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Synthesis and analysis of spherical six-bar mechanisms

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Synthesis and analysis of spherical six-bar mechanisms
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Riddle, Dennis Lee, 1946-
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Axes of rotation ( jstor )
Cosine function ( jstor )
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Dyadic relations ( jstor )
Eggshells ( jstor )
Geometric centers ( jstor )
Kinematics ( jstor )
Kinetics ( jstor )
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Sine function ( jstor )

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SYNTHESIS AND ANALYSIS OF
SPHERICAL SIX-BAR MECHANISMS









By

DENNIS LEE RIDDLE












A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA 1975



























To Fay















ACKNOWLEDGEMENTS


The author would like to express his sincere

thanks for the leadership and guidance of Professor Delbert Tesar,.whose enthusiasm for kinematics has made this work a pleasurable experience. The author would also like to thank Dr. Joseph Duffy for the valuable introduction and support in spatial analysis. The combined leadership of these two mentors has presented the author with exceptional opportunities and challenges.

The author wishes to acknowledge the remainder of his supervisory committee: Drs. R. G. Selfridge, G. N. Sander, and E. K. Walsh for their personal contributions to his academic training.

He is thankful for the opportunity to have shared relevant experiences with all of the students of the machine design group. G. K. Matthew has freely given of his time at critical stages throughout his graduate program.

Finally, he would like to thank his wife, Fay, for her patience and sacrifices. He hopes that he may provide as much encouragement as she finishes her graduate program.



iii















TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .....................................

ABSTRACT ............................................. vi

CHAPTERS

I INTRODUCTION ................................. 1

II SYNTHESIS CONCEPTS ........................... 6

Constraint Systems ....................... 11
Solvable Problems ........................ 15
Synthesis Tools .......................... 19
Spherical Chain .......................... 32
Inversion ................................ 39
Curvature Transformation ................. 43

III DESIGN PROCEDURES ............................ 51

Path Synthesis ........................... 53
Colaminar Synthesis ...................... 54
Angular Coordination of Two-Link Chain ... 156 Angular Coordination of Two Cranks ....... 59 Path Coordination ......... 61
Angular Coordination of a
Chain ..... ** ... ............... 64
Angular Coordination of a Crank with a
Two-Link Chain ......................... 73
Angular Coordination of Three Cranks 77
The Angular Coordination of Three Adjacent Planes .......................... 77
Double Colaminar Synthesis with Common
Knee Joint ............................. 80
Colaminar Motion Coordinated with a
Crank .................................. 82
IV SPHERICAL DYAD ANALYSIS ...................... 85

Prismatic Equivalence .................... 92
Four-Link Chain Equivalence .............. 93


iv









Page


Dyad Collection ........................ 99
Spherical Six-Bars by Dyads ............ 105
Body Rotations ......................... 110
Relative Angles ........................ 113
System Inputs .......................... 116

V DISPLACEMENT ANALYSIS ...................... 119

Watt 1 ................................. 119
Watt 2 ................................. 126
Stephenson 1 ........................... 130
Stephenson 2 ........... ................ 135
Stephenson 3 ........................... 139
Closure ................................ 143
Comparison of Chapters II and III
Analysis Techniques .................. 144

VI APPLICATIONS ............................... 146

Three-Link Chain ....................... 146
TR Suspension ...... 153
Poultry Transfer Device ................. 160
Conclusion ............................. 166

APPENDIX: INTERACTIVE SYNTHESIS AND ANALYSIS
COMPUTER PROGRAMS ....................... 170

REFERENCES ......................................... 180

BIOGRAPHICAL SKETCH ................................ 183


















v











Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


SYNTHESIS AND ANALYSIS OF
SPHERICAL SIX-BAR MECHANISMS


By

Dennis Lee Riddle

August, 1975


Chairman: Delbert Tesar
Major Department: Mechanical Engineering


The first portion of this document deals with the synthesis of spherical mechanisms and is given with the formulation of the necessary tools. These tools are the curvature transformation and the inversion principle which allow closed-form design of multi-linkmechanisms for three, four, and five multiply separated positions. Twelve solvable problems are tabulated with an emphasis on the reduction of parameters for motion specification. The curvature transformation is extended to include geometric or time base motion problems. More than 20 detailed

design procedures are presented for the spherical four- five-, and six-link mechanisms.

The second portion of this document deals with

the analysis of spherical mechanisms and is presented in


vi










terms of two complementary formulations. The first provides for translational and angular evaluation of multilink mechanisms. This analysis is established by means of two-link chains known as dyads. The second treats the input-output equation algebraically. This format yields the evaluation of multiple closure branches simultaneously.

All of the synthesis and analysis tools.are shown to be reducible for use in planar mechanism design.



































vii
















CHAPTER I

INTRODUCTION


The field of kinematics is becoming a major contributor toward advances in industrial automation. Machines are required to simulate motions of human operations as well as motions beyond human capability. It is demanded that these functions be acc-urate, reliable, and efficient. Such demands have created growing interest in the field of machine design. Special spatial motion problems such as transfer mechanisms in automatic processing equipment, function generation between nonparallel axes, and path generation on a sphere cannot be solved by planar mechanisms. The relative importance of the spherical system can be obtained by considering that perhaps 75% of the industrial linkage problems can be satisfied by planar mechanisms, 15% by spherical mechanisms, and the remainder by more complex spatial mechanisms.

Coplanar synthesis, or design of planar linkages, has reached a high level of sophistication, usefulness, and completeness-a development which has occurred chiefly over the past 15 years. Analytical, graphical and special design procedures are now widely availa-ble to create four-





2




five- and six-bar mechanisms. Proper application of the four tools of synthesis (inversion, angular cognates, path

cognates, and curvature theory) allows the design of the geared five-bar, Stephenson six-bar, and Watt six-bar for a wide range of motion problems basic to industrial application.

Since the spherical linkage has properties which are distinct from those of coplanar linkages and represents perhaps 15% of potential industrial application, increased competence by the designer is warranted. Fortunately, much useful work on the spherical four-bar has 0 Ccured, although its general level is approximately that of coplanar synthesis 15 years ago. In particular, few design procedures have been evolved for linkages more com plex than the spherical four-bar, such as the spherical bevel geared five-bar, and equivalent six-bar linkages. The development of the spherical tools of synthesis would fulfill this need for the spherical system. A unified theory of spherical kinematics would not limit itself to the suggested 15% application. Rather it would provide common tools for both the plane and the sphere. Since planar geometry is a proper subset of spherical geometry, the concepts would cover 90% of application problems. Such a unified theory may prove to be the key to a total design package with the aid of dual numbers. Future efforts may extrapolate spherical kinematics to provide unified analytics of planar, spherical and spatial motions.





3




Hence, the objective of this document is to develop general techniques which would make the spherical system equally as accessible to the designer as the coplanar system. Every effort has been expended to insure that the results are immediately useful and valuable to the designer.

Generally the approach to spherical design has

been to extend well-established methods for synthesis in the plane to the sphere. For instance, J. Denavit and R. S. Hartenberg [1] generalized F. Freudenstein's approximate synthesis of planar four-bar linkages to generate functions by means of the spherical four-bar. Also, J. T.

Wilson [2], H. L. Johnson [3], and J. R. Zimmerman [4] obtained design equations for function generation using a rotation matrix.

C. H. Suh and C. W. Radcliffe [5] specified multiple positions on a path in the plane by a displacement matrix. Following this they extended the procedure to the sphere using a rotation matrix. These contributions emphasized the importance of matrix methods for synthesis introduced by J. Denavit and R. S. Hartenberg.

General theories for finitely and infinitesimally separated position motion of a rigid body in space have been developed by B. Roth [6, 7, 8, 9, 10] and G. N. Sandor [11]. These are major contributions in the field of synthesis of spatial motions. In [9, 10], P. Chen and





4



B. Roth developed a general theory which enables the synthesis of mixed displacements (i.e., finite and infinitesimal combined) of a rigid body in space. In [11] G. N. Sandor developed a general method for the synthesis of spatial mechanisms using quaternions.

The theory of coplanar synthesis has been studied exhaustively. The study and development of a corresponding theory for synthesis of spherical motion has been much more limited. G. Dittrich [12], H. J. Kamphuis [13], and K. E. Bisshopp [14] have studied instantaneous spherical kinematics.

The design of the planar geared five-bar for

function generation was treated by Oleksa and Tesar [15]. The extension of this philosophy to the spherical five-bar was this author's first introduction into spherical kinematics. The foundation of this extension was based upon the unified theory of spherical trigonometry by Duffy and Rooney [16]. This theory was generated from an understanding of the work by Todhunter and Leatham[17]. Dowler, Duffy, and Tesar [18-20] combined the theory of spherical trigonometry and the planar Burmester theory to yield the spherical curvature transformation. This tool gave the author the capability to extend the function generation problem to the sphere [21,22]. Duffy, Riddle, and Tesar [23] then established a displacement analysis technique for the spherical five-bar.




5



Eschenback and Tesar [24] expanded the synthesis concept of time dependent motion for the planar system. Mykiebust and Tesar [25,26] extended this effort to yield time dependent motion coefficients for the curvature transformation to establish a uniform approach to treat this new problem. A generalized computer program, MECSYN, was developed for the synthesis of multi-link, planar mechanisms with time state motion specification. A collection of design procedures [27] by Tesar for finitely separated motion specification has been useful for academic instruction. A major portion of the included work will be devoted to establishing similar design procedures for the spherical system.

Analysis provides an evaluation tool of the usefulness of a synthesized mechanism.. Gilmartin and Duffy [28] give a means to identify the type of a four-bar mechanism. Freudenstein and Primrose [20] have stud ied the motion of spherical and spatial five-bars for closure conditions. The most useful analysis tool for planar mechanisms has been established by Pollock and Tesar [30] for dyad-based machinery. This 'philosophy will be considered for the analysis of spherical motion in the second portion of this work.

It will be the object of this work to present

spherical synthesis and analysis tolls in a form usable to industrial designers.















CHAPTER II

SYNTHESIS CONCEPTS


The creative design of machine systems needs to

be based upon a solid foundation of analytical understanding. The technological needs of today have progressed beyond the trial and error evolution of past machine design practice. The development of sophisticated analytical formulations, coupled with the practical convenience of computers, has given the designers tools to accomplish their task with greater efficiency. The synthesis concepts included within are presented with enough detail so that they might be useful to all engineers interested in designing machines with spherical constraints.

It is essential that the designer recognize the

number of parameters {S} that must be specified to satisfy the problem requirements. If the designer selects a mechanism with too few available parameters, {q}, he will be severely limited in the quality of results. On the other hand, if the designer selects a mechanism with too many available parameters, the range of solutions will be too vast for a meaningful evaluation of possible solutions.




6





7



A basic criterion of the design process should be that the available parameters {q} exceed or equal the specified parameters {S} or


{q} > {S}(-i


The synthesis problem involves the satisfaction of a number {F1 of conditional functions in a lim-ited number (k. + 1) of system positions. Suppose the system is to satisfy the condition f(x,y,z) = 0 by the approximating function g(x,y,z) = 0 as in Fig. II-1. The function f(x,y,z) may be a general spatial curve or it may be specialized to spherical motion by the constraint


X+ y2 + Z 2 = constant (11-2)


The function may be further specialized to planar motion by restricting one of the parameters (x,y,z) to be a constant.

Closed form synthesis techniques allow a maximum of five design positions. This does not restrict the actual number of intersections of the two curves, but rather the number of positions free to the designer's specification. These positions may be either finitely or infinitesimally separated. Infinitesimally separated positions yield the ability to match slope, curvature, etc. up to the fourth order. After the choice of design positions, the designer must consider whether it is necessary for





8












f (X, Y, Z) 9(xlylz) Goo










Figure II-I





9




g(x'y'z) to be spatial, spherical or planar to meet the specified design positions. It should be noted that spatial mechanisms include spherical and planar mechanisms as a proper subset.* Planar mechanisms are also a subset of spherical mechanisms. With this in mind, the designer could always choose spatial specifications to satisfy the problem function f(x.,y,z). Spatial mechanisms have many more parameters to be considered and are often unnecessarily complex. In addition, there are no well-established synthesis techniques available for general spatial mechanisms. References include spatial analysis routines that may be considered for trial and error design, but this is recommended only as a last resort. If f(x,y,z) is a general spatial function, it would be most desirable to approximate it with a spherical function g(x,y,z). Spherical mechanisms add little complexity to the analytics, yet allow a type of spatial motion. It will also be seen that spherical and planar mechanisms may share the same body of analytics and add no extra effort in use. This work will continue to emphasize spherical synthesis but

the same philosophy and reduceable analytics follow directly



*Spatial mechanisms have no restrictions on the
rotation axes. Spherical mechanisms have all rotation axes intersecting at common point, the origin of the sphere. Planar mechanisms have all rotation axes parallel, or intersecting at infinity.





10




to the plane. We can choose only a limited number of design positions (x,, y, z k, = 0,1,2, ... to represent f(x,y,z). Should more than one function (F = 1,2,3,

.) need to be satisfied simultaneously, the total number of specified parameters would, be S = F. (11-3)

Note that this result does not include the Z = 0 position. Mechanism constraints can satisfy displacement, but normally not absolute,values. This in no way weakens the synthesis approximation, since any functional reference may be chosen.

Ideally, the synthesis process will not utilize all the system parameters fq} to satisfy the specifications {S}. The remaining parameters


p, = q Fk (11-4)


can be used to meet optimization criteria not included in the specifications. A synthesized mechanism will have zero error at the k + 1 design positions, but no other characteristics are guaranteed. There are over a dozen conditions to be met after synthesis. All of these can be evaluated through present analysis procedures. The designer could be left with thousands of these evaluations, a number much too great to treat realistically. Optimization is the answer, and it is now coming to the surface for planar mechanisms. Concise'and efficient





11



synthesis and analysis procedures are the necessary keys to this type of undertaking. The included work should make a major contribution in this arena for spherical mechanisms. The number of optimization parameters {pJ should be in the range I < p < 3 for good design.


Constraint Systems


The designer has a wide choice of mechanisms to meet the requirements of a given problem. The control system contains a few parameters, usually one or more of which is completely adjustable. The cam system contains an infinite collection of nonadjustable parameters. A linkage is neither adjustablenor does it contain an infinite collection of parameters. The designer must know whether the linkage contains enough parameters {qJ to satisfy his problem. This work will build a family of solutions based upon the four-bar, the geared five-bar, and the six-bar mechanisms. As in the plane, it is felt that these are the building blocks of linkage design and a very large percentage of solutions can be derived from their motions.

The spherical four-bar mechanism (Fig. 11-2) is

the most elementary device of this study and contains the fewest system parameters fqT. If we were to ask the question, "What is the fewest number of parameters necessary





12 x































Figure 11-2





13



to specify the geometric position of this device?", the answer could be given in any number of'terms. The most immediate answer is derived. by considering the location of each pin joint ABCD. If the condition of a spherical mechanism is satisfied, then the axes of each pin joint passes through the origin 0. The arc between any two pin joints is a great circle segment. If we arbitrarily select a unit sphere for convenience, such that


z 2 -(X 2 + Y2 (11-5)

only two parameters would be necessary to describe the position of each pin joint. A-total. of q = 8 parameters is necessary to complete this description. The special case of a prismatic slider can be simulated on the sphere by a 9 00 crank. If, in Fig. 11-2, arc RD is 90', then point C will move on a great circle path. This i s the equivalent of linear motion on the plane-.

The spherical five-bar mechanism (Fig. II-3a) has two degrees of freedom.' Any.two links of the chain can be driven independently to constrain the motion. Alternatively, gear trains can be introduced between two nonadjacent links to yield a one degree of freedom device. Motions derived from the five-bar mechanism family are inherently more complex than those of the four-bar mechanisms. It is expected that a geared five-bar mechanism





14





Bc (A(a)















(d FIEBR 3(i Dik3 FiueB-





15



will solve some problems that a four-bar would not. Note

that there are five pin joints in the five-bar, so that 10 parameters are necessary to describe its location. In order to predict the position of all links relative to a new input position, two additional parameters must be considered. The gear-tra in values M and N (Fig. II-3bcd) are necessary to describe the input dependence of the coupled links. Therefore, a total of q = 12 parameters are necessary for the geared five-bar mechanisms.

The Watt six-bar mechanism (Fig. 11-4) has seven pin joints so that q = 14 parameters is recess ary to describe its position. The Watt 1 and Watt 2 mechanisms are distinct inversions of the general Watt chain that are found by fixing links 1 and 2 respectively. The reader should recognize that the motion in link 5, for example,. is more complex in the Watt 1 than that in the Watt 2. It is true that the motionsof six-bars with two fixed pivots are more complex than those with three fixed pivots.

The Stephenson six-bar mechanism (Fig. 11-5) also requires q = 14 parameters. It has three distinct inversions found by fixing links 1, 2, and 3.


Solvable Problems


With a view of the components in

p = q FX (11-6)





16











F

5
6

G

















,Figure 11-4





17










B

5

c


H
G
















Figure 11-5





18



we are now ready to consider the range of solvable problems and the selection of the proper mechanism for each type. This range of problems can be most efficiently presented by the use of tables with consistent nomenclature.


Let


2. =0,12, .. -the multiply separated 'position counter
p. the number of parameters left for optimization (p2. = q F k ) q -total available parameters in the
constraint system
F number of functions to be specified
simul taneously

a~b~c, ... pin connectors in the solvable
problem statement

A,BqC, ...- pin connectors in the linkage system

m = ...- counter for links in the. solvable
problem statement

m = 1,2,3, counter for links in the linkage
system

e point in the problem tracing a desired point path with curvilinear coordinates (U ,V e)

E(u,v) moving shell

E(U,V) reference shell

IP_ absolute angular motion parameter
m in the solvable problem statement

E_ relative angular motion parameter
m in the solvable problem statement





19


ml M2 ]M3 available system links to repre[in1,sent links in the problem statement

P pin joint available to represent
a general path point e

The tables on the following pages are intended as design sheets for each of the specified problems to assist the designer in the first stages of the design process. The tables are oriented on the base of finitely separated position (FSP) problems but are extrapolated to include guides for the use of multiply separated positions (MSP) and time-state synthesis (TSS). The distinctions of FSP, MSP, and 155 will be made clear in the curvature transformation disc us s ion .


Synthesis Tools


Planar kinematics have four well-established tools of synthesis.

1. path cognates
2. angular cognates
3. inversion
4. curvature transformation

These are fully discussed in the literature [27] for planar

mechanisms but have major distinctions for spherical synthesis. Both the path and angular cognates,that are so helpful in planar design,are based upon properties of parallelograms. Unfortunately, there are no parallel great circles on the sphere, and these two tools must be excluded from the beginning.








20






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(-3 u x 4-3 4-3 (U (D
C> (a ro S-S- :m 3: Q. CL
v I w
cr 4J









(U 4If
.0 .0 11
P't en
E
E a)
4-3 v;;
cu r- K\J
4-3
4J 4-- cli
u E >
c It S=3 IM o
.0 LL. kAp LL0 ic\j
S
ai C\I
.- If
4- U- E

(U 10
a- S"
Cn Col a








29
















V)

4J
r_ 0
.r- 4-3
0


(U


z 0

Ln


R 41 CD
Q fo c -fu 50
cu r-4
4J Ck.
a) (U E LO
Nd C U: L: C C L;



> <
4J

> C) cli
Ln 11 W4- 11 I-gr
&- Cl. In.
E CIQ

r_ co m 0 0
E (A 0

C17 x 4--) (U w w
0 2: U- V) nc 0 Cl. Clci a) w
4-3 4J 4-3
(U V) V) (n



0 0
C3 11

>
K\J =T

4

4j a)

ro
Ln
4-) a tc\j 4)
(a 0 G DD
4..) (D
4-)
u E >
c It OCIJ
=3 a) --=:-I 0
LL- LL. Icli
4
0
S- M
0- u
4- LLCL
V)








30







V)














C.D

M co
o
U 4) (Ll 11
m c cm
cn lz CL


41

Cj
u ce)
W If In I I
V M: I.-J .
(D r4r--i r-oir-l LO m Ln r-I
4J w m I'd, In Ln ko o LO
ro vt .
Lf) Lr) cv) C j In
r" L-.J- S- S- L-JL-j L--i
4A oo 0 00 it- 0
(V E r-l &- r-n
r M In zl* LO LC) t.0 LO C\J Lf) 0 Ul
0 > 4 C\7
0 P C*%j C\j m m 4 C4 C4 C4 cl:
L-i "t L.-it

cli
0 C)

ca. CIL
0 E 04 m
S
m Sco 11 m m 11 0 0 0
to M 1 -.1 co -j Aft (A tA
a) I -P 4-3 Q c
> x 4-) 4J (U a)
C\j .- 1:4- to to = 4-)
E LL- Ln 3: CL ca. CL
cr 4J 4J 4-)
0 cn (f) V)

0


C\j >
E

r
4
F
Qj
4-)
>
0
E tA cl- op
a) C K\J
4-) 0-9- U)
tv C;; >
4-) 4-) '+u E 11
C it I" S- C\j
E = (v -D- o
U- U4- IC\j
KsJ
0 (V Cl) It 11
S- .- 11 Im u
Cl. 4- U- E


CL
441 31







31






(In













4j


in 0
C\j C) Li- I S- u
0 o
w it
CL c:
Ln N4 o0


in
44
Ln fn *-% in
in 0i to r
Cli Cli


(a in Cj in Oct LO 4D CV) in L. S4-) E A
fri : m kn cn Lf) in 00
4u L'i Ict A .
> E C\j m cli in CY)



0 C)
11.4
CL CL
E Cj m
4-) (A S(a S- 4r; C: a c
ca 11 Oll CY) tu 11 C\j 0 0 0
co -j V) in
1 4-) 4-) C: a
4-3 > x 4-) 4-3 4a) 42) (L)
qt to (a -a _rLL- (A) V- r-L CL CL
0 11 a) 0) 4a)
Cr cr 4-) 4-3 4-)
CL V) in (A





0
KNI

kv
to cn
E fro
I C ; C
4-)


4-)
0 --94-3 4-3 4- 0
V) u Qc 11 -1cli E
E =3 (D 0
LL- LL- icy)
4

0 cu cn 11
S- 11 #M u C\j
4


CL
V)





32



Spherical Chain


To appreciate the effect of colaminar inversion we can first study the spherical three-link chain (Fig.
r
11-6). The motion specification set {c,a,y}m for link m in position Z with reference r is a set of Euler rotations about the X,y',z" axes (Fig. 11-7), respectively. From the sine, sine-cosine and cosine laws for the spherical triangle abb', we write


sin 62 = sin ab sin e1



sin a2 cos B2 = sin ab cos eI (11-7)


Cos c2 cos $2 = cos ab



Solving for a2 from Eq. (11-7),

=dk -l
22 dTk [sin (sin b sin el)]T (11-8)



where 2 MSP counter

k ISP counter
T T = _k


T- any geometric or time parameter






33



















3, 0



Ao3


.82
e l --- I
/fll















Fiur 11-61





34














x I x1l
x

















zlzfo -Yo -Yff

a
zo












Figure 11-7





35




The result of Eq. (11-8) for k = 0 will yield a 2 in the first or fourth quadrants, This is sufficient as long as a2 is not restricted to principal quadrants. From Eq.
(11-7), a2 can be found in its proper quadrant by the implicit expressions for sin a 2 and cos a 2. The ratio of the last two expressions in Eq. (11-7) yields





d2 ck ~tan-l 1 sin abCos 6 1 (11-9)




The two parameters a2 and 2 are sufficient to describe the location and higher order properties of point b. The third Euler angle rotation can be seen (Fig. 11-6) to be the sum




=~ dy [(62+P2 )] T (11-10)



The angle 0 2 is assumed to be given as a function of 01 with its appropriate derivatives. The angle P2 can be represented as a function of e01 through the cosine law for the polar triangle abb'

0 = sin P2 cos el cos P2 sin el cos ab





36



or


d k tan- (tan e cos 'ab) (11-12)



Eq. (11-12) can be differentiated to give the higher order properties of Eq. (II-10).

