
SUPPORT ROLLER
As shown in Fig. 9, each roller of the vehicle model used in this investigation
352
consists of two wheels which are rigidly connected. There are four different
possibilities for the roller and track interaction. The first possibility occurs when
a track link and one wheel of the roller are in contact. In this case, a concentrated
contact force is used at the center of the contact surface of the wheel. The contact
force acting on the link is assumed to be equal in magnitude and opposite in
direction to the force acting on the roller. The second possibility occurs when
both wheels of the roller are in contact with the track link. In this case, two
concentrated contact forces are applied to the roller and the track link. The third
and fourth possibilities occur, respectively, when either one wheel or both
wheels are in contact with the edges of track link. In such a case, one or two
concentrated contact forces are applied to the wheel and the edge of the track
link.
1.1.6.2. TRACK CENTER GUIDE AND ROAD WHEEL INTERACTIONS
(a) side wall contact (b) top surface contact
Figure 10. Center guide and wheel interactions
Figure 10 shows a schematic diagram for a track center guide and a road
wheel when they are in contact. As previously pointed out a road wheel of the
vehicle model used in this investigation consists of two wheels, which are rigidly
connected, and therefore, there are four possibilities for the track center guide
and wheel interactions, as shown in the Figure 10. The first possibility is the case
in which the right side plate of the wheel is in contact with the left side wall of
353
the track center guide. In the second possibility, the left side plate of the wheel is
in contact with the right side wall of the track center guide. The third possibility
occurs when one bottom surface of wheel and the top surface of track center
guide are in contact. In these three contact cases, a concentrated contact force is
introduced at the contact surface of the road wheel, and that contact force is
equal in magnitude and opposite in direction to the force acting on the track link.
The fourth possibility occurs when the two road wheels are not in contact with
the track center guide. In this case, no generalized contact forces will be
introduced.
1.1.6.3. INTERACTION BETWEEN THE SPROCKET TEETH AND TRACK LINK
PINS
In this investigation, five tooth surfaces are used to represent the spatial
contact between the sprocket teeth and the track link pins. During the course of
engagement between the sprocket teeth and the track links, several sprocket teeth
can be in contact with several track link pins, as shown in Fig. 11. The sprocket
used in this investigation has ten teeth, and each tooth has five contact surfaces.
These surfaces are the top, the left, the right, front, and back surfaces. A
Cartesian coordinate system is introduced for each surface. The surface
coordinate system is assumed to have a constant orientation with respect to a
selected tooth coordinate system. The tooth coordinate system has a constant
orientation with respect to the sprocket coordinate system. Therefore, the
orientation of a surface coordinate system can be defined in the global system
using three coordinate transformation matrices; two of them are constant and the
third is the time dependent rotation matrix of the sprocket. Using these
coordinate transformations and the absolute Cartesian co-ordinates of the origin
of the sprocket coordinate system, the location and orientation of each tooth
surface can be defined in the global coordinate system. Using the track link
coordinate system, the global position vector of the center of the track link pin
can be defined.
354
(a) sprocket teeth and connector contact (b) teeth side wall and link side wall
contact
(c) teeth top surface and link inner surface contact
Figure 11. Sprocket tooth and track interaction
This vector and the global co-ordinates of the tooth surfaces can be used to
determine the position of the track link pins with respect to the sprocket teeth.
The relative position of the track link pins, with respect to the sprocket teeth can
be used to develop a computer algorithm that determines whether or not the track
link pin is in contact with one of surfaces of the sprocket teeth. The interactions
between the track link pins and the sprocket base circle are also considered in
this investigation. To this end, the distance between the center of the track link
pin and the center of sprocket is monitored. When this distance is less than the
sum of the pin radius and the sprocket base circle radius, contact is assumed and
a concentrated force is applied to the sprocket and the track link pin.
1.1.6.4. GROUND AND TRACK SHOE INTERACTIONS
The track link used in this investigation has a single or double shoe plate, and
therefore, there are one or two surfaces on each track link that can come into
contact with the ground. The global position vectors that define the location of
points on the shoe plates are expressed in terms of the generalized co-ordinates
of the track links and are used to predict whether or not the track link is in
contact with the ground. In this investigation, contact forces are applied at
355
selected six points on the track link shoe when it comes into contact with the
ground. The normal force components are used with the coefficient of friction to
define the tangential friction forces [4, 12].
1.1.7. METHOD OF NUMERICAL INTEGRATION
The equations of motion of a tracked vehicle are formulated as a set of
differential equations, as described in Section 3. The solution of the differential
equations can be obtained by step-by-step numerical integration. There are two
types of integration methods; one is the implicit method and the other is the
explicit method. The implicit method generally has a larger stability region, but
it requires solving a system of nonlinear equations. The explicit method, on the
other hand, has relatively smaller stability region, but it requires solving only a
system of linear equations. In this investigation, an explicit method is employed.
The dynamics of tracked vehicles is characterized by high impulsive forces
resulting from the contact between the track chains and the vehicle components
as well as the ground. Because of the high frequency impulsive forces, the
numerical integration routine is forced to take a small time step, and as a
consequence, the simulation of a complex tracked vehicle model, as the one
described in this paper, represents a challenging task. Nonetheless the high
frequency oscillations may have little influence on the low frequency motion. In
this case, the high oscillations can be damped out to obtain the gross motion of
the track link. Various dissipation algorithms for time integration of structural
systems have been proposed [7,8,13]. In this investigation, the method proposed
by Chung and Lee [7] is considered because of its easy implementation and
computational efficiency. Accuracy and stability conditions must be considered
in carrying out a numerical integration of the tracked vehicle equations. The
accuracy and stability conditions are obtained by using the truncation error and
the error propagation analyses. A variable step algorithm is proposed in the
following subsections.
356
1.1.7.1. ACCURACY ANALYSIS
The following numerical integrator proposed by Chung and Lee [13] is
employed in this research.
)q,N(qMq 1
nnn (18)
])2/3()2/1[(Δ n1nn1n qqqq t (19)
])27/28()54/29[( n1nnn1n qqqqq 2 tt (20)
The 1n
q can be expanded by the Taylor series as follows:
)(O)2/( 3
n
2
nn1n ttt qqqq (21)
where )(O 3t is collection of higher order terms. Subtracting Equation (20)
from Equation (21) yields the truncation error as follows:
)(O))(54/29()( 32
n1nn ttt qq (22)
)(Odt
d)54/29( 33
n tt q (23)
which shows that the proposed integrator achieves the second-order accuracy for
non-linear dynamic systems.
1.1.7.2. STABILITY ANALYSIS
Since it is difficult to analyze the stability condition for a general nonlinear
system, the following linear, undamped, and unloaded system is considered:
02 nn qq (24)
where is a natural frequency. Applying Equation (24) with the integration
formula proposed in this section yields the one step form of the numerical
scheme
nn HXX 1 }1,......,2,1,0{ Nn (25)
357
where T
n
2
nnn ]q,q,q[ tt X (26)
and
00
2/11)2/3(
54/291)27/28(1
2
2
2
H (27)
in which t . The characteristic equation for H is obtained as follows:
0)27/1(})27/2(1{})27/28(2{)det( 2223 IH (28)
where I is the 33 identity matrix and denotes the eigenvalue. The
stability characteristics of the method are determined by the condition that the
roots of the characteristic equation remain in or on the unit circle of the complex
plane as follows:
1 , 321
,,max (29)
where is called the spectral radius. Stability analysis can be assessed by
using the transformation of Eq. 7.9 to map the interior of the unit circle into the
left half-plane and by applying the Routh-Hurwitz criteria to the transformed
characteristic equation. The stability condition for the algorithm is obtained by
applying the Routh-Hurwitz criteria as follows:
0)27/31(4 2 (30)
0≥)27/23(4 2 (31)
which are reduced to
)/8665.1( t (32)
Equation (32) provides a guideline in choosing a step size that satisfies the
stability condition.
358
1.1.7.3. IMPLEMENTATION OF A VARIABLE STEPPING ALGORITHM
Since the governing equations of motion for the tracked vehicle system are
highly nonlinear, the integration step size must be varied so that both the
accuracy and stability conditions are satisfied. For the accuracy condition,
ignoring the higher order terms in Equation (22) yields the local truncation error
formula as follows: 2
n1nn |)(|)54/29()( tt qqτ (33)
The allowable stepsize with a given error tolerance is obtained by solving
Equation (33) for t as follows:
2/1
n1n |}|)(|29/{54| qqτ t (34)
For the stability condition, the apparent frequency method proposed by Park
and Underwood [14] is employed in this research. An apparent frequency is
estimated by substituting q and q into the following equation:
01
2
1
i
napp
i
n qq },......,2,1,0{ qni (35)
where app is the apparent frequency and qn is the number of generalized co-
ordinates. The highest apparent frequency is selected as the reference frequency
in determining the step size.
The step size determination algorithm is shown in Figure 12. Note that the
stability condition of app/ω5.0t instead of app/ω8665.1t in Equation (32)
is used for conservative numerical integration. The integration step size
employed by the variable step integration algorithm used in this investigation,
when the vehicle maximum acceleration, steady state velocity at 50 Km/h and
stiff deceleration of braking, is shown in Figure 13. This figure shows that the
integration step size is relatively depended on the vehicle speed. The increment
of vehicle speed will enlarge impulsive contact forces and oscillation of track
links, and integration step size should be decreased, accordingly.
359
Figure 12. Variable stepsize algorithm
Figure 13. Stepsize of variable step integration algorithm
360
1.1.8. NUMERICAL RESULTS
The high mobility tracked vehicle shown in Figure1 is used as a simulation
model in order to demonstrate the use of the methods proposed in this paper.
Several simulation scenarios, including acceleration, high speed motion, braking
and turning motion, are presented in this investigation. In the simulation of
acceleration, high speed motion, and braking of the vehicle, the same angular
velocity is used for both left and right sprockets in order to obtain straight line
motion. The angular velocities of the sprockets are increased linearly up to -45
rad/s in 10 s, kept constant for 3 s, and then decreased linearly to 0 rad/s in 4 s.
The coefficient of friction between the track links and ground is assumed to be
0.7 in the case of rubber and concrete contact. The double pin track link is used
in the numerical study presented in this section. Figures 14-18 show the
numerical results of simulation of the acceleration, steady-state velocity and
deceleration.
Figure 14. Vertical motion of a track link
361
Figure 15. Radial tension of track link
The vertical displacement of a track link with respect to the global coordinate
system during the constant velocity motion is shown in Figure 14. This figure
clearly shows the effect of three support rollers, idler, six road wheels and
sprocket on the vertical displacement of the track link. The track tension can
have a significant effect on the dynamic behavior of tracked vehicles, such as
preventing the separation of the track chains [18,19] from chassis, distribution of
mean maximum pressure(MMP), power efficiency, and the life of the track chain.
As previously pointed out in Section 2, about 10 percent of the vehicle weight
[15] is used as track pre-tension. Simulation results showed that the track tension
significantly decreases after the start of the motion.
(a) (b)
362
(c) (d)
(e)
Figure 16. Sprocket teeth loading contour: (a) acceleration; (b) cruise at high speed; (c)
braking; (d) turning (right sprocket); (e) turning ( left sprocket)
Figure 15 shows the longitudinal track tension in the bushing between track links,
while Figure 16 shows sprocket teeth loading contour. Heavy duty high mobility
tracked vehicles as one used in this study have, in general, double pin track links.
One of the main advantages of using double pin track is that the shear stress on
the rubber bushings can be significantly reduced as compared to the single pin
track link. In order to compare the loadings on the track bushings in the case of
single or double pins, new driving conditions are examined. The rotational speed
of both sprockets is decreased linearly up to - 9 rad/sec in 2 sec, and then kept
constant velocities.
Figure 17 illustrates the moment on the rubber bushings in the case of the
single and double pin track. The results presented in this figure demonstrate the
significant reduction of the load on the rubber bushings when a double pin track
is used. Figure 18 shows the norm of the contact forces exerted on one of the
363
links of the right track chain as the result of its interaction with the road wheels,
support rollers, idler, sprocket, and ground. Figure 19 shows a road arm angle
and HSU gas pressure of the second road wheel.
Figure 17. Torsional moments of track rubber bushing.
Figure 18. Contact forces of track link
364
Figure 19. Gas pressure of HSU system
The second simulation scenario used in this study is a turning motion. The
turning motion is obtained by providing two different values for the angular
velocities of the sprockets. The angular velocity of the right sprocket is
decreased linearly to -9 rad/s and the angular velocity of the left sprocket is
increased linearly to 9 rad/s in 2 s. The angular velocities are then kept constant
velocities. Using these values for the sprocket angular velocities, the vehicle
rotates counter clock wise as result of opposite rotation directions of right and
left sprockets, the upper part of the right side of the track chain is loose, and the
upper part of the left side of the track chain is tight as shown in Figure 20. Figure
21 shows the forces of contact between side wall of the wheels and center guide
of a track link.
365
Figure 20. Tension adjuster force
Figure 21. Track center guide and wheel contact forces
366
1.1.9. SUMMARY AND CONCLUSIONS
The dynamics of a high speed, high mobility multibody tracked vehicle is
investigated in this paper. Compliant forces are used to define the connectivity
between the links of the track chains instead of an ideal pin joint. Two track link
models are considered in this study. These are the single pin and double pin track
models. In the single pin track model, only one pin is used to connect two track
links in the chain. In the double pin track model, two pins are used with a
connector element to connect two links of the track chain. Rubber bushings are
used between the track links and the pins. The stiffness and damping
characteristics of the contact forces are obtained using experimental testing. By
using experimental data, the generalized contact and bushing forces associated
with the generalized co-ordinates of the tracked vehicle are developed. The
tracked vehicle model used in this investigation includes significant details that
include modeling the chassis, sprockets, idlers, road wheels, road arms, and the
multi-degree of freedom track chains. The vehicle model is assumed to consist of
189 bodies, 36 pin joints, and 152 bushing elements. The model has 954 degrees
of freedom. Because of the high frequency contact forces, numerical difficulties
are often encountered in the simulation of multibody tracked vehicles. An
explicit numerical integration method that has a large stability region is
employed in this study. The method employs a variable time step size in order to
achieve better computational efficiency. It was observed that the time step size
significantly decreases as the vehicle speed increases. Several simulation
scenarios are examined in this investigation. These include accelerated motion,
high speed motion with a constant velocity, braking, and turning motion. The
simulation results demonstrate the significant effect of the bushing stiffness on
the dynamic response of the multibody tracked vehicle. It was also shown that
the use of the double pin track model leads to a significant reduction in the
bushing forces as compared to the single pin track model.
367
REFERENCES
(1) Galaitsis A.G., 1984, “A Model for Predicting Dynamic Track Loads in Military
Vehicles,” ASME, Journal of Vibration,Acoustics, Stress, and Reliability in Design, Vol.
106/289
(2) Bando, K., Yoshida, K., and Hori, K., 1991, “The Development of the Rubber Track
for small Size Bulldozers,” International off-Highway Powerplants Congress and
Exposition, Milwaukee, WI, Sept. 9-12
(3) Nakanishi, T., and Shabana,(1994)"Contact Forces in The Nonlinear Dynamic Analysis
of Tracked Vehicle," International Journal For Numerical Methods in Engineering, 1994
1251-1275.
(4) Choi, J. H., 1996 "Use of Recursive and Approximation Methods in The Dynamic
Analysis of Spatial Tracked Vehicle," Ph. D. Thesis, The University of Illinois at
Chicago
Scholar C. and Perkins N., 1997, “Longitudinal Vibration of Elastic Vehicle Track
System”
(5) Newmark NM. “A method of computation for structural dynamics.” Journal of the
Engineering Mechanics Division, ASCE 1959; 85 (EM3):67-94
(6) J. Chung, J. M. Lee,(1994) “A New Family of Explicit Time Integration Methods for
Linear and Non-linear Structural Dynamics,” International Journal for Numerical
Methods in Engineering, Vol.37, 3961-3976
(7) J. Chung,(1992) “Numerically Dissipative Time Integration Algorithms for Structural
Dynamics,” Ph.D. dissertation, University of Michigan, Ann Arbor
(8) Shabana A,(1989) “Dynamics of Multibody Systems,” John Wiley & Sons, New York
(9) Shabana, A,(1996) “Theory of Vibration, An Introduction,” Second Edition, Springer-
Verlag, New York
(10) Park DC, Seo IS, Choi JH. Experimental study on the contact stiffness and damping
coefficients of the high mobility multibody tracked vehicle. Journal of Korea Society of
368
Automotive Engineers 1999; 7:348-357
(11) Lee, H. C., Choi., J. H., Shabana, A. A., Feb. 1998 "Spatial Dynamics of Multibody
Tracked Vehicles: Contact Forces and Simulation Results," Vehicle System Dynamics, Vol.
29, pp. 113-137
(12) E. L. Wilson,(1968) “A Computer Program for the Dynamic Stress Analysis of
Underground Structures,” SESM Report No. 68-1, Division of Structural Engineering
and Structural Mechanics, University of California, Berkeley
(13) K. C. Park and P. G. Underwood,(1980) “A Varialbe-step Centeral Difference Method
For Structural Dynamics Analysis – Part 1. Theoretical Aspects,” Computer Methods in
Applied Mechanics and Engineering 22, 241-258
(14) Owen J. Guidelines for the Design of Combat Vehicle Tracks. Dew Engineering and
Development Ltd., Ottawa, Canada.
(15) Choi, J. H., Lee., H. C., Shabana, A. A., Jan. 1998 "Spatial Dynamics of Multibody
Tracked Vehicles: Spatial Equations of Motion," International Journal of Vehicle
Mechanics and Mobility, Vol. 29, pp. 27-49
(16) Bruce Maclaurin (1983) “Progress in British Tracked Vehicle Suspension Systems,”
830442 Society of Automotive Engineers(SAE)
(17) Ketting Michael,(1997) “Structural Design of Tension Units for Tracked Vehicles,
especially Construction Machines Under The aspect of Safety Requirements,” Journal
of Terramechanics, Vol. 34, No. 3, pp. 155-163.
(18) Trusty RM, Wilt MD, Carter GW, Lesuer DR. Field measurement of tension in a T-142
tank track. Experimental techniques, 1988.
369
1.2
DYNAMIC TRACK TENSION OF HIGH
MOBILITY TRACKED VEHICLES
1.2.1. INTRODUCTION
The track tension of tracked vehicles plays significant roles for the dynamic
behaviors, such as separation of the track system from the chassis system,
distribution of wheel supporting pressure, power efficiency, vibration and noise,
and the life of the track system. Due to the importance of the track tension in
designing tracked vehicles, study of the dynamic track tension has long been a
subject for many researchers in manufacturers and academia. However, it is very
difficult to clearly understand the nonlinear behaviors of the dynamic track
tension while a vehicle runs, even though both experimental and numerical
works have been attempted [1-7].
Both numerical and experimental investigations are carried out in this paper.
For the experimental investigation, strain gages are attached on track pin-
bushing locations of track shoe body, and signal processing and recording
modules are installed on the inside of track shoe body. Only limited results can
be collected through the experiment due to small installation space inside of a
track shoe body, high impulsive shock and vibration, and high temperature over
150 cent degrees. In order to make up the limitation of experimental results, a
tracked vehicle model developed in [9] is used to obtain various numerical
results. Each track link is modeled as a body which has six degrees of freedom
and is connected by a bushing force element. The numerical results are validated
against the experimental results before they are used for investigations.
Doyle and Workman [1] presented a static prediction of track tension when
the suspensioned-tracked vehicle traverses obstacles using two dimensional
finite element methods. An elastic beam element subjected to tension,
compression and bending loads was utilized to model track links. Galaitsis [2]
370
demonstrated that the analytically predicted dynamic track tension and
suspension loads of a high speed tracked vehicle are useful in evaluating the
dynamic analysis of the vehicle. The predicted track tensions were compared
with the empirically measured track tensions. A detailed track tension
measurement methodology and results are presented by Trusty et al [3]. Strain
gages connected to a portable data acquisition system were installed in the track
link. The flat ground, quick acceleration, traversal of obstacle courses, pivot
turns, moving uphill, and pre and post tension, were used for the tension
investigation scenarios. McCullough and Haug [4] designed a super element that
represents spatial dynamics of high mobility tracked vehicle suspension systems.
The track was modeled as an internal force element that acts between ground,
wheels and the chassis of the vehicle. Track tension was computed from a
relaxed catenary relationship. Empirical normal and shear force formulas based
on constitutive relations from soil mechanics were used to model the soil-track
interface. Choi [5, 6] presented a large scale multibody dynamic model of a
construction tracked vehicle in which the track is assumed to consist of track
links connected by single degree of freedom pin joints. In this detailed three
dimensional dynamic model, each track link, sprocket, roller, and idler is
considered as a rigid body that has a relative rotational degree of freedom.
Scholar and Perkins [7] developed an efficient alternative model of the track
chains considering longitudinal vibrations. The track is assumed to consist of a
finite number of segments, each of which is modeled as a continuous uniform
elastic rod attached to the vehicle wheels. Overall chain stretching effects are
accounted for.
The purpose of this paper is to investigate the dynamic track tensions of a high
mobility tracked vehicle maneuvering under various driving conditions. Both
numerical and empirical methods are employed and the effects of pretensions,
friction forces, interacting proving grounds, vehicle speeds, and driving torque
are explored for the sake of understanding dynamic behaviors of the track system.
371
1.2.2. NUMERICAL MODEL OF A HIGH MOBILITY TRACKED
VEHICLE
Figure 1. High mobility multibody tracked vehicle model
A three-dimensional multibody tracked vehicle model shown in Fig. 1 consists
of a chassis subsystem and two track subsystems. The chassis subsystem
includes a chassis, sprockets, support rollers, idlers, road arms, road wheels and
the suspension units. The sprockets, support rollers, and road arms are connected
to the chassis by revolute joints. The track link subsystem includes a shoe body,
a pin, rubber bushings, and a rubber pad. Rubber bushings and pin are inserted
into the hole of a shoe body with a radial pre-pressure and a rubber pad is
mounted on the ground interaction side of the shoe body. The vehicle model
used in this investigation consists of 189 bodies; 37 bodies for the chassis
subsystem, 76 bodies for each track subsystem, 36 revolute joints and 152
bushing elements and has 954 degrees of freedom.
Track system
Chassis system
Turret
system
372
Figure 2. Hydraulic track tension adjustor system
Suspension systems and tension adjustor: The suspension units of the
vehicle include a Hydro-pneumatic Suspension Unit(HSU), and torsion bar that
are modeled as force elements whose compliance characteristics are obtained
from analytical and empirical methods. The HSU systems are mounted on first,
second, and sixth stations to damp out pitching motion and to decrease an impact
when the vehicle is running over large obstacles. The torsion bars are mounted
on the middle stations for this vehicle model. A simple torsional spring model is
used in this investigation to represent the stiffness of the torsional bars. The
hydraulic passive tension adjustor is installed on the idler to maintain a proper
track tension of the tracked vehicle model. Figure 2 shows the schematic
diagram of the tensioner system of the vehicle. The hydraulic ram of the tension
adjustor is modeled as an equivalent linear spring-damper force element.
Track link connection: Each track subsystem is modeled as a series of
bodies connected by rubber bushings around the link pins which are inserted into
a shoe plate with a radial pressure to reduce rattling of the pin. When the vehicle
runs over rough surfaces, the track systems are subjected to extremely high
impulsive contact forces as the result of their interaction with the vehicle
components such as road wheels, idlers, and sprocket teeth, as well as the ground.
The rubber bushings tend to reduce the high impulsive contact forces by
373
providing cushion and reducing the relative angle between the track links. In this
investigation, a continuous force model is used to represent the pin connections.
This force model is a non-linear function of the coordinates of the two links.
Note that because of the bushing effect, the origins of the joint and pin
coordinate systems do not always coincide.
Contact detection and forces: In this section, the contact force model and the
contact detection algorithms between the track links and the road wheels, rollers,
and sprockets are briefly discussed. A more detailed discussions on the
formulation of the contact force model is presented by Choi, et al, [5, 6, 9]. As
shown in Fig. 1, when a track link travels around vehicle components, its
trajectory is determined by the contact forces. These forces are created by
detecting on contact conditions. The contact detection algorithms monitor the
contacts of, wheel and track link contact, center guide and wheel contact,
sprocket tooth and track link pin contact, and side wall of track link and sprocket
contact. Once a contact condition is satisfied, contact forces are applied at the
contacted position to restitute each other.
Figure 3. Interaction between track shoe body and triangular patch element
374
Interacting ground representations: The ground interacting surface of a track
link can be single or multiple, and therefore, there are one surface or multiple
surfaces on each track link that can come into contact with the ground. The
interacting surface of ground is discretized and each contact node points were
defined. The global position vectors that define the locations of points on the
shoe plates surface of track link are expressed in terms of the generalized
coordinates of the track links and are used to predict whether or not the track link
is in contact with the ground. Since the contact surface of track link consists of
rubber pad and steel shoe plate, the contact forces at each node point are
evaluated by using their own stiffness and damping coefficients. In order to
construct various geometries of tracked vehicle paved proving ground [10], such
as bumping courses, trench course, inclined course, standard cross country
courses, descritized terrain representation methods using triangular patch
element are used in this investigation. The plane equation of interacting ground
profiles for a triangular patch element which has three nodes and a unit normal
vector is employed as illustrated in Fig. 3.
Equations of motion: Since the track system interacts with the chassis
components through the contact forces and adjacent track links are connected by
compliant force elements, each track link in the track system has six degrees of
freedom which are represented by three translational coordinates and three Euler
angles [11]. The equations of motion of the chassis that employs the velocity
transformation defined by Choi [5, 9] are given as follows:
)( r
ii qBMQBqMBBTT (1)
where r
iq , B and q are relative independent coordinates, velocity
transformation matrix, and Cartesian velocities of the chassis subsystem, and M
is the mass matrix, and Q is the generalized external and internal force vector of
the chassis subsystem, respectively. Since there is no kinematic coupling
between the chassis subsystem and track subsystem the equations of motion of
the track subsystem can be written simply as
375
ttt QqM (2)
where tM , t
q and tQ denote the mass matrix; and the generalized
coordinate and force vectors for the track subsystem. Consequently, the
accelerations of the chassis and the track links can obtained by solving Eqs. 1
and 2.
