mathematics and mechanics of solids-2012-lidström-209-42

7/27/2019 Mathematics and Mechanics of Solids-2012-Lidstrm-209-42

1/35

http://mms.sagepub.com/Mathematics and Mechanics of Solids

http://mms.sagepub.com/content/17/3/209The online version of this article can be found at:

DOI: 10.1177/1081286511407111

2012 17: 209 originally published online 9 August 2011Mathematics and Mechanics of SolidsP Lidstrm

On the equations of motion in constrained multibody dynamics

Published by:

http://www.sagepublications.com

can be found at:Mathematics and Mechanics of SolidsAdditional services and information for

http://mms.sagepub.com/cgi/alertsEmail Alerts:

http://mms.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://mms.sagepub.com/content/17/3/209.refs.htmlCitations:

What is This?

- Aug 9, 2011OnlineFirst Version of Record

- May 7, 2012Version of Record>>

by Sudhanwa Kulkarni on December 1, 2012mms.sagepub.comDownloaded from
http://mms.sagepub.com/http://mms.sagepub.com/http://mms.sagepub.com/http://mms.sagepub.com/content/17/3/209http://mms.sagepub.com/content/17/3/209http://www.sagepublications.com/http://mms.sagepub.com/cgi/alertshttp://mms.sagepub.com/cgi/alertshttp://mms.sagepub.com/subscriptionshttp://mms.sagepub.com/subscriptionshttp://mms.sagepub.com/subscriptionshttp://www.sagepub.com/journalsReprints.navhttp://www.sagepub.com/journalsReprints.navhttp://www.sagepub.com/journalsPermissions.navhttp://mms.sagepub.com/content/17/3/209.refs.htmlhttp://mms.sagepub.com/content/17/3/209.refs.htmlhttp://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://mms.sagepub.com/content/early/2011/07/31/1081286511407111.full.pdfhttp://mms.sagepub.com/content/early/2011/07/31/1081286511407111.full.pdfhttp://mms.sagepub.com/content/17/3/209.full.pdfhttp://mms.sagepub.com/content/17/3/209.full.pdfhttp://mms.sagepub.com/http://mms.sagepub.com/http://mms.sagepub.com/http://online.sagepub.com/site/sphelp/vorhelp.xhtmlhttp://mms.sagepub.com/content/early/2011/07/31/1081286511407111.full.pdfhttp://mms.sagepub.com/content/17/3/209.full.pdfhttp://mms.sagepub.com/content/17/3/209.refs.htmlhttp://www.sagepub.com/journalsPermissions.navhttp://www.sagepub.com/journalsReprints.navhttp://mms.sagepub.com/subscriptionshttp://mms.sagepub.com/cgi/alertshttp://www.sagepublications.com/http://mms.sagepub.com/content/17/3/209http://mms.sagepub.com/


2/35

Article

On the equations of motion in

constrained multibody dynamics

Mathematics and Mechanics of Solids

17(3): 209242

The Author(s) 2011

Reprints and permission:

sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286511407111

mms.sagepub.com

P Lidstrm

Division of Mechanics, Lund University, Sweden

Received 11 November 2010; accepted 18 March 2011

AbstractThe equations of motion for a constrained multibody system are derived from a continuum mechanical point of view.This will allow for the presence of rigid, as well as deformable, parts in the multibody system together with kinematicalconstraints. The approach leads to the classical LagrangedAlembert equations of motion under constraint conditions.The generalized forces appearing in the equations of motion are given as expressions involving contact, internal and bodyforces in the sense of continuum mechanics. A detailed analysis of physical constraint conditions and their implicationfor the equations of motion is presented. A precise distinction is made between constraints on the motion on the onehand and resistance to the motion on the other. Transformation properties covariance and invariance under changesof configuration coordinates are elucidated. The elimination and calculation of the so-called Lagrangian multipliers isdiscussed and some useful reformulations of the equations of motion are presented. Finally a Power theorem for theconstrained multibody system is proved.

Keywords

constraints, equations of motion, multibody dynamics

1. Introduction

A central issue in multibody dynamics is the formulation of geometrical or kinematical constraints representing

conditions on the relative motions between parts of the mechanical system and the associated formulation of

the equations of motion. In general, the kinematical constraint will lead to a system of first-order ordinary

differential equations:

g0 +gq = 0m1

where g0 = g0(t, q) Rm1 andg = g(t, q) Rmn are the constraint matrices andq Rn the configurationcoordinates for the multibody. A formulation of the equations of motion for the constrained mechanical system

may, for instance, be obtained by invoking the LagrangedAlembert equations of motion:

d

dt

T

q

T

q Q = 01n

Corresponding author:

P Lidstrm, Division of Mechanics, Lund University, P.O. Box 118, S-22100 Lund, SwedenEmail: [email protected]

http://mms.sagepub.com/http://mms.sagepub.com/http://mms.sagepub.com/http://mms.sagepub.com/


3/35

210 Mathematics and Mechanics of Solids 17(3)

where T = T(t, q, q) is the kinetic energy of the multibody and Q = Q(t, q, q) R1n is the so-called sum ofthe generalized forces acting on the multibody due to internal, contact and body forces. One part of Q is the

force Qkc = Qkc(t, q, q) R1n due to the kinematical constraints. The so-called non-holonomic principlestates that

Qkc = Tg

where the multipliers = (t) Rm1 represent components of constraint forces and moments. A justifica-tion of the non-holonomic principal may be based on the principle of LagrangedAlembert, cf. Bloch et al. [1],

where a discussion, from an historical point of view, on the derivation of the equations of motion is presented.

In Lidstrm [2], the equations of motion for a multibody system, containing rigid as well as elastic parts,

were derived using kinematical concepts and dynamic laws taken from continuum mechanics. Constraints on

multibody systems were, however, not considered. It is the purpose of this paper to include constraints in

the general discussion. The interaction between rigid parts is discussed in terms of kinematical constraints and

the associated constraint forces. Interactions are expressed in coordinate systems compatible or non-compatible

with the constraints. The main issues of this paper are the derivation of the equations of motion for a constrained

multibody system and the elimination and calculations of the constraint forces.

Classical and modern developments concerning constraint enforcement in multibody systems and their

numerical implementation are reviewed by Lalusa and Bauchau [3, 4]. A good reference concerning modernmathematical developments, with applications in control theory, is Bloch et al. [1].

The paper is organized as follows. In Section 2 the basic notations and definitions used throughout the

paper are reviewed. In Section 3 the multibody system is introduced. In Section 4 the interaction between

rigid parts is analysed in terms of contact forces and kinematical constraints. A precise distinction is made

between constraints on the motion on the one hand and resistance on the motion on the other. Constraints on

the motion are maintained by reactions, that is, constraint forces and moments, which are a priori unknown.

They may be restricted by inequalities as, for instance, in the case of contact forces at a unilateral contact

between parts. Forces and moments responsible for resistance on the motion are prescribed constitutively.

The compatibility of the system of generalized coordinates with the imposed kinematical constraints and the

implications for the equations of motion is discussed. In Section 5 the equations of motion under prescribed

bilateral kinematical constraints are derived using the principal of virtual power. This leads to the classical

LagrangedAlembert equations of motion involving multipliers. The multipliers will, as is well known and as

will be further illuminated by the discussion presented in this paper, represent components of constraint forces

and moments. The Lagrangian multipliers correspond precisely to those components of forces and moments

that are responsible for the maintenance of the kinematical constraints. Forces and moments responsible for

resistance on the motion have to be specified separately and then added to the equations of motion. In Section 6

a further analysis of the equations of motion is undertaken. This involves the elimination and calculation of the

Lagrangian multipliers and a reformulation of the equations of motion that decreases the order of the ordinary

differential equations. Finally, in Section 7, a Power theorem for the constrained multibody system is proved.

2. Notation

In this paperR denotes the set of real numbers and Rn the set of n-tuples of real numbers. The set of n-dimensional, real column vectors is denoted by Rn1 and the null vector in Rn1 is written 0n1. R

mn denotes

the set of real matrices of order m n with the null matrix written 0mn. If A Rmn then AT Rnm is the

transpose of A. The rank of a matrix A Rmn is written rank(A). IfA is a square matrix then det(A) denotesthe determinant ofA and if det(A) = 0 then A1 denotes its inverse. Inn denotes the identity matrix in R

nn. If

A Rmn then kernel(A) =

u Rn1 |Au = 0m1

andrange(A) =

v Rm1 | v = Au , u Rn1

.

Let E denote a three-dimensional Euclidean point space with the corresponding translation vector space

V , that is, dim(V) = 3. Points in E are denoted by x,y, . . . ,X, Y, . . . and vectors in V by a, b, . . . ,u, v, . . ..

The space of all second-order tensors A on V , that is, linear mappings V V, is denoted End(V). The setof all rotations on V is denoted SO(V)=

A End(V)

ATA = AAT = 1, detA = 1

. The space of all skew-

symmetric tensors is denoted by Skew(V) = A End(V) AT = A. If A Skew(V) then ax(A) V



4/35

Lidstrm 211

denotes its axial vector, that is, Au = ax(A) u, u V. Given a V, then the tensora 1 Skew(V) isdefined by (a 1)u = a u, u V. IfA End(V) then a A = (a 1)A.

3. Preliminaries

This introductory section follows closely that of Lidstrm [2]. For more information on details and motivations

for the present approach the reader is referred to that paper.

Consider a multibody B consisting of N rigid or elastic bodies: B, = 1, . . . ,N. We write B =N

=1B.