Writing the sine, sine-cosine, and cosine laws for spherical triangle acc*,


sin a3 = c2 Cos 8I y2 sin 01



sin a 3 cos 133 = -(X2 sin 0I + 2 cos e1) (11-13)


cos a 3 cos 03 = Z2 where X2 = sin bc sin 02


Y= -(sin a cos b + cos Ob sin bc cos e2)


2 = cos ab cos bc sin ab sin bc cos 82 Eq. (11-13) can be solved to find





37



dk Cos
3 dTk [sin (X2 cos e 2 s 1 T

(11-14)


Ct dk a -X2 sin Y 2 Cos 6
d -1 2 1 2
d3 k tan
dT [Z2
T



This can be extrapolated to a general n-link chain by the following expressions


dk -I
8n2 dk [sin (Xn cos 6 + Yn nsin 0)]
nt k n-1 ... 32 1 n-1...32 1 T


(11-15)
-Xsi 7 cs

dk Xn- 1..32 sin 1 Yn- 1...32 Cos 1
n tan -Z 1
dr k Zn-1...32 T



The final rotation angle y3 about point c can be expressed as the sum


k R 3 + (11-16)
kY3 d 3 033 T


as depicted in Fig. II-6. For the spherical right triangle acc', the interior angle 63 can be expressed as





38



tan B3
tan = (11-17)


From the cosine law for the same triangle,


Cos ac = Cos a3 Cos B3 (11-18)


Similar to Eq. (11-12), P3 can be expressed implicitly by


tan p3 = tan 4'3 cos ac (11-19)


Substituting Eqs. (11-17, 18) into Eq. (11-19), we find



= d [tan- (sin B3 cot 3)]T (11-20)
P3 Z d Tk33T


Consider the spherical triangle abc; write the sine and sine-cosine laws


2 = sin ac sin a3

(11-21)
Y2 = -sin ac cos a3


where X2 = sin ab sin 82


Y2 = -(sin bc cos 91b + cos R sin ab cos 82)





39



Ratioing Eq. (11-21),


w31 dk tan- T (II-22)
-i Y


Equation (II-16) can be extrapolated for a n-link chain

dk
dk -T n + + pn]T (11-23)
Yn dk [ n T ~


0(k)
where 3k) is specified and



SX1
dk tan -1 23...ndT L 23.. .n-1 T

k

dk -1
P = -- [tanI (sin n cos )]T
dT n T


The computation of {a,8,y}r and {a,,y}3 for the three link chain has been validated by the APL function LINK3 as listed in the Appendix.
Inversion

The motion specification set {a,,ym r for the tI'ree-liflk chain was established with reference to link r. We will see in the design procedures that it is desirable to find the motion specification set {a,a,y}n MR,






40




with respect to another reference link n. The question might be posed, "What is the motion specification set
1 of link 3 with 1 fixed as reference?" To answer this question, consider the transformation between the moving reference x,y,z and the fixed reference X,Y,Z (Fig. 11-7)




Y = M r Y (11-24)




where Mr is the product of successive rotation matrices of
m
Euler angles ar r r about X,Y,Z respectively, i.e.,
m m m



(a)r ( r (a )r
11llm 1a2)m 13 m


Mr (a r ( r (a )r (11-25)
m 21m a22)m 23 (

( r ( r (a )r
31)m 32)m 33 m




where



(allm = cos Bm cos r


(a)r = cos cr sin yr +" sin ar sin Br cos Yr
21 m m m in in i





41



)r r r r r r
(a = sin a sin y m- cos a sin r cos y
(a31m sin mm


r =r r
(a )l2m = -cos 8 sin y
12 m m m

r= r r r r
(a ) = cos cos ym sin a sin sin y
22 m m mm m m


(a r = sin a cos yr + cos r sin sin r
32 m m m m m inm


(a)r = sin r
13 m m


(a-sin Cos r
23 m m m


(a r = Cos Cos r
a33)m m co m


The transformation Mr is orthogonal such that


m = (Mr)T = (Mr)-l (II-26)
r m m


The inverse transformation gives us a means to express
m
the inverse specification set {a,8,y} with the original
reference link r moving and link m fixed as a new reference. Equating terms from Eq. (II-26),





42




m ( )r
1a3 r a31 m


r

a 23a 32(11-27)
a33l r a3/m



r
a,, a 21
al11 r all m



which reduce to


m sin-1 (sin a sin yr cos ar sin r cos ymr r si (snmf m si r)



sin r Cos Yr cos a r sin m cos Y
r= tan-I m m (11-28)
r -Cos rm Cos Bmr






SCos a r sin Yr + sin am sin Br cos mrm
r -Cs r r I
-o m co m




It is also valuable to have the set {a,8,y}n when

shell n is fixed. The set can be obtained by equating components of the matrices





43




Mn = Mn Mr (11-29)
m r m


to obtain



n sn (a n (a) + (a)n (a) + (a) (a)r
$M = sin 11lr 13m 12)r 23)m 13)r 33m]

(a1 n (a_ r +(a n (a r +(a n (a

m:tan-I 1 r 13 a22)r a23)m (23)r (33)m (11-30)
S(a l)n (a r + (a )n (an )r + (a )n (a )r

1 ir 3)m a32)r 23+ (a33r 33)m



(n an- )n ( a2)r + (aI n (a )r + ( n ( r
n 1)r 12 + 2 r (22)m 13 (a 32)m
Ym(all)n (ar +( n ( r ( n (a )r
t1 (a1lm + (al2)r (a21)m (al3)r (a3 1





This transformation allows the reference link

to be temporarily changed so that the appropriate motion specification set {Iay}n can be fed to the curvature transformation.

Curvature Transformation

A detailed description of the 3, 4 and 5 multiply separated position (MSP) transformation is given in references [19, 20]. In any case, the question, "What is the





44



locus of points (x,y,z) in the moving shell E that have three positions on circles about center (Q2 Ql QO) in the reference shell E?" is common and becomes the central question of synthesis. The well-known constraint equation of the plane becomes on the sphere


G(X,Y,Z) : Q2 F + Q1 EY + QO D, = 0 (11-31)


where D. = -G2Y x + GI y- G02 z



E.Z =G 8x + G zG5t (11-32)


FP = G3U x G4Z y + G6Y z



The G m coefficients of Eq. (11-26) are given by


G O 1 cos ctCos


G = sin clcos Y,+ cos asin ksin y. (11-33)


G2 U = cos c sin ,cos y sin a.sin yz



*Note that in the previous references AmZ's were used as Gmk's. The later notation was adopted to distinguish the spherical coefficients from the planar coefficients.





45



G3Z = Cos k Cos Y- 1


G4k = cos aksin y,


G5X = sin tcos at


G6k = sin


G7Z = cos atcos y.- sin atsin .sin y.- 1


= cos sin Y,+ sin asin gcos y,




Equation (11-33) is listed for finitely separated

positions (FSP) only and must be differentiated with respect to the independent parameter y to reflect the role of infinitesimally separated positions (ISP). The Gm notation allows the same body of analytics (and computational tools) to solve Eq. (11-31) for all cases of MSP.

There are four distinctions between the curvature transformation program listed in the Appendix and that covered in reference [19]. First, the original work was centered upon the special reference transformation. This is an effort to reduce the number of parameters for special cases and required that {aa,y} =0 = {0,0,0}. The general reference transformation allows the arbitrary choice of





46



the initial position parameters, and significantly reduces the amount of work required by the program.

Second, the homogeneous cubic cone



a14 z 3 + (a24 zI + a34) z + (a44 z2 + a54 zI + a64) z2


+ (a z 3 + a a z+ a z+ a04)= (11-34)
7 a4 z1 841 9 1 10



is the locus of homogeneous circle points (zl,Z2) in terms of known coefficents a.. [19]. If (x,y) is a known point 13
on the curve such as P019 then Eq. (11-34) can be reduced to a quadratic. Let


z1 = s + x z2 = t + y (11-35)


and substitute into Eq. (11-34). This substitution yields a binomial in s,t whose constant terms collect to zero. Dropping the j=4 subscript for brevity, aIt3 + a2st2 + a4s2t + a7s3 + a 3t2y + a 2(2sty + xt2) + a4(2stx + ys2)


+ a73s2x + a3t2 + a5st + a8s2 + aIty 2 + a2(sy2 + 2txy)


+ a4(tx2 + 2sxy) + a73x 2s + a32ty + a5(sy + tx) + a82sx + a6t

+ a9s = 0 (11-36)





47



Reduce to a quadratic in polar form by the substitutions s = r cos e and t = r sin 0. Collect terms in the form


Ar2 + Br + C = 0 (11-37)


where

A = a'1sin30 + a2sin 20 cos 0 + a 4sin 0 cos 2 + a7 cos3 0


B = sin 2(3aly I a2x + a3) + sin 0 cos 6(2a2y + 2a4x +.a5) + cos20(3a7x + a4y + a8)


C = sin e(3aiy2 +2a2xy + a4x2 + 2a3y + a5x + a6)

+ cos 6(3a7x2 + 2a4xy + a3y2 + 2a8x + a5y + a9) and 0 < 0 < T.


Eq. (11-37) can now be solved for ri and substituted into the homogeneous coordinates


z = ri cos 8i + x; z2 = ri sin 0.i + y (11-38)


The G m coefficients (Eq. 11-33) are functions of y alone since


a f(y) and B : f(y) (11-39)





48




The higher order properties of the coefficients are found by differentiating with respect to y. Now suppose the motion of a system is to be coordinated with a crank parameter 4. This coordination involves three functional constrai nts



ax=f~) f(Op) Y f(f) (11-40)


For ISP it is necessary that the motion be specified in terms of higher order properties of these functions with respect to P.A new class of problems evolves when c is given as a general time function


=f('r)

so that


a f(T), =f(T), y = f(T) (11-41)


Now Eq. (11-34) can represent the motion specification set in coordination with a general input.. The problem is

to specify the time states of a shell, and then synthesize a system to meet those specifications. This is not a normal dynamic response question, since there is no preexisting mechanism responding to an input; rather the constraints of the mechanism are sought from the expanded curvature transformation. This task is achieved by





49



differentiating the G 's (Eq. 11-33) with respect to time in terms of &, A, j, a, 8, etc. To avoid the tedious task of differentiating Eq. (II-33)with respect to time, we can take advantage of the chain rule and the geometric derivations of Eq.(II-33)as shown below:





dk
nk d ) y f(T)
dT


nO E:f(y)


nIi E Y

n2= E 2 + :


n3 = ey + 3e" 7 +


14 = E'"'i 4 + 6E,'" 2 + 3E 2+ 4E + E"y


The prime superscript on c denotes derivatives with respect to y. If we allow E to represent Gmt (y), a(y), or a(y), then n represents Tm (T), U(T), or a(T),respectively. If the independent parameter is y as in the original synthesis problem, then





50




dy + 1
dT dy
(11-42)

dydky= 0 {kJlk > l}
dT k dy k


One computational tool can then be used for either geometry or time dependent problems.

The tables of solvable problems include a column representing MSP (multiply separated position) synthesis

and are representative of the possibility of higher order geometric specification. This is in reference to well-known structural curvature theory. It has a major weakness of not being able to coordinate nonadjacent planes for higher order properties. This class of problems is solvable only by Time State Synthesis (TSS).

The fourth distinction mentioned in the section labeled Synthesis Tools transforms the existing FORTRAN program to a very efficient interactive APL function. This feature greatly enhances the designer's ability to stay in control of the design process and leads to an earlier solution.















CHAPTER III

DESIGN PROCEDURES


The objective of these design procedures is to

present how the spherical four-, five-, and six-bar mechanisms can be determined dimensionally by use of the synthesis tools of Chapter II. There are a few hundred potential design procedures based on the problem charts in Chapter II. This presentation will not be all inclusive but will present at least one solution for each problem type. A similar effort by Prof. D. Tesar for planar mechanisms has proven to be useful in both undergraduate and graduate study as well as in applied industrial practice. The consistent and compact nomenclature allows the designer to approach complex problems with fundamental tools.

Optimization has become the central question of planar kinematics. It will also become the key problem to spherical kinematics as well as all fields of machine design. Optimization must apply weighting factors of design characteristics to a large collection of solutions. The size of this collection can be realized by considering how many sets of motion specifications {aayJ are used in the curvature transformations and how many Z + 1 positions


51




52



are specified. If three positions of the moving plane E are specified, the circle point locus (locus of moving pin joints) is the entire moving shell. This shell can be uniquely represented by a pair of homogeneous coordinates. Each coordinate has a single infinity of choice. Each point in the homogeneous plane maps one-to-one to a center point in the reference shell E. Each pair of circle points and center points construct rigid cranks. For the three-position problem there are a total of -2 cranks. The locus of homogeneous circle points for four positions of E is a cubic curve (Eq. 11-34). There is a single infinity of points on this curve and hence there is a total of cranks. The five-position circle point locus is found by the intersection of two cubics. There is a total of 6 cranks for the five-position problem.

The designer is expected to have started by matching his problem with the problem charts of Chapter II. After selecting an appropriate mechanism based on the number of free optimization parameters pv the nomenclature of the chosen chain should be referenced. The procedures then outline a step by step method of solution.

All of the procedures will be produced for zero order (displacement) positions. Higher order properties are obtainable for each case by differentiating with respect to the generic independent variable T, i.e.,





53




{a Y nk = dk [ 1 (Ill-i




where T can represent y~,or t. The required differentiation is taken automatically in the accompanying interactive computer program. This statement is valid for geometric states or time states. It will be seen that all problems are applicable to the time state, but others may not be specified in the higher order geometric state. The objective of these design procedures is that they be accessible to the designer without any further information than that. given in this document.


1. Path Synthesis


For path synthesis there is only one.(F=l) functional constraint provided by the given path. The motion of a point on a path is independent of a rotation parameter y. The standard curvature transformation requires the set {c,a,y}. The problem statement gives us



a k = Ue O2. = Ve Yk= arbitrary (111-2)


This problem can be treated by the colaminar formulation listed in procedure B by taking y as an arbitrary set of values.





54



2. Colaminar Synthesis


This problem is the most fundamental and the most widely used. Many of the complex design procedures will reference this problem but in no way detract from the significance of its own value.

Some point e of the moving system E(u,v) moves on a path f(U,V)=Q in positions e .with position coordinates (U eV sV e) (Fig. 111-la). The transformation between the rectangular coordinates (X,Y,Z) and the curvilinear coordinates (U,V) is given by


X = si n V


Y = -sin U cos V (111-3)*


Z = Cos U Cos V

The motion of point e along the path is simultaneously coordinated with the angular rotation parameter

The s et {U eY,1 V ek' iP2,} is a set of Euler angles consistent with the motion specification set. Hence the motion specification set**is obtained directly by the relations:



*Note that in the plane Eq. (111-3) reduces to
X=U, Y=V, Z=l. This agrees with the previous planar concepts. Equation (111-3) is solved for the reference system, but is also valid for the moving system.

**All of the illustrations in this chapter use the line codes: solid link line-specified link; long dashspecified shell motion; short dash-constraint found from curvature transformation.





55




x v

E U
E
ee


V Vvi


y
Uei


(a)





v



e
4






b)



Figure III-1





56




L Ue V ~ (111-4)


It is now possible to outline the procedures for the basic mechanisms.


2.1 The Four-Bar


This problem depends directly upon the curvature transformation to determine the dimensional constraints AB and CD (Fig. III-lb).

1. Apply the motion specification set {Uet'
Ve2,s ip Z}l to the curvature transformation.
2. Select pin joints B and C from the tabulated
circle points.

3. Alternately to step 2, select pivots A and D
from the center point tabulation.

4. Selection of either a center point or a circle
point determines the other. Points B and C
are both in the rigid moving shell and can be joined by a rigid link of arc length BC.
It is identical to make the rigid link 3600BC. The pivots A and D are both points of the rigid reference E and provide the base
link of the four-bar ABOD.


3. Angular Coordination of Two-Link Chain


The system of two links, movingg relative to

link T (Fig. III-2a) is completely known if the following data are given


a b, ' O ,1,2,... (111-5)





57











b
















U '004,
0001, 1
14
B

D
0


(b)





Figure 111-2





58



The curvilinear coordinates for the state of b in Z are given by Eqs. (11-8 and 9), where 01 =




dk -1 si ab sin
a2 dT k cos 2 T






dk sin 1 (sin Fh sin ip
U2 d'rk T




The rotation of link about pivot b completes the motion specification of link .


dk + P2




where P2X is given by Eq. (11-12) as



d d tan 1 (tan cos ab) T



3.1 The Four-Bar

1. The four-bar can be used if the symbolic
transfer (Fig. III-2b)

aoA, bcB, us, 2-2, 313 occurs.





59



2. The curvature transformation can be used
to determine crank _CD.


4. Angular Coordination of Two Cranks

The problem of "function generation" deals with synchronizing the angular motion of two cranks rotating about fixed pivots a and b (Fig. III-3a). All of the mechanisms under study are capable of this performance. We shall limit the procedures to the four-bar and five-bar 1. It is felt that these have enough versatility to accomplish most angular motion problems of this type.


4.1 The Four-Bar by Inversion


1. In Fig. 111-2 we have the motion of 72 and -1
given relative to T. If we now invert the current statement by fixing Z and allowing
1 to move we make the symbolic transfer (Fig.
III-3b).
a-*A b-pd
To.2 T-*l, 3-4

2. The motion specification for the two-link
chain (T and 3) is given by



3. From the curvature transformation we find
circle point C and center point B to form
the rigid constraint of link 3.

4. Invert the mechanism back to the original
system by fixing link 1.

Note that the initial positions of 2and iare not met by arcs A-B and CUb. However, the finite displacement values Aip2 and A5are met by moving shells 2 and 4.





60































3 0 c








7r






Figure 111-3





61



4.2 The Five-Bar 1 by Inversion


1. The required symbolic transfer is

a+A, b-F, c-*D 2+2, 3+5

as in Fig. 111-4.
2. The overall gear train value from A through
F to D is Q=MN. Links 2 and 4 are coupled by this train. The influence of 01 and 02
on 03 can be seen by

(e3 e3o) = Q(elk e10) + N(e2Z 020) (111-6) We have no control on 030 SO that we may arbitrarily set it to zero. Eq. (11-6)then
yields 32.

3. The motion specification for link 4 becomes

{aa,y} 2+ Eqs. (11-14,16)

4. The curvature transformation gives pin joint
C as the circle point locus and point B as t_he center point locus. The pair form arc
BC and provide the needed constraint for the
five-bar 1.
5. Invert the mechanism by fixing link 1.


5. Path Coordination

The four-bar has enough parameters to treat the

coordination of a crank with a path (Fig. Ill-5a). Planar four-bars use the path cognate tool that is not available on the sphere. Without this tool we suffer the weakness of using six-bars to solve a problem that a four-bar should be able to treat. The procedure for the six-bars follows:





62












v c e.
3
B 4 D


5


F





.0 ap 82


8 7r
I







Figure 111-4





63




V

e ( U., V, )








(a)
(U, lo








e


b
c I




(b)




Figure 111-5





64



1. Choose ab and find the position of b by

XbY = sin e sin a-b


Yb9 = -cos elz sin ab (111-7)


Z b cos ab


2. From curvilinear coordinates of e(U e,V e)
find the position e(X e,Y e,Z e) by

XeZ = sin Vek


Y e = -sin Vex cos VeZ


Zek = Cos VeZ cos Vek

3. From dyad analysis (Chapter IV), find c(X,Y,Z),
e2k and 03R (Fig. Ill-5b), using dyad bce.

4. Use any of the procedures for the coordination
of a three-link chain to follow.


6. Angular Coordination of a Three-Link Chain


The coordination of a three-link chain is the first problem of this presentation that cannot treat higher order geometric properties by direct use of the Gmt's in the curvature transform. The coordination of two nonadjacent shells for higher order position properties can only be treated by considering another independent parameter other than 0 lz = y1 (Fig. 11-6).

It is more useful in this problem to consider the crank angle 2, = elk = p as independent and specify all other necessary position functions in terms of this parameter.





65








)13 )"C
c




2 b

1182


a.

a.










c


















Figure 111-6





66



If = f(T), the motion is time dependent and the spherical T s must be used in terms of the specification:

P = 61 = f(T), e2Z = f2(3)' 63Z f3(T)

The state of shell 3 can be expressed from Eqs. (II-8,9,10)


dk [an-/sin Eb sin 6 1
b dk tan cos )j
d T








Ybk d k L 82 + Pb T

d
bZt k tan (tan 81 cos )



The state of shell 4 can be expressed from Eqs. (11-14,16),


~c d~k tan 1 ( 2 s in 8 1 y 2 Cos 0 1
dT T



dk 1 i
dk








c k d k sin (X2 cos 81 Y2. sin j) T



Ck dk F 3 + c c
6 + p





67



k ~ iX~
where tan j
whr c dk -Y
ck d Tk 2 T



p tan (sin cos a )
c Y d k c c




6.1 The Stephenson 3, Case 1

1. Coordinate the problem statement and the
Stephenson 3 by the symbolic transfers
(Fig. III-6b)

a+F, b+D, c+C

T+3, 2+1, 3+4, 4+5

2. Find the first motion specification set for
link 4 as


{a,,y} = {bab'b'Ybt}

3. From the curvature transform find the constraint GD (Fig. III-6b).
4. Find the second motion specification set for
link 5 as

3
{a, ,Y}3 = {= ,' ,y } 5 c ci' ci

5. From the curvature transform find the constraint BA (Fig. III-6b).

6. Pin joints C,D, and G form a rigid ternary.
Pin joints B and C form a rigid binary to
complete the Stephenson 3 mechanism.





68




6.2 The Stephenson 3, Case 2

1. Make the symbolic transfer (Fig. III-7a)

a+A, b+B, c-+C

T+3, 2+2, 3+5, 4+4

2. Find the motion specification set for link
4 as

3
{ ', ,y}4 = {ac ,' Yc }

3. From the curvature transform find two constraints GH and DF. Pin joints C, G, and D
form a rigid ternary supported.by four-bar HGDF and driven by binary AB (Fig. III-7a).

6.3 The Stephenson 2


1. Make the symbolic transfer (Fig. III-7b)

a+A, b+F, c+D

T+2, 2+3, 3+1, 4+4

2. Find the motion specification set
2
{a, ,y}41 = {a c t'cyc }

3. From the curvature transform find the constraint CEB.

4. Find the motion specification set for link 3

2
{a ,,y}3 = {0,O,6Ol}

5. Find the inverse motion specification set
when link 3 is fixed as

3
3 = {O,O,_- l#.}




69




V
4
G C
6.d--O>



H 5l


















2
(b)






Figure I-7
3 3lF











(b)




Figure III-7





70




6. Find the motion specification set for link 4
with link 3 fixed by
{ gy3 fnr = 3
1 4,)4z Eq. (11-30) where r 2 Lm =4

7. From the curvature transform with the specification set from step 6, find link 6 from circle
points G and center points H.

8. Invert the mechanism to the original system
by fixing link 2. Note that points A, F,
and H form a rigid ternary as well as points
C, D, and G.

6.4 The Stephenson 1


1. Make the following symbolic transfer (Fig.
III-8a)

a F, b A, c-B

TI, 2 3, 32, W+5

2. Find the motion specification set of link 5
with reference link 1.

l= { cZ c } { 'B',Y}5 z a cY, ', c


3. From the curvature transform find link 4 with
circle point C and tenter point D.

4. The coordinates B(X,Y,Z) can be found from substituting a 5Z and 5. into


X = sin 5Z

Y B5.= cos

ZB Z = cOS U5 Z Cos 5 Z




71




V









(a)
VB 5 C
















(b)
Figure -8
A G 4 6D


U

















(b)




Figure III-8





72




The coordinates of C(X,YZ) in the k + 1 positions can be determined from dyad analysis using dyad BCD. Analysis will also provide
the relative angle ) FDC (eD).

5. The problem now leaves the determination of
link 6 to the procedure of function generation between links 3 and 4 given in Section
4.


6.5 The Watt 1


1. Make the symbolic transfers (Fig. III-8b)

a-A, B D, c-H
I I, Z22, 3S 3, T-5

2. Find the motion specification set of link 4
by
1 = ^b Z


3. From the curvature transform find constraint
link 3 with circle point C and center point
B.

4. The motion specification set of link 2 is
given by
1
12 = {OOe } 2 lk.
5. The motion specification of link 5 is given
by
{ y}1 = { cZ, cZ',YcZ}


6. Invert the mechanism by fixing link 2.
2z = {0,0,- }
' lk





73




7. The motion of link 5 with respect to link 2
is given by

2 Eq. (11-30) wherer*


8. From the curvature transform find the constraint link 6 with circle point F and center point G.

9. Points A, D, and G construct a rigid ternary
link (2). Points C, D and H construct a rigid
ternary link (4). Inverting to the original
system by fixing link 1 satisfies the problem.


7. Angular Coordination of a Crank with a Two-Link Chain


The motion of a two-link chain whose angular state is coordinated with a crank is closely associated with the three-link chain. The problem is depicted in Fig. III-9a. The basic specification follows that of the three-link chain (cab -- abc).


7.1 The Stephenson 2


1. Make the symbolic transfer (Fig. III-9b)





2. Fix link 3 by allowing 61. = qr pyj and 2k.
p39 to find the motion specification for link
4.


4k a9.' az' ad

3. From the curvature transform find constraint
links OF and OH. Points D and G come from
the circle point laws and points F and H come





74















344


3 %o,1b e,5- -r Ck









Figure 111-9





75



from the center point locus. Points C, D and G construct the rigid ternary link 4.

4. Invert the mechanism to the original system
by fixing link 2.


7.2 The Stephenson 1


1. Make the symbolic transfer (Fig. III-lOa)

aD b+C, c-F

T-I, 24, 3+5, 4 3

2. Allow the rotation parameters of the threelink chain to be



i irTi4z

2 = + .



and consider its inversion by fixing link 3.

3. Find the motion specification set for link 4
relative to link 3 as

3 r,
{as, q, )}3k = { a a Ya }


4. From the curvature transform find link 6 with
circle point G and center point H. Note that points C, D, and G form the rigid ternary link
4.

5. Find the motion specification set for link 5,

3
{a, ,y}5 k fabz,6bzybd}





76




V



/ 5 2

6 H 3
4"?--(a)




V
H5 F C 5



r 4;7


2

(b)




Figure III-10





77



6. From the curvature transform find link 2 with
circle points B and center points A. Note that the points H, A, and F form the rigid
ternary link 3.

1. Invert the mechanism back to the original system by fixing link 1.


8. The Angular Coordination of Three Cranks


-This problem can be considered as the double function generator and can be solved directly from applying the four-bar function generator procedures twice. Only three-pivot mechanisms are eligible for this problem.


8.1 The Watt 2


1. Make the symbolic transfers (Fig. I11-lob)

a-A, b-D, c-G

T,2, T2 1, 3-4 T-6

2. Consider the problem parameters lP2z and.j
and use the coordination of two cranks in
Section 4; find link 3 with pin joints C and
B.

3. Consider the problem parameters 1pSz and P4z
and use the coordination of two cranks in
Section 4 to find link 5 with pin joints F
and H .

4. Points 0, C, and H are all on rigid ternary
link 4 to couple the two four-bars together.


9. The Angular Coordination of Three Adjacent Planes


If link T is considered as a reference, the problem is transferred to the type of the three-link chain with (bac abc).





78



9.1 The Watt 1


1. Make the symbolic transfers (Fig. III-11a)

a*A, b+D, c+G

1I1, 22, 3+6, 4 = 4

2. Let O1 = 92z and 02k = q to find the motion
specification of link 4.

1
{a,8,y}41 = {a bz, b ,yb }

3. From the curvature transform find link 3 with
circle point C and center point B.

4. Let

Ilz = t


~2R 7 + 039 = %9+



and find the motion specification set of link 6 with respect to link 4 (considering link 4
as fixed)

4
{a''Y}6 = {act''c}

5. From the curvature transform find link 5 with
circle point F and center point H.

6. Note that points C, D and H form the rigid
ternary link 4.

9.2 The Watt 2


1. Make the symbolic transfer (Fig. III-11b)




79




V F
H








(o)
FF
6H 4
















6/ 4 CB
G 3
2 D 3




















2
A T noB











(b)






Figure I6' 4 C
G 3 s

D o A

2
(b)



Figure I I 1





80



a*D, b+H, c+C

22, 2-+4, 3+5, 4=3

2. Let 01 = T2, 02k = 6 and find the motion
specifications for link 3.
2
{a 3,y}2 = {abt Yb }

3. From the curvature transform find the constraint
link 1 with circle point B and center point A.

4. Let 01 = 9 + T and 02Z = S and find the
motion specification set for link 5.

{ ,B,Y}~2 = {ac t c ,c }


5. From the curvature transform find the constraint
link 6 with circle point F and center point G. 10. Double Colaminar Synthesis with Common Knee Joint

This problem has had several valuable applications in planar kinematics. It is expected that it will also be valuable on the sphere. 10.1 The Stephenson 3

1. Make the symbolic transfer (Fig. III-12a)

e*C, 3+4, 2+5, 1T+3

2. Defind the motion specification set forlink
5 as

3
{a,,y}5 3k= {UcVc' 2





81





vV U4 %











(a)




V4 U5





U4U






(b)





Figure 111-12





82



3. From the curvature transform find the constraint
link 2 with circle points B and center points A.

4. Define the motion specification set for link 4
as

{aB, } 3= {UcVcr}

5. From the curvature transform find the two constraint links 1 and 6. Note that points D,
G, and C form ternary link 4. The resultant
mechanism is depicted in Fig. III-12a.

