SYNTHESIS AND ANALYSIS OF
SPHERICAL SIXBAR 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 TwoLink Chain ... 156 Angular Coordination of Two Cranks ....... 59 Path Coordination ......... 61
Angular Coordination of a
Chain ..... ** ... ............... 64
Angular Coordination of a Crank with a
TwoLink 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
FourLink Chain Equivalence .............. 93
iv
Page
Dyad Collection ........................ 99
Spherical SixBars 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
ThreeLink 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 SIXBAR 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 closedform design of multilinkmechanisms 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 sixlink 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 twolink chains known as dyads. The second treats the inputoutput 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 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 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 completenessa development which has occurred chiefly over the past 15 years. Analytical, graphical and special design procedures are now widely available to create four
2
five and sixbar mechanisms. Proper application of the four tools of synthesis (inversion, angular cognates, path
cognates, and curvature theory) allows the design of the geared fivebar, Stephenson sixbar, and Watt sixbar 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 fourbar 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 fourbar, such as the spherical bevel geared fivebar, and equivalent sixbar 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 wellestablished 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 fourbar linkages to generate functions by means of the spherical fourbar. 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 fivebar for
function generation was treated by Oleksa and Tesar [15]. The extension of this philosophy to the spherical fivebar 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 [1820] 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 fivebar.
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 multilink, 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 fourbar mechanism. Freudenstein and Primrose [20] have stud ied the motion of spherical and spatial fivebars for closure conditions. The most useful analysis tool for planar mechanisms has been established by Pollock and Tesar [30] for dyadbased 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 limited 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. II1. 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 (112)
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 III
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 wellestablished 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. (113)
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 (114)
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 fourbar, the geared fivebar, and the sixbar 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 fourbar mechanism (Fig. 112) 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 112
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 (115)
only two parameters would be necessary to describe the position of each pin joint. Atotal. 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. 112, 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 fivebar mechanism (Fig. II3a) 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 fivebar mechanism family are inherently more complex than those of the fourbar mechanisms. It is expected that a geared fivebar mechanism
14
Bc (A(a)
(d FIEBR 3(i Dik3 FiueB
15
will solve some problems that a fourbar would not. Note
that there are five pin joints in the fivebar, 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 geartra in values M and N (Fig. II3bcd) are necessary to describe the input dependence of the coupled links. Therefore, a total of q = 12 parameters are necessary for the geared fivebar mechanisms.
The Watt sixbar mechanism (Fig. 114) 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 sixbars with two fixed pivots are more complex than those with three fixed pivots.
The Stephenson sixbar mechanism (Fig. 115) 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 (116)
16
F
5
6
G
,Figure 114
17
B
5
c
H
G
Figure 115
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 timestate 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 wellestablished 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
V)
V)
4J (A
S.
0
0
U I= U 0
E 0)
u
U L..j
if r""
wt In Li Li in Lj la* zo
fu kO
> (U In
C 43 0 o 0 0 5(a r_ CL m tt U*) 0
r 0
Ca 0 en CIQ
V)
A, co
E Siv CX. tu m CL r Cj cli
ca co cn
I I C*lj I 4J 4J
fu S co (U C*j M x mt 4J 4J 0 0 0
:3 If > V
u 0 1 1 V) v
w U LL ILI w w
ZE _C
co CL m cx
11 w w 4)
cr +J 4J 4J
m cr
cr
4
0
It
LL 43 43
(A
V)
E .0
(D 4J
u
o S
4
CL Slu
21
V)
CL
V)
kn
4)
it
4J I= U I cc
in 4j
C = CD
0
V) Ili
0. rn
tn "t
U.
LO ko In In o
>1 In
LO
iv 0
0 In
CL
to Lj Lj Lj
>
0 < C)
X: 11 C:t If ::t
S S. CL S CL
nj (a ru en
co co co
I to I r.: I. CIA r_ c C
It (D 11 x it 0 0 0
E A :3 j > i CJ j 4J 43 (n (A V)
u 0 V) 4J 4) C r_ c
LA U to to a)
co cli qr :Z n: _cit rl CL D CL
cr it 11 CD 0) 0)
0' cr 4J 4J
joi
4 44J w
Oj
It
U
4) it
to (A
4) r_ E
tn 0
E 4J
u E
=3 o
0 U LLS
CL
to
u
CL.
