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The analysis of R-S-S-R spatial four-bar mechanism The analysis of R-S-S-R spatial four-bar mechanism

David Perng Chyi

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Recommended Citation Recommended Citation Chyi, David Perng, "The analysis of R-S-S-R spatial four-bar mechanism" (1969). Masters Theses. 5380. https://scholarsmine.mst.edu/masters_theses/5380

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T1-iE ANALYSIS OF R-S-S-R SPATIAL

FOUR-BAR MECHANISM

BY

DAVID FERNG CHYI, 1944-

A

THESIS

submitted to the faculty of the

UNIVERSITY OF MISSOURI - ROLLA

in partial fullfillment of the work required for the

Degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

Rolla, Missouri

Approved by

T2315 c. I 55 pages

183294

ABSTRACT

Although many methods are known for the kinematic

analysis of spatial mechanisms, such as descriptive

geometry, vector analysis, dual number theory, matrices,

screw calculus, and various other methods, a better way

to analyze the specific R-S-S-R four-bar mechanism is

to use tensor methods in order to eliminate the apparent­

ly formidable and tedious tasks of mathematically for­

mulating problems and obtaining solutions.

The analysis described in this paper is developed

basically by using tensor notations and operations, and

the calculations take advantage of the capabilities of

modern digital computers. A FORTRAN program for the IBM

360 Data Processing System which will analyze the dis­

placement of each joint pairs, angular velocity, and

angular acceleration of the output link is included.

This program is available upon request.

Several example problems were solved with the

program and only one is presented in this thesis.

ii

iii

ACKNOWLEDGEMENT

The author is greatly indebeted to Dr. Chung-you Ho,

of the Department of Mechanical Engineering at the Univer­

sity of Missouri at Rolla, for his advice in the formu­

lation of this thesis, and for his guidence and valuable

criticism throughout this research program.

He also wishes to thank Mr. Kuo-Chien Hsei for his

help with problems regarding this study.

Also thanks to the Computer Center of the Univer­

sity of Missouri at Rolla for the use· of their IBM 360

digital computer.

iv

TABLE OF CONTENTS

Page

ABSTRACT ••••••••••••••••• o •••••••••••••••••••••••••••••••••• ii

ACKNOWLEDGEMENT ••••••••••••••••••••••••••••••••••• o •••••• , • iii

TABLE OF CONTENTS. , ••••••••••••••••••••••• , ••••••••••••••••• i v

LIST OF ILLUSTRATIONS ••••••••••••••• o•••••••••••••••••••••••vi

LIST OF TABLES. o ••••••••••••••••••••••••••••••••••••••••••• vii

CHAPTER

CHAPTER

I. INTRODUCTION ••••••••••••••••••••••••••••••• o •••• 1

II. LITERATURE REVIEW•••••••••••••••••••••••••••••••5

A. DEFINITIONS AND BACKGROUND MATERIALS ••••••••• 5

1. KINEMATIC CONCEPTS ••••••••••••••••••••••••• 5

a. MECHANISM••••••••••••••••••••••••••••••o•5

b, KINEMATIC LINK •••••• , ••••••••••••••••• o •• 6

c, KINEMATIC PAIR ••••••••••••••••••••••••••• 6

d, MOTION ••••••••••••••••••• o•••••••••••••••8

e. GRUEBLER'S CRITERION OF MOBILITY ••••••••• 9

2. PRELIMINARY MATHEMATICS•••••••••••••••••••l2

a. NOTATIONAL CONVENTION ••••••••••••••••••• l2

b. DEFINITION OF A TENSOR••••••••••••••••••l4

c. SOME TENSOR PROPERTIES, ••••••••••••••••• 15

CHAPTER III. POSITION ANALYSIS OF THE SPATIAL FOUR-BAR

LINKAGE ••••••••••••••••••••• 0 •• I •••••• 0 •••••••• 17

A. GENERAL DESCRIPTION•••••••••••••••••••••••••l7

B. COORDINATE RELATIONSHIP•••••••••••o•••••••••20

CHAPTER IV. METHOD OF APPROACH AND DERIVATION ••••• o.o••••••22

CHAPTER

v

A. GOVERNING EQUATION OF R-S-S-R MECHANISM ••• 22

B. DERIVATION OF THE OUTPUT FOLLOWER ANGLE

e111 , AND THE FOLAR, AZIMUTHAL ANGLE OF

THE SECOND LINK e11 , %11 ••••••••••••••••••22

C. OUTPUT FOLLOWER ANGULAR VELOCITY, aiii •••• 27 .. III D. OUTPUT ANGULAR ACCELERATION, e ••••••••• 29

V. CONCLUSION ••.••••••••••••••• , •••••••• o ••••••• 31

BIBLIOGRAPHY •••••••••••••••••••••••••••••••••••••••••••••• 32

APPENDICES ••• •••••••• , •••••••••••••••••••••••••••••••••••• 35

APPENDIX A. COMPUTER PROGRAM ••••••••••••••••• 36

APPENDIX B. FIGURES OF SOME VARIABLES OF

R-S-S-R SPATIAL MECHANISM •••••••• 42

VITA. e e a • • a e a • a • a G a • a a • • • • • 1 e • • e e • • • e • • • • e a a • • e a • • • • • 1 1 t a • 48

vi

LIST OF ILLUSTRATIONS

Figures Page

1. General diagram of four-bar spatial linkage ••••••• l8

2. Coordinate relationshiP•••••••••••••••••••••••••••20

3. R-S-S-R four-bar mechanism••••••••o•••••••••••••••23

4. The azimuthal angle of the second link versus

5. The polar angle of the second link versus input

angle •••••••••••••• o ••••••• o ••••••••••• o •••••••••• 44

6. The output angle versus input angle ••••••••••••••• 45

7. The output angular velocity versus input angle •••• 46

8. The output angular acceleration versus input

angle •••••••••••••••••••••••••••••• o •••••••••••••• 47

vii

LIST OF TABLES

Tables Page

1. Values of ri for lower pairs •••••••••••••••••• lO

CHAPTER I

INTRODUCTION

The broad category of devices known as mechanisms

may be divided into the two major categories of planar

mechanisms and spatial mechanisms.