6. Switching the roles of V2 and Ipf by the symbolic transfer 2-4 and 3-5, the steps 2 through 6 can be repeated to double the number of solutions (Fig. III-12b).


11. Colaminar Motion Coordinated with a Crank


This problem involves the coordination of a moving shell with the motion of a crank. It may be the most practically useful procedure of this treatise. 11.1 The Stephenson 2


1. Make the symbolic transfer (Fig. 111-13)

a-A, e-D

T,2, T,3, 3Y,4

2. Define the motion specification set for link
4 as

2 ,
{a,S'Y}49 {UD

3. From the curvature transform find the constraint link 5 with circle point C and center
point B.





83 v









51

















Figure 111-13





84



4. From analysis of dyad DCB find the motion of
link 5 and angular motion $5. Also find the
base angle 42 and angles P4.

5. Let

S P1 =T 2y+ 2k

2 = 5z

O3 = 4


and find the motion of link 4 with respect to
link 3 by considering link 3 fixed.

3
{a,8,y}4z = { ,BcZ ycZ}

6. From the curvature transform find the system
constraints links DF as 1 and GH as 6.
7. Invert the mechanism back to the original reference of link 2.














CHAPTER IV

SPHERICAL DYAD ANALYSIS



Kinematics is based upon developing an understanding of the motion,independent of time, of interconnected rigid bodies. An analysis technique is necessary to describe the motion of any point, shell, or relative angle in a mechanism. This chapter will generate a collection of tools to provide a concise and efficient means of interpreting the analysis problem. It should be noted that analysis is a necessary compliment to synthesis and design, and may well be the final critical step of optimization.

The use of dyads for analysis of planar mechanisms has proven to be the most efficient and versatile closed form tool that is presently available. Referring to Fig. IV-l, a dyad is represented as a pair of rigid links (C,CD) connected by a common joint at point C. If the time states of the two remaining ends (B,D) are known, then the time state of C can be found. Note that BCD forms a spherical triangle. Without loss of generality, we will consider only unit spheres such that 6,c, and a are all unit position vectors of points B,C, and D respectively. The arc RD can be found by the scalar product of 6 and a.


85





86



I X

C
/ i /t




Z C--Y
Ph B 9'\c ofB D ~
BD Pat

(a) ofD
z -Y
X
C'









x
1c






Z B'Y C"



f1



(c)


Figure IV-1





87




BD = cos- (b.d) (IV-l)


The interior angle I CBD (0) can be written in terms of the spherical cosine law

cos CD = cos R cos BD + sin R sin D cos e. (IV-2) Solving for cos e, and taking into account the possible range, we find



-l < cos a = cos CD cos BC cos BD1 <1 (IV-3)
L sin BC sin BD j

Satisfaction of this condition defines whether or not the dyad can be physically connected.

In order to find the position vector c, we must first rotate 6 to (0,0,1) by a rotation u about the

-x-axis and then a rotation v about the -y-axis. This transformation can be described in matrix form by




[ hr (IV-4)



where





88



Cos v Sin U sin V -COS U Sin V

M1 = cos u sin u

sin V -Sin U COS V COS U COS V


However, from the curvilinear to rectangular transformation,

x = sin v

y = -sin u cos v
(IV-5)
Z = COS U COS V

RB=/I-x2 = cos v

Substituting Eq. (IV-5) into Eq. (IV-4), the new position of D is found by

RB -xBYB -xBZB RB RB

a' = ZB -YB a (IV-6)
R B
B B

xB YB zB


Now rotate d' to lie in the (-y,z) plane by a rotation about the -z-axis.

0

YD" = M2a' (IV-7)
z D"





89



where


Cos # sin # 0
M2 = -sin cos 0

0 0 1


However the position of d' can be defined by

xD, = sin BD sin 4

D =-sin B- cos q (IV-8)

zD, = cos BD
and sin RD = = x2 + y2 RD
v D-D 'rx 1+LD'

Substituting Eq. (IV-8) into Eq. (IV-7),


-YDr xD, 0

M2 R1 xD' YD' 0 (IV-9)

0 0 RDThe position of c" is given by the elementary expressions,

XC, = sin BC sin e

YC, =-sin BC cos e (IV-10)

zC = cos C





90



Both transformation matrices M1 and M, are orthogonal such that their inverse is identical to their transpose. (M1 = M1 and M2 = M2) Reversing through the inverse rotations, we find the position vector c of point c in the original system position by



c = MT M 2 (IV-ll)



The assumptions of rigidity of links BC and CD, as well as the constraint of the unit sphere, impose the following set of conditions on points, B,C, and D.


b.c = cos BC constant c.d = cos CD : constant

b.b = cos 0 1 1 (IV-12)

c-c = Cos 0 =1
A a

d.d = cos 0 = 1


Differentiating Eq. (IV-12) yields a set of conditions for the velocity vectors S, and a.
9
+ =

C.a + C.5 = 0 (iV-13)
A A,
beb = 0

c.c 0
.8=0





91



These can be collected in several ways to avoid singularity for the solution of c, e.g.,

XB B B C xB B zB xC


XD D zD YC XD YD ZD C (IV-14)

0 0 0 z C XC YC ZC ZC
M


If the determinant of M(det M) is nonzero then premultiplying Eq. (IV-14) by -M-1 is sufficient to find (xCYc,zc). If det M is zero,then a different collection of Eq. (IV-13) should be assembled in Eq. (IV-14).

To find the acceleration, we can differentiate Eq. (IV-13) again to give the set,

x 11, *
b c + 2 b'c + b8c = 0

d*c + 2 d*c + d*c = 0 (IV-15)

cc + c*c = 0 b*b + b*b = 0 d*d + d*d = 0

These two can be selected in several combinations to find c, e.g.,





92



S 2 B 2y 2z . ""
XB YB 'B XC B;B, XB YB ZB X
D D 2D Yc + 2D 2D 2D ;C xD YD ZD c


0 0 0 zCC C- zJ XC YC Zc Y

M

(IV-16)

If M in Eq. (IV-14) is nonsingular, then Eq. (IV-16) can be premultiplied by -Ml to find (R CC,*C). This procedure could be carried further to find higher order properties if desired.

Now that we have shown how to find the time state of point C in the dyad with points B and D given, an observation can be noted on their extension to points all on the same rigid shell. If this is the case, such that BD and 0 are constant, the same procedure can be utilized to find the time state for a third point C on the same body. This means that one tool can treat both dyads of two rigid links or ternary links of one rigid shell.



Prismatic Equivalence

The slider or prismatic joint is relatively common in planar mechanisms. A slider is generally considered to be a link that is moving along a linear path with a center of rotation at infinity. Links in the plane transform into





93



great circles on the sphere. By rotating a 900 crank on the sphere, any great circle of that sphere can be generated. Hence, any 900 crank on the sphere is the equivalent of a slider in the plane.

The general spherical dyad enables us to illustrate and discuss the four four-bar chains (RRRR, RRRP, RRPP, and RPRP) and all of their inversions by the use of the spherical equivalents to the five planar dyads RRR, RRP, RPR, PRP, PPR).



Four-Link Chain Equivalence

The RRRR chain is the general four-bar linkage. Both the planar and spherical representations are given in Fig. IV-2a. All inversions are the same and all spherical links are different from 900.

The RRRP chain is shown in Fig. IV-2b. One

link (CD) is infinite in the planar case. The offset in the planar chain from pivot A to straight line Z1 is h = AA' The corresponding offset in the spherical chain is A = AA'. In the planar case A-D = D = In the spherical case A'D = CD = 900. No slider needs to be represented physically at C although C always travels on a great circle cI. The travel on k1 is measured by S1. The corresponding measure in the spherical case is a. Note the link AD = h + in the planar case, and




Full Text
terms of two complementary formulations. The first pro
vides for translational and angular evaluation of multi
link mechanisms. This analysis is established by means
of two-link chains known as dyads. The second treats the
input-output equation algebraically. This format yields
the evaluation of multiple closure branches simultaneously.
All of the synthesis and analysis tools are shown
to be reducible for use in planar mechanism design.
vi 1


97
' 2
D
(b)
Figure IV-4


130
Stephenson 1
The Stephenson 1 input-output equation could be found
in at least two different ways. The input-output is
based upon the quadrilateral 1276 and could readily be
written by introducing the constant ternary angles at
vertices 1 and 2. However, it is felt that including as
many sides as possible gives a more pallatable result.
It should be recognized that the numeric results must be
identical.
Consider the hexagon 123765 of Fig. V-4 and write
the cosine law,
Z5123 = C0S 67 (V'16)
where ZJ123 = sin a73(X512 sin *3 V612 cos Tj)+ cosa73 Zj]2
X512 = X51 C0S S2 Y51 Sln 02
Y512 = cos a23(X61 sin 02 + Ygl cos 9.,)- sin a23 Zg,
Z512 = 5in 23(X51 Sin 02 + V51 C0S 02) + C0S a23 Z51
Substituting ^512 anc^ ^512
and collecting terms,
into Eq. (V -16 )


169
formulation of a mu ti-parameter cost function. This
author believes that this is unrealistic due to the num
ber of dependent variables. An alternative approach se
quentially filters undesirable mechanisms. The sequence
is ordered according to computational efficiency. It is
suggested that this philosophy will also apply to spherical
design.


123
In Fig. V-2, let 0'7 = and consider the penta
gon formed by 34567 and v/rite the cosine lav/
Z734 = C0S 56
(V 6)
where Z?34 = sin ac (X77 sin 0 A + Y77 cos 0A) + cos a/ic Z
45 73 u4 73
45
X73 = X 7 cos @3 Y1 sin 03
Y73 = cos a34 (Xy sin 0^ + Yy cos 0j) sin a34 ly
ly3 = sin a34 {Xy sin + Yy cos 0") + cos a34 ly
Xy = sin ag7 sin 0
Y-j = -(sin a37 cos a67 + cos a07 sin aC7 cos 0;)
37 67
7
Z7 = cos a37 cos ag7 sin ao7 sin ac7 cos 0;
37 67
With X73 Y73, and ly3 known, Eq. V-6 becomes a quadratic
in x4 with coefficients,
A
B
C
cos a45 Z73 sin a45 Yy3
- cos a
56
2 sin a45 X73
45
l45 Z73
(V7)
cos a.c Z, + sin a
Y
45 73
- cos a
56
73


154
b. The location of the roll center is fully
the designer's choice.
c. Stationary roll centers should enhance
predictability and stability in high g
turns .
3. Zero acceleration of vehicle during bumps.
a. Most suspensions change the distance be
tween the mass center and the tire center
which creates a fluctuating vehicle ac
celeration.
b. The TR suspension reduces this chugging
action to zero.
4. Anti-lift action during braking.
a. Anti-lift requires that the brake forces
create a moment on the car to keep the
rear down during braking.
Solution
The Mercedes-Benz 450SEL Touring Sedan was chosen
as a reference body to which to fit the TR Suspension.
It may be an example of an automobile with a good balance
of comfort and handling. Its standard suspension has a
fully independent, diagonal pivot rear axle. Items 1 and
2 above may be satisfied by arbitrarily choosing the center
of the sphere to be six inches off the ground at the op
posite wheel (Fig. VI-4a). Items 3 and 4 can be satisfied
by requiring the wheel to also move on a path that is a
constant radius p from the car's mass center (Fig. VI 4 b).
The problem statement is now constrained to spherical mo
tion. However, the sphere has a 60-inch radius and the
wheel motion is in a relatively small area of the sphere.
Initial motion specification can begin by a planar approxi
mation. Assume that we are given the fixed pivot 0, (same
a


16
r


53
{a,e,Y>
nk
m£
{a,B,y}
m
Tk-£
(III-D
where x can represent y,<}>, or t. The required differenti
ation is taken automatically in the accompanying interactive
computer program. This statement is valid for geometric
states or time states. It will be seen that all problems
are applicable to the time state, but others may not be
specified in the higher order geometric state. The objec
tive of these design procedures is that they be accessible
to the designer without any further information than that
given in this document.
1. Path Synthesis
For path synthesis there is only one (F=l) func
tional constraint provided by the given path. The motion
of a point on a path is independent of a rotation parameter
y. The standard curvature transformation requires the set
{a,g,y}. The problem statement gives us
aZ = Ue e£ = Ve Y = arbitrary (III-2)
This problem can be treated by the colaminar formulation
listed in procedure B by taking y as an arbitrary set of
values.


61
4.2 The Five-Bar 1 by Inversion
1. The required symbolic transfer is
a+A, b->F, c-*-D
T-l 2+2, 3+5
as in Fig. 111 4 -
2. The overall gear train value from A through
F to D is Q = MN. Links 2 and 4 are coupled
by this train. The influence of 0-, and 02
on 02 can be seen by
(03£ 930^ = Q(0U 01O^ + N(02JL 02O^ (HI-6)
We have no control on 030 so that we may ar
bitrarily set it to zero. Eq. (II-6)then
yields e3jr
3. The motion specification for link 4 becomes
- Eqs. (11-14,16)
4. The curvature transformation gives pin joint
C as the circle point locus and point B as
tjie center point locus. The pair form arc
BC and provide the needed constraint for the
f i v e b a r 1 .
5. Invert the mechanism by fixing link 1.
5. Path Coordination
The four-bar has enough parameters to treat the
coordination of a crank with a path (Fig. III-5a). Planar
four-bars use the path cognate tool that is not available
on the sphere. Without this tool we suffer the weakness
of using six-bars to solve a problem that a four-bar
should be able to treat. The procedure for the six-bars
fol1ows:


36
or
2*
dx
tan^ (tan 0-j cos b)
(11-12)
Eq. (11-12) can be differentiated to give the higher order
properties of Eq. (11-10).
Writing the sine, sine-cosine, and cosine laws for
spherical triangle acc*,
sin 03 = X 2 cos 01 Y2 sin 0^
sin cos 03 = -(X2 sin 0-j + cos 0-j) (11-13)
cos cos 03 =
where X2 = sin be sin 2
Y2 = -(sin B cos be + cos i"b sin be cos @2)
12 cos STb cos be sin ab sin be cos 0 2
Eq. (11-13) can be solved to find


77
6. From the curvature transform find link 2 with
circle points B and center points A. Note
that the points H, A, and F form the rigid
ternary link 3.
7. Invert the mechanism back to the original sys
tem by fixing link 1 .
8. The Angular Coordination of Three Cranks
This problem can be considered as the double func
tion generator and can be solved directly from applying
the four-bar function generator procedures twice. Only
three-pivot mechanisms are eligible for this problem.
8.1 The Watt 2
1. Make the symbolic transfers (Fig. III-10b)
a-A b-D c-G
T-2, 2-1, 3-4, 4-6
2. Consider the problem parameters anc* ^3i,
and use the coordination of two cranks in
Section 4; find link 3 w i t h pin joints C and
B.
3. Consider the problem parameters and
and use the coordination of two cranks in
Section 4 to find link 5 with pin joints F
and H .
4. Points D, C, and H are all on rigid ternary
link 4 to couple the two four-bars together.
9. The Angular Coordination of Three Adjacent Planes
If link 4 is considered as a reference, the prob
lem is transferred to the type of the three-link chain with
(bac abc).


91
These can be collected in several ways to avoid
A
singularity for the solution of c, e.g.,
ZD
(IV-14)
If the determinant of M(det M) is nonzero then premulti
plying Eq. (IV-14) by -M ^ is sufficient to find
(xc,yc,zc). If det M is zero, then a different collec
tion of Eq. (IV-13) should be assembled in Eq. (IV-14).
To find the acceleration, we can differentiate
Eq. (IV-13) again to give the set,
A a A ^
b c + 2 b c + b c
H A A A A K
d c + 2 d c + d c
c c + c c =
0
0
0
b b + b b = 0

A A A A
d*d + dd = 0
(IV -1 5)
These two can be selected in several combinations
A
to find c, e.g.,


Table 10: Doub 1 e Co 1 emi na r Sy n thes i s with Common Knee Joint
Problem Statement
Specified Functions:
F=3; ue= f(^); ve- f(ip2);
'^3=
Graphical Form:
m-j = 2, m2 = 3
Available Mechanism Constraints
FSP MSP
Mechanism
Five-Bar
{q=12, L=4, p4=0>
1
2
3
Six-Bar
[Knee joint
e >m-j jm£ ]
[0,3,4]
[0,4,5]
[F,1,5]
Y
Y
{q=l4, L-4, p4=2}
Watt 1
Stephenson 1
Stephenson 2
Stephenson 3
[F,5,6],[H,4,5]
[8,2,5]
[D,l,4(or G,4,6)]
[C,4,5]*
TSS
Y
Y
ro


152
deg
deg
Figure VI-3


182
22. Riddle, D., Duffy, J. and Tesar, D. "Kinematic Syn
thesis of Geared Spherical Five-Bar Mechanisms
for Function Generator." ASME Mechanisms Confer
ence 1974, Paper No. 74-DET-74.
23. Duffy, J., Riddle, D., and Tesar, D. "Angular Displace
ment Analysis of the General Planar and Spherical
Geared Five-Bar." To be presented at the IFTOMM
Conference, Newcastle, England, Sept. 1975.
24. Eschenbach, P. and Tesar, D. "The Dynamic State of a
Moving Plane" ASME Mechanisms Conference 1972,
Paper No. 72-Mech-18.
25. Myklebust, A. and Tesar, D. "The Analytical Syn
thesis of Complex Mechanisms for Combination of
Specified Geometric or Time Derivatives up to the
Fourth Order."
26. Myklebust, A. Synthesis of Multi-Link Mechanisms
for Dynamic Specifications. Ph.D. dissertation
presented to the University of Florida, 1974.
27. Tesar, D. "Design Methods for 4, 5, and 6 Bar Link
age Systems." Graduate and undergraduate notes used
at the University of Florida.
28. Gilmartin, M. J. and Duffy, J. "Type and Mobility
Analysis of the Spherical Four-Link Mechanism."
Journal of Mechanisms 1972, 90-97.
29. Freudenstein, F. and Primrose, E. J. F. "Geared
Five-Bar Motion." ASME Mechanisms conference
1962, Paper No. 62-WA-84.
30. Pollock, S. "Dynamic Model Formulation Programmed
for Dyad Based Machines." Master's thesis pre
sented to the University of Florida, 1975.


138
:3 = Sin a73 s1n *3
= -(sin a34 cos a73 + cos a34 sin a73 cos 3)
Z3 cos a34 cos a
73 sin a34 sin a73 cos cf>3
and
8 5 + T5
6 3 + t3
(V 2 8 )
Substituting Eq. (V-28) into Eq. (V 2 7 ) and apply
ing the half-tangent identities, a quadratic in can be
written with coefficients,
A2 cos 67 + Sln 56(X34 Sln T5 Y34 C0S T5)+ C0S 56 Z34
B2 2 sin 56^34 C0S t5 Y34 s,n T5^
C2 = cos 67 si" 56U34 sin T5 Y34 005 T5)+ C0S a56 Z34
(V-29)


Table 11:
Co 1 aminar Motion Coordinated with a Crank
Problem Statement
Available Mechanism Constraints
Specified Functions:
F=3; Ue= f(H>2); Vg= f(^);
^3= f(i|>g)
Graphical Form:
FSP
MSP
Mechanism
Planes
Special
[ro-j
Knee Joint
P = e
m-|= 2^2= 3; Point e in m^
Five-Bar
{q=12, L=4, p4=0)
1
[2,4],[5,3],
[2,3],[5,4]
C
2
[3,5],[1,4],
[3,4],[1,5]
D
3
Six-Bar
[2,5(or 4,1)],
[2,l(or 4,5)]
F
{q=14, L=4, p4
=2}
Watt 1
[2.5],[3,5],
[3.6],[2,6]
F,H
Watt 2
[l,5(or 6,3)]
Stephenson 1
[4,2(or 3,5)],
[4,5(or 3,2)]
B
Stephenson 2
[3,4]*[5,l(or 6)],
[3,1(or 6],[5,4]
D(or G)
Steptheson 3
[1(or 6),5],[2,4],
[2,5]
C
N
N
TSS
Y
Y
OJ
o


35
The result of Eq. (11 8) for k = 0 will yield in the
first or fourth quadrants. This is sufficient as long as
<*2 is not restricted to principal quadrants. From Eq.
(11 7), a2 can be found in its proper quadrant by the im
plicit expressions for sin a2 and cos o^. The ratio of
the last two expressions in Eq. (11 7) yields
(11-9)
T
The two parameters a2 and 0^ are sufficient to describe
the location and higher order properties of point b. The
third Euler angle rotation can be seen (Fig. 11 -6) to be
the sum
a
ll
-
dk
tan-^ i
( sin ab cos 0-j ^
dyk
1
cos Tb
Y
21
dk
71:
dy
(0 2+P2)
J T
(11-10)
The angle 6^ is assumed to be given as a function of 0-|
with its appropriate derivatives. The angle can be
represented as a function of 0-| through the cosine law for
the polar triangle abb'
0 = sin p2 cos 0.| cos p2 sin 0^ cos ib
(11-11 )


98
(a)
(b)
Figure IV-5


158
the final configuration of the four-bar OgABO^. This pro
cedure proved valuable in predicting the spherical solu
tion. The transformation between homogeneous coordinates
and cartesian coordinates is given by
A( z, Z2) A(X Y Z)
Z O + zf + z\)
X = z i Z
Y = z2 Z
(VI-3)
We can now express the four-bar 0cCA0a in cartesian coor
dinates as
0C(X,Y,Z) = (.2838, .9478, .1451)
C(X,Y,Z) = (.1033, 0, .9946)
A(X,Y,Z) = (.1941, -.0435, .98)
0 (X,Y,Z) = (.07, .3333, .9402)
a
Dyad analysis of the four-bar will yield the motion prop
erties of point C. This motion can be related to the ro
tational parameter a and 3 by Eq. (111 3). Further analy
sis yields co and w. The corresponding angular velocity
and acceleration about point C are given by
Y = (w c) sin 8
Y = ( (VI-4)


13
to specify the geometric position of this device?", the
answer could be given in any number of terms. The most
immediate answer is derived by considering the location
of each pin joint A,B,C,D. If the condition of a spheri
cal mechanism is satisfied, then the axes of each pin
joint passes through the origin 0. The arc between any
two pin joints is a great circle segment. If we arbi
trarily select a unit sphere for convenience, such that
z2 = 1 -(x2 + y2), (11-5)
only two parameters would be necessary to describe the
position of each pin joint. A total of q = 8 parameters
is necessary to complete this description. The special
case of a prismatic slider can be simulated on the sphere
by a 90 crank. If, in Fig. 11-2,arc CD is 90, then point
C will move on a great circle path. This is the equiva
lent of linear motion on the plane.
The spherical five-bar mechanism (Fig. 11 -3 a) has
two degrees of freedom. Any two links of the chain can
be driven independently to constrain the motion. Alter
natively, gear trains can be introduced between two non-
adjacent links to yield a one degree of freedom device.
Motions derived from the five-bar mechanism family are
inherently more complex than those of the four-bar mech
anisms. It is expected that a geared five-bar mechanism


Table 3: Angular Coordination of Two-Link Chain
Available Mechanism Constraints
Problem Statement
Specified Functions: F = 1 ;
C -
Mechanism
f Ujr)
= 3
Four-Bar
{q = 4 L = 4 p4 = 0}
Five-Bar
{q 1
2
3
S1 x B a r
{q_<10, L U
Watt 1
Stephenson 1
Stephenson 2
Stephenson 3
FSP MSP
Planes [m^ ,m,,]
[2 (or 4), 3]* Y
Y
[2,3], [5,4]
[1.5], [3,4]
[2,1 (or 4,5)]
Y
[2.6], [3,4]
[4,5 (or 3,2)]
[3 ,1 (or 6) ] [5,4]
[2,5]
TSS
Y
Y
ro
no
Y


132
(X51 Y3 + Y51 V Sin 02 '(X51 X3 Y5l Y3> cos e;
~(Z5l Z3 C0S a67^ 0
(V-
where X3 = sin a^3 sin <*3
= -(sin a33 cos a^2 cos <*23 sin a73 cos T3^
Z3 = cos (*23 cos a^3 + sin a23 sin a^3 cos x3
X51 X5 cos 01 Y51 Sin G1
Y5i= cos a12(x5 sin 6-! + Yg cos 0^)- sin a-j 2 Zg
Z51= sin a-j 2 (X 5 sin e] + Yg cos 0^ + cos a]2 Zg
X5 = Sin a56 Sin T5
Yg = -(sin a51 cos agg cos a51 sin agg cos Tg)
Zg = cos ag cos agg + sin ag sin agg cos x g
If @i is taken to be a known input, then 0^ is
the only unknown in Eq. (V-17). Using the half-tangent
17)


39
Ratio ing Eq. (11-21),
31
di
tan
-1
X
*2
-Y,
Ll
(11-22)
Equation (11-16) can be extrapolated for a n-link chain
r=^-r [6 + a + p]T
n, dTk L n n Mn T
(11-23)
(k 1
where 0^ is specified and
ni 'dTk
tan
-1
X
23. .n-1
-Y,
23... n-1
pru = k t13"'1 dx
V p
The computation of {a,6,y)i anc {a,B,y}g for the three
link chain has been validated by the APL function LINK3 as
listed in the Appendix.
Inversion
The motion specification set {a.B.y}^ for the
ttiree-lifik chain was established with reference to link
r. We will see in the design procedures that it is de
sirable to find the motion specification set {a.B.y}11.,
IIIX/


136
Figure V-5


58
The curvilinear coordinates for the state of in E are
given by Eqs. (11-8 and 9), where 0^ =
a2l
tan
sin ab sin ip^
cos STB
T
s i n
-1
(sin a b sin ip 2 )
T
The rotation of link 3 about pivot b completes the motion
specification of link 3.
dk
Y22'
where p2 is given by Eq. (11-12) as
21
dx
-1
tan" (tan ip2 cos ab)
3.1 The Four-Bar
1. The four-bar can be used if the symbolic
transfer (Fig. Ill- 2b )
a-*A, b-*B 1 +1 2--2 3--3
occurs.


