
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00022474/00001
Material Information
 Title:
 Multiparametric optimization of fourbar and sixbar linkages
 Creator:
 Spitznagel, Kim Loring, 1952
 Publication Date:
 1978
 Language:
 English
 Physical Description:
 vii, 240 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Curvature ( jstor )
Design engineering ( jstor ) Design optimization ( jstor ) Dyadic relations ( jstor ) Engineering ( jstor ) Kinetics ( jstor ) Mechanical transmission ( jstor ) Mechanism design ( jstor ) Necessary conditions ( jstor ) Objective functions ( jstor ) Dissertations, Academic  Mechanical Engineering  UF Links and linkmotion ( lcsh ) Mechanical Engineering thesis Ph.D
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 228239.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Kim Loring Spitznagel.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 022737260 ( ALEPH )
04942749 ( OCLC )

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Full Text 
MULTIPARAMETRIC OPTIMIZATION OF
FOURBAR AND SIXBAR LINKAGES
By
KIM LORING SPITZNAGEL
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 1978
Copyright 1978
by
Kim Loring Spitznagel
ACKNOWLEDGMENTS
I wish to express my gratitude to Professor Delbert Tesar for his guidance and support throughout my graduate studies. My academic endeavors have benefited most from his contributions.
Much appreciation is due to Dr. John M. Vance, Dr. George N. Sandor, Dr. Ralph G. Selfridge, Dr. Martin A. Eisenberg and Dr. Sanjay G. Dhande for their reviewing efforts, suggestions, and especially for their respective influences on my professional development.
I am grateful for the support and aid of my friends and peers, particularly Ms. Barbara Mihatov and Mr. John Elliott. I am also indebted to Ms. Lois Rudloff for her patience and persistance while typing this dissertation.
Finally, my heartfelt thanks go to my family for their tolerance, support and encouragement.
TABLE OF CONTENTS
Page
ACKNOWLEDGI4ENTS . . . . . . . . iii
ABSTRACT . . . . . . . . . . vi
CHAPTER
I INTRODUCTION . . . . . . 1
II A PHILOSOPHY FOR LINKAGE DESIGN AND
OPTIMIZATION . . . . . . . 16
A. Control of Parameters . . . 16 B. The Optimization Procedure . . 17
C. Previous Use of the "Grid"
Philosophy . . . . . . 20
III SOLVABLE PROBLEMS . . . . . 23
A. Linkage Types . . . . . 23
B. Modes of Rotational Cyclicity . 27 C. Motion Specification Types . 35
IV SYNTHESIS OF LINKAGE SOLUTIONS . 39
A. The Four Tools of Synthesis . . 39
B. Coordination of the Tools of
Synthesis . . . . . . 61
C. Grid Dimensions . . . . . 65
D. Dependent Syntheses . . . . 66
V THE NECESSARY CONDITIONS . . . 71
A. FourBar Necessary Conditions . 71 B. SixBar Necessary Conditions . 99 VI THE DESIRABLE CONDITIONS . . . 104
A. The Desirable Conditions . . 104 B. The Objective Function . . . 108
i. V
Page
vii PROGRAMMING FOR FOURBAR AND SIXBAR
OPTIMIZATION . . . . . . . 113
A. Philosophy of Program
Modularity * * * * * 113
B. Control of Input/Output . . . 116 VIII RECOMMENDATIONS AND CONCLUSIONS . . 120
A. Recommendations . . . . . 120
B. Conclusions . . . . . . 124
APPENDIX
A PROBLEM TYPES AND SOLUTION
LINKAGES . . . . . . . . 125
B MODES OF CYCLICITY AND NECESSARY
CONDITION PROCEDURES . . . . . 149
C PROGRAM MODULES . . . . . . 175
D SAMPLE PROGRAMS . . . . . . 212
BIBLIOGRAPHY . . . . . . . . . 228
BIOGRAPHICAL SKETCH . . . . . . . 240
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
MULTIPARAMETRIC OPTIMIZATION OF FOURBAR
AND SIXBAR LINKAGES
By
Kim Loring Spitznayel
December, 1978
Chairman: Delbert Tesar
Major Department: Mechanical Engineering
An approach for fourbar and sixbar linkage optimization relative to numerous design criteria is presented. A linkage solution set is synthesized through an analytical combination of curvature transform theory, inversion, Robert's cognates and angular cognates under the specification of one of twelve possible types of motion specification. Evaluations are made first on the basis of necessary conditions, and then on the basis of assigned weighting factors and design zones relative to the optimization criteria. Subsequent expansions and refinements through resynthesis about the best linkages are used to approach an optimum solution.
The detailed design for an interactive computer program to implement the analytics is presented and discussed. The,
Vi
appendices include synthesis procedures, necessary condition evaluations, program module tables and sample programs.
Chaiviiran
Vi1
CHAPTER I
INTRODUCTION
The level of interest in the application of optimization methods to mechanism design seems to be rising exponentially, with the great majority of effort coming from the past decade. The ultimate goals are means to achieve mechanism designs which are globally optimum relative to all pertinent criteria.
Some early attempts to optimize were those of da Vinci (14521519) and Newton (16421727), who were both involved in specific mechanical design problems, and made modifications of previous designs in order to enhance their performances. Galileo developed a rational mechanical optimization in 1638 by creating an algorithm for the form of a bent beam for uniform strength.
The advent of the differential calculus was fundamental to the development of many modern optimization techniques. The introduction of elegant evaluations of maxima and minima of differentiable functions eliminated the need for discrete evaluations of these functions.
The development of variational techniques is largely credited to Bernoulli, Lagrange and Euler. Chebyshev's [1] involvement in straightline linkage design led him to develop his well known "Chebyshev polynomials"and optimality
1
2
criteria. In 1847, Cauchy [21 contributed the "steepest descent" method.
The German and Russian involvement in mechanism design over the past two centuries produced some extremely sophisticated graphical techniques for design. A graphical approach to fourbar optimization with respect to the transmission angle criterion was presented by Alt [31 in
1925.
The next fundamental influence on optimization was the development of the digital computer. Previously infeasible approaches to optimization became practical and the fields of mechanism design and optimization both changed directions. The birth of modern optimization can probably be credited to the work of Dantzig [41 in the early 1940's, whose work in linear programming included the development of the Simplex method.
In 1948 Svoboda [5] introduced his computational synthesis method of "successive approximations," and in 1954 Levitskii and Shakvazian [6] introduced a method to optimize spatial slidercranks and crankrockers on the basis of a set of linear equations from a finite'number of precision points.
The work of Freudenstein [711] in the latter part of the 1950's formed the basis for modern linkage design procedures. In 1959 Freudenstein and Sandor [10;11] used an IBM 650 and complex number theory to synthesize pathgenerating mechanisms, and Freudenstein [9] developed a
3
procedure to optimally approximate a given function using the precision points as design variables.
In the mid 1960's the mechanical design field began to feel the real impact of modern optimization theory, which involved numerical methods initially developed to solve management science and control system problems. Initial application to mechanical designs was primarily in the fields of structural mechanics and aeronautics, but mechanism design was not far behind. The development of such high level languages as Fortran, Algol, APL and Basic have made the use of the computer much simpler, more effective and less system dependent for the engineering community.
A general expression for the linkage optimization problem is:
Minimize Objective Function F(x)
where x = (xVx 2 . x p
Subject to:
Inequality constraints hk(x) < 0; k=l to m Equality constraints e i (x) = 0; j=l to n
A condition for the existence of an optimization problem is that the number of design parameters must exceed the number of constraining equations, in order to ensure the existence of "free" parameters.
4
The objective function may represent the deviation
between actual and ideal linkage performance, which may be a measure of prescribed path error ("structural error"), dynamic imbalances, error due to elasticity, or some weighted combination of errors relative to pertinent criteria.
Equality constraints have the effect of narrowing the scope of the optimization problem by reducing the number of free parameters. They commonly take the form of a synthesis step in which a linkage solution set is generated to satisfy prescribed precision points or positions.
Inequality constraints may take the form of design constraints placed on available linkage characteristics, such as link length maxima or minima, range of acceptable transmission angles, or geometric bounds on pivot locations or coupler motions.
The range of possible formulations of the linkage optimization problem is extremely large due to the highly variant sets of 1) linkage applications and 2) optimization criteria. It is highly desirable to organize the available approaches in order to get a feel for what is available, help classify past and present efforts, and perhaps enhance decisions in future efforts. Figure 11 shows such a representation.
Esthetically, a closedform approach is preferred for any problem, but the inherently complex and nonlinear
LINKAGE
OPTIMIZATION
CLOSEDFORM ITERATIVE
METHODS METHODS
ALGEBRAIC RANDOM UNCONSTRAINED CONSTRAINED DIRECT
METHODS SEARCH MINIMIZATION MINIMIZATION METHODS
AND SORT
FOigur I I. VLinaEe OIptiizi MIt.h
JCN.GAIET JIT PEAT GADETPO.
Figure 11. Linkage Optimization Methods
6
linkage problem makes the optimization analytics somewhat difficult to implement. Algebraic approaches, generally limited to smallscale problems, are rarely attempted. Sutherland [12] has developed mixed exactapproximate position synthesis in which some points are matched exactly using Burmester theory, and the objective function contains the leastsquares error of approximation of additional precision points. Solutions to the resulting equations can be closed form or iterative depending on the order of magnitude of the problem. Bagci and Lee [13], Bagci [141, and Bhatia and Bagci[151 have developed and expanded on a linear superposition technique in which the error of the loopclosure equations of the linkage system is minimized by partitioning these equations into dyadic loop equations which are linear in terms of the system unknowns.
Random searches are direct and somewhat exhaustive, generally encompassing a synthesis step to generate a large representation of a linkage solution set for a particular problem, and then a rating and sorting step to locate the "best" linkage within this set. As early as 1962, Roth, Sandor and Freudenstein [161 used this approach by generating fourbar linkages satisfying path specifications, and then rating them according to some design criteria (primarily transmission angle). Later Nolle [171 used a random search for function generating linkages. In
7
1968 Tomas [18] discussed the treatment of linkage synthesis as a nonlinear programming problem, and employed the optimization randomsearch method of Garrett and Hall [19]. Eschenbach and Tesar [20] generated a
large set of linkages computationally and then ranked them according to numerous design criteria, the latter making use of limit design zones and design criterion weights. Two and three function problems are handled under the consideration of ten to twelve criteria acting as sequential filters. Eisenstein and Hall [21] approached the optimization of twodegreeoffreedom function generators by generating a small number of "good" designs, and then generating linkages around these to find a "best" design. Spherical fourbar path generators are handled by Sridher and Torfason [22], who synthesize on four precision points, and then search for the linkage which minimizes the maximum deviation from the prescribed path between the precision points using a random sequential search. The author [23] and the author and Tesar [24] have "grid refinement" technique for fourbars which involves generating twodimensional discrete representations of the solution set for four coplanar positions, ranking or eliminating these using a weighted design criteria formulation, and then restarting by generating a new "grid" in the most favorable region. Most of the previously mentioned approaches, as well as the iterative methods, can guarantee true global optimization for convex functions only rarely the case in a linkage problem.
By far most of the emphasis in linkage optimization
has been towards the iterative methods. These methods involve "mathematical" (or "numerical") programming techniques which may be (in order of increasing complexity) linear, quadratic, geometric or dynamic. man techniques exist, Figure 11 showing only those used more commonly in linkage applications.
Constrained minimization techniques involve minimization of the objective function in the absence of inequality constraints, and are generally based on a "stepping" iteration, where each "new" step is a function of the location of the "old" step and some formulation of a new direction and step size.
The methods of Powell [25,26] involve unidirectional orthogonal searches to find local minima for each iteration. It is most applicable for problems with nondifferentiable objective functions. Suh and Mecklenburg [271 have used Powell's method to operate on matrixderived objective functions and constraints for spatial mechanisms.
The conjugate gradient technique bears some resemblance to Cauchy's "steepest descent" method but is much more efficient. Unfortunately, convergence degrades as the optimiia are reached or when the objective function has complicated features or "ridges." Fletcher and Reeves [281 introduced this method in 1964, and later Rees Jones and Rooney [29] employed this method for motion analysis using
9
an analog computer. Necti [301 also used an analog simulation, using a combined relaxation and gradient modification of the conjugate gradient approach to circumvent the problem of instability near the optimum. Sixbar function generators were optimized by Chen and Dalsania [31) by applying a leastsquares gradient method.
Newton's method in its unmodified form is not often
used for linkage problems, largely due to difficulties associated with the selection of initial guesses. One application for this method was found by Rose and Sandor [321, who minimize structural error of fourbar function generators by equalizing error between precision points and at the ends of the specified range. The resulting formulation produces ten nonlinear differential equations to which Newton's method is applied to find a solution.
The socalled "quasiNewton" method, also called the "variable metric" method, is basically a gradient technique which involves the formulation of a differential matrix which approximates the Hessian as the minimum is reached. Davidson, Fletcher and Powell [331 introduced this method in 1963, and MclaineCross [341 used it in 1969 to optimize a crankrocker fourbar to generate solar declination. This formulation, which involved five nonlinear differential equations, allowed for some error tolerance at the precision points. Optimization of mechanisms with flexible links was discussed by Sevak and
10
McLarnan [351 by using the variable metric. Conte, George, Mayne and Sadler (361 optimized with respect to dynamic criteria such as shaking force, input torque fluctuation, shaking moment and bearing reactions, using a penalty function in conjunction with the variable metric.
A fourth unconstrained minimization approach, the
NewtonRaphson method, is very efficient in terms of convergence, but requires formulation of the Hessian of the objective function at each iteration step. Han [371 used this method in 1966 for general mechanism optimization.
All of the unconstrained minimization formulations except for Powell's method require evaluation of the objective function's differential, but finite difference methods are generally applicable.
The second class of iterative methods are the constrained methods using unconstrained minimization. Generally, this means some modification is made to an unconstrained minimization technique to allow the inclusion of inequality constraints. One approach is a simple change of variable which will inherently disallow violation of specified inequalities, but linkage problems are usually too complex for this approach.
Penalty functions are often used to apply constraints to unconstrained minimization techniques. The interior penalty function introduced by Fiacco and McCormick [3840] has received quite a lot of attention recently. It is
sometimes called the Sequence of Unconstrained Minimization Technique (SUMT). Fox and Willmert [41] used this technique to optimize a fourbar pathgenerating linkage using a leastsquares error approach. Tranquilla [42] used this method to optimize a fourbar under the geometric couplerpoint constraints. Spatial mechanisms were considered by Gupta [43] with constraints on closure, mobility and transmissability. Bakthavachala and Kimbrell [44] considered fourbar path generator optimization under clearance, tolerance, transmission angle and link size considerations. Conte, George, Mayne and Sadler [36] used the SUMT for optimization with respect to dynamic criteria. Alizade, Mohan Rao and Sandor [45,46] have considered optimization with respect to structural error for twodegreeoffreedom spatial function generators and offset slidercrank path generators. Alizade, Novruzbekov and Sandor [47] then went on to fourbar function generators with consideration of link length and transmission angle inequality constraints. Kramer and Sandor [48] and Kramer [491 introduced the use of a new type of approximate motion specification in conjunction with an interior penalty function in order to optimally approximate a specified motion. Mariante and Willmert [501 used the interior penalty function to synthesize and optimize a complex convertible top linkage.
Direct methods and linearization methods are used to
12
deal explicitly with inequality constraints. This group includes the method of feasible directions introduced by Zoutendijk, the gradient projection method, and the extensions of Dantzig's simplex method. Fielding and Zanini [51,521 have discussed the use of modified Simplex methods and considered the optimization of an industrial packaging machine. Also, Youssef, Oldham and Maunder [53] have used the modified simplex to optimize multiloop linkages with respect to path error, link length and transmission angle.
The use of variational methods for linkage problems can be traced back to the early work of Freudenstein [9] which later received attention from McLarnan [54], who attempts to minimize structural error by requiring the derivatives at the precision positions to be zero. Huang, Sebesta and Soni [55] and Prasad and Bagci [561 have had more recent applications of variational methods.
Other iterative methods have been introduced by Huang [571, who introduces the concept of sensitivity coefficients, and Lee and Freudenstein [58] and Datseris and Freudenstein [59], who develop the application of Lin's heuristic method (originally developed for the communications field) to mechanism optimization. Moradi and Pappas [60] have developed a "boundary tracking" algorithm for general mathematical programming problems.
Most of the papers discussed so far have dealt primarily with optimization with respect to motion error. If
13
another criterion is to be considered, it is generally related to transmission angle. Lin [61], Hamid and Soni [62,63], Shoup, Staffer and Weatherford [64], Freudenstein and Primrose [65], Sutherland and Roth [66], Bagci [671, and Savage and Suchora [68] have all dealt with optimization relative to transmission angle. Optimization with respect to dynamic criteria is also a field of interest, with contributions coming from Tomas [691, Benedict and Tesar [70], Berkof and Lowen [71,72], Conte, George, Mayne and Sadler [36], Tepper and Lowen [73], Sadler [74], Elliot and Tesar [75], and Sandler [76]. Optimization of more general mechanical systems such as planetary gear reducers and a shaping mechanism are discussed by Golinski [771, and Osman, Sankar and Dukkipati [78] have treated optimization of transmission gears. Optimization using the leastsquares method of Gauss has been done by ChiYeh [79] and Mansour and Osman [80]. In 1971 Bagci and Prasad [81] and Bagci and Parekh [82] dealt with optimization schemes for coplanar and spherical fourbar and sixbars used for rigid body guidance. Speckhart [831 used minimum weight and the cost criterion as his objective function. Khan, Thornton and Willmert have developed "optimality criterion techniques" for minimizing weight in linkages subject to restrictions in stress and natural frequency [84] and stress and deformation [85]. Rao and Ambekar [86] optimized spherical RRRR function generators, and Sallam and
14
Lindholm [87] optimized sixbar Watti function generators. Hobson and Torf son [88] introduced a theoretical method which involves analysis of the centrodes of specified motion.
Graphic synthesis and analysis packages have been developed and bear some consideration in any discussion of linkage optimization due to their high level of human/ computer interaction, which allows convenient adjustment of representations of "good" designs to achieve "better" designs. Reed and Garrett [891 and Smith and Reed [90] have introduced IMAGE (Interactive Mechanism Analysis through Graphic Exchange) and Ricci [911 has introduced SPACEBAR. Perhaps the most powerful is KINSYN III, introduced by Rubel and Kaufman [92]. All of these employ highlevel graphics coupled with linkage synthesis and analysis routines.
The linkage optimization problem has been discussed on a philosophical level by Johnson [933, who discusses stimulants and aids for creativity, such as a building block approach, and systematics of linkages and circuit diagrams. Srivastava and Newcombe [94] introduce a multifactor objective function including such things as pivot location, bearing load limitations, displacement, velocity or acceleration constraints and Grashof constraints. Sutherland and Siddall [95] also use a multifactor objective function with inverse utilics for spherical function
15
and path generators. Fox [961 describes the state of the art of mathematical programming as applied to the optimization of mechanical components such as springs, shafts and bearings. Oderfeld [97], Starr [98] and Chohan [991 have all presented philosophical discussions of optimization of mechanisms with respect to multiple parameters, with emphasis on ordering of design priorities.
Other literature surveys dealing with mechanism optimization have been presented (in chronological order) by Lindholm [100], Seireg [101] Fox and Gupta [1021, Chen [1031 and Root and Ragsdell [104].
The "philosophical" papers [9399] effectively point the way towards the future.. A major concern involves met hods of achieving globally optimal mechanism designs which accomodate conflicting design requirements. Nearly all of the mechanism optimization schemes to date optimize with respect to one or two parameters. An enormous spectrum of mechanism applications exist, and a large number of criteria by which the overall quality of mechanism solution must be evaluated also exists. To thedesign engineer, who must be concerned with a mechanism's total functionality, true mechanism optimization must involve a significant number of these parameters.
CHAPTER II
A PHILOSOPHY FOR LINKAGE DESIGN AND OPTIMIZATION
A. Control of Parameters
In any optimization process, control of the number
of free parameters is a primary consideration for linkage problems, where desired motions are specified at a finite number of precision positions, the number of free parameters is determined by the simple relation:
p = q F
where p = number of free (optimization) parameters. q = number of parameters provided by a solution linkage. = number of "Precision positions" k.
F = number of functional relations defining each precision position. which states that the number of free parameters after synthesis equals the number of parameters provided by the solution linlage minus the number of parameters used to satisfy the precision positions. For a given motion
16
17
specification type (which defines F), one should create a balance between solution linkage type (which defines q) and number of precision positions (which defines z) in order to describe ad equately the desired motion and still leave an acceptable number of free parameters for optimization (between one and three free parameters is desirable).
Naturally there is no guarantee that a design procedure will exist for every combination of qF and R. For example, one cannot expect to satisfy a large number of precision positions simply by using a sufficiently complex linkage.
In this work twelve types of motion are considered, and for each of these at four specified positionsan acceptable number of free parameters is generated by using either a fourbar linkage or one of the sixbar linkages. The synthesis procedures for the solution linkages are well known and have been collected and organized by Tesar (1051.
B. The Optimization Procedure
A synthesis, analysis and resynthesis loop is sufficient as a basis for optimization, and generally allows one to take advantage of the knowledge and insight provided by kinematic synthesis techniques, unlike the more rigid and forceful numerical approaches. The synthesis/
18
analysis/resynthesis loop to be used for this work is shown in Figure 21.
The input to the synthesis step is the precision position description of the desired motion, hereafter termed the motion specification set. A sequence of kinematic synthesis algorithms will produce the output, solution set of likgso antd ,12, or c
(depending upon q, F and k for the particular problem).
Unfortunately, the analytics used in the synthesis process are not sufficient to guarantee that all ( or indeed, any) of the generated linkages will satisfy the specified motion without falling into one or more of a number of physical pitfalls of real linkages, any one of which can make a linkage useless as a solution to the specified problem. For this reason a "necessary condition" step is required. The necessary conditions depend upon the motion specification type, the linkage solution type, and the means by which the linkage is driven. The linkage solution set is mapped onto linear (1dimensional), rectangular (2dimensional) or cubic (3dimensional) spaces, depending upon the magnitude of the solution set space. These representations, termed "grids" due to their discrete, arrayed nature, form the operating environment for all subsequent optimization procedures. Through evaluation of appropriate necessary conditions, portions of the grid representing acceptable linkage solutions are determined and presented to the designer.
19
S Y n thesis (D I (D
Necessary Conditions
Desirable Conditions
Figure 21 The computational steps of a Synthesis/Analysis/
Resynthesis Linkage optimization
20
The third step, evaluation of "desirable conditions," is where the "quality" of the surviving solution linkages is determined by the userdefined objective function. A desirable condition can be any criterion by which a designer might qualitatively judge a linkage. The grid is passed through these desirable conditions sequentially, and for each one grid "topology" is reevaluated. This procedure is known as sequential filtering. Some linkages may perform so poorly relative to one or more of these criteria as to be unacceptable, and will therefore be rejected.
The result is a grid of solution linkagesin which a portion or portions contain acceptable linkages, with scores indicating relative overall quality.
C. Previous Use of the "Grid" Philosophy
The optimization philosophy employing sequential filtering was first implemented by computer by Eschenbach and Tesar [201 in 1969 for crankrocker fourbars satisfying specified coplanar motion for four positions. A very large solution set (62,000 linkages) was passed through the necessary and desirable conditions in a single pass, the optimum linkage possessing the best surviving score. An improvement on this approach was later made by the author [231 and the author and Tesar [241 by introducing the
21.
concept, of grid expansion. Grid expansion is a refinement technique by which the three steps of synthesis, necessary condition evaluation and desirable condition evaluation become part of a continuous loop. A region on the grid about the linkages of higher quality is selected, and resynthesis is performed to generate a new grid with a more finely discretized representation (a conceptual magnification) about the selected region. Two or three passes through the loop is generally sufficient.
Due to the typically high attrition of solution
linkages in the first pass through the necessary conditions and the high computational expense of the desirable condition analysis relative to the necessary condition analysis, a further improvement can be had by looping back just after the necessary conditions on the first pass (as shown in Figure 21). Because desirable conditions are not evaluated initially relatively finely discretized grid can be used on this first pass. All of these techniques are employed in the author's thesis program package SOFBAL (Synthesis and Optimization of Four BAr Linkages), which is capable of synthesizing and optimizing all types of fourbar linkagesfor general coplanar motion.
This successful philosophy is retained for this work,
but now twelve types of motion specification are considered. In the next chapter these motion specification types and
22
the solution linkage types are presented and discussed. Subsequent chapters will treat in detail the analytics behind the steps of synthesis, necessary condition evaluation and desirable condition evaluation.
CHAPTER III
SOLVABLE PROBLEMS
The previous chapter described an optimization philosophy upon which the efforts of this work are based. In this chapter the range of problems which are solvable using this philosophy (and subsequently developed analytics) is discussed. The range of solvable problems is defined by the three concepts of linkage type, mode of rotational cyclicity and motion specification type.
A. Linkage Types
The simplest nonlinear mechanism is the fourbar
linkage, as shown in Figure 31a. It possesses one degree of freedom after one of its links has been grounded, and is conceptually identical in all of its inversions, i.e., regardless of which link is grounded. The fourbar provides eight free parameters.