In planar mechanisms, the motion of all links can

be completely described in a single plane, one single

graphical projection. Two common examples of planar

mechanisms are the four bar linkage and the slider

mechanism.

However, other mechanisms are known whose motion

can not be described in a single projection; they have

three-dimensional motion. Common examples of this type

are the Hooke or Universal Joint, the Bennett mechanism,

and the Bricard mechanism. These form the second broad

category, spatial mechanism.

Recently, more and more emphasis has been placed

on the three dimensional linkages. Because of this extra

dimension habit, the kinematicians have more design pa­

rameters than the traditional two-dimensional mechanisms.

Due to the rapid development of high speed digital com­

puters and various elegant mathematical methods, spatial

mechanisms can be performed with greater ease.

1

In Germany, analysis and synthesis techniques were

* performed by Altman(l) , Beyer(2), Hain(3), Keler(4),

an~ ot~ers, based on graphical and analytical geometry

of dual quantities. In the U.s.s.R., tensor analysis

was used by s. G. Kislitsyn(5) in the theory of spatial

mechanism. While in this country, Denavit and Hartenberg

(6) adopted the matrix calculus; Chace(?), Beyer and

Harrisberger(8) used the vector technique; Yang and

Freudenstein(9) chose matrices with dual-number qua­

ternions and Ho(lO) developed the tensor method.

It seems very convenient if a spatial mechanism is

pursued by tensor methods. In spatial mechanism, almost

every quantity involved can be represented by tensor

notation; such as scalar quantity which is a tensor of

zeroth order, moment of inertia is the tensor of second

order, velo~ity and acceleration are the tensors of

first order, and other complicated quantities can be

expressed as the tensors of higher order.

Graphical and related methods have been important

because they avoid detail computation and provide a

visual perspective. With the advent of digital computers,

the computational advantage of graphical methods become

less significant. And as we know, it has always been

difficult and tedious to apply graphical methods to

* Numbers in the parenthesis refer to the Bibliography at the end of the thesis.

2

three-dimensional analysis.

Analysis and synthesis by conventional complex

mathematics has been very successful for two-dimen­

sional problems. Extension to three-dimensional

analysis was suggested by Raven(ll), but nevertheless

conventional complex mathematics has remained a two­

dimensional tool.

Matrix methods have been developed and applied

by Hartenberg, Denavit, and Uicker(l2). A computer

program, based on this method, will obtain position,

motion and force solutions for the complete motion

cycle of three-dimensional mechanism connected by

lower pairs in any single loopo The method has been

extended to general spatial mechanism. However, the

iteration technique required for the position solutions

is always subjected to the trunctional errors of the

computations, and iterpretation of the matrix equations

is also difficult. Nevertheless, it is a beautifully

formulated solution and it affords the available

numerical solution to an important category of mechanisms.

Dual numbers, quaternions, and vectors have been

applied to this kind of problems. These approaches have

advantages in the representation of spatial problems,

but also have some disadvantages especially in the

representation of complicated linkages not possessing

geometry symmetry.

3

The Instantaneous Screw Axis (ISA) theory adopted

by Skreiner(l3) is a newly developed method which has

a good approach to the problem. However, the positions

of the links relative to the ground link are not easily

determined through this method. It, therefore, is dis­

carded in favor of the more direct tensor methods

introduced here.

The goal of machine design is to find a simple but

effective mechanism for transmitting a desired motion.

Spatial mechanisms have long been avoided by designers

because of their complexities. The construction of a

spatial mechanism is amazingly versatile. Sometimes,

an ingenious arrangement will be able to simplfy a

complex mechanism and accomplish the desired motion.

The spatial four-bar mechanism treated in this

paper for revolute-spherical-spherical-revolute joint

in the order given.

4

CHAPTER II

LITERATURE REVIEW

A. DEFINITIONS AND BACKGROUND MATERIALS

In order to analyze the R-S-S-R spatial four-bar

mechanism, it would be well to review a few important

definitions and concepts to form a general basis on

which to proceed.

1. KINEMATIC CONCEPTS

a. Mechanism

Probably the most important term requiring defini­

tion in the study of kinematics is "mechanism". It is

a term having a definite meaning, but difficult to

define succinctly. A descriptive definition was given by

Reuleaux as "that imaginary concept of a device with

perfect geometry and rigidity••. Although this is true,

it is far from complete. The classical definition states

that a mechanism is an assemblage of kinematic links,

or machine parts, connected together by kinematic pairs,

or joints, used for the purpose of transforming one type

of motion into another. A mechanism is a closed, kine­

matic chain of links. One link is a fixed frame or base,

and all motion is viewed with respect to this link as a

reference.

5

b. Kinematic Link

The term kinematic link was essential to the defi­

nition of a mechanism and so it too must be defined. In

kinematics, the machine members of which a mechanism is

made are called links. In keeping with the purpose of

kinematics, all concepts of deformation due to load and

all stress-strain relationships are neglected. Kine­

matically, the only purpose of the link is to hold a

specific spatial relationship between the several

kinematic pairs or joints, The particular physical

configuration, or the shape, size, weight, and material

are all only incidental in the study of how a mechanism

will move. A link is then a rigid body containing the

elements of at least two kinematic pairs or jointso

c. Kinematic Pair

In order to perform their function of motion,

transformation in a mechanism, the link must be con­

nected by some sort of movable joints. The connections,

or joint, between pairs and links are called kinematic

pairs. Just as the link is the sole criterion of spatial

relationship between its two kinematic pair elements;

the pair is the only determining factor for the type

of motion permitted between the connected links. The

sole function of these pairs is to provide a connection

limiting the relative motion between two links to a

6

certain predetermined ~pe of motion.

Kinematic pairs can take on an infinite variety of

configurations. They are often difficult to recognize

by their physical appearence, but must be categorized

according to their characteristic motions. The categories

are wrapping connectors, lower pairs, and higher pairs.