Let the regular region Bt in Edenote the present (spatial) placementofB, at time t. The function representing

the transplacement of the part B from its reference placement B0 to its present placement Bt is given by the

mapping : B0 Bt , defined by

x = (X, t), (X, t) B0 R (1)

The transplacement (1) is assumed to be a twice-continuously differentiable mapping. Basic kinematical quan-

tities, such as material velocity x, deformation gradientFandGreen-St.Venant strain tensorEetc., belonging

to part B , will be super-indexed by according to x = t , F = X , andE =1

2 (FTF 1). We

assume that int(Bt )

int(Bt )) = , = , reflecting the impenetrability of matter. We also assume, as a

matter of convenience, that B0B

0 = . The transplacement of the multibody (, t) : B0 Bt is defined

by

x = (X, t) = (X, t), X B0 (2)

where B0 =

N=1

B0 andBt =

N=1

Bt . Note that the transplacement, (2), is a twice-continuously differentiable

mapping. We assume that it is possible to introduce configuration coordinates:

q = (q1, q2, . . . , qn) Rn (3)

where is an open set, so that the set of possible placements (configurations) of the parts of the multibody

may (locally) be represented by the mappings

x = q (t, q1, q2, . . . , qn;X) = q (t, q;X), (t, q;X) R B

0 (4)

where q : R B0 E is assumed to be a twice-continuously differentiable mapping in all its arguments

(t, q;X). Then (k, l = 0, 1, . . . , n, q0def= t),

2 q

qlqk=

2 q

qkqland

2 q

Xqk=

2 q

qkX.

The motion of the multibody is locally given by a function q = q(t), t T = ]t1, t2[ R and a mapping

x = q(t, q(t);X)def= q (t, q(t);X), t T, X B

0 (5)

The motion of a rigid partB is given by a function q = q(t), t T = ]t1, t2[ R and the mapping

x = q (t, q;X) = x

A(t, q) +R (t, q)(X XA ), X B

0 (6)

where A B is an arbitrarily selected material point in B with reference place XA and present place x

A =xA(t, q) =

q (t, q;X

A ). R

= R(t, q) SO(V) is the rotation tensorforB. The velocity field of the rigid partB may be written

x = x (t, q, q;x) = xA(t, q, q) + (t, q, q) (x xA), x B

t



5/35


where

xA =

nk=0

xA,kqk and =

nk=0

;kqk

and

x

A,k= x

A,k(t, q)

def=

xA

qkand

;k=

;k(t, q)

def= axR

qkRT , k = 0,1, . . . , n (7)

The vectors xA,k and;k are called the partial velocity vector and the partial angular velocity vectors, respec-

tively. In analogy with definition (7)1 we introduce x

A;k

def=

xA qk

. It follows that, cf. Lidstrm [2], Corollary 4.1,

xA;k = x

A,k and;k =

qk, k = 1, . . . , n.

The kinetic energy for the multibody Bmay be written, cf. Lidstrm [2], Theorem 6.1:

T = T(t, q, q) =1

2(M0 +M1q + q

TM2q)

where M0, M1 andM2 are the mass matrices of the multibody. The equations of motion for the multibody read

qTM2 = Qcif + Qi + Qc + Qb (8)

where

Qcif =

qT

M2

q

1

2

M2

q

tq + qT

M2

t+ skew

M1

q

+

1

2

M1

t

M0

q

(9)

is the complementary inertia force. For a demonstration of (9) cf. Lidstrm [2], where the matrix notation is

explained.

Qi =

N

=1Q,i, Qc =

N

=1Q,c andQb =

N

=1Q,b (10)

are the generalized internal force, the generalized contact force and the generalized body force, respectively,

acting on the multibody. Furthermore

Q,ik =

B

0

P Fq

qkdv(X), Q

,ck =

B

0

q

qk t0 da(X), Q

,bk =

B

0

q

qk b0 dv(X) (11)

Here Fq = Fq (t, q,X) denotes the deformation gradient, t

0 = P

n0 the traction vector on the boundary

surface of part B where P = P (X, t) is the first PiolaKirchhoff stress tensor, n0 = n0 (X, t) the external

unit normal on B0 andb = b (X, t) is the (mass-) specific body force. If the elastic parts are constituted by

St. VenantKirchhoff material, then the generalized internal force may be written, cf. Lidstrm [2]:

Qi

=

Ve,q

q (12)

where

Ve =

N=1

B

0

1

2 (trE)2 + |E |2

dv(X)

(13)

The generalized contact force may be written

Qc = Qec + Qic (14)

where Qec represents the contact force from the exterior ofB, while Qic represents the contact force from all

internal contacts between parts ofB.



6/35

Lidstrm 213

Assume thatB andB are in geometrical contact over the surface St , at time t. We introduce the following

vector fields defined on St :

xk(t, q,x) = q (t, q(t); (

q )

1(t, q;x))

qk, x

k(t, q,x) = q (t, q(t); (

q )

1(t, q;x))

qk, k = 0,1, . . . , n

where ( q )1(t, q; ) is the inverse of q (t, q; ). The generalized force, Q

k , from B on B is then given by

Q

k =

S

t

xk t da(x), = , k = 1, . . . , n (15)

where the contact force t is the traction vector acting on B from B . The corresponding generalized force,

Q

k , from B on B is then given by

Q

k =

S

t

x

k t da(x), = , k = 1, . . . , n (16)

The generalized force Qic, from the internal contacts between all parts of the multibody, may now be written as

Qick =

N, = 1 <

I

k (17)

where

I

k = Q

k + Q

k , k = 1, . . . , n (18)

is called the mechanical interaction between parts B andB . It follows that

Ik =

St

x,k tda(x), k = 1, . . . , n (19)

where x,k = x

k x

k andt = t(x, t) is the contact traction from B on B, cf. Lidstrm [2], Corollary 7.1.

4. The interaction between rigid parts

In Lidstrm [2], a general discussion of the interaction between parts, involving rigid as well as flexible parts,

was undertaken. Here we focus on rigid parts. Constraint conditions are often employed in the description of

the interaction between rigid parts in a multibody system.

Characteristic of multibody systems is the presence ofjoints that impose constraints on the relative motion

between rigid parts of the system. Many joints, in practical applications, may be modelled in terms of so-called

simple joints; examples are the cylindrical, prismatic, screw, revolute, sphericalandplanarjoints, cf. Geradin

and Cardona [5]. These are joints where the mechanical interaction is transmitted by a surface contact. The

relative motion of coincident material points on the contact surface between the two parts B andB are then

similar all over the contact surface. Other types of joints may involve line, as well as point, contacts.

Let St denote the contact surface at time t, let O be an arbitrary point in space and let C be a material

point in B, instantaneously in contact with the material point C B . The contact point in space C, is then

placed at

xC = xC = q (t, q;XC ) = xC =

q (t, q;XC ), XC S

0,t, XC S

0,t

and

xC = x,sysO +

(xC xO) and x

C= x

,sysO +

(xC xO)



7/35


B

B B B

Figure 1. Prismatic and revolute joint.

The relative velocity of the two material points is then given by

x,,sysC = x

,,sysO +

, (xC xO) (20)

where ,def= is the relative angular velocity. The impenetrability condition, n x

,,sysC 0 (cf.

condition (63) in Lidstrm [2]), may be expressed according to

n x,,sys

O + n , (xC xO) 0 (21)

We now give two examples of consequences emanating from condition (21). In the case of a prismatic jointthe

two parts are constrained to translate relative to one another along an axis with direction e = e(t), e e = 1, see

Figure 1. This constraint is then represented by the conditions x,,sys

C = e x

C , x

C R and, = 0. From

this it follows that x,,sysO = e x

C for all points O. The requirement of impenetrability (21) then reads

n x,,sys

O = n e x

C 0 (22)

an inequality that has to be satisfied for all x

C R and this is possible if and only if

n e = 0 (23)

that is, the contact surface normal n has to be perpendicular to the direction of relative translation e.

If the parts are connected by a revolute joint, then they are constrained to rotate relative to one another

around an axis (O, e), e = e(t), e e = 1 where O is a point on this axis, that is, a moving point in E with

place xO = xO(t). This constraint is represented by x,,sysO = 0 and

, = e, R and then, according

to (20), x,,sys

C = e (xC xO), xC S

t . The requirement of impenetrability reads

n e (xC xO) 0, xC St

which has to be satisfied for all R and this is possible if and only if

n e (xC xO) = 0, xC St (24)

The conditions (23) and (24) will put restrictions on the geometry of the contact surface. In order to analyse

this we introduce cylindrical coordinates (r, ,z) at O, with (O, e) defining the z-axis, that is, x = xO + err+ ez,

where er = er(t, ) is a unit vector ander e = 0. The contact surface St is assumed to be locally represented

by a smooth function r = r(t, ,z) > 0, that is

St =

xC E|xC = xC(,z, t)

def=xO(t) + er(t, )r(t, ,z) + e(t)z, 1 < < 2, z1 < z < z2

Proposition 4.1 Condition (23) is equivalent to S

t being a prismatic surface, that is, r = r(t, ). Condition

(24) is equivalent to St being a surface of revolution, with (O, e) as its axis, that is, r= r(t,z).



8/35

Lidstrm 215

Proof: One obtains xC

=

er

r+ er

r

and

xC

z= er

r

z+ e

and consequently

xC

xC

z =er

err

r

z +er

e r+ er e

r

= e r r

z + err+ er e

r

and then, taking H = xC

xCz

= ( r rz

)2 + r2 + ( r

)2 > 0,

n e =1

H

e r

r

z+ err+ er e

r

e =

1

Hr

r

z= 0

r

z= 0 r = r(, t)

proving the first part of the proposition. For the second part we consider

n e =1

He r r

z+ err+ er e

r

e =1

Her e r er r

and then

n e (xC xO) = n e (xC xO) =

1

H

er e r er

r

(err+ ez)

= 1

Hr

r

= 0

r

= 0 r = r(z, t)

and this proves the second part of the proposition.