G-Alpha integrator : Many different types of integration methods can be
employed for solving the equations of motion for mechanical systems. Explicit
methods have small stability region and are often suitable for smooth systems
whose magnitude of eigenvalues is relatively small. Contrast to the explicit
methods, implicit methods have large stability region and are suitable for stiff
systems whose magnitude of eigenvalues is large. One of the important features
of the implicit methods is the numerical dissipation. Responses of mechanical
systems beyond a certain frequency may not be real, but be artificially
introduced during modeling process. In the model used in this investigation, a
contact between two bodies is modeled by compliance elements. Lumped
characteristics of the spring and damper must represent elastic and plastic
deformations, and hysterisis of a material. Such characteristics may include
artificial high frequencies which are not concern of a design engineer. Unless
such artificial high frequency is filtered, an integration stepsize must be reduced
so small that integration cannot be completed in a practical design cycle of a
mechanical system. To achieve this goal, generalized-alpha method [8, 9] has
developed to filter frequencies beyond a certain level and to dissipate an
undesirable excitation of a response. One of the nice advantages of the
generalized-alpha method is that the filtering frequency and dissipation amount
can be freely controlled by varying a parameter in the integration formula. As a
result, the generalized-alpha method is the most suitable integration method for
integrating the equations of motion for stiff mechanical systems. Figure 4 shows
the animation of high mobility tracked vehicle when the vehicle runs over a
trench profile.
376
(a) time = 0.0 sec (b) time = 3.0 sec
(c) time = 6.0 sec (d) time = 9.0 sec
Figure 4. Computer animation of multibody tracked vehicle running
over trench ground profile
1.2.3. INTERACTION GROUNDS
The ground interacting surface of a track chain link can be single or multiple,
and therefore, there are one surface or multiple surfaces on each track link that
can come into contact with the ground. The interacting surface of chain link is
discretized and each contact node points were defined. The global position
vectors that define the locations of points on the shoe plates surface of chain link
are expressed in terms of the generalized coordinates of the track chain links and
are used to predict whether or not the track chain link is in contact with the
ground. Since the contact surface of track chain link consists of rubber pad and
steel shoe plate, the contact forces at each node point are evaluated by using their
own stiffness and damping coefficients. In order to construct various geometries
of tracked vehicle paved proving ground, such as bumping courses, trench
course, inclined course, standard cross country courses, discretized terrain
representation methods using triangular patch element are used in this
investigation. A triangular patch element has three nodes and a unit normal
vector to describe plane equations of interaction grounds [13].
377
(a) Series of triangular patch for generalized virtual body
(b) Triangular patch surface
Figure 5. Discretized terrain representation
Discretized terrain representation: The virtual terrain model used in this
investigation is a general three dimensional surface defined as a series of
triangular patch elements. Figure 5 (a) shows an example of virtual ground using
8 points and 6 elements. Most geometries of various paved proving ground for
tracked vehicle can be represented by using triangular patch elements. The
equation for the plane defined from three nodes can be written as
zxaxaxa 321 (3)
The three coefficients, 1a , 2a , and 3a of the equations of plane can be
obtained by given three locations of triangular patch shown in Fig. 5 (b), and by
using Cramer's rule [14], these coefficients are
378
A
A
det
det k
ka , k = 1, 2, 3 (4)
where
333
332
331
33
][
][
][
][
kkk
kk
kk
kk
zyxA
zxA
yzA
yxA
I
I
I
and T]111[I (5)
Then the unit normal vector of the plane n̂ is defined as
Taaaa
]1[1
1ˆ
212
2
2
1
n (6)
Virtual proving ground : Until recent development of computer simulation model
[6, 7, 9], the development process of tracked vehicle have been depended on
inefficient technologies of repeated procedures ; construction prototype vehicle
based on basic calculations and simple computer simulation, test on proving
ground, then modification. This expensive design procedure can be diminished
by recent developments of computer simulation.
In this investigation, only paved ground models are developed for the virtual
test of dynamic analysis of three dimensional tracked vehicle. The developed
computer models of grounds are stored into the created ground library.
As shown in Figs. 6 and 7 the variety of virtual proving grounds, symmetric
and unsymmetric bump courses, trench and ditch courses, longitudinal and
laternal inclined courses, and standard cross-country courses of RRC9 and
Profile IV, are constructed by using triangular patch elements. When a vehicle
runs over these virtually created proving grounds, the nonlinear behaviors of
track chains resulting from the interacting with the test grounds are obtained in
this numerical investigation.
379
(a) Single bump course (d) Obstacle course
(b) Trench course (e) Grade ability slope
(c) Ditch course (f) Side slope
Figure 6. Various paved virtual proving ground using triangular patch
380
(a) Series of triangular bump
(b) Series of trapezoidal bump
Figure 7. Simulated cross-country course (APG Profile IV)
Methods of finite track chain-ground contact point: Unlike wheel and surface
contact, the interactions between track chain link and ground are very
complicated problems. This is because the track chain link has irregular contact
381
geometry and different material properties. According to large number of track
chain links of each track subsystem, commonly used contact theory of surface to
surface interactions in finite element community can not be employed for this
work. Choi [15] suggested that element free finite contact nodes were distributed
on the contact surface of track chain link, which have their own stiffness and
damping characteristics. The relative indentations of each nodes were monitored
and positions are restored. The use of element free finite contact node methods
demonstrated clearly the computational efficiency for dynamic analysis of track
chain system. Based on the method developed by Choi [15], the interactions
between track chain link surface and triangular patch surface are developed in
this paper. Figure 3 shows the interaction between finite contact nodes of track
link and triangular patch surface. The perpendicular deformation scalar ij
kd of
contact node j of link i on patch plane k can be defined as
ij
k
Pij
kd nr ˆ1 (7)
where P1r is shown in Fig. 3 and unit vector ij
kn̂ is defined in Eq. 6.. The
criterion of necessary condition for the contact to occur of node j , which is
not sufficient, is
separated
contactdij
k
ij
k
0
0 (8)
If this conditions is satisfied, the position vector jk
Br shown in Fig. 3 is used
to compute the node location whether contact point B of node j is on the
patch plane k . The position vector jk
Br can be written as
ij
k
ij
k
ij
p
iijk
B duARr n̂ (9)
where iA is the transformation matrix associated with the orientation
coordinates of link i and ij
pu is the local position vector of node j in the
382
track chain link coordinate system. On the other hand, using scalar triple product
if one of the following conditions is satisfied
0
0
0
331
223
112
n
B
n
B
n
B
urr
urr
urr
or
0
0
0
331
223
112
n
B
n
B
n
B
urr
urr
urr
(10)
then the node j of link i is in contact with patch element k .
If the node j is in contact with patch plane k , the contact force at the
contact node can be computed using the equation as
ij
k
ij
k
ij
k
ij
k
ij
k dCdKF (11)
where ij
kK and ij
kC are, respectively, the stiffness and damping coefficients of
the contact force model at node j of body i on patch plane k . Using the
expression for the contact force as defined by the preceding equation, the contact
force vector can be defined as
ij
k
ij
k
ij
k F nF ˆ (12)
where ij
kn̂ is a unit normal vector shown in Fig. 3. The virtual work of the
contact force at the nodes is given by
n
j
ij
k
i
k WW1
n
j
ij
k
ij
k
ij
k dF1
ˆ n
i
iTiTi
Rθ
RQQ
(13)
where
n
j
ij
k
Ti
R
1
FQ
n
j
ij
k
Tij
p
iTi
1
)~
( FuAQ (14)
383
are the generalized contact forces associated with the Cartesian and orientation
coordinates of link i , and ij
pu~
is the skew symmetric matrix associated with the
vector ij
pu . In order to evaluate the tangential component of these contact forces
for friction effect at each contact nodes, the smooth Coulomb friction model [6]
is employed in this investigation. Figure 8 shows the computer animation of
multibody tracked vehicle running over APG Profile IV test ground.
Note that the proposed element free finite contact node method have several
advantages such as, simple computer implementation, easy contact detecting
algorithm for irregular surface, independent contact coefficients, and distribution
of concentrated contact forces, however, in the penalty function approach used in
this contact force model the determinations of spring and damping coefficients
may be a black art. These coefficients may not correspond to familiar physical
properties that can be measured experimentally. Careful numerical calibration
process is necessary to obtain reliable model, accordingly.
Time = 0.0 sec Time = 9.0sec
Time = 3.0 sec Time = 12.0 sec
384
Time = 6.0 sec Time = 15.0 sec
Figure 8. Computer animation of multibody tracked vehicle running over Aberdeen
profile IV proving ground
1.2.4. MEASUREMENT OF THE DYNAMIC TRACK
The measurement system is composed of strain gages, signal processor, data
storage, and power unit. The system is installed inside of a track shoe body.
When the switch is on, the system will start to measure and store the tensional
forces from the strain gages into data storage processor. The measurement results
are then downloaded into a laptop computer through communication port.
The basic platform of dynamic track instrumentation system is developed by
Kweenaw Research Center at Michigan Technology University [12]. The tension
measurement system records 2 channels, which are track tensions at both ends of
a track link, at the rate of 800 samples per second for 160 seconds. The tension
data in the system memory is offloaded to a computer for storage after the test
vehicle is stopped. A track shoe body was carved to attach full bridge of strain
gages on the outside and inside edges of the body. A track link, as a sensor
system to measure the dynamic track tension of the high mobility tracked vehicle,
was carefully calibrated at the center. A known load, Shunted Engineering Unit
Value, can be simulated by shunting one leg of the strain gage bridge using a
58,900 ohm resistor inside the measurement system. The known load is about
20,000 lb [12]. If any load is applied to the measurement system, the load as an
engineering unit can be determined by a linear interpolation or extrapolation
using the engineering unit value.
Figure 9 shows the comparison of simulation and experimental results when
385
the vehicle runs on flat ground with the velocity of 10 km/h. The figure shows
that there are four disagreement areas between experimental and numerical
results. These disagreements are due to the extra deformation of the strain gage
when the track link moves around sprocket, idler, and first and last road wheels.
The extra deformation makes the track tension look much higher than it actually
is
Figure 9. Dynamic tension of a track link
1.2.5. NUMERICAL INVESTIGATION OF DYNAMIC TRACK TENSION
Extended numerical simulations are carried out to compensate for the
experimental limitations due to space and environment. The track tension is
monitored in two different views of track link following view and chassis fixed
view. In order to acquire the track tension for the chassis fixed view, the track
tensions are recorded until all links pass through one point of the hull. For the
track link following view, the track tension of one selected track link is recorded
when it is moving around vehicle components of idler, road wheels, sprocket and
support rollers.
Key physical quantities influencing the track tension are pre-tension, vehicle
speed, ground profile, traction force, driving torque, and turning resistance,
respectively. The pre-tensions of 25 kN, 50 kN and 100 kN are given to observe
their influences on the dynamics of the vehicle. Three different speeds of 5 km/h,
12 13 14 15 16 170
10k
20k
30k
40k
50k
60k
70k
80k
Time(sec)
Tensio
n(N
)
Experiment
Simulation
386
20 km/h, and 40 km/h are given on both driving sprockets using velocity
constraint equations. Various ground profiles as defined in real proving ground
[10] are developed by using the triangular patch elements. Three friction
coefficients of 0.1, 0.4, and 0.7, between the track shoe body and the ground, are
used for different traction force modeling. The track tensions are observed for a
pivot turning, right and left turning, backward motion, acceleration and braking
motions.
Effect of pre-tension: One of the most critical variables for the dynamic
track tension is the pre-tension. Although an optimal pre-tension has long been a
major subject for academia and industry, researchers only relied on experimental
and field experiences. Most of high mobility suspenioned tracked vehicles,
approximately 10 % of the vehicle total weight is loaded as a track pre-tension.
Figure 10 shows the track tensions of a selected track link in the link following
view with three different pre-tensions. These pre-tensions are 25 kN, 50 kN, and
100 kN, respectively. Both sprockets have constant angular velocity of -17.8
rad/sec which can produce 20 km/h vehicle speed. As illustrated in this figure,
increment of the pre-tension linearly increases the dynamic track tension.
Figure 10. Track tensions of pre-tension effect
0 1 2 3 4 5
0
20k
40k
60k
80k
100k
120k
50kN pre-tension
100kN pre-tension
25kN pre-tension
Te
ns
ion
(N)
Time(sec)
387
Effect of vehicle speed: Like a tire of wheeled vehicles, revolution of a track
system can cause the movement of tracked vehicles. The vehicle speed varies
time to time due to random and irregular vehicle operations. Several numerical
and empirical studies showed that the amplitude of track tension does not change
much as the speed changes. However, the frequency of the track tension changes
significantly. In the case of a bump run, the track tension around contacted
region increases significantly at a higher speed when the vehicle hits a bump. It
is mainly because of large increment of impact force between a track link and the
ground.
Effect of ground profile: In the previous section, a generalized method for
building the proving ground profiles are introduced. Many profiles representing
real testing grounds are developed by using the triangular patch. Since the
ground contact forces are directly transferred to the track links, the track tension
is strongly related to the surface geometry of a given ground profile.
Effect of traction forces: Force transmission of a tracked vehicle can be
understood as three force conversions. When a track system rotates, traction
forces are generated between the track system and the ground in opposite
direction to the velocity of the track system. Contact forces between track link
pins and sprocket teeth are converted to the sprocket moment in the tangential
direction of the pitch circle of the sprocket. The sprocket torque can be converted
again to a translational force acting on the axis of the sprocket center. Finally,
this translational force on the axis of the sprocket can cause movement of a
tracked vehicle. During the force conversion process the traction forces can be
replaced directly by tensional forces of track system. The amount of the traction
forces is determined by a friction model between the track system and the
ground. The track tensions between middle road wheels with different friction
coefficients are shown in Fig. 11. Three different friction coefficients, 0.1, 0.4,
and 0.7 are used in this numerical investigation. In order to show the effect of the
friction, the vehicle is accelerated from zero to 40 km/h in ten seconds. As
shown in this figure, increment of the friction coefficient causes an increment of
the track tension.
388
Figure 11. Track tensions of traction force effect
Effect of sprocket torque : In this investigation power-pack, engine and
transmission, are modeled by using velocity constraints or the sprocket torque. In
the real world there are two major disturbances to keep steady sprocket torque.
These are irregular driver inputs and impacts of transmission shifts. The sprocket
torque is converted to the contact force between the sprocket teeth and track link
pins. The sprocket contact force repeats to pull and push, which makes the track
tension vary. To observe the effects of sprocket torque, step, sinusoidal, and
linear-steady torques are applied on the sprocket. Figure 12 shows the track
tension changes near the sprocket when the step torque is applied.
0.0 0.5 1.0 1.5 2.0 2.5 3.05x10
4
6x104
7x104
8x104
=0.4
=0.7
=0.1
Te
ns
ion
(N)
Time(sec)
389
Figure 12. Track tensions of sprocket torque effect
Figure 13. Track tensions of turning resistance effect
Effect of turning motion: Heading direction of a tracked vehicle is turned by
a speed difference of left and right sprockets, which causes different traction
forces. The traction forces of track systems are converted to the forces at the
center of both sprocket axes. Then the chassis system is rotated with respect to
the vertical axis by a force difference of both sprocket axes. Suppose a track
vehicle is stuck to the ground. If angular velocities of both sprockets are constant
1 2 3 4 5 6 7 8 9 10 110
5k
10k
15k
20k
To
rqu
e(N
m)
Time(sec)
20k
30k
40k
50k
60k
tension
torque
Te
ns
ion
(N)
0 2 4 6 8 10 12
0
50k
100k
150k
200k
250k
road wheels
upper
right track tension
left track tension
sprocket
upper
Te
ns
ion
(N)
Time (sec)
390
with different speeds, significant differences of track tensions may be observed
due to the revolution of the chassis.
Figure 13 shows the dynamic changes of right and left track tensions when the
vehicle makes right turning with angular velocities of right = -4.5 rad/sec and
left = -5.4 rad/sec. It can be shown that the track tension of the upper part of
the left sprocket goes up significantly, while the track tension of the lower part
goes down.
1.2.6. FUTURE WORK AND CONCLUSIONS
The dynamic track tension for a high mobility tracked vehicle is investigated
in this paper. The three dimensional multibody tracked vehicle consists of the
hull, sprockets, road arms, road wheels, support rollers, and sophisticated
suspension systems of hydro-pneumatic and torsion bars. A compliant force
model is used to connect the rigid body track links. The tracked vehicle model
has 189 bodies, 36 pin joints and 152 compliant bushing elements and has 954
degrees of freedom. Various ground profiles are developed by using triangular
patch elements. Numerical results are validated against experimental results.
Numerical simulations have been carried out under various maneuvering
conditions and effects of several conditions are discussed . Numerical results
showed that the optimal track tension may not be necessarily 10 % of the total
vehicle weight as many track vehicle researchers have believed. Further studies
must be carried out to find the optimal track tension.
391
REFERENCES
[1] G. R. Doyle and G. H. Workman, 1979, ''Prediction of Track Tension when Traversing
an stacle'', Society of Automotive Engineers, 790416
[2] A.G. Galaitsis, 1984, "A Model for Predicting Dynamic Track Loads in Military
Vehicles", ASME, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol.
106/289
[3] R.M. Trusty, M.D. Wilt, G.W. Carter, D.R. Lesuer, 1988, ''Field Measurement of Tension
in a T-142 Tank Track'', Experimental Techniques
[4] M. K. McCullough, and E. J. Haug, 1986, ''Dynamics of High Mobility Tracked
Vehicles'', ASME, Journal of Mechanisms Transmissions, and Automation in Design,
Vol.108, pp. 189-196
[5] Choi, J. H., Lee., H. C., Shabana, A. A., Jan. 1998, ''Spatial Dynamics of Multibody
Tracked Vehicles: Spatial Equations of Motion'', International Journal of Vehicle
Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 27-49
[6] Lee, H. C., Choi., J. H., Shabana, A. A., Feb. 1998, ''Spatial Dynamics of Multibody
Tracked Vehicles: Contact Forces and Simulation Results'', International Journal of
Vehicle Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 113-137
[7] C. Scholar and N. Perkins, 1997, "Longitudinal Vibration of Elastic Vehicle Track
System", SAE, 971090, International Congress and Exposition, Detroit, MI, Feb. 24-27
[8] J. Chung, J. M. Lee, 1994 ''A New Family of Explicit Time Integration Methods for
Linear and Non-linear Structural Dynamics'', International Journal for Numerical
Methods in Engineering, Vol.37, 3961-3976
[9] H. S. Ryu, D. S. Bae, J. H. Choi and A. Shabana, 2000 ''A Compliant Track Model For
High Speed, High Mobility Tracked Vehicle'', International Journal For Numerical
Methods in Engineering, Vol. 48, 1481-1502
[10] Changwon Proving Ground Construction Manual, 1996, Agency for Defense
Development, GWSD-809-960634
[11] Shabana A, 1989 ''Dynamics of Multibody Systems'', John Wiley & Sons, New York
392
[12] Glen Simula, Nils Ruonavaara, and Jim Pakkals, 1999 "DTIS operation manual",
KRC, Michigan Technological University.
[13] ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann
Arbor, MI 48105.
[14] E. Kreyszig, 1983 “Advanced Engineering Mathematics”, 5th edition John Wiley &
Sons, New York
[15] Choi, J. H., 1996 “Use of Recursive and Approximation Method in the Dynamic
Analysis of Spatial Tracked Vehicle”, Ph. D. Thesis, The University of Illinois at
Chicago
393
1.3
EFFICIENT CONTACT AND NONLINEAR
DYNAMIC MODELING FOR TRACKED VEHICLES
1.3.1. INTRODUCTION
In the dynamic analysis of vehicle system the mathematical modeling for the
system can be very different according to the objective of analysis. Sometimes
the mathematical modeling methods of vehicle systems are pursuing more
simplified effective models such as for real-time analysis or for the system
design which doesn’t require highly nonlinear effects. Oppositely, due to rapid
developments of computer hardware and numerical technologies, researchers
and engineers can construct super detail nonlinear dynamic models which have
several hundreds, even several thousands of degrees of freedom systems, which
has the same phenomena as physical system. The objective of this research is to
built a reliable tracked vehicle dynamic analysis model so as to design a dynamic
track tensioning control system for high speed tracked vehicle based on
multibody dynamic modeling techniques. One of the key points for the dynamic
analysis of tracked vehicle is to predict the dynamic track tension when the
vehicle operates on various ground. In order to satisfy such objective of the
research, the track links of the track system should be modeled as a rigid body
which has six degrees of freedom connected by bushing force elements.
In early 80’s several dynamic modeling techniques for track systems have
been developed in universities, research institutes and companies. McCullough
and Haug[1] designed a super element that represents spatial dynamics of high
mobility tracked vehicle suspension systems. The track was modeled as an
internal force element that acts between ground, wheels and the chassis of the
vehicle. Track tension was computed from a relaxed catenary relationship.
Empirical normal and shear force formulas based on constitutive relations from
soil mechanics were used to model the soil-track interface. Frank Huck[2]
394
introduced a planar multibody dynamic model of track type tractor by using
DRAM software. In this investigation each track link was modeled as a rigid
body and contact force analyses of sprocket, rollers and soil ground were
represented as a pioneer works. Similar modeling technique was also developed
by Tajima[3] at the similar period. The Komatsu Ltd. in-house multibody
program developed by Tajima is used to simulate planar multibody tracked
vehicles. The contact search mechanics and dynamic analysis of planar rigid
body track system are clearly introduced by Nakanishi and Shabana[4].
Nakanishi’s work was extended for three dimensional analysis by Choi and
Shabana[5, 6], and simultaneously Wehage[7] also developed the full three
dimensional tracked vehicle model, which considers the track link as a rigid
body, under research project in Caterpillar Inc. Whereas Choi used to connect
each rigid tracked link by one degree of freedom pin joint, bushing force
elements were used to connect for rigid track link by Wehage. Choi’s work
shows an possibility of very difficult numerical solution however it fails to give
more freedoms in real world than Wehage’s approach. Ryu et al.[8] extended
previously developed track system modeling techniques for the high mobility
military tracked vehicle which adopts sophisticated suspension and tensioning
systems. In this investigation a new variable step algorithm is implemented into
G-Alpha integrator which gives high numerical damping to integrate smoothly
high frequency and impulsive contact and bushing forces.
There are several reasons why many researchers has tried to develop rigid
multibody track systems even though such modeling techniques burden heavily
for numerical solutions. Unlike tires of wheeled vehicles track system causes
many problems such as separations or failures of connections, etc., furthermore it
is very expensive to maintain and has relatively weak durability. Because of
superiority of track system on very hostile terrain it cannot be replaced by
wheeled system, thus researchers should have solved these difficulties of the
tracked vehicle system. In the beginning of the research several simple modeling
techniques had been introduced however those gave a conclusion that each track
link should be considered as a rigid body to satisfy requirements. For instant, one
of the key issue for tracked vehicle is track tension since track tension has
significant roles for the vehicle maneuverings as focused in this investigation.
395
Very few works have been performed for the analysis of track tension based on
empirical or simple numerical analysis. Doyle and Workman[9] presented a
static prediction of track tension when the suspensioned tracked vehicle traverses
obstacles using two dimensional finite element methods. An elastic beam
element subjected to tension, compression and bending loads was utilized to
model track links. Galaitsis[10] demonstrated that the analytically predicted
dynamic track tension and suspension loads of a high-speed tracked vehicle are
useful in evaluating the dynamic analysis of the vehicle. The predicted track
tensions were compared with the empirically measured track tensions. A detailed
track tension measurement methodology and results are presented by Trusty et
al.[11]. Strain gages connected to a portable data acquisition system were
installed in the track link. The flat ground, quick acceleration, traversal of
obstacle courses, pivot turns, moving uphill, and pre and post tension, were used
for the tension investigation scenarios. Choi et al.[12] predicted and showed the
effect of dynamic track tension for the vehicle by using multibody techniques.
This research focuses on a heavy military tracked vehicle which has
sophisticated suspension and rubber bushed track systems. Various virtual
proving ground models are developed to observe dynamic changes of the track
tension. The predicted dynamic track tensions are validated against the
experimental measurements.
In this investigation for the sake of efficient development of dynamic track
tensioning system for suspensioned high speed military tracked vehicle, detail
nonlinear dynamic modeling methods which can partially replace physical
prototype models are presented. For the multibody dynamic modeling techniques
of the tracked vehicle used in this research several new methods are developed
and suggested. Those are efficient contact detecting kinematics for sprockets,
wheels and track links, parameter extraction techniques from component
experimental test, and a method how to apply Bekker’s[13] soil theory for
multibody track and soil interactions. The simulation results are correlated by
newly developed experimental measurement techniques in this investigation.
396
1.3.2. MULTIBODY TRACKED VEHICLE MODEL AND PARAMETER
EXTRACTIONS
The tracked vehicle model used in this investigation is a military purpose high
speed tank system which has sophisticated suspension system to damp out
impacts from hostile ground. In general this type of vehicle can be divided four
subsystems for overall motion analysis of vehicle dynamics. These subsystems
are two track subsystem with suspension units, main body subsystem with power
pack, and turret subsystem with main gun. The each right and left track
subsystems is composed of rubber bushed track link, double sprockets with
single retainer, seven road wheels and arms, and three upper rollers. The
sprockets, road arms, road wheels, upper rollers and turrets are mounted on main
body by revolute joints which allow single degrees of freedom. Total 38 revolute
joints are used for the vehicle modeling and generate 190 nonlinear algebraic
constraint equations. Two busing force elements to connect each track links and
total 304 bushing forces elements for both track systems are used in this
investigation. The modeled vehicle has 191 rigid bodies and 956 degrees of
freedom. Figure 1 shows a computer graphics model for tracked vehicle used in
this investigation.
Figure 1. Computer graphics of high speed tracked vehicle model
Track system
Chassis system
Turret system
397
1.3.3. EFFICIENT CONTACT SEARCH ALGORITHM
The interactions between the track links and the road wheels, rollers, and
sprockets are explained in this section. When a track link travels around vehicle
components, its trajectory is controlled by contact forces. The contact forces can
be generated computationally by detecting of contact conditions. The contact
collision algorithms are composed of five main routines such as search routines
for, wheel and link contact, center guide and wheel contact, sprocket tooth and
link pin contact, side wall of link and sprocket contact, and ground and link shoe.
The contact points and penetration values are defined from the searching
routines. Then a concentrated contact force is used at the contacted position of
the contact surface of the bodies. A detailed discussion on the formulation of the
contact collision is represented by Choi et al[5,6] and Nakanishi and Shabana[4].
However, it is not efficient for each chassis component such as road wheels and
sprockets to search all track links in detail. Efficient search algorithms and
discretized terrain representation method are investigated, respectively.