ca
S
22
C\i
E d Ln Cdl C\j
rl rI
4.3 (4 ,% LO
0 to A
u C CL Cj
0 (In Lj
Ad (A
w m Ln
z W)
Cj 6J LJ Cj
0
3:
4
to
0 CD
43 CL Cj m
fo CL m
S C: c c
m ra fu to CD 0 0 0
> co lqr Do co ca t 0 4A
S o v I v 41 r a
0 IW j r C"i x j 4) (11 w cu
o > ro
u 0 . (n :29 m m CL
0) U :J" U CC) C) (1) a) 4)
11 v r 4.) 43 4J
tu cr cr v (n Ln V)
Ij cr
lci Im
Cj
E
43 11 Im
LL LJJI lci
P ww
(31
43
m c E
4J
V) .0
4J
E
0
A U LL4
u CL
CL SV) CD
23
CQ
E
C*Q
4J
E
ti to
..Nd 0 CL LO
r_ u 0 Ln
U. (U C14 cl;
r Li
u (U
0 C)
q* C*lj
0 CL
.r. S 0 0
4J to EF to 0
co ll;r co co Co r c
if I v I v I c
i w j cli cn x i cu
> > Q. CL
S 0 (.0 w a)
o R::r U C C) P 4J
0 11 v Ln tn
cr v I
cr
loo
It Im
Im 11
m 4J It
U
E
4J
tz r E
4J 0
(A .11.111. >
4J .0
E
S
o b
0 LL LL0 f CQ
CL
u
4
u CL a
(V eu
CL S(n CD
24
V)
V)
co Cf) LO
In
43 C1
In c to u m UIV S CD
C 0 0
0 E S90 4 rl rl 000
cli M C\i csi
L) *
(D 0 L,) m U U cf cn C3 1=
I*% D I Li a
In
r
to 0 .
.r LO rIrl"
u U r" "rI C'Q ::r
% LO LO In In
4qr M
M cr qr Ln In In
tA r" Li".i LJ" In In L"Li Lo In In o "o 61
(V
C: E S. S L. S S S.0 to rIr7 oo 00 00 0 or"
%j M M
IV r I o'i P
4 Lj en M lt:k .), Ln ko M to mt
>
pA,
C)
4J tr mr
m CL
m o tO r' 0 0
0 4J 41
s _j W _j x P 4) cu ILI
J >
0 u 0 3:
0 ILI U co LA ILI oi
It 4J 41
0 It cr In (n
cr
4J
to
CL
E
a
to (U
4J 0
C"i II (n %ILI 11 U)
E LL(V
43 go
4J E
V)
E 43 4E
0 IN
0 U. 0) LL0
0
cu
4
'Cli
CL
V)
25
0
C
S
4J
rI
u
Ae to
P LO
U en
L,.j
to In to 0
QJ E en wt r 0 Lcl
m In clj
a. Lri I.j LJL.i
4J
Ci
S r Cj
co to rI co lqr
w f C*.j m 1 11 43 0 0 0
u > x j 4)
LL V)
C)
00 r CL
if v 1 0) Qj
cr 43 4J 4)
Ln V) VIP
KJ
If
43
we
E
4) 0
4J 4J 4 CV)
(JO) E
S
E :3 Kn o
W LL LO LA KY3
If
0 (1) to
S . u ClQ
4 ULn
26
Ln
0
Ln Z:
C
IV
CA LO
0
to
j 41
1 0
0 C LO clsi
Cj
A Lc; tr;
4tr Ln Ui A f.
0 to Qct, LO
r_ tn p #.
to
o
u
rw CIQ LO r rn tD 0
(V E A ...f S.
CY) 0
U cl; C4 cl^ Cj CV)
Li
to
4
0
C"i
C) it
0
E CL
4J 4A
to I Sc co Kt ea 11 cli 0 0
to Cj m co j Ln tn tA
4J 41 a C: a
> 41 P
to
LL co CL Cl. Q.