Remark 4.1: Since St is part of a rigid body we may of course conclude that r does not depend on time, that

is, r= r() andr= r(z), respectively.

Rigid parts B andB are said to be rigidly connected, at time t T, if for some point O; x,,sysO (t) = 0and, (t, q(t)) = 0, meaning that the parts, at time t, move as one rigid body.

Proposition 4.2 If the rigid parts B andB are kinematically connected at time t T at three contact points,

not lying on the same straight line, then x,,sysO;k = 0 and

,;k = 0, k = 0,1, . . . , n and the parts are thus rigidly

connected at time t T.

Proof: We have, fork = 0,1, . . . , n, xk = x,sysO;k +

;k (x xO) and then

x,k = x

k x

k = x,sys

O;k x,sys

O;k + (;k

;k) (x xO) = x,,sys

O;k + ,;k (x xO) (25)

Let C,D andEdenote the contact points. Then (cf. Lidstrm [2], Lemma 7.2)

x,,sys

O;k + ,;k (xC xO) = 0, x

,,sys

O;k + ,;k (xD xO) = 0

x,,sys

O;k + ,;k (xE xO) = 0

and from this it follows that ,;k (xD xC) = 0 and

,;k (xExC) = 0. However, since (xD xC)(xExC)

= 0 it may be concluded that ,;k = 0 and subsequently x

,,sys

O;k = 0. The parts are then rigidly connected since

x,,sysO =

nk=0

x,O;kq

k = 0 and, =

nk=0

,;k q

k = 0.

The interaction between rigid parts in mechanical contact is given by the following proposition.



9/35


Proposition 4.3 IfB andB are in mechanical contact over the surface St , at time t T, then

I

k = x,,sys

O;k +

,;k

O , k = 0, 1, . . . , n

where

= St

tda(x) and

O = St

(x xO) tda(x)

are the force and moment sums of the contact forces acting from B on B.

Proof: From (19) and (25) one obtains

I

k =

S

t

x,k t

da(x) =

S

t

(x,,sysO;k +

,;k (x xO)) t

da(x)

= x,,sys

O;k

S

t

t da(x) + ,;k

S

t

(x xO) tda(x)

which proves the proposition.

Remark 4.2: Consider two rigid parts B andB , which are in mechanical contact. Let the contact interaction

be represented by the force sum f and the moment sum MO of the contact forces acting on part B from part

B . The reaction from B on B is given by f andM

O , respectively, where f = f andM

O = M

O .

The contribution to the generalized force, Qick , from this contact may be written as

Q

k = x,sysO;k f

+ ;k M

O , k = 1, . . . , n

where

f =

S

t

tda(x) andM

O =

S

t

(x xO) tda(x)

and then

I

k = x,,sysO;k f

+ ,;k M

O , k = 0,1, . . . , n (26)

A more systematic analysis of the constraint conditions will now follow. Assume that there is a unit direc-tion, c = c(t), such that c x

,,sys

O = 0, that is, the component, in direction c, of the relative system velocity atO is equal to zero. This expresses a bilateral kinematical constraint on the motion of the multibody equivalent

ton

k=0

g

k (t, q)qk = 0 (27)

where g

k (t, q) = c(t) x,,sysO;k (t, q), k = 0,1, . . . , n. Similarly

n

k=0h

k (t, q)qk = 0 (28)

where h

k (t, q) = c(t) ,;k (t, q), k = 0,1, . . . , n, expresses the kinematical constraint that the relative angular

velocity has zero component in the direction defined by c.

As an example, assume that the moving point O E, with place xO = xO(t), is kinematically connected to

both B andB , that is, xO = x,sysO = x

,sysO , t T. Note that O need not be a material point in any of the

two parts. Then we have the constraint

x,,sysO = 0, t T (29)

This is equivalent to the conditions

n

k=0g

ik (t, q)qk = 0, t T, i = 1, 2, 3 (30)



10/35

Lidstrm 217

BB

Figure 2. Point contact.

where g

ik (t, q) = ci(t) x,O;k(t, q) andc = (c1 c2 c3) is a set of basis vectors in V . However, (29) may also be

written

0 =d

dt

(xA(t, q) +R(t, q)(XO X

A ) x

B(t, q) R (t, q)(X

O X

B )) (31)

where XO = X

A + RT(t, q)(xO(t) x

A(t, q)) andX

O = X

B + RT(t, q)(xO(t) x

B(t, q)) are the places corre-

sponding to O in the reference placements andA andB are material points in B andB , respectively. Equation

(31) is equivalent to the condition

C(t, q) = 0, t T, q (32)

where

C(t, q) = xA(t, q) x

B(t, q) +R(t, q)(XO X

A ) R

(t, q)(X

O X

B ) a

= q (t, q;XO) q (t, q;XO) a

where a is a constant vector. Thus the kinematical constraints (30) are equivalent to the geometrical constraint(32).

As a contrast to this we consider two rigid parts B andB with a single contact point C on their boundary

surfaces. Assume that the boundary surfaces have a common tangent plane at the contact point. The relative

angular velocity may be written as

, = np + r ,

r n

= 0 (33)

where p is the pivoting component, r the rolling composant of the relative angular velocity and n

a

normal vector to (see Figure 2). The impenetrability condition, (21), may be expressed according to

n x,,sysC = n

x,,sysO + n

r (xC xO) 0 (34)

a condition that obviously does not involve the pivoting component of the relative angular velocity. The

constraint

x,,sys

C = 0 (35)

represents a situation where B is rolling and pivotingon B without slipping.

By introducing an orthonormal set of basis vectors c = (c1 c2 c3) in V where c3 = n the constraint

condition (34) is equivalent to the three scalar constraints ci x,,sys

C = 0, i = 1, 2, 3. This condition may bewritten as

n

k=0g

ik qk = 0, i = 1, 2, 3 (36)



11/35


where g

ik (t, q) = ci x,,sys

C;k . Note that the impenetrability condition (34) is equivalent to

nk=0

g

3k qk 0 (37)

Since the point C is rigidly connected neither toB

nor toB

it is, in this case, not possible to obtain anequivalent geometric constraint of the type (32). However, the third condition in (36):

c3 x,,sys

C =

nk=0

g

3k qk = 0, t T (38)

representing zero normal relative velocity between the parts, may be shown to be a equivalent to a condition of

the type

C(t, q(t)) = 0, t T (39)

where C: T R is a continuously differentiable scalar valued function. Equation (39) is said to represent

a holonomic constraint on the multibody ifC(t, q)

q= 01n. Now, if C(t, q0) = 0 and if for some 1

l n,C(t, q0)

ql= 0 then, in a neighbourhood of q0 , equation (39) may, as a consequence of the

implicit function theorem, be solved for ql expressed as a function of t, q1, . . . , ql1, ql+1, . . . , qn, that is, ql =f(t, q1, q2, . . . , ql1, ql+1, . . . , qn). The number of configuration coordinates in the equations of motion may then

be reduced by one. This is done by eliminating ql in the kinetic energy, T = T(t, q, q) and in the generalized

forces, Q = Q(t, q, q). This elimination procedure may however lead to equations of motion not optimal from anumerical point of view. From (39) it follows that

d

dtC(t, q(t)) =

nk=0

gkqk = 0 (40)

where gk = gk(t, q) = C(t, q)qk

, k = 0,1, . . . , n and a holonomic constraint, according to (39), thus implies

the kinematical constraint (40), relating the generalized velocity components qk.

A kinematical constraint according to (27) is said to be holonomic if it is equivalent to a condition on the

form (40), that is, precisely if there exists real valued functions

= (t, q) = 0 andG= G(t, q) (41)

such that

gk(t, q) = (t, q)G(t, q)

qk, k = 0,1, . . . , n (42)

This then implies, since = 0,n

k=0

gkqk =

nk=0

G

qkq

k = G= 0 G = 0 Cdef= G c = 0 where

c R is a constant. The function = (t, q) is called an integrating factor to the kinematical constraint. If itis not possible to find functions andG, satisfying (41) and (42), then the kinematical condition (27) is said

to be non-holonomic. There are general integrability conditions giving requirements for conditions of the type

(27) to represent a holonomic constraint. A necessary and sufficient condition is contained in the Frobenius

integrability theorem, see the Appendix. It is well known that the conditions

nk=0

g

ik qk = 0, i = 1, 2 in (36)

are in general non-holonomic.

We now turn to the statement and proof of the holonomic character of condition (38).

Proposition 4.4 The constraint (38) is holonomic.



12/35

Lidstrm 219

Proof: Let C be the contact point with place xC Bt B

t and let NC be a neighbourhood in E of xC.