150
Applying this data set to the curvature transformation
yields the locus of cranks BA. From the locus, circle
point B and center point A are chosen (Fig. VI-2).
B(X Y Z) = (.0564 .2826, .9576)£=Q
A(X Y Z) = (.0878, -.1 993 .9760)
C(X Y Z) = (.4203, -.8400 .3432)£=Q
BA = 27.97
CB = 83.40
The resultant mechanism guarantees the satisfaction of
the initial problem specification, but several additional
quality factors may limit the usefulness of this mechanism
To further evaluate the quality of this mechanism, an anal
ysis must be run. It was found that the crank AB cannot
rotate a full 360 so that a motor could not be attached
directly. Generally, a designer would stop here and se
lect a new crank from the locus satisfying the initial
motion specification. In this case we will pursue the
analysis to demonstrate additional points. The analysis
is summarized in Figure VI-3 for order k = 0,1,2. Note
in this case, that both positions =0 and =3 are satis
fied without reassembling the 1inks in adi fferent config
uration. This is not always true and must be guarded


70
6. Find the motion specification set for link 4
with link 3 fixed by
3 fn = 3
{a, 8, y] ^ + Eq. (11-30) where j r = 2
(m = 4
7. From the curvature transform with the specifi
cation set from step 6, find link 6 from circle
points G and center points H.
8. Invert the mechanism to the original system
by fixing link 2. Note that points A, F,
and H form a rigid ternary as well as points
C, D, and G.
6.4 The Stephenson 1
1. Make the following symbolic transfer (Fig.
Ill-8a)
a+F, b+A, c+B
T+l, 2+3, 3+2, 4+5
2. Find the motion specification set of link 5
with reference link 1.
{o-6+15i = taci-Bc*Tc.>
3. From the curvature transform find link 4 with
circle point C and center point D.
The coordinates B(X,Y,Z) can be found from sub-
s t i t u t i n g
a5£
and
B5£
into
XfU
s i n
e5£
VB* '
sin
LD
3
cos 8
5 £
ZBl
cos
a5£
cos 8
5£
4.


117
B
Figure IV-15


9
g(x,y,z) to be spatial, spherical or planar to meet the
specified design positions. It should be noted that
spatial mechanisms include spherical and planar mechanisms
as a proper subset.* Planar mechanisms are also a subset
of spherical mechanisms. With this in mind, the designer
could always choose spatial specifications to satisfy the
problem function f(x,y,z). Spatial mechanisms have many
more parameters to be considered and are often unneces
sarily complex. In addition, there are no well-established
synthesis techniques available for general spatial mech
anisms. References include spatial analysis routines that
may be considered for trial and error design, but this is
recommended only as a last resort. If f(x,y,z) is a
general spatial function, it would be most desirable to
approximate it with a spherical function g(x,y,z). Spher
ical mechanisms add little complexity to the analytics,
yet allow a type of spatial motion. It will also be seen
that spherical and planar mechanisms may share the same
body of analytics and add no extra effort in use. This
work will continue to emphasize spherical synthesis but
the same philosophy and reduceabl eanalytics follow directly
Spatial mechanisms have no restrictions on the
rotation axes. Spherical mechanisms have all rotation axes
intersecting at common point, the origin of the sphere.
Planar mechanisms have all rotation axes parallel, or in
tersecting at infinity.


CHAPTER VI
APPLICATIONS
Three-Link Chain
The three-link chain is a valuable aid to several
synthesis procedures. A complete sample solution will
follow to validate the use and operation of the chain
problem. This example will follow the design procedures
of Section 6.1 in Chapter III. The understanding of this
problem has become the key to understanding basic principles
of spherical synthesis.
Consider the three-link chain in which the grounded
link is driven by a motor at a constant angular velocity.*
Assume that the relative angles 02 and 63 (Fig. VI -1 a)
are functions of the input crank. The desired discrete
values to be satisfied are given by
*Inputs are not restricted to only constant veloc
ities and may be general input functions. This fact is
due to the time state formulation and was a previous re
striction of geometric synthesis.
146


94
C(R)
(a) RRRR Four
Link Chain
B(R)
B(R)
(b) RRRP Four Link Chain
Figure IV-2


129
Substituting 0l = ty + 7 into Eq. (V-11 )
gives
cos a34 = sin a47 {X2(sin(x7+?7)cos 07-cos(x7+T7)sin 07)
(V-13)
+ Y2(cos(t7+T7jcos 07+sin(t7+T7)sin 07)}+ cos a47
Equation (V-3) can be written as a quadratic in
xlj with coefficients,
A2 = cos a47 Z2 cos a34 sin a47(X2 sin(x7+x7)+ Y2 cos(x7+T7))
B2 = 2 sin a47(Y2 sinx^+x^)- X^ cos(x7+x7)) (V-1 4)
C2 = cos a47 Z2 cos a34 + sin a47(X2 sin(x7+?7)+ Y2'cos(x7+?7))
Eliminating x7 between quadratic Eqs. (V-12) and
(V-14) yields a quadratic in x2 by the expansion of the
resultant,
0
(V -1 5 )


179
[24] ZD+Tll;2]x(1 *.oL)xloT[1;2 ]
[25] i4[2;2 ]-*-((( 2of[ i ; 1 ] )xXD-Y*Tl 2 ; 1 ] ) (loT[ 1; 1 ] ) *YD+X*T[
2 ; 1 ] ) 2 Oil [ 1 ;2]
[2 6] /;-H((2or[l ;1] ) Y x 7jD) Z Y Di- X* 2 ;1] ) + (lo?[l ;1] )*^(^
xZP )-Zx^d-7x!T[ 2 ; 1 ]
[2 7] 4[2 ;1]-/7*£N-Zx2*( 204 [ 1 ; 1 ] ) *2
[2 8] YOZMiT 3;2]x(loL[ 2])x2o?[i ; 2 ] ) (!T[ 2 ; 2 ] 2 ) x (lo5[
2])xio7[l ;2]
[29] YDD+(2 1 x.oL)xZDP-e-(7[3;2]xio2[l ;2] ) + (!T[2;2]*
2 )x2or[1;2]
[30] ZDD+( 1 x oL ) x 0 + Z.7D
[31] 4[3 ;2>(( 207[ 1 ;l] )xXDD-{ Yx7[3 ;l] ) + ( Yx 7[ 2 ; 1 ] *
2 ) + 2xYDx?[2 ;1] )-(lorri ;1] )xYZ7iO + (2xYZ?x2*[2;l]) + (Yx2'[
3;1])-Yx7[2 ; 1 ] 2
[32] 4[3;2]-(4[3;2] + (4[2;2]*2)x 104 [l;2])v204[l;2]
[33] tfZM2or[l ;l])x(Zp/}xy)-(Znx;;x7[2;l] ) + (Zx.Y/)xf[
2 ; 1] ) + (2x Jx r[3;l]) + (Z xYDD)-C*Tl2;1]
[34] ND*-NDl-(. 107[ 1 ; 1 ] ) x ( Z55 x X ) + ( ZD x Yx 7[ 2 ; 1 ] ) + ( 2 x YDx 2[
2;l])+(ZxYx?[3;l])-(Z*XDP)+B*T2;1]
[3 5] £>ZM 2 x Zt ( 204 [ 1 ; 1 ] ) 2 ) x ZD+A [ 2 ; 1 ] x Zx 304 [ 1 ; 1 ]
[36] 4[3;lX (Dxr!D)-nD*N)iD*D
[37] P-(/X lx moL )xio7[ 1 4 ; 2 ] ) VATAN X 205[1 ] ) (
20L[2])xx/204[1 4 ; 1 2]
[38] Ail 4 ; 3 ]+T[ 1 4 ;3]+P+5+((104[1 4 ;2])x2o4[l
4 ; 1 ] ) VATAU 104[l 4 ;l]
[39] ND+(1x,OL)x7[2;2]x207[1;2]
[40] DD-*-( 2oL [2])x(4[2;l]x l 2 x.o4[l; 1 2])+4[2;2]x
2 1 x 04 [ 1 ; 1 2 ]
[41 ] PZX ( 2oP+l + P)*2 )x( ( DxtlD) -DD*ll+-1 hV)iOxfl+l + 3
[42] 4 [ 2 ; 3 ]+7[ 2 ; 3 ] + PP+5P-( ( 205-*-1 t5)*2)x(4[2;2]x 2
3 t.04[1; 2 1] )-4[2 ;1]x(104[1;2])+ (104[1;1])*
2
[4 3] NDD+( 1 x .oL )x(7f3 ; 2 lx20 7[ 1 ; 2] )-( 7[ 2 ; 2] *2 )xlo7[ 1 ;
2]
[44] DDD+-C 20L[2])x(4[ 3;l]x l 2 x 04 [ 1 ; 1 2]) + (4[3;
2 ] x 2 1 x 04[1 ; 1 2 ] ) + ((4[2 ; 1 2]+.*2)*2x.o4[l;
1 2])-x/2,4[2; 1 2],lo4[l; 1 2]
[4 5] P+( -2xPD*PD* 3op) + ( ( 20P)*2)x(,(D*(DxnnP)-N*DnD)-
2 xDDx(Dxfjn)-fxDD) iD* 3
[46] 4 [ 3 ;3]-7[ 3 ; 3 ] + P+( -2 x5 P x 5 P x 3 05) + ( ( 205) *2 )x ( ( 204 [ 1 ;
2] )x(4[3 ; 2]*304[1;l])-2x4[2;l]x4[2;2]*(lo4[l;l])*
2 )
[47] 4 [ 3 ; 3 ] -*-4 [3;3] + ( ( 2o5)*2)x( lo4ri;2])x(2x(4[2;l]*
2)t 1 1 3 x.o4[1;1])-(4[3;1]t(1o4[1;1])*2)+(4[
2 ; 2 ] 2 ) t 3 04 [ 1 ; 1 ]
[ 4 G ] +5 0
V


118
of coordinates of B' with A temporarily held at (0,0,1)
are given by
V
yB.
ZB
= X2 cos 0 Y2 sin 0
= )T2 sin 0 + Y2 cos 0
(IV 30)
where
= sin A'B sin A
Y? = -(sin AA cos A'B + cos AA' sin A1B cos A)
^ (IV 31)
Z2 = cos AA' cos A'B sin AA' sin A'B cos A
A = AQ + N (0-0Q).
The pivot may be moved to a general location A, by
Eq. (IV-29).
All of the above tools have been collected into a
group of compact and efficient APL functions to provide
interactive access to the spherical and planar analysis
problems. This set of tools create the desired output
for all input positions simultaneously. The use and list
ing of these programs are described in the Appendix.


38
tan 8-
tan d>
^3 sin a.
(II-17)
From the cosine law for the same triangle,
cos ic = cos cig cos B3 (11-18)
Similar to Eq. (11-12), p3 can be expressed implicitly by
tan p3 = tan 3 cos ac (11-19)
Substituting Eqs. (11-17, 18) into Eq. (11-19), we find
Hk l
p3£ = k [tan (sin B3 cot a3)]T (11-20)
dx
Consider the spherical triangle abc; write the sine and
sine-cosine laws
= sin ic sin a3
(II-21)
Y2 = -sin ic cos a3
where = sin a~b sin
Y2 ~ -(sin be cos ab + cos be sin ab cos 0^)


TOP
FRONT
Figure VI-6
SIDE
I 59


165
point c as a design position. Analysis will yield the
angular motion of link AB. This motion is to be coordi
nated with an input crank. The problem is now reduced
to a function generation problem where the solution must
yield a four-bar drag link so that a continuous motor
drive will provide the input. The cycle time of 1.8
bird per second translates into an angular velocity of
the input crank of 11.3 rad/sec. This problem can be
solved by any of the mechanisms listed in Table 4 and by
any procedure in Section 4 of Chapter III. We will seek
a drag link by choosing a short base link (AB) of ten
degrees. The angular parameters of the input link (GH)
and the analyzed link (AB) complete the necessary data
for problem specification.
l
k
02
units
0
0
3.14159
1 .0724
rad
1
1
11.2923
-2.344
rad/sec
2
0
4.1888
0
rad
3
1
11.2923
-2.344
rad/sec
This
data
set can
be used in the
APL function
LINK3 to
yield
the
locus of
link GF. The
coordinates
of the
four-bar drag link solution are


Table 2: Colaminar Motion Synthesis
Problem Statement
Available Mechanism Constraints
Specified Functions: F=2; Ug = f(ij^-);
Ve =
Graphical Form: m = 2
FSP
MSP
TSS
Mechanism
Plane [m]
Special Knee
Joint P = e
Four-Bar
[33*
Y
N
{q=8, L=4, p4=
=0}
-
Five-Bar
Y
N
{q=12, 1=6, p4=
=4}
1
[3], £4]
C
2
[4], [5]
D
3
[1 (or 5)]
F
Six-Bar
Y
N
{q=l4, L=7, p4=
=6}
Watt 1
[53, [63
F
Watt 2
[3 (or 5)3
-
Stephenson 1
[2 or 5)3
B
Stephenson 2
[1 (or 6)3,
[4]
G (or D)
Stephenson 3
£53
C


1 33
identities, Eq. (V-17) can be written as a quadratic
with coefficients,
A = X51X3 Y51Y3 Z51Z3 + C0S a67
B = 2(X51Y3 + Y51X3) (V -18 )
C = "X51X3 + Y51^3 Z51Z3 + C0S a67
For completeness, the other angles can be found
by the appropriate use of sine, sine-cosine, and cosine
laws. Writing the cosine law
Z4 = Z12 (V -19 )
where Z12 = sin sin 02 + Y] cos e2) + cos 23 Z1
X-j = sin a5l sin 01
Y-j = -(sin a-|2 cos 5] + cos sin a5i cos 0-|)
Z-| = cos a-j2 cos a¡.j sin a^2 sin cos 6^
Z4 ~ cos a 3 ^ cos ct4^ sin c*34 sin a ^ 3 cos 0 ^


181
11. Sandor, G. N. "Principles of a General Quaternion-
Operator Method of Spatial Kinematic Synthesis."
Journal of Applied Mechanics V. 90, Series E,
March 1 968 40-46 .
12. Dittrich, G. "On the Instantaneous Geometry of Move
ment of a Rigid System in Spherical Motion." (Ger
man). Ph.D. Dissertation. Mechanical Engineer
ing Faculty of the Rhine-Westphelia Technical In
stitute, Aachen, July 1964.
13. Kamphuis, H. J. "Application of Spherical Instantan
eous Kinematics to the Spherical Slider-Crank
Mechanism." Journal of Mechanisms V. 4, 1969, 43-
56.
14. Bisshopp, K. E. "Note on Spherical Motion." Journal
of Mechanisms V. 3, 1969, 159-166.
15. Oleksa, S. A. and Tesar, D. "Multiply Separated Posi
tion Design of the Geared Five-Bar Function Gener
ator." ASME Mechanisms Conference 1970, Paper No.
70-Mech-16.
16. Duffy, J. and Rooney, J. "Notes on the Development
of a Unified Theory for the Analysis of Spatial
Mechanisms Using Spherical Trigonometry." Infor
mal discussion at Cranfield Institute of Technol
ogy, May 1962.
17. Todhunter, I. and Leathan, J. G. Spherical Trigo
nometry Macmillan and Co. Ltd., London, 1932.
18. Dowler, H. J. Curvature Transformations for Spheri
cal Synthesis. Dissertation to be presented at
Liverpool Polytechnic in 1974.
19. Duffy, J., Dowler, H. J., and Tesar, D. "The Gen
eralized Theory of 3MSP in Spherical Motion." Re
port No. 1, NSF Grant GK-35335.
20. Duffy, J., Dowler, H. J., and Tesar, D. "The Gener
alized Theory of 4 and 5 MSP in Spherical Motion."
Report No. 2, NSF Grant GK-35335.
21. Tesar, D., Riddle, D., and Duffy, J. "Kinematic Syn
thesis and Analysis of Geared Spherical Five-Bar
Mechanisms for Function Generation." Report No.
4, NSF Grant GK-35335.




90
Both transformation matrices M-j and M0 are orthogonal
such that their inverse is identical to their transpose.
(M^= and M^= M^) Reversing through the inverse
rotations, we find the position vector c of point c in
the original system position by
c = m{ c" (IV-11)
The assumptions of rigidity of links BC and CD,
as well as the constraint of the unit sphere, impose the
following set of conditions on points, B,C, and D.
b*c = cos
BC =
constant
A /\
c d = cos
CD =
constant
A /\
b* b = cos
0 =
1
A A
c*c = COS
0 =
1
A A
d *d = cos
0 =
1
(IV-12 )
Differentiating Eq. (IV- 12) yields a set of con-

ditions for the velocity vectors 6 ,c and 3.
6 c + Bc = 0
c* 3 + c* 3 = 0
A A
b b = 0
c c = 0
3*3 = 0
(i V -1 3 )


1 03
A'B=BC=CD=DF=90
BA' = CA"= CD= 90
(b) RPR Dyad
Figure IV-8


Table 8:
Problem Statement
Angular Coordination of Three Cranks
Available Mechanism Constraints
Specified Functions:
F=2; tyy f(ip2); fy* f(^) Mechanism
Graphical Form:
=2; = 3; m^ = 4
Six-Bar
{q<10, L=4, p4=2)
Watt 2
Stephenson 3
FSP
Planes
[m^
[1,4,5]*
[1,6,2]
MSP
Y
TSS
Y
rsi


Table 1: Path Synthesis
Problem Statement
Available Mechanism Constraints
FSP
Specified
Graphical
Functions: F=1; V = f(U )
e e
Form: Point e in m
Mechanism
Plane [m]
containing
Point e
Special
Knee Joint
P = e
Four-Bar
[3]
(q=8, L=8, p4=4>
-
Five-Bar
MSP
Y
Y
{q=12, L<12, Pi
1
2
3
Six-Bar
(q=14, L<14, p
Watt 1
Watt 2
Stephenson 1
Stephenson 2
Stephenson 3
[3], [4]
[4], [5]
[1 (or 5)]
>}
[5], [6]
[3 (or 5)]
[2 (or o}]
[1 (or 6)], [4]
[5]
C
D
F
F
B
G (or D)
TSS
N
N
N
PO
o


30
a+D, b+H, c+C
T+2, 2+4, 3+5, 4 = 3
2. Let 6-| = 42,> 02 = £4, and find the motion
specifications for link 3.
{a,3,Y>3£ = {abeb,Yb}
3. From the curvature transform find the constraint
link 1 with circle point B and center point A.
4. Let 0]£ = ip j? % + t and 0 21 = and find the
motion specification set for link 5.
{a,B,y}5 = {aC£0c/Yc}
5. From the curvature transform find the constraint
link 6 with circle point F and center point G.
10. Double Colaminar Synthesis with Common Knee Joint
This problem has had several valuable applications
in planar kinematics. It is expected that it will also be
valuable on the sphere.
10.1 The Stephenson 3
1. Make the symbolic transfer (Fig. III-12a)
e+C, 3+4, 2+5, T+3
2. Defind the motion specification set forlink
5 as
(exB 5 y} 5£
{Uc,Vc^2}


73
7.The motion of link 5 with respect to link 2
is given by
Eq- (II_3) where -
8. From the curvature transform find the con
straint link 6 with circle point F and cen
ter point 6.
9. Points A, D, and G construct a rigid ternary
link (2). Points C, D and H construct a rigid
ternary link (4). Inverting to the original
system by fixing link 1 satisfies the problem.
7. Angular Coordination of a Crank with a Two-Link Chain
The motion of a two-link chain whose angular state
is coordinated with a crank is closely associated with the
three-link chain. The problem is depicted in Fig. 111 9 a.
The basic specification follows that of the three-link
chain (cab ++ abc).
7.1 The Stephenson 2
1. Make the symbolic transfer (Fig. 111 -9 b)
a+B, b+C, c+A
T+2, T+5, 3+4, 4+3
2. Fix link 3 by allowing 0] = tt and 2a =
4>3 to find the motion specification for link
4.
{cx.B.y}^ tai.6arYat}
3. From the curvature transform find constraint
links DF and GH. Points D and G come from
the circle point laws and points F and H come


104
H(R)
K(P)
777777
FG = JK= 90
EF= e
EB = 90+ A
EC=90+A
Figure IV-9


CHAPTER I
INTRODUCTION
The field of kinematics is becoming a major con
tributor toward advances in industrial automation. Ma
chines are required to simulate motions of human operations
as well as motions beyond human capability. It is demanded
that these functions be accurate, reliable, and efficient.
Such demands have created growing interest in the field
of machine design. Special spatial motion problems such
as transfer mechanisms in automatic processing equipment,
function generation between nonparallel axes, and path
generation on a sphere cannot be solved by planar mecha
nisms. The relative importance of the spherical system
can be obtained by considering that perhaps 75% of the
industrial linkage problems can be satisfied by planar
mechanisms, 15% by spherical mechanisms, and the remainder
by more complex spatial mechanisms.
Coplanar synthesis, or design of planar linkages,
has reached a high level of sophistication, usefulness,
and completeness a development which has occurred chiefly
over the past 15 years. Analytical, graphical and special
design procedures are now widely available to create four-
1


86
(O
Figure IV- 1


67
where o
c£
dx
I X0
ta n
c£
dx
tan (sin 3c cos ac )
J T
6.1 The Stephenson 3, Case 1
1. Coordinate the problem statement and the
Stephenson 3 by the symbolic transfers
(Fig. 111 6 b)
a-+F, b->-D, c-*C
1 -*3 2-> 1 3->-4 4->-5
2. Find the first motion specification set for
link 4 as
{ a > 6 j Y £ {ab£,2b£Yb£}
3. From the_curvature transform find the con
straint GD (Fig. 111 6b ) .
4. Find the second motion specification set for
link 5 as
ta>B'Y}5l
'yc2
5. From the__curvature transform find the con
straint BA (Fig. 111 6 b).
6. Pin joints C,D, and G form a rigid ternary.
Pin joints B and C form a rigid binary to
complete the Stephenson 3 mechanism.


111
cu
O =
O =
cu x a
Qx 6
6 h
£ 6
Figure IV-I3


74
Figure 111-9


134
After solving the input-output Eq. (V-17), then Eq.
(V-19) can be written as a quadratic in x 4 with coeffi
cients,
A = cos a 4 cos + sin ou4 sin
or A = cos(a34-a45)- Z]2
B = 0
C = cos(x34+a45)- Z12
Writing the half-tangent laws for 03 0^, 0g,
x
3
+ V4>
(x 12 + ^4 ^
x
5
(Y 21 + Y4^ X21 X4
(x21 + X4) = Y21 Y4
- Z
12
(V-20)
and 0^
(V 21 )
(V 22)
Y5123 ~ Sin a67
X 51 2 3
'5123
Y5123 + Sin a67
(V-23)


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
ABSTRACT vi
CHAPTERS
I INTRODUCTION 1
II SYNTHESIS CONCEPTS 6
Constraint Systems 11
Solvable Problems 15
Synthesis Tools 19
Spherical Chain 32
Inversion 39
Curvature Transformation 43
III DESIGN PROCEDURES 51
Path Synthesis 53
Colaminar Synthesis 54
Angular Coordination of Two-Link Chain ... 56
Angular Coordination of Two Cranks 59
Path Coordination 61
Angular Coordination of a Three-Link
Chain 64
Angular Coordination of a Crank with a
Two-Link Chain 73
Angular Coordination of Three Cranks 77
The Angular Coordination of Three Ad
jacent Planes 77
Double Colaminar Synthesis with Common
Knee Joint 80
Colaminar Motion Coordinated with a
Crank 82
IV SPHERICAL DYAD ANALYSIS 85
Prismatic Equivalence
Four-Link Chain Equivalence
92
93


178
V Z*-!l QD n
[1] Z-(( 2t//,0) QP 0Cl],*D[2])tlHD CP P[l],-P[2]
V
v z-y £P y
Cl] z-(-/yxy) ,+//x^y
v
Cl]
C 2 ]
[3]
Cl]
C2]
[3]
[4]
[5]
[6]
C7]
C8]
[9]
CIO]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
V Z*-N CR XiTiR
ZMo2)| Y[2] ATAN Y[1 ]
P-(+/Y*2)*0.5
Z+(P**A/)x$(2,A/)p ( 207*) ,102M iA/)xy+(o2)xl + iAr
V
V T LINK3 LiKiNiAiBiCiDiXiYiZiRiRDiXDilDiZDiXDDiYDDiZDD
;ZZ ;DD;ND
HpDpO '
4 [A/; 2 ]-*-~lo( ioL[ 1 ])xior[ A'- 1 4 ; 1 ]
4[tf ;1>(( loL[l])x(2or[-V;l] )*2o4[A/;2] ) VATAH
20L[l])*2o4[//;2]
4 [ 2 ; 2 ]*-[ 2 ;l]x(loZi[l] )x(20Z'[l ; 1 ] )t 204 [ 1 ;2]
4[ 2 ;l]-*--r[2;l]x(3o£,[l] )x(io!T[l ;l])x( 204 [ 1 ;l] )*
2
4[3;2]-(((l OL [l])x(2[3;l]x2o2[l;l])-x/2,[2; 1 1],loP[1
;!]) + (104 1 s 2] ) x 4 [ 2 ; 2 ] 2 ) 2 OA [ 1 ; 2 ]
4[3;l]-*-(3oL[l])x(x/2,7'[2;l],4[2;l], 1 2 1 oTl 1 ; 1 ] A [ 1
; 1 1] )-(2x ,o4[1 ; 1 1])x(T[3;1]xlOTl1;1]) + x/TL
2; 1 1],2o7[i;1]
Ail 4 ;3>m 4 ;2]+P^-( ( 2oL[ 1] )xioT[i 4 ; 1 ]) VATAH
20T1 4 ;1]
4 [ 2 ; 3 ]-*-2[ 2 ; 2 ] +RD+TL 2 ; 1 ] x5--( 2oL[ l ] ) x ( ( 20P-i +P ) *
2or[1;1])*2
A [ 3 ; 3 >r[ 3 ; 2 ] + ( T 3 ; 1 ] *li ) + 2 x p[ 2 ; 1 ] x ( 2oL [ 1 ] ) x ( /
20J? T 1 ; 1 ] ) x ( ( r[ 2 ; 1 ] x ( 20i? ) x lore 1 ; 1 ] ) -RDx (loji) x
2or[1;1])i(2o2T 1 ;1 ] ) *2
L0:CflsP-ZZ-4[l;l] -AS 4[1;2];CP
P- 0 12 0 MSP 4
P^P,DP(7x("20 + /P[ ;x3]xp[ ; 4 5 6 ] ) ( ~20ZZ+ x$fl[ ; i
3]),[1.1] "20ZZ+.xfijPC ; 4 5 6]
3 pen
-*0 x \ 2 *p ,/
Y-( loL[2] ) x ioT[ y+--(l 2 x .oL ) t ( 2 1 x .OL )x 2or[A/ ; 2 ]
Z-( 2x .OL ) (lx ,oL ) x2o7[P ;2 ]
4 [ .V ; 2 >~lo(yx2or[ //; 1 ] ) y x 1 oP[ A7; 1 ]
4[ff;lX(-Uxio27[7/;l])i-yx202,[//;l])i204[A7;2]) VATAH Z*
204[A7; 2]
yzm y-*-1 + y) ,(x*-itx) ,z-i tz
.YP-PC 2 ; 2 ] x (1 oL[ 2 ] )x2or[l ; 2 ]
yp+-y[2;2]x( 2 1 x.oL)xlor[l;2]


47
Reduce to a quadratic in polar form by the substitutions
s = r cos 0 and t = r sin 0. Collect terms in the form
Ar2 + Br + C = 0 (11-37)
where
3 2 2 3
A = a^sin 0 + a2sin 0 cos 0 + a4sin 0 cos 0 + a7 cos 0
O
B = sin 0(3a-jy + a2x + a3) + sin 0 cos 0(2a2y + 2a4x + .a5)
p
+ cos 0(3a7x + a4y + ag)
2 2
C = sin 0 (3 a -j y + 2a2xy + a4x + 2agy + agx + ag)
2 2
+ cos 0(3ayX + 2a4xy + agy + 2agx + agy + ag)
and 0 £ 0 £ it ;
Eq. (11-37) can now be solved for r. and substituted into
the homogeneous coordinates
= r. cos 0. + x; z2 = r.. sin 0. + y (11-38)
The Gm coefficients (Eq. 11-33) are functions of
Y a 1 one si nee
a
= f(y) and 6 = f(Y)
(11-39)


137
Applying the half-tangent identities to Eq.
(V-25), a quadratic in can be written with coeffi
cients,
A1 cos a51 Z34 Sin a51 Y34 C0S al2
B1 = 2 sin a51 X34 (V-26)
C1 = cos a5i Z34 + sin agl Y34 cos ot-^
At this point either 63 or 04 may be chosen as an input,
but this would leave the other, as well as 0^, unknown
and hence cannot be solved yet.
Writing the cosine law for the pentagon loop 67345,
Z
345
cos a67
(V 2 7 )
where Z^45 = sin a56(x^4 sin (J>5 + Y^4 cos 5)+ cos a56 Z^4
Y '
*34
X3
cos
CD
Y3
s i n
04
Y'
T 34
= COS
a45
s i n
9 4 +
V3 C0S
64)
sin a45 Z'
7'
*34
= sin
a45
(X3
s i n
9 4 +
V3 C0S
-f
CD
cs a45 Z;


145
Dyad Analysis
1. Based upon a combination
of vectors and spherical
trigonometry
2. Treats one closure at a
ti me
3. Treats all states with
respect to input posi
tions simultaneously
4. Is easily accessible to
higher order states
5. Finds translational and
angular states in either
an absolute or relative
reference
6. No iteration necessary
per mechanism
7.Dead centers are handled
but not predicted
Displacement Analysis
1.Based upon spherical
trigonometry
2. Treats all closures simul
taneously
3. Treats one input position
at a time
4. Is awkward for higher
order states
5. Finds relative angular
displacement
6. Solution of polynomials
(above quartic) is itera
tive, but polynomial
development is closed form
7. Easily develops an under
standing of special proper
ties such as dead center
predic tions
The user must decide what type of information is
necessary in his examination. Most often the dyads ap
proach will prove more appropriate, but the displacement
analysis can give a higher level of understanding of the
nature of closure loops.