The wrapping connectors usually take the form of belts

or chain and may be recognized by the fact that they

transmit motion in only one direction, tension. The

lower pairs, as identified by Reuleaux, are recognizable

by their clearly defined pair variable. They are six in

number: the revolute pair (R), which permits rotational

motion about an axis and is defined by a single variable

angle e; the prismatic pair (P), which permits translation

motion in one direction and is defined by a single dis­

placement variableS; the screw pair (H), which permits

helical motion and is defined by either the rotation

angle e or translation displacement S, related through

9/2~ = S/h, where h is the lead of the screw (advance per

revolution); the cylindrical pair (C), which permits a

translation motion parallel to an axis and also a rota­

tion motion about the same axis independently; the

spherical pair (S), which permits a globular rotation

about a point; and the last one is the planar pair (F),

7

which permits planar motion and is defined by two dis­

placement variables and one rotation variable. Note that

the spherical pair may be viewed as wrapping pair for its

hollow element is wrapped around its full element, but it

8

is not a wrapping connector which transmits motion in only

one direction. The higher pairs, according to the definition

by Reuleaux, are the pairs with the surface elements so

shaped that only line or point contact are possible between

elem~nts.

d. Motion

ordinarily, there are two kinds of motion, one is

translation and the other is rotation. Both of them

include at least three distinctly different yet related

characteristics, i.e., displacement, velocity, and

acceleration, requiring a certain amount of time for

their completion.

Whenever a motion is described, the reference frame

with which the motion is related must be defined first,

In mechanism, it is tha very fact that motion exists that

implies reference frames of some sort on the moving parts

or a certain specified fixed coordinate. If the reference

frame of one machine part moves with respect to the

reference frame of another, we speak of relative motion.

When the reference frame of one part is fixed with respect

to the ground and the motions of the other parts are

referred to it, then these particular relative motions

may be termed absolute motions. Absolute motion is thus

a special case of relative motion, in which the reference

frame is fixed.

e. Gruebler's Criterion of Mobility

Before a spatial mechanism is established and motion

analysis started, the first question is whether or not

the mechanism will move. Gruebler's criterion (14) is a

quick and easy test for that. It is derived as follows:

since any rigid body has six degrees of freedom in space,

a mechanism of n links considered independently in space,

should have a total of 6n degrees of freedom. If these

links were combined to form a linkage, one link would

be taken to be a stationary frame of reference. Thus the

total number of degrees of freedom, f, with respect to

that stationary frame would become

f = 6(n - 1) (2-1)

when these links are connected by the various pairsf they

are imposed by certain restraints on the movability. The

total number of degrees of freedom which remains in the

mechanism is now k .

f = 6(n - 1) - ~ r~ ~1

(2-2)

where: ri is the number of restraints imposed by the i's

pair

9

k is the total number of pairs in the mechanism.

Equation (2-2) is essentially Gruebler's criterion

of movability. It shows the number of degrees of free­

dom of a mechanism as a function of the number of links

and the number of restraints imposed by pairs.

The value of ri are different for different pairs

involved. Those for the lower pairs are listed in

Table I.

Table I

Values of ri for Lower Pairs

Pair Symbol ri

Revolute R 5

Prismatic p 5

Screw H 5

Cylindrical c 4

Planar F 3

Spherical s 3

It should be noted that this criterion has several

exceptions. In these cases some of the restraints become

redundent and the degrees of freedom may be more than

that predicted by equation (2-2), Some known exceptions

are the Bennett and the Bricard mechanism, Hooke's joint,

this R-S-S-R spatial mechanism, and all planar mechanisms.

In the case of planar mechanisms, another form of the

10

criterion should be used, that is k .

f = J(n - 1) - E (r1 - 3) i=l

The parameters in equation (2-J) are the same as

defined for equation (2-2).

11

(2-J)

Usually a mechanism is more interesting if it has one

degree of freedom. A single input will then completely

determine the motion of the entire mechanism. This

condition is known as constrained motion.

In a simple closed-loop chain, composed entirely

of binary links, k is equal to n and for constrained

motion n .

1 = 6(n - 1) - E (r 1 ) i=l

Since any kind of pair is permitted in a closed-n .

loop chain, E r 1 is always less than 5n. This yield i=l

the following inequality

1 ~ 6(n - 1) - 5n ~ n - 6

that is n ~ 7.

(2-4)

(2-5)

This means the number of links in a simple closed­

loop chain for a constrained motion can not be more than

seven.

12

2. PRELIMINARY MATHEMATICS

Tensor calculus came to prominence with the development

of general theory of relativity by Einstein in 1916. It

provides the only suitable mathematical language for

general discussion of that theory. But actually the tensor

calculus is older than that. It was invented by the Italian

mathematicians Ricce and Levi-Vivita, showing its appli­

cations in geometry and classical mathematical physics in

1900. Thus tensor calculus comes near to being a universal

language in mathematical physics. Not only does it enable

us to express general equations very compactly, but it

also guides us in the selection of physical variables,

by indicating, automatically, invariance with respect

to the transformation of coordinates.

a, Notational Conventions

A point in three-dimensional space located with

respect to a Cartesian coordinate system Xi , may also

be located with respect to another Cartesian coordinate • system, X. , by the equations ~

• t I

xl = Allxl + A2lx2 + A31x3 +

• I • x2 = Al2xl + AzzX2 + A32x3 +

• t 1

XJ = AlJxl + A23x2 + A33x3 +

• where the Aij s are constants of

Bl

Bz (2-6)

B3

rotation between the

axes of the two coordinate systems and the Bi's are

constants of translation between the origins of the

system.

Expression like Eq. (2-6) may be expressed more

concisely by adopting the following notational conven-

tions:

Range Convention:

When an index (su~script) occurs unrepeated in

a term of an expression, it is understood to

take, in turn, each value in the range of that

index. In this paper the values will always be

1, 2, J.

Summation Convention:

When an index is repeated in a term, summation

over the range of that index is implied. Using

these conventions, equation (2-6) is written

as, t

X. = A .• X. +B. ~ J~ J ~

(2-7)

and the inverse transformation is written as,

t I

X. = A .. X. + B. ~ ~J J ~

(2-8)

The range and summation conventions will be used

throughout this paper. No confusion should occur if

the reader remembers that they are implicitly present

in the notation henceforth.