11 v (D cu w
0 cr 4J 41 4)
to
KJ
4
it
4J E
Im
F 43 0
ro
4) 4it
U LO LL(V Cj to
4 Ljloi
CL 0
tn
27
V)
4)
S
4J
rl
E 4c
rn rlI
E
CL r d j
cu V) fj3
4J
o CL
0
V)
to it C\j 0
ra tu cn i
1
x 4)
ai
M ClC V I (U
cr 4)
CAI
co
No
0 KNJ k"V
4
1;,,
4) t
LS co Ito
E
E
4J C K\I
fu 0 D4)  11
V) 4E
E 11 sKO) o
U _4 LL IN
0
S (V C\j
4 LLCD
28
4J
4J
4J
Ill
u
en So
V) Cj m In
cli
0i It q:r en
4J
CL 04
S
o fu 11 C\j 0 0
0 m co j W
J 1 43 4> c
(3 u x 43 43 (U (D
C> (a ro SS :m 3: Q. CL
v I w
cr 4J
(U 4If
.0 .0 11
P't en
E
E a)
43 v;;
cu r K\J
43
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 43
0
(U
z 0
Ln
R 41 CD
Q fo c fu 50
cu r4
4J Ck.
a) (U E LO
Nd C U: L: C C L;
> <
4J
> C) cli
Ln 11 W4 11 Igr
& 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
43 4J 43
(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 r4ri roirl LO m Ln rI
4J w m I'd, In Ln ko o LO
ro vt .
Lf) Lr) cv) C j In
r" L.J S S LJLj Li
4A oo 0 00 it 0
(V E rl & rn
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:
Li "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 43 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) 09 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
43 > x 4) 43 4a) 42) (L)
qt to (a a _rLL (A) V rL CL CL
0 11 a) 0) 4a)
Cr cr 4) 43 4)
CL V) in (A
0
KNI
kv
to cn
E fro
I C ; C
4)
4)
0 943 43 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 threelink chain (Fig.
r
116). 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. 117), respectively. From the sine, sinecosine and cosine laws for the spherical triangle abb', we write
sin 62 = sin ab sin e1
sin a2 cos B2 = sin ab cos eI (117)
Cos c2 cos $2 = cos ab
Solving for a2 from Eq. (117),
=dk l
22 dTk [sin (sin b sin el)]T (118)
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 1161
34
x I x1l
x
zlzfo Yo Yff
a
zo
Figure 117
35
The result of Eq. (118) 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.
(117), 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. (117) yields
d2 ck ~tanl 1 sin abCos 6 1 (119)
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. 116) to be the sum
=~ dy [(62+P2 )] T (1110)
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) (1112)
Eq. (1112) can be differentiated to give the higher order properties of Eq. (II10).
Writing the sine, sinecosine, 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) (1113)
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. (1113) can be solved to find
37
dk Cos
3 dTk [sin (X2 cos e 2 s 1 T
(1114)
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 nlink chain by the following expressions
dk I
8n2 dk [sin (Xn cos 6 + Yn nsin 0)]
nt k n1 ... 32 1 n1...32 1 T
(1115)
Xsi 7 cs
dk Xn 1..32 sin 1 Yn 1...32 Cos 1
n tan Z 1
dr k Zn1...32 T
The final rotation angle y3 about point c can be expressed as the sum
k R 3 + (1116)
kY3 d 3 033 T
as depicted in Fig. II6. For the spherical right triangle acc', the interior angle 63 can be expressed as
38
tan B3
tan = (1117)
From the cosine law for the same triangle,
Cos ac = Cos a3 Cos B3 (1118)
Similar to Eq. (1112), P3 can be expressed implicitly by
tan p3 = tan 4'3 cos ac (1119)
Substituting Eqs. (1117, 18) into Eq. (1119), we find
= d [tan (sin B3 cot 3)]T (1120)
P3 Z d Tk33T
Consider the spherical triangle abc; write the sine and sinecosine laws
2 = sin ac sin a3
(1121)
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. (1121),
w31 dk tan T (II22)
i Y
Equation (II16) can be extrapolated for a nlink chain
dk
dk T n + + pn]T (1123)
Yn dk [ n T ~
0(k)
where 3k) is specified and
SX1
dk tan 1 23...ndT L 23.. .n1 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'reeliflk 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. 117)
Y = M r Y (1124)
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 (1125)
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
(asin 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 (II26)
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. (II26),
42
m ( )r
1a3 r a31 m
r
a 23a 32(1127)
a33l r a3/m
r
a,, a 21
al11 r all m
which reduce to
m sin1 (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= tanI m m (1128)
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 (1129)
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:tanI 1 r 13 a22)r a23)m (23)r (33)m (1130)
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 wellknown constraint equation of the plane becomes on the sphere
G(X,Y,Z) : Q2 F + Q1 EY + QO D, = 0 (1131)
where D. = G2Y x + GI y G02 z
E.Z =G 8x + G zG5t (1132)
FP = G3U x G4Z y + G6Y z
The G m coefficients of Eq. (1126) are given by
G O 1 cos ctCos
G = sin clcos Y,+ cos asin ksin y. (1133)
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 (1133) 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. (1131) 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)= (1134)