Consider the sub-surface st = Bt NC of the material boundary surface B

t . Let it be (locally) represented

by the equation h(t, q,x) = 0, that is, x st h(t, q,x) = 0 where the mapping h : T N R

is a continuously differentiable function. The corresponding sub-surface st = B

t NC is assumed to be

represented by the equation h (t, q,x) = 0. Obviously h (t, q,xC) = h (t, q,xC) = 0 and we may assume that

xh

(t, q,x) = 0, x NC andxh

(t, q,x) = 0, x NC. Note that with

h(t, q,x) =h (t, q,x)

H(t, q,x)where H(t, q,x) = |xh

(t, q,x)|

one obtains

x St xh(t, q,x) =

1

Hxh

h xH

H2=

xh

H

xh (t, q,x) = 1Thus, h andh may be chosen so that x st |xh

| = 1 andx st

xh = 1 and, since the surfaceshave a common tangent plane with normal vectorc3 at xC:

c3 = xh (t, q,xC) = xh

(t, q,xC) (43)

We haveh(t, q, (X, t)) = 0, X s0 (44)

where s0 is the referential placement of st , that is, s

t =

(s0 , t). Differentiating (44), with respect to time,

one obtains

th (t, q, (X, t)) + qh

(t, q, (X, t))q + xh(t, q, (X, t)) x(X, t) = 0, X s0

and by taking X = XC ,xC = q (t, q;XC) one obtains, using (43) and leaving out the arguments (t, q,xC):

th + qh

q + xh x (XC , t) = th

+ qh q + c3 x

(XC , t) = 0, XC s0

Similarly th + qh

q + c3 x (XC , t) = 0, XC s

0 and from this it follows

0 = c3 x,,sysC = (th + qh q (th + qh q)) = (th, (t, q) + qh, (t, q)q) =

d

dth, (t, q(t))

where h, (t, q) = h(t, q,xC) h (t, q,xC). This proves the proposition.

Constraint conditions, such as (27) and (28), will in general have to be accounted for as additional differen-

tial equations along with Lagranges equations. If, however, q-coordinates are suitably chosen, these equations

may be identically satisfied. A q-coordinate system is said to be compatible with the constraints (27) and (28)

ifn

k=0g

k (t, q)qk = 0 and

n

k=0h

k (t, q)qk = 0, (t, q) T , q Rn

that is, if the constraint conditions are identically satisfied for all (t, q) T , q Rn.

Proposition 4.5 The q-coordinate system is compatible with the kinematical constraints (27) and (28) if and

only if

g

k (t, q) = 0, h

k (t, q) = 0, (t, q) T , k = 0,1, . . . , n.

Proof: For constraint (27), if

nk=0

g

k (t, q)qk = 0, (t, q) T , q Rn1 then, by taking q = 0n1, we

conclude that g

0 (t, q) = 0, (t, q) T . Thus

n

k=1g

k (t, q)qk = 0, (t, q) T , q Rn1. Then, by



13/35


taking qi = 1, 1 i n, qk = 0 k = i, we conclude that g

i (t, q) = 0, (t, q) R . The correspondingproof for constraint (28) is identical.

Given a direction, defined by the unit vectorc, we may decompose the interaction force and moment sums

according to

f = cf +f

andM

A = cM

O +M

O,

where

f = c f,f

= (c f) c and M

O = c M

O ,M

O, = (c M

O ) c

Note that c f

= 0 andc M

O, = 0.

We now have the following proposition.

Proposition 4.6 For a q-coordinate system compatible with the constraints given by (27) and (28)

I

k = x,,sysO;k f

+ ,;k M

O,, k = 0,1, . . . , n.

Proof: From (26)

Ik = x,,sysO;k f + ,;k MO = x,,sysO;k (cf +f ) + ,;k (cMO +MO,)

= x,,sys

O;k cf +

,;k cM

O + x,,sys

O;k f

+ ,;k M

O, = gk(t, q)f + hk(t, q)M

+ x,,sys

O;k f

+ ,;k M

O, = x,,sys

A;k f

+ ,;k M

O,

where we have used Proposition 4.5

A constraint condition is a geometrical or kinematical condition of the type given in (27). A constraint mech-

anism is a physical arrangement that, by giving rise to necessary constraint forces and moments (reactions),

guarantees that a certain constraint condition is maintained. It should be noted that a specific constraint con-

dition may be realized using different kinds of constraint mechanisms. The constraint mechanism maintaining

the constraints (27) and (28) is equilibratedif, for instance, f

= 0 andM

O, = 0, which means that the force

f

and moment M

O acting on B

from B

are both parallel to c, that is

f = cf andM

O = cM

O (45)

where f andM

O are scalar components. The mechanical interaction between B andB is said to be equili-

brated ifI

k = 0, k = 1, . . . , n for some system of configuration coordinates q = (q1, q2, . . . , qn) Rn,

cf. Lidstrm [2].

Corollary 4.1 For constraint conditions (27) and (28) compatible with the q-coordinates and with a constraint

mechanism satisfying (45) the interaction is equilibrated.

Proof: This is a direct consequence of Proposition 4.6 and (45).

We now look at a situation where the q-coordinates are not necessarily compatible with the constraints. Weintroduce an orthonormal set of basis vectors c = (c1 c2 c3), ci = ci(t) and define the functions

gik(t, q) = ci(t) x,,sys

A;k (t, q), hik(t, q) = ci(t) ,;k (t, q), i = 1,2,3, k = 1, . . . , n

We do not here make any specific assumption about the orthonormal basis c. It may, in a specific situation, be

rigidly attached to one of the parts B orB .

Proposition 4.7 The interaction may be written as

I

k =

3

i=1(g

ik f

i + h

ik M

i ), k = 0, 1, . . . , n



14/35

Lidstrm 221

where f

i = ci f and M

i = ci M

O , i = 1, 2, 3 are components of the constraint forces and momentsrelative to the basis c.

Proof: From I

k = x,,sys

O;k f +

,;k M

O , x,,sys

O;k =

3i=1

g

ik ci and,;k =

3i=1

h

ik ci one obtains

I

k =

3

i=1

g

ik ci

f +

3

i=1

h

ik ci

M

O

=

3i=1

(g

ik ci f + h

ik ci M

O ) =

3i=1

(g

ik f

i + h

ik M

i )

and this proves the proposition.

Remark 4.3: A coordinate system not compatible with the constraints may usually be obtained by performing

an efficient coordinate change q = q(t, q) with n > n.

The net powerof the mechanical interaction, Pnet = P

net(t, q, q), expendedover the contact surface between

parts B andB is defined by Pnet = P

+P , cf. Lidstrm [2], where

P =

S

t

x tda(x),P =

S

t

x t da(x), , = 1, . . . ,N, =

In Lidstrm [2], Proposition 7.3, it is demonstrated that

Pnet =

nk=0

I

k qk (46)

Proposition 4.8 The net powerP

net

is given by

Pnet =

3i=1

n

k=0

g

ik qk

f

i +

n

k=0

h

ik qk

M

i

Proof: Equation (46) in combination with Proposition 7 gives

Pnet =

nk=0

I

k qk =

nk=0

3

i=1

(g

ik f

i + h

ik M

i )

qk =

3i=1

n

k=0

g

ik qk

f

i +

n

k=0

h

ik qk

M

i

It is clear from Proposition 4.5 that if constraint forces and moments are wanted as a result of the analysis ofthe multibody dynamics, then a system of configuration coordinates non-compatible with the constraint should

be used. As an example we may consider the ideal revolute joint, see Figure 1. We assume that the axis of

revolution is (O, c3). Then the kinematical constraints are

x,,sysO = 0, c1

, = 0, c2 , = 0 (47)

This is equivalent to (we partially drop indices , whenever this is possible without endangering the

intelligibility)n

k=0gikq

k = 0, i = 1,2, 3 and

n

k=0hikq

k = 0, i = 1, 2 (48)



15/35


B

B

Figure 3. Screw joint.

Then from (47) Pnet = (

nk=0

h3kqk)M3 = 0, (t, q) T , q R

n1, since the joint is assumed to be ideal.

Thus, c3 , =

nk=0

h3kqk = 0 M3 = 0 and then

Ik =

3i=1

gikfi + h1kM1 + h2kM2

where we here have five a priori unknown reaction components: f1, f2, f3,M1,M2. If the configuration coor-

dinates are non-compatible with the constraints (48) then gik = gik(t, q) and hik = hik(t, q) will not all be

identically zero and constraint forces will appear in the interaction I

k . Note that in the general case five

unknowns f1,f2,f3,M1,M2 will appear in the equations of motion by the addition of the interaction. This is in

addition to the n unknown q-coordinates. The additional unknowns are balanced by the addition of the five

differential equations (48).

Next consider the ideal screw joint (see Figure 3). We assume that the axis of the screw is ( O, c3). The

kinematical constraints are

x

,,sys

O = c3

,

2 L,

,

= c3

,

,

,

R

(49)

where L is the lead of the screw.

Conditions (49) are equivalent to

nk=0

gikqk = 0, i = 1, 2,

nk=0

hikqk = 0, i = 1, 2 (50)

nk=0

g3kqk =

,

2L,

nk=0

h3kqk = ,

For an ideal screw joint it follows from Proposition 8 and (50)1,2 that

Pnet =

n

k=0

g3kqk

f3 +

n

k=0

h3kqk

M3 = 0, (t, q) T , q R

n1

which, according to (50)3,4, is equivalent to,

2Lf3 +

,M3 = 0. Then, if , = 0, this is equivalent to

L

2f3 +M3 = 0 and

I

k =

3

i=1(gikfi + hikMi) = g1kf1 + g2kf2 +

g3k h3k

L

2

f3 + h1kM1 + h2kM2



16/35

Lidstrm 223

and we have five a priori unknown reaction components: f1, f2, f3,M1 andM2. Note that by eliminating , in

(50) one gets five constraint equations.