1.3.3. 1. ROAD WHEEL-TRACK LINK CONTACT
Each road wheel is usually composed of two wheels. The interactions between
road wheel and track link can be divided into two types contact, as shown in Fig.
4. One is road wheel-track link body contact and the other is wheel side-track
link center guide contact. Each track subsystem has 6 road wheels and 76 track
links. In order to search wheel-track link contact efficiently, the pre-search and
post-search algorithm is applied. In the pre-search, bounding circle relative to
road wheel center is defined. All of track links are considered to detect a starting
link and ending link which has a possibility of wheel contact. Post-search means
a detailed contact inspection for track links in a bounding circle. Once a starting
and an ending link are found at one time through pre-search prior to analysis,
only detailed search is carried out by using the information of starting link and
ending link from the next time step.
398
Figure 4. Wheel and track link interactions
1.3.3. 2. SPROCKET-TRACK LINK CONTACT
The interactions between sprocket and track link can be divided into two types
contact, as shown in Fig. 5. One is sprocket-track link pin(end connector) contact
and the other is sprocket side-track link side. Each track subsystem has 1
sprocket and 76 track links, moreover a sprocket has many teeth. For the
efficient search of the sprocket-track link contact, contact search algorithm is
composed of the pre-search and post-search. In the pre-search, bounding circle
relative to sprocket center is defined. All of track links are employed to detect a
starting link and ending link which has a possibility of sprocket contact. Then,
track links from starting link are investigated the engagement with sprocket
valley. Post-search means a detailed contact inspection for track links in a
bounding circle. Once a starting and ending link is found at one time through
pre-search prior to analysis, only detailed search is carried out by using the
information of starting link and ending link from the next time step.
starting link
ending link
Bounding circle
Search direction
Search direction
399
Figure 5. Sprocket and track link interactions
1.3.3. 1. GROUND-TRACK LINK CONTACT
The ground interacting surface of a track link can be single or multiple, and
therefore, there are one surface or multiple surfaces on each track link that can
come into contact with the ground. The interacting surface of track link is
discretized and each contact node points were defined. The global position
vectors that define the locations of points on the shoe plates surface of track link
are expressed in terms of the generalized coordinates of the track links and are
used to predict whether or not the track chain link is in contact with the ground.
In order to construct various geometries of tracked vehicle paved proving ground,
such as bumping courses, trench course, inclined course, standard cross country
courses, discretized terrain representation methods using triangular patch
element are used in this investigation. A triangular patch element has three nodes
and a unit normal vector to describe plane equations of interaction grounds [16].
starting link
ending link
starting engagement
Bounding circle
Search direction
Search direction
ending engagement
400
(1) DISCRETIZED TERRAIN REPRESENTATION
The virtual terrain model used in this investigation is a general three dimensional
surface defined as a series of triangular patch elements. Figure 6 shows an
example of obstacle course created by triangular patch surfaces. Most geometries
of various paved proving grounds for tracked vehicle can be easily represented by
using triangular patch.
Figure 6. Terrain representation (obstacle course)
The equation for the plane defined from three nodes can be written as
zayaxa 321 (3)
The three coefficients, 1a , 2a , and 3a of the equations of plane can be
obtained by given three locations of triangular patch, and by using Cramer's rule,
these coefficients can be obtained[16].
Then the unit normal vector of the plane n̂ is defined as
Taaaa
11
1ˆ
212
2
2
1
n (4)
(2) METHODS OF FINITE CONTACT NODES FOR GROUND INTERACTIONS
Unlike wheel and surface contact, the interactions between track link and ground
are very complicated problems. This is because the track link has irregular
401
contact geometry and different material properties. Due to large number of track
links of each track subsystem, commonly used contact theory of surface to
surface interactions in finite element community can not be employed for this
work. Choi[17] suggested that element free finite contact nodes were distributed
on the contact surface of track link, which have their own stiffness and damping
characteristics. The relative indentations of each node were monitored and
positions are restored. The use of element free finite contact node methods
demonstrated clearly the computational efficiency for dynamic analysis of track
system. Based on the method developed by Choi[17], the interactions between
track link surface and triangular patch surface are developed in this investigation.
Figure 7. Interaction between track shoe body and triangular patch element
Figure 7 shows the interaction between finite contact nodes of track link and
triangular patch surface. The perpendicular deformation scalar kd of contact
node j of link i on patch plane k can be defined as
k
Pij
kd nr ˆ1 (5)
402
where P1r is shown in Fig. 7 and unit vector kn̂ is defined in Eq. (4). The
criterion of necessary condition for the contact to occur of node j , which is not
sufficient, is
seperated
contactdij
k
ij
k
:0
:0 (6)
If this conditions is satisfied, the position vector jk
Br shown in Fig. 7 is used
to compute the node location whether contact point B of node j is on the
patch plane k . The position vector jk
Br can be written as
kk
ij
p
iijk
B d nuARr ˆ (7)
where iA is the transformation matrix associated with the orientation
coordinates of link i and ij
pu is the local position vector of node j in the
track link coordinate system. On the other hand, using scalar triple product if one
of the following conditions is satisfied
0ˆ
0ˆ
0ˆ
0ˆ
0ˆ
0ˆ
331
223
112
331
223
112
k
B
k
B
k
B
k
B
k
B
k
B
or
nrr
nrr
nrr
nrr
nrr
nrr
(8)
, then the node j of link i is in contact with patch element k .
1.3.4. EQUATIONS OF MOTION
In this investigation, the relative generalized coordinates are employed in
order to reduce the number of equations of motion and to avoid the difficulty
associated with the solution of differential and algebraic equations. Since the
track chains interact with the chassis components through contact forces and
403
adjacent track links are connected by compliant force elements, each track chain
link in the track chain has six degrees of freedom which are represented by three
translational coordinates and three Euler angles. Recursive kinematic equations
of tracked vehicles were presented by [8] and the equations of motion of the
chassis are given as follows :
)( r
i
Tr
i
TqBMQBqMBB (9)
where r
iq and B are relative independent coordinates, velocity
transformation matrix, and M is the mass matrix, and Q is the generalized
external and internal force vector of the chassis subsystem, respectively. Since
there is no kinematic coupling between the chassis subsystem and track
subsystem, the equations of motion of the track subsystem can be written simply
as
tttQqM (10)
where tM , t
q and tQ denote the mass matrix, the generalized coordinate
and force vectors for the track subsystem, respectively. Consequently, the
accelerations of the chassis and the track links can be obtained by solving Eqs. (9)
and (10)..
1.3.5. EXTENDED BEKKER’S SOIL MODEL FOR MULTIBODY TRACK
SYSTEM
The interactions between track link and soil used in this investigation consist
of the normal pressure-sinkage and shear stress-shear displacement relationships.
Bekker[13] developed the bevameter technique to measure terrain characteristics
by the plate penetration and shear tests. He also proposed the equation for
pressure-sinkage relationship, given by
nc zkb
kzp )()( (11)
where p is pressure, b is the width of a rectangular contact area, z is
404
sinkage, ck is the soil cohesive modulus, k is the soil frictional modulus and
n is the exponent of soil deformation. The value of ck , k , and n can be
obtained from empirical test. From the experimental observations[13], the range
between unloading and reloading can be approximated by a linear function in the
pressure-sinkage relationship.
)()( zzkpzp uuu (12)
where p and z are the pressure and sinkage, respectively, during unloading
or reloading; up and uz are the pressure and sinkage, respectively, when
unloading begins; and uk is the average slope of the unloading-reloading line.
The slope of the unloading-reloading represents the degree of elastic rebound. If
the slope is vertical, there is no elastic rebound. That means the terrain
deformation is entirely plastic.
The shear stress-shear displacement relationship proposed by Janosi and
Hanamoto[13] is used for tangential shear forces, given by
)1)(tan(),( / Kjepczj (13)
where is the shear stress, p is the normal pressure, j is the shear
displacement, c and are the cohesion and the angle of internal shearing
resistance of the terrain, respectively, and K is the shear deformation modulus.
In summary, the proposed equations are applied for track system and soil
interactions as;
Loading condition ( pzz ) :
nc zk
b
kzp )()( (14)
)1)(tan(),( / Kjepczj (15)
Unloading, Reloading condition ( uzz ) :
ur zzzif
405
)()( zzkpzp uuu (16)
)1)(tan(),( / Kjepczj (17)
rzzif
0)( zp (18)
0),( zj (19)
Loading condition after reloading ( uzz )
nc zk
b
kzp )()( (20)
)1)(tan(),( / Kjepczj (21)
where pz is sinkage at the previous time step and rz is sinkage when the
plastic effect of terrain is started during unloading. Figure 8 shows the
simulation response to normal load of a track link on dry sand terrain when the
vehicle is accelerated from the rest. The soil conditions for simulation are1/95.0 n
c mkNk , 2/43.1528 nmkNk , c = kPa04.1 , = o28 , and n =1.1. The
pattern of result agrees to the experimental result shown in reference[13].
Figure 8. Simulation response to normal load of a dry sand terrain
uz
rz
0.00 0.03 0.06 0.09 0.12 0.15 0.18
0.0
5.0x104
1.0x105
1.5x105
2.0x105
2.5x105
reloading
unloading
loading
Pre
ssure
(N
/m2)
sinkage (meter)
406
Figure 9. Mesh areas and detect nodes of a track link
As shown in the Fig. 9, if the node j of link i is in contact with triangular
patch ground, the contact force at the contact segment area can be computed
using the equation as
)( segmentthjofareapF ijij
p (22)
)( segmentthjofareaF ijij
s (23)
where ijp and ij are the normal pressure and shear stress, respectively.
Using the expression for the contact force as defined by the preceding equation,
the contact force vector can be defined as
k
ij
sk
ij
p
ij FF tnF ˆˆ (24)
where kn̂ and kt̂ are a unit normal vector and tangential vector of ground
patch k . The virtual work of the ground force at a track link, which has the
number of n rectangle surface, is given by
n
j
ijTijn
j
iji WW11
Fr (25)
where ijr is a j th node position vector of link i defined by inertia
reference frame.
Track link i
Contact detect node
Contact segment area
407
1.3.6. SUMMARY AND CONCLUSION
For the sake of efficient component development of tracked vehicle at early
design stage, it is clearly proved that the multibody dynamic simulation methods
can be very useful tool. The presented three dimensional multibody tracked
vehicle consists of the hull, sprockets, road arms, road wheels, support rollers,
and sophisticated suspension systems of hydro-pneumatic and torsion bars. A
compliant force model is used to connect the rigid body track links. The tracked
vehicle model has 191 bodies, 38 pin joints and 304 compliant bushing elements
and has 956 degrees of freedom. The suspension, contact and bushing
characteristics are extracted by empirical measurements and implemented into
the simulation model. The efficient kinematic contact search algorisms between
track system and chassis components are suggested and implemented. Two
methods are developed for the interactions between track shoe body and ground.
When the distributed node points on shoe body surface detect contact condition,
direct forces are calculated based on the contact deformation on node points, or
pressure and shear forces on each segment areas of the contact surface are
calculated based on pressure-sinkage relationship and shear stress-shear
displacement relationship. In order to validate and construct the simulation
database, positions, velocities, accelerations and forces of the tracked vehicle are
measured empirically. The simulation results show very good agreements with
experimental measurements. Therefore, the suggested methods by using the
multibody dynamic technologies can be used efficiently for tracked vehicle
developments.
408
REFERENCES
[1] M. K. McCullough, and E. J. Haug, 1986, ''Dynamics of High Mobility Tracked
Vehicles'', ASME, Journal of Mechanisms Transmissions, and Automation in Design,
Vol.108, pp. 189-196.
[2] F.B. Huck, ''A Case for Improved Soil Models in Tracked Machine Simulation'',
Caterpillar, Inc.
[3] Tajima, and T. Nakanishi “Technical discussions” Komatsu Ltd.
[4] Nakanishi, T., and Shabana, 1994 "Contact Forces in the Nonlinear Dynamic analysis
of Tracked Vehicles", International Journal for Numerical Methods in Engineering,
Vol.37, pp. 1251-1275.
[5] Choi, J. H., Lee., H. C., Shabana, A. A., Jan. 1998, ''Spatial Dynamics of Multibody
Tracked Vehicles: Spatial Equations of Motion'', International Journal of Vehicle
Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 27-49.
[6] Lee, H. C., Choi., J. H., Shabana, A. A., Feb. 1998, ''Spatial Dynamics of Multibody
Tracked Vehicles: Contact Forces and Simulation Results'', International Journal of
Vehicle Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 113-137.
[7] R. Wehage, F. Huck “Technical discussions” Caterpillar Inc.
[8] H. S. Ryu, D. S. Bae, J. H. Choi and A. Shabana, 2000 ''A Compliant Track Model For
High Speed, High Mobility Tracked Vehicle'', International Journal For Numerical
Methods in Engineering, Vol. 48, 1481-1502.
[9] G. R. Doyle and G. H. Workman, 1979, ''Prediction of Track Tension when Traversing
an stacle'', Society of Automotive Engineers, 790416.
[10] A.G. Galaitsis, 1984, "A Model for Predicting Dynamic Track Loads in Military
Vehicles", ASME, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol.
106/289.
[11] R.M. Trusty, M.D. Wilt, G.W. Carter, D.R. Lesuer, 1988, ''Field Measurement of
Tension in a T-142 Tank Track'', Experimental Techniques.
[12] J. Choi, D. Park, H. Ryu, D. Bae, K. Huh, 2001 “Dynamic Track Tension of High
409
Mobility Tracked Vehicles” Proceedings of DETC’01, ASME Third Symposium on
Multibody Dynamics and Vibration, Pittsburgh, PA, USA.
[13] J. Wong, 2001, “Theory of Ground Vehicles” 3rd Ed. John Wiley & Sons.
[14] Shabana A. 1996 “Theory of Vibration, An Introduction, 2nd Ed.” Springer: New York.
[15] Berg, M., 1998 ''A Non-Linear Rubber Spring Model for Rail Vehicle Dynamics
Analysis'', International Journal of Vehicle Mechanics and Mobility, Vehicle System
Dynamics, Vol. 30, pp. 197-212.
[16] ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann
Arbor, MI 48105.
[17] Choi, J. H., 1996 “Use of Recursive and Approximation Methods in The Dynamic
Analysis of Spatial Tracked Vehicle”, Ph. D. Thesis, The University of Illinois at
Chicago.
2. Chain
411
2.1
NONLINEAR DYNAMIC MODELING OF
SILENT CHAIN DRIVE
2.1.1. INTRODUCTION
Chain drives are widely used in the power transmission applications in the
automotive field for a long time because they are capable of transmitting large
power at high efficiency and low maintenance cost. However, the noise and
vibrations created by chain drives have always been major problems, especially
for higher speed, lighter weight, and higher quality. Noise and vibrations in
chain systems are largely caused by chordal(polygonal) action and impacts
between chain and sprocket. The links of the chain form a set of chords when
wrapped around the circumference of the sprocket. As these links enter and leave
the sprocket, they impart a jerky motion to the driven shaft by chordal action.
The chordal action causes chain span longitudinal and transverse vibrations.
Whereas, impact between sprocket and link excites high frequency vibration and
is a major source of noise in chain drives at high speeds. In order to minimize
such problems, silent chains are introduced in many camshaft drives of
motorcycle/automobile engines and the primary drive between the engine and
transmission, as well as in other high-speed applications. It is also used with the
object of increasing chain life. However, in spite of the widespread use of silent
chain drives, surprisingly little works have been published about their dynamic
analysis. This may be due to three major difficulties; the first is the complexity
of the contact algorithms among components, the second is small integration step
size resulting from the impulsive contact forces and the use of stiff compliant
elements to represent the joints between the chain links, and the third is the large
number of the system equations of motion to solve.
411
Figure 1. Silent Chain Drive Model of Automotive Engine
Chen and Freudenstein [1] presented a kinematic analysis of chain drive
mechanism with the aim of obtaining insight into the phenomena of chordal
action, with the associated impact and chain motion fluctuation. Veikos and
Freudenstein [1] developed a lumped mass dynamic model based on Lagrange’s
equations of motion and showed chain drive dynamics and vibrations. Wang [3,
4] investigated the stability of a chain drive mechanism under periodic sprocket
excitations and studied the effect of impact intensity in their axially moving
roller chains. Kim and Johnson [5, 6] developed a detailed model of the roller-
sprocket contact mechanics that allowed the first determination of actual
pressure angles and a multi-body dynamic simulation. This investigation is based
on Kane’s dynamic equations. Choi and Johnson [7, 8] investigated the effects of
impact, polygonal action, and chain tensioners into the axially moving chain
system and showed the transverse vibration of chain spans. Quite recently Ryu et
Crankshaft
Sprocket
Fixed Guide
Pivot Guide
Camshaft Sprocket
(Idler Sprocket)
Tensioner
412
al [9] developed very detailed chain models including contact forces for links,
sprockets and idlers with special application to large-scaled civilian and military
tracked vehicles. There has been some design analysis in the view of dynamic
behaviors of silent chain in powertrain industry and commercial software [10].
However they showed some primitive dynamic analysis and design of silent
chain system because it has high frequency contact forces, speedy revolution and
large number of bodies.
The purpose of this work is to investigate and suggest the dynamic modeling
and analysis of silent chain drive mechanism with high speed revolution using
multibody dynamic techniques. In this investigation, numerical skills of
multibody chain dynamic analysis are employed and showed very good
agreement of physical phenomenon of silent chain system. Dynamic tension,
impact forces, and vibration of chain links are explored for the sake of
understanding dynamic behaviors of the chain system.
2.2.2. MULTIBODY MODELING OF SILENT CHAIN DRIVE
As shown in Fig. 1, in general a chain drive mechanism has four main
components, which are sprockets, chain links, guides and tensioner element. The
sprockets can be recognized as drive sprockets and idler sprockets. The chain
link can includes link plates, guide plates, and pins. The tensioner element
maintains stable tension during operation by adjusting pressure force to the chain
link system. While roller chain mechanism has engagements between pins and
sprocket, since silent chain mechanism engages between chain link teeth and
sprocket teeth, there is much less chodal vibrations and can transmit the power
more quietly. In this investigation the dynamic analysis and numerical modeling
techniques are presented by using multibody methods.
2.2.2.1. SPROCKET
The sprockets of the chain system are interacted by the introduced contact and
friction forces acting on between the chain and the sprocket teeth. The crank
sprocket of the system is driven by motion constraint. This motion constraint can
be constant or time dependent. In this investigation the sprocket is modeled as a
X
Z
Y
ir
iy
ix
iz
iO
Oif
ig
ih
413
rigid body and attached on ground by revolute joint. The geometry of sprocket
teeth profiles consists of a series lines and arcs with different length and radii as
shown in Figure 2. The sprocket of silent chain is shaped more like a gear than
one of roller chain.
Figure 2. Geometry of Silent Chain Sprocket
2.2.2.2 SILENT CHAIN LINK
Roller chains, although having excellent wear and strength capability, are
inherently noisy and oscillatory. As a result, inverted-tooth chain mechanisms
were developed in order to reduce the forcing function of the noise-producing
mechanism. The difference in noise performance between silent and roller chains
can be attributed to the manner in which they engage and disengage the sprocket
teeth. After the sprocket tooth initially contacts the chain link, and as the
engagement proceeds, a combination of rolling and sliding motion occurs
between the tooth and link contacting surfaces. Such an engagement mechanism
effectively spreads the engagement time over a significant interval, thereby
minimizing tooth/link impact and its inherent noise generation.
A silent chain consists of several layers of links connected with pins. Since
there is no advantage for the modeling of pins and multi layer links as separate
components, in this investigation these multi layer links are treated as a rigid
body with mass and inertia property which takes into account the effects of the
pins. An individual silent chain link looks much different comparing to a roller
chain link. The geometry of link profile, which resembles a tooth, consists of
414
several lines and arcs in a complex arrangement as shown in Fig. 3. As used in
the roller chain from previous work, the connections between links are modeled
with bushings to account for the flexibility in this investigation. Though the
sprockets of the silent chain serve in the same function of the rolling chain
system, however, they are designed to engage specifically with the links of the
silent chain with different tooth contour as illustrated.
Figure 3. Components of Silent Chain Link System
2.2.2.3 TENSIONER AND CHAIN GUIDE
In a chain drive system, the chain guide ensure that the chain remains on the
path, while tensioner try to keep constant tension of chain system. Usually the
chain guide directs the tight chain portion which runs from the driven sprocket to
the driving sprocket. Conversely, the chain arm directs the slack portion of the
chain which runs opposite (from the driving to the driven). The pivot guide also
serves to distribute the force on the chain from the hydraulic tensioner to
maintain certain level of chain tension. In this investigation hydraulic tensioning
force model is used which is offered from hydraulic tensioner manufacturer.
The chain guide and the chain arm are both modeled as separate rigid body
parts. The geometric profiles of the guides consist of a series arcs with different
radii. If desired, the chain guides can be modified so that they are constructed as
flexible bodies for the calculation of vibrations, stresses and bending moments,
etc.
2.2.2.4. EQUATIONS OF MOTION AND INTEGRATION
Since the chain system interacts with the frame component through the contact
415
forces and adjacent chain links are connected by compliant force elements, each
chain link in the chain system has six degrees of freedom which are represented
by three translational coordinates and three Euler angles. The equations of
motion of the frame structure such as sprockets that employs the velocity
transformation defined by Choi [9] are given as follows :
)( r
i
r
i qBMQBqMBBTT (1)
where r
iq and B are relative independent coordinates and velocity
transformation matrix of the engine chassis subsystem, and M is the mass matrix,
and Q is the generalized external and internal force vector of the frame structure
subsystem, respectively. Since there is no kinematic coupling between the frame
structure subsystem and chain subsystem, the equations of motion of the chain
subsystem can be written simply as
ttt QqM (2)
where tM , t
q and tQ denote the mass matrix, the generalized coordinate
and force vectors for the chain subsystem, respectively. Consequently, the
accelerations of the frame structure components and the chain links can be
obtained by solving Eqs. (1) and (2).
Many different types of integration methods can be employed for solving the
equations of motion for mechanical systems. Explicit methods have small
stability region and are often suitable for smooth systems whose magnitude of
eigenvalues is relatively small. Contrast to the explicit methods, implicit
methods have large stability region and are suitable for stiff systems whose
magnitude of eigenvalues is large. In the model used in this investigation, a
contact between two bodies is modeled by compliance elements. Lumped
characteristics of the spring and damper must represent elastic and plastic
deformations, and hysterisis of a material. Such characteristics may include
artificial high frequencies which are not concern of a design engineer. Unless
such artificial high frequency is filtered, an integration stepsize must be reduced
so small that integration can’t be completed in a practical design cycle of a
mechanical system. To achieve this goal, the implicit generalized-alpha method
416
[9, 11] has been employed to filter frequencies beyond a certain level and to
dissipate an undesirable excitation of a response in this investigation. One of the
nice advantages of the generalized-alpha method is that the filtering frequency
and dissipation amount can be freely controlled by varying a parameter in the
integration formula. As a result, the generalized-alpha method is the most
suitable integration method for integrating the equations of motion for stiff
mechanical systems.
2.2.3. CONTACT FORCE ANALYSIS
The contact collision algorithms for a silent chain drive used in this
investigation are composed of three main routines such as search routines for,
sprocket teeth and chain link contact, chain guide and chain link contact, and
side guide of chain link and sprocket contact. The contact positions and
penetration values are defined from the kinematics of components in searching
routines. Thereafter a concentrated contact force is used at the contacted position
of the contact surface of the bodies. A detailed discussion on the formulation of
the contact collision is represented in this section, respectively. Efficient search
algorithms should be considered seriously because there are large number of
chain link bodies and sprocket which take long time to search all the bodies
whether they are in contact or not.
2.2.3.1 STRATEGE OF CONTACT SEARCH
For the efficient search of the sprocket-chain link contact kinematics, the
contact search algorithm is divided by pre-search and post-search. In the pre-
search, bounding circle relative to sprocket center is defined. All of chain links
are employed to detect a starting link and ending link which has a possibility of
sprocket contact. Then, chain links from starting link are investigated the
engagement with sprocket valley. Post-search means a detailed contact
inspection for chain links in a bounding circle. Once a starting and ending link is
found at one time through pre-search prior to analysis, only detailed search is
carried out by using the information of starting link and ending link from the
next time step. There are four contact possibilities such as, arc-line, arc-point,
417
arc-arc and line-point contact for interaction between the sprocket teeth and chain
link.
2.2.3.2. LINE-ARC CONTACT
Figure 4. Line-Arc Contact Kinematics
The contact conditions between the sprocket teeth line segment and the chain
link arc segment can be determined. A coordinate system i
t
i
t
i
t ZYX is attached
to each of the sprocket surfaces shown in Fig. 4. The surfaces of the tooth line
are approximated by plane surfaces and the i
tX axis of each surface coordinate
system is assumed to be parallel to the tooth surface. The surfaces of the chain
link arc segment are approximated by plane surfaces and thej
pX axis of each arc
origin coordinate system is assumed to be directed to the starting arc point from
arc origin. The orientation of the tooth surface k coordinate system with
respect to the global system is defined by
i
k
ii
t AAA (3)
where iA is the transformation matrix that defines the orientation of the
coordinate system of the sprocket i and i
kA is the transformation matrix that
X
Y
Z
iX
iY
iZ
i
tX
i
tY
i
tZ
iR
i
tu
j
pXjY
jZ
jR
j
pu
p
t
ij
kua
j
pY
jX
Global coordinate
system
Sprocket
coordinate system
Chain link
coordinate system
Tooth
coordinate system
Tooth line
Link arc
418
defines the orientation of the tooth line surface k coordinate system i
t
i
t
i
t ZYX
with respect to the sprocket coordinate system. The orientation of the link arc
coordinate system with respect to the global system is defined by
j
l
jj
a AAA (4)
where j
lA is the transformation matrix that defines the orientation of the chain
arc surface l coordinate system j
p
j
p
j
p ZYX with respect to the chain link
coordinate system.
The global position vector of the coordinate system of the tooth surface k is
defined as
i
t
iii
t uARr (5)
where iR is the global position vector of the coordinate system of the
sprocket i and i
tu is the position vector of point t with respect to the origin
of the sprocket coordinate system iii ZYX .
The global position vector of the center of the chain link arc segment, denoted
as point p , can be defined as
j
p
jjj
p uARr (6)
where jR is the global position vector of the origin of chain link j , j
A is
the transformation matrix of chain link j and j
pu is the position vector of
point p defined in the chain link coordinate system jjj ZYX .