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. (1134) can be reduced to a quadratic. Let
z1 = s + x z2 = t + y (1135)
and substitute into Eq. (1134). 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 (1136)
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 (1137)
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. (1137) can now be solved for ri and substituted into the homogeneous coordinates
z = ri cos 8i + x; z2 = ri sin 0.i + y (1138)
The G m coefficients (Eq. 1133) are functions of y alone since
a f(y) and B : f(y) (1139)
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) (1140)
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) (1141)
Now Eq. (1134) 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. 1133) with respect to time in terms of &, A, j, a, 8, etc. To avoid the tedious task of differentiating Eq. (II33)with respect to time, we can take advantage of the chain rule and the geometric derivations of Eq.(II33)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
(1142)
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 wellknown 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 sixbar 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 onetoone to a center point in the reference shell E. Each pair of circle points and center points construct rigid cranks. For the threeposition problem there are a total of 2 cranks. The locus of homogeneous circle points for four positions of E is a cubic curve (Eq. 1134). There is a single infinity of points on this curve and hence there is a total of cranks. The fiveposition circle point locus is found by the intersection of two cubics. There is a total of 6 cranks for the fiveposition 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 (Illi
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 (1112)
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. 111la). 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 (1113)*
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. (1113) reduces to
X=U, Y=V, Z=l. This agrees with the previous planar concepts. Equation (1113) 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 linespecified link; long dashspecified shell motion; short dashconstraint found from curvature transformation.
55
x v
E U
E
ee
V Vvi
y
Uei
(a)
v
e
4
b)
Figure III1
56
L Ue V ~ (1114)
It is now possible to outline the procedures for the basic mechanisms.
2.1 The FourBar
This problem depends directly upon the curvature transformation to determine the dimensional constraints AB and CD (Fig. IIIlb).
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 fourbar ABOD.
3. Angular Coordination of TwoLink Chain
The system of two links, movingg relative to
link T (Fig. III2a) is completely known if the following data are given
a b, ' O ,1,2,... (1115)
57
b
U '004,
0001, 1
14
B
D
0
(b)
Figure 1112
58
The curvilinear coordinates for the state of b in Z are given by Eqs. (118 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. (1112) as
d d tan 1 (tan cos ab) T
3.1 The FourBar
1. The fourbar can be used if the symbolic
transfer (Fig. III2b)
aoA, bcB, us, 22, 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. III3a). All of the mechanisms under study are capable of this performance. We shall limit the procedures to the fourbar and fivebar 1. It is felt that these have enough versatility to accomplish most angular motion problems of this type.
4.1 The FourBar by Inversion
1. In Fig. 1112 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.
III3b).
a*A bpd
To.2 T*l, 34
2. The motion specification for the twolink
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 AB and CUb. However, the finite displacement values Aip2 and A5are met by moving shells 2 and 4.
60
3 0 c
7r
Figure 1113
61
4.2 The FiveBar 1 by Inversion
1. The required symbolic transfer is
a+A, bF, c*D 2+2, 3+5
as in Fig. 1114.
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) (1116) We have no control on 030 SO that we may arbitrarily set it to zero. Eq. (116)then
yields 32.
3. The motion specification for link 4 becomes
{aa,y} 2+ Eqs. (1114,16)
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
fivebar 1.
5. Invert the mechanism by fixing link 1.
5. Path Coordination
The fourbar has enough parameters to treat the
coordination of a crank with a path (Fig. Ill5a). Planar fourbars use the path cognate tool that is not available on the sphere. Without this tool we suffer the weakness of using sixbars to solve a problem that a fourbar should be able to treat. The procedure for the sixbars follows:
62
v c e.
3
B 4 D
5
F
.0 ap 82
8 7r
I
Figure 1114
63
V
e ( U., V, )
(a)
(U, lo
e
b
c I
(b)
Figure 1115
64
1. Choose ab and find the position of b by
XbY = sin e sin ab
Yb9 = cos elz sin ab (1117)
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. Ill5b), using dyad bce.