Next we consider a revolute joint with friction present. We assume that the axis of revolution is (O, c3). The

kinematical constraints are

x,,sysO = 0, c1

, = 0, c2 , = 0

c3

, = 0 if |M3

| M3,c

(51)

where M3,c = M3,c(f1, f2, f3,M1,M2) is a given function determined by the properties of the contact surfaces ofthe revolute joint. In this case we have six a priori unknown reaction force components: f1, f2, f3,M1,M2,M3and

I

k =

3i=1

(gikfi + hikMi), k = 1, . . . , n (52)

If, however, it turns out that |M3| > M3,c then one has to assume that the constraint c3 , = 0 is no longer

valid and one may then take

M3 = M3,r(,3 ;f1, f2, f3,M1,M2) (53)

as a constitutive assumption for the resistance to the motion, where

,

3 = c3

,

. The condition on the netpower is then given by

Pnet = (

nk=0

h3kqk)M3 =

,3 M3 0, (t, q) T , q R

n1

and the constitutive function M3,r must then satisfy

M3,r(,3 ;f1,f2,f3,M1,M2) =

< 0 if

,3 > 0

> 0 if ,3 < 0

One way to accomplish this is to assume that

M3,r(,3 ;f1,f2,f3,M1,M2) < 0 if

,3 > 0

M3,r(,3 ;f1,f2,f3,M1,M2) = M3,r(

,3 ;f1,f2,f3,M1,M2)

In this case there are five a priori unknown reaction components: f1,f2,f3,M1,M2 and

I

k =

3i=1

gikf+ h1kM1 + h2kM2 + h3kM3,r

Now consider the case of a geometrical point contact C between two rigid parts B andB with the constraint

condition given by

ci x,,sysC =

nk=0

g

ik qk = 0, i = 1, 2 (54)

where c = (c1 c2 c3) is an orthonormal basis set with c3 = n. If there is no rolling or pivoting resistance,

that is, ifM

C = 0, we have the interaction I

k =

3i=1

gikfi. In this case there are three a priori unknown reaction

force components: f1, f2 and f3. For a unilateral constraint of Coulomb type we have the following additional

conditions

f3 0, f2

1 +f2

2 sta |f3| (55)



17/35


where sta 0 is the coefficient of static friction. The net power of the mechanical interaction may be writtenas

Pnet =

3i=1

((

nk=1

gikqk)fi + (

nk=1

hikqk)Mi) = (

nk=1

g3kqk)f3 (56)

where we have used the fact that M1 = M2 = M3 = 0 and the constraint conditions (54).

From (37), (55)1 and (56) it follows that Pnet 0. However, according to Lidstrm [2], Equation (82),

Pnet 0 and then it follows that P

net = 0. From this we conclude that

f3 < 0

nk=1

g3kqk = 0 and

nk=1

g3kqk < 0 f3 = 0

Thus if the parts are mechanically in contact then they are kinematically connected and if they are in geometrical

contact and separating then they are not in mechanical contact. If it turns out that

f21 +f

22 > sta |f3|, then

one has to assume that the constraint (54) is no longer valid and that x,C, = x

,C c3(c3 x

,C ) = 0. Following

Coulomb, one may then take

f =kin x,C,x,C, + c3

f3where kin 0 is the coefficient of kinetic friction.

5. The equations of motion involving kinematical constraints

The distinction between constraints on the motion on the one hand and resistance to the motion on the other is

fundamental in multibody dynamics. Constraints on the motion are maintained by reactions, that is, constraint

forces and moments that are a priori unknown. They may be restricted by inequalities as, for instance, in the

case of contact forces at a unilateral contact between parts. Forces and moments responsible for the resistance

to the motion have to be prescribedconstitutively.

Consider a multibody with a motion described by (5) and with the following m, (1 m n) prescribed

bilateral kinematical constraints:

g0 +

nk=1

gkqk = 0, = 1, . . . , m (57)

wheregk = gk(t, q), = 1, . . . , m, k = 0,1, . . . , n. As was demonstrated in the previous section, constraintsinvolving rigid parts are of this type. A holonomic constraint according to (39) is assumed to be represented by

its kinematical counterpart, that is:

C

t+

n

k=1C

qkqk = 0

It should be noticed that, within the description adopted here, using a finite number of configuration coordinates,

these constraints will also appear in connection with some constraints connecting rigid and elastic parts. If the

parts B andB are geometrically connected over the surface St during the time interval T

q (t, q(t);XC ) = q (t, q(t);XC ), XC S

0,t,XC S

0,t, t T

where S0,t andS

0,t are the surfaces in contact in their reference placements, then the corresponding kinematical

constraint is written as

n

k=0( q,k(t, q(t);XC )

q,k(t, q(t);XC ))qk(t) = 0, XC S

0,t,XC S

0,t, t T (58)



18/35

Lidstrm 225

Now, if a discretization in the sense of the finite element method is employed and a number of so-called nodal

points are introduced on the contact surfaces:

Z1C , . . . , ZnnodeC

S0,t and

Z1C , . . . , Z

nnodeC

S

0,t

where nnode is the number of nodes, then (58) is replaced by a finite number of constraints according to

n

k=0

( q,k(t, q(t); Z

C

)

q,k(t, q(t); Z

C

))qk(t) = 0, = 1, . . . , nnode

Remark 5.1: In (3)(5) it is assumed that (t, q, q) T Rn. The constraint (57) implies that the generalized

velocity q at (t, q) T is required to belong to a hyper-plane in Rn. The discussion in this paper is limited tobilateral constraints. The introduction of unilateral constraints will increase the complexity of the mathematical

model, since then constraints may become inactive and active as a result of a dynamics which thus has to

involve impacts. A discussion of the equations of motion under unilateral constraints is presented by Ltstedt

[6] and Ballard [7].

If the constraint equations (57) are independent in the sense that the constraint matrix, g = (gk) Rm(n+1),

satisfies rankg = m then g is said to have full rank. The constraint matrix may be written g =

g0 g

Rm(n+1) where

g0 =

g10...gm0

Rm1 andg = g11 g1n... . . . ...

gm1 gmn

Rmnand then (57) obtains the compact format

gq = g0 +gq = 0m1 (59)

The constraint (59) is said to be homogeneous (in the generalized velocities) ifg0 = g0(t, q) = 0m1, (t, q) T. The constraint conditions in (59) may be given an equivalent formulation using other constraint matrices.

Let A = A(t, q) Rmm be a non-singular matrix and define 0 = Ag0, = Ag. Then =

0

has full

rank if and only if g = g0 g has full rank. Furthermore 0 + q = 0m1 g0 +gq = 0m1.Remark 5.2: Consider two rigid parts B andB , which are constrained to have the same velocity at three

contact points A, B and C, not lying on a straight line, that is, x,,sys

A = x,,sys

B = x,,sys

C = 0. Then, if D is

a fourth contact point, the constraint x,,sys

D = 0 will, according to Proposition 4.2, not be independent of theprevious ones. This kind of redundancy, giving rise to a constraint matrix not having full rank, may usually be

eliminated from multibody systems. The presence of redundant constraints in multibody systems gives rise to

mathematical complication. A treatment of this case is presented by Ltstedt [6].

The equations of motion for the multibody now read

qTM2 = Q

cif + Qi + Qc + Qb

g0 +gq = 0m1(60)

where Qcif andQi are defined by (9) and (12), respectively. The generalized contact force is given by the sum

of the internal and external contact forces, that is:

Qc = Qic + Qec (61)

where Qec represents the contact force from the exterior of B. The generalized internal contact force is,

according to Proposition 4.7, given by

Qick =

N, = 1 <

3i=1

(g

ik f

ik + h

ik M

ik ), k = 1, . . . , n (62)



19/35


This may be written

Qick = Qkc + Qrick , k = 1, . . . , n (63)

where

Qkck =

m

=1F k, k = 1, . . . , n (64)

is the generalized contact force due to the kinematical constraints, F , = 1, . . . , m are the componentsof forces and moments responsible for the maintenance of the constraints (57) and Rmn is the

constraint matrix extracted from (62). Thus Qric represents that part of Qic not containing the constraint

forces. The generalized force Qric is called the residual internal contact force. With the matrix notation

F =

F1 F2 . . . FmT

Rm1 (64) may be written as

Qkc = FT (65)

and the equations of motion then read

qTM2 = Q

cif + Qi +FT + Qic + Qec + Qb

g0 +gq = 0m1(66)

The linear space of virtual velocity fields on B is given by

Wq =

w : B0 V| w = wq(t, q, w;X) =

nk=1

q (t, q;X)

qkwk, X B0 , w

k R

cf. Lidstrm [2].

We introduce the following space of constrained virtual velocity fields:

Wcq =w : B0 V| w Wq, X B

0 , gw = 0m1, w =

w1 . . . wn

T

Rn1

(67)

This is a linear subspace ofWq.

Lemma 5.1 If = Agwhere A = A(t, q) Rmm is non-singular then

Qicw = Qricw, w Rn

satisfying gw = 0m1

Proof: Ifw Rn and gw = 0m1 then, according to (63) and (65):

Qicw = (FT + Qric)w = (FTAg+ Qric)w = Qricw

We now retake the derivation of Lagranges equations presented in Lidstrm [2]. In the application of

the principal of virtual power the space of virtual velocities, Wcq , satisfying the constraint conditions, is nowadopted. The equations obtained do not contain anything new compared to (66). In fact these sets of equations

will turn out to be identical if the constraint matrix has full rank. The derivation will, however, lead to an

automatic inclusion of the appropriate reaction forces. Remarkably enough, these reactions are now obtained

without an explicit analysis of the interactions. The proof of the following theorem originates with dAlembert

[8] and Lagrange [9] and versions of it have appeared in, for instance, Hertz [10], Pars [11], Lanczos [12] and

Neimark and Fufaev [13].

Theorem 5.1 (LagrangedAlembert Equations of Motion)

qTM2 = Q

sum

g0 +gq = 0m1(68)



20/35

Lidstrm 227

The sum of generalized forces on the multibody, Qsum, is given by

Qsum = Qres + (69)

where

Qres = Qcif + Qi + Qec + Qric + Qb (70)

is called the residual generalized force. Furthermore

k =

m=1

gk, k = 1, . . . , n (71)

where 1, . . . , m is a set of real variables, i = i(t).