CHAPTER V
DISPLACEMENT ANALYSIS
The previous chapter discussed methods of analyz
ing a specified closure branches of a mechanism including
those with inputs that cannot rotate 360 due to closure
limitations. A higher level of understanding can be
achieved by treating algebraically all angular input-
output relations of a given mechanism simultaneously.
This section will seek the development of input-
output functions as well as all significant relative
angles of six-link spherical mechanisms. A similar treat
ment for four- and five-link mechanisms is fully treated
in references [21, 23].
Watt 1
For the Watt 1 mechanism (Fig. V-l), the cosine
law for quadri 1 atera 1 1 237 yields,
Z
1 2
cos a37
(V-l)
where


Figure VI-5
1 56


75
from the center point locus. Points C, D
and G construct the rigid ternary link 4.
4. Invert the mechanism to the original system
by fixing link 2.
7.2 The Stephenson 1
1. Make the symbolic transfer (Fig. III-10a)
a+D, b->-C, c+F
T+l, 2+4, 3+5, 4+3
2. Allow the rotation parameters of the three-
link chain to be
u
TT -
*4t
21
TT +
*21
31
^3£
and consider its inversion by fixing link 3.
3. Find the motion specification set for link 4
relative to link 3 as
4. From the curvature transform find link 6 with
circle point G and center point H. Note that
points C, D, and G form the rigid ternary link
4.
5. Find the motion specification set for link 5,
(ot ,y)5£
{ctb£eb£,Yb£}


52
are specified. If three positions of the moving plane 2
are specified, the circle point locus (locus of moving
pin joints) is the entire moving shell. This shell can
be uniquely represented by a pair of homogeneous coordi
nates. Each coordinate has a single infinity of choice.
Each point in the homogeneous plane maps one-to-one to a
center point in the reference shell 2. Each pair of circle
points and center points construct rigid cranks. For the
2
three-pos i ti on problem there are a total of cranks.
The locus of homogeneous circle points for four positions
of 2 is a cubic curve (Eq. 11-34). There is a single
infinity of points on this curve and hence there is a
total of 00 cranks. The five-position circle point locus
is found by the intersection of two cubics. There is a
total of 6 cranks for the five-position problem.
The designer is expected to have started by match
ing his problem with the problem charts of Chapter II.
After selecting an appropriate mechanism based on the
number of free optimization parameters p, the nomen
clature of the chosen chain should be referenced. The
procedures then outline a step by step method of solution.
All of the procedures will be produced for zero
order (displacement) positions. Higher order properties
are obtainable for each case by differentiating with re
spect to the generic independent variable x, i.e.,


171
7 Z+W ABI A
[1] Z-(p/l)pO
[2] Zil 4 ; 2 3-*-"*10.4 [ 1 4 ; 1 ]
[33 Zil 4 ;l3-*-(-4[l 4 ¡2]f2oZCl 4 ; 2 ] ) VAT AN Ail
4 3]*20Z[1 4 ;2]
[4] Z[ 2 ; 2 3-*-4 [ 2 ; 1 J Oo4 [ 1; 1 ]
[5] Z[2;l3<*-(4[1 ; 2 33-.x4[2; 3 2])*[1;3]*2
[63 Z[3;2 3-*-((4[3;l3x0O4[l;l])+4[l;l3*4[2;l3*2)*l-4[l;l3*
2
[7] Z[3sl3-*-((>tCl ;33x>l[ls 2 3]-.x4[3; 3 2 ] ) 2 x [ 2 ;
3 3 *4[1; 2 3 3 Xi4[ 2 ; 3 2 3 ) *A i 1 ; 3 ] *3
7
V Z-rB AUG CU i A \X \ AD
[13 JN-(l+pB), 3 3
[2] A-JYpO,fl[;3],(-fl[;23).fl[; 4 5 fi 3 Ci ; 2 3 .-Cl ; 1 3 [
1.130
[3] 4-*-4+(tfpB[ ; 4 5 63 ,C[; 4 5 63,(-B[ ; 33 ) 0 [
1.13B[ ;l3) + 7/pCC ; 4 5 6 3 0 Ci ; 3 ] (-Cl ; 2 3 ) ,B [ ;
23,-B[il3.[l.l] o
[43 X*-(N- 0 0 2 )p(0,0,[1.53 fl[ ;53) + (0,i7C ;43,C
1.53 B[j63)+BC ¡43.0.C1.53 CC;63
[53 Z+X QUADIV A
[63 X-*-(N- 0 0 2)p(0,0.[1.53 B[ ;83) + (0,i7[ ;73,C
1.53 B[;93)+B[;7].0,[1.53 7 [ ; 9 3
[73 AD+Np 0,fl[;63,(-B[ ; 5 3 ) ,B[ ; 7 8 9 3 Ci ; 5 3 ,-<7[ ; 43 C
1.130
[83 AD+AD + (NpBl; 7 8 93 .CC; 7 8 93 (-B[ ; 63 ) 0 [
1.13B[;43)+BpC[; 7 8 9],0tCl;63.(-Ci;53),fl[;
53.-Bi ; 4 3 .[1.13 0
[93 Z-*-Z,U-(7Y- O O 2)p + /£x2,r23 Z,[
1.13Z) QUADIV A
[103 Z-Z,((Z[;i33+.*2)*0.5),[1.13(Z[; 4 5 63+.*2)*
0.5
7
7 B-4 CRANK X\AB\V\T\N ;7?;7?
[13 4*-3 + ,4
[23 r^(i?*-O+180)x^[33+^[5 3x"l + \Af^l + L( XI 4 3-*C 3 3 )**[
53
[33 B+UB+RxXlll) ,W*-n*Xl2l
[4 3 £-*-( ( loAB)*loT) ( (104B )x2or) ,[
1.1320AB
[53 B++/(.T+ 13 2 K.V, 3 3)pr.(-x/4[l 2 3 *T) ( x/A [
1 33. *T) .0. (4[ 3HT) ( -A 2 3*r-0o4[ 13 ),3t4)x£,[
23 S_, [ 1.1 3 2.
[6 3 B*-5,+/2,x^,[ 2 3 7?, [1.1 3 7?-,Vx( -£[ ; 2 3 ) .£[ ; 1 3 ,C
1.130
[73 B+B +/T*R ,[23 7?, [ 1.1 3 7?+i/x ( -/?[ ; 2 3 ) ./?[ i 1 3 C
1.130
7


45
G
G
G
G
G
G
31
4 i
5 a
6 i
11
81
cos cos y£- 1
cos 6sin
sin a^cos B
sin 6
cos acos y sin asin Bsin y 1
cos asin Y£+ sin a^sin Bcos y
Equation (II-33 ) is listed for finitely separated
positions (FSP) only and must be differentiated with respect
to the independent parameter y to reflect the role of in
finitesimally separated positions (ISP). The G notation
allows the same body of analytics (and computational tools)
to solve Eq. (11-31) for all cases of MSP.
There are four distinctions between the curvature
transformation program listed i n the Appendix and that cov
ered in reference [19]. First, the original work was cen
tered upon the special reference transformation. This is
an effort to reduce the number of parameters for special
cases and required that {a,B,Y}£=0 = {0,0,0}. The general
reference transformation allows the arbitrary choice of


106
FOUR-BAR
SIX-BAR
ADD DYAD BETWEEN
WATT I
WATT 2
STEPHENSON I
STEPHENSON 3
e, ^e2 or e3 ,e2
e3,g or e, ,g
el e3
e2 ,g
Figure IV-10


163
last two describe the matched velocity. The curvature
transformation yields six real solutions of which any two
may be chosen to form the four-bar constraint. The chosen
constraints are
A(X Y Z) = (-.9046, -.2563 .3406)
B(X Y,Z) = (-.6259 -.6441 .4398)
C(X Y Z ) = (-.7070, .7072 0)
D(X Y Z ) = (0,0,1)
B = 28.23
BC = 90.74
CD = 90
Since CD = 90, this mechanism is the equivalent of the
planar slider-crank mechanism (Fig. VI-8). This device
will constrain the specified point path but it is not
yet coordinated to the input drive of the picking line.
Analyzing the four-bar ABCD will yield the position,
slope, and curvature of any point b along the path. Ar
bitrarily choose a point b along the path from the pickup
to release and choose a relatively slow velocity magnitude
in the direction of the slope at that point. The cuspidal
motion is satisfied by the four-bar geometry, so the
second velocity point can replace the cuspidal motion of


2
five- and six-bar mechanisms. Proper application of the
four tools of synthesis (inversion, angular cognates, path
cognates, and curvature theory) allows the design of the geared
five-bar, Stephenson six-bar, and Watt six-bar for a wide
range of motion problems basic to industrial application.
Since the spherical linkage has properties which
are distinct from those of coplanar linkages and repre
sents perhaps 15% of potential industrial application, in
creased competence by the designer is warranted. Fortu
nately, much useful work on the spherical four-bar has oc
curred, although its general level is approximately that
of coplanar synthesis 15 years ago. In particular, few
design procedures have been evolved for linkages more com
plex than the spherical four-bar, such as the spherical
bevel geared five-bar, and equivalent six-bar linkages.
The development of the spherical tools of synthesis would
fulfill this need for the spherical system. A unified
theory of spherical kinematics would not limit itself to
the suggested 15% application. Rather it would provide
common tools for both the plane and the sphere. Since
planar geometry is a proper subset of spherical geometry,
the concepts would cover 90% of application problems. Such
a unified theory may prove to be the key to a total design
package with the aid of dual numbers. Future efforts may
extrapolate spherical kinematics to provide unified ana
lytics of planar, spherical and spatial motions.


12
2
Figure 11-2


89
where
M,
cos 4>
-sin 0
sin (p 0
cos

0 1
However the position of d' can be defined by
Xq. = sin BD sin yq =-sin BD cos <£
zD, = cos BD
sin BD = Vi -zD = VV
and
D + yD
RD '
Substituting Eq. (IV -8) into Eq. (I V-7) ,
(IV-8)
M =
2 Rn.
~7nt
'XD1
0
XD'
-yD
0
0
0
RD'.
(IV 9)
The position of c" is given by the elementary
expressions,
x^ = sin BC sin 0
y^ =-sin BC cos 0
z^ = cos BC
(IV -1 0)


122
Let 0^ = + 6^, then the half-tangent laws yield
= S1n a37 Y12
x3 X
12
12
si" c37 + Y]2
(V 4)
where
X12 = X-j cos @2 Yi sin 02
Y12 = cos a23 (X^ sin 02 + Y-j cos 02) -sin a23 Z-|
The angle 0y can be found by another set of half-tangent
laws of the form
x7 '
sin a37 Y21
21
k21
si" a37 + Y21
(V-5)
where
X21 = X2 Cos 61 ^2 sin 91
Y21 = cos a41 (X2 sin 0-| + Y2 cos 0-j) -sin al Z
41
X2 = sin a23 sin @2
Y9 = -(sin a^2 cos a0Q + cos a19 cos a9-, cos 09)
23
12
23
Z2 = cos cos a23 -sin sin t23 cos


87
BD = cos'1(bd ) (IV-1 )
The interior angle ) CBD (6) can be written in terms of
the spherical cosine law
cos CD = cos BC cos BD + sin BC sin BD cos 0. (IV- 2)
Solving for cos 0, and taking into account the possible
range, we find
-1 <
cos 0
cos CD cos BC cos BD
sin BC sin BD
(IV 3 )
Satisfaction of this condition defines whether or not the
dyad can be physically connected.
In order to find the position vector c, we must
first rotate 6 to (0,0,1) by a rotation u about the
-x-axis and then a rotation v about the -y-axis. This
transformation can be described in matrix form by
0
0
1
(IV 4)
where


160
The time state motion specification is now complete for
the 3 ISP case (PPP). The curvature transformation yields
B(X,Y Z) = (.01 324, .0475 .9988)
0b(X,Y,Z) = (.2551, .1769, .9506);
Fig. VI-6 illustrates this solution. It is felt that the
TR suspension is a superior suspension in accordance with
the initial objective. The most significant point of the
solution is that the arbitrary decisions could be modified
to suit an experienced suspension designer. This example
illustrates the creative capabilities of well-formulated
analytics translated into efficient computer programs.
Poultry Transfer Device
Like most industries, the poultry industry faces
the need to automate processes that have become undesirable
to human labor. The poultry preparation process is divided
by a break between the picking line and the eviscerating
line. Current practice connects the two lines by a con
veyor carrying birds with random orientation. Workers
pick up each bird and place it in a shackle. This task
is messy and unrewarding. Early automation attempts have
focused on picking up the irregular objective and rehanging
the bird by machine hands. The uncontrolled variation of
too many parameters cancelled the reliability of such an


43
M
n
m
Mn Mr
r m
(11-29)
to obtain
B sin 1 [(an)" (a13)J¡ + (a]2)r (^m + ^a13^r ^a33^m ^
m
n -1
a tan
m
(a21 V ^a13^m + ^a22^r (a23^m + ^23^^33^
(a3l>r (al3>m + (a32>r ? ¡¡
(11-30)
= tan 1
m
(an)rh2):+h2>nr:
(all (all^m + ^a12^r (a21)m + ^a13^r (a31 ^rri
This transformation allows the reference link
to be temporarily changed so that the appropriate motion
specification set (a,8 ,y}^ can be fed to the curvature
transformation.
Curvature Transformation
A detailed description of the 3, 4 and 5 multiply
separated position (MSP) transformation is given in ref
erences [19, 20]. In any case, the question, "What is the


8
/
Figure II-l


83
U
Figure 111 1 3


105
The spherical configuration is built from four RRR dyads
with the following link lengths,
1.
BCD
BC,
CD
2.
BEC
O
O
O'*
+
3.
EFG
V
90
4.
HJK
HJ ,
90
If the arc lengths AB BC ,CD A-| ,A£ >HJ are quite
small (largest ~.5) then a planar approximation with
errors less than 10 ^ can be expected. The error can be
further refined by reduced magnitudes on the largest link
length that is not 90.
Spherical Six-Bars by Dyads
Figure IV-10 illustrates a collection of common
mechanisms that can be solved directly by dyads. Each
mechanism is built from a four-bar substructure and then
combinations of dyads applied to the ternary points e.¡
and to fixed pivot g to provide the analysis of a large
percentage of mechanisms. The primary exception to this
analysis procedure is the Stephenson 2 mechanism.
If (or cog) is the input to the Stephenson 2
shown in Fig. IV-11 then the input loop is based upon a
pentagon rather than the quadrilateral loop common to all
other six-bar mechanisms. This means that the Stephenson 2


147
Figure VI 1


CHAPTER III
DESIGN PROCEDURES
The objective of these design procedures is to
present how the spherical four-, five-, and six-bar mecha
nisms can be determined dimensionally by use of the synthe
sis tools of Chapter II. There are a few hundred potential
design procedures based on the problem charts in Chapter
II. This presentation will not be all inclusive but will
present at least one solution for each problem type. A
similar effort by Prof. D. Tesar for planar mechanisms
has proven to be useful in both undergraduate and graduate
study as well as in applied industrial practice. The
consistent and compact nomenclature allows the designer
to approach complex problems with fundamental tools.
Optimization has become the central question of
planar kinematics. It will also become the key problem
to spherical kinematics as well as all fields of machine
design. Optimization must apply weighting factors of de
sign characteristics to a large collection of solutions.
The size of this collection can be realized by considering
how many sets of motion specifications {a,B,y} are used in
the curvature transformations and how many + 1 positions
51


173
[5] Z-Z,[l.l](B+lo: )*(+/*[ ; 3+J]xyi[ ; 18+1] ) + + //.[ llx/ic ;
21+13
[63 Z+-Z,(-(Z[ ;23*2) + SOZ[ ;13)+(fl*loZ[ ;l3)x(+//l[ ;6+l3x[ ;
18+J3) + (2x + //[ ;3+J3M[ ;21+J3 ) + +//}[ ; J3x[ ; 24+13
7
V Z-STIHV AiM\B\C\K\L\MZ\bV\\V\R
[ 1 3 R+A [ 1 ; 1 2 3 + 2
[2 3 M+(3 ,C ,Ot( -C*-(A [ 1 ; 1 3 x<4 [ ; 2 3 )-/![ 1 ; 23X[ ; 1 3 ), [ 1 ; 1 3x
Ai ;13)+4[12]x[ ;23).0.0,0,[1.13U-1+P
[33 M K*-( 6 xK ) 3 3)p 2 1 3 4 $(6 ,K 3 3 )ptf
[43 Z- + /tfx 13 2 fi?Kp£,£,[2.53 7>-4[ ;L+- 1 2 3 10 11
12 19 20 21 28 29 30 37 38 39 46 47 4e3
[53 !I3+(B,C,0 ,(-£>U[l;l3*[;53)-[l;23x/l[;43). (fl-U[ 1; 13
x/1 [ ;43)+/1[1;23x4[ ;5l),0,0,0,[l.l3 Kp0)if
[63 M3 + (K+( 6xK) 3 3)p 2 1 3 4 ?(6,K, 3 3)p:*f3
[73 Z-*-Z ,(+/M3 x D ) + +/:!* V*- 1 3 2\KpV,V,
2.23 V-Al ;L + 3 3
[83 //4+-(5,C,0t(-C-*-(/l[lil3x4[ ;83)-4[l;23x4[ ;73).(S^-(/l[l;l3
xA ;73)+4[l;23x/l[ ;83),0,0,0,[l.l3 IpO)iR
[93 M4+-(iM BxK ) 3 3 ) p 2 13 4 ? ( 6 ,K, 3 3 )pM4
[103 Z+-Z ( + /i'/4x£) ) + ( 2x+/',3x V) + + ///x 13 2 ^p,4,[
2.23 A+Al;£ +63
[113 Z+- 2 1 3 fi? C A'. 6 9 ) p Z
V
7 Z+-Sn /I ;*5 ;Y5;Z5 ;Y3;Y3 ;Z3 ;*51 ;Y51 ;Z5 1 ;T
[13 Y5-l x 0/1 [ 4 83
[23 Y5+--(l 2 x.4[ 3 43 )+ 2 1 2 x.o[3 4 83
[33 Z5-*-(2x.O/[3 43 )- 1 1 2 x o/l [ 3 4 83
[43 *3-lx.O>l[6 73
[53 Y3-- ( l 2 x o A [ 2 63 )+ 2 1 2 x 04 [ 2 6 73
[63 Z3+-( 2x ,OA 2 63 )- 1 1 2 x.oA 2 6 73
[73 T-*- (o + lfl)xi + \3G
[83 Y51-(^5x207)-Y5X10T
[93 Y51-(( 20/1 [ 13 )x>( Y5xio?) + y5x207)-Z5xio/l[l3
[103 Z51-( ( 10/1 [ 1 3 ) > ) + Z5x2o/l[ 1 ]
[113 D+( *3**5l)+( 2Cvl[5 3 )-( Y3xYS1 )+Z3xZ51
[123 S-2x(*3x751)+Y51xZ3
[13 3 ^(Y3xY5l) + (20/l[53)-(Y3xY51) + Z3xZ51
[143 K+( XS*-1E~ 25 K )/ '+(J *2 ) -^*DxK
[15 3 Z-*-(X5/T) , 0.5 )+2x4,A+X5/D
7
7 Z+A XPR 3
[13 Z^((-4[23xB[33)-4[33xB[23 ) .( (4[l3x5[33 )->![33xB[l3) ,(/!
[13xS[23)-[23xB[13
7


Table 9:
Angular Coordination of Three Adjacent Planes
Problem Statement
Specified Functions:
F=2; £3= ffag); £5= f(^)
Graphical Form:
m1 = 2; m^ = 3; m^ = 4
Available Mechanism Constraints
FSP MSP TSS
Mechanism Planes
[m^ ,1^2 *01^]
Six-Bar
{q<10, L*4, p4=2}
Watt 1
Watt 2
Stephenson 1
Stephenson 2
[2,4,6]*
[4,5,3]*
[4,6,5(or 3,2,6)]
[3,6,1]
Y Y
rv>
cc


4
To Fay
'


155
/
/
/
(a)
Figure V I -4


102
The PRP dyad (Fig. IV-8a) has both pin joints B
and D at infinity in the planar case. This configuration
has similar characteristics to the RRP dyad.
The PPR dyad has two linear constraints. In the
plane, straight lines £-| and can be considered to lie
in the same link, in this case link 1. Also pin joints
B and C lie at infinity, perpendicular to lines and
2 respectively. Because of this, pin joints B and C
may be considered to both lie in link 1 (as always has
been the case for all dyads). Point A is offset by dis
tance h = AA1 from line The lines correspond
to great circles c-j ,c^ in link 1 of the spherical repre
sentation. Offset AA1 becomes A-j and the dyad becomes
BCD where AB = A-j + 90, and CD = 90.
As is apparent in all cases of the spherical
dyads, BCD, all joints are pin joints. The only distinc
tion occurs when 90 is the arc length between these pin
joints. Hence only one analysis routine is sufficient
to interpret the very large class of planar and spherical
mechanisms that can be constructed by dyads. To illustrate
an example of an assemblage of dyads, consider the eight-
link mechanism of Fig. IV 9. The planar configuration is
built from four dyads,
1. BCD (RRR)
2. BEC (RPR)
3. EFG (PPR)
4. HJK (RRP)


65
Figure 111 6


13]


78
9.1 The Watt 1
1.Make the symbolic transfers (Fig. III-lla)
a+A, b+D, c+G
T+l 2+2, 3+6, 4 = 4
2.Let 0i5 = ib?n and 0?f = Oo to find the motion
specification of link 4.
t'6+141 = tabiBb4Ybl}
3. From the curvature transform find link 3 with
circle point C and center point B.
4. Let
eu
II
=3
1
d2i
+
it
9 31
o*
ICO
UJ'
II
and find the motion specification set of link
6 with respect to link 4 (considering link 4
as fixed)
{c.,$,y)w = {oci6cn-Yci}
5. From the curvature transform find link 5 with
circle point F and center point H.
6. Note that points C, D and H form the rigid
ternary link 4.
9.2 The Watt 2
1. Make the symbolic transfer (Fig. III-llb)