13

With the orthogonal property of the coordinate

systems, the rotational coefficients have the relation

where

0 if i f j = A .A = 6 •. = [

kl kj lJ 1 if i = j

6 .. is the so-called Kronecker delta. lJ

(2-9)

In the paper the orthogonal transformation is always

positive; i.e., IAijl = 1.

b. Definition of a Tensor

14

If each number, Ti , of a set of quantities associated

with a Cartesian coordinate system, Xi , and with a point, • P, can be transformed to any other coordinate system Xi ,

according to the equation

• T. =A .• T. J Jl l

(2-10)

then is said to be a component of a tensor of the first

rank.

It will be seen that a Cartesian tensor of the first

rank is equivalent to an ordinary Cartesian vector.

Tensor of higher rank can also be defined. That is,

quantities Tab which transform according to

are called tensors of the second rank. A tensor of the

nth rank has n indices and transforms through its multi­

plication by n coefficients:

' Tij •••• k = Ai~jb•••••••AkcTab •••• c

c. Some Tensor Properties

A tensor is said to be symmetric in two indices,

j and k, if the value of any component is not changed

by interchanging the positions of j and k. That is,

if

T. 'k 1e•••J •••• m = T. . 1 •••• kJ •••• m

then the tensor is symmetric in j and k. The tensor

is completely symmetric if its components retain the

same value when any two indices are interchanged.

Similarly, a tensor is said to be screw-symmetric in

tho indices j and k, if

s. 'k 1 •••• J •••• m = - s. . 1. • • .kJe ••• m

The tensor is completely screw-symmetric if its

components retain the same value, but are changed in

sign when any two indices are interchanged.

say

say

The product of any completely symmetric tensor,

T .. , and any completely screw-symmetric tensor, 1J

S. . , is 1J

T •. S •. = 0 1J 1J

This property will be used to great advantage in the

application of tensor operations discussed in this

paper.

15

(2-12)

(2-13)

(2-14)

(2-15)

The Kronecker delta defined in equation (2-9)

is an example of a symmetric tensor. It is also

called an isotropic tensor because its components

retain the same values in any coordinate system.

Another tensor which will be very useful in

the present application is the permutational symbol,

y .. k • This tensor is both isotropic and completley lJ . screw-symmetric; the values of its components are

obtained as follows;

0 if any two indices have the same value.

1 if the values of the indices ijk repre-

Yijk= sent an even permutation of the sequence

1,2,3.

-1 if the values of the indices ijk represent

an odd permutation of the sequence 1,2,3.

An important relation between o .. and y .. k is lJ lJ given by

(2-16)

16

17

CHAPTER III

POSITION ANALYSIS OF THE SPATIAL FOUR-BAR LINKAGE

A. General Description

In order to demonstrate the application of the

tensor method to the analysis of spatial mechanisms,

a closed-loop, four-bar spatial linkage has been studied.

Figure 1 represents the general diagram of the linkage

and relationship of the links to the coordinate frames.

It is convenient that each link can be determined with

respect to its local coordinate frame qy a set of spheri-

cal polar coordinates. The so-called ground link, Cc., 1.

has a length of C units and is directed along a vector

of unit length, c .• 1.

The components of ci with respect

to the ground coordinate frame are defined by

cl = sin% cos a

c2 = sin% sine (3-1)

c3 = cos%

Where }6 and e represent the polar and azimuthal angles,

respectively.

Likewise, the first link, Rri1 ' has magnitude R

and is directed along a unit vector ri1 which origi­

nates at the first joint. The component of ri1 with

I Rr. ~

xrrr 2

COORDINATE SYSTEM

Figure 1. General diagram of four-bar spatial

linkage

18

respect to its local coordinate frame are defined by

r 1I = sin%I cose 1

r 2I = sin%I sinei

r 3I = cos%I

(3-2)

where %I and e1 represent the polar and azimuthal

angles of the link in the first frame. In the same

manner, the components of the second and third link

vectors, s 1II and t 1II can be expressed in term of the

polar coordinates of the second and the third frames,

respectively.

s II 1 = sin,0II coseii

s II 2 = sin,0'II sine II (3-3)

s II 3 = cos,0II

t III 1 = sin,0III coseiii

t III 2 = sin,0III sine III (3-4)

t III 3 = cos_eiii

To derive relationships between the link vectors,

it is desirable to specify an arbitrary point to which

all the vectors may be referredo Through this paper

the ground joint will serve as the reference point.

The transformation matrix Ai~ can be written as the

19

20

direction cosines between the axes; that is

[All Al2 Al3 A •• = A21 A22 A23 1J

A31 A32 A33

0 0 0

= [

cos(X1 ,x1 ) cos(X1 ,x2 ) cos(X1 ,x3 )

cos(x2 ,xi) cos(x2 ,x;) cos(X2 ,xj) (3-5)

cos(x3 ,xi) cos(x3 ,x~) cos(x3 ,xj)

Bo Coordinate Relationship

Whenever one of the axes of the prime reference frame

is specified, the prime frame system is conveniently se-

lected in following way, as shown in Figure 2. xo

3

~~-----------+------~----~~x2 I //

-----4/

xo 2

Figure 2o Coordinate Relationship

If xj is the specified axis of the prime frame

system in xl ' x2 • x3 , reference system with given

polar and azimuthal angles ( ~o , 90 ) then it can be

expressed as

(3-6)

The other two prime reference axes are defined by

(X~)i = Yik3<x;)kx3 = Yik3wk

<x;)i = Yij~<x;)j(X~)k = Yijkwj Ykl3wl

= (oil oj3 - oi3 ojl) wjwl

= - 0i3 + w3wi

(3-7)

(3-8)

Equations (3-6) and (3-8) can be expanded and formulated

to be the unit tensor that

(3-9)

(3-10)

From (3-6), (3-9), and (3-10), the relation of

transformation can be readily got as

(3-11)

where

[ sine, -coseo 0

Amn = cose. cos%. sine. cos%0 -sin%0

sin%o cose. sin%. sine. cos% 0

( 3-12)

21

CHAPTER IV

METHOD OF APPROACH AND DERIVATION

A, Governing Equation of R-S-S-R Mechanism

Figure 3 is a diagram of the R-S-8-R four-bar

mechanism with revolute pairs on the ground and first

joints, and with spherical pairs on the second and

third joints.