Two types of singledegree of freedom sixbar chain exist, the Stephenson and the Watt, as shown in Figure 3lbc. They are both composed of two ternary links and four binary links, and have different conceptual inversions, as shown in Figures 32 and 33. Each one provides
23
24
B 3 C
2 4
A D
1
a. The FourBar Chain
D 4 C
5
G
B
1
6
2
F
3
b. The Stephenson SixBar Chain
C 4 H
3 5
B F
1 6
A 2
c. The Watt SixBar Chain
Figure 31 FourBar and SixBar Chains
25
D 4 C
15
6 B
/H
:C~25
F 3 A
a. The Stephenson 1
D 4 C
5
1 B
F 3 A
b. The Stephenson 2
D 4 C
5
1 G
F tA
3
c. The Stephenscn 3
Figure 32 Inversions of the Stephenson SixBar Chain
26
C 4 H
3 5
B F
D
6
1
A 2 G
a The Watt 1
C 4 H
3 5
B F
D
6
A G
2
b. The Watt 2
Figure 33 Inversions of the Watt Chain
27
fourteen free parameters. A "Stephenson 1" is produced if link 1 or 6 of the Stephenson chain is grounded, a "Stephenson 2" if link 2 or 5 is grounded, and a "Stephenson 3" if link 3 or 4 is grounded. A "Wattl" results from grounding link 1, 3, 5 or 6 of a Watt chain, and a "Watt 2" if link 2 or 4 is grounded. These are available linkage solutions to the problems in this work, and the notation introduced in Figure 31 will be adhered to.
Other simple linkages exist, such as the slidercrank and inverted slidercrank fourbars, the geared fivebars and sixbars with sliders and oscillating blocks. They have less general applicability and except for the geared fivebar are simply subsets of the linkages already under consideration.
B. Modes of Rotational Cyclicity
A linkage chain can have a number of distinct types of motion depending upon linkage geometry and the nature of the input drive. These motion characteristic types are hereafter termed "modes of rotational cyclicity."
The possible modes of rotational cyclicity are displayed for the fourbar chain in Figure 34, and for the sixbar chains in Figures 35 and 36. The information in these figures is central to this work. This unfortunate
28
2 4 2 4
3 3
2 4 2 4
1 i 1
3 3
24 4 2 4
3 3
2 4 2 4
1 1 4
44
43 3
3 2 4 3 2 4
1 1
I4 2 4
3 Notation:
3 2 4
1 / Driven
/ Link
2 4 /
1 Complete Incomplete
Relative Relative
Link Link
Rotation Rotation
Figure 34 Modes of Rotational Cyclicity of the FourBar
Chain
29
Stephenson SixBars With No Cyclic Links
4
K5 A& Any pair of links
+ may be driven (15
1 6 t distinct possi2 bilities)
3
Stephenson SixBars With CyclIic Links
Figure 35 Modes of Rotational Cyclicity of the
Stephenson SixBar Chain
30
Watt SixBars With No Cyclic Links
4
5 Any pair of links
+may be driven (15 distinct possi1 6 bilities)
2
Watt SixBars With Cyclic Links
Figure 36 Modes of Rotational Cyclicity of the
Watt SixBar Chain
31
and exhaustive analysis is necessary because the nature of the necessary condition analysis depends upon the mode of cyclicity. For example, the pivots within a linkage solution at which change point positions (i.e. adjacent link collinearities) can be tolerated depe nds upon how the linkage is driven.
For our purposes it often becomes useful to consider each chain as free in space with two links driven  one taking the place of the previously grounded link. In essence, the grounding and driving of links becomes conceptually identical and interchangeable.
At this point a restriction is made in order to eliminate modes of cyclicity for which special knowledge of velocities and system inertias are needed to define motions through change point positions. It will be assumed that when links with two cyclic pivots exist in a chain, one of these links and an adjacent link are driven.
This restriction eliminates from further consideration linkages such as the rockerdriven crankrocker with complete crank rotation, and the Grashof doublerocker with complete coupler rotation. These linkages must rely upon link momenta or switching of inputs to maintain continuous motion, and are beyond the consideration of this work.
Any given fourbar or sixbar problem will have a
subset of the possible modes of cyclicity, depending upon the grounded link and the conceptual equivalence or
32
nonequivalence of links (a function of motion specification type). As an example, the fourbar solution to the twolink chain problem (Figure 37) has seven modes of cyclicity. Modes number 3 and 4 would otherwise be identical except that constraint links 2 and 4 are not conceptually the same due to the nature of the motion specifications: link 2 is contained within the motion specifications, link 4 is not. The same is true for modes 5 and 6. The necessary condition analyses will depend upon 1) required Grashof type for those modes which include cyclic links, and 2) the pivots which are not allowed to pass change points. For example, in mode number 6 a change point is permissible at pivots A, C and D,but not at B, regardless of Grashof type. These considerations are presented in detail in Chapter V.
Figure 38 shows some examples of modes of rotational cyclicity which are unacceptable or geometrically impossible. Figures 38a and 38b might represent the cyclic couplerdriven Grashof double rocker and the cyclic rockerdriven crankrocker, respectively, which have already been discussed. In Figure 38c, control of motion through the change points of pivots r and s is impossible. The motion shown for the sixbar linkages of Figures 38d and 38f simply cannot exist, and the motion shown for the Watt sixbar linkages of Figure 38e would require a linkage with very unusual geometry and an external control of motion for both fourbar chains.
33
Problem Solution
33
A D
Applicable Nodes of Rotational Cyclicity
2(2)
Figure 37 Applicable Modes of Rotational Cyclicity of the
FourBar Solution to the TwoLink Chain Problem
34
B 3 C
22
2 4
A D
B 3 C
(a)
2 4
D
A 1 D
5 (b)
A 6 B
2
FA
(c) D 4 C
1 G6 B
2
F C4k 3 C 4 H (d)
3 \5
B F
A 2
(e) c 4
15
B D F
A~ 42
(f)
Figure 38 Some Unacceptable Modes of Rotational Cyclicity
35
C. Motion Specification Types
The types of coplanar problems which are solvable and optimizable using the procedures developed in this work are shown in Figure 39. The first and second problems, coplanar path and motion syntheses, are the subject of most linkage optimization efforts.
1) Path Synthesis. The motion of a point is guided by attaching the point to the coupler of an appropriate fourbar linkage.
2) Motion Synthesis. The motion of a body is guided by attaching the body to the coupler of an appropriate fourbar linkage.
3) Angular Coordination of a TwoLink Chain. The angular motion oil two connected bodies, one of which is connected to ground, is controlled by attaching these bodies to an appropriate fourbar linkage.
4) Angular Coordination of TwoCranks. The angular motions of two unconnected bodies, both of which are connected to a common ground, are coordinated by attaching these bodies to an appropriate fourbar linkage.
5) PathCrank Coordination. The motion of a point is coordinated with the rotation of a body connected to ground by attaching the point and body to an appropriate fourbar linkage.
6) Angular Coordination of ThreeLink Chain. The
36
I.) 2.) 3.) Angular Coordination
Path Synthesis Coplanor Motion of Two Link
Synthesis Choir
4.)C5a) 6.)
4.) Angular Coordin Path Crank Angular Coordination
otion of Two Coordination of Three Link Chain
Cranks
7)
7.)Angular Coordin .) Angular Coordin 9.) Angular Coordination motion of a Croank and action of Three Cranks of Three Adjacent Planes
a Two Link Chain 1
IO.) Coplanar Molion I.) Copranar Motion 12.) Point Path Point Path
Synt hcsis of Two Crank Coordination Coordinated with Two
Adjocent Planes C
Fiure 39 Tes of Motion SCrecifications ns
ILI
Figure 39 Types of Motion Specifications
37
motion of three bodies connected in a chain (one connected to ground), is guided by attaching these bodies to the links of an appropriate sixbar linkage.
7) Angular Coordination of a Crank and a TwoLink
Chain. The motion of two connected bodies (one connected to ground) is coordinated with the rotation of another body which is connected to ground by attaching these bodies to the links of an appropriate sixbar linkage.
8) Angular Coordination of ThreeCranks. The rotation of three nonadjacent bodies, all connected to ground, is coordinated by attaching these bodies to the links of an appropriate sixbar linkage.
9) Angular Coordination of Three Adjacent Planes. The motion of two nonadjacent bodies and a third body which is connected to ground and to which the first two bodies are connected, is controlled by attaching these bodies to the appropriate links of an appropriate sixbar linkage.
10) Coplanar Motion.._Synthesis of Two Adjacent Planes. The general motion of two connected bodies is controlled by attaching these bodies to the links of an appropriate sixbar linkage.
11) Coplanar Motion Crank Coordination. The
motion of a body is coordinated with the rotation of a body connected to ground by attaching these bodies to the links of an appropriate sixbar linkage.
38
12) Point Path Coordinated with Two Cranks. The motion of a point and the rotation of two nonadjacent bodies which are connected to a common ground are coordinated by attaching the point and two bodies to the links of an appropriate sixbar linkage.
These problems and their possible linkage solutions are presented in the charts of Appendix A. The steps involved in an actual synthesis procedure are discussed in Chapter IV. Examples of synthesis procedures and necessary condition procedures are presented in Appendix B, and the necessary conditions will be discussed in detail in Chapter V.
CHAPTER IV
SYNTHESIS OF LINKAGE SOLUTIONS
In the last chapter the types of problems under consideration were presented (Figure 39). The synthesis solution to each consisted of attaching these bodies with specified motions to the links of an "appropriate" linkage. The means by which this "appropriate" linkage is found is the subject of this chapter.
A. The Four Tools ofSynthesis
The synthesis procedure for the problems shown in Figure 39 depends on the problem type and the solution linkage type. In each case some combination of four synthesis techniques is employed. These techniques are termed the."four tools of synthesis" and are depicted graphically in Figure 41.
1.) Synthesis Tool #1. The Curvature Transform
Probably the single most important concept in the
field of kinematic synthesis is the curvature transform. All of the synthesis procedures used in this work will use this tool of synthesis at least once to generate constraint links (which maybeoperated upon by the other tools of
39
40
\3/
CURVATURE TRANSFORM PATH COGNATES
ANGULAR COGNATES INVERSION
0
I '
I "
Figure 41 The Four Tools of Synthesis
41
synthesis) in order to produce a functional linkage. For the first three problems the curvature transform alone is sufficient.
Since no linkage provides an infinite number of design parameters the number of functional motion specifications must be finite. If the precision positions are specified in such a way that an independent parameter, such as the coupler angle, undergoes a finite displacement between positions, the specifications are termed finitely separated positions. These are shown for a plane in coplanar motion in Figure 42a.
Finitely separated position synthesis of fourbars
was first done by Burmester [106] in 1888, using graphical techniques for five positions. In 1964 the problem was solved analytically by Bottema [107] and Primrose, Freudenstein and Sandor [108].
If higher order derivatives of motion at a precision position are specified, they are referred to as infinitesimally separated positions (ISP) as depicted by the conceptual combination of positions in Figure 42b. Mueller 11091, Allievi [110], Krause [111], Wolford [112] and Veldkamp [113] performed graphical studies of infinitesimally separated positions for five positions.
Mixtures of finitely separated positions and inf initesimally separated positions are termed multiply separated positions (MSP) (Figure 42c). The unified theory for MSP was developed by Tesar [114] in 1967.
E=0,1, 2,3,4
b, Infinitesimally Separated Positions (ISP)
2 E
a. Finitely Separated Positions c. Multiply Separated Positions
(FSP) (MSP)
Figure 42 Finitely, Infinitesimally and Multiply Separated Positions
43
All of the previously mentioned work involving ISP
and MSP involved geometric derivatives, where the independent parameter is the angular orientation of the body. The specifiable higherorder motions are consequently geometric (slope, curvature, inflection, etc.). In 1974 Myklebust [115] presented a unified theory making dynamic (i.e., time dependent) higher order properties specifiable. This is termed "time state synthesis." For the fourbar problems the dynamic properties are equivalent to the geometric properties, but for a sixbar linkage the use of an equivalent independent parameter for the decoupled subsections of a linkage solution is a necessity. Use of time state synthesis allows true "kinematic synthesis" to be carried out, as conditions can be prescribed for position and its time derivatives (velocity,acceleration, etc.) or combination of inertia and momenta effects, up to the fourth order.
Formulation of the curvature transform
Consider the moving plane in Figure 43 where motion
is described by the coordinates a~, and y. E is the fixed ref eZence with orthogonal coordinates U and V attached to it, and E is the moving reference with orthogonal coordinates u and v attached. They are related by the "coordinate transform":
U = u cos y vsin y+ a
(4.1)
V = u CosY v Cos Y+~
44
V
JE
Fixed Plane U
Figure 43 Parameters Relating Fixed and Moving Planes
45
and the "inverse coordinate transform":
u = (Ua) cos y + (V3) sin y
(4.2)
v = (Va) cos y (Ua) sin (
The constrained motion of E can be described in
terms of two functional relationships. Letting y be the independent parameter, let a = f1(Y) and a = f2(y). Links with moving pivots attached to E and fixed pivots attached to E might constitute physical constraints. If so, the functional description of the circular motion of a moving pivot (attached at say, point A in E) relative to E can take the form F(U,V) = 0. Combining this with the coordinate transform (4.1), the constraint equation can be represented by:
F = f (u,v,a,a.y,Q0,QIQ2 . .) 0
where Qi's are constants. For finitely separated specifications of the motion of plane E the above constraint equation takes the form:
F. f (u,v,Cjj, jy,Q,Q,Q2 ... ) = 0 j = 0,1,2 . .
For infinitesimally separated positions this relationship takes the form:
dk
F d [f(u,v,CC,,y,Q0,Q1Q2* .. )] = 0
k k
46
For multiply separated positions these two expressions would be mixed. Now, the circular constraint equation can be expressed as:
Q0(U 2+V2 ) + 2Q1U + 2Q2V + Q3 = 0
For a point A on plane E whose path describes a circle for multiply separated positions, this condition will take the form:
g.(UV) = dkQ0(U 2+V 2) + 2Q1U + 2Q2V + Q3] = 0 (4.3)
dYk =
Y=
If the zeroth position (go=O) is subtracted, these equations will reduce to: k 2 22 2
G (U,V) = go= d kQ0(U +V U0V 2 + 2Q(UU
dYk (4.4)
+ 2Q2 (Vv0)]l = 0 Y=Yk
for Z = 1,2,3
and coefficient Q3 is eliminated.
Now, if the specifications are normalized so that
O0 = 0 = Y0 = 0, then U0=u and V0=v. Substituting the coordinate transform into (4.4): U 2+V 2_u2v2 2 + 2+u(2ccosy+2 siny)+v(2 cosy2asiny)
47
and the generalized circular constraint function (divided by 2) takes the form: Gz = dk Q0 (X 2 2)+ u(ccosY+6sinY) + u(asinY+cosY))
dYk 2
+ Q1(u(cosY1) usinY +0) + Q2(usinY
+ u(cosY1) + 8)] = 0
Y=Y
which may be written as:
G = Q0(A09 + uAi + vA2Z) + QZ (uA3Z + VA4Z + A5)
+ Q2(uA49, + vA3Z + A 6Z) =0 (4.5)
k = 1,2. . .
where Am coefficients are listed in Table 41. They are a function of the motion specifications only.
Now, let:
D = A0t + uAlZ + vA2Z E = uA3 VA4 + A5Z F = uA4 + vA3 + A6P
and the constraint equations now take the form:
G = QODY + QE + Q2F9 = 0 (4.6)
48
For 4L4SP motion specifications (k 1,2,3):
QO D1 + Q 1E 1 + Q 2F 1 = 0 QO D2 + Q 1 E2 + Q 2F 2 = 0 QO D3 + Q 1E 3 + Q 2 F3 = 0
The matrix
D1 1 1
D2 E2 F2 (4.7)
D3 E3 F3
must be singular if the constraint equations are to
be meaningful since these equations are linear and homogeneous with respect to Q01 Q1 and Q 2'
If this matrix (4.7) is expanded, the result is a cubic in terms of E coordinates u and v. This is the circlee point cubic" and represents the locus of permissable moving pivot locations on E.
The formulation for time state specification synthesis is quite similar. Equation (4.3) will now take the form:
GP d k [Q (U 2+V 2)+2QU + 2Q2 v+Q 3 1 = 0 (4.8)
d tkI
t=t9
for k. = 0,1,2
49
and the A 'z s in equation 4.5 will be replaced by timestate motion coefficients termed D 'ils. The D 'PZ s are listed in Table 42.
In timestate synthesis y becomes a dependent parameter, but the D 'zIs reduce to A 'z s by replacing independent parameter t by y so that:
t~y d tY ~ dL k 1; d ky = 0 k>l
dt dy dt k dy k
and the timestate synthesis formulation reduces to the geometric synthesis formulation.
2.) Synthesis Tool #2. Inversion
The inversion concept essentially involves a change
of reference. Two basic types of inversion are of interest. In the first, the motions of planes N and M (Figure 44) are defined relative to a fixed plane R, the reference is to be changed to plane N, and the motion of plane M. relative to plane N is to be defined (formulations #1, 2 and 3, below). In the second type of inversion the motion of N is defined relative to fixed plane R, the motion of plane M is defined relative to plane N, and the motion of M relative to R is to be defined (formulations #4, 5 and 6, below).
Motion specifications for the two planes which have
some ISP content must share a common independent parameter,
50
n r
n nI
r r
I\n
mI \\
I
I m \
n j I r Ir
nj I r n
Figure 44 Inversion Notation
51
or have a defined relationship between independent parameters. The types of motion specification sets which correspond to the formulations that follow are displayed in Table 43.
a.) Inversion formulation number 1.
Given: Motion of N relative to R; M relative to R.
Find: Motion of M relative to N ( for R0 reference ).
Independent parameter for given MSP: time t.
Independent parameter for derived MSP: time t.
(k) n (k)r
a m = Sk cos C + Tk sin C + a n0
(k)n (k)r
_mj =Sk sin C + Tk cos C + 3nO
(k)n (k)r (k)r (k)r ]mj Y mj Y nj + nO
Where: j = FSP counter,
(k)
k = ISP counter ( for example, for k=2, a = a ).
so A r r0 mj nj
T B r r
T0 = mj nj
= ,r r
nj nO
52
S =A + BC
1
6*
T1 = AC + B
S2 = 2AC AC + B BC
*2 **
T = A AC + 2BC + BC
2
.... *2 + * "3 **
S = A 3ACC 3AC2 + 3BC + 3BC BC + BC
3
T = 1B 3BCC 3BC 3AC 3AC + A AC
3
b.) Inversion formulation number 2.
Given: Motions of N relative to R; M relative to R.
Find: Motion of M relative to N ( for R0 reference ).
Independent parameter for given MSP: yr
n
Independent parameter for derived MSP: y
m
This formulation is identical to that of (2a), except that the dots in the S and Tk formulations should be replaced with primes ('), and now A', A", A", B', B", B"'and C', C", and C"'must be defined.
53
r ,. n
Let a n an; an dyr ;etc. (same for 13n and y~ n
d ,r
a *r a *
an a~m i etc. (same for1 andy )
Now,
a, a,
A'/ m n f
A" =n 2' n 3 m
1), (y'
in m
(a"'/ a".) (/ ) 2 + (a" a" (3 y"(y
A"= m n in i n M
1(a/ a') (3y"/2 + y ""(y' )
in n in in i (y' 5
B', B//, B"'are similar (substitute a3's for a's )
C/= 1
(ym
54
y"(y' 1) + 2
C "" = m
(y 5
c.) Inversion formulation number 3.
Given: Motions of N relative to R; M relative to R.
Find: Motion of M relative to N ( for R0 reference ).
Independent parameter for given MSP: yr, yr must be
related ).
Independent parameter for derived MSP: n
Again, this formulation is identical to that of (2a),
except that the dots in the Sk and Tk formulations should be replaced with primes ('), and now A', A", A', B*, B", B", C', C"' and C" must be defined.
d~r
r / n
Let an =X n a =n ; etc. ( same for n and y
d~r
am a m da y etc. ( same for and y
M m m dr m m
dyr 2 r
dn n
Ynm r Ynm r2 ;etc.
dYm dym
The formulations for A', A", A"', B', B" and B"'are
equivalent to those of section (2b), under the above notation, except that y yn Ym y/ etc., and C', C1" and C"/ must
M nm m nm
55
be redefined.
Now,
C nm
1 ynm
nm
C, = nm
1C
'(1 y/ ) + 3y"2
Gnm nm nm
C _
2
(1 Y)
d.) Inversion formulation number 4.
Given: Motions of N relative to R, M relative to N.
Find: Motion of M relative to R.
Independent parameter for given MSP: time t.
Independent parameter for derived MSP: time t.
(k) r (k)
r r
Smj = S cos C T sin C + a
m] k k nj
(k) r (k)
r r
S= Sk sin C + Tk cos C + nj
(k) r (k) (k)
r m Y m + Y nj
mj mijn
56
Where,
s n
0 mj
C r
n Ynj
0 mj
on n r
s 1 Ct mj amjynj
T n Or + ; n
1 mjynj mj
#on n r n or 2
s 2 Ot mj 2 mi ni mjynj amjynj
0.
n r n *or n r 2+ n
T 2 2a mi ni + a mi Yni mj ni mj
on ..n Or on or n ... r on Or 2
s 3 =ct mj 3 mjynj 3 mjynj mj y n3 3a mj y nj
n or r n or 3
3a mi y ni ni + mjynj
*on *r on or 2 n *or r n or
T 3 = 30C mjynj 3 mjynj 3a mi Yni ni + 3a mjynj
n or 3 n I *or ...n
ot mj'ynj + a mjynj + m 3
57
e.) Inversion formulation number 5.
Given: Motions of N relative to R, M relative to N.
Find: Motion of M relative to R.
rIndependent parameter for given MSP:y
Independent parameter for given MSP:yn
Independent parameter for derived MSP: yr
This formulation is identical to that of (2d), except
that the dots in the Sk and Tk formulations should be
replaced with primes (').
der
Let a a ; n; etc. ( same for and y )
n Ln n r n n
dy
n
da n
am a ; a d ; etc. ( same for 8m and y
m m m rmm
n
Now,
Now, dan a /
n m m
a
m dy 1 + y
m m
a'(l + ym) a'y
/'n m m m
m /3
am( + ym)
m
(1 + 2y' + 2) a"(y' + 2y" + 2y"y' + y2)
a =m m m m m m mm
+ a'y"/(2y' 1)
mm m
(1 + Y)5
'n "nd
6 h and are similar ( substitute B's for a's).
m m m
58
_r 1
I
r Ym
"r Yi//
Yn i+y/) 3
3 y y (1 + Ym)
r = i m in
Yn (1+ y)5
f.) Inversion formulation number 6.
Given: Motions of N relative to R, M relative to N.
Find: Motion of M relative to R.
Independent parameters for given MSP: yr, yn ( must be
related).
Independent parameter for derived MSP: yr
This formulation is identical to that of (2d), except
that the dots in the S k and Tk formulations should be replaced with primes (').
dar
r / n
Let a = n a  ; etc. ( same for 8and
n n n dy rnn dan
inn
a dy ; etc. ( same for am dyM dyr y r
yn Y/ dyndn etc.
Ymn n ;n mn dyn 2 ;ec
dm dm
59
/ In
The formulatiortM for ,n and their derivatives are
equivalent to those of section (2c), under the above notation, except that y mn' m Ym'/ etc.
M mn m mn Now,
Y
Ir Ymn
Yn =
mn
"r Ymn
= (ly 3
(I + Y/)
mn
"' (1 + y' ) 3y, 2
Ym n Ymn ) mn
Yn +Y 5
Y 5
( +mn
3.) Synthesis Tool #3. Path Cognates
The Roberts Chebyshev theorem states that three different coplanar fourbar mechanisms will trace identical couplercurves. An example is shown in Figure 45b, where coupler point e has the same motion whether carried by fourbar O1dfO2, fourbar OlgjO3 or fourbar 02 hkO3. An additional property is that the angular parameters of those links with identical hash marks in Figure 45b are identical for any reachable position or position derivative of point e.
These properties can be used to solve the problem of
60
V
U
a. Prescribed Path,and Input Crank 03
k
e
g h
d
b. Robert's Cognates
Figure 45 Use of Path Cognates to Solve the PathCrank Coordination Problem
61
pathcrank coordination (Figure 45a). The synthesis procedure is: 1) specify the motion of coupler def by using the translation of point e and the rotation 0, use the curvature transform to synthesize fourbar 0 1 dfO 2" and find its path cognate 0 lgj03 or 02 hk03 Either cognate will coordinate the rotation of a crank parameter 0 (attached to link 0 1 d or 0 2 h) with the motion of point e (attached to coupler link gej or hek, respectively).
4.) Synthesis Tool #4. Angular Cognates
Consider the parallelogram formed by links BCD and
BCID in Figure A6c. Regardless of the motion of point B (guided by crank AB for this example) the angular parameters of links BC and CID are identical. So too for links BC' and CD. Figure 46 outlines the use of the concept of angular cognates to solve a crankcoordination problem. The second angular parameter 0 2 is used to define the motion of coupler plane BC, the curvature transform is used to find a second constraint link CD, and the angular cognate of dyad BCD is taken to provide solution linkage ABCID.
B. Coordination of the Tools of Synthesis
Appendix A contains tables of linkage solutions and recommended synthesis procedures for each of the twelve problem types. These synthesis procedures consist entirely
62
a. Input Output Crank3
I 4D
b. Curvature Transform
C' e C
e2~ee
c. Angular Cognate
Figure 46 Use of Angular Cognates to Solve the
Function Generation Problem
63
of appropriate combinations of the four tools of synthesis just presented, and are generally not unique for any given problem type and linkage solution.