176
[15] J-+- 43p234134124123 xJ-l+tf-0
[16] L2: H*-$( A yi[4 1 2 ;?]),(( A /1[4 1 3 ;P])+A Al 5
1 2 ;P]),((A Al5 1 3 ;P]) + A /l[6 1 2 ;P]),A Al
6 13 ; P-*-J[ I; ] ]
[17] £>E+(l*I)xU[7;I]xl+//) ,U[7 8 ; I] + x "2 + P ) (A [
7 8 9 ;J]+.x~3tP) ,(i4[7 8 9 1 0 ; I ]+ xtf ) (A [ 8
9 10 ;J]+.x3+ff),U[g 10 ; I] +. x 2 iH ) ,Al 10 ;I ]x l
[18] -*-L2xi 5>J^J+1
[19] -*-0x Zl-i 1 >N+-p Z2*-( 0 = Z 2[ ; 2] ) / ( Z2+-PBRT £)[;!]
[20] Z--( ( 7/, 3 )p 4 [ 1 ;7 ] ) [ J-l ] ( Zl/1 [ 2 3 ; J ] ) [ 1 ] ( Z xA [
4 5 6 ;7]),[0.7](Z--(//,l)pZ2)j.i4[7 8 9 1 0 ;J+\
3]
[21] Z1-Z1,-(A Z[1 2 4 ;!;])*A Z[i3;J¡]
[22] -+(~l+x2e>)*\NZl4-I+l
[23] L3:Z-(ZxZi),[i](ZxZ2),[0.5] Z-( 1 + (Z 1*2 ) +Z2* 2 ) *
0.5,0 + Q-*- 0 2 p7-*- 0 0
[24] D+lfZtiJlxAMtZ 2 1 ; J] ) ( -/Z[ 1 3 2 ; 9 6 8 ;J]),[1.5]-/Z[ iJ*-J + l]xA'1lB 5 7 ;I<- 2 3]
[25] (l + l26)xiJ7>ltpQ^O,Cl](-Z?[ ;l])fP[; 2 3]
[26] -*0 , Z-*-( FTJ Z ), ) ,0-( 1+ 0.5
[27] FOUR:X+{-/AI5 7 ;l]x/17[6 8 ; 1 ] ) + ."[ 4 5 ;l]xy};*/[
8 9 ; 1 ]
[28] Y+-+/AM17 4 ;1]xA.VT9 6 ;!])*£>
[2 9] C-*-(.Al 1 ]xP*3 ) + (l[2]xflxflxF) + (4[4]xEx:xP-202,)+4[
7 ] *ExE*E-*-loT+-( O f 1 8 )x 11 n
[30] C+C.l1.l](dC3]xPxp)+(4[5]xpxF)+(4[8]xFxF)+(4[l]x
3x^'xPxP)t(d[2]x(2xA'xPxF) + yxpx/}) + (i[4]x(2xyxPxF) + JxFxF
)+3x[7]xyxFxF
[31] J?-([6]xP)+(d[9]xF)+2xd[8]xyxF
[32] C^P,P + (3x4[i]xyxyxp) + (d[2]xyx(yxF)t2xyxP ) + (d[
4]xyx( yxD ) + 2xyx£) + ( 3x1[ 7]xyxyx,F) + ( 2xd[ 3]xyxP) + (d[
5]x(yx£)+yxp)
[33] P-*-( 0. 5*C,C-I7ff[ ;1 ] )*(D %D+-I/Cl i 2 ] ) +D (I*-D 0 ) / D+ ( C
[;2]*2)-4xx/C[; 1 3])*0.5
[34] Z l^(Px 20T*-T, T*-I / T) + X
[35] -+-L3 U-*-Q Z2-*-(Pxio2) + y
V
V Z+-PBBT P\B \ C \ N \ TQL \A ;Z?; J
[1] Z,02>lfflO,O + ,Z- 1 2 p0
[2] -*-( 5 4 3 =p P)/CT,CB %QD
[3] B+C-*- ( //-p /4 0 0 )p 0
[4] J--3,0 + £/-( 1 l x -2 +4 ) t 1 + ~3 +j4
[5] -+5xi //> J--J + 1,0+C[J]+-(P[J ]--i4 [ J ] + /+ xP[J- l 2] ) + £/+.xC[J-
1 2]
[6] U+U+D+UBW- 0 l]-.x(7[;/- 3 2]),P[P- 1 0]-.x<7[//-
1 2] )*£[//- 2 1 ] x (7 [ 7/ 2 3]
[7] -*-(2+I-3)x ir£L [8] Z-Z,[l] P7MP U-l,-U
[9] -*-2 ,P+P PDIV U
[10] OD:-+0,,Z+ 1 0 +Z,[1] OVAD 73
[11] C3:-y0,,Z* 1 0 +Z.T1] CUBIC P
[12] QT:-0,tZ<- 1 0 +Z,[1] QUART P
V


124
7\5
Figure V-2


120
Figure V-l


50
P P- = 1
dx dy
(11-42)
d ky ^ dky = 0 {k | k > 1}
dTk dyk
One computational tool can then be used for either geometry
or time dependent problems.
The tables of solvable problems include a column
representing MSP (multiply separated position) synthesis
and are representative of the possibility of higher order
geometric specification. This is in reference to well-known
structural curvature theory. It has a major weakness of
not being able to coordinate nonadjacent planes for higher
order properties. This class of problems is solvable only
by Time State Synthesis (TSS).
The fourth distinction mentioned in the section
labeled Synthesis Tools transforms the existing FORTRAN
program to a very efficient interactive APL function. This
feature greatly enhances the designer's ability to stay in
control of the design process and leads to an earlier solu
tion.


174
V // ASSIGN L\A ;3
[1] ->(3 4 =N)/L1,L2
[2] -*0x i 0=N, 0U.CG+2OA ) {SG+IOA+Ut L-N ; 3 ] ) CB+-20A ) (S3+-104
4-MlL-Nl 2] ) (CA+-20A ) tSA*-loA<-MLL-N; 11
[3] +0* 11-N ,0tUl-/[L+l-tf ;1 ) ,Bl+-:1lL + l-N ; 2 ] ti4-.7[L+1 -N ;
3]
[4] ->0, (B2-(7[ 3;2]-31x/1)t3) ,>12^(//C 3 ; 3 ;
3] ) t 3 -//[ 2 ; 3 ] 2
[5] LI : ->-0 ( 3 3-( //[ 4 ; 2 ] (3 1 x./[ 4 ; 3 ] ) + 3 x3 2 x3 ) *4 ) ,/13-<-(7[
4 ; 1 ] (/11 *i7[ 4 ; 3 ] ) + 3 *4 2 x3--x /'f[ 2 3 ; 3 ] ) *A+-Ml 2 ; 3 ] *
3
[6] L2 :-0, (4 4*-7[ 5 ; 1 ] ) ,3 4-*-.f[ 5 ; 2 ]
7
7 Z+CASE K it!
[1] K+ 1 2[1+P'*K1,Z-iO
[2] Z+Z%~l + \~1+H4-K\2
[ 3 ] *2 x 0 V
V Z+CUBIC XtA ;3 ;C;B
Cl] -*-4xi(Z = 0)v0<*-((Z<-UxC,)-3*2)*3) + (£>--( -3*3 ) + _
0.5(3x/lx( B-a[ 2 ]t3 ) xCV*[ 3 ]*3 )-4x (4-*-l + *) x3--l + ) *
2
[2] Z-( 3 1 p( ( 2xxD)x( Z*0.5 )x2o( ( ^3 )x"20( |3 )t( Z^--Z)*
1.5)+ 0 12 xo 2 t 3 ) ) 0
[3] -*0, Z*-( Z- 3 2 pS ,0)+-4
[4] Z+- 3 2 p(t4)x(-3- + /7),0,Z,3,(Z*-(-(t2)x + /C)-3), -3*- ( *
2 )x( 3*0.5 )*-/C 7
7 Y+FI* XiMFiSAiSB ;CA -,CB \SG\CG
[1] 0 ASSIGN 1
[2] UF+ 3 3 p(C3*CG) A-C3xSG) ,S13 ,((CA*SG)+SAxSBxCG) .(.(.CA*
CG)-SA*S3xSG),(-SAxCB),((SAxSG)-CAxSBxCG),((SAxCG)+C4
xSdxSG ) C>1 xC3
[3] Y+WF+.xX
7
7 A+G'fL K-,AliA2\A?,iA^\Bl\B2\B2\Bi\tSA',CA\SBiCB\SG,CG\SiT
[1] 0 ASSIGN L-(F = 0)/ip/C
[2] /-( C3] AIL i ]-*-( 1 -CBxCA ) ( (SGxCAxSB )+SAxCG) ( ( CA xSBxCG ) -SGxSA )
, ( ~1 +C3 x CG ) ( CBxSG ) (SAxCB) ,SB,((-SBxSAxSG)-l-CAxCG),
[ 1.5](C/ixSff)+i7GxSBxSA
[4] i4[L;]-*-/lCL;]-((pL ) 9 )p//0-4 [ 1 ;]
[5] +OUTx\~\zK
[6] 1 ASSIGN L-(X = 1 )/ i pK
C7] AIL-, 1 2]-( UlxSA*C3)+BlxCAxSP) ,C
1 .l](SGx( -SA + (.AlxSA*Sn) -BlxCAxCP ) ) +CG*CA x SB+A1
[8] 4[L; 3 4]-( (Sffxi7/lx-33+/ll )+CGx( -SAxl+AlxSB)+BlxCAxCB) ,
C1.5]-(SffxC3)+CGxSBx31


81
(b)
Figure 111 12


72
The coordinates of C(X,Y,Z) in the l + 1 po
sitions can be determined from dyad analysis
using dyad BCD. Analysis will also provide
the relative angle $ FDC (e^).
5. The problem now leaves the determination of
link 6 to the procedure of function genera
tion between links 3 and 4 given in Section
4.
6.5 The Watt 1
1. Make the symbolic transfers (Fig. 111 -8 b )
a+A B+D c+H
T+l, 2+2, 3+3, 4+5
2. Find the motion specification set of link 4
by
(a 3,y}4£ = ab£, 6b,Yb£)
3. From the curvature transform find constraint
link 3 with circle point C and center point
B.
4. The motion specification set of link 2 is
given by
{a,B,y} 2£ = (0,0,01
5. The motion specification of link 5 is given
by
c,b,y>5* = Invert the mechanism by fixing link 2.
{a,3,Y)i& = {0,,0-|{
6.


109
Figure IV-12


76
Figure II-10


127
Figure V-3


48
The higher order properties of the coefficients are found
by differentiating with respect to y. Now suppose the
motion of a system is to be coordinated with a crank para
meter . This coordination involves three functional con-
straints
a = fU), 6 = fU), y = f(<|>) (11-40)
For ISP it is necessary that the motion be specified in
terms of higher order properties of these functions with
respect to . A new class of problems evolves when is
given as a general time function
= f(x)
so that
a f(x), B = f(x), y = f(x) (11-41)
Now Eq. (11-34) can represent the motion specification
set in coordination with a general input. The problem is
to specify the time states of a shell, and then synthesize
a system to meet those specifications. This is not a
normal dynamic response question, since there is no pre
existing mechanism responding to an input;rather the con
straints of the mechanism are sought from the expanded
curvature transformation. This task is achieved by


95
AD = A + 90 in the spherical case. There are two unique
inversions of the RRRP chain. The first occurs when link 1
(or link 4) is fixed. It is called the slider-crank
mechanism (Fig. IV 3 a ). The second inversion (Fig. IV 3 b)
occurs when you fix link 2 (or link 3).
The third common four-link chain is the RRPP
(Fig. IV-4a), which has three unique inversions. Note
that link 1 contains straight line constraints j¡,-¡ and
in the planar case. This means that in the spherical
configuration and become great circles c-j and C2
which have a constant relative orientation angle 0 .
The corresponding angle in the plane is 0p. Great cir
cles have their centers at the center of the sphere, and
normals to the planes of the great circles intersect the
sphere at points A for c-| and point D for c2. Since these
points (D,A) correspond to c-] and c2, both in link 1,
points D and A are in link 1. Hence DA is a constant arc
length equal to the orientation angle 9 Fixing link 1
makes the double-slider mechanism which generates ellipti
cal motion on the plane. Fixing link 2 (or link 4) forms
the Scotch Yoke Mechanism which is well known as a harmonic
input output function generation. Fixing link 3 makes the
double oscillating block mechanism and is the last of the
unique inversions for this chain. In this case, pin
joints B and C become fixed pivots. All points in link 1
generate cardiod motion.


115
where BC and CD are given and BD can be found by Eq.
(IV-1 ). The result of Eq. (IV 2 4 ) is an angle between 0
and tt, but in reality 0^ may range from -tt to tt. The
appropriate sign can be found from the sign of c (bx3).
Note that only the position of point C is needed to find
the proper sign of 0^, and no higher order properties
are used. Differentiating Eq. (IV 2 4) with respect to
time yields,
BD sin BD
sin 0q sin BC sin CD
(IV 2 5)
Differentiating Eq. (IV-1) and substituting into Eq. (IV-
25) yields

/S A a
b-d + b-3
C ^
sin 0C sin BC sin CD
(IV 2 6)
Differentiating Eq. (IV 2 5) once more yields the relative
angular acceleration,
n b-d + 2b-d + b-d
9r ~ TT
sin 0£ sin BC sin CD
cot 0C
(IV-27 )
This process could be continued without end as long as the
higher order properties of points B and D are known.


19
[m,, m?, m ] available system links to repre-
3 sent links in the problem state
ment
P pin joint available to represent
a general path point e
The tables on the following pages are intended as design sheets
for each of the specified problems to assist the designer
in the first stages of the design process. The tables are
oriented on the base of finitely separated position (FSP)
problems but are extrapolated to include guides for the
use of multiply separated positions (MSP) and time-state
synthesis (TSS). The distinctions of FSP, MSP, and TSS
will be made clear in the curvature transformation dis
cussion.
Synthesis Tools
Planar kinematics have four well-established tools
of synthesis.
1 path cognates
2. angular cognates
3. inversion
4. curvature transformation
These are fully discussed in the literature [27] for planar
mechan i sms but have major distinctions for spherical synthesis.
Both the path and angular cognates,that are so helpful in
planar design,are based upon properties of parallelograms.
Unfortunately, there are no parallel great circles on the
sphere, and these two tools must be excluded from the begin
ning.


114
y = a*a) = IcjI cos \\
y QU) = la)l cos X2
Figure I V 1 4


143
A, C,
B, C,
A 2 C2
B £ C 2
A, B,
A, C,
A2 B2
A 2 C 2
(V 3 7 )
Closure
The degree of the input-output equation is a
direct means of indicating the maximum number of real
closures for the mechanism. If a mechanism is degree
eight in the input-output, then there is a maximum of
eight geometric configurations possible in which to assem
ble the links. For the Stephenson 1, there are two pos
sible ref 1 ecti ons (2-6 2-5) for a given input 0-|. This
yields a total of four configurations or closures and is
identical if 02 is chosen as input. The Stephenson 2
input-output relation is based upon the pentagon loop
34567 and has a maximum of eight real configurations.
This was clearly established earlier in reference [29]. The
Stephenson 3 closures can be explained in a similar manner
considering the input loops 1276 and 73456. If 0 ^ is
considered as a known input then the six-link mechanism
is being driven by a quadrilateral loop and has a maximum
of four real closures. If 0^ is considered as a known
input then the mechanism is being driven by a pentagon
loop and has a maximum of eight real closures.


7
A basic criterion of the design process should be that the
available parameters {q} exceed or equal the specified
parameters {S} or
{q} > (S> (11 -1 )
The synthesis problem involves the satisfaction of
a number {F} of conditional functions in a limited number
[Z + 1) of system positions. Suppose the system is to
satisfy the condition f(x,y,z) = 0 by the approximating
function g(x,y,z) = 0 as in Fig. 11 -1 The function
f(x,y,z) may be a general spatial curve or it may be spe
cialized to spherical motion by the constraint
x^ + y^ + = constant (11 -2)
The function may be further specialized to planar motion
by restricting one of the parameters (x,y,z) to be a con
stant.
Closed form synthesis techniques allow a maximum
of five design positions. This does not restrict the ac
tual number of intersections of the two curves, but rather
the number of positions free to the designer's specifica-
tion. These positions may be either finitely or infin
itesimally separated. Infinitesimally separated positions
yield the ability to match slope, curvature, etc. up to
the fourth order. After the choice of design positions,
the designer must consider whether it is necessary for


96
Figure IV-3


140
Figure V-6


139
Equations (V 2 6) and (V-29) represent a pair of quadratics
in xc of the form
A1 x5 + B1 x5 + C1 = 0
(V 3 0)
B2 xB + B£ Xg + C2 = 0
Eliminating x^, the resultant can be expressed
as
A1 B1
A2 B2
A1 C1
A2 C2
A1 C1
a2 c2
B1 C1
8 2 C 2
= 0
(V 31)
Expanding Eq. (V-31 ) yields an eighth degree
polynomial in x^ and x^. Selecting either as input, the
other can then be solved for.
Stephenson 3
For the Stephenson 3 in Fig. V-6, write the cosine
law for hexagon 123765,
3215
cosa
67
(V 3 2)


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SYNTHESIS AND ANALYSIS OF
SPHERICAL SIX-BAR MECHANISMS
By
Dennis Lee Riddle
August, 1975
Chairman: Delbert Tesar
Major Department: Mechanical Engineering
The first portion of this document deals with the
synthesis of spherical mechanisms and is given with the
formulation of the necessary tools. These tools are the
curvature transformation and the inversion principle which
allow closed-form design of multi-link mechanisms for
three, four, and five multiply separated positions.
Twelve solvable problems are tabulated with an emphasis
on the reduction of parameters for motion specification.
The curvature transformation is extended to include geo
metric or time base motion problems. More than 20 detailed
design procedures are presented for the spherical four-,
five-, and six-link mechanisms.
The second portion of this document deals with
the analysis of spherical mechanisms and is presented in
vi


33
Figure 11-6


166
H ( X Y Z ) = (0,0,1)
A'(X,Y,Z) = (-.0868, .1504, .9848)
F'(X,Y,Z) = (.0870, -.0256, .9959)
G'(X,Y,Z) = (-.2244, -.0898, .9703)
The relative location of the references is arbitrary for
function generation. This allows the transfer of the de
vice (Fig. VI-9) so that points A and A' coincide, and
points B and F are in the same moving shell. Hence the
dyad HGF drives the four-bar ABCD with the proper dynamic
motion. The solution shown in Fig. VI-9 could be further
optimized by repeated computations of the above procedures.
Conclusion
This work has met the major proposed objectives.
The analytical tools of spherical synthesis and analysis
have advanced to a level of understanding approaching
that of planar kinematics. The past works of Drs. Tesar
and Duffy have allowed a coordinated mesh of planar syn
thesis and spatial analysis to yield comprehensive tools
of spherical mechanism design. The combined works of D.
Tesar, J. Duffy, H. Dowler, and D. Riddle have accomplished
the following objectives of spherical kinematics:


141
where Z3215 sin a56(X321 sin T5 Y321 cos x5)+cos 55 Z321
X321 X32 C0S 91 Y32 Sin 91
Y321 C0S a5l^X32 Sin 91 + Y32 C0S 91)' sin a51 Z32
Z321 = sin ot5l(X32 sin e1 + Y32 cos 6-| )+ cos agl Z
32
X32 = X3 C0S 02 Y3 sin 92
32
= cos a 12 (X sin Q0 + Y cos 0 9) sin an Z
2 3
12 3
Z32 = sin a12(X3 sin + Y3 cos 6,,)+ cos a12 Z3
Applying the half-tangent identities to Eq. (V-32), a
quadratic in x-j can be written with coefficients
A1 = X32X1 Y32Y1 Z32Z1 + cos a67
B1 = 2(X32Y1 + Y32X1>
(V-33)
C1 X32X1 + Y32Y1 Z32Z1 + C0S a67
If @2 is a known input, then 0^ is the only
unknown since X3, Y3, and Z3 are constants included in
Eq. (V 17 ).


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[8] ->-0xp Z-*-Zx


5
Eschenback and Tesar [24] expanded the synthesis
concept of time dependent motion for the planar system.
Myklebust and Tesar [25,26] extended this effort to yield
time dependent motion coefficients for the curvature trans
formation to establish a uniform approach to treat this new
problem. A generalized computer program, MECSYN, was de
veloped for the synthesis of multi-link, planar mechanisms
with time state motion specification. A collection of de
sign procedures [27] by Tesar for finitely separated motion
specification has been useful for academic instruction. A
major portion of the included work will be devoted to estab
lishing similar design procedures for the spherical system.
Analysis provides an evaluation tool of the use
fulness of a synthesized mechanism. Gilmartin and Duffy
[28] give a means to identify the type of a four-bar
mechanism. Freudenstein and Primrose [20] have studied
the motion of spherical and spatial five-bars for closure
conditions. The most useful analysis tool for planar
mechanisms has been established by Pollock and Tesar [30]
for dyad-based machinery. This philosophy will be con
sidered for the analysis of spherical motion in the second
portion of this work.
It will be the object of this work to present
spherical synthesis and analysis tolls in a form usable
to industrial designers.


Table 7: Angular Coordination of a Crank with a Two-Link Chain
Problem Statement
Specified Functions:
F=2; Ky f^) f(^)
%
Graphical Form:
m.| = 2; rr^ = 3; m^ = 4
Available Mechanism Constraints
FSP MSP
Mechanism Planes
[rri-j
Five-Bar
N
{q=8, L=4, p4=0}
Stephenson 1
Stephenson 2
Stephenson 3
[2,3,5],[5,4,2]
[3,4,1], [1,5,3]
[2,1,4(or 4,5,2)]
[2,6,3],[3,4,2]
[1,2,6(or 6,5,1)]
[2,3,4(or 4,5,3)]*
[3,1(or 6,5)],[5,4,3]*
[1(or 6),4,2],[2,5,1(or 6)]*
TSS
Y
Y
ro


APPENDIX
INTERACTIVE SYNTHESIS AND ANALYSIS
COMPUTER PROGRAMS


162
Figure VI-7


92
? If
XB yB ZB
r
o
X
l
2xb 2yB 2zb
- f -
XC
"" XB yB ZB"
XC
** tf
XD yD ZD
yc
+
2xd 2yD 2zd
yc
= -
XD yD ZD
yC
1
o
o
o
1
_zc_
_ XC yC ZC -
-ZC_
_ xc yc ZC_
_zc_
M
(IV-16)
If M in Eq. (IV-14) is nonsingular, then Eq. (IV -16) can
be premultiplied by -M~^ to find (x^ ,y^ ,z^). This pro
cedure could be carried further to find higher order
properties if desired.
Now that we have shown how to find the time state
of point C in the dyad with points B and D given, an ob
servation can be noted on their extension to points all on
the same rigid shell. If this is the case, such that BD
and 0 are constant, the same procedure can be utilized to
find the time state for a third point C on the same body.
This means that one tool can treat both dyads of two rigid
links or ternary links of one rigid shell.
Prismatic Equivalence
The slider or prismatic joint is relatively common
in planar mechanisms. A slider is generally considered to
be a link that is moving along a linear path with a center
of rotation at infinity. Links in the plane transform into



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116
System Inputs
A simple crank (Fig. IV -15a) is the most common
input to a linkage system. Given the location of pivot
A and the arc length B, the time state of point B can be
found for the chosen values of 0 while being driven at a
constant velocity w. The coordinates of B' for A' =
(0,0,1) are given by
xDi = sin 6 sin AB
D
yB, = -cos e sin AB (IV-28)
zBi = cos AB
The rotation matrix M-j in Eq. (IV-4) can be used to rotate
B' to the original system position B. Let the B subscript
of Eq. (IV 6) be replaced by A and find,
6 = mJ £' (IV-29)
Equation (IV-2 9) can be differentiated to yield the higher
order properties of point B. If AB is equal to 90, then
the same formulation describes the slider input on a
great circle.
The cycloidal crank (Fig. IV-15b) offers a second
type of input to the dyad system. Given the location of
A with links AA' and A1B and gear ratio N, the coordinates


172
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
7
C-A DYAD X iB ; D i T ; P13CDD iCP iR iK;N iMi?
K+(BC*-P*X 1] ) ( CD-*-X 2 ]+O 18 0),
20 + /(fl-/i[ ; 1 2 3 ] )x D-*-A [ ; 10 11 12]
13 ; ¡14 ;C 1
3 3 ,V-p BD+-
-+REDx\0eK*-lZ | 2M ( 2OC0 ) ( 20BC ) x 200 D ) ( lo3C ) x\OBD
PA+-1 (Ot2)-~20( ( 2oBD)-( 20BC)*20CD)i- ( 103 C) x 1OC.0
P-+/[ 2](£>,P .[ 2 5] P)x 13 2 $A/3-:/pP, (-P[ ;1 ]x/?[ ;
2]t?),(-S[ ;l]xfl[ ;3]*J?) ,0,<3[ ;3]*P) ,( -3[ ; 2 ] *P-0Oi?[ ;1] )
,S
MH+Np{ -PC ;2]*?) (P[ ;l]vP) ,0. (-P[ ; 1 2]*P.[
1.1] P^(P[; 1 2]+.*2)*0,5),0,0,0,[
1.1] tfpl
Ct- + /[ 2] (i? ,P.C 2.2] /?*-( (Io"2o7)xio57) ( -T*loBC ) [
1.1] 20BC ) x 74 PZir .73
Cl-- + /[2](C,C,[2.2] <7)*(1 3 2 14 15] ,0,0,[1 .1] 'fp0) DPT 13 2 W-MINV rpB.D,
2*C
C*-C,Cl,- + /T*P,2l P, [ 1 5 ] ?-( +/(NpA ; 7 8 9
16 17 18],0,0,[1.1] 7p 0)x 0,[2] C [
1.1] 0 + 2* + / (NpA ; 4 5 6 1 3 14 1 5 ] C1 ) xC1, [ 2 ] C1 [
1.1] C1
[9] ->0
[10] RED :*!,( ,A + ( CLOSE+X )/A ) 0+P-' *** CLOSURE REDUCTION **
* *
7
7 Z+MINV 3iM;I
[1] Z+- 0 3 3 pM-*-( !*! ) +pB
[2] Z-*-Z,[1]EB[J;;J
[3] +2*MiI+I+l
7
7 Z+M PDT N
[ 1] Z*((2+pM)p 0 "l "2 )4>(+/;'xO) ,(+/.7xi[2] N),
2.2 ] + //*"l[ 2] U- 13 2
7
7 QUADIV BiDiE-.I
[1] 0-*-pBx J<- i 0pP+-( 0 ( pA ) [ 2 ] )p 0
[2] R+R.Z 1 j ( (2+p/l )pAI ; ; ] )ffi( 2 tZ? )p3[I; ; ]
[3] -+-2xiO[i]>J+-/ + l
7
7 Z+-RELA A \B ; J
[1] 3-2o( +//1 [ ;9 + J]x^[ ;18+J]),(+/i4[ ;J]x/}[ ;18+J]),[
1 1 ] + / /l [ ;J]x/l[ ; 9 +I+-\ 3 ]
[2] Z*-_20(-(2oOr2])-x/2O0[; 1 3])*x/lo3[; 1 3]
[3] Z+(o2) |Zx"l*0> + //r ;9+J]x (-//.[ ; 20 2l]x[; 3
21),(--/Ai ; 19 2l]x/l[ ; 3 1]),[
1.H-/A ; 19 2 0 ] x/1 [ ; 2 1]
[4] B-* x/ioor ; 1 3]
[5] Z-Z,[l.l](B*loZ)x(+/[;3+J]x/1[ ;18 + J] ) + + /4[ ;J]x/?[ ;