In this mechanism. a single closed-loop vector

equation has been used:

Cc. + Rr. + Ss. + Tt. = 0 2 2 2 1 (4-1)

Components of unit vectors c. , r. , s. , and t. 1 2 1 1

are all taken with respect to the ground frame by the

transformations

r. = AI I 2 mi rm

s. = AI~ sii (4-2) 2 m2 m t. = 2

AI~Itiii m2 m

I II III where A., A ., A . , are the transformation matrices m1 m1 m2

relating the ground frame and the first, second, and

the third frames, respectively. Equation (4-1) actu­

ally represents three individual equations (for i = 1,

2, 3).

B. Derivation of the output follower angle e11I, and the

polar, azimuthal angle of the second link e11 , %II.

22

/

xi 1

FOLLOWER

/

//

//

/ //

Rr. / Cc. l. / l.

/ /

,.., ... xy/

~

Figure 3, R-S-S-R four-bar mechanism

Xz

!\) \,.)

The ground link is Cc.; its magnitude and direction ~

are always given. In usual, the magnitude and direction

of the ground frame are chosen to designate the relative

positions of the other links, input angle ei, the polar

angle %I, and the magnitude R. AI. and r~ are known, m~ ~

magnitude S, polar angle %III between the ground frame

and ti' and the magnitude T are given.

In this paper, the auther wants to use the given

informations to find and analyze the components of the

unit vector si with respect to the ground frame; and

24

the polar and azimuthal angles of the link Ssi' %II and

eii with respect to the ground frame; output azimuthal

angle eiii, anglar velocity aiii and angular acceleration

eiii with respect to the ground frame.

From Eqs. (4-1) and (4-2), the unknown quantities

can be determined by solving

where

t. = ~

Letting K. = Cc. + RAI~rmi ~ ~ m ....

any given input crank angle ei,

expanded as:

(4-3)

a constant vector for

equation (4-3) can be

25

Kl + s sin,0'11cose 11 + T sin}?JIIIcoseiii = 0 (4-4a)

Kz + s sin,011sine 11 + T sin,0IIIsineiii = 0 (4-4b)

K3 + s cos%II + T cos%III = 0 (4-4c)

Equations (4-4) are the general scheme for analyzing

the R-S-S-R mechanism. In the R-S-S-R mechanism shown in

Figure 3, the axis x3 of the ground frame is the axis of

rotation of the follower. The angle, %III, between the

x3 axis and the follower T is a right angle.

For ,0III= 1T/2 , equations (4-4) now become

Kl + s sin,0I1 cosa 1I + T cose 111 = 0 (4-5a)

K2 + s sin,0I1sine 11 + T sinaiii = 0 (4-5b)

K3 + s cos,0II = 0 (4-5c)

The solution of equation (4-5c) is

,0 = arc cos( -K 3 /S )

= arc sin ( J S 2 - K ~ /S ) (4-6)

substituting Eq. (4-6) into Eqs, (4-5a) and (4-5b)

Kl + Jsz_ K2 + )sz_

(4-?a)

( 4-Tb)

From Eqs. (4-?a), and (4-?b), it becomes clear

that the o·utput follower angle e111 has two possible

solutionso This confirms the statement by Harrisberger

(8) that the R-S-S-R mechanism has two degrees of

freedom.

26

Rewrite equations (4-?a) and (4-7b) as

Js2 K2 - 3 cose 11 = - T cosa 111 - Kl (4-8a)

js2 K2 - 3 sine 11 = - T sina 111 - K2 (4-8b)

square (4-8a), and (4-8b) and add them together

(s2- K~)(cos2 a 11+ sin2e11 ) = T2cos2e111+ T2sin2e111

2 2 + K1 + K2 + 2K1T cosa 111+ 2K T sine111 (4-9) 2

s2- K~ = T2 + Ki + K~ + 2T(K cose 111+ K sine111 ) 1 2

K1cose 111+ K2sina 111= l/2T (S 2- T2- Ki- K~- K~) 2 2 2 2 2 letting B = l/2T (S - T - K1- K2- K3 )

(4-10)

(4-11)

(4-12)

where B is another constant vector for any given crank

angle e1 , equation (4-11) can be changed in form as

(4-13)

From equation (4-13), sina111 and cose 111 can be

solved easily as follow:

sine 111 = ----~--~--~---=-------BK2 ± K1JKi + K~ - B2

K2 + K2 1 2

(4-14)

BKl + K2JKi + K~ - B2

K2 + K2 1 2

cosa 111 = ----~--~--~~-=------- (4-15)

cose 11 and sine11 can be obtained by substituting the

value of sine 111 , cose 111 , and equation (4-6) into

equations (4-5a), (4-5b).

27

) (4-16)

) ( 4-17)

c. Output Follower Angular Velocity,

For K. = Cc. + Rr. ~ ~ ~

where r. = A. r , therefore ~ ~m m

sine. cose. cos%o cose. sin.f4 cosei

r. = -cos eo sineo cos%o sin,0. sine sine 1 ~ 0 0

0 -sin%. cos%. 0

and the derivative of r 1 , r 2 , and r 3 with respect to

time are

• ( . . I cos90 cos%0 cose I) •I rl = -s~neo s~ne + e

• ( cose. sinei sineo cosfd.cosei) er (4-18) r2 = + 0

( -s in%o cose I) •I r3 = e

and Kl = Cc1 + Rr1

K2 = cc2 + Rr2 (4-19)

K3 = cc3 + Rr3

R, C, cl' c2 • and c3 are all constants, so the time

derivative of K. ~

(Eq. 4-19) is

28

• • K.= Rr. J. J.