A Stephenson 2 solution to problem number twelve is shown in Figure 47. The synthesis procedure utilizes the tools of inversion, path cognates, and (as always) the curvature transform. The procedure begins with a combination of the translational motion specifications of point e and the rotational motion specifications of crank K to create a new motion specification set describing general coplanar motion (Figure 47a). Operating on these specifications with the curvature transform will produce constraint links as shown in Figure 47b (actually a set of constraint links is produced). Operating on this preliminary linkage solution with path cognate analytics will produce constraint links 7' and 5. These constraint links will move the coupler, link 4, in a way such that coupler point e will move on its specified path in coordination with the specified angular motion of crank link 5. Now, the motion specifications exist for links 4', 5 and 3 relative to fixed link 2. Changing the reference to link 3, it is possible through inversion analytics to define a new motion specification set for plane 4 relative to 3 (Figure 47d). Operating on these motion specifications with the curvature transform produces constraint links 1 and 6, with the circle point cubic defining pivots D and G and the
64
e
D 4 C
Jz
35 5 F 3 A
STEPHENSON I
\(a) (b) f3
+ CURVATURE e
e TRANSFORM
+
PATH
COGNATES
e C (d) (c)
e55 3
4' B 5
B
2 INVERSION + 7' '2
A A
+
CURVATURE
TRANSFORM
e (e) (f) e
D 4 C D 4 C
5 5
SB (+ REINVERSION)i B
3 2
F A F A
Figure 47 A Stephenson2 Solution to Point/Path
Coordination with Two Cranks
65
center point cubic defining pivots F and H. A conceptual reinversion (which requires no computational activity) produces a Stephenson 2 sixbar linkage which satisfies the motion sQecifications.
C. Grid Dimension
If 4MSP motion specifications are used, every operation involving a curvature transform generates an infinity of constraint links. If one constraint link is to be selected from this set, a one dimensional grid is required to discretely represent the solution set. Likewise, if two constraint links are to be selected from the set, the grid will have two dimensions. A curvature transform followed by a path cognate operation will generate (2) twodimensional grids, since each fourbar (which occupies a place on a twodimensional grid) has two path cognates, a "left" cognate and a "right" cognate.
The example discussed in the last section produces two (2) twodimensional grids representing link 5 and 71 and another twodimensional grid representing links 1 and
6. The total, if these grids are combined, is two (2) fourdimensional grids. This problem lies at one extreme of the problems considered in this work. The opposite extreme occurs for problem Vs 3 and 4, whose entire linkage solution sets can be represented by onedimensional grids. The nature of the component grids is a
66
function of problem type and linkage solution, but the shape of the overall grid is a function of problem type alone.
D. Dependent Syntheses
The previous example demonstrates an undesirable
property which will be hereafter termed "dependent syntheses." Essentially, this property exists whenever a synthesis procedure involves two (or more) curvature transform operations in such a way that the order in which they are performed cannot be switched. In the previous example, the second curvature transform uses the motion specifications of plane 4, which does not exist until plane 4 has been generated by (among other things) the first curvature transform. Thus, every plane 4 generated by the first synthesis will have a unique motion specification set for the second synthesis to operate upon.
This property greatly complicates subsequent analysis and optimization steps, and at this point these difficulties are circumvented by performing a complete optimization on the results of one portion of a dependent synthesis before proceeding to the second synthesis (see Appendix D, problem 12).
Fortunately, not all sixbar synthesis procedures
involve dependent syntheses. In fact, all of the problems
67
except number twelve can be solved by at least one linkage solution without dependent syntheses. These solutions must be considered to be the more desirable from a computational standpoint, and are indicated *in Appendix A with the t symbol.
Table 41. Motion Coefficients, Am
k A Oz A l A2Z A 3Z A~ 4ZA5 ,
0 cxX 2 Cos y. ax. sin y. Cos y.l sin y. a. .
+ sin yj +i Cos y.
1 a~!6S (cx + Cos y. ($!~a.)Cos yj sin yj Cos y. ax!
+(r3!ct!) sin y. (Ox!+. sin yj 2 (Ux6) +N%) (X "+otO
0 0~2x l00x
3 3cx6co+3F13" Ot 1 Ncx 1 3%30t I 0 1 a i 1
For k > 2, the position j=0, is assumed as the initial position. When k. j
a 0 =30 =Y0 = 0. The primes denote derivatives with respect to y.
Table 42. Motion Coefficients, D mt
k D D 3 P4 D 5 D 6
0 cosf 4 S i. nY P cosy ,A Cos Y C' i'( sinY .sinY
2 0 0 j j
0 c 0 SY 0 0 S i rl"f 0 0 CDSYOkl 0 S in'f 0 L (,tj0y) siny Y) silly YS. Ln ;cosy
+ + cosy
1*6*) Giny 25y) siri,( 3 i n y Sin'(
+ 2 +2 2 COSY
Cosy +(.,ty2uy y +13 2 COS,, 4COS y
:,LL t 3 + 01 c ;% 3,*,'Y* 3 3,, y) z; i n,( (3yy)sin),
3 Y S I n Y 6y3al 36y) s in,, +(3y'f)C0SY +(y ty)cosy
+ .y 3 4 y
+ 0 3 3,x+3 6j 11,T 3 3,; 3 1!' 6)cosy
70
Table 43
Related Motion Specification Sets for Two Planes M and N (2ISP)
(Notation: Figure 44)
Inversion Formulation #1 Inversion Formulation #4
(Independent parameter t) (Independent parameter t)
Plane N: Plane M: Plane N: Plane M:
r r r r r r r r r n n n
an n Yn a Yi n n'n n am' m m
dar d r dy r dar d r dy d r d fr dyr dan d n dn
n n n dm m m n n m minm
dt' dt' dt dt' dt' dt dt' dt' dt dt' dt' dt
Inversion Formulation #2 Inversion Formulation #5
(Independent parameter r (Independent parameter r
coupler angle y) coupler angle y)
Plane N: Plane M: Plane N: Plane M:
r r r r r r r r r n n n
an' n n am' m m an n Y n m' m m
r r r r r r r n dam dy
da dB da dr dy da dB da d$ dy
n n m m m n n II m in
r I 1
r r r r r r r d r dy
dy r ddy r ddy r ddy r dy rdy
n n n n n n n n n n
Inversion Formulation #3 Inversion Formulation #6
(Independent parameters (Independent parameters
coupler angles Yr Y m (related)) coupler angles yn' ym (related))
n' nni
Plane N: Plane M: Other Plane N: Plane M: Other
r r r r r r r r n n
dor dr dr dr dyr dr dpr dA n dBn dyr
n n in in n n n in in n
rr r dy I, n, n1
d'fdyr dy dyr dy dy d 1' d dym
n n m m m n n m m i
CHAPTER V
THE NECESSARY CONDITIONS
Linkages generated through the use of the kinematic
synthesis techniques of Chapter IV are definitely capable of satisfying the specified positions. Unfortunately, the analytics presented there take no consideration of linkage performance in positions other than those specified. Real linkages may fall into a number of physical pitfalls, any one of which is sufficient to eliminate the linkage as a viable solution to the specified motion problem. The analytics now presented are for "Inecessary conditions"I which must be satisfied before a given solution linkage can be considered for further optimization analysis. The conditions which apply to a given problem are dependent upon the solution linkage type, the means by which the linkage is to be driven and the nature of its motion (i.e., mode of cyclicity). It is primarily because of the dependence of the necessary conditions upon mode of cyclicity that the concept of the latter has been developed.
A. FuurBar Necessary Conditions
1.) Order
It is entirely possible that a linkage solution will
71
72
satisfy the specified positions in an improper sequence,
as displayed in Figure 51 for a fourbar linkage. Six possible orders exist for 4FSP:
0 1 2 3 0 1 3 2 0 2 1 3
0 3 2 1 0 2 3 1 0 3 1 2
Only those on the left would be acceptable, assuming the linkages can be driven in either direction.
Fortunately, relatively few linkages are eliminated by order considerations. First of all, motion specifications placed in a relatively smooth and continuous sequence tend to be satisfied naturally. Secondly, the specified cases with higher ISP content diminish the order problem. Finally, problems which have motions coordinated with a crank automatically satisfy specified orders.
The most direct way to analyze order is to use the method of Modler [1161, which divides the circle point cubic into segments which represent constraint links of the same order. These segments are delimited by the image poles and the Ball point. This is a powerful tool for graphical analysis, but for discrete computational analysis it is simpler and more direct to inspect exhaustively the sequence at specified positions of all constraint links.
2.) The Grashof Criterion
A fourbar chain may have either zero pivots with complete rotation, or two adjacent pivots with complete
73
0
2
a. Improper Position Sequence
21v/
b. Order Analysis Crank Rocker)
Figure 51 Order
74
rotation. The latter situation occurs if the linkage satisfies the Grashof condition:
kZ+ s < p+ q
where
Z= length of longest link
s = length of shortest link p,q = length of other two links Grounded fourbar chains which satisfy this relationship are known as "Grashof fourbars," and take the forms shown in Figure 52 a,b,c. They are simply different inversions of one another, and are generally more useful than nonGrashof linkages. The crankrocker, for example, can be used to generate an arbitrary shape with a coupler point while its crank is driven continuously, and the draglink is often used for continuous function generation. In a Grashof fourbar only those pivots which are cyclic will ever enter change points.
The nonGrashof fourbars take only one form (regardless of inversion), the nonGrashof doublerocker. This type of linkage tends to dominate a linkage solution grid generated through the curvature transform, but has the unfortunate property of being able to reach "change points" (discussed in next section) at any of its pivots. Control of linkage motion through these change points requires special considerations.
75
(a) CRANKROCKER
(b) DRAGLINK
(c) GRASHOF DOUBLE ROCKER
(d) NON GRASHOF DOUBLE ROCKER
Figure 52 FourBar Grashof Types
76
3.) Branching and Change Points
A change point for a given pivot is that position at which attached links become collinear. This position, which might lie between specified positions, may or may not be acceptable depending upon the nature of the linkage and how it is driven.
a.) Dyad change points
In most cases change points in the dyad of a Stephenson or Watt sixbar indicate simultaneous disconnection (or a limit in motion of the rest of the linkage, Figure 53b). At this point the dyad could pass from one dyad "type" (relative angular orientation, Figure 52c) to the other. One generally wishes to verify that the specified positions are satisfied while the dyad maintains its type, although one exception is that of the cyclic dyaddriven sixbar linkage (Figure 53d).
There are two times at which dyad type can be checked. If both links of the dyad are defined by the motion specifications, such as the Stephenson 3 solution to a threelink chain problem (dyad links 2 and 5 are the first two links of the chain), it is a simple matter to simply check the motion specifications before synthesis. More often, however, one of the two links of the dyad is generated in the synthesis procedure, and dyadtype must be checked during necessary condition analysis.
77
2 0/
I
2 5
2
5/
d5
Figure 53 Dyad Change Points
78
b.) Change points in fourbar chains
All of the solution linkages can be considered as
single fourbar chains, a combination of fourbar chains and dyads, or a combination of two fourbar chains. A detailed analysis of change points within fourbar chains is therefore very worthwhile. It involves considerations of both synthesis procedure and mode of cyclicity.
If a fourbar chain (or any Grashof type) is driven as shown in Figure 54a through a change point at the indicated pivot, the dyad portion of the chain connected to this pivot may go into either type. This is beyond the direct control of the driven members. If specified positions are to be satisfied in a positive manner, such a change point must be avoided. Similarly, fourbar chains driven through opposite links (Figure 54b) cannot tolerate change points at any pivots, for the same reason.
It may be assumed that at least one of the links in the chain has been synthesized through use of the curvature transform. Waldron [117, 1181 has developed graphical techniques which operate directly on Burmester cubics, identifying sections which represent usable constraint links (for fourbar chains) on the basis of changepoint considerations. These techniques are used extensively in the necessary condition analysis of this work, so detailed descriptions now follow.
i.) Determination of"permissible links. Let the name
79
4W
Figure 54 Change Point Considerations in a
FourBar Chain
80
"trailer" signify constraint links with limited rotation at the moving pivot and the name "rocker" signify constraint links with limited rotation at the fixed pivot (Figure 55). For example, the follower of a crankrocker fourbar is both a trailer and a rocker, but a constraint link of a draglink four bar is a trailer only. Trailers
def ined by circle and centerpoint cubics which have moving pivot rotation less than 7T for the 4MSP are termed "permissible". The first technique of Waldron is used to determine sections of the circle point cubic which represent permissible constraint links.
Let T.. be the rotation of the coupler relative to
the constraint link in moving from positions i to j. This angle will be defined between Tr and N, clockwise positive. Let n denote the position with the smallest or most negative value of T., so that it defines one end of the angular
13
range. if T nj is > 0 for all possible values of j, the angular range must be less than T. If it can be established that for some value k, T jk is always either positive or negative, the constraint link under consideration must be permissible.
The following property can be used to find the sign of 'Y jk :
A line through the image pole P.. and
13
circle point A bisects the angle if ij
(see Figure 56a).
This property applies to any T ik' where k is another position, so the angle T jk can be found from:
81
Limited relative rotation at the moving pivot a. Trailer
Limited relative rotation at the fixed pivot b. "Rocker
Figure 55 Trailer and Rocker
82
2
Pii
p Image pole for positions i,j SkOa
a.
y A jk
jk2 2 1Ik Ij
(Vij Yik) '
2 Pik
Oa
b.
Figure 56 Determination of Relative Moving Pivot
Rotation for 2FSP
83
T jk = (T 13 T ik) (see Figure 56b)
2 2
Now the location of circle points with positive and negative values of the T in terms of P!. and P! (Figure 57c) jk 1] 1k
can be determined by using the property:
The three lines through diametral
points and a third point on a circle
will define a right triangle.
For circle points lying on the right side of P! P! and ii 1k
outside the circle, 0 < T jk /2 < ff/2, so that 0 < T jk < 7T and for circle points on the same side but inside the circle, 7T/2 < T jk /2 < R (Figure 57a). On the opposite side of P! P! the signs will be reversed. Figure 57b
ii ik
shows this construction overplayed onto the corresponding circle point curve.
Two values of i could be chosen: the only requirement is that i be distinct from j and k. Interestingly enough, if evaluations were made using both possible values of i some spaces in the plane would have conflicting signs, but the circle point cubic will occupy none of these spaces. Another interesting property is that the signs will not change as the circle point cubic passes through an image pole, but will change as the circle point cubic crosses the circle or line segment singly.
The portions of the circle point cubic which represent permissible constraint links will be defined by those pottions for which all the components of any of the strings
84
4Pij
0.
+ Circle Point
+Curve /
b.
Figure 57 Determination of Permissible Trailers for 2FSP
85
T 01, T 02 T 03
T 10, T 12' T 13 T 20' T 21' T 23 T 30' T 31' T 33
have the same sign.
ii.) Determination of "compatible" links. A "compatible" constraint link is one which can be used in combination with a permissible constraint link so that the 4MSP will be satisfied on the same branch. The previous analysis is used to eliminate portions of the cubic which represent constraint links which will definitely pass through a change point at the moving pivot while satisfying the 4MSP (regardless of the other constraint link chosen). A permissible constraint link which has been selected from a portion of the permissible circle point cubic might still pass through a change point if coupled with an "incompatible" constraint link.
Let 0 a A be a selected permissible constraint link. Figure 58a shows one in the four positions in which the motion specifications are satisfied. T 03 defines the relative coupler rotation in reaching position j from position
0. Figure 59b shows 0 a A in the zeroth position with angles T 01, f 02 and T 03 drawn through point A. In this example, T 01 and 11' 03 represent the extremes of the angular range. Now, if a circle point representing a possible second constraint link lies in region Z, it must cross
86
IQ~~2 A01, Position of
=0 point A when coupler is in
00 specified pos.
a.
Fgr 5 C t C
b,
Figure 58 Determination of, compatible Constraints
87
line 0 a A to reach the extremal positions. As this happens a change point occurs at point A. Therefore, the portions of the circle point cubic which lie in the Z regions represent constraint links which are incompatible with the selected permissible link.
It is quite possible that the circle point cubic will not pass into the compatible regions denoted by Y. If so, another permissible link must be selected. Note that the Y regions disappear altogether for nonpermissible links, having no possibility of finding compatible constraint links.
c.) General branching analyses of fourbarchains in sixbar linkages
Displayed in the lefthand column of Figure 59 are
the possible forms of fourbar chain which may be needed as all or part (for a sixbar) of a linkage solution. The chains on the right are the different inversions in which the chain on the left may actually be synthesized. It is necessary to treat these inversions as they stand. A number of "necessary condition procedures" have been developed to handle the range of possible synthesis inversions. (The conditions under which order considerations can be ignored have been discussed in part A of this chapter).
Necessary
Condition
Procedure
case
FinalForm: As Synthesized:
NCA NOPO NCPA NCPC
IbaI Id
NCPB NCPA NCPC NCPA
NPD NCPD NCPG NCPG
___________ lMa Mlb Mlc flld
NCPH NCPH NOPH NOPH
I~ lb WVc
Figure 59 Necessary Condition Procedures for
FourBar Chains
89
Necessary Condition Procedure A (for cases Ia, Ic, Ib and IId)1) Choose a followerconstraintlink to be a
trailer/rocker.
2) Choose a crankconstraintlink of proper order
which is compatible with the follower.
3) Test Grashof type for a crankrocker.
Necessary Condition Procedure B (for cases Ib and IIa)1) Choose a constraint link to be a trailer of
proper order.
2) Choose another constraint link to be a trailer
which is compatible with the first link.
3) Test Grashof type for a draglink.
Necessary Condition Procedure C (for cases Id and IIc)1) Invert to change reference to the coupler link,
i.e. generate the motion specifications of the
fixed plane relative to the moving plane, interchange the circlepoint and centerpoint
cubic curves, etc.
2) Follow procedure B.
Necessary Condition Procedure D (for case IIIa)1) Choose a followerconstraintlink to be a
trailer.
2) Choose a crankconstraintlink of proper order
which is compatible with the follower.
Necessary Condition Procedure G (for case IIIc)1) Invert to change reference to the coupler link.
2) Follow procedure D.
Necessary Condition Procedure H (for cases IVa,b, c and d)1) Choose a constraint link which is a trailer
of proper order.
2) Choose a second constraint link which is a
trailer which is compatible with the first
link.
3) Check mutual compatibility.
4) Invert to change reference to the coupler link.
5) Repeat 1), 2) and 3) (no need to check order
again).
90
In some cases, tests for order and trailer/rocker are unnecessary if Grashof type are known, but are included because they improve the chances of producing a desired Grashof type. The Grashof type test is relatively expensive computationally because constraint link combinations, not just single constraint links, are under evaluation. d.) Analysis of branching in pathcognates
For motion specification types #5 and #12 (Figure 39) fourbar chains are produced by performing a curvature transform followed by a generation of path cognate fourbars.
We assume fourbar I in Figure 510b ha's been produced by the curvature transform and its path cognates, fourbars II and III, are to be usable as all or part of a linkage solution. What tests can be made on fourbar I. to ensure proper performance relative to change points in fourbars II and III? This question is central to establishing practical necessary condition analyses for pathcognate problems.
A study of fourbar II will now be made in detail. All comments made hereafter regarding "left" cognate fourbar II also apply to "right" cognate fourbar III. The modes of cyclicity which II may take are shown in Figure 512. Each of these modes will involve a separate analysis (necessary condition procedures I N) to be performed on fourbar I
to ensure proper performance in II.
91
V
e(Ue,Ve)
U
a. Prescribed Path,and Input Crank
03
lll
k
I
e
9 h
f
.02 '01
b. Robert's Cognates
Figure 510 Use of Path Cognates to Solve the PathCrank Coordination Problem
92
Necessary condition procedures I N take advantage of some unusual properties of the "cognate 10bar" (Figure 5l0b):
i.) A cognate 10bar will be made up of one of three sets of three fourbars: either three draglinks, three nonGrashof doublerockers, or a Grashof doublerocker and two crankrockers.
ii.) When 0 3 in fourbar II is at a change point, 0 2 in fourbar I is also at a change point.
iii.) When g in fourbar II is at a change point, d in fourbar I is also at a change point.
iv.) When 0 1 in fourbar II is at a change point, links df and 0 1l02 in fourbar I become parallel.
v.) When j in fourbar II is at a change point, links
0 1d and 0 2f in fourbar I become parallel.
Proof of (ii): Referring to Figure 513a,
11 101021
Now, from [105;79j
Z' =AXe ia(27) and R=Xe (R 0)
Therefore, when Y and K coincide, and R' will also coincide, and change points will occur simultaneously in fourbar I at 0 2 and in fourbar II at 0 3.
93
Proof of (iii): Referring to Figure 513b,
c(x + + + (D Bx + e 2 7T
A A because 0lged is a parallelogram,
A = 0 at simultaneous change
points for g in fourbar II and d in fourbar I.
Proof of (iv): Referring to Figure 513c, it follows from pathcognate property (ii) that change points occur simultaneously at 01in fourbar II and at 0 2 in fourbar III. In this position 0 2h and 0 203 become collinear. Therefore 6 2' + 0 = lG and 0 Te+ Oa3
where e refers to "the angle of ..
Since 02 hef is a parallelogram, fe and UF2 are prl lel. It follows that:
0102 df 6' 1
Proof of (v): Referring to Figure 513d, it follows from path cognate property (iii) that pivot j in fourbar II and pivot k in fourbar III will pass through change points simultaneously. In this position 0 3j and 03q are collinear, as are 0 3k and O H Due to the parallelograms and similar triangles, it follows:
ge and ehare collinear and
ge 0Old and eh 11 0 2f 0d 11 02f

Full Text 
166
Problem #9
Angular Coordina
tion of Throe Ad
jacent Planes
Linkage Solution Synthesis Procedure
1) CURVATURE TRANS
FORM to find 1.
2) CURVATURE TRANS
FORM to find 6.
WATT 2
Pivots IT,C,D
prespecified
Modes of Cyclicity
Necessary Condition Procedures
a)
Necessary Condition
cedure D for 1234.
Pro
b)
Necessary Condition
cedure D for 2456.
Pro
a)
Necessary
Condition Pro
cedure II
for
1234 .
b)
Necessary
Condition Pro
cedure D
for
2456 .
a)
Necessary
Condition Pro
cedure D
for
1234.
b)
Necessary
Condition Pro
cedure D
for
2456.
a)
Nocoss.
ary Condition
Pro
cedure
A for 1234.
b)
Nocoss;
ary Condition
Pro
cedure
D for 2456.
c)
Verify
Connection of
56
235
75. Elliott, J., and D. Tesar, "The theory of torque,
shaking force, and shaking moment balancing of four
link mechanisms," Journal of Engineering for Industry,
Trans ASME, Series B, Volume 99, No. 3, August 1977,
p. 715.
76. Sandler, B.A., "The use of a random algorithm for
dynamic optimization of mechanisms," Journal of
Engineering for Industry, ASME Paper No. 76WA/DE22,
1976.
77. Golinski, J., "Optimal synthesis problems solved by
means of nonlinear programming and random methods,"
Journal of Mechanisms, Volume 5, No. 3, 1970,
pp. 287309.
78. Osman, M.O.M., S. Sankar, and R.V.Dukkipati, "De
sign synthesis of a multispeed machine tool gear
transmission using multiparametric optimization,"
Journal of Mechanical Design, Trans ASME, Volume
100, April 1978, pp. 30310.
79. ChiYeh, H., "A general method for the optimum
design of mechanisms," Journal of Mechanisms,
Volume 1, 1966, pp. 30113.
80. Mansour, W.M., and M.O.M. Osman, "The method of
residues for the.synthesis of coupler curve generat
ing mechanisms," ASME Paper 70Mech53.
81. Bagci, C., and K.N. Prasad, "Minimum error synthesis
of plane mechanisms for rigid body guidance," Journal
of Engineering for Industry, Trans ASME, Volume 91, No.l
February 1971.
82. Bagci, C., and K.C. Parekh, "Minimum error synthesis
of the spherical fourbar and Watt's type spherical
sixbar mechanisms," Communications of the Third
World Congress on the Theory of Machines and Mech
anisms, Volume C, Paper No. 1, Dubrovnik, Yugoslavia,
1971.
83. Speckhart, F.H., "Calculation of tolerance based on
a minimum cost approach," Journal of Engineering
for Industry, Trans ASME, Series B, Volume 94, No. 2,
May 1972, pp. 44753.
84. Khan, M.R., W.A. Thornton, and K.D. Willmert, "Op
timality criterion techniques applied to nechanical
design, Journal of Mechanical Design, Trans ASME,
Volume 100, April 1978, pp. 319327.
36
Figure 39
Types of Motion Specifications
204
Module Chart 27
Name Link length Ratio (Desirable Condition)
Label DLKLNRT
Description
Evaluates the solution set
of linkages on the basis of
link length ratio. User
specifies links of interest.
+(03) (02) DLKNRT (11) ,( 12) (I ) ....
where:
(1^) = Previous scoring grid.
(I.), (I_); (I .) (I_);... = Constraint link
pivots (1 pair =
1 link).
(0^) = Output scoring grid, cumulative.
(O2) = Output scoring grid, this condition
only. Optional.
31
and exhaustive analysis is necessary because the nature
of the necessary condition analysis depends upon the mode
of cyclicity. For example, the pivots within a linkage
solution at which change point positions (i.e. adjacent
link collinearities) can be tolerated depends upon how
the linkage is driven.
For our purposes it often becomes useful to consider
each chain as free in space with two links driven  one
taking the place of the previously grounded link. In es
sence, the grounding and driving of links becomes concep
tually identical and interchangeable.
At this point a restriction is made in order to elim
inate modes of cyclicity for which special knowledge of
velocities and system inertias are needed to define mo
tions through change point positions. It will be assumed
that when links with two cyclic pivots exist in a chain,
one of these links and an adjacent link are driven.
This restriction eliminates from further considera
tion linkages such as the rockerdriven crankrocker with
complete crank rotation, and the Grashof doublerocker with
complete coupler rotation. These linkages must rely upon
link momenta or switching of inputs to maintain continuous
motion, and are beyond the consideration of this work.
Any given fourbar or sixbar problem will have a
subset of the possible modes of cyclicity, depending upon
the grounded link and the conceptual equivalence or
147
b.) Crank J
Point f
Crank K
Point e
> Link 3 and
= Pivot B.