Proof: Starting from the principle of virtual power one obtains the following condition on the generalized forces

Qa, Qi, Qc andQb, cf. Lidstrm [2]:

w = 0, w Rn

where =Qa Qi Qic Qec Qb. Let =Qa Qi Qric Qec Qb then from Lemma 5.1 it follows that

w = w (72)

for all w Rn1 satisfying gw = 0m1. Let 1, . . . , m be a set of real variables. Then, from (72), it follows that

0 = w Tgw = ( Tg)w =

nk=1

(k

m=1

gk)wk (73)

if gw = 0m1. Now, since 1 m n, n m of the components in w may be arbitrarily and independently

chosen. If 1 m < n let us take these to be wm+1, . . . , wn. We then consider the equations

k =

m

=1

gk, k = 1, . . . , m (74)

This is a linear system of m equations in the m unknowns 1, . . . , m. Since the kinematical constraints are

assumed to be independent the sub-matrix g = (gk), , k = 1, . . . , m is non-singular. Equations (74) will thenhave a unique solution = (t, q, q, q), = 1, . . . , m. This inserted into (73) gives the equation

nk=m+1

(k

m=1

gk)wk = 0 (75)

However now, since wm+1, . . . , wn may be independently chosen and (75) is satisfied for all wm+1, . . . , wn, we

have to conclude that

k

m

=1

gk = 0, k = m + 1, . . . , n (76)

These conditions, together with those in (74), may be summarized as

k

m=1

gk = 0, k = 1, . . . , n (77)

Ifm = n then (74) reads

k =

n=1

gk, k = 1, . . . , n

and this proves the theorem.



21/35


The n + m ordinary differential equations in (68) contain n + m unknowns: q1, . . . , qn, 1, . . . , m consideredas functions of time. The functions = (t) are usually calledLagrangian multipliers and the relation (71)

may be written as

= Tg (78)

In the following proposition it is demonstrated that the Lagrangian multipliers are equal to the components of

forces and moments responsible for the maintenance of the constraints (57). From (66) 1 and (68)1 it followsthat

= Qkc (79)

Proposition 5.1 If = Agthen = ATF.

Proof: Using (65), (78) and (79) it follows that FT = Tg (ATF )Tg = 01n = ATF since

rank(g) = m.

The physical interpretation of the multipliers, as components of constraint forces and moments, is thus,

according to the previous proposition, closely related to the actual mathematical representation of the physical

constraint condition. This fact is clearly underlined by the discussion given in Section 4. In many practical

situations it might be that A = diag 1 . . . n , where k = 1 or + 1, k = 1, . . . , n corresponding to apossible change in sign gk gk for some of the values: = 1, . . . , m. Then, for the corresponding

,one has = F .

Remark 5.3: Note that for a holonomic constraint, according to (39), the corresponding constraint force

component is equal to C

qk.

Remark 5.4: The constraint conditions given by (59) are affine in the generalized velocity components. As was

demonstrated in Section 4, these types of constraints naturally arise from assumptions concerning the physical

character of the mechanical interaction between the parts of a multibody. Theoretically there is a possibility of

considering more general constraints of the type g(t, q, q) = 0m1 where g : T Rn Rm1 is some

given non-linear function of q. However, these so-called non-linear constraints do not seem to play a major

role in the modelling of multibody systems, although there may be some applications to the control theory ofsuch systems. Furthermore, in the non-linear case, the principle of virtual power cannot be used to set up the

equations of motion. In the perspective of its fundamental status this seems to be a major drawback. There is a

vast literature dealing with non-linear constraints, see for instance Neimark and Fufaev [13], Papastavridis [14]

and Udwaida and Kalaba [15]. In these papers the principle of virtual power is replaced by some other principle

that does not seem to be equally well founded. In Weber [16] the problem is discussed using the canonical

formalism and this approach appears to avoid some of the obscurities.

The equations of motion may be written asqTM2 Q

res Tg = 01ng0 +gq = 0m1

(80)

The initial conditionsq(0) = q0, q(0) = q0 (81)

have to satisfy the constraint conditions, that is:

g0(0, q0) +g(0, q0) q0 = 0m1 (82)

Remark 5.5: If we write

Qres = Qcif + Qcon + Qnon (83)

where Qcon is a conservative force with potential V = V(t, q), that is:

Qcon = V

q(84)



22/35

Lidstrm 229

then (80) is conventionally written

d

dt

L

q

L

q Qnon Tg = 01n

g0 +gq = 0m1

(85)

where L = L(t, q, q) = T(t, q, q) V(t, q) is the Lagrangian andQnon is the non-conservative generalized forcecontaining the non-conservative parts of Qres. If the specific body force is given by the acceleration of gravity

b(X, t) = g, where gis a constant vector, then V = V(t, q) will contain the sum

N=1

(Ve + V

g ), where Ve is the

elastic energy according to (13) and Vg = (xc xO) gm

is the potential energy in the gravitational field,

where xc is the place of the centre of mass of part B in its present placement:

xc = xc (t, q) = xO +

1

m

B

0

(q(t, q;X) xO)0 (X) dv(X))

Note that ifB is rigid then one may take Ve = 0.

For a conservative multibody system, that is, Qnon

= 01n, equation (85)1 may be derived using the so-calledLagrangedAlembert principle. Given a mechanical system defined by the Lagrangian L = L(t, q, q) and theconstraint conditions g0 +gq = 0m1, where rank(g) = m . Let t1, t2 R, t1 < t2 be two fixed points in time

and let q1, q2 Rn1 be two fixed points in q-space. Introduce the space of trajectories in q-space with fixed

end-points and satisfying the constraints, that is:

U =

q C1([t1, t2]) q(t1) = q1, q(t2) = q2, g0(t, q(t)) +g(t, q(t)) q(t) = 0m1, t [t1, t2]

and, forq U, the space of variations:

W(q) =

w C1([t1, t2])

w(t1) = w(t2) = 0n1, g(t, q(t))w(t) = 0m1, t [t1, t2]

Note that W(q) is a linear subspace of C

1

([t1, t2]). The mapping I : U R, defined by

I(q) =

t2t1

L(t, q, q)dt

is calledthe action of the mechanical system. Then (see Rund [17])

I(q + w) I(q) = I(q, w) +R(q, w)

where the variation of the action, I, is given by

I(q, w) = t2

t1

L

q(t, q(t), q(t))

d

dt

L

q(t, q(t), q(t))w(t)dt, q U, w W(q)

and

R(q, w)|w|

0, |w| 0.

Theorem 5.2 (LagrangedAlembert principle). The trajectory q U satisfies

I(q, w) = 0, w W(q)

if and only if there is a continuous function : [t1, t2] Rm1 such that

d

dt

L

q(t, q(t), q(t))

L

q(t, q(t), q(t)) = (t)Tg(t, q(t)) (86)



23/35


For the proof of this theorem we need the following lemma.

Lemma 5.2 Let a : [t1, t2] R1n andb : [t1, t2] R

mn be continuous functions.

Assume that rank(b(t)) = m, t [t1, t2] and that

t2t1

a(t)w(t)dt = 0, w : [t1, t2] R1n

such that b(t)w(t) = 0m1, t [t1, t2] (87)

Then there exists a continuous function : [t1, t2] Rm1 such that a(t) + (t)Tb(t) = 01n, t [t1, t2].

Proof: We prove this lemma by contradiction. Assume that there is a point t0 [t1, t2] such that

a(t0) + (t0)Tb(t0) = 01n

Then there is a vector k0 Rn1 such that b(t0)k0 = 0m1, but a(t0)k0 = 0. Let us assume that a(t0)k0 > 0.

Introduce the matrix B(t) = b(t)b(t)T Rmm. Since rank(b) = m it follows that B is symmetric and positivedefinite and thus non-singular. Define

w(t) = k0 b(t)TB(t)1b(t)k0

This is a continuous function equal to k0 at t0 and

b(t)w(t) = b(t)k0 b(t)b(t)TB(t)1b(t)k0 = b(t)k0 b(t)k0 = 0m1

The same holds for (t)w(t), where : [t1, t2] R is a continuous function. Since a(t0)k0 > 0 anda and w

are continuous, we have a(t)w(t) > 0 for all t close to t0. If we choose (t) 0 such that (t) = 1 when t isclose to t0 anda(t)w(t) > 0 where (t) > 0 we have

t2t1

a(t)(t)w(t)dt > 0

This is a contradiction, so for all t [t1, t2] there is a vector (t) such that a(t) + (t)Tb(t) = 01n. That is

continuous follows from

a(t)b(t)T = (t)Tb(t)b(t)T = (t)TB(t) (t) = B(t)1b(t)a(t)T

This proves the lemma.

Proof of Theorem 5.2: Put

a(t) =L

q

(t, q(t), q(t)) d

dt

L

q

(t, q(t), q(t)) andb(t) = g(t, q(t))

then from I(q, w) = 0, w W(q) it follows that (87) is satisfied and consequently (86) is valid. On the otherhand, from (86) it follows that

I(q, w) =

t2t1

L

q(t, q(t), q(t))

d

dt

L

q(t, q(t), q(t))

w(t)dt =

t2t1

(t)Tg(t, q(t))w(t)dt = 0

w W(q) and this proves the theorem.