Table 6: Angular Coordination of Three-Link Chain
Problem Statement
Specified Functions:
F=2; iy ffifjgjs Ky f(^)
Graphical Form: m^ = 2,
m^ = 3; m^ = 4
Available Mechanism Constraints
FSP
Mechanism Planes
[m^.m^
Five-Bar
{q=8, L=4, p4=0}
1
2
3
Six-Bar
{q Watt 1
Stephenson 1
Stephenson 2
Stephenson 3
[2.3.4],[5,4,3}
[3.4.5],[1,5,4]
[2,1,5{or 4,5,1)]
[2,6,5],[3,4,5],[2,4,5]*
[4,5,2(or 3,2,5)]*
[3,1(or 6),43*
[5,4,1(or 6)]
[1(or 6),4,5]*
[2,5,4]*


93
great circles on the sphere. By rotating a 90 crank on
the sphere, any great circle of that sphere can be gen
erated. Hence, any 90 crank on the sphere is the
equivalent of a slider in the plane.
The general spherical dyad enables us to illus
trate and discuss the four four-bar chains (RRRR, RRRP,
RRPP, and RPRP) and all of their inversions by the use of
the spherical equivalents to the five planar dyads
RRR, RRP, RPR, PRP, PPR).
Four-Link Chain Equivalence
The RRRR chain is the general four-bar linkage.
Both the planar and spherical representations are given
in Fig. IV 2 a All inversions a re the same and all
spherical links are different from 90.
The RRRP chain is shown in Fig. IV-2b. One
link (CD) is infini te i n the planar case. The offset in
the planar chain from pivot A to straight line is
h = AA*. The corresponding offset in the spherical chain
is A = AA'. In the planar case A'D = CD = . In the
spherical case A'D = CD = 90. No slider needs to be
represented physically at C although C always travels on
a great circle c-| The travel on £-| is measured by S-j .
The corresponding measure in the spherical case is a.
Note the link AD = h + =
in the planar case, and


101
C
Dyad
C at ao
C
Dyad
(c) RRP Dyad
Figure IV-7


71
(a)
(b)
Figure 111 8


57
Figure 111-2


107
cannot be built from dyads directly, but is dependent upon
the higher level Assur group shown in Fig. IV-12. No
closed form analysis tool is available for this class,
and in order to avoid iterative techniques, an alterna
tive closed form solution will be developed.
Consider the Stephenson 3 inversion by fixing link
3 and allowing link 2 to be free to rotate about point A
as shown in Fig. IV-11. If we now drive link 1, the
Stephenson 3 mechanism can be completely analyzed by the
use of dyads. Note in the Stephenson 2, that point B is
a fixed pivot and has no motion. If a transformation
matrix could operate on point B of the Stephenson 3 to
rotate it back to Bg, then this same transformation could
be applied to points, C,D,F,G, and H to give their inverted
motion in the Stephenson 2. This tactic is valid for posi
tion and all higher order properties as well. Assume,
without loss of generality that point A is at (0,0,1), then
the rotation matrix from B to Bg is given by
(IV-1 7 )
where
-1
2
0
0


18
we are now ready to consider the range of solvable problems
and the selection of the proper mechanism for each type.
This range of problems can be most efficiently presented
by the use of tables with consistent nomenclature.
Let
It
o
*0
ro
#

- the multiply separated position
counter
h
- the number of parameters left for
optimization (p = q F)
q
- total available parameters in the
constraint system
F
- number of functions to be specified
simultaneously
a j b j c j ...
- pin connectors in the solvable
problem statement
A, 8 C ...
- pin connectors in the linkage
system
m = T,2,3, . .
- counter for links in the solvable
problem statement
m 1,2,3, ...
- counter for links in the linkage
system
e
- point in the problem tracing a de
sired point path with curvilinear
coordinates (U ,V )
E(u,v)
- moving shell
E(U,V)
- reference shell
- absolute angular motion parameter
in the solvable problem statement
i
- relative angular motion parameter
in the solvable problem statement


164
Figure VI-8


40
with respect to another reference link n. The question
might be posed, "What is the motion specification set
{a,g,Y>3 of link 3 with 1 fixed as reference?" To answer
this question, consider the transformation between the
moving reference x,y,z and the fixed reference X,Y,Z
(Fig. 11-7)
(11-24)
where is the product of successive rotation matrices of
Euler angles a^, 3^, about X,Y,Z respectively, i.e.,
M'
m
: :
(a31'm
a22>:
:
^a23^m
i
(11-25)
where
m
cos
cos y
r
m
/ \ r r.r .r.r r
UonJm = cos an, sin + Sln a sin 3 cos y
I m m m m m 1 m


CHAPTER IV
SPHERICAL DYAD ANALYSIS
Kinematics is based upon developing an under
standing of the motion,independent of time, of intercon
nected rigid bodies. An analysis technique is necessary
to describe the motion of any point, shell, or relative
angle in a mechanism. This chapter will generate a collec
tion of tools to provide a concise and efficient means of
interpreting the analysis problem. It should be noted that
analysis is a necessary compliment to synthesis and design,
and may well be the final critical step of optimization.
The use of dyads for analysis of planar mechanisms
has proven to be the most efficient and versatile closed
form tool that is presently available. Referring to
Fig. IV-1 a dyad is represented as a pair of rigid links
(BC,CD) connected by a common joint at point C. If the
time states of the two remaining ends (B,D) are known,
then the time state of C can be found. Note that BCD
forms a spherical triangle. Without loss of generality,
we will consider only unit spheres such that 6,c, and 3 are
all unit position vectors of points B,C, and D respectively.
The arc BD can be found by the scalar product of 6 and 3.
85


82
3. From the curvature transform find the constraint
link 2 with circle points B and center points A.
4. Define the motion specification set for link 4
as
{a > 8 jY}^ = {UCVC^3}£
5. From the curvature transform find the two con
straint links 1 and 6. Note that points D,
6,and C form ternary link 4. The resultant
mechanism is depicted in Fig. III-12a.
6. Switching the roles of V2 and 'Pj by the sym
bolic transfer 2+4 and 3+5, the steps 2 through
6 can be repeated to double the number of solu
tions (Fig. 111 12b ) .
11. Colaminar Motion Coordinated with a Crank
This problem involves the coordination of a moving
shell with the motion of a crank. It may be the most prac
tically useful procedure of this treatise.
11.1 The Stephenson 2
1. Make the symbolic transfer (Fig. 111 13)
a+A, e+D
T+2, 2+3, 3+4
2. Define the motion specification set for link
4 as
{a,B,y}4 = upvd>^3^£
3. From the curvature transform find the con
straint link 5 with circle point C and center
point B.


142
For the pentagon 12345, we have the cosine law,
Z^21 ~ cos (V-34)
where Z^] is given within Eq. (V-32). Eq. (V-34) may
also be expressed as a quadratic in x-| with coefficients,
A2 = C0S a51 Z32 sin a5l Y32 cos a45
B2 = 2 sin a51 *22 (V-35)
C2 = C0S a5l Z32 + Sin a51 Y32 ~ cos a45
Eliminating x ^ from the quadratic Eqs. (V 33)
and (V-35) yields the resultant,
A1 B1 A1 C1
A £ B2 A2 C2
A1 C1 B1 C1
A2 C2 B2 C2
= 0
(V 3 6 )
Expanding Eq. (V-36) gives a multinomial which is
eighth degree in Q2 an(i fourth degree in 0^. For any
specified value of 02 we obtain a maximum of four values
of @2 and vice versa. Corresponding values of 0-| can be
found by


49
differentiating the G s (Eq. 11-33) with respect to time
m ^
in terms of , 3 y > a, 3 etc. To avoid the tedious
task of differentiating Eq (I I-33) wi th respect to time,
we can take advantage of the chain rule and the geometric
derivations of Eq. (11-33) as shown below:
K dx
nQ = e=f(y)
n-j = e 'y
r¡2 = e y2 + e y
03 = c'"y^ + 3c" YY + e'y
= e"" y^ + 6e"'Y^Y + 3c" y 4e" y Y + e'y*
The prime superscript on e denotes derivatives with respect
to y. If we allow e to represent G^Cy), a(y) > or B(y)>
then n represents Tm(x), a(x), or 8(x), respectively: If
the independent parameter is y as in the original synthesis
problem, then


68
6.2 The Stephenson 3, Case 2
1. Make the symbolic transfer (Fig. 111 7 a)
a-A, b-B, c-C
T-3, 2-2, 3-5, 4-4
2. Find the motion specification set for link
4 as
3. From the curvatui^e transform find two con
straints GH and DF. Pin joints C, G, and D
form a rigid ternary supported by four-bar
HGDF and driven by binary AB (Fig. 111 7 a) .
6.3 The Stephenson 2
1. Make the symbolic transfer (Fig. 111 7 b )
a-A, b-F, c-D
T-2, 2-3, 3-1, 4-4
2. Find the motion specification set
(a,8,Y}= {aC£3c£Yc£}
3. From the_curvature transform find the con
straint CB.
4. Find the motion specification set for link 3
{ot,B >Y>|£ = {0.0,eu}
5. Find the inverse motion specification set
when link 3 is fixed as
(ot > 8 >Y >2£
{ 0 1 £ }


CHAPTER II
SYNTHESIS CONCEPTS
The creative design of machine systems needs to
be based upon a solid foundation of analytical understand
ing. The technological needs of today have progressed
beyond the trial and error evolution of past machine design
practice. The development of sophisticated analytical
formulations, coupled with the practical convenience of com
puters, has given the designers tools to accomplish their
task with greater efficiency. The synthesis concepts in
cluded within are presented with enough detail so that
they might be useful to all engineers interested in de
signing machines with spherical constraints.
It is essential that the designer recognize the
number of parameters {S} that must be specified to satisfy
the problem requirements. If the designer selects a mech
anism with too few available parameters, {q}, he will be
severely limited in the quality of results. On the other
hand, if the designer selects a mechanism with too many
available parameters, the range of solutions will be too
vast for a meaningful evaluation of possible solutions.
6


157
position in existing Mercedes system), and the direction
to the center of gravity. For the planar approximation
assume the axle to be infinite in length. Figure VI-5
illustrates this planar approximation. At the curb con
dition, it is desired that point C move perpendicular to
the direction of the mass center. Further choose 0cC to
be very long so that the corresponding point on the in
flection circle is given by Jc< Arbitrarily choose the
pin joint A to be near the perimeter of the rim. The in
tersection of 0,A and 0rC locates the instant center P.
a
The Euler-Savary equation
JaA
PA2
0aA
(VI-1)
allows the calculation of a third point on the inflection
circle. The circle may be drawn through points P,Ja, and
Jc< Now any point B may be chosen to locate a new con
straint that will replace 0cC.
A line through points P and B intersects the in
flection circle at J^. Rearranging the Euler-Savary
equation,
PR2
V = (vi-2)
allows the calculation of fixed pivot 0^. About six quick
iterations on the choice of points A and B resulted in


34
-y
-y,-y
ii
Figure II-7


41
cos ar sin Br cos y r
m m m
sin ar sin 81 sin yr
m m m
cos ar sin Br sin y r
m m ,m
a i = -sin a cos
2 3m m
(a__) = cos a cos B
33 m m m
The transformation is orthogonal such that
m = The inverse transformation gives us a means to express
the inverse specification set {a,B>y} with the original
reference link r moving and link m fixed as a new reference.
Equating terms from Eq. (11-26),
(a01)m = sin a' sin ym
31 m m m
(a)
12m
r r
-cos 8 sin y
m m
:
r r
cos B cos y
m m
(a o o)
32 m
sin ar cos yr +
m m
= sin B
m
x r


125
Finally angles 0^, and 0^ + can be found by the half
tangent laws
X 1 =
x 5
sin a56 Y?34 X734
X ~ c -i n a +Y )
X734 S1 56 734
where X734 =
= X73 cos @4 Y73 sin 04
Y
T 734
= cos a45 (X73 sin 04 + Y73 cos @4) sin a45 Z?3
and
x' =
x6
sin r /- Y.
56 43/ 437 /y n\
X437 sin a56 + Y437
where X437 =
= X43 cos 0' = Y43 sin 07
Y 4 3 7
= cos a67 (x43 Sin e) + Y43 cos 0j) sin ag7 Z43
X43
= X4 cos 03 Y4 sin @3
Y43
cos a37 (X4 sin @3 + Y4 cos 0^) sin a3? Z4
Z43 *
sin a37 (X4 sin 63 + Y4 cos 03) + cos a37 Z4
II
X
sin a34 sin 04


42
, x m
^a 1 3 r
(a31
( a23^
m
a32 '
\ a33J
r
i'33/
( a12
m
a21
Ia",
r
la"i
(II-27)
which reduce to
3m = sin"1 (sin ar sin yr cos ar sin $r cos yr)
r 'mm mm m'
m -
= tan
r
sin a.
m
cos yr cos ar sin gr cos yr
m m jn m
-cos ol cos el
m m
(11-28)
tan
r
cos a
m
sin yr + sin ar sin 6r cos yr
m m m m
-cos b! cos yl*
m m
It is also valuable to
shell n is fixed. The set can
have the set {a,8,y}n when
m
be obtained by equating com
ponents of the matrices


REFERENCES
1. Denavit, J. and Hartenberg, R. S. "Approximate Syn
thesis of Spatial Linkages." Journal of Applied
Mechanics, V. 2 7, Series E, No. 1, 1960, 201-206.
2. Wilson, J. T. "Analytical Kinematic Synthesis by Fi
nite Displacements." Journal of Engineering for
Industry, V. 87, Series B, 1965, 161-169.
3. Johnson, H. L. "Path Generation in Space." ASME
Paper No. 65-WA/MD-13, 1965.
4. Zimmerman, J. R. "Four-Precision-Point Synthesis of
the Spherical Four-Bar Function Generator." Jour
nal of Mechanisms, V. 2, No. 2, 1967, 133-139.
5. Suh, C. H. and Radcliffe, C. W. "Synthesis of Spher
ical Linkages with Use of the Displacement Matrix."
Journal of Engineering for Industry, Vol. 89,
Series B, No. 2, May 1967, 215-222.
6. Roth, B. "The Kinematics of Motion Through Finitely
Separated Positions." Journal of Applied Mechan-
i_c^, Series E, September 1 967 591 -598.
7. Roth, B. "Fini te-Pos ition Theory Applied to Mechan
ism Synthesis." Journal of Applied Mechanics,
Series E, Septemeber 1967, 599-605.
8. Roth, B. "On the Screw Axes and Other Special Lines
Associated with Spatial Displacements of a Rigid
Body." Journal of Engineering for Industry, Feb
ruary 1967, 102-110.
9. Chen, P. and Roth, B. "Design Equations for the Fi
nitely and Infinitesimally Separated Position Syn
thesis of Binary Links and Combined Link Chains."
ASME Mechanisms Conference 1968, Paper No. 68-
Mech-7.
10.Chen, P. and Roth, B. "A Unified Theory for the Finite
ly and Infinitesimally Separated Position Prob
lems of Kinematic Synthesis." ASME Mechanism Con
ference 1968, Paper No. 68-Mech-8.
180


84
4. From analysis of dyad DCB find the motion of
link 5 and angular motion <¡>5. Also find the
base angle 2 and angles 4*4.
5. Let
0U = v ^h+ Si,
62£ = Si,
6 3£ = Si,
and find the motion of link 4 with respect to
link 3 by considering link 3 fixed.
6. From the curvature^_transform fi_nd the system
constraints links DF as 1 and GH as 6.
7. Invert the mechanism back to the original ref
erence of link 2.


54
2. Colaminar Synthesis
This problem is the most fundamental and the most
widely used. Many of the complex design procedures will
reference this problem but in no way detract from the sig
nificance of its own value.
Some point e of the moving system E(u,v) moves on
a path f(U,V)=0 in positions e with position coordinates
(UeS.Ve£,) (Fi9- 111 ~ 1 a ). The transformation between the
rectangular coordinates (X,Y,Z) and the curvilinear coordi
nates (U,V) is given by
X = sin V
Y = -sin U cos V (111-3)*
Z = cos U cos V
The motion of point e along the path is simultan
eously coordinated with the angular rotation parameter
The set {Ue, Ve, 1 s a set of Euler angles con
sistent with the motion specification set. Hence the motion
specification set**isobtained directly by the relations:
*Note that in the plane Eq. (111 3) reduces to
X=U, Y=V, Z=1. This agrees with the previous planar con
cepts. Equation (111 3) is solved for the reference system,
but is also valid for the moving system.
**A11 of the illustrations in this chapter use the
line codes: solid link line-specified link; long dash-
specified shell motion; short dash-constraint found from
curvature transformation.


113
found for the pole axis, then y and y about point A have
the same direction and magnitude as w and to respectively.
This is due to the fact that the direction cosine between
parallel axes is unity. However, the sphere does not
share this characteristic,and the absolute rotation proper
ties about a point other than the pole must account for
the direction cosines of the angle between the position
vectors of point A and the pole axis (Fig. IV-14).
Relative Angles
In planar and spherical mechanisms, the relative
angle between adjacent links is often of significant
interest. This interest may be due to the consideration
of transmission angles, or for function generation. Con
sider in Fig. IV-14, the relative displacement angle be
tween links BC and CD about point C. Writing the cosine
law for the spherical triangle BCD,
cos BD = cos BC cos CD sin BC sin CD cos 0^ (IV-23)
Solving for 0^ we have
6C= cos
-1
cos BC cos CD cos BD
sin BC sin CD
(IV 2 4 )


BIOGRAPHICAL SKETCH
Dennis Lee Riddle was born on June 17, 1946, to
Donald L. and Hazel M. Riddle of Madison, Wisconsin. In
June, 1964, he was graduated from Monona Grove High School.
He then won an appointment to the United States Air Force
Academy. In the fall of 1966, he entered Tulane Univer
sity. On June 20, 1970, he married the former Leita Fay
Aycock of Atlanta, Georgia. While in attendance at Tulane,
he participated in intercollegiate basketball and re
ceived the degree of Bachelor of Science in Mechanical
Engineering in May, 1971. Immediately following this he
joined the Deering Milliken Research Corporation in Spart
anburg, South Carolina, where he performed duties as a
machine design engineer. In September, 1972, he entered
the Graduate School of the University of Florida in Me
chanical Engineering with a specialization in machine de
sign. In April, 1974, he received the degree of Master
of Engineering. To date, he has continued this study in
pursuit of the degree of Doctor of Philosophy.
Dennis Lee Riddle is a member of the American
Society of Mechanical Engineers and of the engineering
honorary fraternity, Tau Beta Pi.
183


153
against. Note also that the position specifications for
03 are not satisfied. Only the relative displacement be
tween the £=0,3 positions are met. In other words, the
same locus would have been generated if 83q = 0 and 833 =
18.
TR Suspension
The state of the art of automobile suspension has
evolved over the past 50 years. Race cars and passenger
cars have mixed objectives of handling and comfort. Most
of the comfort problems can be satisfied by the proper
application of energy devices such as springs and dampers.
The handling problem is also dependent upon these devices,
but it is more heavily dependent upon the suspension geome
try. The position of the roll center and the reaction of
the body mass to braking and bumping have lead to a wide
variety of suspension geometries. If this problem were
to be solved with spherical motion, several advantages
could be anticipated.
Advantages of Tesar-Riddle (TR) Suspension
1. Virtually zero tread width change during large
bump motion.
a. Reduce scrubbing to zero.
b. Reduces potential break-away due to scrubbing
during bump.
2. Virtually stationary roll center.
a. Most roll centers tend to raise or lower
during bump action or during high g turns.


32
Spherical Chain
To appreciate the effect of colaminar inversion
we can first study the spherical three-link chain (Fig.
11 6) The motion specification set for link
m in position l with reference r is a set of Euler rota
tions about the X,y',z" axes (Fig. 11-7 ), respecti vely.
From the sine, sine-cosine and cosine laws for the spheri
cal triangle abb', we write
sin $2 = sin a b sin 9i
sin a2 cos 82 2 sin Tb cos 0^ (11 -7)
cos a2 cos $2 = cos ab
Solving for from Eq. (11 7 ) ,
j k
62j = j< [sin (sin ab sin 0 -j) 3 T (II-8)
dx
where 2, -* MSP counter
k ISP counter
T *T = TJi-k
t -* any geometric or time parameter


99
The last four-link chain is the RPRP chain (Fig.
IV-6). It has only one unique inversion which is the
slide-guide mechanism. In this case, there are two
straight line constraints SL^ and each of which may
be considered in each of the four links of the chain.
Dyad Collection
The RRR dyad (Fig. IV-7a) is the generic dyad dis
cussed in the earlier sections. In both the plane and the
sphere it is constructed of rigid links 1 and 2 with
revolute joints B,C, and D.
The RPR dyad (Fig. IV-7b) has joint C at infinity
in the plane. There is one linear constraint, straight
line i in link 1 with offset h-j from B. This corre
sponds with great circle c-j in link 1 for the spherical
case. The offset is measured from pin joint B. Note
that point C could be taken on either side of the great
circle.
The RRP dyad (Fig. IV-7c) has joint D at infinity
in the planar configuration. There is one linear con
straint 2 offset from pin joint F by h^ = FF'. In the
spherical configuration, the line &2 1S represented by
great circle c2 in link 2. Point F is offset by A2 = FF1
from c2 and CD = 90.


14
(a)
C
(b) FIVE-BAR I (fix link I)
(c) FIVE-BAR 2 (fix link 2)
(d) FIVE-BAR 3 (fix link 3)
Figure 11-3


63
Figure 111-5


46
the initial position parameters, and significantly re
duces the amount of work required by the program.
Second, the homogeneous cubic cone
a14 z2 + *a24 Z1 + a34 ^ z2 + *a44 Z1 + a54 Z1 + a64^ Z2
+ (a74 Z1 + a84 Z1 + a94 Z1 + a104)=0 t1134)
is the locus of homogeneous circle points (z.|,z2) in terms
of known coefficents a.. [19]. If (x,y) is a known point
^ J
on the curve such as Pg-| > then Eq. (11-34) can be reduced
to a quadratic. Let
z^=s+x z2=t+y (11-35)
and substitute into Eq. (11-34). This substitution yields
a binomial in s,t whose constant terms collect to zero.
Dropping the j=4 subscript for brevity,
a^t3 + a2st2 + a^s2t + a^s3 + a^3t2y + a2(2sty + xt2) + a^(2stx + ys2)
+ a73s2x + agt2 + agst + ags2 + a^3ty2 + a2(sy2 + 2txy)
2 2
+ a^(tx + 2sxy) + ay3x s + ag2ty + ag(sy + tx) + ag2sx + agt
+ ags = 0
(11-36)


1 51
X
Figure VI-2


121
if e1
known
into
input
Z-J2 = sin a23 (X-j sin 02 + Y-| cos e2) + cos a23
X-j = sin a7i sin 0^
Y-j = -(sin a-|2 cos + cos a-|2 sin ay-| cos0^)
Z-j = cos a-|2 cos sin a-j2 sin-j cos 0^
is known as an input then Y^ and Z-j are all
Substituting the half-tangent identities
2x. 1-x?
sin 0. = \ ; cos 0- = \ (V-2)
1+xf 1+xf
(where x^ = tan ~)
Eq. (V-l), terms can be collected in a quadratic
-output equation of the form
Ax^ + Bx2 + C = 0 (V-3 )
A = cos a23 -sin a23 Y-j -cos a37
B = 2 sin a23 X^
C = cos a23 Z^ + sin a23 Y-j -cos a37
whe re


144
All of the above formulation is readily reducible
to planar results and all closure conditions are directly
analogous.
Comparison of Chapters IV and V
Analysis Techniques
The analysis tools of the preceding chapters have
presented the designer with two basic alternatives to
iterative analysis. If iteration is required at the level
of examination of each mechanism, and then again to loop
through a collection of mechanisms, the cost and accumula
tive error can become impressively large. The designer
must be able to treat his problem in the most direct and
efficient means available. Closed form analysis is
necessary whenever possible,and iterative solutions should
be pursued as a last resort. This leads to expanded
efforts in the development stages but one should take
the attitude that once it is done, it's done.'
All of the included techniques are developed with
this spirit. It may be helpful to readers of this work
to outline a comparison of the approaches in Chapters IV
and V.


I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor
of Philosophy.
Professor of Mechanical
Engineering
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor
of Philosophy.
F! (T Sel fridge !/'*
Professor with Computer and
Information Science
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor
of Philosophy.
E. K. Walsh
Professor of Engineering
Science


126
Y4 = -(sin a^g eos + eos a^g sin eos 0^)
Z4 = C0S a45 cos 34 sin a45 sin a34 cos e4
(e6+T6>
= tan ^
Note that since quadratics were solved in and x^, there
are four possible closures for an input 0-j.
Watt 2
The Watt 2 mechanism can be broken into two
four-bars of quadrilateral loops I and II as in Fig. V-3.
Write the cosine law for loop I,
Z17 = C0S a56
(V 1 0)
where Z^ = sin a75 (X-| sin 0^ + Y-j cos 0^) + cos a75 Z1
Xi = sin a6l sin 0|
Y i = -(sin a 17 cos ag-| + cos a17 sin agi cos 0^)
Z i = cos a-| 7 cos a^i sin a-j7 sin agi cos 0^)


108
STEPHENSON 2
STEPHENSON 3
Figure IV-11


4
B. Roth developed a general theory which enables the synthe
sis of mixed displacements (i.e., finite and infinitesi
mal combined) of a rigid body in space. In [11] G. N.
Sandor developed a general method for the synthesis of
spatial mechanisms using quaternions.
The theory of coplanar synthesis has been studied
exhaustively. The study and development of a correspond
ing theory for synthesis of spherical motion has been much
more limited. G. Dittrich [12], H. J. Kamphuis [13], and
K. E. Bisshopp [14] have studied instantaneous spherical
kinematics.
The design of the planar geared five-bar for
function generation was treated by Oleksa and Tesar [15].
The extension of this philosophy to the spherical five-bar
was this author's first introduction into spherical kine
matics. The foundation of this extension was based upon the
unified theory of spherical trigonometry by Duffy and
Rooney [16]. This theory was generated from an understand
ing of the work by Todhunter and Leatham[17]. Dowler,
Duffy, and Tesar [18-20] combined the theory of spherical
trigonometry and the planar Burmester theory to yield the
spherical curvature transformation. This tool gave the
author the capability to extend the function generation
problem to the sphere [21,22]. Duffy, Riddle, and Tesar [23]
then established a displacement analysis technique for the
spherical five-bar.