* Link 2 .
on Link 6.
CURVATURE TRANSFORM and PATH
COGNATES to find 2,6, dummy 7
INVERSION to 2.
CURVATURE TRANSFORM to find 4
INVERSION to 4.
CURVATURE TRANSFORM to find 5
[4MSP Grid:L,R 2D, +(2)xlD]
Watt 2 Link 2 is grounded.
Crank J * Link 1 and
Point f = Pivot A..
Crank K Link 4 .
Point e on Link 5.
[4MSP Grid:L,R2D,+1D]
* CURVATURE TRANSFORM and PATH
COGNATES to find 6,4.
INVERSION to 1.
CURVATURE TRANSFORM to find 3
Stephenson 1 Link 1 is grounded.
Crank
J
Link 4 and
CURVATURE
TRANSFORM
and PATH
Crank
K
>
Link 3 .
COGNATES to find 4,
dummy 7'.
Point
g
=
Pivot F.
INVERSION
to 3 .
Point
e
on Link 5.
CURVATURE
TRANSFORM
to find 6
CURVATURE
TRANSFORM
to find 2
[4MSP Grid
:L,R 2D,
+(2)xlD]
Stephenson 2
i _
Link 2 is
grounded.
a. )
Crank
J
>
Link 3 and
CURVATURE
TRANSFORM
and PATH
Point
f
Pivot A.
COGNATES to find H,
dummy 7 '.
Crank
K
>
Link 5.
INVERSION
to 3 .
Point
e
on
i Link 4.
CURVATURE
TRANSFORM
to find
1 ,6 .
[4MSP Grid
:L,R 2D,
+2D]
b.)
Crank
J
y
Link 3.
CURVATURE
TRANSFORM
and PATH
Crank
K
y
Link 5 and
COGNATES to find 3 ,
dummy 7 '.
Point
g
=
Pivot B.
INVERSION
to 5 .
Point
e
on
i Link 1.
CURVATURE
TRANSFORM
to find 4
INVERSION
to 3 .
CURVATURE
TRANSFORM
to find 6
I 4MSP Grid
:L,R 2D,
+ (2)xlD]
154
Probleni_lf 3
Angular Coordi
nation of a Two
Link Chain
Linkage SoluLion
Pivots pre
specified .
Synthesis Procedure
1) CURVATURE
TRANSFORM
to find 4 .
Necessary Condition Procedures
s
a) Necessary Condition Pro
cedure D for 1234.
a) Necessary Condition Pro
cedure H for 1234.
a) Necessary Condition Pro
cedure D for 1234.
a) Necessary Condition Pro
cedure A for 1234.
20
The third step, evaluation of "desirable conditions,"
is where the "quality" of the surviving solution linkages
is determined by the userdefined objective function.
A desirable condition can be any criterion by which a
designer might qualitatively judge a linkage. The grid
is passed through these desirable conditions sequentially,
and for each one grid "topology" is reevaluated. This
procedure is known as sequential filtering. Some link
ages may perform so poorly relative to one or more of
these criteria as to be unacceptable, and will there
fore be rejected.
The result is a grid of solution linkages in which a
portion or portions contain acceptable linkages, with
scores indicating relative overall quality.
C. Previous Use of the "Grid" Philosophy
The optimization philosophy employing sequential fil
tering was first implemented by computer by Eschenbach and
Tesar [20] in 1969 for crankrocker fourbars satisfying
specified coplanar motion for four positions. A very large
solution set (~62,000 linkages) was passed through the
necessary and desirable conditions in a single pass, the
optimum linkage possessing the best surviving score. An
improvement on this approach was later made by the author
[23] and the author and Tesar [24] by introducing the
Ill
p2
Figure 61 Desirable Condition Scoring Curves
207
(O^) = Output scoring grid, cumulative
(all desirable conditions).
(O2) (Oj) ... = Output scoring grids, sepa
rate conditions. Optional
115
coordinating linkage synthesis and optimization, cam syn
thesis and analysis, mechanism shaking force, shaking moment
and torque balancing, spring, energy and dashpot synthesis,
dynamic analysis and eventually spatial and spherical
analytics. Just to meet the goals of synthesis and opti
mization, however, the package must be capable of handling
the analytics for an estimated 500 different problem/link
age/mode of cyclicity combinations.
A most feasible, practical, flexible and marketable
approach to the optimization package and its proposed ex
pansion can be based on a programming technique used by
Pollock [123] for a dynamic linkage analysis package. The
package consists of a pool of modules of equivalent pro
gram hierarchy from which a user can select and coordinate
those needed for a particular application.
The optimization package has been broken up into such
a pool of modules (Table 71) consisting of an input and
output module for each computational block, five synthe
sis modules, fourteen necessary condition modules, four
desirable condition modules and four general support rou
tines. The function and input/output of each module is
presented in Appendix C.
The advantages of the Pollock approach to program
structure pertain to both the optimization package and its
proposed expansion:
1.) A central control program is not necessary. The
152
a) Necessary Condition Proce
dure H for 1234.
a) Necessary Condition Proce
dure A for 1234.
a) Noor'ksary Condition Proce
dure' H Cor 1234.
6
linkage problem makes the optimization analytics somewhat
difficult to implement. Algebraic approaches, generally
limited to smallscale problems, are rarely attempted.
Sutherland [12] has developed mixed exactapproximate po
sition synthesis in which some points are matched exactly
using Burmester theory, and the objective function con
tains the leastsquares error of approximation of addi
tional precision points. Solutions to the resulting equa
tions can be closed form or iterative depending on the
order of magnitude of the problem. Bagci and Lee [13],
Bagci [14], and Bhatia and Bagci[15] have developed and
expanded on a linear superposition technique in which the
error of the loopclosure equations of the linkage system
is minimized by partitioning these equations into dyadic
loop equations which are linear in terms of the system
unknowns.
Random searches are direct and somewhat exhaustive,
generally encompassing a synthesis step to generate a
large representation of a linkage solution set for a par
ticular problem, and then a rating and sorting step to
locate the "best" linkage within this set. As early as
1962,Roth, Sandor and Freudenstein [16] used this approach
by generating fourbar linkages satisfying path specifica
tions, and then rating them according to some design
criteria (primarily transmission angle). Later Nolle [17]
used a random search for function generating linkages. In
230
22. Sridher, B.N. and L.E. Torfason, "Optimization of
spherical fourbar path generators," ASME Paper
70Mech46, 1970.
23. Spitznagel, K., "Nearglobal optimum of synthesized
fourbar mechanisms by interactive use of weighted
kinematic sequential filters," M.S. Thesis, Univer
sity of Florida, 1975.
24. Spitznagel, K. and D. Tesar, "Multiparametric op
timization of fourbar linkages," to be presented
at the ASME Design Engrg. Tech. Conf., Minneapolis,
Minn., September 2427, 1978.
25. Powell, M.J.D., "A Method for minimizing a sum of
squares of nonlinear functions without calculating
derivatives," Computer Journal, Volume 7, 19 64 ,
pp. 15562.
26. Powell, M.J.D., "An efficient method for finding the
minimum of a function of several variables without
calculating derivatives," Computer Journal, Volume 7,
No., 4, 1964, pp. 30307.
27. Suh, C.H. and A.W. Mecklenburg, "Optimal design of
mechanisms with the use of matricies and least
squares," Mechanisms and Machine Theory, Volume 8,
No. 4, Winter, 1973, pp. 47995.
28. Fletcher, R., and C.M. Reeves, "Function minimization
by conjugate gradients," Computer Journal, Volume 7,
1964, pp. 14954.
29. Rees Jones, J., and G.T. Rooney, "Motion analysis of
rigid link mechanisms by gradient optimization on the
analogue computer," Journal of Mechanisms, Volume 5,
1970, pp. 191201.
30. Nechi, J.A., "A relaxation and gradient combination
applied to the computer simulation of a plane four
bar chain," Journal of Engineering for Industry,
Trans ASME, Series B, Volume 93, No. 1, Feburary
1971, pp. 11319.
31. Chen, F.Y., and V. Dalsania, "Optimal synthesis of
planar sixlink chains using leastsquares gradient
search," Trans Canadian Soc. of Mechanical Engineers,
Volume 1, No. 1, March, 1972, pp. 3136.
32. Rose, R., and GkSandor, "Direct analytical synthesis
of fourbar function generator with optimal structural
error," Journal of Engineering for Industry, Trans
ASME, Series B, Volume 95, No. 2, May, 1973.
67
except number twelve can be solved by at least one linkage
solution without dependent syntheses. These solutions must
be considered to be the more desirable from a computational
standpoint, and are indicated in Appendix A with the t
symbol.
77
(a)
/
/
/
r
(b)
(d)
Figure 53 Dyad Change Points
82
2
Figure 56 Determination of Relative Moving Pivot
Rotation for 2FSP
219
Program Chart 5
Problem if 5 Path Coordination
with a Crank.
Linkage Solution FourBar
Mode of Cyclicity #5
FOURBAR
Synthesis Procedure
1) CURVATURE TRANSFORM and
PATH COGNATES to find
2, 4.
Necessary Condition Procedure
1) Necessary Condition Pro
cedure J for 1234.
Sample Program
VPROB5FBMODE5
[1] INITIAL
[2] * 31 SYNINP
[3] 32 33 34 35 SCRVTRN 31
[4] > 36 37 38 39 SPTHCOG 31 32 33 34 35
[5]
>
NECINP
[6]
40
NCPJ 31+ 18
[7]
>
NECOUT 40
[8]
V
41
DESINP
40
[9]
>
41
DPVTLCA
41
[10]
>
41
DLKLNRT
41
[11]
V
41
DLKSIZE
41
[12]
41
DESDYAD
41
1,
[13]
V
DESOUT 41
V
37
36
38
39
36
38
36
37
38
39
3 7
39
38
39
36
37 0 0 0 0
\6
Files
31)
4MSP(3/1)
32)
pivot B'
33)
pivot A'
34)
pivot C
35)
pivot D'
36)
pivot B
37)
pivot A
38)
pivot C
39)
pivot D
40)
NC Grid(2x4)
41)
DC Grid (2x4)
184
Module Chart 7
Name Angular Cognate (Synthesis Tool)
Label SANGCOG
Description
Generates the angular cog
nate of dyad abc.
,,>(01) SANGCOG (I1) (I2) (I3) "
where:
(1^) = pivot a
d2) = pivot b
(1^) = pivot c
(0^) = pivot b'
124
B. Conclusions
A unified approach to the design and optimization of
fourbar and sixbar linkages has been presented. By iso
lating and defining the synthesis procedures and necessary
condition procedures required (Appendix A,B), optimization
relative to a broad range of design criteria can be per
formed .
A computational design based on a pool of modules of
equivalent program hierarchy (Appendix C) has been devel
oped and presented. The optimization routine for a par
ticular problem is created through an appropriate coordina
tion of these modules, as displayed by the examples of
Appendix L).
Hopefully, the analytical and computational methodol
ogy introduced herein will provide the impetus for future
efforts towards practical mechanism design and optimiza
tion.
197
Module Chart 20
Name Necessary Condition Procedure L
Label NCPL
Description
Evaluates synthesized four
bar I on the basis of change
point considerations within
path cognate fourbars II and
III as shown. Generates a
necessary condition grid
(2x2D) indicating acceptable
cognates.
APL Call
where:
,'>(01) NCPL (IL) (I2) /
d1), d2)
d3}' d4)
d5),d6)
d7), d8)
(ox)
= link dO.
= link fO.
link gO^ (and h02)
link jO^ (and KO^)
Output grid.
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.
December, 1978
(Tt
V4C // C/l
Dean, College of Engineering
Dean, Graduate School
222
Program Chart 8
Problem #8 Angular Coordination
of Three Cranks.
Linkage Solution Watt 2
Mode of Cyclicity #2
Synthesis Procedure
1)
INVERSION
to 4.
2)
CURVATURE
find 3.
TRANSFORM
to
3)
CURVATURE
find 5.
TRANSFORM
to
Necessary Condition Procedure
1)
Necessary Condition
cedure H for 1234.
Pro
2)
Necessary Condition
cedure D for 2456.
Pro
Sample Program
Files
VPROB8WATT2MODE2
(1]
4
INITIAL
(2]
V
31 34 SYNINP
[3]
V
32 40 SYNINP
(4)
>
33 35 SYNINP
[5]
*
41 SINVERT 31
32
1
[6]
V
42 SINVERT 33
32
1
(7]
V
34 40 39 38 SCRVTRN
41
34
40
(8]
V
35 40 36 37 SCRVTRN
42
35
40
[9]
V
SYNOUT 39 38
34
40
[10]
V
SYNOUT 36 37
35
40
[11]
V
NECINP
[12]
V
43 NCPH 39 38
34
40
[13]
V
4 4 NC.PD 3 6 3 7
35
40
[14]
V
NECOUT 43 44
[15]
V
45 DES INP 43
44
[16]
4
45 DPVTLCA 45
34
39
38
40
36
35
[17]
 V
45 DI.KLNRT 45
37
40
38
40
[18]
V
45 DLKSIZE 45
37
38
[19]
4
45 DESDYAD 45
39
34
38
40
37
36
31
2,
i 6
[20]* DESOUT 45 V
31) 4MSP (1/2)
32) 4MSP (4/2)
33) 4MSP (6/2)
34) pivot A
35) pivot G
36) pivot F
37) pivot H
38) pivot c
39) pivot B
40) pivot F
41) 4MSP (1/4)
42) 4 MSP (6/4)
43) NC Grid (3)
44) NC Grid (5)
45) DC Grid (3,5)
28
Figure 3
3
1
Notation:
*
/ Driven
/ Link
0
Complete
Relative
Link
Rotation
Incomplete
Relative
Link
Rotation
4 Modes of Rotational Cyclicity of the FourBar
Chain
202
Modulo Chart 25
Name Desirable Condition Input
Label DESINP
Description
Given a grid of acceptable
linkages from the necessary
conditions, a scoring grid is
initialized and placed in
files.
Necessary
Condition
0 r i d
AFL Call
"MC^) DESINP (I ) (I2)
where:
(1^), (I2) .... = Necessary condition grids,
(If more than one, combines
them).
(01) = Initialized scoring grid for desir
able conditions.
238
108. Primrose, E.J.F., F. Freudenstein, and G.N. Sandor,
"Finite Burnester theory in Plane Kinematics,"
Journal of Applied Mechanisms, Volume 31, Series E,
1964, pp. 683693.
109. Mueller, R., "Papers on geometrical theory of motion"
(translated by D. Tesar, et al.), Special Report
No. 21, Kansas Engineering Experiment, Manhattan,
Kansas, June 1960.
110. Allievi, L., "Cinemtica della biella piaa,"
Giannini and Figli, Naples, Italy, 1895.
111. Krause, M., "Analysis der ebener bewegung," Walter
de Gruyter, Berlin, 1920.
112. Wolford, J.C., "An analytical method for locating
the Burmester points for five infinitesimally sepa
rated positions," Journal of Applied Mechanics,
Volume 25, Trans ASME, Series E, 1960, pp. 18287.
113. Veldkamp, G.R., Curvature Theory in Plane Kinematics,
Dissertation, Walters Groningen, Delft, 1963.
114. Tesar, D., J.W. Sparks, and W.T. Walters, "Multiply
separated position synthesis: part 1 point syn
thesis, part 2 function generation," Transactions
of the Tenth Conference on Mechanisms, Georgia
Institute of Technology, Atlanta, Georgia, 1968.
115. Myklebust, A., "Synthesis of multilink mechanisms
for dynamic specifications," Ph.D Dissertation,
University of Florida, 1974. (Diss. Abstract #7516426),
116. Modler, K.H., "Reihenfolge der homologen puntke,"
Maschinenbautechuck, Volume 21, 1972, pp. 25865.
117. Waldron, K.J., "Range of joint rotation in planar
fourbar synthesis for finitely separated positions:
part 1 the multiple branch problem," ASME Paper
No. 74DET108, Design Engineering Technical Con
ference, New York, N.Y., October 59, 1974.
118. Waldron, K.J., "Range of joint rotation in planar
fourbar synthesis of finitely separated positions:
part 2 elimination of unwanted Grashof configura
tions," ASME Paper No. 74DET109, Design Engineer
ing Technical Conference, New York, N.Y., October
59, 1974.
90
In some cases, tests for order and trailer/rocker are
unnecessary if Grashof type are known, but are included be
cause they improve the chances of producing a desired Gras
hof type. The Grashof type test is relatively expensive
computationally because constraint link combinations, not
just single constraint links, are under evaluation.
d.) Analysis of branching in pathcognates
For motion specification types #5 and #12 (Figure 39)
fourbar chains are produced by performing a curvature
transform followed by a generation of path cognate four
bars .
We assume fourbar I in Figure 5IDb has been produced
by the curvature transform and its path cognates, fourbars
II and III, are to be usable as all or part of a linkage
solution. What tests can be made on fourbar I. to ensure
proper performance relative to change points in fourbars
II and III? This question is central to establishing
practical necessary condition analyses for pathcognate
problems.
A study of fourbar II will now be made in detail. All
comments made hereafter regarding "left" cognate fourbar
II also apply to "right" cognate fourbar III. The modes of
cyclicity which II may take are shown in Figure 512. Each
of these modes will involve a separate analysis (necessary
condition procedures I N) to be performed on fourbar I
to ensure proper performance in II.
210
Module Chart 32
Name Analyze
Label ANALYZE
Description
Given a crank in zeroth
position, 4MSP of coupler,
generates 4MSP of rotating
crank.
APL Call
,,v(01) ANALYZE (I x) (12 ) (13) "
where:
(1^),(I2) = moving, fixed pivots of crank
in zeroth position.
(1^) = 4MSP of coupler.
(0^) = 4MSP of crank.
232
45. Alizade, R.I., A.V. Mohan Rao, and G.N. Sandor,
"Optimum synthesis of twodegreeoffreedom planar
and spatial function generating mechanisms using the
penalty function approach," ASME Paper No. 74DET51,
1974.
46. Alizade, R.I., A.V. Mohan Rao, and G.N. Sandor,
"Optimum synthesis of fourbar and offset slider
crank planar and spatial mechanisms using the penalty
function approach with inequality and equality con
straints," ASME Paper No. 74DET30, 1974.
47. Alizade, R.I., I.G. Novruzbekov, and G.N. Sandor,
"Optimization of fourbar function generating mech
anisms using penalty functions with inequality and
equality constraints," Mechanism and Machine Theory,
Volume 10, 1975, pp. 32736.
48. Kramer, S.N., and G.N. Sandor, "Selective precision
synthesis: a general method of optimization for
planar mechanisms," ASME Paper No. 74DET68, J,
Eng. Ind., Transactions of the ASME, Volume 97, Series
B, No. 2, May 1975, pp. 689701.
49. Kramer, S.N., "Using the selective precision synthesis
technique to optimize planar mechanisms," Fourth
Applied Mechanisms Conference, November 35, 1975,
Oklahoma State University, Stillwater, Oklahoma.
50. Mariante, W. and K.D. Willmert, "Optimum design of
a complex planar mechanism," ASME Paper No. 76DET47,
Trans ASME, 1976.
51. Fielding, J.B., and J.C. Zanini, "The assessment of
an optimization technique used for the design of a
packaging machine," Proceedings of IFTOMM, Interna
tional Symposium, Machines and Mechanisms, Univer
sity Research Work and Its Application to Industry,
1974, p. 315.
52. Fielding, B.J.and J.C. Zanini, "Coplanar synthesis
using an optimisation technique," Fourth World
Congress on the Theory of Mach, and Mech., September
812, 1975, pp. 38792.
53. Youssef, A.H., K. Oldham, and L. Maunder, "Optimal
kinematic synthesis of planar linkage mechanisms,"
Fourth World Congress on the Theory of Mach, and
Mech., University of Newcastle upon Tyne, England,
September 812, 1975, pp. 39398.
52
S1= A + BC
Tx = AC + B
S2= 2AC AC + B BC
T2 = A AC + 2BC + BC
S3= A 3ACC 3AC2 + 3BC + 3BC BC3 + BC*
T3 = B 3BCC 3BC2 3AC 3AC + AC3 AC*
b.) Inversion formulation number 2.
Given: Motions of N relative to R; M relative to R.
Find: Motion of M relative to N ( for Rq reference ).
Independent parameter for given MSP: yn
Independent parameter for derived MSP:
This formulation is identical to that of (2a), except
that the dots in the S^ and formulations should be re
placed with primes ('), and now A', A", A"', B' B", B"' and
CC", and C'"must be defined.
103
position. If the chain is cyclically driven through a dyad
link 2 or 5, the linkage is driven through the fourbar
chain's full range of possible motion between dyad dis
connections. A third possibility occurs if the chain is
cyclically driven through a fourbar chain link, in which
case the entire linkage is driven through a complete cycle.
139
Problem #9: ANGULAR COORDINATION OF THREE ADJACENT PLANES
The coplanar motions
of planes J and K
are coordinated with
the rotation of L.
SOLUTION LINKAGES
WATT
D 4 C
Watt 1 Link 1 is grounded.
Synthesis Procedures
Plane J v
Point e =
Plane K *
Point f =
Plane L =
Point g =
Link 4 and
Pivot D.
Link 6 and
Pivot G.
Link 2 and
Pivot A.
1 CURVATURE TRANSFORM to find 3
INVERSION to 6.
CURVATURE TRANSFORM to find 5
[4MSP Grid: (2)xlDj
133
Problem #6: ANGULAR COORDINATION OF A THREELINK CHAIN
The coplanar motion
of planes J and K and
crank L are coordi
nated. '
SOLUTION LINKAGES
WATT
D 4 C
Watt 1 
a.) Plane
Point
Plane
Point
C rank
Point
Link 1 is
J v Link
e = Pivot
K = Link
f = Pivot
L = Link
g = Pivot
grounded.
5 and
F.
6 and
G.
2 and
A.
b.) Plane J
Point e
Plane K
Point f
Crank L
Point g
Link 5 and
Pivot H.
Link 4 and
Pivot C.
Link 3 and
Pivot B.
Synthesis Procedures
INVERSION to 2.
CURVATURE TRANSFORM to find 4
INVERSION to 1.
CURVATURE TRANSFORM to find 3
l4MSP Grid:(2)xlD]
CURVATURE TRANSFORM to find 2
INVERSION to 2.
CURVATURE TRANSFORM to find 6
t 4MSP Grid: (2)xlDj
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.
Delbert Tesar, Chairman
Professor of 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.
c_John M. Vance
Associate Professor of 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.
George N. Sandor
Professor of 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.
Ralph G. Selfri^ge
Professor of Mathematics
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.
>2
/
X
Martin A. Eisenberg
/Professor of Engineering
/
155
a) Necessary Condition Pro
cedure A for 1234.
a) Necessary Condition Pro
cedure B for 1234 .
a) Necessary Condition Pro
cedure B for 1234 .
CHAPTER VI
THE DESIRABLE CONDITIONS
Chapter IV presented the synthesis procedures by which
linkage solution sets of one to four dimensions are gen
erated. The necessary conditions, which serve to eliminate
from the linkage solution set those linkages with some un
acceptable physical characteristics, were treated in Chap
ter V. Now the desirable conditions, which serve to sort
and rank the remaining linkages relative to a userdefined
objective function, will be discussed. These conditions
constitute the final step in the synthesis/analysis/resyn
thesis loop by which optimization occurs.
A desirable condition may be any criterion by which
the quality of a linkage solutiom may be judged. The range
of application for linkages is enormous, and so too is the
range of possible desirable conditions. Nine generally use
ful desirable conditions have been considered for this work.
A. The Desirable Conditions
1.) Pivot Location
In many applications the designer has restrictions
related to moving pivot locations or mounting locations
relative to the moving bodies. Depending on the problem
type and the linkage solution type selected, some of the
104
Module Chart 33
Name Analyze PathCognate
Label ANLZPCG
Description
Given 4MSP of coupler of a
synthesized fourbar and pivot
locations in the zeroth posi
tion of one of its path cog
nates, generates (3) sets of
4MSP representing motion of
path cognate's links.
APL Call
"MOj.) (02) (03) ANLZPCG (I1) (I2) (I3) (I4) (I5) "
where:
(V
=
4 MSP
of coupler of four
u2)
=
g in
zeroth position.
d3)
=
CL in zeroth position.
(V
=
j in
zeroth position.
a5)
=
0^ in zeroth position.
(Oj.)
=
4MSP
of g01
(0?)
=
4MSP
of Jg
,03>
=
4MSP
la
44
0
81
a. "Trailer"
Figure 55 Trailer and Rocker
186
Module Chart 9
Name Necessary Condition Input
Label NECINP
Description
Presently has no function, is re
tained only for reasons of consistency
and possible use in future expansion.
APL Call
"NECINP"
233
54. McLarnan, C.W., "On linkage synthesis with minimum
error," Journal of Mechanisms, Volume 3, 1968,
pp. 101105.
55. Huang, M., H.R. Sebesta and A.H. Soni, "Design of
linkages using dynamic simulation and optimization
techniques," ASME Paper No. 72Mech84, 1972.
56. Prasad, K.N., and C. Bagci, "Minimum error synthesis
of multiloop plane mechanisms for rigid body
guidance," Journal of Engineering for Industry, Trans
ASME, Series B, Volume 96, No. 1, 1974, pp. 10716.
57. Huang, M., "Optimal design of linkages using sensi
tivity coefficients," ASME Paper No. 74DET59,
1974 .
58. Lee, T.W., and F. Freudenstein, "Heuristic combina
torial optimization in the kinematic design of mech
anisms: Part 1 theory; part 2 applications,"
ASME Paper No. 76DET25, Trans ASME, 1976.
59. Datseris, P. and F. Freudenstein, "Optimum synthesis
of mechanisms using heuristics for decomposition and
search," to be presented at the ASME Design Engrg.