The position vector of the center of the arc of chain link j with respect to the
origin of the tooth line surface coordinate system can be defined in the global
coordinate system as
i
t
j
p
ij
k rru (7)
The components of the vector ij
ku along the axes of the tooth line surface
coordinate system are determined as
ij
k
Ti
t
Tij
z
ij
y
ij
x
ij uuu uAu (8)
419
Necessary but not sufficient conditions for the contact to occur between the
chain link arc and the sprocket tooth line surface k are
k
ij
x lu 0 (9)
pt
ij
zpt wwuww (10)
ru ij
y (11)
where kl is the length of the tooth line surface k , tw is half width of the
tooth and pw is half width of the chain link outer plate and r is the radius of
the chain link arc. If the above conditions are satisfied, it has to be checked if
contact point is existed in the arc range for the next step.
gd ji
k , where ][ hgfA i
k (12)
ji
k
Tj
a
Tji
z
ji
y
ji
x
ji
k ddd dAd (13)
),(atan2 ji
x
ji
yk dd (14)
ak 0 (15)
where ji
kd is the opposite signed normal vector of the tooth line surface k ,
k is the angle of ji
kd with respect to the link arc segment coordinate system
and a is the angle of arc segment.
If the above conditions are satisfied, the penetration ij is evaluated as ij
y
ij ur (16)
420
2.2.3.3. ARC-POINT CONTACT
Figure 5. Arc-Point Contact Kinematics
There are two arc-point contact possibilities such as convex arc vs. point and
concave arc vs. point contact for arc-point interaction between the sprocket teeth
and chain link. Figure 5 shows a convex arc-point contact kinematics. The arc-
point contact conditions between the sprocket teeth and the chain link can be
determined. A coordinate system i
t
i
t
i
t ZYX is located at the center point of the
sprocket arc surfaces.
The position vector of the point p of chain link j with respect to the center
point of the tooth surface coordinate system can be defined in the global
coordinate system such as in Eqs. (7) and (8)
Necessary but not sufficient conditions for the contact to occur between the
chain link point and the sprocket tooth surface k are
ruu ij
y
ij
x 22 )()( (17)
pt
ij
zpt wwuww (18)
where r is the radius of the sprocket arc segment, tw is half width of the
X
Y
Z
iX
iY
iZ
i
tXi
tY
i
tZ
iR
i
tu
jY
jZ
jR
j
pu
p
t
ij
kua
jX
Sprocket
coordinate system
Tooth
coordinate system
Global
coordinate system
Chain link
coordinate system
Tooth arc
Link point
421
tooth and pw is half width of the chain link outer plate.
If the above conditions are satisfied, it has to be checked if contact point is
existed in the arc range for the next step.
),(atan2 ij
x
ij
yk uu (19)
ak 0 (20)
where k is the angle of ij
ku with respect to the sprocket arc segment
coordinate system and a is the angle of arc segment.
If the above conditions are satisfied, the penetration ij is evaluated as
22 )()( ij
y
ij
x
ij uur (21)
2.2.3.4. ARC-ARC CONTACT
There are four arc-arc contact possibilities such as convex vs. convex, convex
vs. concave, concave vs. convex, concave vs. concave arc contact for arc-arc
interaction between the sprocket teeth and chain link. Since the radius and angle
of each arc are given at geometry, the contact kinematics between arcs can be
calculated by expanding arc-point contact logic. At the center of the arc a marker
is attached and X axis is fixed to the starting point of arc. The monitoring vector
between arc centers can be easily detected whether they are in contact boundary
or not using the arc angles with respect to the X axis of the marker. If the vector
is in contact boundary and the length between the centers of arcs is less than the
sum of the radii of arcs, they are considered in contact situation.
2.2.3.5. LINE-POINT CONTACT
The search kinematics of line-point contact is one of the most simple search
algorithms in contact analysis. An axis of marker can be attached on the line and
the vertical vector from the point to line can be evaluated whether the point is in
contact with line, respectively.
422
2.2.3.6. CONTACT FORCE MODEL
In the field of multi-body dynamics, one of the most popular approximation of
the dynamic behavior of a contact pair has been that one body penetrates into the
other body with a velocity on a contact point, thereafter the compliant normal
and friction forces are generated between a contact pair. In this compliant
contact force model, a contact normal force can be defined as an equation of the
penetration, which yields
nm
n ckf (22)
where and are an amount of penetration and its velocity, respectively.
The spring and damping coefficients of k and c can be determined from
analytical and experimental methods. The order m of the indentation can
compensate the spring force of restitution for non-linear characteristics, and the
order n can prevent a damping force from being excessively generated when the
relative indentation is very small. As it happens, the contact force may be
negative due to a large negative damping force, which is not realistic. This
unnatural situation can be resolved by using the indentation exponent greater
than one. A friction force can be determined as follows.
nf fvf )( (23)
2.2.4. NUMERICAL STUDY OF AN AUTOMOTIVE SILENT CHAIN
SYSTEM
Four cylinder DOHC (double overhead cam) engine valve drive mechanism is
employed for the sake of numerical verification of proposed methods as shown
in Figure 1. A silent chain drive system has 1 crankshaft sprocket, 2 camshaft
sprockets, 1 fixed guide, 1 pivot guide, tensioner element, and 135 chain links.
The crank sprocket of the system is rotated by motion constraint. Resistance
torque is applied at each camshaft sprockets. Hydraulic tensioning force model is
used which is offered from manufacturer.
Figure 6 shows the computer simulation model of automotive silent chain
system in computer graphic environment. The system consists of 143 rigid
423
bodies, 270 bushing force elements to connect chain link bodies, 4 revolute
joints, 2 resistance torque and a hydraulic force element of tensioner. It has 815
degrees of freedom.
Figure 7 and 8 demonstrate the trajectory and velocity of the chain link during
the cycle around the system when the engine runs 4000 rpm, respectively. The
X-Y trajectory of the links agrees the defined path of the chain motion and the
magnitude of link velocity with respect to system inertia reference frame reflect
the linear velocity of 4000rpm as clearly shown in Fig. 8. Figure 9 shows the
contact force between a chain link and the sprockets or the chain guides and
figure 10 shows the dynamic chain tension measured between chain links during
simulation. Since the hydraulic auto tensioner is attached on guide arm, the
dynamic tension of the chain is controlled not to have excessive or be loosened.
Dynamic analysis of the silent chain system is performed for 200 milli-sec. It is
found that the CPU simulation times is 4039 sec on a Pentium 1.8 GHz platform
personal computer. Note that since the numerical results from the proposed
methods are almost showing the real physical behaviors and dynamic
characteristics of the chain mechanism, the proposed methods using multibody
dynamic techniques can be valid and suitable for the design of the silent chain
system, accordingly.
Figure 6. Simulation Model of Automotive Silent Chain System
424
Figure 7. Trajectory of the Chain Link
Figure 8. Velocity of the Chain Link
425
Figure 9. Contact Forces of the Chain Link at 4000 rpm
Figure 10. Dynamic Tension of the Chain Link at 4000 rpm
Cam Sp. Cam Sp. Crank Sp. Cam Sp.
Pivot
Guide Fixed
Guide
Pivot
Guide
426
2.2.5. FUTURE WORK AND CONCLUSIONS
It is clearly proved in this investigation that using the multibody dynamic
simulation methods the dynamic analysis of silent chain mechanisms can be
achieved clearly. While previous works showed rough estimations of the silent
chain system, the proposed methods in this paper show the possibility of the
replacement of real prototype at early design stage. The presented three
dimensional silent chain consists of the driving sprocket, idle sprockets, pivot
guide, fixed guide, tensioner, and chain links. Pre and post contact search
algorithms are employed in order to increase the simulation speed significantly.
For the sprocket teeth and link teeth, guide and link contacts, line-arc, arc-point,
arc-arc, and line-point kinematic interactions are presented in this investigation.
A compliant force model is used to connect the rigid body chain links. The silent
chain model has 143 bodies, 4 pin joints, tensioner element and 270 compliant
bushing elements and has 815 degrees of freedom. The numerical study of
automotive silent chain system shows that the tendency of the chain motion and
tensions are close as real system and it shows the characteristics of silent chain
comparing to roller chain with less oscillation.
REFERENCES
1. C. K. Chen and F. Freudenstein, ''Towards a More Exact Kinematics of Roller
Chain Drives”, ASME Journal of Mechanisms, Transmission, and Automation in
Design, Vol.110, No.3, 123-130 (1988)
2. N. M. Veikos and F., Freudenstein, "On the Dynamic Analysis of Roller Chain
Drives: Part1 and 2", Mechanism Design and Synthesis, DE-vol 46, ASME, NY,
431-450 (1992)
3. K. W. Wang, ''On the Stability of Chain Drive Systems Under Periodic Sprocket
Oscillations'', ASME Journal of Vibration and Acoustics, Vol. 114, 119-126 (1992)
4. K. W. Wang, et al, ''On the Impact Intensity of Vibrating Axially Moving Roller
Chains'', ASME Journal of Vibration and Acoustics, Vol. 114, 397-403 (1992)
427
5. M. S. Kim and G. E. Johnson, Advancing Power Transmission into the 21st
Centrury, DE-vol. 43-2, ASME, NY, 689-696 (1992)
6. M. S. Kim and G. E. Johnson, Advances in Design Automation, DE-vol. 65-1
(B. J. Gilmore et al., eds), ASME, NY, 257-268 (1993)
7. W. Choi and G. E. Johnson, Vibration of Mechanical Systems and the History of
Mechanical Design, DE-vol. 63 (R. Echenpodi et al., eds), ASME, NY, 29-40
(1993)
8. W. Choi and G. E. Johnson, Vibration of Mechanical Systems and the History of
Mechanical Design, DE-vol. 63 (R. Echenpodi et al., eds), ASME, NY, 19-28
(1993)
9. H. S. Ryu, D. S. Bae, J. H. Choi and A. Shabana, ''A Compliant Track Model For
High Speed, High Mobility Tracked Vehicle'', International Journal For
Numerical Methods in Engineering, Vol. 48, 1481-1502 (2000)
10. “Phased Chain System Quietly Transmits Power”, Automotive Engineering,
Dec. (1995)
11. J. Chung, J. M. Lee, ''A New Family of Explicit Time Integration Methods for
Linear and Non-linear Structural Dynamics'', International Journal for
Numerical Methods in Engineering, Vol.37, 3961-3976 (1994)
428
2.2 THE RESEARCH OF MULTIPLE AXES
CHAIN COUPLER METHOD FOR AUTOMOTIVE ENGINE SYSTEMS
2.2.1. INTRODUCTION
Modern design process of automotive systems is required to reduce the overall
development time. That point became most significant issue in automotive area and
it will be so in future. The engine system is composed a lot of component modules.
Among them, timing chain system is one of the most important systems. To
analyze timing chain systems in multi-body dynamics (MBD), Ryu and Choi
presented efficient contact search algorism using compliance contact model
between sprockets and chain links, which are modeled by arcs and line [1]. Chen
and Freudenstein presented a kinematic analysis of chain drive mechanism with the
aim of obtaining insight into the phenomena of chordal action, with associated
impact and chain motion fluctuation [2]. Veikos and Freudenstein developed a
lumped mass dynamic model based on Lagrange’s equations of motion and showed
chain drive dynamics and vibrations [3]. Wang investigated the stability of a chain
drive mechanism under periodic sprocket excitations and studied the effect of
impact intensity in their axially moving roller chains [4]. Kim and Johnson
developed detailed model of the roller-sprocket contact mechanics that allowed the
fires determination of actual pressure angles and a multi-body dynamic simulation
[5],[6]. This investigation is based on Kane’s dynamics equations. Choi and
Johnson investigated the effects of impact, polygonal action, and chain tensioners
into the axially moving chain system and showed the transverse vibration of chain
spans [7],[8]. In this paper, the most focused point is primary to propose proper
method to get reasonable numerical results using minimum engineering parameters,
secondary, to spend less time to analyze timing chain system using proposed
method.
429
2.2.2 THE CONNECTIVITY OF MULTIPLE AXES CHAIN COUPLER (MACC)
Figure 1 represents connectivity of MACC in coupling system using timing
chain. Screw forces couplers (SFC) are used to represent reaction force and torque
acting on chain sprockets in a timing chain system. Two SFCs are attached for one
set of MACC. One SFC consists of two screw forces for a drive body and a driven
body.
Figure 1. The connectivity of MACC in a Timing Chain System.
The chain sprockets can be drive body or driven body according to acting cases of
SFC. In figure 1, symbol A and B are screw forces for the chain sprocket #1 as a
drive body and for the chain sprocket #2 as a driven body. Symbol C and D are
also screw forces for the chain sprocket #1 as a driven body and for the chain
sprocket #2 as a drive body.
2.2.3 MODELING OF MACC
A set of MACC is modeled as shown figure 2. D denotes distance between center
of two bodies. 1aR , 2aR denote radius of drive body and radius of driven body in
Screw Force Coupler #2
(For right side tension)
Chain Sprocket #1
Becomes drive body in case of Screw Force Coupler #1
Becomes driven body in case of Screw Force Coupler #2
Chain Sprocket #2
Becomes driven body in case of Screw Force Coupler #1
Becomes drive body in case of Screw Force Coupler #2
Screw Force Coupler #1
(For left side tension)
A
B
C
D
430
SFC #1, respectively. 1bR , 2bR denote radius of drive body and radius of driven
body in SFC #2, respectively. 0 is defined in the following form.
D
RR aa 211
0 sin or
D
RR bb 121
0 sin
In figure1, k , c denote stiffness and damping coefficient, respectively. In the case
of SFC #1, the coordinate system aRMX _ , aRMY _ will be reference frame for
reaction force and torque. In the case of SFC #2, the coordinate system bRMX _ ,
bRMY _ will be reference frame for reaction force and torque.
Figure 2. Modeling of MACC.
2.2.4. FORMULATION OF MACC
In figure 3, the drive body (driven body in SFC #2) in SFC #1 is frozen with
ground body by a fixed joint not to rotate. And the driven body (drive body in SFC
#2) in SFC #1 is connected with ground body using a revolute joint with motion.
0
D
1aR
2aR
aRMx _
aRMy _
Drive Body
Driven Body
0D
1bR
2bR
bRMx _
bRMy _
Drive Body
Driven Body
SFC #1 SFC #2
[ A set of MACC ]
kc k c
(1)
431
MACC can be simply represented with two examples, which are positive rotating
(+rad/s) and negative rotating (-rad/s). First, when the body, which is driven body
(drive body in SFC #2) in SFC #1, is rotating with positive direction (+rad/sec),
reaction force and torque are generated.
Figure 3. 1st Example Model using MACC
In figure 3, L denotes distance between tangential points of two bodies. It is
defined in the following form.
)cos( 0DL
reF , which is reaction force, is defined in the following form.
)()(
LcLkFre
Where k , c denote stiffness coefficient and damping coefficient, respectively.
And L ,
L denote change of L and velocity of L , respectively. L and
1aR
2aR
11, aa
aRMX _
aRMY _
Drive Body
Driven Body 22 , aa
0
1bR
2bR
11, bb
bRMX _
bRMY _
22 , bb
Drive Body
Driven Body
SFC #1 SFC #2
0
L
reF
1_ areT
reF2_ areT
D
(2)
(3)
432
L are defined as following.
)( 2211 aaaa RRL
)( 2211 aaaa RRL
In (3), (4), 1aR , 2aR can be replaced 1bR , 2bR In case of SFC #2 as shown
figure 3. 1a , 1a
, 2a , 2a
can be also replaced 1b , 1b
, 2b , 2b
in case
of SFC #2 as shown figure 3. 1_ areT , 2_ areT , which are reaction torques, are
defined in the following form.
11_ areare RFT
22_ areare RFT
If L is less than 0 in equation (3), the reaction force becomes negative value. In
this case, the negative reaction force is dealt with as zero according to an
assumption of MACC in this paper. Accordingly, the reaction force and torque are
not generated in SFC #2 as shown figure 3. Finally, generalized force and torque
are generated referred by coordinate system aRMX _ , aRMY _ as shown figure 4.
Figure 4. The Result of 1st Example Model from Figure 3
zareT _1_
xreF _
yreF _
zareT _2_
(4)
(5)
(6)
(7)
433
xreF _ , yreF _ denote generalized force. Those are defined as following.
)cos( 0_ rexre FF
)sin( 0_ reyre FF
Second example with negative rotating (-rad/s) is represented in figure 5. The
driven body (drive body in SFC #1) in SFC #2 is frozen with ground body by a
fixed joint not to rotate. And the drive body (driven body in SFC #1) in SFC #2 is
connected with ground body using a revolute joint with motion.
Figure 5. 2nd Example Model using MACC
In figure 5, reF is defined in (3). L ,
L are defined using (4), (5), which have
1bR , 2bR , 1b , 1b
, 2b and 2b
for SFC #2. Finally, generalized force and
torque are generated referred by coordinate system bRMX _ , bRMY _ as shown
figure 6.
1aR
2aR
11, aa
aRMY _
aRMY _
Drive Body
Driven Body 22 , aa
0
1bR
2bR
11, bb
bRMX _
bRMY _
22 , bb
Drive Body
Driven Body
SFC #1 SFC #2
0L
reF
2_breT
reF1_breT
D
(8)
(9)
434
`
Figure 6. The Result of 2nd Example Model from Figure 5
2.2.5. CONSIDERATION OF PRE-LOAD AND DISTANCE CHANGE BETWEEN CHAIN SPROCKET BODIES
2.2.5.1. PRE-LOAD
Pre-load is applied with equation (4) in the following form.
kpreloadRRL aaaa /)( 2211
Pretension effect can be considered through (10) as shown figure 7
xreF _
yreF _
zbreT _1_
zbreT _2_
(10)
435
Figure 7. Pre-Load Effect
prereF _ denotes reaction force caused by pre-load in figure 7.
2.2.5.2. DISTANCE CHANGE BETWEEN CHAIN SPROCKETS
In real timing chain system, chain sprocket can have tiny translational movement
by vibration from a crank shaft and a cam shaft. Distance change between chain
sprockets is considered to represent the phenomenon as shown figure 8.
Figure 8. Distance Change between Chain Sprockets.
C
D
D
1
Vertical + Parallel Movement
CD
1
D
0L1L
A
B
Vertical Movement
expL
AB
2R
1R
RMX
RMY
discreF _
discreF _
discreF _
discreF _
prereF _prereF _
prereF _
0
prereF _
436
In figure 6, denotes distance change between chain sprockets. 1 is defined in
the following form.
D
RR 121
1 sin
1L is defined as following
)cos( 11 DL
expL denotes chain expansion caused by translational movement of chain
sprocket. expL is defined in the following form.
01 LLLexp
Distance change effect is applied with equation (4) in the following form
)( 2211 expLRRL
discreF _ denotes reaction force caused by distance change as shown figure 8.
Generalized forces are defined in the following form.
)cos( 1___ discrexdiscre FF
)sin( 1___ discreydiscre FF
Where is angle, which appears when distance change is occurred. The angle is
measured based on coordinate system RMX , RMY . is defined as following.
(11)
(12)
(13)
(14)
(15)
(16)
437
ABCD
ABCD1cos
Through the combination of equation (4), (10) and (14), the form of L is
complete including pre-load and distance change effect. The final form of L is
defined as following.
kpreloadLRRL exp /)( 2211
2.2.6. KINEMATICS AND EQUATION OF MOTION FOR RECURSIVE FORMULAS
The proposed method makes use of the relative position and orientation matrix.
This section presents the relative coordinate kinematics for proposed method as
well as for joints connecting two bodies. Translational and angular velocity of the
body coordinate system with respect to the global coordinate system are
respectively defined as following.
wr
Their corresponding quantities with respect to the body coordinate system are
defined as following.
wArA
YT
T
Where, Y is the combined velocity of the translation ans rotation. The recursive
velocity and virtual relationship for a pair of contiguous bodies are obtained in [9]
as following.
1)i(i1)i2(i1)(i1)i1(ii qBYBY
where 1)i(iq denotes the relative coordinate vector. It is important to note that
matrices 1)i1(iB and 1)i2(iB are only functions of the 1)i(iq . Similarly, the
(17)
(18)
(19)
(20)
(21)
438
recursive virtual displacement relationship is obtained in the following form.
1)i(i1)i2(i1)(i1)i1(iiδ δqBδZBZ
If the recursive formula in (21) is respectively applied to all joints, the following
relationship between the Cartesian and relative generalized velocities can be
obtained in the following from.
qBY
Where B is the collection of coefficients of the 1)i(iq and
T1nc
TT
2
T
1
T
0 nY,,Y,Y,YY
T1nr
T
)1(
T
12
T
01
T
0 nnq,,q,q,Yq
Where nc and nr denote the number of the Cartesian and relative coordinates,
respectively. Since q in (23) is an arbitrary vector in nrR , (21) and (23), which
are computationally equivalent, are actually valid for any vector nrRx such that
xBX
1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX
Where ncRX is the resulting vector of multiplication of B and x . As a
result, transformation of nrRx into ncRBx is actually calculated by
recursively applying (27) to achieve computational efficiency in this research.
Inversely, it is often necessary to transform a vector G in ncR into a new vector
GBg T in nrR . Such a transformation can be found in the generalized force
computation in the joint space with a known force in the Cartesian space. The
virtual work done by a Cartesian force ncRQ is obtained in the following form.
QZW Τδδ
(22)
(23)
(24)
(25)
(26)
(27)
(28)
439
Where Zδ must be kinematically admissible for all joints in a system.
Substitution of qBZ δδ into (29) yields
*TTT δδδ QqQBqW
Where QBQ T* .
The equations of motion for constrained systems have been obtained in the
following form.
0)QλΦYMBF ΤΖ
T (
Where the λ is the Lagrange multiplier vector for cut joints [10] in mR and Φ
represents the position level constraint vector in mR . The M and Q are the mass
matrix and force vector in the Cartesian space including the contact forces,
respectively. The equations of motion and the position level constraint can be
implicitly rewritten by introducing vq as
0λv,vqF ),,(
0qΦ )(
Successive differentiations of the position level constraint yield
0υvΦvqΦ q ),(
0γvΦvvqΦ q ),,(
Equation (31) and all levels of constraints comprise the over determined
differential algebraic system (DAS). An algorithm for the backward differentiation
formula (BDF) to solve the DAS is given in [11] as following.
(29)
(30)
(31)
(32)
(33)
(34)
440
0
βvvUβvqU
vvqΦvqΦ
qΦ)λv,vq(F
pH
)β(
)β(
,,
,
,,
)(
20
T
0
10
T
0
Where TTTTT λ,v,v,qp , 0β , 1β and 2β are determined by the
coefficients of the implicit integrators and 0U is an m)(nrnr matrix such that
the augmented square matrix
qΦUT
0 is nonsingular. The number of equations and
the number of unknowns in (35) are the same, and so Eq. (35) can be solved for p .
Newton Raphson method can be applied to obtain the solution p .
HΔpHp
1,2,3,...i,i1i Δppp
0
0UU000UU0ΦΦΦ00ΦΦ000Φ
FFFF
H
T0
vvq
vq
q
λvvq
p
T
00
T
00
T
0
β
β
Recursive formulas for pH and H in (36) are derived to evaluate them
efficiently.
2.2.7. NUMERICAL RESULTS
The timing chain model, which has compliance contact between sprockets and
chain link segments, is prepared for the sake of validation of proposed MACC
method as shown figure 9. The model has 3 chain sprockets, 2 guide rails and 122
(35)
(36)
(37)
(38)
441
chain links segments. The chain links segments are connected with compliant
bushing elements. The timing chain model is built using MACC as shown figure 10.
The model using MACC consists of 3 chain sprockets and 3 SFCs. In figure 9, 10,
the all chain sprockets are constrained by revolute joints with ground body
respectively. The revolute joint, which connects sprocket #1 with ground body, has
driving motion. In figure 11, the driving motion is defined as sine wave to clearly
compare chain tensions between two models. End time is 0.2 sec. Integrator is used
so called Implicit G-alpha.
Figure 9. Timing chain model for validation of proposed MACC method.
Sprocket #1
Sprocket #2
Sprocket #3
A
B
C
442
Figure 10. Timing Chain Model using MACC method.
The Chan tensions are respectively compared between area A, B and C in figure 9
and chain tension of SFC #1, #2 and #3 in figure 10. The comparison result is
shown in figure 12, 13 and 14, respectively. The results are compared in specific
time range from 0.025 to 0.2 sec. There is too noisy behavior at early time. That is
why comparing time range is modified as shown figure 12, 13 and 14. According
to the results, the tendency of the results is coincident with two models from
systemic point of view. In detail design point of view, MACC method is hard to
catch noise problems caused by complex characteristics including nonlinear contact
and bushings. The timing chain model has 746 DOF and calculation time is 4329
sec. On the other hand, the model using MACC method has 3 DOF (depends on
modeling) and calculation time is 3 sec.
SFC #1
SFC #2
SFC #3
Sprocket #1
Sprocket #2
Sprocket #3
443
Figure 11. Angular Velocity of Sprocket #1
Figure 12. Comparison of chain tensions acting on Area A and SFC #1
Figure 13. Comparison of chain tensions acting on Area B and SFC #2
444
Figure 14. Comparison of chain tensions acting on Area C and SFC #3
2.2.8. CONCLUSION
The proposed method using screw force couplers was presented and analyzed. This
proposed method enables fast analysis with small DOF and minimum engineering
parameters. Comparison between two models was implemented. One is the model
of timing chain system, which is modeled with detail components close to real. The
other is simplified model using MACC method. From the previous investigation,
even though MACC method is hard to catch highly oscillated problems caused by
nonlinearity, the tendency of the results is coincident each other. And it is expected
from systemic point of view that time for development of automotive timing chain
systems is more reduced before.
REFERENCES
1. H. S. Ryu, H. J. Cho, J. H. Choi, K. S. Park, “Efficient Contact and
Nonlinear Dynamic Modeling of Automotive Silent Chain Drive”,
MULTIBODY DYNAMICS 2003, IDMEC/IST, Lisbon, Portugal, July 1-4,
(2003)
2. C. K. Chen and F. Freudenstein, “Towards a More Exact Kinematics of
Roller Chain Drives” ASME Journal of Mechanisms, Transmission, and
Automation in Design, Vol.110, No.3, 123-130 (1988).