• • K1 = Rr1 • • K? = Rr2 (4-20)

"--

• • K3 = Rr3

From equation (4-11)

K1 coseiii+ K2 sineiii= 1/2T (S2- T2- Ki - K~ - K~)

In this equation, T and S are constants, so the

time derivative is:

(4-21)

• • • • III • III e I I I = _-_l_/_T_(K......;l;;;;...K...;;l;;;..+_K_2_K....;.2_+K~JK::-:'!J::-)--_K..-1.._c_o_s-:e ==---K..;..;2_s_i_n_e --so.

where

-K1sin9III+ K2coseiii

[ sineiii == (BK2 ± K1 jKi + K~- B2 )/(Ki + K~)

coseiii (BK1 + K2 )Ki + K~ - B2 )/(Ki + K~)

(4-22)

D. ••III Output Angular Acceleration, 9 •

In order to find the output angular acceleration, •• •• we have to calculate the K. and r. by taking an addi-~ ~

• • tional derivative from Ki and ri. Therefore,

•• R"" K. = r. l ~

[ K1

•• = Rr1 •• .. K2 = Rr2 •• •• KJ = Rr 3

(4-23)

•• I I ••II I r 1=(-sine0 sine +cos90 cos.¢ocose )e +(-sin9.cose-

cose., cos%" sinei)(ei) 2

•• I d. I .. I I r 2=(cos90 sine +sin90 cos;u0 cose )e +(cose., cose -

sineo cos%o sinai) (ei )2

r 3=( -sin%., sine I )ei+( sin,0'0 sine I)( a I )2 (4-24)

The 9III can be obtained the same way as before by

taking another derivative to eii (Eq. 4-22) •

where

.. III •2 "2 •2 •• .. .,. III 9 = -1/T (K1+K2+KJ+K1K1+K2K2+K3K3 )+(K1cos9 +

K2sine 11I)(eiii) 2+(2K1sine 111-2K2cose 111 )

(eiii)-K1cose 111-R2sineiii /(-K1sineiii+

K2cose 11I) (4-25)

[ Kl = cc1 + Rr1

Kz = Cc2 + Rr2

KJ = cc3 + Rr3

29

• I I • I

[

~1= R(-sine. sine + cosa. cos,0'0 cosa ) a

K2:::: R(cose. sinai + sine. cos,0'. cosai) ai

KJ= R(-sin,ef. cosai) ai

K1 = R (-sine. sine I+ cos a. cos%. cos a I )a I+ R ( -s ina. cos a I­

cos90 cos,0'. sine I)( e I ) 2

K2== R(cosa. sinai+ sine. cos%. cosa 1 )ei+ R(cose. cosa 1-

s ina. cos,0'. sinaI)( a I) 2

K3== R(-sin,ef. sine 1 )ai+ (sin9f. sine 1 )(ai) 2

[ sine 111

cose 111 =

A specific example of this analysis has been

demonstrated in Appendix B.

30

CHAPTER V

CONCLUSION

The work in this paper is basically an extensional

analysis of the work of Ho (10) who developed the tensor

analysis of spatial mechanism.

Tensor notation offers a convenient and compact

means for expressing relationship in spatial mechanismo

but other mathematical methods such as matrices, vectors,

etc, still have merit in the analysis of spatial mechan­

ism problems. Futher, the tensor transformation used

in spatial mechanism have great advantages in relieving

the burdens of the tedious and confusing reference of

the coordinate frame.

This paper has shown that tensor notation combined

with matrix and vector methods provide a suitable

addition· to the existing methods of exploring the spatial

domain of linkage, and also lends itself well to pro­

gramming for computer solution of problems in spatial

kinematics.

In practical use, the R-S-S-R mechanism offers

merit over Hook's joints and bevel gears for the purpose

of indirect transmission, because the transmission

torque is not dependent on the rotating angle as Hook's

joints and is evenly distributed on the connecting

links instead of overloading a single gear tooth pair

of the bevel gears.

31

BIBLIOGRAPHY

(1) Altman, F. G.

a. "Uber Rambliche Sechsgliedrige Koppelgetriebe"

z. VDI., Vol. 96, March 11, 1954, pp. 245-249.

b. "Raumgetriebe", Feinwerktechnik, Vol. 60, No. 3,

1956, pp. 83-92.

( 2) Beyer, R.

a. "Kinematische Getriebesynthese .. , Springer-Verlag,

Berlin, Germany, 1953. pp. 189-191.

b. "Space Mechanisms", Transaction of the Fifth

Conference on Mechanisms, Purdue University,

w. Lafayette, Ind., 1958, pp. 141-163.

(3) Hain, K.

"Das Spektrum der Gelenkvierechs bei Veraender­

licher Gestel1-Laenge", Forch Ing.-Wes., Vol 30,

No. 2, 1964, pp. 33-42.

(4) Keler, M. L.

"Analysis and Synthesis of Spatial Crank Linkages

32

by means of Spatial Line Geometry and Dual Values",

Forsching auf dem Gebiete des Ingenieurwegens,

Vol. 25. 1959, PP• 26-32, 55-63.

(5) Kislisyn, s. G.

"Use of the Method of Tensors in the Theory of

Spatial Mechanisms", Trudii Sem. Theorii Mash.

Mekh., Vol. 14, 1954, pp. 51.

(6) Hartenberg, R. s., and Denavit, J.

a. "Approximate Synthesis of Spatial Linkage",

ASME paper A-24, 1959, No. 59. Kinematic Syn­

thesis of Linkages. McGraw-Hill Book Co,,

New York, 1964, pp. 343.

b. "A Kinematic Notation for Lower-Pair Mechanisms

Based on Matrices", ASME Trans., Journ, Appl.

Mech. 22, 215 (1955).

(7) Chase, M. A.

"Solutions to the Vector.Tetrahedron Equation",

ASME Trans., Journ. Eng. for Ind., 87, 228

(1963).

(8) Harrisberg, L.

"Mobility Analysis of Three-Dimensional Four­

Link Mechanisms", ASME Conference Paper 64-

WA/MD-16 (1965).

(9) Yang, A. T., and Feudenstein, F.

"Application of Dual-Number Quaternion Algebra

to the Analysis of Spatial Mechanisms", Journ.

of Applied Mechanics, Vol. 31, ASME Trans.,

Vol. 86, Series E, June, 1964, pp. 300-308.

33

(10) Ho, C. Y.

"Tensor Analysis of Spatial Mechanisms", IBM

Journ. of Research and Development, Vol. 10,

No. 3, May, 1966.

(11) Raven, F. H.

"Position, Velocity, and Acceleration Analysis

Synthesis of Plane and Spatial Mechanism",

Ph. D. Thesis, Connell University, pp. 59,

1956.