Tech. Conf., Minneapolis, Minn., September 2427,
1978.
60. Moradi, J.Y. and M. Pappas, "A boundary tracking al
gorithm for constrained nonlinear problems," Journal
of Mechanical Design, Trans ASME, Volume 100 April
1978, pp. 292296.
61. Lin, Y.L., "Use of computer methods to facilitate
synthesis of fourbar mechanisms processing specific
minimum transmission angles," ASME Paper 70DEll,
1970.
62. Hamid, S., and A.H. Soni, "Design of an RSSR crank
rocker mechanism for optimum force transmission"
Proceedings of Second Appl. Mechanisms Conference,
Oklahoma State University, 1971, Paper No. 17.
63. Hamid, S., and H.A. Soni, "Design of a spacecrank
RSSP mechanism with optimum force transmission,"
ASME Paper No. 72Mech82, 1972.
64. Shoup, T.E., J.R. Steffer, and R.E. Weatherford,
"Design of spatial mechanisms for optimal load
transmission," ASME Paper No. 72Mech88, 1972.
Use of Appendix C
Appendix C is intended to serve as a guide to the pro
gram modules needed to perform the analytics presented in
this work. Each chart contains a module's functional de
scription and input/output. As such Appendix C could serve
as an outline for a program package on nearly any system
providing adequate storage. At the present time the pack
age is probably most effectively implemented in interactive
APL, and the subroutine calls, "gotos" and input/output
conventions used are those of APL.
Here the flow of data between modules utilizes user
specified files with the restriction that file numbers 1 to
30 are reserved for operating storage. Occasionally a
module's specified file arguments are not entirely appli
cable (usually for fourbar applications using four/six
bar routines), in which case reference should be made to
file zero. Once an optimization routine has been created
and module I/O coordinated, all human/computer interfacing
is interactive and handled by each module independently.
176
74
rotation. The latter situation occurs if the linkage
satisfies the Grashof condition:
Â¡L + s < p + q
where
Z = length of longest link
s = length of shortest link
p,q = length of other two links
Grounded fourbar chains which satisfy this relation
ship are known as "Grashof fourbars," and take the forms
shown in Figure 52 a,b,c. They are simply different in
versions of one another, and are generally more useful than
nonGrashof linkages. The crankrocker, for example, can
be used to generate an arbitrary shape with a coupler
point while its crank is driven continuously, and the
draglink is often used for continuous function genera
tion. In a Grashof fourbar only those pivots which are
cyclic will ever enter change points.
The nonGrashof fourbars take only one form (regard
less of inversion), the nonGrashof doublerocker. This
type of linkage tends to dominate a linkage solution grid
generated through the curvature transform, but has the un
fortunate property of being able to reach "change points"
(discussed in next section) at any of its pivots. Control
of linkage motion through these change points requires
special considerations.
189
Modulo Chart 12
Name Necessary Condition Procedure B
Label NCPB
B
/
c
U/
*
*
A
Description
Evaluates fourbar chains
on the basis of changepoint
considerations for the case
shown. Creates a necessary
condition grid indicating ac
ceptable linkages.
APL Call
where:
,,(01) NCPB (I1) (I2) (I3) (I4) "
, d2)
(i3>'(i4>
Pivots of first constraint link
(moving, fixed pivots).
Pivots of second constraint link
(moving, fixed pivot).
(o1)
= Output grid.
194
Module Chart 17
Name Necessary Condition Procedure I
Label NCPI
Description
Evaluates synthesized four
bar I on the basis of change
point considerations within
path cognate fourbars II and
III as shown. Generates a
necessary condition grid
(2x2D) indicating acceptable
cognates.
APL Call
"* (01) NCPI
(I1) (I2) .... (Ig) "
where:
d1), d2)
d3), d4)
d5),(i6)
(I 7) (Ig)
= link dO^
= link fC>2
= link gO^ (and hC^)
= link jO^ (and KO^)
(0^) = Output grid.
APPENDIX A
PROBLEM TYPES AND SOLUTION LINKAGES
118
5) With the use of the program charts of Appendix
C, coordinate input and output files between modules.
These modules are completely compatible with existing
Pollock dyadanalysis routines and any others following a
similar input/output format. For example, an optimization
routine could be immediately followed by a dyadanalysis
routine within the same program to perform a detailed
dynamic analysis of the optimal linkage.
132
Problem #5: PATH COORDINATION WITH A CRANK
The coplanar motion
of point e is coor
dinated with the ro
tation of crank J.
SOLUTION LINKAGE
FourBar
Synthesis Procedure
Link 1 is
Point e +
Crank J =
Point f =
grounded
Link 3.
Link 2.
Pivot A.
*.i.
CURVATURE TRANSFORM and
PATH COGNATES to find
2.4
[4MSP Grid: L,R 2D]
102
as the gfunction (i.e., relative angular velocity) of
link 2 relative to 3 approaches zero and link 2 comes to
a halt. On either side of this position ^2/3 assumes op
posite values. If dyad link 5 is the driven member in the
original Stephenson 2, a critical position occurs when g^^
approaches zero. In either case, the specified positions
must be satisfied without the linkage passing through a
critical position.
4.) Encirclement and Nonencirclement
In a Stephenson chain, if the coupler point C encir
cles point A, as shown in the example of Figure 514c,
pivots C and A both have complete relative rotation. If
encirclement does not occur (and the chain is not cyclicly
dyad driven), pivots C and A will have only relative rota
tion. In either case the dyad will not change type.
Either of these motion characteristics (or its exclusion)
may be desirable in a particular application.
The necessary conditions connection, disconnection,
gfunction reversal, encirclement and nonencirclement are
evaluated simultaneously during a single dyadbased motion
analysis. The range through which the Watt 2 or Stephenson
3 inversion of the sixbar linkage is driven depends upon
the mode of cyclicity. If the mode of cyclicity involves
no links with complete rotation, the linkage will be driven
in small increments from the zeroth to the third specified
44
Figure 43 Parameters Relating Fixed and Moving Planes
123
Other subsets of allowable modes of cyclicity can and
should be defined and compiled, like those of Figures 34,
35 and 36 for the geometric restrictions. For example,
there might be "dynamic" restrictions, "twoinput" restric
tions, and so on.
4.) Multiple Modes of Cyclicity
In putting together the computational modules of Ap
pendix C for a particular problem, the user is constrained
to select a single mode of cyclicity in order to define the
appropriate necessary condition procedure. For many appli
cations more than one of the available modes of cyclicity
would be of interest.
It is sufficient to generate separate necessary con
dition grids for each of the selected modes of cyclicity,
and combine these at the appropriate times (generally, be
fore the necessary condition output, or the necessary con
dition dyad routine). The present modules have this capa
bility (NECOUT and NECDYAD of Appendix C).
A better approach would consider the minimum range of
conditions to be satisfied, and generate a single necessary
condition procedure for the selected modes of cyclicity.
This would be considerably more efficient due to the elim
ination of computational duplications at overlapping grid
regions.
Table 71
Program Modules
Synthesis
Necessary Conditions
SYNINP
SYNOUT
NECINP
NECOUT
SCRVTRN
SINVERT
SPTHCOG
SANGCOG
SDYDTYP
NCPA
NCPB
NCPC
NCPD
NCPG
NCPH
NCPI
Support Routines
NCPJ
NCPK
INITIAL
ANALYZE
ANLZPCG
EXTRACT
NCPL
NCPM
NCPN
NDYDTYP
NECDYAD
Connection,
Disconnection
gfn. reversal,
Encirclement,
Nonencirclement
Desirable Conditions
DESINP
DESOUT
DPVTLCA
DLKLNRT
DLKSIZE
DESDYAD
Overall Boundary,
gfunction
Vhfunction
hrmsfunction
VTorque
Shock Level
119
99
Necessary Condition Procedure N.
1) Perform necessary condition procedure L on
fourbar I to indicate change points at g in cog
nate II and h in cognate III.
2) Invert coupler and ground,swap constraint
links in I, and execute necessary condition pro
cedure L again, to indicate change points at 0^
in cognates II and III.
3) Execute necessary condition procedure M on
fourbar I to indicate change points at j in cog
nate II or K in cognate III.
4) Test the distance of d and f relative to OjC^
for the specified positions. If the "altitude" of
d is always less or greater, no change points will
occur at 0^ in cognate II or 02 in cognate III.
B. Sixbar Necessary Conditions
The necessary conditions presented thus far are suf
ficient for the necessary condition analysis of fourbar
linkages and the constituent parts (i.e., dyads and four
bars) of sixbar linkages. Several conditions involving
the combination of these parts into complete sixbar
linkages are still needed.
1.) Dyad Connection
The fact that a sixbar linkage is capable of reach
ing specified positions without change point or order dif
ficulties is not enough to ensure nondisconnection of a
linkage member while moving between the specified positions.
Conceptually one can imagine a fourbar linkage whose
coupler pivot passes outside (or inside) of the bounds of
possible motion of dyad pivot q (Figure 514a). Verifica
tion of this condition is made using a dyadbased analy
sis of the entire linkage at discrete intervals.
64
+
PATH
COGNATES
Figure 47 A Stephenson2 Solution to Point/Path
Coordination with Two Cranks
89
Necessary Condition Procedure A (for cases la, Ic,
lib and Ild)
1) Choose a followerconstraintlink to be a
trailer/rocker.
2) Choose a crankconstraintlink of proper order
which is compatible with the follower.
3) Test Grashof type for a crankrocker.
Necessary Condition Procedure B (for cases lb and Ila)
1) Choose a constraint link to be a trailer of
proper order.
2) Choose another constraint link to be a trailer
which is compatible with the first link.
3) Test Grashof type for a draglink.
Necessary Condition Procedure C (for cases Id and IIc)
1) Invert to change reference to the coupler link,
i.e. generate the motion specifications of the
fixed plane relative to the moving plane, in
terchange the circlepoint and centerpoint
cubic curves, etc.
2) Follow procedure B.
Necessary Condition Procedure D (for case Illa)
1) Choose a followerconstraintlink to be a
trailer.
2) Choose a crankconstraintlink of proper order
which is compatible with the follower.
Necessary Condition Procedure G (for case IIIc)
1) Invert to change reference to the coupler link.
2) Follow procedure D.
Necessary Condition Procedure H (for cases IVa,b,
c and d)
1) Choose a constraint link which is a trailer
of proper order.
2) Choose a second constraint link which is a
trailer which is compatible with the first
link.
3) Check mutual compatibility.
4) Invert to change reference to the coupler link.
5) Repeat 1), 2) and 3) (no need to check order
again).
108
9.) Overall Boundary
This condition is used to rank solution linkages
relative to overall space required. Some designs will
place "soft" constraints on working volumes in the form
of preferred regions, while others may have "hard" ab
solute restrictions created by other system components.
B. The Objective Function
The linkages which have survived to the desirable
conditions are ordered using some userdefined objective
function. This function may take the form:
N = ENk = EWkSk (6.1)
k k
k = 1,2,3 . .
where N represents a cumulative score attached to a par
ticular linkage, k is the desirable condition counter,
Nk is a linkage score for condition k, Wk is a weight
signifying relative importance of condition k, and Sk is
a score signifying performance of the linkage relative to
condition k.
The weighting factor Wk is set by the designer, who
must make this decision relative to the application using
his own experience and judgement. In this work both
weights and scores may be assigned within a range of 0
to 10. A weight of 10 signifies a condition of primary
12
deal explicitly with inequality constraints. This group in
cludes the method of feasible directions introduced by Zouten
dijk, the gradient projection method, and the extensions of
Dantzig's simplex method. Fielding and Zanini [51,52] have
discussed the use of modified Simplex methods and consid
ered the optimization of an industrial packaging machine.
Also, Youssef, Oldham and Maunder [53] have used the mod
ified Simplex to optimize multiloop linkages with respect
to path error, link length and transmission angle.
The use of variational methods for linkage problems
can be traced back to the early work of Freudenstein [9]
which later received attention from McLarnan [54], who at
tempts to minimize structural error by requiring the
derivatives at the precision positions to be zero. Huang,
Sebesta and Soni [55] and Prasad and Bagci [56] have had
more recent applications of variational methods.
Other iterative methods have been introduced by Huang
[57], who introduces the concept of sensitivity coeffi
cients, and Lee and Freudenstein [58] and Datseris and
Freudenstein [59], who develop the application of Lin's
heuristic method (originally developed for the communica
tions field) to mechanism optimization. Moradi and Pappas
[60] have developed a "boundary tracking" algorithm for
general mathematical programming problems.
Most of the papers discussed so far have dealt pri
marily with optimization with respect to motion error. If
231
33. Fletcher, R., and M.J.D. Powell, "A rapidly conver
gent descent method for minimization," The Computer
Journal, Volume 6, 1963, pp. 163168.
34. MclaineCross, I.L., "A numerical method of Chebychev
optimum linkage design," Journal of Mechanisms,
Volume 4, 1969, pp. 3141.
35. Sevak, N.M. and C.W. McLarnan, "Optimal synthesis of
flexible link mechanisms with large static deflec
tions," ASME Paper No. 74DET83, 1974.
36. Conte, F.L., G.R. George, R.W. Mayne, and J.P. Sadler,
Optimum mechanism design combining kinematic and
dynamic force considerations," ASME Paper No.
74DET55, 1974.
37. Han, C.Y., "A general method for the optimum design
of mechanisms," Journal of Mechanisms, Volume 1,
No. 4, 1966, pp. 30113.
38. Fiacco, A.V. and G.P. McCormick, "Programming under
nonlinear constraints by unconstrained minimization;
A primaldual method," RACTD96, Bethesda, Maryland:
The Research Analysis Corporation, September, 1963.
39. Fiacco, A.V. and G.P. McCormick, "Computational
algorithm for the sequential unconstrained minimiza
tion for nonlinear programming," Management Science,
Volume 10, 1964, pp. 60117.
40. Fiacco, A.V. and G.P. McCormick, Nonlinear Program
ming: Sequential Unconstrained Minimization Tech
niques, New York: Wiley, 1968.
41. Fox, R.L., and K.D. Willmert, "Optimum design of
curve generating linkage with inequality constraints,"
Journal of Engineering for Industry, Trans ASME,
Series B, Volume 89, No. 1, February 1967, pp. 14452.
42. Tranquilla, M., "Optimum design of a fourbar linkage
whose coupler path has specified extremes," ASME Paper
No. 7lVibr109, 1971.
43. Gupta, V., "Computer aided synthesis of mechanisms
using nonlinear programming," Journal of Engineer
ing for Industry, Trans ASME, Series B, Volume 95,
No. 7, February 1973, pp. 33944.
44. Bakthavachala, N., and J.T. Kimbrell, "Optimum syn
thesis of pathgenerating fourbar mechanisms," ASME
Paper No. 74DET6, 1974.
CHAPTER VIII
RECOMMENDATIONS AND CONCLUSIONS
There are several logical extensions of the presented
work. For the most part, they involve expansions of the
underlying theories to reduce the levels of constraint in
the choices of motion specification type and solution link
age type.
A. Recommendations
1.) Alternate Synthesis Procedures
As has been previously mentioned, it is desirable to
keep the number of free parameters to 1, 2 or 3. With con
ventional 4MSP synthesis procedures (Chapter IV, Appendix A)
this range is exceeded for problems #1 and #12. In problem
#1 the number of free parameters is five, but is reduced to
two by arbitrarily selected coupler angles in the motion
specifications set. The number of free parameters in prob
lem #12 is four, resulting in a fourdimensional grid and
escalated computational complexity and cost.
All of the synthesis procedures presented in Chapter IV
and Appendix A involve at least one application of the first
tool of synthesis, the curvature transform, for 4MSP. Al
ternate synthesis procedures have been developed which allow
120
25
D 4 C
D 4 C
b. The Stephenson 2
D 4 C
c. The Stephenson 3
Figure 32 Inversions of the Stephenson SixBar Chain
7
1968 Tomas [18] discussed the treatment of linkage syn
thesis as a nonlinear programming problem, and employed
the optimization randomsearch method of Garrett
and Hall [19]. Eschenbach and Tesar [20] generated a
large set of linkages computationally and then ranked
them according to numerous design criteria, the latter
making use of limit design zones and design criterion
weights. Two and three function problems are handled
under the consideration of ten to twelve criteria acting
as sequential filters. Eisenstein and Hall [21] approached
the optimization of twodegreeoffreedom function gener
ators by generating a small number of "good" designs, and
then generating linkages around these to find a "best"
design. Spherical fourbar path generators are handled
by Sridher and Torfason [22], who synthesize on four pre
cision points, and then search for the linkage which min
imizes the maximum deviation from the prescribed path be
tween the precision points using a random sequential
search. The author [23] and the author and Tesar [24]
have "grid refinement" technique for fourbars which in
volves generating twodimensional discrete representa
tions of the solution set for four coplanar positions,
ranking or eliminating these using a weighted design cri
teria formulation, and then restarting by generating a
new "grid" in the most favorable region. Most of the pre
viously mentioned approaches, as well as the iterative
methods, can guarantee true global optimization for convex
functions only rarely the case in a linkage problem.
50
Figure 44 Inversion Notation
223
P_rogram Char L 9
Problem H 9 Angular Coordination
of Three Adjacent Pianos.
Linkage Solution Watt 2
Mode of Cyclicity #9
Synthesis Procedure
1) CURVATURE TRANSFORM to
find 1.
2) CURVATURE TRANSFORM to
find 6.
Necessary Condition Procedure
1) Necessary Condition Pro
cedure B for 1234.
2) Necessary Condition Pro
cedure A for 2456.
Sample Program
VPROB9WATT2MODE9
[1] > INITIAL
[2] * 31 4 0 SYNINP
[3] 32 38 SYNINP
[4] > 33 37 SYNINP
[5] 38 40 34 39 SCRVTRN 32 38 40
[6] 37 40 36 35 SCRVTRN 33 37 40
[7]  SYNOUT 34 39 38 40
[8] *> SYNOUT 36 35 37 40
[9] NECINP
[ 10] * 41 NCTB 34 39 38 40
[11] > 4 2 NCPA 36 35 37 40
[12] NECOUT 41 42
[13]
4 3
DESINP
4 1
42
[14]
4 3
DPVTLCA
43
3 4
35
3 6
37
38
[15]
4 3
DF.KLNRT
4 3
37
36
3 4
3 9
[16] *
43
UI.KS I ZE
4 3
38
4 0
37
40
39
35
[171
43
DCSDYAD
4 3
34
39
38
40
37
36
35
0
1 ,
i 6
[18] DESOUT 4 3
Files
31)
4MSP (4/2)
32)
4MSP (3/2)
33)
4MS P (5/2)
34)
pivot A
35)
pivot G
36)
pivot F
37)
pivot H
38)
pivot C
39)
pivot B
40)
pivot D
41)
NC Grid (1)
42)
NC Grid (6)
43)
DC Grid (1,6)
178
Module Chart 1
Name Initialize
Label INITIAL
Description
Creates and initializes
files, and collects informa
tion related to problem type.
First module of every program.
APL Call
If
INITIAL"
216
Program Chart 2
Problem If 2 Motion Synthesis,
Linkage Solution FourBar
Mode of Cyclicity # 2
Synthesis Procedure
1) CURVATURE TRANSFORM to
find 2; 4.
FOURBAR
Necessary Condition Procedure
1) Necessary Condition Pro
cedure B for 1234,
Sample Problem
VPROB2FBMODE4
Files
[1]
>
INITIAL
[2]
V
31
SYNINP
[3]
35
32 34 33 SCRVTRN
31
[4]
SYNOUT 35
32
34
33
[5]
V
NECINP
[6]
>
36
NCPB 35
32
34
33
[7]
*
NECOUT 36
[8]
>
37
DESINP
36
[9]
>
37
DPVTLCA
37
32
35
34
(10)
y
37
DLKLNRT
37
34
33
35
ill J
37
DLKSIZE
37
32
33
3 5
[12]
y
37
DESDYAD
37
3 5
32
34
31
1,
i 6
[13]
>
DESOUT 37V
31)
4MSP(3/1)
32)
pivot
A
33)
pivot
D
34)
pivot
C
35)
pivot
B
36)
NC Gri
.d ( 4x2)
37)
DC Grid(4x2)
33
32
34
33 0 0 0
182
Mod ul c Ch art 5
Name Inversion (Synthesis Tool)
Label SINVERT
Description
Generates a new 4MSP set
representing a change of ref
erence .
Case HI Given N relative to
R, M relative to R, generate
M relative to N.
Case it 2 Given N relative to
R, M relative to N, generate
M relative to R.
Case ft 3 Given M relative to
R, generate R relative to M.
APL Call
SINVERT (1^ (I2) (I3) "
where:
(1^) = 4MSP associated with M.
(I2) = 4MSP associated with N.
1^ = 1,2 or 3, case number
(0,) = Generated 4MSP set (M relative to
1) N or 2) R,or 3) R relative to N).
83
Y., (y. V.. )
3k = _il
2 2 2
(see Figure 56b)
Now the location of circle points with positive and nega
>
ik
tive values of the in terms of P^ and P!v (Figure 57c)
can be determined by using the property:
The three lines through diametral
points and a third point on a circle
will define a right triangle.
For circle points lying on the right side of PjP^ and
outside the circle, 0 < V ^/2 < tt/2, so that 0 < < 77
and for circle points on the same side but inside the
circle, tt/2 < V.,/2 < tt (Figure 57a) On the opposite
3 K
side of PjP^ the signs will be reversed. Figure 57b
shows this construction overlayed onto the corresponding
circle point curve.
Two values of i could be chosen: the only requirement
is that i be distinct from j and k. Interestingly enough,
if evaluations were made using both possible values of i
some spaces in the plane would have conflicting signs, but
the circle point cubic will occupy none of these spaces.
Another interesting property is that the signs will not
change as the circle point cubic passes through an image
pole, but will change as the circle point cubic crosses the
circle or line segment singly.
The portions of the circle point cubic which represent
permissible constraint links will be defined by those por
tions for which all the components of any of the strings
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
ABSTRACT vi
CHAPTER
I INTRODUCTION 1
II A PHILOSOPHY FOR LINKAGE DESIGN AND
OPTIMIZATION 16
A. Control of Parameters 16
B. The Optimization Procedure .... 17
C. Previous Use of the "Grid"
Philosophy 20
III SOLVABLE PROBLEMS 23
A. Linkage Types 23
B. Modes of Rotational Cyclicity ... 27
C. Motion Specification Types .... 35
IV SYNTHESIS OF LINKAGE SOLUTIONS .... 39
A. The Four Tools of Synthesis .... 39
B. Coordination of the Tools of
Synthesis 61
C. Grid Dimensions 65
D. Dependent Syntheses 66
V THE NECESSARY CONDITIONS 71
A. FourBar Necessary Conditions ... 71
B. SixBar Necessary Conditions ... 99
VI THE DESIRABLE CONDITIONS 104
A. The Desirable Conditions 104
B. The Objective Function 108
IV
34
Figure 38 Some Unacceptable Modes of Rotational Cyclicity
234
65. Freudenstein, F., and E.J.F. Primrose, "The clas
sical transmission angle problem," Conference Proc.
on Mechanisms, Paper No. C96/72, The Institution of
Mechanical Engineers, London, 1972, pp. 10510.
66. Sutherland, G., and B. Roth, "A transmission index
for spatial mechanisms," Journal of Engineering for
Industry, Trans ASME, Series B, Volume 95, No. 2,
May 1973, pp. 58997.
67. Bagci, C., "Design of spherical crankrocker mech
anism with optimal transmission, "Journal of Engi
neering for Industry, Trans ASME, Series B, Volume
95, No. 2, May 1973, pp. 57783.
68. Savage, M., and D.H. Suchora, "Optimal design of
fourbar crank mechanisms with prescribed extreme
velocity ratios," ASME Paper No. 73WA/DE13, 1973.
69. Tomas, J., "Optimum seeking methods applied to a
problem of dynamic synthesis in a loom," Journal of
Mechanisms, Volume 5, No. 4, 1970, pp. 495504.
70. Benedict, C.E. and D. Tesar, "Optimal torque balance
for a complex stamping and indexing machine," ASME
Paper No. 70Mech82, 1970.
71. Lowen, G.G., andR.S. Berkof, "Determination of
forcebalanced fourbar linkages with optimum shak
ing moment characteristics," Journal of Engineering
for Industry, Trans ASME, Series B, Volume 93, No. 1,
February 1971, pp. 3946.
72. Berkof, R.S., and G.G. Lowen, "Theory of shaking
moment optimization of forcebalanced fourbar
linkages," Journal of Engineering for Industry,
Trans ASME, Series B, Volume 93, No. 1, February
1971, pp. 5360.
73. Tepper, F.R., and G.G. Lowen, "Shaking force opti
mization of fourbar linkages with adjustable con
straints on ground bearing forces," ASME Paper No.
74DET53, 1974.
74. Sadler, J.D., "Balancing of sixbar linkages by non
linear programming," Fourth World Congress on the
Theory of Mach, and Mech., September 812, 1975,
pp. 13944.
209
Module Chart 31
Name Extract
Label EXTRACT
Description
Allows user to select a
particular linkage frotn a
grid; compresses pivot files
to represent only the selected
linkage. Useful for dependent
synthesis problems.
APL Call
EXTRACT (I.)"
where:
(1^) = Scoring grid (previously outputted)
188
Modu_l_o _Char I; 11
Nano Necessary
Label NCPA
Condition
Procedure A
Description
Evaluates fourbar chains
on the basis of changepoint
considerations for the case
shown. Creates a necessary
condition grid indicating ac
ceptable linkages.
APL Call
where
"*(0^ NCPA
(11) (12) (13) 14) (15) "
(Il) d2)
(i3), d4)
Pivots of first constraint link
(moving, fixed pivots).