445
3. N. M. Veikos and F. Freudenstein, “On the Dynamics Analysis of Roller
Chain Drives: Part 1 and 2”, Mechanism Design and Synthesis, DE-vol 46,
ASME, NY, 431-450 (1992)
4. K. W. Wang, “On the Stability of Chain Drive Systems Under Periodic
Sprocket Oscillations”, ASME Journal of Vibration and Acoustics, Vol. 114,
119-126 (1992)
5. M. S. Kim and G. E. Johnson, Advancing Power Transmission into the 21st
Century, DE-vol. 43-2, ASME, NY, 689-696 (1992)
6. M. S. Kim and G. E. Johnson, Advances in Design Automation, DE-vol.
65-1 (B. J. Gilmore et al., eds), ASME NY, 257-268 (1993)
7. W. Choi and G. E. Johnson, Vibration of Mechanical Systems and the
History of Mechanical Design, DE-vol. 63 (R. Echenpodi et al., eds),
ASME, NY, 29-40 (1993)
8. W. Choi and G. E. Johnson, Vibration of Mechanical Systems and the
History of Mechanical Design, DE-vol. 63 (R. Echenpodi et al., eds),
ASME, NY, 19-28 (1993)
9. Angeles, J., 1997, “Fundamentals of Robotics Mechanical Systems”,
Springer
10. Wittenburg, J., 1997, “Dynamics of Systems of Rigid Bodies”, B. G.
Teubner, Stuttgart
11. Yen, J., Haug, E. J. and Potra, F. A., 1990, “Numerical Method for
Constrained Equations of Motion in Mechanical Systems Dynamics”,
Technical Report R-92 Center for Simulation and Design Optimization,
Department of Mechanical Engineering, and Department of Mathematics,
University of Iowa, Iowa City, Iowa.
446
2.3
SYSTEMATIC ENVIRONMENT
CONSTRUCTION FOR EFFICIENT
TIMING CHAIN ANALYSIS OF
MOTORCYCLE’S ENGINE
2.3.1. INTRODUCTION
Chain drive systems have been used as weight parts for power transmission and
matching sequential timing among driving components in motorcycle area, since
they are capable of transmitting large power and precise timing with high
efficiency and low maintenance cost is required as well. However, since the
complexity of modeling and difficulties of simulation of various types of chain
drive system, in spite of widespread use of timing chain, little works have been
implemented only regarding their own mechanism in the field of multi body
dynamics,. Especially, in timing chain drive system of motorcycle’s engine, more
accurate shape of parts, higher speed and lighter weight of all components are
required than others. Thus, modeling and simulation of a timing chain drive system
are considered as hard tasks without any special environment including convenient
modeling method and suitable integrator for stiff mechanical system. In this paper,
for the modeling and simulation of the timing chain system of motorcycle’s engine,
systematic environment for modeling for dynamic analysis is constructed and
demonstrated. The environment provides four main functions. First, an automatic
assembly function is employed to avoid initial interference between sprockets and
chain links. Second, an approach of quasi-flexible method is applied to express
bending behavior and vibration characteristic of guide components. Third, contact
search is implemented with projected method to reduce computational time of
dynamic analysis. Forth, in order to define the driving motion of rotating parts such
as a crank shaft and cam shafts, measured data from real timing chain model is able
to be used by the input function as boundary condition.
Yamaha Motor Co,.LTD [1] suggested to define the main construction of
systematic modeling environment for timing chain drive system of motorcycle’s
engine. Ryu and Choi [2] developed detail modeling method for assembled links
including compliant contact forces for links and other parts. Chen and Freudenstein
[3] presented a general kinematic analysis of chain drives with the aim of obtaining
447
insight into the phenomena of chordal action, and the associated impact and chain
motion fluctuation. Veikos and Freudenstein [4] developed a lumped mass dynamic
model based on Lagrange’s equations of motion and also studied chain drive
vibrations. Wang [5] investigated the stability of a chain drive under periodic
sprocket excitations and Wang et al [6] studied the effect of impact intensity in their
axially moving material model. There has been some design analysis in the view of
primitive dynamics behaviors of silent chain in power train industry and
commercial software [7].
2.3.2 CONSTRUCTION OF SYSTEMATIC ENVIRONMENT
FOR MODELING OF THE TIMING CHAIN DRIVE
SYSTEM OF MOTORCYCLE’S ENGINE
As shown in figure 1, in this investigation, the systematic environment consists
of mainly four steps to define the process of modeling efficiently. During the
process in the environment, entire construction of the timing chain system is
specified with engineering parameters.
Figure 1. THE PROCESS OF SYSTEMATIC ENVIRONMENT OF TIMING CHAIN
DRIVE SYSTEM OF MOTORCYCLE ENGINE
As for the first step of the systematic modeling process, in the step 1, main
layout of the timing chain drive system is defined by special parametric markers
that represent the position and orientation to handle all components of timing chain
model with parameterized relationship. In this step, the two types of timing chain
drive system are supported by alternative option as predefined global data, such as
448
a type of roller timing chain and a type of silent timing chain. In this paper, the type
of silent type is mainly introduced.
In the second step of the process, all components that are required for the
construction of timing chain model are selected and created based on special
parametric marker generated in first step. Added to it, the connectivity between
created components is able to be set by joint constraints, force elements and
compliant contact force. For instance, sprockets are connected with ground body by
revolute joints or bearings entities defined by bushing force element. And
compliant contact force is applied between sprocket teeth and chain links that are
defined by different arc segments having various radii. For the contact, furthermore,
during simulation, projected method is performed to reduce the calculation time as
two- dimensional approach. In this step, one of the main functions of the systematic
modeling process is applying experimental data measured from real timing chain
model. In order to give driving motion to the sprockets with the experimental data,
the function for applying boundary condition is developed to simulate more closely
to the real behavior compared to real timing chain model. In this paper, the analysis
results applied boundary condition are shown in section 6.
As for the third step, a method to find smooth tangential path automatically
along to the edge of components is implemented considering the number of teeth
and assembly radius of sprockets for engagement with chain links. The main
purpose of the automatic assembly function is to avoid interference between
sprockets and chain links before starting dynamic analysis.
Finally, the constructed model is simulated with suitable integrator that is
generalized-alpha, which has good performance regarding stiff mechanical system
in last step of the process. In section 7, it is explained in detail.
The all functions are handled by control panels during the process of systematic
environment.
2.3.3. AUTOMATIC ASSEMBLY FOR ENGAGEMENT
BETWEEN SPROCKETS AND CHAIN LINKS
As shown in figure 2, in this function of automatic assembly, according to the
shape of components, finding the smooth path is started as primary. For the finding
smooth path, The path is considered as intervals defined by geometric tangential
points that are named by number in figure 2.
449
Figure 2. FINDING SMOOTH TANGENTIAL PATH FOR THE ASSEMBLY OF CHAIN
LINKS ON SPROCKET
Besides, as shown in figure 3, secondary searching for the tangential point
considering position of sprocket’s teeth is performed to avoid conflict situation
between sprockets teeth and chain links at initial time. Finally, the chain links are
well engaged on the sprocket with proper position in figure 4.
Figure 3. FINDING TANGENTIAL POINT OF SPROCKET
Figure 4. WELL ENGAGED CHAIN LINKS WITH A SPROCKET
450
2.3.4. CONTACT SEARCH USING PROJECTED METHOD
The contact collision algorithms for a timing chain drive system are composed of
three kinds of contact type in this investigation. The types are for sprocket teeth
and chain links contact, chain guides and chain links contact, and side contact
between sprockets and chain links. The contact positions and penetration values are
defined from the kinematics of components in the contact searching. Thereafter a
concentrated contact force is used at the contacted position of the contact surface of
the bodies. For the timing chain drive, efficient contact searching algorithms should
be considered seriously because there are large number of chain link bodies and
sprockets, which take long time to search all the bodies whether they are in contact
or not. Generally, it assumed that contact situation is occurred in three-dimensional
space. Therefore the searching algorism employs depth direction once again, which
is generally z-direction. While on the other, in this investigation, depth direction is
not a concerned anymore. The contact situation is defined on projected plane that is
considered as two-dimensional plane. As for the projected method, computational
time consuming to consider the depth direction is much more saved. As shown in
figure 5, the computational time is decreased by 35%.
Figure 5. COMPARISON THE COMPUTATIONAL TIME BETWEEN TWO METHOD
OF CONTACT [Unit: Hour]
2.3.4.1. CONTACT SEARCH BETWEEN A SPROCKET AND A CHAN
LINK
There are several types of contact condition, such as line to arc, arc to arc and
point to arc etc. In this investigation, line to arc is represented for the explanation
of contact search between a sprocket and a chain link. The contact conditions
between line segments of the sprocket teeth and arc segments of the chain link are
able to be determined. In this section, three-dimensional contact search is explained
451
first. Thereafter the two dimensional method, which is projected method is
explained for easy understanding compared with both contact search algorisms.
A coordinate system i
t
i
t
i
t ZYX is attached to each of the sprocket surfaces shown in
figure 6. The surfaces of the tooth line are approximated by plane surfaces and the
axis of each surface coordinate system is assumed to be parallel to the tooth surface.
The surfaces of the chain link arc segment are approximated by plane surfaces and
the axis of each arc origin coordinate system is assumed to be directed to the tooth
arc point from arc origin. The orientation of the tooth surface k coordinate system
with respect to the global system is defined as
i
k
ii
t AAA ,
where iA is the transformation matrix that defines the orientation of the coordinate
system of the sprocket i . i
kA is the transformation matrix that defines the origin’s
orientation of the tooth surface k coordinate system i
t
i
t
i
t ZYX with respect to the
sprocket coordinate system. The global position vector of the coordinate system of
the origin of the tooth surface k is defined as
i
t
iii
t uARr
where iR is the global position vector of the coordinate system of the sprocket i
and i
tu is the position vector of point t with respect to the origin of the sprocket
coordinate system iii ZYX
The global position vector of the center of the chain link arc segment, denoted as
point p , can be defined as
j
p
jjj
p uARr ,
where jR is the global position vector of the origin of chain link j , j
A is the
transformation matrix of chain link j and j
pu is the position vector of point p
defined in the chain link coordinate system jjj ZYX
The position vector of the center of the arc of chain link j with respect to the
origin of the tooth line surface coordinate system can be defined in the global
coordinate system as
i
t
j
p
ij
k rru
The components of the vector ij
ku along the axes of the tooth line surface
coordinate system are determined as
(1)
(2)
(3)
(4)
452
ij
k
Ti
k
Tij
z
ij
y
ij
x
ij uuu uAu
Necessary but not sufficient conditions for the contact to occur between the chain
link arc and the sprocket tooth line surface k are
k
ij
x lu 0 ,
p
ij
zp lbulb ,
and
ru ij
y
In above equations, kl is the length of the tooth line surface k , b is half thickness
of the tooth and pl is half the length of the chain link pin and r is the radius of
the chain link arc. If the above conditions are satisfied, it has to be checked if
contact point is existed in the arc range for the next step.
j
l
jj
a AAA
In Eq. (9), jA is the transformation matrix that defines the orientation of the
coordinate system of the chain link j and j
lA is the transformation matrix that
defines the orientation of the chain arc surface l coordinate system j
p
j
p
j
p ZYX with
respect to the chain link coordinate system.
Ti
k
i
k
i
k
ji
k )6()5()4( AAAd ,
ji
k
Tj
a
Tji
z
ji
y
ji
x
ji
k ddd dAd ,
),(atan2 ji
x
ji
yj dd ,
and
ejj _0 ,
where jid is the opposite signed normal vector of the tooth line surface k . The j
is the angle of jid with respect to the orientation of chain link arc segment and
ej _ is the angle of arc segment as shown in figure 7.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
453
If the above conditions are satisfied, the penetration δ is evaluated as
ruδ ij
y .
As for the difference between three-dimensional contact search and contact search
using projected method, the contact search is repeated twice to consider depth
direction in three-dimensional contact search. However, in projected method, the
contact search is performed only once on the projected plane without consideration
of depth direction in figure 7. In this method, the projected plane is located in the
middle of body. As mentioned in the section 4, according to decreasing the time
and repetition for contact search, total computational time is also decreased.
Figure 6. COORDINATE SYSTEM FOR A SPROCKET AND A CHAIN LINK
Figure 7. COORDINATE SYSTEM ON PROJECTED PLANE.FOR CONTACT
SEARCH
(14)
454
2.3.4.2. CONTACT FORCE MODEL
In the field of multi-body dynamics, one of the most popular approximations of
the dynamic behavior of a contact pair has been that one body penetrates into the
other body with a velocity on a contact point, thereafter the compliant normal and
friction forces are generated between a contact pair. In this compliant contact force
model, a contact normal force can be defined as an equation of the penetration,
which yields
nm
n ckf
In Eq. (15), and are an amount of penetration and its velocity, respectively.
The spring and damping coefficients of k and c can be determined from analytical
and experimental methods. The order m of the indentation can compensate the
spring force of restitution for non-linear characteristics, and the order n can prevent
a damping force from being excessively generated when the relative indentation is
very small. As it happens, the contact force may be negative due to a large negative
damping force, which is not realistic. This unnatural situation can be resolved by
using the indentation exponent greater than one. A friction force can be determined
as
nf fvf )( ,
where, nf and )(v are a contact normal force and a friction coefficient,
respectively. 2.3.5 QUASI-FLEXIBLE BODY METHOD FOR THE
MODELING OF GUIDE COMPONENTS In structural analysis, in order to obtain deformable behavior, vibration
characteristic and load history by external excitation, finite element method is used
in general. However, according to increasing the DOF of the flexible body, it
carries a restraint of analysis caused by long calculation time and burden regarding
capability of machine. As for the multi body dynamics area, quite recently it was
started for the analysis of flexible body from dynamics point of view. However, it
has same restraint for the analysis based of DOF as well. In this paper, quasi-
flexible body method is introduced to obtain advantages especially in the guide
components of timing chain drive model. As for the advantage of this approach,
small DOF to be free from the restraint is required, besides it is possible to consider
flexibility in some measure. In timing chain drive system of motorcycle’s engine,
the guide components are excited directly by external contact force from chain
links that has extremely high speed. For that reason, the bending behavior and
(15)
(16)
455
vibration characteristic of the guide components are considerably important. In this
investigation, as shown in figure 8, the guide components are modeled by rigid
body segments, which are connected by revolute joints and rotational spring
elements. According to the connectivity for rigid segments, the grouped guide
presents flexibility in simulation of multi body dynamics. As shown in figure 9, this
approach of quasi-flexible body gives good correlation compared with
experimental data. This method also gives better correlation compared with the
numerical result of figure 10. The comparison data is the displacement of guide
segment edge as shown in figure 11.
.
Figure 8. GROUP GUIDE USING QUASI-FLEXIBLE MODELING METHOD.
Figure 9. COMPARISON RESULTS BETWEEN NEMERICAL RESULT OF QFB
GUIDE COMPONENTS AND EXPERIMENTAL DATA ON 9000RPM IN
CASE OF SINGLE-ENGINE
456
Figure 10. COMPARISON RESULTS BETWEEN NUMERICAL RESULT OF
ONLY ONE RIGID GUIDE COMPONENTS AND EXPERIMENTAL DATA ON
9000RPM IN CASE OF SINGLE-ENGINE
Figure 11. THE MEASURED POSITON OF TENSIONER GUIDE FOR THE
COMPARISON
2.3.6 APPLYING BOUNDARY CONDITION
In this investigation, the function to apply boundary condition for driving motion of
a crank shaft and cam shafts is introduced. In order to define position and
orientation of a body in three-dimensional space, six independent coordinates are
required. For that reason, three revolute joints and three translational joints are used
with five dummy bodies. It means that the five dummy plus two more bodies that
ground body and a sprocket components are connected by the six joint constraints.
In the timing chain drive, the sprockets are rotated with shafts, of which speed
reaches over 10,000rpm. Using this function, six kinds of constraint motion, which
are defined by experimental data from real timing chain drive system, are applied
to define the dynamic behavior the sprockets. In general, driving constraint motion
457
along to the driving axis is mainly applied to define rotational sped of sprockets.
The figure 12 shows the comparison results with good agreement, which are
flapping amplitude of chain locus at the specified position according to the engine
speed respectively. In figure 13, in addition, it shows that if the experimental data is
used using the function of boundary condition, an actual characteristic is
expressible. The comparison data in figure 13 is measured based on the fixed point
as shown in figure 14.
Figure 12. FLAPPING AMPLITUDE OF CHAIN LOCUS AT THE SPECIFIED
POSITION IN CASE OF FOUR-CYLINDER ENGINE
Figure 13. FLAPPING AMPLITUDE OF CHAIN LOCUS BASED ON FIXED
POSITION ON 9000RPM IN CASE OF SINGLE-ENGINE
458
Figure 14. THE FIXED POSITION TO MEASURE FLAPPING AMPLITUDE OF
CHAIN LOCUS
2.3.7 EQUATION OF MOTION AND INTEGRATOR
The engaged chain links interact with the frame component, which are sprockets
and guides, through the contact forces and adjacent chain links are connected by
compliant force elements. Therefore, each chain link in the timing chain system has
six degrees of freedom, which are represented by three translational coordinates
and three Euler angles. The equations of motion of the frame structure that employs
the velocity transformation defined by Choi [2] are given as
)( r
ii qBMQBqMBBTT ,
where r
iq , B and q are relative independent coordinates, velocity transformation
matrix, and Cartesian velocities of the engine chassis subsystem, and M is the mass
matrix, and Q is the generalized external and internal force vector of the frame
structure subsystem, respectively. Since there is no kinematic coupling between the
frame structure subsystem and chain subsystem, the equations of motion of the
chain subsystem can be written simply as
ttt QqM ,
where tM , t
q and tQ denote the mass matrix; and the generalized coordinate
and force vectors for the chain subsystem simply since each chain links are
connected by bushing force elements. Consequently, the accelerations of the frame
structure components and the chain links can be obtained by solving Eqs. (15) and
(16).
Many different types of integration methods can be employed for solving the
(17)
(18)
459
equations of motion for mechanical systems. Explicit methods have small stability
region and are often suitable for smooth systems whose magnitude of eigenvalues
is relatively small. Contrast to the explicit methods, implicit methods have large
stability region and are suitable for stiff systems whose magnitude of eigenvalues is
large. In the timing chain model used in this investigation, a contact between two
bodies is modeled by compliance elements. Lumped characteristics of the spring
and damper must represent elastic and plastic deformations, and hysteresis of a
material. Such characteristics may include artificial high frequencies which are not
concerns of a design engineer. Unless such artificial high frequency is filtered, an
integration stepsize must be reduced so small that integration cannot be completed
in a practical design cycle of a mechanical system. To achieve this goal, the
implicit generalized-alpha method [2, 8] has been employed to filter frequencies
beyond a certain level and to dissipate an undesirable excitation of a response. One
of the nice advantages of the generalized-alpha method is that the filtering
frequency and dissipation amount can be freely controlled by varying a parameter
in the integration formula. As a result, the generalized-alpha method is the most
suitable integration method for integrating the equations of motion for stiff
mechanical systems.
2.3.8. CONCULSIONS
In this paper, the systematic environment for modeling of timing chain drive
system of motorcycle’s engine is developed in multi body dynamics point of view.
Four main functions are employed for more efficient modeling and simulation such
as automatic assembly function, contact search using projected method, group
guide modeling using quasi-flexible body method and the function of boundary
condition. This study demonstrated each implementation procedure of four the
main functions and numerical results compared to experimental data measured
from real timing chain model. The numerical study shows good agreement and
tendency compared with experimental results. The timing chain model used in this
study has 775 degree of freedom and 236 compliant bushing forces. As a result of
this proposed modeling method, it shows possibility to replace for real prototype
model at early design stage.
REFERENCES
1. Technical report of Yamaha Motor CO,.LTD, 2005
2. H.S Ryu, D.S.Bae, J.H.Choi and A.Shabana, 2000 “A compliant Track
Model For High Speed, High Mobility Tracked Vehicle”, International
Journal For Numerical Methods if Engineering, Vol. 48, 1481-1502.
460
3. C. K. Chen and F. Freudenstein, 1988, ''Towards a More Exact Kinematics
of Roller Chain Drives”, ASME Journal of Mechanisms, Transmission,
and Automation in Design, Vol.110, No.3, pp.123-130
4. N. M. Veikos and F., Freudenstein, 1992, "On the Dynamic Analysis of
Roller Chain Drives: Part1 and 2", Mechanism Design and Synthesis, DE-
vol 46, ASME, NY, pp 431-450
5. K. W. Wang, 1992, ''On the Stability of Chain Drive Systems Under
Periodic Sprocket Oscillations'', ASME Journal of Vibration and Acoustics,
Vol. 114, pp.119-126
6. K. W. Wang, et al, 1992, ''On the Impact Intensity of Vibrating Axially
Moving Roller Chains'', ASME Journal of Vibration and Acoustics, Vol.
114, pp.397-403
7. “Phased Chain System Quietly Transmits Power”, Automotive Engineering,
1995, Dec
8. J. Chung, J. M. Lee, 1994 ''A New Family of Explicit Time Integration
Methods for Linear and Non-linear Structural Dynamics'', International
Journal for Numerical Methods in Engineering, Vol.37, 3961-3976
3. Belt
462
3.1
HYDRAULIC AUTO TENSIONER (HAT) FOR
BELT DRIVE SYSTEM
3.1.1. INTRODUCTION
The hydraulic auto tensioner is a device that automatically adjusts the tension
for engine belt drive system. By reducing the noise due to play that occurs if the
tension on the belt drive system is insufficient and by holding the tension
constant, an auto tensioner extends the product life of the belt drive system and
is an indispensable part for improving engine reliability [1]. It is important to
analyze and to predict the dynamic behavior and the characteristics of the
hydraulic auto tensioner for design of the system. At this, numerical simulation
models can provide significant advantages in early design stage referred in [2]
and [3]. A simple simulation technique of HAT is applied for the initial design of
belts and chains using commercial multibody software [7]. Figure 1 shows the
hydraulic auto tensioner system. The plunger is connected to the belt drive
system. The spring force and the hydraulic force of the pressure chamber create
the damping force and are balanced with the load that is from belt drive system.
The check ball has the function of the check valve for control the oil flow
through orifice between the plunger and the cylinder.
463
Plunger
Cylinder
Check Ball
Pressure Chamber
Plunger
Spring
Check Ball
Spring
Figure 1. Hydraulic auto tensioner system
Figure 2 shows the schematic diagram of the operating principle of HAT. As
the tension of the belt drive system decreases, and the pressure of chamber
decreases, the check ball moves down and the check valve opens. Afterward due
to the plunger moves up by the spring force and the plunger pushes the belt, and
then the tension of belt system is increased. As the tension of the belt drive
system increases, the plunger moves down by the load and the plunger pushes
the pressure chamber, and it leads that the pressure of chamber increases, finally
the check ball moves up and the check valve closes. As a result, the oil flows
through the leakdown and the plunger moves down slowly.
Tension Decreasing
Tension Increasing
Oil
Flow
Figure 2. Operating principle
464
Since the tension of the belt drive system is oscillated over 200~300Hz, the
hydraulic auto tensioner must be a reciprocating hydraulic device that can
respond to frequencies up to 300Hz.
The multibody simulation model of the hydraulic auto tensioner is presented
in the following sections. The differential equations are used to describe the
function and damping characteristics of the hydraulic auto tensioner, and the
circle to curve contact model is used for the movement of the check ball. In this
investigation, the developed HAT model is tested numerically for multibody belt
drive system.
3.1.2. MULTIBODY SIMULATION MODEL
The hydraulic auto tensioner consists of cylinder, plunger and check ball. The
spring force and the damping force of the plunger relative to the cylinder balance
these bodies. The spring force is built up by the spring preload and the spring
rate multiplied by the spring stiffness. The damping force is a friction force and a
hydraulic force that is proportional to the relative velocity of plunger and
cylinder [3].
The schematic diagram of analysis model is shown in Figure 3. When the
plunger is loaded from belt drive system, the spring force and the hydraulic force
react against the motion of the plunger. The hydraulic force from the check ball
is ignored in this investigation since it is relatively small amount. The motion of
the plunger is assumed to have the parallel direction to the motion of the cylinder.
The check ball has the spring force and the hydraulic force from the plunger. The
motion of the check ball is also assumed to have the parallel direction to the
motion of the plunger. The check ball is contacted between plunger and retainer.
465
Joint
Tra
nsla
tion Jo
int
Sprin
g F
orc
e
Hydra
ulic
Forc
e
Tra
nsla
tion Jo
int
Check
ball
Sprin
g F
orc
e
Hydra
ulic
Forc
e
Plunger
Conta
ct
Cylinder
Belt Drive System
Figure 3. Schematic diagram of analysis model
3.1.3. THE EQUATION OF MOTION
When the external load is forced to the plunger, the equation of the plunger
motion is following [3].
loadriccp
cpphydraulicppp
Ffxx
xxKFxm
)sgn(
)(.
(1)
where, px, px
, and pxare the displacement of the plunger, and its first and
466
second time derivatives, and pm
, cx, cx
, and pKare the mass of plunger, the
displacement of the cylinder, its first time derivative, and the stiffness coefficient
of the plunger spring, and hydraulicpF . , ricf, and loadF
are the hydraulic force, the
friction force and the load form belt drive system, respectively.
In the case of check ball, since its motion is forced by the hydraulic force, the
spring force, and contact force, the equation of motion of the check ball can be
written as [3]
contactBOpBB
cBBhydraulicBBB
FFxxK
xxFxm
)(
)(.
(2)
3.1.4. HYDRAULIC FORCES
The hydraulic forces that interact with the check ball and the plunger are
obtained from the pressure of the pressure chamber. The pressure is caused by
the volume variation of the pressure chamber and the oil flow rate. The volume
variation of the pressure chamber can be described by relative velocity between
plunger and cylinder. The rate change of the chamber volume is given by the
following equation [5].
pBairoilcppchamber QQVVxxSV )( (3)
where oilV is the compressed volume rate of pure oil and airV is the compressed
volume rate of air component in the oil. Qp is leak oil flow rate out of the high
compression chamber at high pressure phase, QB is the oil flow rate through
check valve, and Sp is the effective area of hydraulic force, respectively.
467
3.1.4.1. OIL FLOW RATES THROUGH THE CHECK VALVE
Accordingly to the check ball moves between the plunger and the retainer, the
check valve opens or closes and the oil flows. When the check valve opens, the
oil flow through the check valve is shown in Figure 4.