Remark 5.6: For a conservative mechanical system Theorem 5.2 demonstrates that the LagrangedAlembert

principle produces equations of motion equal to those derived by the principle of virtual power. Note, however,



24/35

Lidstrm 231

that the incorporation of non-conservative forces into the LagrangedAlembert principle is not straightfor-

ward. These forces usually have to be added afterwards. This seems to be a major drawback for the Lagrange

dAlembert principle as compared to the principle of virtual power. The formulation of a LagrangedAlembert

principle in this context is non-trivial and will usually result in equations of motion different from the ones

derived from the principle of virtual work. See Bloch et al. [1] and Rund [17] for a discussion on the history

and validity of various variational principles.

A change of configuration coordinates:

q = (q1, q2, . . . , qn) Rn q = (q1, q

2, . . . , q

n

) Rn

is given by a continuously differentiable mapping q : T

q = q(t, q), (t, q) T (88)

Obviously, the following relationship between the generalized velocities, q andq, holds:

q =q

t+

q

qq (89)

Then we have for the constraint condition (59):

0m1 = g0 +gq = g0 +g(q

t+

q

qq) = g

0 +g

q (90)

where

g0(t, q) = g0(t, q(t, q)) +g(t, q(t, q)) q(t, q)

t Rm1 (91)

g(t, q) = g(t, q(t, q)) q(t, q)

q Rmn

The change of configuration coordinates (88) is said to be efficientif rank( q

q) = min(n, n), cf. Lidstrm [2].

Proposition 5.2 If a specific q-coordinate system is compatible with the constraint (27) then any q-coordinatesystem related to q by (88) will be compatible with the constraints (27). On the other hand if the q-coordinate

system is compatible with the constraint, n n, and if the coordinate change (88) is efficient then theq-coordinate system is compatible with the constraint.

Proof: From g0 = 0m1 andg = 0mn it follows, from (91), that g0 = 0m1 andg

= 0mn . This proves thefirst part. The second part follows from

g(t, q)q(t, q)

q= g(t, q) = 0mn g(t, q) = 0mn

since rank( q

q) = n.

Proposition 5.3 Let the coordinate change (88) be efficient with n n and let the constraint matrix g havefull rank, then g has full rank.

Proof: Ifu Rm1 then uTg = 01n uTg

q

q= 01n u

Tg = 01n u = 0m1.

In the next proposition we demonstrate that the Lagrangian multipliers are invariant under efficient

coordinate changes.

Proposition 5.4 Let the coordinate change (88) be efficient with n n and let the constraint matrix g havefull rank. Then the Lagrangian multipliers are invariant under this change of configuration coordinates, that is:

(t, q, q) = (t, q(t, q),q(t, q)

t+

q(t, q)

qq)



25/35


Proof: We have the transformation properties Qic = Qic qq

, Qric = Qric qq

. Using (62) one may then conclude

that

Qkc = Qic Qric = (Qic Qric) q

q= Qkc

q

q Tg = Tg

q

q(92)

Thus, according to (92)2, and the assumption that g has full rank:

(Tg Tg) q

q= 01n (

T T)g = 01n = 0m1


The set of linearly independent kinematical constraints in (57) is said to be holonomic if there are functions

: T R, , = 1, . . . , m with the matrix (t, q) = ( (t, q)) Rmm being non-singular and

functions G : T R, = 1, . . . , m such that

n

k=0 gk(t, q)qk =

m

=1 (t, q)G(t, q, q), = 1, . . . , m, (t, q) T , q Rn (93)

where G(t, q, q) =

nk=0

G(t, q)

qkqk; otherwise they are non-holonomic. The following result is well known

Theorem 5.3 Take a set of linearly independent holonomic kinematical constraints, according to (57). Then

there exists an efficient change of configuration coordinates to a q-coordinate system compatible with these

constraints.

Proof: If the constraints in (57) are holonomic then according to (93)

n

k=0

gk

qk = 0 G = 0 G = c , = 1, . . . , m

where c are constants. From (93) it follows that

gk(t, q) =

m=1

(t, q)G(t, q)

qk, = 1, . . . , m, k = 1, . . . , n, (t, q) T

This implies that rank

G(t, q)

q

= m where G(t, q) =

G1(t, q) c1 . . . Gm(t, q) cm

Rm. Introduce

the mapping F : R1+n T r s Rm defined by F(t, q) = G(t, q) where q = (r,s), r r

R

nm

, s s

Rm

is a re-arrangement of the q-coordinates so that the matrixG

s

Rmm

is non-

singular. From the implicit function theorem it follows that there is a differentiable mapping f : R1+nm T r Rm such that F(t, (r,f(t, r))) = 0. Put n = n m and introduce the coordinate transformationq = q(t, q), q

r Rn

defined by r = q, s = f(t, q). It then follows that the constraint conditionsG = c, = 1, . . . , m are identically satisfiedq

. This coordinate change is efficient since

q(t, q)

q=

Inn

( Fs

)1(q,f(q))Fr

(q,f(q))

Rnn

and consequently rank

q(t, q)

q = n = n m = min(n, n).



26/35

Lidstrm 233

Remark 5.7: A necessary and sufficient condition for the constraint (57) to be holonomic is given by the

Frobenius integrability theorem, see the Appendix.

Remark 5.8: If the constraint conditions in (57) are organized in such a way that the first h; 0 h m

constraints are holonomic and the following m h constraints are non-holonomic, then h is an invariant underregular coordinate changes.

Remark 5.9: Since it is not possible to choose a q-coordinate system that is compatible with a non-holonomic

kinematical constraint, the corresponding reaction components will always have to appear in the mechanical

interaction.

Remark 5.10: The q-coordinate system in Theorem 5.3 is called a minimal coordinate system for the multibody.

The numbern = n m is called the number of degrees of freedom for the multibody.

6. Elimination and calculation of the constraint reactions

First consider the case of holonomic constraints and a change of configuration coordinates, q = q(t, q),

where the starred configuration coordinates q are compatible with the constraints. Then, according toProposition 4.5

g0 = 0m1 and g = 0mn (94)

From Lidstrm [2], Proposition 6.3 and Lemma 6.1, it follows that

M2 = ( q

q)TM2

q

qandQi + Qc + Qb = (Qi + Qc + Qb)

q

q(95)

Proposition 6.1. For a coordinate change, leading to configuration coordinates q that are compatible with the

constraints, the equations of motion may be written as

qTM2 Q

res = 01n (96)

where Qres = Qcif + Qi + Qc + Qb.

Proof: By introducing the coordinate transformation q = q(t, q) into (80) and using (95), the equations ofmotion in the starred coordinates read

qT M2 Q

res Tg = 01n

g0 +g q = 0m1

which, according to (94), is equivalent to (96).

Remark 6.1: Note that if the coordinate change is efficient, n n and M2 is non-singular then M2 is non-

singular, cf. Lidstrm [2].

Remark 6.2: It follows that Qres = Qresq

q qT

d

dt

q

q

TM2

q

q.

In the differential equation (96), the multipliers have been eliminated due to a special choice of coordinates.

The initial conditions, accompanying (96), are given by

q(0) = q0, q(0) = q0 (97)

where q0 andq0 are determined from the equations q0 = q(t, q0) andq0 = q(0, q0)

t+

q(0, q0)

qq0 andq0

andq0 satisfy (82).



27/35


Below an explicit expression for the multipliers = (t, q, q) will be given. With this expression, and ifq = q(t) is solved for in (96), (97), we may calculate the constraint forces and moments from

= = (t, q(t, q(t)), q(t, q(t))

t+

q(t, q(t))

qq(t))

We will now look at a method where the multipliers are eliminated from the differential equations but never-theless retained as a result of the procedure. A reformulation of the equations of motion (96) and (97) may be

found using the following scheme: Firstly, take the time derivative of the constraint conditions (57), obtaining

nk=l

gvkqk = (t, q, q), = 1, . . . , m (98)

Then, solve forq in (80)1 as a function oft, q, q and, that is:

qk = k(t, q, q, ), k = 1, . . . , n (99)

This is always possible if the q-coordinate system is regular. Note that the dependence of qk on , in (99), is

linear. By substituting (99) into (98) one obtains a system of linear equations in , which can be solved if the

constraint equations are supposed to be independent. We then get

= (t, q, q) (100)

Inserting this expression for into (80)1 one obtains a set of differential equations:

qk = fk(t, q, q)

Solving these equations for q, under the given initial conditions (81), results in the motion q = q(t) and thisinserted into (100) will give the Lagrangian multipliers = (t). The method sketched above may always, at

least in principle, and under the assumptions made, be used to eliminate the Lagrangian multipliers. We now

formalize the discussion above starting with the following lemma.

Lemma 6.1 For the function = (t, q, q) in (98) we have

(t, q, q) = gv0

t

nl=1

gv0

qlql

nk=1

gvk

tqk

nk,l=1

gvk

qlqkql =

nk,l=0

gvk

qlqkql (101)

Proof: The constraint conditions (57) imply ( = 1, . . . , m)

nk=0

(gkqk +gkq

k) = 0 =

nk=0

gkqk =

nk=0

gkqk (102)

However, gk =

n

l=0

gk

ql

ql and this inserted into (102) proves the lemma.

Remark 6.3: In matrix format, (101) may be written as

= g0 gq = g0

t

g0

qq

g

tq

g

qq

q =

g0

qq

g

qq

q =

g

qq

q

where =

1 2 . . . mT

Rm1. For the matrix notation used here we refer to Lidstrm [2],Appendix A.1.

Lemma 6.2 IfM2 is non-singular and ifghas full rank then the matrix = g M12 g

T Rmm is symmetric and

non-singular. Furthermore, is invariant under a regular coordinate change, that is, (t, q) = (t, q(t, q)).