Pivots of second constraint link
(moving, fixed pivots).
1 or 2, identifies constraint
link with changepoint restric
tions .
(CL ) = Output grid.
200
Modulo Chart: 23
Name Necessary Condition Dyad Analysis
Label NECDYAD
Description
Performs a dyadbased
analysis, evaluates the solu
tion set of linkages on the
basis of:
 Connection
 Disconnection
 Encirclement
 Nonencirclement
 gfunction reversal
"(C^) NECDYAD .
where:
For sixbars,
Stephenson 3
or Watt 2
inversion.
(I1),(I2) = Crank pivots (moving,fixed)
(13) (I4) = Follower pivots (moving, fixed)
(1^) (Ig) (1^) = Dyad pivots ( = 0 for fourbar)
(Ig) = MSP set, coupler or crank relative to
ground. (Can be 0 if Ig = 1)
Ig = Mode of drive.
= 1 for cyclic drive.
= 2 from position 0 to 3.
= 3 through range of disconnection.
I^q+ = Necessary conditions
= 1 for Connection
= 2 for Disconnection (TTlT
 3 for Disconnection drpTg
= 4 for Encirclement
 5 for Nonencirclement.
= 6 for gfunction reversal
relative to Igl7).
= 7 for gfunction reversal
relative to I_I_) .
O /
(O^) = Output grid.
short)
short)
(Ground
(Ve
APPENDIX D
SAMPLE PROGRAMS
143
Problem #11: COPLANAR MOTION COORDINATED WITH A CRANK
The coplanar motion
of Plane K is coordi
nated with the rota
tion of crank J.
SOLUTION LINKAGES
C 4 H
WATT
Watt 1 Link 1 is grounded
Synthesis Procedures
Crank
J
Link 2 and
INVERSION
to 2 .
Point
e =
Pivot A.
CURVATURE
TRANSFORM
to
find
Plane
K 
Link 5.
4,6.
INVERSION
to 1 .
CURVATURE
TRANSFORM
to
find
3
[ 4MSP Grid
1: 2D+lD]
C rank
J 
Link 3 and
INVERSION
to 3.
Pn.i n t
o =
Pivot B.
CURVATURE
TRANSFORM
to
f i nd
4
Plane
K >
Link S.
INVERSION
to 1 .
CURVATURE
TRANSFORM
to
find
2
INVERSION
to 2.
CURVATURE
TRANSFORM
to
find
6
[4MSP Grid:(3)xlD]
27
fourteen free parameters. A "Stephenson 1" is produced
if link 1 or 6 of the Stephenson chain is grounded, a
"Stephenson 2" if link 2 or 5 is grounded, and a
"Stephenson 3" if link 3 or 4 is grounded. A "Watt1"
results from grounding link 1, 3, 5 or 6 of a Watt chain,
and a "Watt 2" if link 2 or 4 is grounded. These are
available linkage solutions to the problems in this work,
and the notation introduced in Figure 31 will be adhered
to.
Other simple linkages exist, such as the slidercrank
and inverted slidercrank fourbars, the geared fivebars
and sixbars with sliders and oscillating blocks. They
have less general applicability and except for the geared
fivebar are simply subsets of the linkages already under
consideration.
B. Modes of Rotational Cyclicity
A linkage chain can have a number of distinct types
of motion depending upon linkage geometry and the nature
of the input drive. These motion characteristic types
are hereafter termed "modes of rotational cyclicity."
The possible modes of rotational cyclicity are dis
played for the fourbar chain in Figure 34, and for the
sixbar chains in Figures 35 and 36. The information in
these figures is central to this work. This unfortunate
Mt* Vo'0
a o^'
30
Watt SixBars With No Cyclic Links
4
Any pair of links
may be driven (15
distinct possi
bilities )
Watt SixBars With Cyclic Links
Figure 36 Modes of Rotational Cyclicity of the
Watt SixBar Chain
221
Program Chart 7
D 
^V.4^ ~
Problem if I Angular Coordination
of a Crank and a TwoLink Chain.
Linkage Solution Stephenson 1
Mode of Cyclicity H9
1
s
s
J _
J 1 3 ^A
STEPHENSON i
Synthesis Procedure
Necessary
Condition Procedure
1) INVERSION to 4.
1) Necessary Condition
2) CURVATURE TRANSFORM to
find 6.
Dyad Type (25).
2) Necessary Condition Pro
3) CURVATURE TRANSFORM to
find 5.
cedure
3) Verify
4) Verify
A for 1346
Connection of 25.
Encirclement (AC).
Sample Program
VPROB7STEPH1MODE9
[1] INITIAL
[2]
V
31
34 SYNINP
[3]
V
32
35 SYNINP
[4]
V
33
38 SYNINP
[5]
+
41
SINVERT 33
31
1
[6]
V
42
SINVERT 33
32
1
[7]
V
34
38 40 38 SCRVTRN
41
34
38
[8]
>
36
37 SCRVTRN
42
(9)
V
SYNOUT 40 39
34
38
[10]
V
SYNOUT 36 37
[11]
NECINP
[12]
V
44
NDYDTYP 37
36
35
1
[13]
4
4 3
NCFA 40 39
3 4
38
1
[14]
V
45
NECDYAD 34
38
40
39
35
t
36
37
41
1
1
4
[15]
V
NECOUT 45
[16]
 V
4 6
DESNT 45
[37]
V
4 6
DPVTLCA 46
35
36
37
38
34 39 40
[18]
46
47 DLKLNRT
45
36
37
36
35
[19]
V
46
DLKSIZE 46
35
3 4
37
38
[20]
V
46
DESDYAD 46
34
38
40
39
t
35
36
37
41
1,
i 6
(21]
DESOUT 46 V
Files
31) 4MSP (3/1)
32) 4MSP (2/1)
33) 4 MSP (4/1)
34) pivot F
35) pivot A
36) pivot B
37) pivot C
38) pivot D
39) pivot G
40) pivot B
41) 4 MSP (3/4)
42) 4MSP (2/4)
43) NC Grid (6)
44) NCGrid (5)
45) NC Grid(6,5)
46) DC Grid (6,5)
47) DC Grid (LLR
only)
72
satisfy the specified positions in an improper sequence,
as displayed in Figure 51 for a fourbar linkage. Six
possible orders exist for 4FSP:
0123 0132 0213
0321 0231 0312
Only those on the left would be acceptable, assuming the
linkages can be driven in either direction.
Fortunately, relatively few linkages are eliminated
by order considerations. First of all, motion specifica
tions placed in a relatively smooth and continuous se
quence tend to be satisfied naturally. Secondly, the
specified cases with higher ISP content diminish the order
problem. Finally, problems which have motions coordinated
with a crank automatically satisfy specified orders.
The most direct way to analyze order is to use the
method of Modler [116], which divides the circle point
cubic into segments which represent constraint links of
the same order. These segments are delimited by the image
poles and the Ball point. This is a powerful tool for
graphical analysis, but for discrete computational analysis
it is simpler and more direct to inspect exhaustively the
sequence at specified positions of all constraint links.
2.) The Grashof Criterion
A fourbar chain may have either zero pivots with com
plete rotation, or two adjacent pivots with complete
76
3.) Branching and Change Points
A change point for a given pivot is that position at
which attached links become collinear. This position,
which might lie between specified positions, may or may
not be acceptable depending upon the nature of the linkage
and how it is driven.
a.) Dyad change points
In most cases change points in the dyad of a Stephen
son or Watt sixbar indicate simultaneous disconnection
(or a limit in motion of the rest of the linkage, Figure
53b). At this point the dyad could pass from one dyad
"type" (relative angular orientation, Figure 52c) to the
other. One generally wishes to verify that the specified
positions are satisfied while the dyad maintains its type,
although one exception is that of the cyclic dyaddriven
sixbar linkage (Figure 53d).
There are two times at which dyad type can be checked.
If both links of the dyad are defined by the motion specifi
cations, such as the Stephenson 3 solution to a threelink
chain problem (dyad links 2 and 5 are the first two links
of the chain), it is a simple matter to simply check the
motion specifications before synthesis. More often, how
ever, one of the two links of the dyad is generated in the
synthesis procedure, and dyadtype must be checked during
necessary condition analysis.
151
"find 2"
INVERSION
Motion of link B is derivable for
4MSP, even though (if B is ternary)
link shape may not yet be com
pletely definable.
to 3" 3 is the new reference link.
174
a) Necessary Condition Pro
cedure I for 2456.
b) Necessary Condition Pro
cedure C for 1234.
a) Necessary Condition Pro
cedure I for 2456.
b) Necessary Condition Pro
cedure A for 1234.
a) Necessary Condition Pro
cedure I for 2456.
b) Necessary Condition Pro
cedure A for 1234.
a) Necessary Condition Pro
cedure K for 2456.
b) Necessary Condition Pro
cedure C for 1234.
a) Necessary Condition Pro
cedure K for 2456.
b) Necessary Condition Pro
cedure C for 1234.
Use of Appendix B
Each part of Appendix B contains a problem type, a
linkage solution and a corresponding synthesis procedure.
The solutions have been selected so as to show at least
one example of a solution for every type of solution linkage.
The contents of Appendix B are designed to provide a
guide for the writing of the necessary condition part of
an optimization program, for all twelve problem types. For
an implicit understanding of the procedures, the reader is
referred to Chapter 5.
Notation
Driven link.
The links attached to the circled pivot may
have complete relative rotation.
The links attached to the pivot cannot have
complete relative rotation.
The pivot may not pass through a change point.
Link is synthesized and selected from a
circle point cubic curve.
"CURVATURE TRANSFORM" Tools of synthesis are capitalized.
150
229
12. Sutherland, G.H., "Mixed exactapproximate planar
mechanism position synthesis," Trans ASME, ASME Paper
No. 76DET29, 1976.
13. Bagci, C., and I.P.J. Lee, "Optimum synthesis of plane
mechanisms for the generation of paths and rigidbody
positions via the linear superposition technique,"
ASME Paper No. 74DET10, 1974.
14. Bagci, C., "Optimum synthesis of planar function
generators by the linear partition of the dyadic loop
equations," Mechanism and Machine Theory, Volume 11,
1976, pp. 3346.
15. Bhatia, D.H., and C. Bagci, "Optimum synthesis of
multiloop planar mechanisms for the generation of
paths and rigidbody positions by the linear parti
tion of design equations," Trans ASME, February
1977, pp. 11623.
16. Roth, B., G.N. Sandor, and F. Freudenstein, "Synthesis
of fourbar path generating mechanisms with optimum
transmission characteristics," Transactions of the
Seventh Conference on Mechanisms, Purdue University,
October, 1962.
17. Nolle, H., "On capability of four bar mechanisms as
function generators," Trans Instu. Engrs. Australia,
Mech. and Chem. Eng., VMC3, No. 2, November 1967,
pp. 25968.
18. Tomas, J., "The synthesis of mechanisms as a nonlinear
programming problem," Journal of Mechanisms, Volume 3,
1968, pp. 11930.
19. Garrett, R.E., and A.S. Hall, "Optimal synthesis of
randomly generated linkages," Journal of Engineering
for Industry, Trans ASME, Series B, Volume 90, No. 3,
August 1968, pp. 47580.
20. Eschenbach, P.W., and D. Tesar, "Optimization of four
bar linkages satisfying four generalized coplanar po
sitions," Journal of Engineering for Industry, Trans
ASME, Series B, Volume 91, No. 1, February 1969,
pp. 7582.
21. Eisenstein, I., and A.S. Hall, "Synthesis of Randomly
generated twodegreeoffreedom plane linkages for
function generation," Proceedings of the Second Inter
national Congress on the Theory of Machines and Mech
anisms, Poland, 1969, pp. 30008.
45
and the "inverse coordinate transform":
u = (Ua) cos y + (V$) sin y
(4.2)
v = (V6) cos y (Ua) sin y
The constrained motion of E can be described in
terms of two functional relationships. Letting y be the
independent parameter, let a = f^(y) and 6 = f2(y). Links
with moving pivots attached to E and fixed pivots attached
to Â£ might constitute physical constraints. If so, the
functional description of the circular motion of a mov
ing pivot (attached at say, point A in E) relative to Â£
can take the form F(U,V) = 0. Combining this with the co
ordinate transform (4.1), the constraint equation can be
represented by:
F = f (u,v,a,B.y,Q0,Q1,Q2 . .) = 0
where 's are constants. For finitely separated speci
fications of the motion of plane E the above constraint
equation takes the form:
Fj = f (u,v,aj,8j,Yj,Qq,Q1,Q2 ....)= 0
j = 0,1,2 . .
For infinitesimally separated positions this rela
tionship takes the form:
F = d_ [f(u,v,o,Bfy,Q ,Q1,Q2. .
k dYk
. .) ] = 0
140
Watt
2 
Link 2 is grounded.
Plane
J
Link 3 and
+* CURVATURE
TRANSFORM
to
find 1
Point
e
=
Pivot C.
CURVATURE
TRANSFORM
to
find 6
Plane
K
v
Link 5 and
Point
f
=
Pivot H.
[4MSP Grid
[: (2)xlD]
Plane
L
=
Link 4 and
Point
g
=
Pivot D.
Stephenson 1 Link 1
is grounded.
Plane
j
>
Link 6 and
CURVATURE
TRANSFORM
to
find 4
Point
e
=
Pivot H.
INVERSION
to 2.
Plane
K
>
Link 2 and
CURVATURE
TRANSFORM
to
find 5
Point
f
=
Pivot A.
Plane
L
=
Link 3 and
[4MSP Grid
:(2)xlD]
Point
g
=
Pivot F.
Stephenson 2 Link 2
is grounded.
Plane
j
Link 1 and
INVERSION
to 1.
Point
e
=
Pivot F.
CURVATURE
TRANSFORM
to
get 4 .
Plane
K
*
Link 6 and
INVERSION
to 2.
Point
f
=
Pivot H.
CURVATURE
TRANSFORM
to
get 5.
Plane
L
=
Link 3 and
Point
g
=
Pivot A.
[4MSP Grid
: (2)xlD]
148
Stephenson 3 Link 3 is grounded.
a.) Crank J > Link 1 and
Point f = Pivot F.
Crank K > Link 2.
Point e on Link 5.
b. ) Crank J * Link 1.
Crank k ^ Link 2 and
Point g = Pivot
Point e on Link 4.
CURVATURE TRANSFORM and PATH
COGNATES to find 2, dummy 7'.
INVERSION to 1.
CURVATURE TRANSFORM to find 4.
INVERSION to 3.
CURVATURE TRANSFORM to find 6.
[4MSP GRID: L,R 2D, + (2)xlD]
CURVATURE TRANSFORM and PATH
COGNATES to find 1,6.
INVERSION to 2.
CURVATURE TRANSFORM to find 5.
[4MSP Grid: L,R 2D, +1D]
158
a) Necessary Condition Pro
cedure J for 1234.
a) Necessary Condition Pro
cedure K for 1234.
a) Necessary Condition Pro
cedure K for 1234.
21
concept of grid expansion. Grid expansion is a refine
ment technique by which the three steps of synthesis,
necessary condition evaluation and desirable condition
evaluation become part of a continuous loop. A region on
the grid about the linkages of higher quality is selected,
and resynthesis is performed to generate a new grid with
a more finely discretized representation (a conceptual
magnification) about the selected region. Two or three
passes through the loop is generally sufficient.
Due to the typically high attrition of solution
linkages in the first pass through the necessary condi
tions and the high computational expense of the desirable
condition analysis relative to the necessary condition
analysis, a further improvement can be had by looping
back just after the necessary conditions on the first
pass (as shown in Figure 21). Because desirable condi
tions are not evaluated initially,a relatively finely
discretized grid can be used on this first pass. All of
these techniques are employed in the author's thesis
program package SOFBAL (Synthesis and Optimization of
Four BAr Linkages), which is capable of synthesizing and
optimizing all types of fourbar linkages for general co
planar motion.
This successful philosophy is retained for this work,
but now twelve types of motion specification are considered.
In the next chapter these motion specification types and
Table 42
Motion Coefficients, D
mÂ£
k
uo
2
D 3
U4
U5
0
(fi .cosY .t'ii s
J ] 3
inY )
3
(0.cosY.a sinY )
3 3 3 3
cos 1.cosY0
3 ^
sinY sinY.
3 0
aj"o 3j
(VOSV30
sinY )
o
(3ocosYo"aosinYo)
L
uj. v 3
(atfty) siny
(aBy)siny
ysiny
YCOSY
a 8
+ (a * fs y) c o s y
+(ay+6)cosy
,
ii L 3 ti B
9
(uy2ay3y2
i
i B)siny
4 1 #
(ay2uSy23y)siny
ysinY
y 2siny
mm
a Â£
+ {ay 2 f,x+3y+23'() cosy
+ (ay2ay0y J+B) cosy
y 2cosy
+ycos y
3
3'iataf 3.s + bd
, M *
(ay 5ay3ay3uy
, 
3yyj3y + 0)siny
(3ayy+3ay2afy
 t y 3 0 y J B y > smy
# *
(y JyJ siny
+(~3yy)cosy
*
(3y y)siny
+ (Y + Y> cosy
a B
( 3uyy3ay 2
i
+u3y
14* M
+ (uy a 3c*y 3uY
+By+33a+33y) cost
3 ty y3fty HJ) cos y
157
Modes
1)
of Cyclicity
Necessary Condition Procedures
a) Necessary Condition Pro
cedure M for 1234.
a) Necessary Condition Pro
cedure N for 1234,
a) Necessary Condition Pro
cedure L for 1234.
4)
a) Necessary Condition Pro
cedure I for 1234.
183
Module Chart 6
Name Path Cognate (Synthesis Tool)
Label SPTHCOG
Description
Given a fourbar (I),
generates pathcognate four
bars II and III.
APL Call
,,>(o1), (o2), (o3), (o4)
SPTHCOG (IJL) (I2) (I3) (I4) (I5) "
where:
(I.) = 4MSP
d2), d3) =
(o1),(o2) =
(03),(04) =
of def (identifies e)
link dO^ pivots
link f02 pivots
gO^ (and h02)
j03 (and K03)
1UO
Modes of Cyclicity Necessary Condition Procedures
1)
*
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure D for 1346.
c) Verify Connection of 25.
a) Necessary Condition Dyad
Type (25) .
b) Necessary Condition Pro
cedure H for 1346.
c) Verify Connection of 25.
a) Necessary Condition Dyad
Type (25 ) .
b) Necessary Condition Pro
cedure H for 1346.
c) Verify Connection of 25.
d) Verify Nonqfunction Re
versal (2 relative to 3).
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure H for 1346.
c) Verify Connection of 25.
11
sometimes called the Sequence of Unconstrained Minimization
Technique (SUMT). Fox and Willmert [41] used this tech
nique to optimize a fourbar pathgenerating linkage using
a leastsquares error approach. Tranquilla [42] used this
method to optimize a fourbar under the geometric coupler
point constraints. Spatial mechanisms were considered by
Gupta [43] with constraints on closure, mobility and trans
missability. Bakthavachala and Kimbrell [44] considered
fourbar path generator optimization under clearance,
tolerance, transmission angle and link size considera
tions. Conte, George, Mayne and Sadler [36] used the SUMT
for optimization with respect to dynamic criteria. Alizade,
Mohan Rao and Sandor [45,46] have considered optimization
with respect to structural error for twodegreeoffreedom
spatial function generators and offset slidercrank path
generators. Alizade, Novruzbekov and Sandor [47] then
went on to fourbar function generators with consideration
of link length and transmission angle inequality con
straints. Kramer and Sandor [48] and Kramer [49] intro
duced the use of a new type of approximate motion specifica
tion in conjunction with an interior penalty function in
order to optimally approximate a specified motion.
Mariante and Willmert [50] used the interior penalty func
tion to synthesize and optimize a complex convertible top
linkage.
Direct methods and linearization methods are used to
192
Module Chart 15
None Necessary Condition Procedure G
Label NCPG
Description
Evaluates fourbar chains
on the basis of changepoint
considerations for the case
shown. Creates a necessary
condition grid indicating ac
ceptable linkages.
APL Call
where
(0 x) NCPG
(ix) d2) (i3) d4) d5)"
d1), d2)
d3), d4)
Pivots of first constraint link
(moving, fixed pivots).
Pivots of second constraint link
(moving, fixed pivots).
1 or 2, identifies constraint
link with changepoint restric
tions.
(o1)
= Output grid.
29
Stephenson SixBars With No Cyclic Links
Any pair of links
may be driven (15
distinct possi
bilities)
Figure 35 Modes of Rotational Cyclicity of the
Stephenson SixBar Chain
237
96. Fox, R.L., "Optimization applied to mechanisms and
mechanical components," ASME AMD Volume 7, 1974,
for meet. New York, N.Y., November 1721, 1974.
97. Odenfeld, J., "On optimum synthesis of machines,"
Fourth World Congress on the Theory of Mach, and
Mech., University of Newcastle upon Tyne, England,
September 813, 1975.
98. Starr, P.J., "Optimization and the design process:
similarities, differences, and needs," Fourth Ap
plied Mechanisms Conference, University of Chicago,
Ill., November 35, 1975.
99. Chohan, S.M., "Optimizing designs of machines and
mechanisms by functional approach," ASME Paper
No. 76DET15, Trans ASME, 1976.
^OO. Lindholm, J.C., "Synthesis and optimization of four
bar and sixbar mechanisms literature survey,"
Second Applied Mechanisms Conference, Oklahoma State
University, Stillwater, Oklahoma, November, 1971.
101. Seireg, A., "A survey of optimization of mechanical
design," Journal of Engineering for Industry, Trans
ASME, Series B, Volume 94, No. 2, 1972, pp. 49599.
102. Fox, R.L., and K.C. Gupta, "Optimization technology
as applied to mechanism design," Journal of Engineer
ing for Industry, Trans ASME Series B, Volume 95,
No. 2, May 1973, pp. 65762.
103. Chen, F.Y., "A survey of computer use in mechanism
analysis and synthesis," First Des. Eng. Tech. Conf.,
New York, N.Y., October 59, 1974.
104. Root, R.R., and K.M. Ragsdell, "A survey of optimiza
tion methods applied to the design of mechanisms,"
ASME Paper No. 75DET95, Trans ASME, August ]976,
pp. 10361041.
105. Tesar, D., "Design methods for four, five aid six
bar linkage systems," class notes, University of
Florida, Gainesville, FI., January, 1973.
106. Burmester, L., "Lehrbuch der kinematik," A. Felix,
Leipzig, 1888.
Bottema, 0., "On the determination of Burmester points
for five positions of a moving plane," Proceedings,
Koninklijke Nederlandse Akademic van Wettenschappen,
Series A,67, 1968, pp. 310318.
107.
APPENDIX B
MODES OF CYCLICITY AND
NECESSARY CONDITION PROCEDURES
Module Chart 30
Name Desirable Condition Output
Label DESOUT
OUTPUT
Description
Outputs scoring grid from
desirable condition analysis.
Collects expansion input for
the selected portion of the
grid.
APL Call
"DESOUT (I ) (I ) . .
where:
(1^), (12) = Scoring grids to be displayed.
(If more than one, add them).
165
a) Necessary Condition Pro
cedure B for 1234.
b) Necessary Condition Pro
cedure B for 2456.
a) Necessary Condition Pro
cedure A for 1234.
b) Necessary Condition Pro
cedure A for 2456.
a) Necessary Condition Pro
cedure A for 1234 .
b) Necessary Condition Pro
cedure A for 2456.
a) Necessary Condition Pro
cedure A for 1234 .
b) Necessary Condition Pro
cedure B for 2456 .
9)
a) Necessary Condition Pro
cedure A for 1234.
b) Necessary Condition Pro
cedure B for 2456.
51
or have a defined relationship between independent param
eters. The types of motion specification sets which cor
respond to the formulations that follow are displayed in
Table 43.
a.) Inversion formulation number 1.
Given: Motion of N relative to R; M relative to R.
Find: Motion of M relative to N ( for RQ reference ).
Independent parameter for given MSP: time t.
Independent parameter for derived MSP: time t.
(k) (k)
an = S. cos C + T. sin C + a rri
m3 k k nO
(k).
n
mj
(k)r
S, sin C + T. cos C + S A
k k nO
oon (k)r
Y Y Y + Y
mj mj nj 1 nO
Where: j = FSP counter,
(k)
k = ISP counter ( for example, for k=2, a = a )
T0 = B
c =
r
a .
r
 a .
m3
n3
er.
 er.
m3
03
r
r
Ynj
Yn0
191
Module Cha_rt _14
Name Necessary Condition Procedure D
Label NCPD
Description
Evaluates fourbar chains
on the basis of changepoint
considerations for the case
shown. Creates a necessary
condition grid indicating ac
ceptable linkages.
APL Call
o
t
NCPD
d1) d2) d3) (i4) d5)"
where:
(Ij_) /
<12> 
Pivots of first constraint link
(moving, fixed pivots).
d3)'
II
M
Pivots of second constraint link
(moving, fixed pivots).
*5 =
1 or 2, identifies constraint
link with changepoint restric
tions .
O
t*
11
Output grid.
185
Module Chart 8
Name Synthesis Output
Label SYNOUT
Description
Displays 4MSP set, gener
ated cubics, focus, Ball point,
inverse Ball point, axes.
OUTPUT
APL Call
"+SYNOUT (I1),(I2),(I3),(I4)"
where:
dx), d2) = Circle, center points of first
constraints.
d3), d4) = Circle, center points of second
constraints. (Optional for
Stephenson dyad links).
LINKAGE
OPTIMIZATION
Figure 11. Linkage Optimization Methods
236
85. Thornton, W.A., K.D. Willmert, and M.R. Khan, "Mech
anism optimization via optimality criterion tech
niques," to be presented at the ASME Design Engrg.
Tech. Conf., Minneapolis, Minn., September 2427,
1978 .