Po
Pi
d
r
QB
Figure 4. Oil flow rate through check valve
As the resistance of the oil through the check valve depends on the orifice area,
in this investigation, the dynamic resistance is considered for the turbulent flow
of the oil flow through the check valve [4], which yields,
ioiodB PPg
PPACQ
2)sgn(
(4)
where, dC is discharge coefficient of check valve, A is the orifice area, g is
gravity acceleration, and is weight density of oil, respectively. QB represents
the oil stream flowing rate through the opened check valve into or out of the
pressure chamber. The area of orifice is obtained such as
cossin2 drA (5)
468
3.1.4.2. OIL FLOW RATES THROUGH THE LEAK BETWEEN PLUNGER AND
CYLINDER
As the pressure of chamber is different comparing to the air pressure, the oil
flow through the gap between plunger and cylinder is shown in Figure 5. The oil
flowing between the plunger and the cylinder is laminar flow. The oil speed is
faster than the plunger speed. As shown in Figure 5, variation of the oil speed is
fully depended on the pressure difference between the pressure chamber and
reserver, and it is not affected by the plunger speed. The oil flow rate between
the gap of plunger and cylinder, Qp , can be written as [4]
)(12
2 3
ioP
p PPl
hrQ
(6)
where is the coefficient of viscosity of oil.
Po
Pi Qp
l
rp h
Figure 5. Oil flow rate through leak As shown in the Figure 4 and 5, we can consider about the relationship
between the plunger speed and the flow rate. It is assumed that inflow does not
induce any outflow from the pressure chamber by considering compressibility,
and expansion and compression processes are isentropic. It is also assumed that
469
there is no cavitation caused by negative pressure. The volume of air in chamber is
obtained as.
0.
1
airi
oair V
PP
V
(7)
where is the ratio of specific heat and Vair.0 is the initial air volume. Since air
can be compressed, the volume rate is achieved by using the following equation,
such as
i
i
airair P
P
VV
(8)
, and the volume of oil in chamber can be approximated by
pcpoil SxxV )( (9)
In the case of oil, since it can also be compressed with high pressure, the
volume rate of oil is written as.
ioil
oil PK
VV
(10)
where K is the bulk modulus.
The equations (4), (6), (8) and (10) are substituted into the equation (3), and
the differential equation for the pressure of the camber can be obtained,
accordingly;
470
K
V
P
V
PPg
PPACPPl
hrxxS
poil
i
air
ioioioP
CPP
i
2)sgn()(
12
2)(
3
(11)
The hydraulic force to the plunger and to the check ball yield as
)(. iophydraulicp PPSF (12)
)(. ioBhydraulicB PPSF (13)
where SB can be obtained from Figure 4 as following.
2cos rSB (14)
3.1.5. CONTACK OF THE CHECK BALL The contact analysis of the check ball employs the circle to curve contact
method [6] in this investigation. This method is very efficient algorithm in
contact detection and force generation of the check ball contact.
471
Figure 6. Concept of circle to curve contack
The candidate lines on the plunger body have been selected for the contact of
the check ball. For the candidate lines, it is necessary to compute the amount of
penetration to generate the contact forces, as shown in Figure 6.
The relative position pnd of a check ball with respect to the contact reference
frame is obtained as follows.
1pcnpn sdd (15)
where the vector pnd is projected into the contact reference frame as
pn
T
ppn dCd (16)
where Cp is the orientation matrix of the contact reference frame. The
penetration of the node into the patch is calculated by
472
pn
T
p- dn r (17)
where is always positive. The pn is a normal vector of a line and a constant
vector with respect to the contact reference frame. Thus, the contact normal force
is obtained by
32
1 mm
m
c o n t a c t ckF
(18)
where k and c are the spring and damping coefficients which are determined by
assumed numerical experiences, or experimental methods, respectively and the
δ is time differentiation of δ . The exponents 1m and 2m generates a non-
linear contact force and the exponent 3m yields an indentation damping effect.
When the penetration is very small, the contact force may be negative due to a
large negative damping force, which is not realistic. This situation can be
avoided by using the indentation damping exponent greater than one.
3.1.6. BELT DRIVE SYSTEM
An automotive belt drive system is used for the simulation of HAT in order to
test numerically. This system is consisted of 5 pulleys and a belt system. A
continuous belt system can be modeled using series of a single body that has six
degrees of freedom and has a matrix (6x6) force element to connect the belt
bodies. Contact forces between the belt and pulleys are defined clearly.
473
Disturbance roller
Drive
Pulley
HAT
Sensing belt tension
Belt
Figure 7. Belt drive system
As shown in Figure 7, there are a drive pulley, a disturbance roller, four idle
pulleys, and an idle roller equipped with HAT.
3.1.7. NUMBERICAL RESULTS
The hydraulic auto tensioner must be a reciprocating hydraulic device that can
respond to frequencies up to 300Hz. When the reciprocating load is applied to
plunger with 300Hz, Figure 8 shows the result of the pressure in chamber and
Figure 9 shows the result the displacement of the check ball. The numerical
results show that the proposed modeling of HAT is acting to the reciprocating
load with 300Hz. As the load increases, the check valve closes and the oil flows
out only through the leak. As the load decrease, the check valve opens and the
oil flows in through the check valve.
474
Figure 8. Pressure in chamber [300Hz]
Figure 9. Displacement of check ball [300Hz]
The proposed modeling method of hydraulic auto tensioner is applied for the
belt drive system as shown in Figure 7. The drive pulley rotates with 100 rpm.
As the disturbance roller increases the belt length, the belt tension around HAT
increases such as shown in Figure 10. Due to the belt tension increases, the
475
pressure in chamber arises as shown in Figure 11 and the oil flows out through
the leak as shown in Figure 12. As a result, the plunger is pushed back and the
belt tension decreases. Figure 10 shows less increase of tension of the belt with
HAT comparing to without it.
Figure 10. Tension
Figure 11. Chamber pressure
477
Figure 12. Oil flow rate through leak
As the disturbance roller decreases the belt length, the tension around HAT
decreases as shown in Figure 13. Due to the belt tension decreases, the pressure
in chamber decreases as illustrated in Figure 14. Figure 15 shows the oil flow
rate through the check valve. As a result, the plunger is pushed to the direction
for increasing the tension by the plunger spring, and therefore the tension
increases. The tension drop can be quickly recovered with proposed HAT
element as shown in Figure 13.
478
Figure 13. Tension
Figure 14. Chamber Pressure
479
Figure 15. Oil flow rate through check valve
3.1.8. CONCLUSIONS
In this investigation, in order to design automotive power transmitting system
at early design stage, modeling and simulation methods of HAT, which is
necessary component for the tension adjusting system, are presented. The
multibody simulation model is proposed using three rigid bodies, which are
plunger, check ball and cylinder. The plunger and the cylinder bodies can be
connected by constraints and mechanical force elements. The plunger and the
cylinder are interacted by hydraulic force and spring force. The forces between
plunger and check ball are modeled by contact, hydraulic, and spring forces. The
circle to curve contact analysis is employed for the plunger and the check ball
contact efficiently. The differential equations of motion of the components and
the hydraulic force equations are developed in this investigation. It can be
assured that the proposed HAT model is able to respond to frequencies up to
300Hz. The proposed methods of HAT are simulated in different ways,
component level simulation with reciprocating forces, and with automotive belt
system. Both numerical results show reasonable responses as expected. Though
it is necessary to be correlated by experimental results. Therefore the proposed
480
numerical method of HAT shows the possibility of simulation for automotive
power transmitting system, which has been challenging works for long period.
REFERENCES
1. http://www.ntn.co.jp/english/corp/news/news/20011001_2.html
2. NTN TECHNICAL REVIEW No. 61
3. NTN TECHNICAL REVIEW No. 67
4. Frank M. White, "Fluid Mechanics", 5th edition, McGraw-Hill International
Editions, 1999.
5. E. Sonntag , Richard, Claus Borgna, kke, and Gordon J. Van Wylen,
"Fundamentals of Thermodynamics", 5th Edition, John Wiley & Sons, Inc., 1998.
6. B. O. Roh, H. S. Anm, D. S. Bae, H. J. Cho, H. K. Sung, "A Relative Contact
Formulation for Multibody System Dynamics", KSME International Journal, Vol.
14, No. 12, pp. 1328-1336, 2000.
7. www.fev.com
4. Gear
482
4.1
DYNAMIC ANALYSIS OF CONTACTING SPUR
GEAR PAIR FOR FAST SYSTEM SIMULATION
4.1.1. INTRODUCTION
All Geared systems are commonly used in many mechanical power
transmitting systems, such as robot manipulator, automotive transmissions, etc.,
so as to transmit motion and power from one shaft to another. One of important
factors in the gear design is the dynamic transmission error, which gear vibration,
noise and other performance can be predicted by. When two mating gear is
operated, the dynamic transmission error is generated by gear dynamic forces.
These forces are caused by contact between meshing teeth. In other words,
contact mechanics between meshing teeth, considering backlash and tooth
geometric profile, is very important in the dynamic analysis of geared systems. A
lot of numerical and experimental works have been published about their
dynamic analysis. One of main topics in these studies is the conventional finite
element analysis. Traditional finite element methods are effective for calculating
quantities such as mesh stiffness, tooth deformations, and stress distributions
under static conditions. But it requires refined meshes to represent the tooth
contact and precise tooth surface shape for gear mechanics. Also, it takes
amazingly long time to analyze the dynamics effects of contacting gears.
Moreover, it is not suitable for analysis of entire system with the sets of gear
pairs as well as other components [1, 2]. Another topic is that concerning the
single degree of freedom(sdof) models of a pair of gears. It is because sdof
model can give relatively accurate results and computational efficiency despite
its simplicity. The sdof model approach in terms of entire system dynamic
analysis with gear pairs is desirable from research and design perspectives. In
sdof model, primitive approach is to model gear pairs with simple constraint or
483
force element using speed ratio, pressure angle and rotational angles. Gear
systems can be analyzed with fast computational time, but detailed inputs such
as tooth profiles and distance between gears are not considered directly because
it is not real gear teeth contact. More advanced approach is considered by contact
between teeth profile of gears. It enables designer not only to obtain gear contact
position and force exactly but also to simulate with entire system in various
operating conditions [3, 4].
A review of the mathematical models used in gear dynamics was given by
Ozguven and Houser [5], and T. Shing et al. presented an improved model for
the dynamics of spur gear systems with backlash consideration [6]. The torsional
vibration behavior was investigated experimentally by Kahraman and
Blankenship [7, 8, 9]. In the recent studies, a sdof model was proposed, which
considers a time-varying stiffness and backlash of the meshing tooth pairs with
similar formulations. However, most gear models in these numerical
investigations have been used the kinematic relations between the rotational
angles of each gear. It is not real contact model between bodies and needs some
limitation that gear shafts have no translational displacement.
The main purpose of this paper is to develop efficient contact algorithm
between meshing teeth in geared system for better understanding of the dynamic
behavior of entire system. Externally specified dynamic forces, or assumptions
about modeling the mesh forces by time-varying stiffness and static transmission
error are not required since dynamic mesh forces are obtained by contact
analysis at each time step. A simple spur gear pair modeled by using proposed
methods is compared and verified with the measurement results represented by
reference [7]. The dynamic modeling techniques are suggested and efficient &
fast dynamic analysis of a set of complex geared mechanical system is presented
in this investigation.
4.1.2. TOOTH PROFILE OF SPUR GEAR
The gear teeth profile is usually defined a special profile called an involute
curve for constant speed ratio. However, it is not efficient to use the exact
involute profile in the contact search algorithm because of its complexity of
484
contact search kinematics. In order to approximate the involute profiles, biarc
curve fitting method which is proposed by Bolton[11] is employed in this
investigation. The optimum biarc curve passing through a given set of points
along involute curve can be determined by this approximation technique. The
more arcs are used to describe the involute profiles, the less numerical error is
occurred in approximation, but the more calculation time will be required for
contact search of tooth profiles. Consequently, the real geometry of involute
tooth profiles in this investigation is represented by 5 arcs with different radii as
shown in Figure 1, since the error is acceptably small.
Fig. 1 Involute curves by 5 arcs
Arc segment Absolute error (mm) Relative error (%)
1 0.000229 0.00147
2 0.000349 0.00152
3 0.000388 0.00165
4 0.000409 0.00168
5 0.000461 0.00182
Table 1. Absolute and relative error
Table 1 shows the difference between exact involute curve and approximated
485
arc segment in spur gear with 24 teeth, 2 mm module, and 20 pressure angle.
Absolute error is an average distance between points on exact involute curve and
points on arc segment from gear center. Relative error is an average difference
percentage that absolute error is divided by average distance of points on
involute curve from gear center. Since the main purpose of the research is to
understand the dynamic behaviors of system with the gear pairs, these kinematic
errors might affect very small for the highly oscillating nonlinear dynamics of
gear system, accordingly.
4.1.3. EFFICIENT CONTACT SEARCH ALGORITHM AND CONTACT
FORCE MODEL
The contact algorithms for a gear pair are investigated in this section. The
contact positions and penetrated values are defined from the kinematics of
components in searching routines. Thereafter, a concentrated contact force is
generated at the contacted position of the contact surface of the bodies. A
detailed discussion on the formulation of the contact collision is represented in
this section.
4.1.3.1. ARC-ARC CONTACT
Since the radius and angle of each arc are given at geometry, the contact
kinematics between arcs can be calculated by contact logic. A marker is attached
at the center of the arc and X axis is fixed to the starting point of arc. The
monitoring vector between arc centers can be easily detected whether they are in
contact boundary or not using the arc angles with respect to the X axis of the
marker. If the vector is in contact boundary and the length between the centers of
arcs is less than the sum of the radii of arcs, they are considered as contact
candidate.
486
Fig. 2 Arc-arc contact kinematics
The contact conditions between the gear tooth convex arc segment and the
pinion tooth convex arc segment can be determined as follows. A coordinate
system i
t
i
t
i
t ZYX and j
p
j
p
j
p ZYX is attached to each arc origin coordinate system
shown in Fig. 2. The surface of the gear tooth arc segment is approximated by
plane surfaces and the i
tX axis of each surface coordinate system is assumed to be
directed to the starting arc point from arc origin. The surface of pinion tooth arc
segment is approximated by plane surfaces and thej
pX axis of each arc origin
coordinate system is assumed to be directed to the starting arc point from arc
origin. The orientation of the gear tooth arc k coordinate system with respect to
the global system is defined by
ik
iit AAA (1)
where iA is the transformation matrix that defines the orientation of the
X
Y
Z
iX
iY
iZ
i
tXi
tY
i
tZ
iR
i
tu
j
pX
jY
jZ
jR
j
pup
t
ij
ku
jj
pY
jX
Global coordinate system
Gear
coordinate system
Pinion
coordinate system
Gear tooth
coordinate system
Pinion tooth
coordinate system
i
487
coordinate system of the gear i and i
kA is the transformation matrix that
defines the orientation of the gear tooth arc k coordinate system i
t
i
t
i
t ZYX with
respect to the gear coordinate system. The orientation of the pinion tooth arc l
coordinate system with respect to the global system is defined by
jl
jjp AAA
(2)
where jA is the transformation matrix that defines the orientation of the
coordinate system of the pinion j and j
lA is the transformation matrix that
defines the orientation of the pinion tooth arc l coordinate system j
p
j
p
j
p ZYX
with respect to the pinion coordinate system.
The global position vector of the center of the gear arc segment, denoted as
point t , is defined as
it
iiit uARr (3)
where iR is the global position vector of the origin of the gear i and i
tu is
the position vector of arc center point t with respect to the origin of the gear
coordinate system iii ZYX .
The global position vector of the center of the pinion arc segment, denoted as
point p , can be defined as
jp
jjjp uARr
(4)
arc center point p defined in the pinion coordinate system jjj ZYX .
The position vector of the center of the arc of pinion with respect to the origin
of the gear tooth arc can be defined in the global coordinate system as
it
jp
ijk rru
(5)
488
The components of the vector ij
ku with respect to the gear and pinion tooth
coordinate system are determined, respectively, as
ijk
Tit
Tiijz
iijy
iijx
iij uuu uAu ,,,,
(6)
ijk
Tjp
Tjjiz
jjiy
jjix
jji uuu uAu ,,,,
(7)
Necessary but not sufficient conditions for the contact to be occurred between
the gear and pinion arc segment are
ptiij
yiij
x rruu 2,2, )()( (8)
ptijzpt wwuww
(9)
where tr and pr are the radius of the gear and pinion arc segment respectively,
tw is half width of the gear tooth and pw is half width of the pinion tooth.
If the above conditions are satisfied, it has to be checked if contact point is
existed in the arc range for the next step.
),(atan2 ,, iijx
iijym uu
, ),(atan2 ,, jji
xjji
yn uu (10)
km 0 , ln 0 (11)
where m and n are the angle of ij
ku with respect to the gear and pinion
tooth arc segment coordinate system and k and l are the angle of gear and
pinion arc segment, respectively.
If the above conditions are satisfied, the penetration ij is evaluated as
22 )()( ij
yijxpt
ij uurr (12)
489
4.1.3.2. ARC-POINT CONTACT
The arc-point contact conditions between the gear and the pinion can be
determined. A coordinate system i
t
i
t
i
t ZYX is located at the center point of the
gear arc surfaces.
The position vector of the point p of pinion j with respect to the center point
of the gear tooth arc is defined in the global coordinate system such as in Eqs. (5)
and (6).
Necessary but not sufficient conditions for the contact to occur between the
pinion point and the gear tooth k are
ruu ijy
ijx 22 )()(
(13)
ptijzpt wwuww
(14)
where r is the radius of the gear arc segment, tw is half width of the gear
tooth and pw is half width of the pinion tooth.
If the above conditions are satisfied, it has to be checked if contact point is
existed in the arc range for the next step.
),(atan2 ijx
ijym uu
(15)
km 0 (16)
where m is the angle of ij
ku with respect to the gear arc segment coordinate
system and k is the angle of arc segment.
If the above conditions are satisfied, the penetration ij is evaluated as
22 )()( ijy
ijx
ij uur (17)
4.1.3.3. CONTACT FORCE MODEL
490
In the field of multi-body dynamics, one of the most popular approximations
of the dynamic behavior of a contact pair has been that one body penetrates into
the other body with a velocity on a contact point, thereafter the compliant normal
and friction forces are generated between a contact pair. In this compliant
contact force model, a contact normal force can be defined as an equation of the
penetration, which yields
32
1 mm
mn δδ
δ
δckδf
(18)
where k and c are the spring and damping coefficients which are determined,
respectively and the is time differentiation of penetrated value . The
exponents 1m and 2m generates a non-linear contact force and the exponent
3m yields an indentation damping effect. When the penetration is very small,
the contact force may be negative due to a negative damping force, which is not
realistic. This situation can be overcome by using the indentation damping
exponent greater than one. The friction force is obtained by
nf ff (19)
where μ is the friction coefficient and its sign and magnitude can be
determined from the relative velocity of the pair on contact position.
4.1.4. KINEMATICS AND EQUATION OF MOTION FOR SYSTEM
DYNAMICS USING THE RECURSIVE FORMULAS
Recursive formulas using relative coordinates are very useful for gear system
dynamic analysis since gears in geared systems are usually rotated to one axis
direction. This section presents the relative coordinate kinematics for a contact
pair as well as for joints connecting two bodies.
Translational and angular velocities of the body coordinate system with
491
respect to the global coordinate system are respectively defined as
w
r
(20)
Their corresponding quantities with respect to the body coordinate system
are defined as
wA
rAY
T
T
(21)
where Y is the combined velocity of the translation and rotation. The recursive
velocity and virtual relationship for a pair of contiguous bodies are obtained in
[16] as
1)i(i1)i2(i1)(i1)i1(ii qBYBY (22)
where 1)i(iq denotes the relative coordinate vector. It is important to note that
matrices 1)i1(iB and 1)i2(iB are only functions of the 1)i(iq . Similarly, the
recursive virtual displacement relationship is obtained as follows
1)i(i1)i2(i1)(i1)i1(iiδ δqBδZBZ (23)
If the recursive formula in Eq. (22) is respectively applied to all joints, the
following relationship between the Cartesian and relative generalized velocities
can be obtained:
qBY (24)
where B is the collection of coefficients of the 1)i(iq and
T1nc
TT2
T1
T0
nY,,Y,Y,YY (25)
492
T1nr
T
)1(
T
12
T
01
T
0 nnq,,q,q,Yq (26)
where nc and nr denote the number of the Cartesian and relative coordinates,
respectively. Since q in Eq. (24) is an arbitrary vector in nrR , Eqs. (22) and
(24), which are computationally equivalent, are actually valid for any vector nr
Rx such that
xBX (27)
and
1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX (28)
where ncRX is the resulting vector of multiplication of B and x . As a
result, transformation of nr
Rx into ncRBx is actually calculated by
recursively applying Eq. (28) to achieve computational efficiency in this
research. Inversely, it is often necessary to transform a vector G in ncR into
a new vector GBgT in nr
R . Such a transformation can be found in the
generalized force computation in the joint space with a known force in the
Cartesian space. The virtual work done by a Cartesian force nc
RQ is
obtained as follows.
QZWΤδδ (29)
where Zδ must be kinematically admissible for all joints in a system.
Substitution of qBZ δδ into Eq. (29) yields
*TTT δδδ QqQBqW (30)
where QBQT* .
The equations of motion for constrained systems have been obtained as
follows.
493
0)QλΦYMBFΤΖ
T ( (31)
where the λ is the Lagrange multiplier vector for cut joints [17] in mR and
Φ represents the position level constraint vector in mR . The M and Q are
the mass matrix and force vector in the Cartesian space including the contact
forces, respectively.
4.1.5. NUMERICAL RESULTS
A spur gear pair system is analyzed for the sake of numerical verification of
proposed methods as shown in Fig. 3. The shafts of the two gears are assumed to
be rigid and the only the compliance of contact force between meshing teeth is
considered in this model. The gear pare model is composed of 2 spur gears, 2
revolute joints, and a gear contact element. Rotational dampers are used for
resistance torque at revolute joints. A gear is driven by steady torque of 10 Nm.
Fig. 3 Gear pair model
Gear/Pinion
Module 3 mm
Number of teeth 50
Contact element
Applied torque
Revolute joint & Rotational spring
damper
11 , r 22 , r
494
Pressure angle 20
Radius of pitch circle 75 mm
Radius of outside circle 78 mm
Radius of base circle 70.477 mm
Radius of root circle 71.25 mm
Tooth width 20 mm
Elasticity modulus 29 /10200 mN
Density 33 /1085.7 mkg
Center distance 150 mm
Table 2 Design parameters of gear and pinion
Table 2 shows design parameters of the spur gear sets which are the inputs of
numerical simulation. Dynamic analysis of a spur gear pair is simulated during
0.08 sec. Gear speed is increased up to 500 rad/sec (4800 rpm) almost linearly as
shown in Fig. 4(a). It is found that the CPU simulation time is just 15 sec on a
Pentium IV 3.0 GHz platform personal computer. Figure 4(b) demonstrates the
dynamic transmission error (DTE= 2211 rr ) with respect to time domain when
a gear is driven at the constant torque of 10 Nm. As rotating speed of gear is
increased, dynamic transmission error (DTE) is changed by gear teeth contact.
Figure 5(a) and 5(b) show the time-domain DTE around mesh frequency of 1900
Hz and 3000 Hz. Magnitude and waveform of DTE are different in each mesh
frequency. Magnitude of DTE is around 30 and 3 micro meter, respectively.
These results show similar magnitude and exact dynamic pattern as compared to
experimental measurement results (in the reference Fig. 6 and 7) introduced by
Blankenship and Kahraman [7]. The minor differences between the proposed
method and referenced [7] might be expected from the dimensions, measurement
settings and noises.
495
(a) Rotational velocity of driven gear
(b) Oscillating DTE with respect to time
Fig. 4 Rotational velocity and DTE
496
(a) DTE at the mesh frequency of 1900Hz
(b) DTE at the mesh frequency of 3000Hz
Fig. 5 Oscillating DTE time history
497
The key advantage of the proposed method is the fast & efficient system
simulation of geared multibody dynamic system without losing the system
dynamic characteristics caused by gear pair contacts and their flexibility. An
Engine system with multi gear sets is illustrated as another geared system
example model. The system has 4 degrees of freedom, which has 13 bodies, 6
revolute joints, one translational joint, 14 fixed joints, and 2 sets of contacting
spur gear pairs. Crankshaft in this model is rotated by gas force and gear sets are
driven by rotation of crankshaft as shown in Fig. 6. In order to examine the
effect of gear contact dynamics, the proposed gear contact force model is
compared by constraint coupler model which should be ideal solution but not
realistic. Figure 7 shows well the difference of output velocity from the final
gear between proposed method and conventional dynamic anaysis using
constraint only. Dynamic analyses of both models are performed for 0.01 sec. It
is found that the CPU simulation time is just 85 sec for the proposed method on a
Pentium IV 3.0 GHz platform personal computer.
Fig. 6 Engine model with multi gear set
498
Fig. 7 Rotational velocity in output gear
4.1.6. CONCLUSION
This research proposes an efficient implementation algorithm of spur gear
contact mechanisms for the fast system dynamic analysis. Externally specified
dynamic forces, or assumptions about modeling the mesh forces by time-varying
stiffness and static transmission error are not required since dynamic mesh forces
are obtained by contact analysis directly at each time step. Arc-Arc and arc-point
kinematic interactions are presented and a compliant force model is used in this
investigation. The relative coordinate formulation is employed to generate the
equations of motion. Two numerical examples, a simple spur gear pair and an
engine transmission system, are illustrated and simulated numerically in this
investigation. A simple spur gear pair model shows the validation of the
proposed method with measurement results illustrated by reference, and engine
transmission system shows the advantages of the proposed method, respectively.
Consequently it is possible to simulate the entire geared system dynamic analysis
without losing its important dynamic characteristics, such as vibration and noise,
etc., with reasonable CPU time as represented in this investigation.
499
REFERENCES
1. ANSYS User Manual, ANSYS Inc., PA., USA
2. ABAQUS User Manual, ABAQUS Inc., RI., USA
3. Kahraman, A. and Singh, R., ''Non-Linear Dynamics of a Spur Gear Pair'',
Journal of Sound and Vibration, Vol.142, No.1, pp.47-75, 1990.
4. Amabili, M. and Rivola, A., ''Dynamic Analysis of Spur Gear Pairs: Steady-State
Response and Stability of the SDOF Model with Time-Varying Meshing
Damping'', Mechanical Systems and Signal Processing, Vol.11, No.3, pp.375-390,
1997.