(12) Uicker, J. J., Jr., Denavit, J., and Hartenberg,

R. s.

"An Iterative Method for the Displacement Analysis

of Spatial Mechanism", Journ. of Applied Mechanics,

Vol. 31, Trans. ASME, Vol, 86, Series E. June,

1964, PP• 309-314.

(13) Skreiner, M.

"Acceleration Analysis of Spatial Linkage Using

Axodes and the Instantaneous Screw Axis", Journ,

of Eng. for Ind., ASME Trans., Vol, 89, Series

B. 1967, PP• 97-101.

(14) Greubler, M.

"Civilingenieur", Vol. 29, 1883, PP• 187 and

"Encyklopadie der Mathematischen Wissenschaften",

Vol. IV, 1901, PP• 127.

35

------------- APPENDICES

APPENDIX A

COMPUTER PROGRAM

36

NAMES OF FORTRAN VARIABLES

The following is a list of the important variables

of the program and what they represent.

1) AA's

2) BB's

3) CA's

4) CITO

5) CIT

6) CI2

7) CI)

8) CI4

9) DCIT

10) DCIJ

11) DDCIJ

components of the direction vector of

the driving link with respect to the

input coordinate system.

matrix of transformation for the input

coordinate system.

components of the direction vector of

the ground link.

azimuthal angle of the unit vector of

the input shaft, eo. I input angle, e , or azimuthal angle of

the driving link with respect to the

input coordinate system.

azimuthal angle of the second link,

9II.

azimuthal angle of the direction vector

of the driven link, aiii.

azimuthal angle of the ground link, e1v. angular velocity of the driving link, ·I a •

angular velocity of the output link, •III a • angular acceleration of the output link,

••III e •

37

12) DDCIT

13) GA's

14) PAO

15) PA2

16) PA4

17) c 18) R

19) s 20) T

21) DGA(I)

22) DDGA(I)

23) RA's

angular acceleration of the driving link ••I or the input link, a •

components of the direction vector of

K. • ~

polar angle of the input shaft, ~o•

polar angle of the second link, ~rr.

polar angle of the ground link, ~rv.

length of the ground link.

length of the input crank.

length of the second link.

length of the output link, or the

follower. • the first time derivative of GA, or Ki.

the second time derivative of GA, or •• Ki •

components of the direction vector of

the driving link with respect to the

ground coordinate system.

38

FORTRAN IV G LEVEL 1, MOD 2 MAIN DATE = 69092 19/15/55

0001

0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 00.34 0035

c C THE ANALYSIS OF R-S-S-R SPATIAL FOUR-BAR MECHANISM c

DIMENSION AA(3),BB(3,3),CA(3),GA(361), RA(36l),X(36l),CI3(36l), 1CI2(361),PA2(36l),DCI3(361),DDCI3(.361) C=l. R=0.2 S=1,6 T=1.23 DEG=3.141593/l80. PA0=60.*DEG CI0=60.*DEG PA4=30. *DEG CI4=75.*DEG CIT0=60.*DEG DCIT=1. DDCIT=O. SICO=SIN ( CIO) COCO=COS(CIO) SIPAO=SIN (PAO) CO PAO=COS ( PAO) BB(1,1)=SIN(CITO) BB(1,2)=-COS(CITO) BB(1,3)=0. BB(2,1)=COS(CITO)*COS(PAO) BB(2,2)=SIN(CITO)*COS(PAO) BB(2g3)=-SIN(PAO) BB(3,1)=COS(CITO)*SIN(PAO) BB(3,2)=SIN(CITO)*SIN(PAO) BB(3,3)=COS(PAO) CA(1)=SIN(PA4)*COS(CI4) CA(2)=SIN(PA4)*SIN(CI4) CA ( 3) =COS ('PA4) DO 3 1=1,361 X(I)=1.*(I-1) CIT=(I-1)*1.*DEG AA(1)=COS(CIT) AA ( 2 ) =SIN (CIT) AA(.3)=0o

w '-.()

0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 00.50 0051 00.52 0053 00.54 005.5 0056 0057 0058 00.59 oo6o 0061 0062 0063 0064 006.5 0066 0067 0068 0069 0070 0071 0072 0073

DO 2 J=1,3 SM=O. DO 1 K=1,3

1 SM=SM+AA(K)*BB(K,J) RA(J)=SM

2 GA(J)=C*CA(J)+R*RA(J) B=(S**2-T**2-GA(1)**2-GA(2)**2-GA(3)**2)/(2.*T) D=GA(1)**2+GA(2)**2 E=SQRT(D-B**2) SIC3=(B*GA(2)+GA(1)*E)/D CIC3=(B*GA(1)-GA(2)*E)/D F=SQRT(S**2-GA(3)**2) SIC2=-(GA(2)+T*SIC3)/F CIC2=-(GA(1)+T*CIC3)/F PA2(I)=ARCOS(-GA(3)/S)/DEG DGA1=R*(-SICO*AA(2)+COCO*COPAO*AA(l))*DCIT DGA2=R*(COCO*AA(2)+SICO*COPAO*AA(1))*DCIT DGA3=-R*(SIPAO*AA(1))*DCIT TN2=SIC2/CIC2 CI2(I)=ATAN(ABS(TN2))/DEG IF(CIC2) 11, 16~ 14

11 IF(SIC2) 13, 16, 12 12 CI2(!)=180,-CI2(I)

GO TO 16 13 CI2(I)=180,+CI2(I}

GO TO 16 . 14 IF(SIC2) 15, 16, 16 15 CI2(I)=360.-CI2(I)

RUTH·=ABS ( (B*GA ( 1 )-GA (2 )*E)/D) 16 CI3(I)=(ARCOS(RUTH))/DEG

IF(CIC3) 21, 26, 24 21 IF(SIC3) 23, 26, 22 22 CI3(I)=180.-CI3(I)

GO TO 26 23 CI3(I)=180,+CI3(I)

GO TO 26 24 IF(SIC3) 2.5, 26, 26 25 CI3(I)=360,-CI3(I) .{:::"