86. Rao, S.S., and A.G. Ambekar, "Optimum design of
spherical 4R function generating mechanisms," Mech
and Mach. Theory, Volume 9, No. 3, Autumn 1974,
pp. 405410.
87. Sallam, M.M., and J.C. Lindholm, "Procedure to syn
thesize and optimize the sixbar Watt1 mechanism
for function generation," ASME Paper No. 74DET18,
1974.
88. Hobson, Douglas A., and L.E. Torfason, "Computer
optimization of polycentric prosthetic knee mecha
nisms." U.S. Veterans Adm. Dep. Med. Surg. Bull.
n 1023, Spring 1975, pp. 187201.
89. Reed, W.S., and R.E. Garrett, "A graphical man/com
puter interface for the design of planar mechanisms
First Design Eng. Tech. Conf., New York, N.Y.,
October 59, 1974, pp. 8591.
90. Smith, W.D., and W.S. Reed, "Interactive mechanism
optimization employing computer graphics," Fourth
World Congress on the Theory of Mach, and Mech.,
University of Newcastle upon Tyne, England, Septem
ber 812, 1975, pp. 627632.
91. Ricci, R.J., "SPACEBAR: kinematic design by com
puter graphics," Computer Aided Design, Volume 8,
No. 4, October 1976, pp. 21926.
92. Rubel, A.J. and R.E. Kaufman, "KINSYN III: a new
humanengineered system for interactive computer
aided design of planar linkages," ASME Paper No.
76DET48, Trans ASME, 1976 .
93. Johnson, R.C., "Optimum design by synthesis," ASME
Paper No. 70DE8 for meet. May 1114, 1970, p. 11.
94. Srivastava, A.K., and W.R. Newcombe, "Synthesis of
coupler curves multifactor optimization methods,"
ASME Paper No. 73WA/DE14, 1973, p. 7.
95. Sutherland, G., and S. Siddall, "Dimensional syn
thesis of linkages by multifactor optimization,"
Mech. and Machine Theory, Volume 9, 1974.
2
criteria. In 1847, Cauchy [2] contributed the "steepest
descent" method.
The German and Russian involvement in mechanism de
sign over the past two centuries produced some extremely
sophisticated graphical techniques for design. A graph
ical approach to fourbar optimization with respect to the
transmission angle criterion was presented by Alt [3] in
1925.
The next fundamental influence on optimization was
the development of the digital computer. Previously in
feasible approaches to optimization became practical and
the fields of mechanism design and optimization both changed
directions. The birth of modern optimization can probably
be credited to the work of Dantzig [4] in the early 1940's,
whose work in linear programming included the development
of the Simplex method.
In 1948 Svoboda [5] introduced his computational syn
thesis method of "successive approximations," and in 1954
Levitskii and Shakvazian [6] introduced a method to opti
mize spatial slidercranks and crankrockers on the basis
of a set of linear equations from a finite number of pre
cision points.
The work of Freudenstein [711] in the latter part of
the 1950's formed the basis for modern linkage design pro
cedures. In 1959 Freudenstein and Sandor [10;11] used an
IBM 650 and complex number theory to synthesize pathgen
erating mechanisms, and Freudenstein [9] developed a
112
Onedimensional desirable conditions (e.g., link lengths)
will have singledimensional regions. Twodimensional
desirable conditions (e.g., pivot locations) will have
twodimensional regions, and these regions may now take
the form of circles or rectangles.
CHAPTER II
A PHILOSOPHY FOR LINKAGE DESIGN
AND OPTIMIZATION
A. Control of Parameters
In any optimization process, control of the number
of free parameters is a primary consideration for linkage
problems, where desired motions are specified at a finite
number of precision positions, the number of free param
eters is determined by the simple relation:
p = q FÂ£
where p = number of free (optimiza
tion) parameters,
q = number of parameters pro
, vided by a solution linkage.
Â£ = number of "precision posi
tions" Â£ .
F = number of functional
relations defining each
precision position.
which states that the number of free parameters after
synthesis equals the number of parameters provided by the
solution linkage minus the number of parameters used to
satisfy the precision positions. For a given motion
16
79
Figure 54 Change Point Considerations in a
FourBar Chain
49
and the A^'s in equation 4.5 will be replaced by time
state motion coefficients termed D 's. The D 's are
mi. mi,
listed in Table 42.
In timestate synthesis y becomes a dependent param
eter, but the D^^'s reduce to A^'s by replacing indepen
dent parameter t by y so that:
t>y ; dy_ dY
dt dy
.k ,k
i.
^.4k a k
dt dy
0
k>l
and the timestate synthesis formulation reduces to the
geometric synthesis formulation.
2.) Synthesis Tool #2. Inversion
The inversion concept essentially involves a change
of reference. Two basic types of inversion are of interest.
In the first, the motions of planes N and M (Figure 44)
are defined relative to a fixed plane R, the reference is
to be changed to plane N, and the motion of plane M rela
tive to plane N is to be defined (formulations #1, 2 and 3,
below). In the second type of inversion the motion of N
is defined relative to fixed plane R, the motion of plane
M is defined relative to plane N, and the motion of M
relative to R is to be defined (formulations #4, 5 and 6,
below).
Motion specifications for the two planes which have
some ISP content must share a common independent parameter,
APPENDIX C
PROGRAM MODULES
162
6)
7)
8)
9)
10)
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure II Cor 1346.
c) Verily Connection of 25.
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure A for 1346.
c) Verify Connection of 25.
d) Verify Nonencirclement
(AC) .
a) Necessary Condition Dyad
Type (25 ) .
b) Necessary Condition Pro
cedure A for 1346.
c) Verify Connection of 25.
d) Verify Nonencirclement
(AC ) .
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure A for 1346.
c) Verify Connection of 25.
d) Verify Encirclement (AC).
a) Necessary Condition Dyad
Type (25 ) .
b) Necessary Condition Pro
cedure A for 1346,
c) Verify Connection of 25.
d) Verify Encirclement (AC).
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure C for 1346.
c) Verify Connection of 25.
d) Verify Nonencirclement
(AC) .
70
Table 43
Related Motion Specification Sets for Two Planes M and N (2ISP)
(Notation: Figure 44)
Inversion Formulation #1
Inversion Formulation #4
(Independent
parameter t)
(Independent
parameter t)
Plane
N:
Plane M:
Plane
N :
Plane M:
r or
V Sn'
r
Yn
r nr r
a 3 Y.
m m m
r or
v v
r
Yn
n n n
am' m' Ym
dar d3r
n n
dYr
n
dc d3^ dY,^
m m m
dar dBr
n n
dYr
1 n
dan dBn dYn
m m. 1 m
dt' dt'
dt
dt' dt' dt
dt' dt'
dt
dt' dt' dt
Inversion Formulation #2
Inversion Formulation #5
(Independent parameter 
coupler angle Yn)
(Independent parameter 
coupler angle y )
Plane
N:
Plane M:
Plane
N:
Plane M:
r or
V V
r
Yn
r nr r
m' ^m' Ym
r
Yn
n Qn ,n
am' m' Ym
dar dgr
n n
r' r'
dY dY
n n
1
dm dem dYm
t r' r
dY dY dY
n n n
K K
r' r'
dY dY
n n
1
dan dBn dYn
m m m
, r' r' j r
dY dY dY
1 n n n
Inversion Formulation #3
Inversion Formulation #6
(Independent parameters 
coupler angles Y Ym (related))
Plane N:
n m
Plane M:
(Independent parameters 
coupler angles Yn Ym (related))
Other
Plane N:
Plane M:
Other
r
r r
r
r
r
r
,x
r
n
n
V
pn' Yn
%
lm'
Ym
n
n'
Yn
Um'
3m'
Ym
dar
n
dBn
n 1
dar
m
dBr
m
1
dYr
'n
drar
n
dPn
1
dan
m
d3n
m
1
dYr
n
d^n'
a r'
dy
n
a r'
dy
'm
a r'
dY
'm
dYr
m
dYn
d^'
, n'
dy
m
a
dY
m
dY
'm
98
(a) (b)
Figure 5 13 Path Cognate Change Point Derivations
156
Modes of Cyclicity
Necessary Condition Procedures
a) Necessary Condition Pro
cedure D for 1234.
a) Necessary Condition Pro
cedure II for 1234.
a) Necessary Condition Pro
cedure B for 1234.
a) Necessary Condition Pro
cedure A for 1234 .
87
line O^A to reach the extremal positions. As this happens
a change point occurs at point A. Therefore, the portions
of the circle point cubic which lie in the Z regions
represent constraint links which are incompatible with the
selected permissible link.
It is quite possible that the circle point cubic will
not pass into the compatible regions denoted by Y. If so,
another permissible link must be selected. Note that the
Y regions disappear altogether for nonpermissible links,
having no possibility of finding compatible constraint
links.
c.) General branching analyses of fourbar chains in six
bar linkages
Displayed in the lefthand column of Figure 59 are
the possible forms of fourbar chain which may be needed as
all or part (for a sixbar) of a linkage solution. The
chains on the right are the different inversions in which
the chain on the left may actually be synthesized. It is
necessary to treat these inversions as they stand. A num
ber of "necessary condition procedures" have been developed
to handle the range of possible synthesis inversions. (The
conditions under which order considerations can be ignored
have been discussed in part A of this chapter).
203
Module Chart: 2Â£
Name Pivot Location (Desirable Condition)
Label DPVTI.CA
Description
Evaluates the solution set
of linkages on the basis of
pivot location. User speci
fies pivots of interest.
"^(01)(02) DPVTLCA (I1),(I2)
where:
Six bars
only
'V
=
Previous scoring grid
(i2)
=
A
pivot
(I3>
=
B
pivot
=
C
pivot
(I5>
=
D
pivot
=
F
pivot
(I?)
=
G
pivot
(I8)
H
pivot
(ot)

Output scoring
grid, cumulative.
<2>
=
Output scoring
grid, this condition
only. Opt iona 1.
10
McLarnan [35] by using the variable metric. Conte,
George, Mayne and Sadler [36] optimized with respect to
dynamic criteria such as shaking force, input torque fluc
tuation, shaking moment and bearing reactions, using a
penalty function in conjunction with the variable metric.
A fourth unconstrained minimization approach, the
NewtonRaphson method, is very efficient in terms of con
vergence, but requires formulation of the Hessian of the
objective function at each iteration step. Han [37] used
this method in 1966 for general mechanism optimization.
All of the unconstrained minimization formulations
except for Powell's method require evaluation of the ob
jective function's differential, but finite difference
methods are generally applicable.
The second class of iterative methods are the con
strained methods using unconstrained minimization. Gen
erally, this means some modification is made to an uncon
strained minimization technique to allow the inclusion of
inequality constraints. One approach is a simple change
of variable which will inherently disallow violation of
specified inequalities, but linkage problems are usually
too complex for this approach.
Penalty functions are often used to apply constraints
to unconstrained minimization techniques. The interior
penalty function introduced by Fiacco and McCormick [3840]
has received quite a lot of attention recently. It is
47
and the generalized circular constraint function (divided
by 2) takes the form:
= dk [Q0(a2+g2)+ u(cosY+BsinY) + u(asinY+BcosY))
dY
+ (u(cosY1) usinY +a) + Q2 (usinY
+ u(cosY1) + 6)]
= 0
Y = Y
Â£
which may be written as:
G VA0i + uA + vA22> + Qi(uA3t + VA4l + A5l
+ Q2(uA4J + vA3l + A6i> = 0
(4.5)
Â£ = 1,2 . .
where AmP coefficients are listed in Table 41. They
are a function of the motion specifications only.
Now, let:
D
E
F
Â£
Â£
Â£
0Â£
uA
3Â£
uA
4 Â£
+ uA + vA
1 Â£
 vA4ji + A
+ vA3Â£ + A
2 Â£
5 Â£
6 Â£
and the constraint equations now take the form:
GÂ£ Q0DÂ£ + QÂ£EÂ£ + Q2FÂ£
0
(4.6)
227
[ / ) 'If) S INVERT 4 5 4 4 1
(1H 37 49 47 48 SCRVTRN 46 37 49
119) SVNOUT 47 48 37 49
[20 NEC IIJP
[21J 50
NCPB 47
4 8
37
49
(22) 51
NECDYAD
4 8
49
4 7
37
36,
38
39
0
1
1
[23] NECOUT 51
[24] 52
DES INP
51
[25] 52
DPVTLCA
52
49
39
38
36 47 48 37
[26] 52
DLKLNRT
52
36
38
48
4 9
[ 27 J 52
DLK.S I ZE
52
36
38
4 7
48
(28] 52
DESDYAD
52
48
49
47
37 36,
38
39
0
1,
t 6
l 2 9 J DESOUT 52 V
19
,
...
Necessary
Conditions
Desirable
Conditions
Figure 21 The computational steps of a Synthesis/Analysis/
Resynthesis Linkage optimization
136
Watt 2 Link 2 is grounded.
Plane
J
Link 3 and
CURVATURE
TRANSFORM
to
find
4
Point
e
=
Pivot B.
INVERT to
6.
Crank
K
=
Link 1 and
CURVATURE
TRANSFORM
to
find
5
Point
f
=
Pivot A.
Crank
L
>
Link 6 and
[4MSP Grid
(2 ) xlD ]
Point
g
=
Pivot G.
Stephenson 1
 Link 1 is
grounded.
Plane
j
>
Link 2 and
* +
INVERSION
to 4 .
Point
e
=
Pivot A.
CURVATURE
TRANSFORM
to
find
6
Crank
K
=
Link 3 and
CURVATURE
TRANSFORM
to
find
5
Point
f
=
Pivot F.
Crank
L
Link 4 and
[4MSP Grid
:(2)xlD]
Point
g
=
Pivot D.
Stephenson 2
i
 Link 2 is
grounded.
a.)
Plane
j
>
Link 1 and
INVERSION
to 1.
Point
e
=
Pivot F.
CURVATURE
TRANSFORM
to
find
4
Crank
K
=
Link 3 and
INVERSION
to 3.
Point
f
=
Pivot A.
CURVATURE
TRANSFORM
to
find
6
Crank
L
>
Link 5 and
Point
g
=:
Pivot B.
[4MSP Grid
.: ( 2)xlD]
b.)
Plane
j
Link 4 and
t
INVERSION
to 3.
Point
e
=
Pivot G.
CURVATURE
TRANSFORM
to
find
Crank
K
=
Link 5 and
1,6.
Point
f
Pivot B.
Crank
L
>
Link 3 and
[4MSP Grid
: 2D ]
Point
g
=
Pivot A.
Stephenson 3
 Link 3 is
grounded.
a.)
Plane
J
>
Link 4 and
t
CURVATURE
TRANSFORM
to
find
6
Point
e
=
Pivot D.
INVERSION
to 2.
Crank
K
=
Link 1 and
CURVATURE
TRANSFORM
to
find
5
Point
f
=
Pivot F.
Crank
L
>
Link 2 and
[4MSP Grid
:(2)xlD]
Point
g
=
Pivot A.
b.)
Plane
J
~>
Link 5 and
INVERSION
to .1.
Point
e
=
Pivot B.
CURVATURE
TRANSFORM
to
find
4
Crank
K
=
Link 2 and
INVERSION
to 3 .
Point
f
=
Pivot A.
CURVATURE
TRANSFORM
to
find
6
Crank
L
Link 1 and
Point
g
=
Pivot F.
[4MSP Grid
: ( 2)xlD]
177
Notation
File number of file containing data x.
APL goto. The outputted global argument
of each module is the next line number to
be executed.
The links attached to the circled pivot may
have complete relative rotation.
A change point cannot be allowed at the
pivot.
135
Problem H7: ANGULAR COORDINATION OF A CRANK AND A TWO
LINK CHAIN
The coplanar motion
of twolink chain JK
is coordinated with
the rotation of
crank L.
SOLUTION LINKAGES
WATT
Watt 1 Link 1 is grounded.
Synthesis Procedures
Plane J 
Point e =
Crank K =
Point f =
Crank L 
Point g =
Link 6 and
Pivot A.
Link 2 and
Pivo t A.
Link 3 and
Pivot B.
INVERSION to 2.
CURVATURE TRANSFORM to find 4
INVERSION to 6.
CURVATURE TRANSFORM to find 5
[4MSP Grid: (2)xlD]
137
Problem #8: ANGULAR COORDINATION OF THREE CRANKS
The rotations of
cranks J,K and L
are coordinated.
SOLUTION LINKAGES
WATT
Watt 2 Link 2 is grounded.
Synthesis Procedures
Crank J *
Point e =
Crank K >
Point f =
Crank L 
Point g =
Link 1 and
Pivot A.
Link 4 and
Pivot D.
Link 6 and
Pivot G.
*T INVERSION to 4.
CURVATURE TRANSFORM to find 3
CURVATURE TRANSFORM to find 5
[4MSP Grid: (2)xlD]
15
and path generators. Fox [96] describes the state of the
art of mathematical programming as applied to the optimiza
tion of mechanical components such as springs, shafts and
bearings. Oderfeld [97], Starr [98] and Chohan [99] have
all presented philosophical discussions of optimization of
mechanisms with respect to multiple parameters, with empha
sis on ordering of design priorities.
Other literature surveys dealing with mechanism op
timization have been presented (in chronological order)
by Lindholm [100], Seireg [101] Fox and Gupta [102], Chen
[103] and Root and Ragsdell [104].
The "philosophical" papers [9399] effectively point
the way towards the future. A major concern involves
methods of achieving globally optimal mechanism designs
which accomodate conflicting design requirements. Nearly
all of the mechanism optimization schemes to date optimize
with respect to one or two parameters. An enormous spec
trum of mechanism applications exist, and a large number
of criteria by which the overall quality of mechanism
solution must be evaluated also exists. To the design engi
neer, who must be concerned with a mechanism's total func
tionality, true mechanism optimization must involve a sig
nificant number of these parameters.
199
Modulo Chart 22
Name Necessary Condition Procedure N
Label NCPN
Description
Evaluates synthesized four
bar I on the basis of change
point considerations within
path cognate fourbars II and
III as shown. Generates a
necessary condition grid
(2x2D) indicating acceptable
cognates.
APL Call
n*(o1) ncpn d1),d2), .... dg)"
where:
d1), d2)
d3), d4)
s'V
(I7),
link dO^
link fO^
link gO^ (and h02)
link jO^ (and KO^)
(0^) = Output grid.
Ib4
Problem Â¡I8
Angular Coordina
tion of Three
Cranks
Solution Linkage
C 4 H
WATT 2
Pivots A,G,D
prespecified.
SyiitJesis Procedure
1) INVERSION to 4.
2) CURVATURE TRANS
FORM to find 3.
3) CURVATURE TRANS
FORM to find 5.
Modes of Cyclicity Necessary Condition Procedures
a)
Necessary
Condition
Pro
cedure
G
for
1234 .
b)
Necessary
Condition
Pro
cedure
D
for
2456.
a)
Necessary
Condition Pro
cedure H
for
1234 .
b)
Necessary
Condition Pro
cedure D
for
2456.
a)
Necessary
Condition Pro
cedure D
for
1234 .
b)
Necessary
Condition Pro
cedure D
for
2456 .
a)
Necess
ary Condition
Pro
cedu re
C for 1234.
b)
Necess
ary Condition
Pro
cedu re
D for 2456.
c)
Verify
Connection of
56 .
4)
131
Problem #4: ANGULAR COORDINATION OF TWO CRANKS
The rotation of
crank J is coordi
nated with the ro
tation of crank K
SOLUTION LINKAGE
FOURBAR
FourBar
Synthesis Procedure
Link 1 is
Crank J *
Point e =
Crank K v
Point f =
grounded.
Link 2 and
Pivot A.
Link 4 and
Pivot D.
*1 INVERSION to 2.
CURVATURE TRANSFORM TO
find 1,3.
[4MSP Grid: 1D]
62
c. Angular Cognafs
Use of Angular Cognates to Solve the
Function Generation Problem
Figure 46
92
Necessary condition procedures I > N take advantage
of some unusual properties of the "cognate 10bar" (Fig
ure 510b):
i.) A cognate 10bar will be made up of one of three
sets of three fourbars: either three draglinks, three
nonGrashof doublerockers, or a Grashof doublerocker and
two crankrockers.
ii.) When in fourbar II is at a change point, 02
in fourbar I is also at a change point.
iii.) When g in fourbar II is at a change point, d
in fourbar I is also at a change point.
iv.) When 0^ in fourbar II is at a change point,
links df and 0^C>2 *n furfrar I become parallel.
v.) When j in fourbar II is at a change point, links
O^d and C^f in fourbar I become parallel.
Proof of (ii): Referring to Figure 513a,
Lec A df I ge I j12 
Now, from (105;79]
Z' = Aeia(Z) and R' = Aeia(R)
de
Igi
12
Therefore, when Z and R coincide, Z' and R' will also
coincide, and change points will occur simultaneously in
fourbar I at C>2 and in fourbar II at 03>
153
2)
3)
4)
a) Necessary Condition Pro
cedure H for 1234.
a) Necessary Condition Pro
cedure A for 1234.
a) Necessary Condition Pro
cedure H for 1234 .
48
For 4MSP motion specifications (Z = 1,2,3):
Vi
+
Qlfl
+
Q2Fi
= 0
v2
+
1*2
+
Q2F2
= 0
Q03
+
Qlf3
+
Q2fr3
= 0
The matrix
must be singular if the constraint equations are to
be meaningful since these equations are linear and homogen
eous with respect to Qq, and Q2.
If this matrix (4.7) is expanded, the result is a
cubic in terms of E coordinates u and v. This is the
"circle point cubic" and represents the locus of permis
sable moving pivot locations on E.
The formulation for time state specification syn
thesis is quite similar. Equation (4.3) will now take
the form:
GJ?,
_d^[Q0(U2+V2)+2Q1U + 2Q2V+Q3] = 0
dt*
t=tÂ£
for = 0,1,2
(4.8)
78
b.) Change points in fourbar chains
All of the solution linkages can be considered as
single fourbar chains, a combination of fourbar chains
and dyads, or a combination of two fourbar chains. A
detailed analysis of change points within fourbar chains
is therefore very worthwhile. It involves considerations
of both synthesis procedure and mode of cyclicity.
If a fourbar chain (or any Grashof type) is driven
as shown in Figure 54a through a change point at the in
dicated pivot, the dyad portion of the chain connected to
this pivot may go into either type. This is beyond the
direct control of the driven members. If specified posi
tions are to be satisfied in a positive manner, such a
change point must be avoided. Similarly, fourbar chains
driven through opposite links (Figure 54b) cannot toler
ate change points at any pivots, for the same reason.
It may be assumed that at least one of the links in
the chain has been synthesized through use of the curva
ture transform. Waldron [117, 118] has developed graph
ical techniques which operate directly on Burmester cubics,
identifying sections which represent usable constraint
links (for fourbar chains) on the basis of changepoint
considerations. These techniques are used extensively in
the necessary condition analysis of this work, so detailed
descriptions now follow.
i.) Determination of "permissible links. Let the name
Each chart in Appendix D contains a sample program
demonstrating a coordination of the modules of Appendix C
to create an optimization routine for a particular prob
lem/linkage/mode of cyclicity set. There is an example
for each of the twelve problem types, each one using an
Appendix B solution.
Three types of data are channeled between modules
through files. They are motion specification sets, pivot
coordinates and grids. The nomenclature used to represent
these quantities is explained below.
Notation
"4MSP (3/4) "
Motion specification set of four multiply
separated positions describing the motion
of plane 3 relative to plane 4.
pivot A"
The coordinates of pivot A. May repre
sent a set of pivots or a single pivot.
pivot A'"
In pathcognate problems, the pivots
generated by the curvature transform are
denoted by a prime ('), and cognate
pivots (representing solutions) are
unprimed.
213
214
NC Grid (3)
DC Grid (3,5
"36+17"
Necessary condition grid for a set of
#3 links (1dimensional).
Desirable condition grid for sets of
#3 and #5 links (2dimensional)
"i7" in APL represents 1234567.
36+i7 represents 37 38 39 40 41 42 43.
APL goto. Each module generates as
an argument the next line to be executed.
Driven link.
Cyclic pivot. Attached links have com
plete relative rotation.
A change point cannot be allowed at
this pivot.
Encircled link is synthesized using the
curvature transform (circle point cubic
shown).
33
Problem Solution
Applicable Modes of Rotational Cyclicity
Figure 37 Applicable Modes of Rotational Cyclicity of the
FourBar Solution to the TwoLink Chain Problem
41
synthesis) in order to produce a functional linkage. For
the first three problems the curvature transform alone is
sufficient.
Since no linkage provides an infinite number of design
parameters the number of functional motion specifications
must be finite. If the precision positions are specified
in such a way that an independent parameter, such as the
coupler angle, undergoes a finite displacement between po
sitions, the specifications are termed finitely separated
positions. These are shown for a plane in coplanar motion
in Figure 42a.
Finitely separated position synthesis of fourbars
was first done by Burmester [106] in 1888, using graphical
techniques for five positions. In 1964 the problem was
solved analytically by Bottema [107] and Primrose, Freuden
stein and Sandor [108].
If higher order derivatives of motion at a precision
position are specified, they are referred to as infinites
imally separated positions (ISP) as depicted by the concep
tual combination of positions in Figure 42b. Mueller
[109], Allievi [110], Krause [111], Wolford [112] and
Veldkamp [113] performed graphical studies of infinitesi
mally separated positions for five positions.
Mixtures of finitely separated positions and infin
itesimally separated positions are termed multiply sepa
rated positions (MSP) (Figure 42c). The unified theory
for MSP was developed by Tesar [114] in 1967.
129
Problem #2: MOTION SYNTHESIS
The coplanar motion
of plane J is con
trolled .