5. Ozguven, H. N. and Houser, D. R., ''Mathematical Models used in Gear
Dynamics – a Review'', Journal of Sound and Vibration, Vol.121, pp. 383-411,
1988.
6. Shing, T., Tsai, L., and Krishnaprasad, P., "An Improved Model for the Dynamics
of Spur Gear Systems with Backlash Consideration ", ASME-PUNLICATION-
DE, Vol.65-1, pp. 235-244, 1993.
7. Blankenship, G. W. and Kahraman, A., “Gear dynamics experiments, Part-I:
Characterization of forced response”, ASME, Power Transmission and Gearing
Conference, San Diego, 1996.
8. Kahraman, A. and Blankenship, G. W., “Gear dynamics experiments, Part-II:
Effect of involute contact ratio”, ASME, Power Transmission and Gearing
Conference, San Diego, 1996.
9. Kahraman, A. and Blankenship, G. W., “Gear dynamics experiments, Part-III :
Effect of involute tip relief”, ASME, Power Transmission and Gearing
Conference, San Diego, 1996.
10. Parker, R. G. and Vijayakar, S. M. and Imajo, T., ''Non-linear Dynamic
Response of a Spur Gear Pair : Modeling and Experimental Comparisons'',
500
Journal of Sound and Vibration, Vol.237, pp. 435-455, 2000.
11. Bolton, K. M., “Biarc curves”, Computer Aided Design, Vol.7, No.2, pp.89-92,
1975.
12. Parkinson, D. B. and Moreton, D. N., ''Optimal biarc curve fitting”, Computer
Aided Design, Vol. 23, No.6, pp.411-419, 1991.
13. Ryu, H. S., Huh. K. S., Bae, D. S. and Choi, J. H., ''Development of a Multibody
Dynamics Simulation Tool for Tracked Vehicles, Part I : Efficient Contact and
Nonlinear Dynamic Modeling'', JSME International Journal, Series C, Vol.46,
No.2, pp.540-549, 2003.
14. Lankarani, H. M., "Canonical Impulse-Momentum Equations for Impact
Analysis of Multibody System", ASME, Journal of Mechanical Design, Vol. 180,
pp. 180-186, 1992.
15. Bae, D. S., Han, J.
5. Bearing
502
5.1
CONTACT AND NONLINEAR DYNAMIC
MODELING OF A BALL BEARING FOR
MOTORCYCLE ENGINE SYSTEM
5.1.1. INTRODUCTION
Ball bearings are commonly used machine elements. They are employed to
permit rotary motion of shafts not only in simple commercial devices such as
bicycles, roller skates, and electric motors but also in complex engineering
mechanisms such as aircraft gas turbines, rolling mills, and power transmissions.
Especially, ball bearings have been often used in the motorcycle engine, shown
in Fig. 1, because of high efficiency and low maintenance cost. However, the
noise and vibrations caused by ball bearings have always been a problem, since
higher speed, lighter weight, and higher durability are required in recent
motorcycle engine. To ascertain the effectiveness of ball bearings in modern
motorcycle engine applications, it is necessary to obtain a firm understanding of
how these bearings perform and influence in the entire system under varied and
extremely demanding conditions of operations.
However, in spite of the widespread use of ball bearings, surprisingly a little
works have been published about modeling methods considering their basic
mechanics in viewpoint of entire system level. In other words, the most of
published papers[1-6] related to analysis of ball bearing have been focused on
analysis for ball bearing by itself. These techniques have a critical limitation in
that they cannot be extended with other entire system. It is because all waviness
of geometric entities such as inner/outer raceway and rolling balls should be
given in a form of boundary condition before analysis.
The purpose of this paper is to propose modeling method for the dynamic
characteristics of ball bearings rotating under high speed in system level. The
numerical methods are employed in this investigation. Dynamic impact forces of
ball and race are explored for the sake of understanding dynamic behaviors of
503
ball bearing.
5.1.2. BALL BEARING MODEL
A three-dimensional ball bearing, shown in Fig. 2, is composed of an inner
race, cage, balls, and outer race. In this paper, outer race is modeled as a series of
segments and they are connected with beam elements in order to consider its
flexibility. The inner race of the bearing is fixed rigidly with the shaft and the
outer race of the bearing is supported by surface contact element with housing
body. As balls are rotated, rotations of cage and ball can be determined from
kinematic relations and be represented to rotational constraint motions. Dummy
bodies are used for movement of balls in the radial and axial direction. Balls can
be in contact with inner and outer race. Figure 3 shows connections among
geometric entities that are used in this model.
Figure 1. Motorcycle Engine System with Ball Bearing
Figure 2. A Ball Bearing Model
riD roD
baD
inner
race
outer
race
cag
e
ball
504
Figure 3. Connection diagram of Geometric Entities
(1) Rotational Motion of Cage and Balls
As shaft is rotated, cage and balls are rotated with kinematic relation as
following equations, respectively.
baba
osri
i
DD
2)(
2)1(
baba
oro
o
DD
22)1(
where
s : rotational angle of shaft
ba : rotational angle of ball by itself
o : rotational angle of cage
i : a slip ratio between inner race and ball
o : a slip ratio between ball and outer race.
Rotational angles of cage and balls can be represented as a function of shaft
Shaft
Inner race
Cage
Fixed joint
Revolute joint + rotational motion
Orientation + inplane constraint
Contact element
Revolute joint + rotational motion
i-th dummy ball
i-th ball
j-th outer race segment
Housing
Contact element
Contact element
(1)
(2)
505
angle from the above equation (1) and (2).
sroorii
riio
DD
D
)1()1(
)1(
sroorii
rioi
ba
roba
DD
D
D
D
)1()1(
)1)(1(
5.1.3. BEAM MODEL
Figure 4. Mathematical Model of Outer Race
(1) Beam Element Between Outer Race
Outer ring is composed of n segments and n elastic beam elements as shown in
Fig.4. All segments have 6 degrees of freedom. In this section, beam force will
be explained in two-dimensional problem for the sake of simplicity. Figure 5
defines the segment coordinate system x - y . The -axis of the element
coordinate system is taken to be collinear with the line connecting segment i and
i+1. Referring to figure 6, element force vectors can be defined with respect to
the element coordinate system as
Tiiii NRYRXR R
Tiiii NQYQXQ 1111 Q
1iRiQ iQ iR
iR
1iQ
segment
beam i-1 i-1
i
i+
1
beam i
(3)
(4)
(5)
(6)
(7)
506
TTi
Tii 1 QRS
Also, the small deformation vector can be defined as
Tiiii vu δ
TTi
Tii 1 δδε
Beam forces can be calculated as
niiii ,...,2,1, εKS
where
iK : stiffness matrix of beam element i
iS : element force vector of beam element i
iε : segment displacement vector of beam element i
The stiffness matrix iK is given as
izzzz
zzz
zz
z
i
lIlIlIlI
IlII
AlAl
lIlI
symmetryI
Al
l
E
22
22
2
2
3
460260
1206120
00
460
120
K
where
E : Young’s elastic modulus
A : cross sectional area
zI : moment of inertia of cross section
l : length of a beam element
(8)
(9)
(10)
(11)
507
Since the -axis of the element coordinate system is chosen collinear to the
line which connects segment i and i+1, the segment displacement vector iε , that
is, iδ and 1iδ can be written as
Tiii 00δ
Tiioiii ll 11 0δ
where
il : length of the i-th beam element
oil : free length of the i-th beam element
The element forces iR and 1iQ can be obtained by solving Eq(10).
YX : inertial Coordinate System
yx : segment Coordinate System
: beam Element Coordinate System
Figure 5. Angular Displacement
i
i
i
1i
2i
ix
iyi
i
i
1i
1i
1i
1ix
1iy
X
Y
(12)
(13)
508
Figure 6. Beam Element Forces on the basis of the Beam Element i-th Coordinate System
(2) Beam Deflection
In order to obtain deformation shape of a beam, bending moments, iNR and
1iNQ , are used as a significant forces. The rest of the components for the force
vector are not taken into consideration, because the deformation is very small.
Deflection curve can be obtained from below differential equation.
EI
M
d
d
2
2
where M is a moment at the position .
From free-body diagram of Fig.7,
0/)( 1 LNQNRNRM iii
LNQNRNRM iii /)( 1
Deflection curve iv at the position can be calculated as follows from Eq.(14) and
Eq.(16).
iXR
iYR
iNR
1iXQ
1iYQ
ii
1iNQi
1i
(14)
(15)
(16)
509
iiiiiiiiiii
i LNQNRLNRNQNREIL
v 21
231 )2(3)(
6
1
This calculated deflection is be used for contact between ball and outer race.
Figure 7. Free-body Diagram of Beam and Segment Beam
5.4.4. BALL CONTACT
Balls can be in contact with inner and outer race. Contact method with outer
race is explained in this section. Figure 8 shows the flow chart of contact
algorithm. At first, corresponding beam element is found from ball position, then,
the contact plane can be defined from ball position and housing center as shown
in Fig.9. Using beam theory, deformation of the beam in the ball position is
obtained from the continuous deformed shape of beam in order to calculate the
center position of contact arc. And the contact arc of outer race and the contact
circle of ball can be defined in the contact plane. Contact search is performed
whether a contact between contact arc and ball is detected or not. If a contact is
detected, contact force is generated at contact position and derived contact force
is distributed to segment bodies that are connected with beam.
(17)
L
iNR
1iNQ
LNQNR ii /)( 1
MiNR
510
Figure 8. Flow Chart of Contact algorithm
(1) Contact Search
The contact conditions between ball and outer race can be determined as
follows. A coordinate system iii ZYX and jjj ZYX are attached to ball and
housing body coordinate system, respectively, as shown in Fig. 9. The position
vector of ball center with respect to the origin of the housing body can be
defined in the global coordinate system as
jiji rrd
The components of the vector jid with respect to the housing coordinate
system are defined as
TzjiyjixjijiTjji ddd ___ dAd
Calculate ball position
Find i-th beam element corresponding to each ball
Define Contact plane and Calculate to get beam deflection
Calculate beam deflection at from beam forces
Determine whether ball is in contact with outer race or not
When contact is detected, contact force is distributed to segment bodies that are connected with beam
(18)
(19)
511
2_
2_ )()( yjixji ddl
Arc coordinate system for arc center on the contact plane with respect to the
reference frame of the housing body, shown in Fig.10, is defined in the housing
body as
hgfD j
where
Tyjixji ldld 0// __ f
T100g
gfh
And position of arc center can be calculated from arc radius and beam deflection.
fs )( 1 vR jj
where 1jR is distance between housing center and groove center of outer race.
The position vector of ball center with respect to the center of arc can be defined in
the housing body coordinate system as
jji sdk
The components of the vector k with respect to the coordinate system jD are
defined as
Tzyx
T
j kkk kDk
Necessary but not sufficient condition for the contact to be occurred between ball
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
512
and outer race are
ijyx RRkk 2
22 )()(
where iR is a radius of ball and 2jR is a groove radius of outer race.
If ball is in range of arc angle or is in contact with arc end point, the penetration
is evaluated as
ijyx RRkk 222 )()(
(2) Contact Force Model
In the field of multi-body dynamics, one of the most popular approximation of the
dynamic behavior of a contact pair has been that one body penetrates into the other
body with a velocity on a contact point, thereafter the compliant normal and
friction forces are generated between a contact pair. In this compliant contact force
model, a contact normal force can be defined as an equation of the penetration,
which yields
32
1 mmmn δδ
δ
δckδf
where k and c are the spring and damping coefficients which are determined,
respectively and the is time differentiation of . The exponents 1m and 2m
generates a non-linear contact force and the exponent 3m yields an indentation
damping effect. When the penetration is very small, the contact force may be
negative due to a negative damping force, which is not realistic. This situation can
be overcome by using the indentation damping exponent greater than one. The
friction force is obtained by
nf ff
where μ is the friction coefficient and its sign and magnitude can be
determined from the relative velocity of the pair on contact position.
(28)
(29)
(30)
(31)
513
Figure 9. Contact Plane
Figure 10. Arc coordinate system
5.1.5. NUMERICAL RESULTS
A ball bearing system is analyzed for the sake of numerical verification of
proposed methods. This example model has 8 balls and 16 outer race segments.
Shaft is driven by initial velocity of 100 rad/sec. Table 1 shows dimension of a ball
bearing system that is analyzed numerically. Figure 11 shows ball contact forces
f
g k
f
hjs
jX
jY
X
Y
jX
jY
iX
iY
jid
Contact plane
ir
jr
Contact force
Inner
race Outer race
514
between ball and inner/outer race with respect to ball position. Contact forces
between outer ring segments and housing are shown in Fig.12. Load distribution in
a ball bearing is shown in Fig.13.
Number of balls 8
Number of outer race
segments
16
Inner raceway diameter 80.0 mm
Outer raceway diameter 112.0 mm
Ball diameter 16.0 mm
Inner groove diameter 16.0 mm
Outer groove diameter 16.0 mm
Shaft diameter 62.5 mm
Housing diameter 120.0 mm
Bearing thickness 20.0 mm
Center distance 150 mm
Table 1. Dimension of Ball Bearing System
Figure 11. Ball Contact Forces
0 60 120 180 240 300 360
-2
0
2
4
6
8
10
12
inner race contact
outer race contact
Ball
Conta
ct F
orc
e (
New
ton)
Ball Position (degree)
515
Figure 12. Contact Forces between Outer Ring Segments and Housing
Figure 13. Load Distribution in a Ball Bearing
5.1.6. CONCLUSION AND FUTURE WORK A three-dimensional ball bearing model is proposed in this paper. It consists of
inner race, cage, balls and outer race. Outer race is modeled as a series of segments
with beam element for its flexibility. Contact algorithm between ball and
inner/outer race considering the deflection of outer race is represented. Numerical
simulations of ball bearing model with 118 degrees of freedom has been carried out.
Contact forces between ball and race and between outer race and housing are
investigated. For the sake of better validation of the proposed methods, empirical
gravity
Segment
1 2
3
4
5
6
7
8 9
10
11
12
13
14
15
16
0.00 0.04 0.08 0.12 0.16 0.20
-1
0
1
2
3
4
5
6
segment 3
segment 5
segment 7
segment 9
Conta
ct F
orc
e (
New
ton)
Time (sec)
516
measurements of the system are further recommended.
REFERENCES
[1] S. Harsha and P. Kankar, 2004, ''Stability Analysis of a Rotor Bearing System
due to Surface Waviness and number of Balls'', International Journal of
Mechanical Sciences, Vol.46, pp. 1057-1081
[2] G. Jang and S. Jeong, 2004, ''Vibration Analysis of a Rotating System due to
the Effect of Ball Bearing Waviness'', ASME, Journal of Sound and Vibration,
Vol.269, pp. 709-726
[3] N. Akturk, M. Uneeb, and R. Gohar, 1997, ''The Effect of Number of Balls and
Preload on Vibrations Associated with Ball Bearings'', ASME, Journal of
Tribology, Vol.119, pp. 747-753
[4] J. Yang and S. Chen, 2002, ''Vibration Predictions and Verifications of Disk
Drive Spindle System with Ball Bearings'', Computers and Structures, Vol.80, pp.
1409-1418
[5] G. Hagiu and M. Gafitanu, 1997, ''Dynamic Characteristics of High Speed
Angular Contact Ball Bearings'', WEAR, Vol.211, pp. 22-29
[6] M. Tiwari and K. Gupta, 2000, ''Effect of Radial Internal Clearance of a Ball
Bearing on the Dynamics of a Balanced Horizontal Rotor'', Journal of Sound and
Vibration, Vol.238, pp. 723-756
[7] S. Sugiyama and T. Otaki, 1992, ''Mathematical Model for Brake Hose Layout'',
Society of Automotive Engineers, 922123
[8] T. Harris, 2001, "Rolling Bearing Analysis", 4th
Edition, John Wiley & Sons,
Inc.
517
5.2 NUMERICAL MODELING AND ANALYSIS OF JOURNAL BEARING WITH COUPLED
ELASTOHYDRODYNAMIC LUBRICATION AND FLEXIBLE MULTIBODY DYNAMICS
5.2.1. INTRODUCTION
The journal bearings, which is the one of the widely used machine elements,
transmit the power while reducing the friction and resisting the external loads. In
particular, in the internal combustion engine which is frequently used for power
generation, the various and lots of journal bearings are used between the piston,
piston pin, connecting rod, crankshaft, and engine block. These journal bearings,
which is under the alternating loads caused by the gas forces of the internal engines,
guarantee the smooth operation of the engine and are tightly related to the
durability of the engine system. Recently, in order to achieve the high-performance
output and to reduce the engine weight, the importance of the bearing lubrication
analysis is increasing (Taylor 1993, Oh and Goenka 1985, Labouff and Booker
1985).
The research on the lubrication characteristics and performance for journal
bearing has been studied widely in the area of tribology (Nair, Sinhasan, and Singh
1987, Makino and Koga 2002). The lubrication study for the bearing is based on
the Reynolds equations (Reynolds 1886) which is related to the thickness and
pressure of fluid film generated by the relative motion of objects. In particular, the
study on the trajectory by the relative motion of the journal bearing such as the
engine bearing which is resisting the alternating loads was first tried by Ott (1948)
and Hahn (1957). Dowson and Higginson (1959) had studied about the numerical
solutions for the elastohydrodynamic problems. And Hamrock and Dowson (1976)
had studied about the oil film thickness and the relations between contacts. In order
to estimate the lubrication film characteristics such as the oil film thickness,
pressure, power loss, and flow rate, the analysis using the elastohydrodynamics
lubrication is needed. In particular, in order to calculate the relative displacements
between bearing and journal, the theory for flexible multibody dynamics (or
MFBD) is also needed (Peiskammer et al. 2002, Riener et al. 2001, Choi 2009).
Generally, elastohydrodynamic lubrication can be classified by two types, which
518
are based on the relation of surface roughness and oil film thickness. One is the
full-film lubrication. It has been widely used when the lubricant film is sufficiently
thick so that there is no significant asperity contact. In this case, the pressure is
only governed by Reynolds equation which is first established by Reynolds (1886).
The other is the mixed lubrication. When the lubricant film is not enough to thick,
the asperity contacts between two bodies can be occurred (Zhu and Cheng 1988,
Greenwood and Tripp 1970-1971). As a result, the pressure by the fluid flow and
the pressure by the asperity contact should be considered together. Therefore, in
mixed lubrication region, the total pressure can be treated by the sum of the
pressures induced by the fluid flow and the asperity contact.
In order to consider the full-film and mixed regions together, this paper uses the
Greenwood and Tripp’s asperity contact model (1971) and Reynolds equation to
obtain the hydrodynamic pressure. The oil hole and groove effects are considered
by applying the pressure boundary conditions. Also the dynamic viscosity of oil is
considered as the function of the pressure by using the Barus law (Dowson and
Higginson 1977).
In the Section 2, the MFBD theory used in this study is introduced. The EHD
governing equations are introduced in Section 3 and the analysis procedure for
fluid-structure interactions is explained in Section 4. A numerical example is
discussed in Section 5, and finally the conclusions are in Section 6.
5.2.2. MULTI-FLEXIBLE-BODY DYNAMICS (MFBD)
The MFBD formulation which is used in this study is described well in Choi
(2009). In this section, the brief formulations for MBD and MFBD are introduced.
5.2.2.1. MBD Formulation
The coordinate systems for two contiguous rigid bodies in 3D space are shown in
Fig. 1. Two rigid bodies are connected by a joint, and an external force F is acting
on the rigid body j . The X-Y-Z frame is the inertial or global reference frame and
the x -y -z is the body reference frame with respect to the X-Y-Z frame. The
subscript i means the inboard body of body j in the spanning tree of a recursive
formulation (Bae et al. 2001). And, in this section, the subscript j can be replaced
with the subscript ( 1)i .
Velocities and virtual displacements of the origin of body reference frame x -y -z
with respect to the global reference frame X-Y-Z , respectively, defined as
rω
(1)
519
and
rπ
(2)
Figure 1. TWO CONTIGUOUS RIGID BODIES
Their corresponding quantities with respect to the body reference frame x -y -z are,
respectively, defined as T
T
r A rY
ω A ω (3)
and T
T
r A rZ
π A π (4)
where A is the orientation matrix of the x -y -z frame with respect to the X-Y-Z
frame.
The recursive velocity equations for a pair of contiguous bodies is obtained as 1 2
j ij i ij ij Y B Y B q (5)
where Y is the combined velocity of the translation and rotation as defined in Eq.
(3) and 1
ijB and 2
ijB are defined as follows:
T T
1
T
ij iij iiijj ij ji ij
ij
ij
A 0 I s d A s AB
0 A 0 I (6)
and
ri
X
Z
Y
xi1’’
yi1’’zi1’’
Inertial Ref. Frame
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Rigid Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
rj
F
di1(j1)
Joint
si(i1)
sj(j1)
ri
X
Z
Y
X
Z
Y
xi1’’
yi1’’zi1’’
Inertial Ref. Frame
Ai
xi’
yi’
zi’ Aj
xj’
yj’
zj’
Rigid Body i
Rigid Body jAi1
Aj1
Aj2
xj1’’
yj1’’zj1’’
xj2’’
yj2’’zj2’’
rj
F
di1(j1)
Joint
si(i1)
sj(j1)
520
TT
2
Tij
ij ij ji ij ijij
ij
ij
qI d A s A HA 0
B0 A 0 I
(7)
It is important to note that matrices 1
ijB and 2
ijB are only functions of ijq . Similarly,
the recursive virtual displacement relationship is obtained as follows: 1 2
j ij i ij ij Z B Z B q (8)
If the recursive formula in Eq. (5) is respectively applied to all joints along the
spanning tree, the following relationship between the Cartesian and relative
generalized velocities can be obtained:
Y Bq (9)
where B is the collection of coefficients of the ijq and
T
T T T T
0 1 2nc? 1
, , , , n Y Y Y Y Y (10)
And
T
T T T T
0 01 12 ( 1)nr? 1
, , , , n n q Y q q q (11)
where nc and nr denote the number of the Cartesian and relative generalized
coordinates, respectively. The Cartesian velocity ncY R with a given nrq R can
be evaluated either by using Eq. (9) obtained from symbolic substitutions or by
using Eq. (5) with recursive numerical substitution of jY .
It is often necessary to transform a vector G in ncR into a new vector Tg B G in nrR . Such a transformation can be found in the generalized force computation in
the joint space with a known force in the Cartesian space. The virtual work done by
a Cartesian force ncQ R is obtained as follows:
Τ W Z Q (12)
where Z must be kinematically admissible for all joints in a system. Substitution
of Z B q into Eq. (12) yields
T Τ T * W q B Q q Q (13)
where * TQ B Q .
521
The equations of motion for a constrained mechanical system (García de Jalón et al.
1986) in the joint space (Wittenburg 1977) have been obtained by using the
velocity transformation method as follows:
( T ΤΖF B MY Φ λ Q ) 0 (14)
where Φ and λ , respectively, denote the cut joint constraint and the
corresponding Lagrange multiplier. M is a mass matrix and Q is a force vector
including the external forces in the Cartesian space.
5.2.2.2. MFBD Formulation
The equation of motion for the rigid body can be expanded from the Eq. (14) as
follows:
T T T 0r rr rr er er r z zF B MY Φ λ Φ λ Q (15)
where, the superscript r means the quantity for the rigid body. The superscripts rr
means the quantities between rigid bodies and the superscript er means the
quantities between a flexible nodal body and a rigid body. The constraints
equations between rigid bodies are expressed as the function of rigid body
generalized coordinates rq as follows:
,rr rr r tΦ Φ q (16)
Similarly, we can derive the equations of motion for the flexible body as follows:
T T 0e e
e e e ee ee er er e q q
F M q Φ λ Φ λ Q (17)
where, the superscript e means the quantities for the flexible nodal body and eq is
the generalized coordinate for the flexible nodal bodies. The superscript ee means
the quantities between flexible nodal bodies and the superscript er means the
quantities between a flexible nodal body and a rigid body. The forces eQ between
flexible nodal bodies can be expressed as the sum of the element forces and applied
forces such as gravity or contact forces as follows:
e element applied Q Q Q (18)
Also, for the flexible body joint constraints between a flexible nodal body and a
virtual rigid body, we can express the erΦ as follows:
522
, ,er er e r tΦ Φ q q (19)
Similarly, the constraint equations eeΦ between flexible nodal bodies can be
expressed as Eq. (20).
,ee ee e tΦ Φ q (20)
Finally, we can make the whole system matrix for the MFBD problems as Eq. (21)
and we can solve Eq. (21) using the sparse matrix solver for the incremental
quantities.
T T
T T T T
e e
e
r
e r
e eee er
e ree e
eeee ee
r r rr rrr er
e r rr rr
rr er er
er er
q q
q
z z
q
q q
F FΦ 0 Φq q q F
Φ 0 0 0 0 λ ΦF F q F0 B Φ B Φq q λ Φ0 0 Φ 0 0 λ Φ
Φ 0 Φ 0 0
5.2.3. ELASTOHYDRODYNAMIC (EHD)
5.2.3.1 Governing Equation of Hydrodynamics
Fig. 2 shows a schematic diagram for the relative motion and dimensions between
a bearing and a journal.
Figure 2. THE SCHEMATIC DIAGRAM OF A JOURNAL BEARING
R
H
e
Cr
x,u
y,v 0
Bearing
Journal
R
H
e
Cr
x,u
y,v 0
Bearing
Journal
(21)
(21)
523
If we consider the journal radius R and the clearance rC in the journal bearing
lubrication problems of the laminar flow, following assumptions can be used.
2
1rC
R
Re 1 , Re 0.001r rC CGenerally O
R R
Under the above assumptions, the governing equation for the fluid flow becomes
the Couette-Poiseuille flow equation (Sabersky et al. 1989, Gohar 2001). And then,
if the mass or flow rate conservation law is applied, the Reynolds’ equation for the
hydrodynamic problems can be expressed as follows:
3
12 6 6p p H H
V U Wx x z z x z
H
Here, U , V , and W are x , y , and z relative velocities of the journal surface
(at y H ) based on the bearing, respectively. H and are the oil film thickness
and the dynamic (or absolute) viscosity, respectively. In this study, the oil film
thickness is defined as follows:
( ) cos sinr x yh C e e
Also, as shown in Eq. (23), the dynamic viscosity can be varied in space, it
depends on the oil pressure. So, in order to consider the pressure-viscosity relations,
this paper uses the Barus Law (Dowson and Higginson 1977) as Eq. (25).