28/35

Lidstrm 235

Proof: The symmetry is obvious and the non-singularity follows from the following argument. If u Rm thenuT u = uTg M12 g

T u = (gT u)TM12 gT u > 0 gT u = 0 u = 0m1 since the constraint matrix g has full

rank andM12 is positive definite. The invariance of is demonstrated by

= gM12 gT = g

q

q q

q

T

M2 q

q

1

g qq

T

= g q

q

q

q

1M12

q

q

T q

q

TgT = gM12 g

T =

and this proves the lemma.

We may now formulate the main theorem of this section. In what follows we assume that det M2 = 0 and

rank(g) = m.

Theorem 6.1 (The initial value problem; reformulation A). The initial value problem (80)(82) is equivalent to

M2 q Qres Tc +g

T1 (g

qq) q = 0n1

qk

(0) = qk0, q

k

(0) = qk0

g0(0, q0) +

nk=1

gk(0, q0)qk0 = 0, = 1, . . . , m

(103)

where

Qresc = QresRT,R = Inn P and P= g

T1g M12 Rnn

Qresc is called the constrained residual generalized force. The matrices R and P are projections, that is, R2 =

P2 = Inn andP = 0nn. Furthermore, range(gT) is a linear subspace of range(P). The Lagrangian multipliers

are given by

=

1g M12 Q

resT + 1

g

qq

q

(104)

Remark 6.4: A similar result, but of less generality, has appeared in Hemami and Weimer [18] and Ltstedt [6].

Proof: From (80)1 it follows, after transposing M2 q QresT gT = 0n1. Since M2 is non-singular this is

equivalent to q = M12 QresT +M12 g

T. However, according to (98), one then obtains = gq = g M12 QresT +

g M12 gT = g M12 Q

resT + and, since according to Lemma 6.2 the matrix is non-singular, we may solvefor the Lagrangian multipliers:

= 1g M12 QresT + 1 = 1g M12 Q

resT 1 ( g

qq) q

and then eliminate the Lagrangian multipliers from the equations of motion thus obtaining

M2 q (Inn gT1g M12 )Q

resT +gT1 ( gq

q) q = 0n1

orM2 q RQresT +gT1

g

qq

q = 0n1 where

R = Inn P,P= gT1g M12 R

nn (105)

P is a projection since P2 = gT1g M12 gT1g M12 = g

T11g M12 = gT1g M12 = P where we

have used Lemma 6.2, and then R is a projection as well. Furthermore, if u range(gT) then u = gTv, v R

m1 and

Pu = PgTv = gT1g M12 gTv = gT1v = gTv = u



29/35


and consequently, u range(P). Since rank(gT) = m 1 it is concluded that P = 0nn. Now, to prove theconverse, we just have to show that ifq = q(t) satisfies (103) then (80)2 is fulfilled. From (103)1 it follows that

gq = gM12 Qres Tc +gM

12 g

T1 = gM12 RQresT + .

From (105) one finds that gT1g M12 R = PR = 0nn and this implies g M12 R = 0mn, since rank(g) = m.

Consequently

gq = gq + gq + g0 = 0m1 ddt

(g0 +gq) = 0m1 g0 +gq = 0m1

where we have invoked (103)3. This proves the theorem.

Remark 6.5: It follows from standard theorems on ordinary differential equations that for sufficient regularity

of the participating functions in (103)1 and the assumption that M2 is non-singular and g has full rank, the

initial value problem defined by (103) has a unique solution in a neighbourhood of t = 0, cf. Coddington and

Levinson [19].

Remark 6.6: A direct proof of the invariance of the Lagrangian multipliers, under a regular coordinate change,

is obtained by considering (104) and using the fact that 1g M12 (Qi + Qc + Qb)T and1g M12 Q

cifT + 1are invariant.

The use of so-calledlinear coordinates will result in a constant mass matrix, that is, a matrix not dependingon t andq. This will greatly simplify the equations of motion.

Corollary 6.1 IfM2 is a constant matrix, then by performing a regular change of coordinates according to

q = M1/22 z (106)

the equations of motion may be written as

z+ QresTM,c +M1/22 g

T1(g0,z

t

g0,zz

z gzt

M1/22 z (

gzz

z)M1/22 z) = 0n1

zk(0) = zk0, zk(0) = zk0

g0,z(0,z0) +gz(0,z0)z= 0m1

where gz(t,z) = g(t,M1/22 z)M

1/22 , Q

resM,c = Q

resc M

1/22 = Q

resM1/22

T, = Inn and = M1/22 PM

1/22 .

Furthermore, in this case

Qcif = qTskew(M1

q)

1

2(

M1

t (

M0

q)T)

Proof: From (106) it follows, since M2 is a constant matrix, that q = M1/22 zand this inserted into (103)1 gives

M2 q + QresTc +g

T1 (g0

t

g0

qq

g

tq (

g

qq)q) = M

1/22 z+ Q

resTc +

gT1(g0,z

t

g0,z

zz

gz

tM

1/2

2

z (gz

zz)M

1/2

2

z) = 0n1

and then after pre-multiplying this equation with M1/22 one obtains

z+M1/22 Q

resTc +M

1/22 g

T1(g0,z

t

g0,z

zz

gz

tM

1/22 z (

gz

zz)M

1/22 z) = 0n1

where gz(t,z) = g(t,M1/22 z)M

1/22 . Now

M1/22 Q

resTc = M

1/22 (Q

resRT)T = M1/22 RQ

resT = M1/22 (Inn P)M

1/22 M

1/22 Q

resT

= (Inn )M1/22 Q

resT = (QresM1/22

T)T = QresTM,c

The constraint conditions (80)2 readg0 +g M1/22 z= 0m1 and this proves the corollary.



30/35

Lidstrm 237

The following lemma will be useful in the forthcoming discussion.

Lemma 6.3 Let 1 m < n and L Rnm with rank(L) = m. Then L(LTL)1LT Rnn is an orthogonalprojection and range(L(LTL)1LT) = range(L).

Proof: See Lancaster and Timetsky [20] or Lidstrm and Olsson [21].

Proposition 6.2 = M1/22 PM1/22 is an orthogonal projection and range() = range(M

1/22 g

T).

Proof: By taking L = M1/22 g

T it follows that rank(L) = m and = LTL. Consequently

= M1/22 PM

1/22 = M

1/22 g

T1g M12 M1/22 = M

1/22 g

T1g M1/22 = L(L

TL)1LT

and then the proposition follows by invoking Lemma 6.3.

Proposition 6.3 RT is a projection onto kernel(g) andM2RT = RM2 = RM2R

T.

Proof: We have

RT = Inn M12 g

T1g = M12 (Inn gT1g M12 )M2 = M

12 R

TM2 (107)

Theng RT = gg M12 g

T1g = g 1g = 0mn (108)

and thus RT is a projection into kernel(g). Furthermore, from Proposition 6.2

rank(RT) = rank(R) = n rank(P) = n rank() = n rank(gT) =

n rank(g) = dim(kernel(g)) (109)

and thus RT is a projection onto kernel(g). Finally, from (107), it follows that M2RT = M2M

12 RM2 = RM2


Remark 6.7: Since (RTu)TM2v = uTRM2v = u

TM2RTv, the matrix RT represents an orthogonal projection with

respect to the scalar product (u, v)M2 = uTM2v on R

n1.

By post-multiplying equation (80)1 with RT we may eliminate the multipliers, since then

qTM2RT QresRT TgRT = qTRM2R

T QresRT = 01n

where we have used Proposition 6.3. We shall now take this a bit further by assuming that the generalized

velocity may be represented by

q = RTr+s (110)

where r Rn1 is considered as a new variable ands = s(t, q) Rn is a suitably selected function. According

to Proposition 6.3, RTr kernel(g). Consequently gq + g0 = g(RTr+ s) +g0 = gs + g0. Since 1 m < n

and rank(g) = m we may find a function s = s(t, q) Rn1 such that g(t, q)s(t, q) + g0(t, q) = 0m1 andthe constraint condition is then identically satisfied. Note that the function s is not uniquely defined, since if

g s +g0 = 0m1 then g(s + u) +g0 = 0m1, u kernel(g). Thus q = RTr+s = RTr +s where

r = r u ands = s + u, u kernel(g) (111)

Now, from (110), it follows that q = RTr+RTr+ s and this inserted into (80)1 gives

( RTr+RTr+ s)TM2 Qres Tg = rTRM2 + r

T RM2 + sTM2 Q

res Tg = 01n

Post-multiplying with RT, one obtains

rTRM2RT + rT RM2R

T + sTM2RT QresRT = 01n

where we have used Proposition 6.3. We may summarize this in the following theorem.



31/35


Theorem 6.2 (The initial value problem; reformulation B). The equations of motion (80) imply thatrTRM2R

T + rT RM2RT + sTM2R

T QresRT = 01nq RTrs = 0n1

(112)

where g s + g0 = 0m1. The appropriate initial conditions are q(0) = q0, r(0) = r0, where q0 and r0 have to

satisfy q0 = R(t, q0)Tr0 +s(t, q0) andq0 andq0 have to fulfil (82). Equation (112) is a system of 2 n first-orderordinary differential equations in the 2n unknowns q and r. Once this system has been solved one may use

(104) to calculate the multipliers. Note that the equations (112) are invariant under the transformation (111).

Remark 6.8: The mass matrix RM2RT, in equation (112), is not positive definite since dim(kernel(RT)) = m >

0. This may eventually cause problems when solving (112) numerically.

As a remedy for the problem noticed in the previous remark let us once again consider the projection RT.

Since, according to (109), rank(RT) = n m > 0, it is possible to construct a matrix S Rn(nm) from RTbyselecting n m columns from RT to build up S so that rank(S) = n m. Obviously, g S = 0mm. Let a R

nm

and

mathematics and mechanics of solids-2012-lidström-209-42

Documents