56
(111 4)
It is now possible to outline the procedures for the basic
mechanisms.
2.1 The Four-Bar
This problem depends directly upon the curvature
transformation to determine the dimensional constraints
AB and CD (Fig. III-lb).
1. Apply the motion specification set {Ug,
Vg£, t0 the curvature trans
formation.
2. Select pin joints B and C from the tabulated
circle points.
3. Alternately to step 2, select pivots A and D
from the center point tabulation.
4. Selection of either a center point or a circle
point determines the other. Points B and C
are both in the rigid moving shell and can
be joined by a rigid lnk of arc length BC.
It is identical to make the rigid link 360-
BC. The pivots A and D are both points of
the rigid reference Z and provide the base
link of the four-bar ABCD.
3. Angular Coordination of Two-Link Chain
The system of two links, 2,3 moving relative to
link 1 (Fig. 111 2 a) is completely known if the following
data are given
l = 0,1,2
9
(111 5)


Table 12
Problem Statement
Specified Functions:
f=3; Ue= f(ip2); Ve = f(^)
^3-
Graphical Form: m^ = 2;
it, = 3; Point e in m^
U
Point Path Coordinated With Two Cranks
Available Mechanism Constraints
FSP
Mechanism
Planes
[m1,m2,m3]
Five-Bar
{q=12, L=4, P4=0}
1
2
3
Six-Bar
{q=14, L=4, p4=2}
Watt 1
Watt 2
Stephenson 1
Stephenson 2
Stephenson 3
[2.5.3],[2,5,4]
[3.1.4],[3,1,5]
[2,4,5(or 1)]
[2,3,6],[2,3,5]
[1,6,3(or 5)]
[4,3,5(or 2)]
[3,5,1(or 6)],
[3,5,4]
[1(or 6),2,5],
[1(or 6),2,4]
Special
Knee Joint
P = 2
C
0
F
B
D(or G)
MSP
TSS


SYNTHESIS AND ANALYSIS OF
SPHERICAL SIX-BAR MECHANISMS
By
DENNIS LEE RIDDLE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975


no
where
constant
Differentiating Eq. (11-17)
f f
bQ=0=Mb+Mb
bQ = O = M b + 2M b + M b
CIV-18)
This technique is supported by an APL function listed in
the Appendix, and has all of the planar implications that
apply to previous tools.
Body Rotations
If the time states of two points on a rigid body
are known, the velocity and acceleration poles can be
determined. The velocity pole is the instantaneous axis
about which pure rotation exists. Referring to Fig.
IV -13, this rotation axis o can be related to the transla
tion of a point on the body by the vector cross product.
a = a) x a (IV -1 9)
This can be broken into three components and then expressed
as a matrix of simultaneous equations
y
z
0
L z J
y
-X
~
y
ux
X
wy
0 _
CO
L z J
A
(IV-20)


10
to the plane. We can choose only a limited number of de
sign positions (x, y, z), £ = 0,1,2, ... to repre
sent f(x,y,z). Should more than one function (F = 1,2,3,
...) need to be satisfied simultaneously, the total number
of specified parameters would be
S = F i (11 3)
Note that this result does not include the £ = 0 position.
Mechanism constraints can satisfy displacement, but normally
not absolute, values. This in no way weakens the synthesis
approximation, since any functional reference may be chosen.
Ideally, the synthesis process will not utilize all
the system parameters {q} to satisfy the specifications {S}.
The remaining parameters
P = q F£ (11-4)
can be used to meet optimization criteria not included in
the specifications. A synthesized mechanism will have
zero error at the £ + 1 design positions, but no other
characteristics are guaranteed. There are over a dozen
conditions to be met after synthesis. All of these can
be evaluated through present analysis procedures. The
designer could be left with thousands of these evalu
ations, a number much too great to treat realistically.
Optimization is the answer, and it is now coming to the
surface for planar mechanisms. Concise and efficient


3
Hence, the objective of this document is to de
velop general techniques which would make the spherical
system equally as accessible to the designer as the co-
planar system. Every effort has been expended to insure
that the results are immediately useful and valuable to
the designer.
Generally the approach to spherical design has
been to extend well-established methods for synthesis in
the plane to the sphere. For instance, J. Denavit and
R. S. Hartenberg [1] generalized F. Freudenstein' s approx
imate synthesis of planar four-bar linkages to generate
functions by means of the spherical four-bar. Also, J. T.
Wilson [2], H. L. Johnson [3], and J. R. Zimmerman [4] ob
tained design equations for function gene rat ion using a
rotation matrix.
C. H. Suh and C. W. Radcliffe [5] specified mul
tiple positions on a path in the plane by a displacement
matrix. Following this they extended the procedure to
the sphere using a rotation matrix. These contributions
emphasized the importance of matrix methods for synthesis
introduced by J. Denavit and R. S. Hartenberg.
General theories for finitely and infinitesimally
separated position motion of a rigid body in space have
been developed by B. Roth [6, 7, 8, 9, 10] and G. N.
Sandor [11]. These are major contributions in the field
of synthesis of spatial motions. In [9, 10], P. Chen and


175
[9]
CIO]
[11]
C12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
AlL\ 5 6 T]+UCBxCG)-SBxSGxPi) ,((-31xSAxSB)i-AlxCAxCB)
,[1.1] PlxCB
4[L; 8 3l-*-((SGx-C)+CGxD) ,[ 1. l](5Gxp*SA xSB +41 ) + CG*C*-(
B1 xSAxCB ) +CA +41 *CA *SB
+0UT* \~2eK
2 ASSIGN 3
A [ 3 ; 1 >(£4x( -2x41 xfi 1x5??) +A2xCB )+(D 2*CA*SP )+CA*CB*C4-(
41 ,B1 )+.*2
4[3;2]+-(5f7x-(i4 2*SA *SB)+( 2xAl*BlxSAxC3 ) + (CA*SB ) + (. C*CA*
SB) + (2xAlxCA)-B2xCA*CB)
4 [ 3 ; 2 >4 [ 3 ; 2 ] + CG x ( 2 xA \ xSA xSB ) +4 + ( SA xA 1 2 ) (
2xBlxCAxCB)+A2xCA
At3;3']*r(SGx(SAxl+Al*2) + (2xAlxSAxSP)-(A2xCA) +
2xBlxCAxCB)+CGx(B2xCAxCB ) ( 2x41 x£4 ) + (4 2x54 xSB ) +(
2xAlxBlxSAxCB)+CAxSBxl+C
Ai 3; 4 5 6 7 ]-*-( (SGx 2xB1 SB )-CGx(CPxl+Bl*2 )+B2xSB ) ( (
SGx-(B2xSB ) + CBx+31*2)~2xBlxS3xCG), ( (5/1 x ( 3 2 xSB ) + C*C3
) + C4x(2xAlxB\xSB)-A2xCB).(-SBxB1*2)+ B2xCB
[3; 8 9] + ((SG*-S) + CGxT),(.SGxT<--(2xB1xSAxCB) + U2xSA)-
(2xAlxCAxSB)+CAxl+Al*2)+55x5-(-SAxSPxi+C)+(B2xSAxCB) +
( 2xA lxSA )+54 x(4 2 xSB ) + 2 xA 1 x£? 1 x CB
OUT: A 1 0 +4
V Z+K MSP MiiNi3;CiDiEiF;GiHiAiPiQiRiQiM0iAr>iIiJiL;T;U
;Z1 ;Z2
AM+A+§TML GUI 0x7-l
L1 : B-*-( A /I [ 3 9 ;0]),(A /I [ 8 2 iff]), (A 4[9 2 ;ff])+A /I [
3 8 ;ff- 2 1 1 2 3 2 3 3[7,7+4]]
S-B,((A All 9 ;ff])+A 4[6 3 ;ff]),((A 4[2 6 ;ff])+A 4[
1 8 ;A]),A 4[6 1 ;ff]
MA 4 [ 3 4 ;ff]),(A /I [ 2 5 sff]),(A 4[4 2 ;ff])+A 4[
5 3 ;ff]
C+C t ( ( A 4[1 4 ;ff])+A /I [ 3 7 ;//]), ((A /I [ 5 1 ;ff])+A Al
7 2 ;ff] ) ,A All 7 ;ff]
ZMA 4 [ 4 9 ;//]),(A 4 [ 3 5 ; ff ] ) ( A 4[9 5 ;ff])+A 4[
4 8 ;ff ]
D+-D, ( (A 4 [ 6 4 ;//]) + A 4[7 9 ;ff]),((A 4[5 £ ;ff])+A 4[
7 8 ; TV ] ) A 4 [ 6 7 ;ff]
H+-+ / ( 6 3 p4[ 4 9 3 ¡I-*- 1 4 4 4 [ J ] ] ) xB ,-C [
1.5] D
H-*-H [ 0.5 ] + / ( 6 3 p4 [ 2 5 8 ¡:])xt-B,[
1.5] C
ff-ff,[1]+/(6 3 p4[7 6 1 ;J])xS,C,[
1.5]-D
P<-ff[2;2],(ff[2;5]+ff[3;2]),(ff[2;6]+ff[3;5]).P[3;
6]
+,nl 1 ; 1 ] (//[ 1; 3 ] + "[ 2 ; 1 ] ) (//[ 1 ; 4] +//[ 3 ; 1 ] ) (//[ 1 ;
2]+//[2;3]).(fl[l ;5]+ff[2;4]+5[3;3]),(ff[l;6]+P[
3 ; 4 ] ), P
-*(( 3=pA') ,5>W + 1 )/F0U,L 1
A*-§ 4 10 p4


64
1.Choose ab and find the position of b by
Xbl = s1n 9u sin rb
Yb = cos eU s^n ^ (111 7 )
hi= cos 71
2.From curvilinear coordinates of e(U ,V )
find the position e(Xe,Ye,Ze) by
s i n
V
e£
Y
eZ
-sin Ve cos
V
eZ
1
eZ
cos V cos V
eZ eZ
3. From dyad analysis (Chapter IV), find c(X,Y,Z),
02£ and 03£ (Fig. IIl-5b), using dyad bee.
4. Use any of the procedures for the coordination
of a three-link chain to follow.
6. Angular Coordination of a Three-Link Chain
The coordination of a three-link chain is the first
problem of this presentation that cannot treat higher or
der geometric properties by direct use of the G 1s in the
curvature transform. The coordination of two nonadjacent
shells for higher order position properties can only be
treated by considering another independent parameter other
than 91^ = yi (Fig. 11 6).
It is more useful in this problem to consider the
crank angle = 0 -| =

other necessary position functions in terms of this parameter,


44
locus of points (x,y,z) in the moving shell E that have
three positions on circles about center (Q2 Q-j Qq) in
the reference shell £?" is common and becomes the central
question of synthesis. The well-known constraint equation
of the plane becomes on the sphere
G(X ,Y Z) = Q2 F£ + Q1 E£ + Q0 D£ = 0 (II-31)
where D,
-G
Zl X + GU y G0i 1
G8l x + G7 y G
5i
(11-32)
Fi ~ G3i X G4i y + G6£ Z
The G^ coefficients of Eq. (11-26) are given by
G0i ~ 1 cos aicos \
¡1 = sin a^cos y+ cos asin B^sin (11-33)
G2 = cos a^sin B cos y sin asin y
*Note that in the previous references Ami's were
used as GmJ1 s. The later notation was adopted to dis
tinguish the spherical coefficients from the planar co
efficients.


Table 4: Angular Coordination of Two Cranks
Problem Statement
Available Mechanism Constraints
Specified Functions: F = 1 ; Mechanism
Graphical Form: m, = 2 i m = 3
U
Four-Bar
{q = 4, L = 4 P4=0}
F i v e B a r
{qf_8 L<8 P4 = 0>
1
2
3
S i x B a r
{q Stepehnson 2
Stephenson 3
FSP
Planes [m1,m2]
[2.4]
MSP
TSS
[2,5]*
[1.3]
[2.4]
[3,5]
[1 (or 6), 2]
PO
CO


60
Figure 111-3


17
Fi9u,e


37
33 = ~k tsin_1 (X2 cos 01 Y2 Sin 01)]T
dx
(11-14)
31 dTk
tan
^ -X2 sin 91 Y2 cos 0
This can be extrapolated to a general n-link chain by the
following expressions
Bni = fk [sin 1 (Xn-1...32 cos 01 + Yn-1...32 sin 01)]T
dx
an = i-
dx
tan
(II-15)
-1
-X 0O sin 9, Y cos 0,
n-1...32 1 n-1...32
'n-1...32
The final rotation angle y^ about point c can be
expressed as the sum
y3£= 7~k ^0 3 + a3 + P3^T (11-16)
dx
as depicted in Fig. 11 6. For the spherical right triangle
acc', the interior angle



Page
Dyad Collection 99
Spherical Six-Bars by Dyads 105
Body Rotations 110
Relative Angles 113
System Inputs 116
V DISPLACEMENT ANALYSIS 119
Watt 1 119
Watt 2 126
Stephenson 1 130
Stephenson 2 135
Stephenson 3 139
Closure 143
Comparison of Chapters II and III
Analysis Techniques 144
VI APPLICATIONS 146
Three-Link Chain 146
TR Suspension 153
Poultry Transfer Device 160
Conclusion 166
APPENDIX: INTERACTIVE SYNTHESIS AND ANALYSIS
COMPUTER PROGRAMS 170
REFERENCES 180
BIOGRAPHICAL SKETCH 183
v


161
attempt. It is apparent to this author that control of
the bird must be maintained during the transfer. This
means that the device must match velocities before it is
released from the picking line, and transfer with matched
velocities of the second line. The object of this design
will be to synthesize a mechanism to transfer from the
picking line to a transfer slide. The same mechanism can
be used on the other end of the slide to transfer to the
eviscerating line.
The hock cutter is the last operation of the pick
ing line and produces 1.8 birds/sec with a linear velocity
of 10.8 in/sec. Inspection of the machine shows that a
40-inch sphere would keep the volume in proportion to the
frame and maintain an approximate straight line while
matching velocities. The bird will be delivered to a
slide by an in and out cuspidal motion (Fig. VI-7).
The 5 MSP motion specification set to describe
the desired path is given by
point c
point a
1
0
1
2
3
4
k
0
1
2
0
1
a
866
0
8726
0
0
3
.5
0
1.511
1
0
Y
0
0
0
0
.54
where the first three positions describe the cusp and the


135
Y3215 ~ S1P a67
X 3 21 5
3215
Y3215 + Sin a67
(V-24)
Stephenson 2
In the Stephenson 2 in Fig. V-5 write the cosine
law for the pentagon loop 12345,
Z345 C0S al2
(V 2 5)
where Z345 = sin agl(X34 sin 05 + Y34 cos 05)+ cos agl Z34
X34 = x3 cos 04 Y3 sin @4
Y34 = cos a45(X3 sin Q^ + Y3 cos 04)- sin a45 Z3
Z34 = sin a45(X3 sin @4 + Y3 cos 04 + cos c*45 Z3
X3 = sin a23 sin @3
Y3 = (sin a34 cos a23 + cos a34 sin a23 cos @3)
X3 = cos a34 cos a23 sin a34 sin a23 cos @3


69
U
(a)
Figure 111-7


62
u
a =ir-ife
Figure 111-4


149
of e3. Selection of crank GH from the locus provides
a four-bar FDGH that is sufficient to control the motion
of 02 with system input 0l£. From the locus, circle
point G and center point H are chosen (Fig. Vl-lb).
G ( X Y Z) = (-.0364 .2409 .9699) = Q
H(X,Y Z) = (.3658 .1 21 1 .9228)
D( X Y Z) = (.25 -.433, .866)£ = Q
F(X Y Z) = (0,0,1)
GH = 24.38
DG = 43.41
CG = 83.38
The state of shell 4 (Fig. VI -1 a) is given by the motion
specification set ^ac"Yc) as listed in Section 6.
This computation is also an integrated part of LINK3.
aC £
ec£
units
67.77
24.85
17.09
deg
-16.66
79.19
59.89
deg/sec
-73.55
-76.30
-21.87
deg/sec
61 .28
39.26
33.97
deg
3
0


88
sin u sin v -eos u sin v
eos u sin u
-sin u eos v eos u eos v_
However, from the curvilinear to rectangular transforma
tion,
x = sin v
y = -sin u cos v
(IV 5 )
z = cos u cos v
Rb = >/1 -x^ = cos v
Substituting Eq. (IV-5) into Eq. (IV-4), the new position
of D is found by
3 (IV 6)
Now rotate d' to lie in the (-y,z) plane by a rotation
about the -z-axis.
'Vb
"XBZB
Mi -
COS V
0
sin v
(IV 7 )


15
will solve some problems that a four-bar would not. Note
that there are five pin joints in the five-bar, so that
10 parameters are necessary to describe its location. In
order to predict the position of all links relative to a
new input position, two additional parameters must be con
sidered. The gear-train values M and N (Fig. 11 -3 b c d )
are necessary to describe the input dependence of the
coupled 1 inks. Therefore, a total of q = 12 parameters are
necessary for the geared five-bar mechanisms.
The Watt six-bar mechanism (Fig. 11 -4) has seven
pin joints so that q = 14 parameters is necessary to de
scribe its position. The Watt 1 and Watt 2 mechanisms are
distinct inversions of the general Watt chain that are
found by fixing links 1 and 2 respectively. The reader
should recognize that the motion in link 5, for example,
is more complex in the Watt 1 than that in the Watt 2. It
is true that the motions of six-bars with two fixed pivots
are more complex than those with three fixed pivots.
The Stephenson six-bar mechanism (Fig. 11 5) also
requires q = 14 parameters. It has three distinct in
versions found by fixing links 1, 2, and 3.
Solvable Problems
With a view of the components in
p = q fa
(11-6)


100
RPRP Four Link Chain
Figure IV-6


112
Matrix A is singular and cannot be used alone to
solve for go. Writing Eq. (IV 2 0) for each of the two
given points A,B gives six equations from which three
could be chosen to give a nonsingular matrix A. Two more
equations can be added by noting that the linear and angu
lar velocities are perpendicular such that

a go = 0 (IV 21 )
For computational purposes, the singular condition can
be most easily avoided by summing three sets of three
equations e. g ,
r~ . --i
yB
0
o
I
<
0
+
0
+
xc
>
i
SI .
o
l
- yB .
-ZB-
(IV-22)
/
~ 0
ZB
1
<<
DO
1
XB
CQ
>>
ZB_
xc
yc
h
GO
X
<
XB
yB
ZB
+
xc
yc
zc
+
0
zc
-yc
>
GO
y
- yc
'xc
0 _
--zc
0
xc -
-yB
'XB
0 _
/
L 03z J
\
The angular acceleration go can be readily found by differ
entiating Eq. (IV-22) with respect to time.
Planar motion possesses the characteristic of equal
angular velocities and accelerations for all lines through
any point in a rigid body. In Fig. IV 9, if go and go are


128
9i = 91 + T1
Write the cosine law for loop II,
Z27 = C0S 34
(V-ll)
where Z27 = sin a^7 (X2 sin 0^ + Y2 cos 9^) + cos a^7
X2 = sin a23 sin 0£
Y2 = -(sin acos a 2 3 + cos a27 sin a 2 3 cos 0 2)
Z2 = cos a27 cos a23 sin a27 sin a23 cos 02
0 2 = @2 + "^2
If 0-| is the input, Eq. (V -10) can be written as
a quadratic in x7 with coefficients
A, = cos a7C Z-i sin a,, Yn cos arc
751 751 56
Bi = 2 sin a75 X .j
(V -1 2 )
C1 C0S a75 Z1 + Sin a75 Y1 C0S a56


Table 5: Path Coordination with a Crank
Problem Statement
Specified Functions: F=2;
ue = f%); ve = f(^)
Graphical Form: m^ = 2; Point e in it^
Available Mechanism Constraints
FSP
MSP
TSS
Mechanism
Planes
Special Knee
Joint P at
e, m1
Four-Bar
[3,2 (or 4)]
Y
Y
{q=8, L=4, p-=0>
Five-Bar
Y
Y
II
CL
*
to
II
1
*
C\J
II
cr
i
[3.2], [3,5],
[4.2], [4,5]
[C,2], [C,5]
2
[4.3], [4,1],
[5.3], [5,1]
[D,3], [D,l]
3
[1 (or 5), 2],
[1 (or 5), 4]
[F,2 (or 4)]
Six-Bar
Y
Y
(q=14, 1=7, p4=6}
Watt 1
[6.2], [5,2],
[6.3], [5,3]
[F,2], [F,3]
Watt 2
[3 (or 5) ,
6 (or 1)]
-
Stephenson 1
[2,3 (or 5,4)],
[2,4 (or 5,3)]
[B,4 (or 3)]
Stephenson 2
[1 (or 6),3]
[1 (or 6),5]
[4,3], [4,5]
[D (or G), 3]
[D (or G), 3]
[D (or G), 5]


ACKNOWLEDGEMENTS
The author would like to express his sincere
thanks for the leadership and guidance of Professor Del
bert Tesar, whose enthusiasm for kinematics has made this
work a pleasurable experience. The author would also like
to thank Dr. Joseph Duffy for the valuable introduction
and support in spatial analysis. The combined leadership
of these two mentors has presented the author with excep
tional opportunities and challenges.
The author wishes to acknowledge the remainder of
his supervisory committee: Drs. R. G. Selfridge, G. N.
Sandor, and E. K. Walsh for their personal contributions
to his academic training.
He is thankful for the opportunity to have shared
relevant experiences with all of the students of the
machine design group. G. K. Matthew has freely given of
his time at critical stages throughout his graduate pro
gram.
Finally, he would like to thank his wife, Fay, for
her patience and sacrifices. He hopes that he may provide
as much encouragement as she finishes her graduate program.


I certify that I have read-this study and that in
my opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
6. N. .Sandor
Professor of Mechanical
Engineering, Rensselaer
Polytechnic Institute
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate Council,
and was accepted as partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
August, 1975
Dean, Graduate School


66
If (j) = f(x), the motion is time dependent and the spheri
cal T 's must be used in terms of the specification:
^ = 6u 02a f2^T^ 63£ f3^T^
The state of shell 3 can be expressed from Eqs. (11-8,9,10)
a
b£
dx
tan
1 i sin a b sin 0 -j
cos a"b
bn dTk
sin1 (sin a b sin 0 ^ )
' dTk
02 + pb
where p
b£ k
dx
tan 1 (tan 0^ cos ab)
The state of shell 4 can be expressed from Eqs. (11-14,16),
a
cl k
dx
tan
/ -7
X0 sin 0.| Y^ cos 0^
cl k
dx
sin"1 (X2 cos 0-j Y2 sin 0-j )
cl k
dx
0 3 + + P,
c "c


148
l
k
9U
CD
ro
03£
units
0
0
30
-5
10
deg
1
1
120
-50
50
deg/sec
2
2
0
-100
0
deg/sec2
3
0
60
-25
28
deg
Let arc ab =
30 and arc be = 40
. The
problem is now
fully
specified. The
state of shell 3 is given by the
motion
specification set (a, ,8, .
b£ b£
,yb£} as listed in Sec-
t i o n 6
of the
design procedures.
This formulation has
been coded in
the APL
function LINK3.
l
k
ab£
6b£
^b£
units
0
0
25.56
14.48
21 .57
deg
1
1
-27.71
53.67
60.85
deg/sec
2
2
-113.94
-51.91
-46.38
deg/sec2
3
0
16.10
25.66
31 .31
deg
This motion set can be
treated by
the curvature transfor-
mation
for 4
MSP. This is automatic in
the LINK3 function
The resultant
is a set
of cranks
satisfying the conditions
of the
motion
Recall
that these
motion
specifications
are generated from the 0^ and 0££ data, and are independent


79
Figure 111 11


11
synthesis and analysis procedures are the necessary keys
to this type of undertaking. The included work should
make a major contribution in this arena for spherical
mechanisms. The number of optimization parameters {p>
should be in the range 1 1 P 1 3 for good design.
Constraint Systems
The designer has a wide choice of mechanisms to
meet the requirements of a given problem. The control
system contains a few parameters, usually one or more of
which is completely adjustable. The cam system contains
an infinite collection of nonadjustable parameters. A
linkage is neither adjustable, nor does it contain an in
finite collection of parameters. The designer must know
whether the linkage contains enough parameters {q} to
satisfy his problem. This work will build a family of
solutions based upon the four-bar, the geared five-bar,
and the six-bar mechanisms. As in the plane, it is felt
that these are the building blocks of linkage design and
a very large percentage of solutions can be derived from
their motions.
The spherical four-bar mechanism (Fig. 11 2) is
the most elementary device of this study and contains the
fewest system parameters {q}. If we were to ask the ques
tion, "What is the fewest number of parameters necessary


55
(a)
Figure 111 -1


168
1. Curvature transformation for 3, 4 and 5 mul
tiply separated positions
2. Time based curvature transformation
3. Colaminar Inversion
4. Classification of twelve solvable problems
5. Synthesis Procedures of four-, five-, and six-
bar mechanisms
6. Closed form angular displacement analysis of
all four-, five-, and six-bar mechanisms
7. Time state analysis of translational and ro
tational motion up to the k-th order for all
four-, five-, and six-bar mechanisms
The design of spherical mechanisms will reach a
parallel level with planar mechanisms after the completion
of two additions to this work. First, a comprehensive
user's manual of interactive APL functions is needed. A
few of the functions listed in the Appendix could be fur
ther generalized (e.g. LINK3 is coded only for the case
PPP-P). Most of the existing functions make efficient use
of the central processing unit, but greater care could be
observed in storage requirements. Second, the analysis
tools for systems with mass, springs, and dashpots could
be added for general dynamic analysis. This would lead to
a basis of shake balancing.
Optimization is the most significant missing ele
ment in spherical design. It is also needed in all other
areas of machine design including planar kinematics. Planar
optimization is currently being examined by several re
searchers. Classical methods generally require the


167
Figure VI-9


59
2.The curvature transformation can be used
to determine crank CD.
4. Angular Coordination of Two Cranks
The problem of "function generation" deals with
synchronizing the angular motion of two cranks rotating
about fixed pivots a and b (Fig. III-3a). All of the mecha
nisms under study are capable of this performance. We
shall limit the procedures to the four-bar and five-bar
1. It is felt that these have enough versatility to ac
complish most angular motion problems of this type.
4.1 The Four-Bar by Inversion
1. In Fig. 111-2 we have the motion of 2 and T
given relative to T. If we n^ow invert the
current statement by fixing 2 and allowing
1 to move we make the symbolic transfer (Fig.
111 3b).
a-* A b-d
2-2 T-l, 3-4
2. The motion specification for the two-link
chain (1 and 3) is given by
- Eqs. (11-8,9,10,12)
3. From the curvature transformation we find
circle point C and center point B to form
the rigid constraint of link 3.
4. Invert the mechanism back to the original
system by fixing link 1.
Note that the initial positions of and ^3 are not met
by arcs AB and CD. However, the finite displacement values
Ai^2 and are met by moving shells 2 and 4.