0

0074

0075 0076 0077 0078

0079 0080

0081 0082 0083 0084 0085 0086 0087 0088

26 DCI3(I)=(DGA1*CIC3+DGA2*SIC3+(GA(1)*DGA1+GA(2)*DGA2+GA(3)*DGAJ) 1/T)/(GA(1)*SICJ-GA(2)*CICJ)

DDGA1=DGA1*DDCIT/DCIT-R*(SICO*AA(1)+COCO*COPAO*AA(2))*DCIT**2 DDGA2=DGA2*DDCIT/DCIT+R*(COCO*AA(1)-SICO*COPAO*AA(2))*DCIT**2 DDGA3=DGA3DDCIT/DCIT+R*(SIPAO*AA(2))*DCIT**2 DDCI3(I)=(-(DGA1**2+DGA2**2+DGA3**2+GA(l)*DDGAl+GA(2)*DDGA2+GA(3)

1*DDGA3)/T+(GA(l)*CIC3+GA(2)*SICJ)*DCI3(I)**2+2.*(DGA1*SIC3-DGA2* 1CICJ)*DCI3(I)-DDGA1*CICJ-DDGA2*SICJ)/(-GA(l)*SIC3+GA(2)*CIC3)

3 WRITE(3,102) X(I),CI2(I),CI3(I),DCIJ(I),DDCI3(I),PA2(I) 101 FORMAT(5X,'INPUT ANGLE',4X,'CIT2',5X,'OUTPUT ANGLE',JX,'OUTPUT AN

lGVEL',4X,'OUTPUT ANGACC',4X,'PA2',////) 102 FORMAT (3X, F5,1,5(5X,F8.4))

CALL PPLT(X,CI2,361) CALL PPLT(X,PA2,36l) CALL PPLT(X,CI3,361) CALL PPLT(X,DCI3,36l) CALL PPLT(X,DDCI3,361) SWP END

.{::" 1-'

APPENDIX B

FIGURES OF SOME VARIABLES OF R-S-S-R

SPATIAL MECHANISM

For a specific R-S-S-R spatial four-bar mechanism

with a constant input angular velocity, ai=l.O rad/sec,

and the known properties are chosen as,

a) length of the ground link, C=l.o~.

b) length of the crank, R=0.2".

c) length of the second link, R=l.6".

d) length of the follower, T=l.2}".

e) azimuthal angle of the ground link, eiV=75°,

) 1 1 t . aiV o f po ar ang e of he ground l1nk, ? =30 • ..

g) azimuthal angle of the reference frame, 9 .. =60 •

h) polar angle of the reference frame, % .. =60".

The unknown quantities,

a) azimuthal angle of the second link, aii.

b) polar angle of the second link, ¢1I.

c) azimuthal angle of the follower, a11I.

d) azimuthal angular velocity of the follower, alii.

42

e) azimuthal angular acceleration of the follower, aiii,

can be determined by Eqs. (4-14), (4-15), (4-16), (4-22),

and (4-25).

These unknown quantities are plotted versus the

input angle, ai (from o" to 360" for each cycle), as

follows:

349 CONSTANT INPUT ANGULAR VELOCITY = 1,0 (rad/sec)

-~ ~

ffi ~ Q

JJ8 ...........

H H

(I)

-~ z ~ 328 Q z 0 0 ~ t:ll

~

~ 317 lit 0

f:il

~ ~ 307 ...:;~ <(

g:; :::> :;E H lSI <(

296 108 144 . INPUT ANGLE (DEGREE)

FIGURE 4. THE AZIMUTHAL ANGLE OF THE SECOND LINK (a~~) VERSUS INPUT ANGLE l9-J ~ \......) -

-~ ~

ffi ~ Q ..._,

H

~ • ~ z H ~

Q z 0 u ll:l Cll

~

~ ll:t 0

~ z <

~ &'!

CONSTANT INPUT ANGULAR VELOCITY = 1.0 (rad/sec) 131 ~

128

119

INPUT ANGLE (DEGREE) FIGURE 5. _THE POLAR ANGLE OF THE SECOND LINK (~II) VERSUS INPUT ANGLE (ei) ~

-«="

CONSTANT INPUT ANGULAR VELOCITY = 1,0 (rad/sec) 192r-----------------------------~~~------------------

......... ~ 181 t§ f:il Q ........

H H H

(t) 170 • ~ eJ

~ 8 ~ 159 8 ::> 0

f:il

~ 149

144 180 INPUT ANGLE (DEGREE)

FIGURE 6. THE OUTPUT ANGLE (alii) VERSUS INPUT ANGLE (e 1 ) .{:::"" \J\

0.395

.......... 0

~ 0.193 .......,_ 'd ro f...l -

H H ':!:n -0.00923

~ H 0

s ~-0.311

ll::

~ 5 ~ 8 -0.413 ~ E-1 :::> 0

CONSTANT INPUT ANGULAR VELOCITY = 1.0 (rad/sec)

-0.616 I I I I J I I 0 16 72 108 144 180 21

INPUT ANGLE (DEGREE) •• FIGURE 7. THE OUTPUT ANGULAR VELOCITY ( eiii) VERSUS INPUT ANGLE (ei) -{::"

0\

CONSTANT INPUT ANGULAR VELOCITY = 1.0 (rad/sec) o. 5441 ________________ __:._:._~-----::; ........ ;.:--"l

......... N

('.) Q)

~ "d 0.30 ro S... .._.

H H H :<t> .. z 0.07) 0 H 8 <I! ~ r:q H r:q 0 ~ -0.162

~ 8 ~ E-1 -0.398 fi! E-1 ::> 0

-0.633~----~----~----~~--~~--~~----~----~~--~----~----~ 0 16 72 108 144 180 216 252 288 324 360

INPUT ANGLE (DEGREE) FIGURE 8. THE-OUTPUT ANGULAR ACCELERATION (alii) VERSUS INPUT ANGLE (a 1 )

-t::--..J

VITA

David Perng Chyi was born in Shenyang, China, on

July 9, 1944o His high school education was received

at Taiwan, Republic of China. Later, he attended

National Taiwan University, and received the Bachelor

of Science in Mechanical Engineering in June 1966. He

served in the Chinese Army as a 2nd Lt. for one year

immediately after graduation from college.

In September, 1967, he enrolled in the graduate

school of the University of Missouri - Rolla, Rolla,

Missouri, to pursue the degree of Master of Science

in Mechanical Engineeringo

48

1S3~94