0
pe
Here, 0 is a dynamic viscosity at the atmosphere state, is the pressure-
viscosity coefficient which is related with the lubricant properties.
(22)
(23)
(24)
(25)
524
5.2.3.2 Asperity Contact Model
As shown in Fig. 3, when the oil film thickness is not enough to thick compared
to the surface roughness, the contact pressure resulting from the asperities between
bodies should be considered.
Figure 3. AN EXAMPLE OF ASPERITY CONTACT
In this paper, the asperity contact model by Greenwood and Tripp (1971) is used
to consider the mixed lubrication region. In the Greenwood and Tripp’s model, the
asperity contact pressure ap can be calculated as follows:
5 / 2
6.804
5
5 / 2
( ) ( / )
4.4086 10 4 , 4
0 ,
a s
s ss
p H KE F H
H HifH
F
otherwise
Here, K is the elastic factor and s is the root mean square (rms) of the asperity
summit heights and E is the composite elastic modulus, which is defined from the
material properties of contacting surfaces, as Eq. (27).
2 2
1 2
1 2
1 11
E E E
where, the subscripts 1 and 2 mean the bearing and journal bodies, respectively.
E is the Young’s modulus and is the Poisson’s ratio.
5.2.4. FLUID-STRUCTURE INTERACTIONS
In this study, as shown in Fig. 4, EHD and MFBD solvers are used together to
Asperity Contact PointsAsperity Contact Points
(26)
(27)
525
analyze the lubrication and flexible multibody dynamics characteristics of the
journal bearing. First, the pressure distributions are evaluated from the
hydrodynamic lubrication analysis. Then, the calculated pressure field and resulting
forces and torques are transmitted into the MFBD solver. In the MFBD solver, the
transmitted pressure, force and torque data are used as the external forces or
torques acting on the journal and bearing bodies. Then, from the MFBD analysis,
the positions and velocities of all the related bodies are calculated. From these
position and velocity data, the oil film thickness is evaluated and transferred to the
EHD solver. Like these procedures, in order to analyze the lubrication and dynamic
characteristics of the journal bearing, EHD and MFBD solvers are used iteratively.
Figure 4. Fluid-Structure Interactions between EHD and MFBD Solver.
Also, in order to support the general-purpose EHD solution, the groove and oil
hole effects, which are treated as the pressure boundary condition in EHD solver,
are implemented.
5.2.5. NUMERICAL EXAMPLES
In order to implement the EHD module with MFBD solver together, this study
used the RecurdynTM
(2010) MFBD environment.
To validate the numerical results of this study, the experimental and numerical
analysis results of Nakayama et al. (2003) are used. The detailed explanation about
the numerical model is described well in Nakayama et al. (2003). Fig. 5 shows the
numerical model and measurement points of the oil film thickness. In the numerical
model, in order to check the effects on the external loads, various external loads
such as 100N, 500N, 1000N, 1500N, and 2000N are applied. Also, 3570 rpm is
used for the rotational speed of crank shaft.
Table. 1 shows the simulation parameters used in the numerical model. And the
Fig. 6 shows the numerical results and the results are compared with the results of
EHL solutions of Nakayama et al. (2003). As shown in Fig. 6, the numerical results
of current study also shows a good agreement with the EHL results at center
526
position.
Figure 5. NUMERICAL MODEL FOR JOURNAL BEARING BETWEEN
THE CONNECTING ROD AND CRANK SHAFT.
Table 1. THE PARAMETERS OF NUMERICAL MODEL.
Parameters Values
Mesh size (circum. ×
depth) 60×5
Journal Diameter 45 mm
Bearing Width 15.6 mm
Lubrication Gap 0.068 mm
Dynamic Viscosity 3.5e-2 Pa·s
Pressure-Viscosity Coeff. 1.28e-8 Pa-1
Roughness 5.0e-4 mm
Dummy Body
Ground
Conrod
TranslationalJoint
Spherical Joint
Force
Shaft
Ground
RevoluteJoint
Motion3570rpm
Conrod
Shaft
RD/EHD
100N, 500N, 1000N, 2000N
Dummy Body
Ground
Conrod
TranslationalJoint
Spherical Joint
Force
Shaft
Ground
RevoluteJoint
Motion3570rpm
Conrod
Shaft
RD/EHD
100N, 500N, 1000N, 2000N
Measure Point
Center
Edge
Measure Point
Center
Edge
527
Composite Elastic
Modulus 206000 MPa
Elastic Factor 3.e-3
Figure 6. COMPARISON WITH THE RESULTS OF EHL (NAKAYAMA ET AL. 2003)
5.2.6. CONCLUSIONS
In this study, the elastohydrodynamic lubrication system coupled with flexible
multibody dynamics (or MFBD) is developed in order to analyze the dynamic
bearing lubrication characteristics such as the pressure distribution and oil film
thickness. In order to solve coupled fluid-structure interaction system, this study
uses two main parts. The one is the MFBD solver and the other is
elastohydrodynamic module. The elastohydrodynamic lubrication module
developed in this study transmits the force and torque data to the MFBD solver
which can solve general dynamic systems. And then, the MFBD solver analyses the
positions and velocities of the flexible multibody system with the pressures, forces
and torques results of the elastohydrodynamic module. And the MFBD solver
transmits the position and velocity data, which can evaluate the oil film thickness,
to the EHD solver. These kinds of procedures are used iteratively between MFBD
and EHD solvers. Moreover, other functions such as mesh grid control and oil hole
and groove effects are implemented. Finally, the numerical results are validated and
compared with other experimental and numerical solutions by using the journal
bearing example between the connecting rod and the crank shaft.
0
5
10
15
20
25
30
35
100N 500N 1000N 1500N 2000N
EHL(Center)
EHL(Edge)
NEW(Center)
Oil
Fil
m T
hic
knes
s (μ
m)
Load (N)
0
5
10
15
20
25
30
35
100N 500N 1000N 1500N 2000N
EHL(Center)
EHL(Edge)
NEW(Center)
Oil
Fil
m T
hic
knes
s (μ
m)
Load (N)
528
REFERENCES
1. Bae D. S., Han J. M., Choi J. H., and Yang S. M., A Generalized Recursive
Formulation for Constrained Flexible Multibody Dynamics, International
Journal for Numerical Methods in Engineering, Vol. 50, pp.1841-1859,
2001.
2. Choi, J., A Study on the Analysis of Rigid and Flexible Body Dynamics
with Contact, PhD Dissertation, Seoul National University, Seoul, 2009.
3. Dowson, D., and Higginson, G. R., A numerical solution to the
elastohydrodynamic problem, J. Mech. Eng. Sci., Vol. 1, pp.6-15, 1959.
4. Dowson, D., and Higginson, G. R., Elastohydrodynamic Lubrication, SI
Edition, Chapter 6, Pergamon Press, Oxford, 1977.
5. García de Jalón, D. J., Unda, J., and Avello, A., Natural coordinates for the
computer analysis of multibody systems, Computer Methods in Applied
Mechanics and Engineering, Vol. 56, pp.309-327, 1986.
6. Gohar, R., Elastohydrodynamics, Second Edition, Imperial College Press,
2001.
7. Greenwood, J. A., and Tripp, J. H., The Contact of Two Nominally Flat
Rough Surfaces, Proc. Instn. Mech. Engrs., Vol. 185, Part 1, No. 48,
pp.625–633, 1970–1971.
8. Hahn, H. W., Das Zulindrische Gleitlager endlicher Breite unter zeitlich
veranderlicher Belastung, Diss. TH. Karlsruhe, 1957.
9. Hamrock, B. J., and Dowson, D., Isothermal Elastohydrodynamic
Lubrication of Point Contacts, Part I, Theoretical Formulation, ASME J.
Lubr. Technol., Vol. 98, pp.223-229, 1976.
10. Labouff, G. A., and Booker, J. F., Dynamically loaded journal bearings: a
finite element treatment for rigid and elastic surfaces, ASME, Journal of
tribology, Vol. 107, No. 4, pp.505-515, 1985.
11. Makino, T., and Koga, T., Crank Bearing Design Based on 3-D
Elastohydrodynamic Lubrication Theory, Mitsubishi Heavy Industries, Ltd.
Technical Review, Vol. 39, No. 1, pp.16-20, 2002.
12. Nakayama, K., Morio, I., Katagiri, T., and Okamoto, Y., A Study for
Measurement of Oil Film Thickness on Engine Bearing by using Laser
Induced Fluorescence (LIF) Method, SAE International, 2003.
13. Nair, K. P., Sinhasan, R., and Singh, D. V., A study of elastohydrodynamic
effects in a three-lobe journal bearing, Tribology international, Vol. 20, No.
3, pp.125-132, 1987.
14. Oh, K. P., and Goenka, P. K., The elastohydrodynamic solution of journal
bearings under dynamic loading, ASME, Journal of tribology, Vol. 107, No.
3, pp.389-395, 1985.
15. Ott, H. H., Zylindrische Gleitlager unter instationarer Belastung, Diss.
529
ETH. Zurich, 1948.
16. Peiskammer, D., Riener, H., Prandstotter, M., and Steinbatz, M.,
Simulation of motor components : intergration of EHD - MBS - FE -
Fatigue, ADAMS User Conference, 2002.
17. RecurdynTM
Manual, http://www.functionbay.co.kr, FunctionBay, Inc.,
2010.
18. Reynolds, O., On the Theory of Lubrication and its Application to Mr.
Beauchamp Tower’s Experiments, Including an Experimental
determination of the Viscosity of Olive Oil, Phil. Trans. Roy. Soc., Vol. 177,
pp.157-234, 1886.
19. Riener, H., Prandstotter, M., and Witteveen, W., Conrod Simulation:
Integration on EHD - MBS - FE - Fatigue, ADAMS User Conference, 2001.
20. Sabersky R. H., Acosta, A. J., and Hauptmann, E. G., Fluid Flow : A First
Course in Fluid Mechanics, Third Edition, Maxwell Macmillan
International Editions, 1989.
21. Taylor, C. M., Engine Tribology, Elsevier science publishers B. V., pp.75-
87, 1993.
22. Wittenburg J., Dynamics of Systems of Rigid Bodies, B. G. Teubner,
Stuttgart, 1977.
6. Media Transport
System
531
6.1
DYNAMIC ANALYSIS AND CONTACT
MODELING FOR TWO DIMENSIONAL
MEDIA TRANSPORT SYSTEM
6.1.1. INTRODUCTION
Recently the media transport systems, such as printers, copiers, fax, ATMs,
cameras, film develop machines, etc., have been widely used and being
developed rapidly. Especially, in the development of those systems, the media
feeding mechanism for paper, film, money, cloth etc., is an important key
technology for the design and development of the media transport systems.
Tedious and iterative experimental trial and errors methods have been essential
way to determine kinematic mechanisms of parts dimensions, and materials, etc
for the media machine developers. Since the iterative trial & error methods are
truly inefficient, in order to shorten the time, reduce the cost, and improve the
machine performance, it has been absolutely required to develop the computer
simulation tool, which analyses the paper feeding and separation process.
Cho and Choi [1] developed a computational modeling techniques for two
dimensional film feeding mechanisms. The flexible film is divided by several
thin rigid bodies which are connected by revolute joints and rotational spring
dampers. The primitive computer implementation methods for contact search
algorithms are presented. Diehl [2, 5] presented the local static mechanics of
electrometric nip system for media transport system. The nonlinear finite
element method and experimental measurement techniques are used to
investigate the large deformable rollers. Several unique phenomena, such as
skewing sheet, etc., of nip feeding system are well described in this research.
Ashida [3] suggested the computer modeling techniques for the design and
analysis of film feeding mechanisms. The primitive dynamic analysis of two
532
dimensional film feeding models are presented by using commercial computer
program. The paper feed mechanism with friction pad system is investigated by
Yanabe [4] by using commercial nonlinear FEA program. It show the local
separation phenomenon between papers and roller, and proved very good
agreement with experimental measurements. Shin [6, 7] developed web
simulation and design tools using roll tensions. They show that the control of
tensions of each segment is the key design factors for web system.
In this investigation, a numerical modeling method and dynamic analysis of
the two dimensional flexible sheet for thin flexible media materials such as paper,
film, etc., and their roller and guide contacts are suggested by using multibody
dynamic techniques. Since the flexible sheet undergoes large deformation with
assumed linear material properties, the flexible sheet has been modeled as a
series of thin rigid bars connected by revolute joints with rotational spring
dampers force elements. It shows good visual appearance of the sheet under
severe bending conditions. An efficient contact search and force analysis
between sheet and rollers, and guides are developed and implemented
numerically. The sheet is fed by contact and friction forces when it contacts with
rotating rollers or guides. In order to detect a contact phase efficiently, the
bounding box method is used in this contact search algorithm. The method has
an advantage that the number of contact search can be smaller than conventional
methods for a system in which the position of rollers and guides are fixed on a
point of a base body. The proposed numerical models for media transport
systems will make it possible to confirm the potential problems of jamming by
given different sheet size, weight, stiffness, temperature, humidity extremes,
sheet velocity due to misalignment of drive-driven roller sets, and roller
velocities due to gap, wear or etc.
533
6.1.2. TWO DIMENSIONAL FLEXIBLE MULIBODY SHEET
In general, there are two methods to build a thin 2-D flexible sheet for
dynamic analysis. One is to employ beam element at discretized sheet body, and
the other is small rigid bar interconnected by revolute joint with rotational
spring-damper forces. In this investigation, the second method is used and
proposed the modeling techniques.
Figure 1 Modeling definition of a two dimensional flexible sheet
Several research works show that the most efficient way to model two-
dimensional approximation of the proper behavior of a sheet can be a series of
rigid bars connected by revolute joints and rotational spring-dampers as shown
in Figure 1 [1, 3]. The sheet is divided into a number of rigid bars with mass.
The mass and inertia moment of each rigid bar can be defined as follows
ss tLm (1)
12
)( 22
sszz
LtmI
(2)
where, is a sheet density per unit depth, st is thickness, and
sL is length
of each rigid bar. The leading body is connected to a ground by a planar joint to
guarantee an in-plane motion. The planar joint has one rotational and two
translational degrees of freedom. The i body is connected to the )1( i body
by a revolute joint and rotational spring damper. The revolute joint has one
534
rotational degree of freedom between two rigid bars. The relative angle of )1( ii
is directly integrated. The torque of the rotational spring-damper is computed as
following
)1()1( iiii ck (3)
s
s
L
tEk
12
3
(4)
kc (5)
where, )1( ii and
)1( ii are relative angles and angular velocities of the
revolute joints, and E and are the young’s modulus and the structural
damping ratio of a sheet.
Figure 2 Contact geometry of two-dimensional sheet
The contact geometry of a sheet is described as a box and two circles as
shown Figure 2. The x-axis of the body reference frame of each rigid bar is
defined along longitudinal length direction and the y-axis is defined by right
hand rule. The mass center of each rigid bar is located at the center point of box.
In order to generate a continuous contact force, two circles are located on both
sides of the box. Even thought the proposed assumed method for flexible sheet
has an excellent visual appearance of the sheet under severe bending conditions,
this approach shows the lack of continuity between rigid bodies, which can cause
noise problems when the sheet is contact with rollers. It has also rigid leading
and trailing effect of the sheet. Problems can be overcome with introducing a
535
circular edge at leading and trailing points of each rigid bar.
There can be another approach to assume flexible sheet in dynamic analysis,
which employs a series of beam forces, and for the contact definitions, a rigid
bar can be attached simply. One of the advantages of this approach is a natural
definition of the flexible properties using the beam elements. However this
approach can cause problems with the contact definitions since it has possible
gaps and the lack of continuity between rigid contact bodies. The contact forces
on the edges of the rigid bodies are amplified as torques applied where the rigid
body is connected to the junction of two beams, and the rigid leading and trailing
edges of the sheet cause unnatural behaviors.
6.1.3. CONTACT FORCE ANALYSIS
In the field of multi-body dynamics, one of the most popular approximation of
the dynamic behavior of a contact pair has been that one body penetrates into the
other body with a velocity on a contact point, thereafter the compliant normal
and friction forces are generated between a contact pair. Figure 3 shows the
schematic diagram of contact force analysis used in this investigation.
Figure 3 Contact forces between a contact pair
In this compliant contact force model, a contact normal force can be defined
as an equation of the penetration [9], which yields
536
nm
n ckf (6)
where and are an amount of penetration and its velocity, respectively.
The spring and damping coefficients of k and c can be determined from
analytical and experimental methods. The order m of the indentation can
compensate the spring force of restitution for non-linear characteristics, and the
order n can prevent a damping force from being excessively generated when
the relative indentation is very small. As it happens, the contact force may be
negative due to a large negative damping force, which is not realistic. This
unnatural situation can be resolved by using the indentation exponent greater
than one. The phenomenon is very important for the case of sheet contact
interaction since it is very thin and light. A friction force can be determined as
follows.
nf fvf )( (7)
where, nf and )(v are a contact normal force and a friction coefficient,
respectively.
6.1.3.1. KINEMATICS NOTATIONS
The YX coordinate system is the inertial reference frame and the single
primed coordinate systems are the body reference frames, and the double primed
coordinate system is the contact reference frame in order to define contact
conditions as shown in Figure 4. The orientation and position of the body
reference frame are denoted by A and r , respectively.
537
Figure 4 Kinematic notations of a contact pair
6.1.3.2. SHEET AND ROLLER INTERACTIONS
In this investigation, two kinds of rollers are defined for the system. One is a
fixed roller with one rotational degree of freedom. The fixed roller is linked to
the ground with a revolute joint. The other is a movable roller, which has two
degrees of freedom for a translational and a rotational motion. The movable
roller is linked to rotational axis retainer (RAR) with a revolute joint and the
retainer is linked to the ground with a translational joint. The contact geometry
of rollers is described as a circle as shown in Figure 5
Figure 5 Definition of rollers
538
Two different interactions between roller and sheet are introduced in this
investigation. Since the proposed flexible sheet is constructed by linear part and
circular part, these are interactions between linear part and rollers, and circular
part and rollers, as clearly illustrated in Figure 6
Figure 6 Sheet and roller interaction
In the case of linear part contact with rollers, the contacted penetration is
determined as follows:
rR ysr,d , )r(rAd sr
T
ssr (8)
where, sA is the orientation matrix of a rigid bar, and
rR is the radius of a
contacted roller, respectively. The location of contact between rigid bar and
roller can be defined as follows:
0
2/)( sys tsign sr,
xsr,
c d
d
s , (9)
and
)ds(AAs srscs
T
rrc (10)
where, rA and
st are the orientation matrix of a roller and the thickness of
the sheet. The relative velocity at the contact point can be determined as
539
scrsrcrrr
T
n
rsrcrr
T
n
swArswAru
sArsAru
s
dt
d
~~
(11)
c
T
ndu (12)
and tangential relative velocity is
c
Tdu
ttv (13)
where, rw and
sw are the angular velocities of a roller and a rigid bar with
respect to each body reference frame, and nu and
tu are the normal and
tangent vectors of relative position between rigid bar and roller, respectively.
6.1.3.3. ROLLERS INTERACTIONS
A circle to circle contact is used to describe the interactions between circular
rollers. In this circle to circle contact, the positive normal direction is same in the
direction of the relative position vector between two roller center points. The
tangent direction vector is determined by the right hand rule. The relative
velocity and the contact forces at the contact point can be computed similarly as
the sheet and roller interactions.
6.1.3.4. SHEET AND GUIDE INTERACTIONS
Guide has three types. Commonly used sheet guides for media transport
machines can be divided into three different types, which are an arc guide with
radius and angle, a linear guide with two points, and a circle guide similar to a
roller. In order to avoid the complex contact detect algorithms. It is assumed that
the arc and line guide are interacted with the circular part of rigid bars of the
sheet. However, in the case of circle guide, both linear and circular parts of the
sheet are interacted with.
540
Figure 7 Sheet and arc guide interactions
As shown in Figure 7, the relative displacement between a circular edge of
rigid bar and arc guide can be determined as
gggsssgs sArsArd (14)
where, gr and
gA are the center position and the orientation of the guide,
and the vectors of gs and
ss are positions of the arc reference frame and the
circular edge center position with respect to each body reference frame,
respectively. If the vector gsd is projected into the arc reference frame, the
resultant vector can be represented as follows
gsgs dCAd gg
T)( (15)
where, gC is the orientation matrix of the arc reference frame. The relative
angle between x-axis of the arc reference frame and the resultant vector of Eq.
15 is within an arc angle, which can be written as
g
)(cos0 1
gs
g
T
gs
d
fd (16)
541
where, g is the arc angle and
gf is a constant unit vector of T001 . If
the condition of Eq. 16 is satisfied, the penetration between circular part of sheet
and arc can be defined as follows
gsgs Rt d (17)
where, gR is a radius of the arc guide. The contact positions can be computed
as follows.
ngg R us c (18)
cgg
T
c
cc
sCAAs
dss
s
g
ss
gsgs
g
(19)
where, nu is the normal direction vector and determined
gs
gs
d
dun
(20)
The tangent direction vector is determined by the right hand rule, and the
relative velocity at the contact point is defined as follows.
)(~)(~
)()((
gcggggssss
gcgggsssdt
d
sCswArsswAr
sCsArssArd
gsc
gscc
(21)
where, gw and
sw is the angular velocities of guide and a bar with respect
to each body reference frame, respectively. The contact forces at the contact
point can be computed similarly as described in the sheet and roller interactions.
542
Figure 8. Sheet and line guide interactions
The sheet and line guide interactions are clearly illustrated in Figure 8. If the
x component of the vector gsd defined in the double primed line guide
reference frame is the range of guide length, simple circle and line contact
algorithm is used in this investigation. After definitions of penetration and its
derivative, the contact force is created to restitute each body as similar as
previous interactions between sheet and guides.
6.1.4. EQUATIONS OF MOTION
Figure 9 Kinematic relationships between rigid bars of the sheet
543
Since the multibody sheet system interacts with the roller and guide
components through the contact forces and adjacent rigid bars are connected by
revolute joint and rotational spring damper forces as shown in Figure 9, each
sub-rigid bar in the sheet system has one degree of freedom which is represented
by one rotational coordinates and the leading body has three free coordinates.
The equations of motion of the sheet system that employs the velocity
transformation defined by Bae [8] are given as follows:
)( r
ii qBMQBqMBBTT (22)
where riq , B and q are relative independent coordinates, velocity
transformation matrix, and Cartesian velocities of the media feeding system, and
M is the mass matrix, and Q is the generalized external and internal force
vector of the media feeding system, respectively. The velocity transformation
matrix B of the sheet is more explicitly as
1)n2(n2321)n1(n1221)n1(n0121)n1(n
232122231012121231
122012121
012
BBBBBBB
0BBBBBB
00BBB
000B
B
where the recursive velocity and virtual relationship for a pair of rigid bars are
obtained [8] as
1)i(i1)i2(i1)(i1)i1(ii qBYBY (23)
and 1)i(iq denotes the relative coordinate vector. It is important to note that
matrices 1)i1(iB and
1)i2(iB are only functions of the 1)i(iq .
544
6.1. 5. NUMERICAL RESULTS
The proposed algorithm is implemented and a film-feeding problem is solved
to demonstrate the efficiency and validity of the proposed method.
Figure 10 Film feeding machine
The system has 29 degrees of freedom and consists of four fixed rollers and
three movable rollers, five line guides, one arc and circle guide and one sheet of
film shown in Figure 10. The sheet is modeled by using 20 rigid bars. The
density and Young’s modulus of sheet are 2.2e-6( 3/ mmkg ) and 2250( 2/ mmN ),
respectively. And the thickness and length of sheet are 0.5( mm) and 200( mm),
respectively.
545
Figure 11 Slip between rollers and sheet
The film goes through a path while contacting the roller pairs. The
circumferential speed of each driving roller is 300( sec/mm ). The slip velocities
between driving rollers and the sheet are shown in Figure 11. The path of first,
second and third segment bodies of the thin film are plotted as shown in Figure
12. The x and y axes of the plot are displacements measured in the directions of
x and y axes in the global reference frame, respectively.
Figure 12 Path of segmented bodies of film
The analysis was performed on an IBM compatible computer (PIII-933Mhz)
546
and took about 60 sec. per 1 sec. for simulation.
6.1. 6. CONCLUSIONS
The dynamics and modeling techniques of two-dimensional media transport
system is investigated in this paper. The flexible sheet is divided by finite
number of rigid bars. Linear motions are constrained in order to allow rotations
between the rigid bars of the sheet. Rotational spring damper force is applied for
the reflection flexible stiffness of the sheet. From previous empirical
measurements in manufacturing process effective stiffness and damping
coefficients are substituted in this investigation. Compliant contact force model
is used for the interactions between sheet rollers, and guides. Kinematics
notations of the contact search algorithms for the media transport system are
clearly represented. A simple film feeding example is represented in this
investigation and manufacture [3] confirms that simulation results have very
good agreement with experimental measurements. The media transport system
manufactures have rely on trial error techniques for the design of their core
mechanisms, however the proposed method by employing multibody dynamics
in this paper can reduce many difficulties at the early design stage.
REFERENCES
1. H. J. Cho, and J. H. Choi, 2001, “2DMTT development specification” Technical
report, FunctionBay Inc.
2. Ted Diehl, 1995, “Two dimensional and three dimensional analysis of nonlinear
nip mechanics with hyper elastic material formulation” Ph. D. Thesis,
University of Rochester, Rochester, New York
3. Tsuyoshi Ashida, 2000, “The meeting material of The Japan Society for
Precision Engineering” Japan
4. http://www.yanabelab.nagaokaut.ac.jp
547
5. http://www.me.psu.edu/research/bension.html
6. http://www.engext.okstate.edu/info/WWW-WHRC.htm
7. Shin, K. H., 1991, “Distributed Control of Tension in Multi-Span Web
Transport Systems “, Ph. D. Thesis Oklahoma State Univ.
8. D. S. Bae, J. M. Han, and H. H. Yoo, 1999, “A Generalized Recursive
Formulation for Constrained Mechanical System Dynamics”, Mech. Struct. And
Machines, Vol. 27, No 3, pp 293-315
9. Lankarani H. M. and Nikravesh P. E., 1994, “Continuous Contact Force
Models for Impact Analysis in Multibody Systems”, Journal of Nonlinear
Dynamics, Kluwer Academic Publishers, Vol. 5, pp 193-207
547