4. Use any of the procedures for the coordination
of a threelink chain to follow.
6. Angular Coordination of a ThreeLink Chain
The coordination of a threelink 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. 116).
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 1116
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. (II8,9,10)
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. (1114,16),
~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. III6b)
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. III6b).
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. III6b).
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. III7a)
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 fourbar HGDF and driven by binary AB (Fig. III7a).
6.3 The Stephenson 2
1. Make the symbolic transfer (Fig. III7b)
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.dO>
H 5l
2
(b)
Figure I7
3 3lF
(b)
Figure III7
70
6. Find the motion specification set for link 4
with link 3 fixed by
{ gy3 fnr = 3
1 4,)4z Eq. (1130) 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.
III8a)
a F, b A, cB
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 III8
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. III8b)
aA, B D, cH
I I, Z22, 3S 3, T5
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. (1130) 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 TwoLink Chain
The motion of a twolink chain whose angular state is coordinated with a crank is closely associated with the threelink chain. The problem is depicted in Fig. III9a. The basic specification follows that of the threelink chain (cab  abc).
7.1 The Stephenson 2
1. Make the symbolic transfer (Fig. III9b)
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 1119
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. IIIlOa)
aD b+C, cF
TI, 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 III10
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 fourbar function generator procedures twice. Only threepivot mechanisms are eligible for this problem.
8.1 The Watt 2
1. Make the symbolic transfers (Fig. I11lob)
aA, bD, cG
T,2, T2 1, 34 T6
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 fourbars 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 threelink chain with (bac abc).
78
9.1 The Watt 1
1. Make the symbolic transfers (Fig. III11a)
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. III11b)
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. III12a)
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 11112
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. III12a.
6. Switching the roles of V2 and Ipf by the symbolic transfer 24 and 35, the steps 2 through 6 can be repeated to double the number of solutions (Fig. III12b).
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. 11113)
aA, eD
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 11113
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. IVl, 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 CY
Ph B 9'\c ofB D ~
BD Pat
(a) ofD
z Y
X
C'
x
1c
Z B'Y C"
f1
(c)
Figure IV1
87
BD = cos (b.d) (IVl)
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. (IV2) Solving for cos e, and taking into account the possible range, we find
l < cos a = cos CD cos BC cos BD1 <1 (IV3)
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
xaxis and then a rotation v about the yaxis. This transformation can be described in matrix form by
[ hr (IV4)
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
(IV5)
Z = COS U COS V
RB=/Ix2 = cos v
Substituting Eq. (IV5) into Eq. (IV4), the new position of D is found by
RB xBYB xBZB RB RB
a' = ZB YB a (IV6)
R B
B B
xB YB zB
Now rotate d' to lie in the (y,z) plane by a rotation about the zaxis.
0
YD" = M2a' (IV7)
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 (IV8)
zD, = cos BD
and sin RD = = x2 + y2 RD
v DD 'rx 1+LD'
Substituting Eq. (IV8) into Eq. (IV7),
YDr xD, 0
M2 R1 xD' YD' 0 (IV9)
0 0 RDThe position of c" is given by the elementary expressions,
XC, = sin BC sin e
YC, =sin BC cos e (IV10)
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 (IVll)
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 (IV12)
cc = Cos 0 =1
A a
d.d = cos 0 = 1
Differentiating Eq. (IV12) yields a set of conditions for the velocity vectors S, and a.
9
+ =
C.a + C.5 = 0 (iV13)
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 (IV14)
0 0 0 z C XC YC ZC ZC
M
If the determinant of M(det M) is nonzero then premultiplying Eq. (IV14) by M1 is sufficient to find (xCYc,zc). If det M is zero,then a different collection of Eq. (IV13) should be assembled in Eq. (IV14).
To find the acceleration, we can differentiate Eq. (IV13) again to give the set,
x 11, *
b c + 2 b'c + b8c = 0
d*c + 2 d*c + d*c = 0 (IV15)
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
(IV16)
If M in Eq. (IV14) is nonsingular, then Eq. (IV16) 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 fourbar 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).
FourLink Chain Equivalence
The RRRR chain is the general fourbar linkage. Both the planar and spherical representations are given in Fig. IV2a. All inversions are the same and all spherical links are different from 900.
The RRRP chain is shown in Fig. IV2b. 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 AD = 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