SOLUTION LINKAGE
FourBar
Synthesis Procedure
*! CURVATURE TRANSFORM to
find 4,2.
Link 1 is grounded.
Plane J > Link 3.
[ 4MSP Grid: 2D]
167
7)
a) Necessary Condition Pro
cedure B for 1234.
b) Necessary Condition Pro
cedure B for 2456.
a) Necessary Condition Pro
cedure B for 1234.
b) Necessary Condition Pro
cedure B for 2456.
a) Necessary Condition Pro
cedure A for 1234 .
b) Necessary Condition Pro
cedure A for 2456 .
a) Necessary Condition Pro
cedure B for 1234.
b) Necessary Condition Pro
cedure A for 2456.
a) Necessary Condition Pro
cedure B for 1234.
b) Necessary Condition Pro
cedure A for 2456.
Page
VII PROGRAMMING FOR FOURBAR AND SIXBAR
OPTIMIZATION 113
A. Philosophy of Program
Modularity 113
B. Control of Input/Output 116
VIII RECOMMENDATIONS AND CONCLUSIONS 120
A. Recommendations 120
B. Conclusions 124
APPENDIX
A PROBLEM TYPES AND SOLUTION
LINKAGES 125
B MODES OF CYCLICITY AND NECESSARY
CONDITION PROCEDURES 149
C PROGRAM MODULES 175
D SAMPLE PROGRAMS 212
BIBLIOGRAPHY 228
BIOGRAPHICAL SKETCH 240
v
17
specification type (which defines F), one should create
a balance between solution linkage type (which defines q)
and number of precision positions (which defines a) in
order to describe adequately the desired motion and still
leave an acceptable number of free parameters for optimi
zation (between one and three free parameters is desirable).
Naturally there is no guarantee that a design pro
cedure will exist for every combination of q,F and i .
For example, one cannot expect to satisfy a large number
of precision positions simply by using a sufficiently
complex linkage.
In this work twelve types of motion are considered,
and for each of these at four specified positions, an ac
ceptable number of free parameters is generated by using
either a fourbar linkage or one of the sixbar linkages.
The synthesis procedures for the solution linkages are
well known and have been collected and organized by Tesar
[105] .
B. The Optimization Procedure
A synthesis, analysis and resynthesis loop is suf
ficient as a basis for optimization, and generally allows
one to take advantage of the knowledge and insight pro
vided by kinematic synthesis techniques, unlike the more
rigid and forceful numerical approaches. The synthesis/
122
blocks. In addition, fourbars have been designated as
solution linkages for the first five problem classes, but
more complex linkages could conceivably be used to provide
more free parameters for these five classes.
3.) Alternate Modes of Cyclicity
It has been explained that the necessary condition
procedure for a particular problem depends upon the de
sired type of motion in the solution linkage, and this
dependency is the reason for the creation of the concept of
mode of cyclicity. Restrictions are made in Chapter III
which have the effect of limiting the modes of cyclicity
under consideration to those for which there need be no
supplementary knowledge of system dynamics in order to
ensure control of motion through deadcenter positions.
These restrictions might be termed "geometric restrictions."
Many more modes of cyclicity are conceivable, some of
which cannot exist for physical reasons, and some of which
can exist but violate the geometric restrictions. Most of
the physically realizable modes of cyclicity can be treated
by the computational modules already presented. For exam
ple, a Grashof doublerocker with fully rotating coupler
violates the geometric restrictions and is therefore not
considered here as an acceptable mode of cyclicity, but a
designer who could control this type of mechanism in a
particular application could use Necessary Condition Pro
cedure C (NCPC, Appendix C) to design one.
240
BIOGRAPHICAL SKETCH
Kim L. Spitznagel was born July 20, 1952, in Charles
ton, West Virginia. He attended primary school in Camp
Hill, New Jersey, and York, Pennsylvania. In 1969 he be
gan studies in mechanical engineering at Lehigh Univer
sity and graduated with a Bachelor of Science in Mechan
ical Engineering in June, 1973. In September, 1973, he
began attending the University of Florida and received a
Master of Science in Engineering in December, 1975, and a
Bachelor of Science in Computer and Information Science
in March, 1978. He is a member of Pi Tau Sigma, Tau Beta
Pi and Phi Kappa Phi honorary fraternities.
ACKNOWLEDGMENTS
I wish to express my gratitude to Professor Delbert
Tesar for his guidance and support throughout my graduate
studies. My academic endeavors have benefited most from
his contributions.
Much appreciation is due to Dr. John M. Vance, Dr.
George N. Sandor, Dr. Ralph G. Selfridge, Dr. Martin A.
Eisenberg and Dr. Sanjay G. Dhande for their reviewing
efforts, suggestions, and especially for their respective
influences on my professional development.
I am grateful for the support and aid of my friends
and peers, particularly Ms. Barbara Mihatov and Mr. John
Elliott. I am also indebted to Ms. Lois Rudloff for her
patience and persistance while typing this dissertation.
Finally, my heartfelt thanks go to my family for
their tolerance, support and encouragement.
ii i
171
a) Necessary Condition Pro
cedure H for 1346.
b) Verify Disconnection of
25 (2Short).
a) Necessary Condition Pro
cedure H for 1346.
b) Verify Disconnection of 25
(2short).
a) Necessary Condition Pro
cedure H for 1346.
b) Verify Disconnection of 25
(5short) .
18
analysis/resynthesis loop to be used for this work is shown
in Figure 21.
The input to the synthesis step is the precision
position description of the desired motion, hereafter
termed the motion specification set. A sequence of
kinematic synthesis algorithms will produce the output,
2 3
a solution set of linkages of magnitude 00 00 or 00
(depending upon q, F and Â£ for the particular problem).
Unfortunately, the analytics used in the synthesis
process are not sufficient to guarantee that all ( or
indeed, any) of the generated linkages will satisfy the
specified motion without falling into one or more of a
number of physical pitfalls of real linkages, any one of
which can make a linkage useless as a solution to the
specified problem. For this reason a "necessary condi
tion" step is required. The necessary conditions depend
upon the motion specification type, the linkage solution
type, and the means by which the linkage is driven. The
linkage solution set is mapped onto linear (1dimensional),
rectangular (2dimensional) or cubic (3dimensional)
spaces, depending upon the magnitude of the solution set
space. These representations, termed "grids" due to their
discrete, arrayed nature, form the operating environment
for all subsequent optimization procedures. Through
evaluation of appropriate necessary conditions, portions
of the grid representing acceptable linkage solutions are
determined and presented to the designer.
96
Figure 511
Detection of Change Points in Cognate
FourBar Linkages
75
(a) CRANKROCKER
DOUBLEROCKER
(b) DRAGLINK
DOUBLEROCKER
Figure 52 FourBar Grashof Types
53
Let a = a
n n
a = a
m m
da
a/ =
n
n
r
n
r
, m
a =
m r
dy
da
dy
etc. ( same for 3 and y )
n n
etc. ( same for 3 and y )
m m
m
Now,
a' a'
A' = m
y/ 1
'm
A" =
m n
(y' 1)
m
(gm an^m
A" =
K > ~ 1)1
~(a/ a/)(3y// + y "'(y' 1) )
m n 'm 'm 'm
(y 1)
'm
B', B/y, B are similar (substitute 3's for a's )
C' =
y' 1
'm
y
c" =
m
(y' 1)
'm
109
importance and a weight of 0 signifies a condition of no
importance. A condition score of 0 signifies ideal per
formance (almost never attained) and a score of 10 delim
its the bound of acceptable performance. In this way a
linkage receives a cumulative score on a demerit system:
the linkage completing the analysis with the lowest score
is the optimum one in the solution set.
There are a number of ways to correlate score
and linkage performance. The simplest way is to use a
linear scale between ideal performance and the acceptable
limit. The score is then generated by:
Sk = 10
where P2
(6.2)
difference between ideal
and actual performance.
= difference between ideal and
unacceptable performance.
An alternative used by Eschenbach and Tesar [20] used
ranking zones in which "best", "good" and "acceptable" re
gions are defined and assigned score values. A better ap
proach was also used in the same paper, where the ratio of
actual to maximum value is operated upon by an exponent n
assigned by the designer:
P
n
S
2
k P
3
(6.3)
161
1)
2)
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure H for 1346
c) Verify Connection of 25.
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure G for 1346.
c) Verify Connection of 25.
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure D for 1346.
c) Verify Connection of 25.
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure II for 1346 .
c) Verify Connection of 25.
4)
appendices include synthesis procedures, necessary con
dition evaluations, program module tables and sample pro
grams .
vii
Use of Appendix A
Each part of Appendix A contains a drawing and brief
description of a motion specification type, a drawing
(with link and pivot notation) of solution linkage(s) and
a small section describing each possible linkage solution.
Each of these sections contains a movingbody/linkagecom
ponent matchup on the left and a corresponding synthesis
procedure on the right (other matchups are sometimes pos
sible by switching conceptually identical bodies in the
motion specification). Synthesis procedures are generally
not unique.
The solution linkages considered here are those which
best provide an acceptable number of free parameters for
optimization after a 4MSP synthesis. Geared fivebar and
sixbar solutions to the simpler problems are not included
for more information on these types of solutions reference
[105] is recommended.
Notation
. . is attached to . ."
. . is equivalent to . .
126
190
Modulo Chari 13
Name Necessary Condition Procedure c
Label NCPC
Description
Evaluates fourbar chains
on the basis of changepoint
considerations for the case
shown. Creates a necessary
condition grid indicating ac
ceptable linkages.
APL Call
where:
"dO^ NCPC (1^ (I2) (I3) (I4) "
(I1) d2)
d3), d4)
= Pivots of first constraint link
(moving, fixed pivots).
= Pivots of second constraint link
(moving, fixed pivot).
(o1)
= Output grid.
32
nonequivalence of links (a function of motion specifica
tion type). As an example, the fourbar solution to the
twolink chain problem (Figure 37) has seven modes of
cyclicity. Modes number 3 and 4 would otherwise be iden
tical except that constraint links 2 and 4 are not concep
tually the same due to the nature of the motion specifica
tions: link 2 is contained within the motion specifica
tions, link 4 is not. The same is true for modes 5 and 6.
The necessary condition analyses will depend upon 1) re
quired Grashof type for those modes which include cyclic
links, and 2) the pivots which are not allowed to pass
change points. For example, in mode number 6 a change
point is permissible at pivots A, C and D, but not at B,
regardless of Grashof type. These considerations are pre
sented in detail in Chapter V.
Figure 38 shows some examples of modes of rotational
cyclicity which are unacceptable or geometrically impos
sible. Figures 38a and 38b might represent the cyclic
couplerdriven Grashof double rocker and the cyclic rocker
driven crankrocker, respectively, which have already been
discussed. In Figure 38c, control of motion through the
change points of pivots r and s is impossible. The motion
shown for the sixbar linkages of Figures 38d and 38f
simply cannot exist, and the motion shown for the Watt
sixbar linkages of Figure 38e would require a linkage
with very unusual geometry and an external control of mo
tion for both fourbar chains.
MULTIPARAMETRIC OPTIMIZATION OF
FOURBAR AND SIXBAR LINKAGES
By
KIM LORING SPITZNAGEL
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
163
a)
Necessary Condition
Type (.25) .
Dyad
*
b)
Necessary Condition
cedure C for 1346.
Pro
c)
Verify Connection of
25 .
d)
Verify Encirclement
(AC )
a)
Necessary Condition
Type (25).
Dyad
*
b)
Necessary Condition
cedure B for 1346.
Pro
c)
Verify Connection of
25 .
d)
Verify Nonencirclement
(AC) .
a)
Necessary Condition
Type (25).
Dyad
*
b)
Necessary Condition
cedure B for 1346.
Pro
c)
Verify Connection of
25 .
d)
Verify Encirclement
(AC )
180
Module Chart 3
Name Dyad Type (Synthesis)
Label SDYDTYP
Description
Evaluates type for prespe
cified dyads. Generates a
program interrupt prior to
synthesis if dyad type is in
consistent for the 4MSP's.
APL Call
"vSDYDTYP (IJL) (I2) (I3) "
where:
(11) = 4MSP
(12) = 4MSP
(13) = 4MSP
associated with
associated with
associated with E^
58
'r
Yn =
1 + Y7
'm
"r
Yn =
_y//
'm
(1 +y')
m
///
r
Yn
n 2 n/., / ,
3y y (1 + Y )
'm m_ 'm
(l + y')5
m
f.) Inversion formulation number 6.
Given: Motions of N relative to R, M relative to N.
Find: Motion of M relative to R.
3T n
Independent parameters for given MSP: yn/ Ym ( must be
related).
Independent parameter for derived MSP: Ym
This formulation is identical to that of (2d), except
that the dots in the S^ and T^. formulations should be replaced
with primes (').
Let
a
n
a
n
n
a = a
m m
a
n
/
a =
m
da
dy
da
r
ri
r
n
n
m
dy
dy
/ n
mn
dy
etc. ( same for 8 and y )
n 'n
etc. ( same for 8 and y )
m m
Y
// =
mn
,2 r
d Y
1 n
dY
n 2
m
; etc.
57
e.) Inversion formulation number 5.
Given: Motions of N relative to R, M relative to N.
Find: Motion of M relative to R.
Independent parameter for given MSP:yn
2T
Independent parameter for derived MSP: ym
This formulation is identical to that of (2d), except
that the dots in the and formulations should be
replaced with primes (').
Let a
n
a
m
a
n
n
a
m
or
n
a
m
da
n
dy
da
n
n
m
dy
n
; etc.
; etc.
( same for $n and yn )
( same for 3 and ym )
m m
Now,
*n
a
m
da
n
"m
dy
m
a'
m
1 + y
/
m
n
a =
m
+ y'J 
u + Y;>3
"n a (1 + 2y/ + y' ) a" (y' + 2y" + 2y//y/ + y/ )
a = m 'm 'm m 'm m 'm'm 'm
m
+ a'Y^(2y' 1)
mm m
d + y')
m
'n "n
Â£ 6 and B are similar ( substitute B's for a's)
m m m
101
2.) Dyad Disconnection
In contrast to dyad connection, it may be necessary
to verify dyad disconnection. This occurs when a Stephenson
sixbar linkage is driven by means of a dyad link (Figure
514b). Since a dyad analysis cannot be performed on a
Stephenson chain with a dyad link as crank, it becomes
necessary to drive the dyad through the fourbar through
its full range of motion (defined by dyad disconnection)
to make sure the dyad link .is cyclic. At the same time
the fourbar is checked throughout its range of motion for
change points. One additional check of relative dyad link
length is made to ensure that the appropriate dyad link is
rotating completely.
3.) NongFunction Reversal
Peculiar to Stephenson 2 sixbar linkages which are
driven by one of the two links connected to ground is a
necessary condition involving the first order influence
coefficient, the gfunction. It occurs because the driven
link may reach a limit in angular motion which is defined
by a higher order geometry constraint. Since it is impos
sible to analyze a Stephenson 2 directly using a dyad
based analysis, the test is performed by inverting to
a Stephenson 3 and driving the linkage at the fourbar
chain. If ternary link 3 is the driven member in the orig
inal Stephenson 2, the offending position occurs
218
Program Chart 4
Problem #4 Angular Coordina
of Two Cranks.
Linkage Solution FourBar
Mode of Cyclicity #3
Synthesis Procedure
1) INVERSION to 2.
2) CURVATURE TRANSFORM to
find 3.
Necessary Condition Procedure
1) Necessary Condition Pro
cedure B for 1234.
Sample Problem
Files
VPR0B4 FBM0DE3
[1J INITIAL
[2] 31 3 3 SYNINP
[3] 32 34 SYNINP
[4] 37 SINVERT 32 31 1
15] 34 33 35 36 SCRVTRN 37
[6] SYNOUT 34 33 35 36
[7] NECINP
[8] 
38
NCPB 34
33
35
36
[9] 
NECOUT 38
[10] v
39
DESIMP
38
[ID 
39
DPVTLCA
39
33
36
35
34
[12]
39
DLKLNRT
39
33
34
34
35
35 36
36 33
[13]
39
DLKSIZE
39
33
36
[14] v
39
DESDYAD
39
36
33
35
34
0 0 0
0
1,
i 6
[15]* DESOUT 39V
31)
4MSP (2/1)
32)
4MSP (4/1)
33)
pivot A
34)
pivot D
35)
pivot C
36)
pivot B
37)
4MSP(4/2)
38)
NC Grid(1x3)
39)
DC Grid(1x3)
142
Stephenson 1 Link 1 is grounded.
Plane
J *
Link 5.
CURVATURE
TRANSFORM
to
find
3
Plane
K >
Link 2.
CURVATURE
TRANSFORM
to
find
4
Point
e =
Pivot B.
INVERSION
to 3 .
CURVATURE
TRANSFORM
to
find
6
[4MSP Grid
i: (3) xlD]
Stephenson
i 2
 Link 2 is
grounded.
Plane
J >
Link 1.
CURVATURE
TRANSFORM
to
find
3
Plane
K
Link 4.
CURVATURE
TRANSFORM
to
find
5
Point
e =
Pivot D.
INVERSION
to 3 .
CURVATURE
TRANSFORM
to
find
6
[4MSP Grid
( 3) xlD ]
Stephenson
3
 Link 3 is
grounded.
v
Plane
J *
Link 4.
*+ CURVATURE
TRANSFORM
to
find
Plane
K >
Link 5.
1,6
Point
e =
Pivot C.
CURVATURE
TRANSFORM
to
find
2
[4MSP Grid: 2D+lD]
38
12) Point Path Coordinated with Two Cranks. The
motion of a point and the rotation of two nonadjacent
bodies which are connected to a common ground are coordi
nated by attaching the point and two bodies to the links
of an appropriate sixbar linkage.
These problems and their possible linkage solutions
are presented in the charts of Appendix A. The steps in
volved in an actual synthesis procedure are discussed in
Chapter IV. Examples of synthesis procedures and neces
sary condition procedures are presented in Appendix B,
and the necessary conditions will be discussed in detail
in Chapter V.
66
function of problem type and linkage solution, but the
shape of the overall grid is a function of problem type
alone.
D. Dependent Syntheses
The previous example demonstrates an undesirable
property which will be hereafter termed "dependent syn
theses." Essentially, this property exists whenever a
synthesis procedure involves two (or more) curvature trans
form operations in such a way that the order in which they
are performed cannot be switched. In the previous example,
the second curvature transform uses the motion specifica
tions of plane 4, which does not exist until plane 4 has
been generated by (among other things) the first curvature
transform. Thus, every plane 4 generated by the first
synthesis will have a unique motion specification set for
the second synthesis to operate upon.
This property greatly complicates subsequent analysis
and optimization steps, and at this point these difficul
ties are circumvented by performing a complete optimiza
tion on the results of one portion of a dependent synthe
sis before proceeding to the second synthesis (see Appen
dix D, problem 12).
Fortunately, not all sixbar synthesis procedures
involve dependent syntheses. In fact, all of the problems
56
Where,
n
a .
mj
3n
m j
C = yr.
T nJ
.n Dn *r
S, = a w 6 .Y
1 mj mj'nj
n r *n
T, = a .y + 3
1 mj'nj mj
**n *n *r nn **r n *r 2
Sn = a 23 y 3 Y a .y .
2 mj mj'nj mjnj mj'nj
_ *r n *r 0n r 2, *n
T~ = 2a .y + a .y 3 Y + 3
2 mj'nj mj'nj mj'nj mj
"*n
S3 a
mj
0**n *r *n **r n ***r *n *r 2
33 Y 33 Y B Y 3a .Y .
mj'nj mj'nj mj nj mj'nj
n **r *r cn *r 3
3a .y .y + 3 Y
mj nj nj mj'nj
m ,**n *r *n *r 2 oran **r r , n r
T. = 3a .y 33 Y 33 Y Y + 3a .y .
3 mj'nj mj'nj mj'nj'nj mj'nj
n *r 3 n "*r ** n
a.y. + a y + 3
mj nj mj nj rnj
54
y* (y 1) +
O'/// rom ro
 !>
c.) Inversion formulation number 3.
Given: Motions of N relative to R; M relative to R.
Find: Motion of M relative to N ( for Rq reference ).
Independent parameter for given MSP: Yn^ Ym ( must be
related ).
Independent parameter for derived MSP: y^
Again, this formulation is identical to that of (2a),
except that the dots in the S^ and formulations should be
replaced with primes ( ) and now A' A", A"', B' B", B'",
C' 0." and must be defined.
Let a =
n
a =
m
r ^an
a ; a' = ; etc. ( same for $ and y )
n n j r Mn 'n
n
r
; etc. ( same for 3 and y )
r m m
m
dy
da
a ; a/ =
m m
*Y,
, r j2 r
dy d y
. / t n 1 n
^nm r ^nm r2 e c*
dy
m
dy
m
The formulations for A' A", Pt", B', B" and B'" are
equivalent to those of section (2b), under the above notation,
except that Ym+ Ynm, y'> y/_, etc., and C', C^and C^must
195
Module Chart 18
Name Necessary Condition Procedure J
Label NCPJ
Description
Evaluates synthesized four
bar I on the basis of change
point considerations within
path cognate fourbars II and
III as shown. Generates a
necessary condition grid
(2x2D) indicating acceptable
cognates.
APL Call
where:
(01) NCPJ
(ix), d2),
a8r
(i^, d2)
d3), d4)
d5),d6)
(I7) (Ig)
(0^
= link dO.
= link fO.
= link gO^ (and h02)
= link j02 (and KO3)
= Output grid.
141
Problem 10: DOUBLE COPLANAR SYNTHESIS OF TWO
ADJACENT PLANES
The coplanar motions
of Plane J and Plane
K (connected at pivot
e) are coordinated.
SOLUTION LINKAGES
C 4 H
WATT
Watt 1 Link 1 is grounded.
Synthesis Procedures
a.) Plane J
Plane K
Point e
 Link 4.
* Link 5.
= Pivot H.
CURVATURE TRANSFORM to find
2,3.
INVERSION to 2
CURVATURE TRANSFORM to find 6.
b.) Plane J
Plane K
Point e
y Link 5.
> I,ink 6.
= Pivot F
( 4MSP Grid: 2D+lD]
CURVATURE TRANSFORM to find 2.
INVERSION to 2.
CURVATURE TRANSFORM to find 4.
INVERSION to 1.
CURVATURE TRANSFORM to find 3.
( 4MSP Grid: (3)xlD]
center point cubic defining pivots F and H. A concep
tual reinversion (which requires no computational activity)
produces a Stephenson 2 sixbar linkage which satisfies
the motion specifications.
C. Grid Dimension
If 4MSP motion specifications are used, every opera
tion involving a curvature transform generates an infinity
of constraint links. If one constraint link is to be
selected from this set, a one dimensional grid is required
to discretely represent the solution set. Likewise, if
two constraint links are to be selected from the set, the
grid will have two dimensions. A curvature transform fol
lowed by a path cognate operation will generate (2) two
dimensional grids, since each fourbar (which occupies a
place on a twodimensional grid) has two path cognates,
a "left" cognate and a "right" cognate.
The example discussed in the last section produces
two (2) twodimensional grids representing link 5 and 7'
and another twodimensional grid representing links 1 and
6. The total, if these grids are combined, is two (2)
fourdimensional grids. This problem lies at one extreme
of the problems considered in this work. The opposite
extreme occurs for problem #'s 3 and 4, whose entire
linkage solution sets can be represented by onedimen
sional grids. The nature of the component grids is a
198
Module Chart 21
Name Necessary Condition Procedure M
Label NCPM
Description
Evaluates synthesized four
bar I on the basis of change
point considerations within
path cognate fourbars II and
III as shown. Generates a
necessary condition grid
(2x2D) indicating acceptable
cognates.
APL Call
"(0^ NCPM (I1),(I2), .... (Ig)"
where:
(I1) d2)
d3), d4)
u7), d8)
= link dO^
= link f02
= link gO^ (and h02)
= link jO^ (and KO^)
(0^) = Output grid.
100
N
Figure 5 14 DyadRelated Necessary Conditions
145
b.) Crank J > Link 2.
Point e = Pivot A.
Plane K Link 4.
t CURVATURE TRANSFORM to find
1,6.
INVERSION to 2.
CURVATURE TRANSFORM to find 5.
[4MSP Grid:2D+lD]
106
filter out inferior linkages before more expensive desir
able conditions are evaluated.
The five following desirable conditions are also
geometric linkage parameters which act as indexes of the
dynamic characteristics of linkages. They are the result
of detailed studies of cam system dynamics by Tesar and
Matthew [120] and were recently applied to Geneva mech
anisms by Taat [121] and Taat and Tesar [122].
4.) gfunction
The gfunction is a first order geometric derivative
relating output and input motions. It is a reliable in
dicator of dynamic properties related to output veloci
ties, kinetic energies, force magnifications and input
torque requirements. For sixbar problems it often be
comes desirable to monitor the maximum values of the g
functions of two output links.
5.) Ah
The hfunction is a second order geometric derivative
relating output and input motion. It is a good predictor
of dynamic properties related to inertia forces and in
ternal deformation. Here the interest is in monitoring
peaktopeak values.
6.) hrms
The rootmeansquare of the hfunction over a cycle
93
Proof of (Hi) : Referring to Figure 513b,
a + $ + 0 = a+ $ + 0 = 2 tt
A A B B
9 = 0 because 0,ged is a parallelogram,
_L
= 0 at simultaneous change
points for g in fourbar II and d in fourbar I.
Proof of (iv): Referring to Figure 513c, it follows
from pathcognate property (ii) that change points occur
simultaneously at 0^ in fourbar II and at 02 in fourbar
III. In this position C^h and 0203 become collinear.
Therefore 0^^ + 6 = and 0^ + 6 = 0^
where 0 refers to "the angle of ..."
Since C^hef is a parallelogram, fe and O^h are paral
lel. It follows that:
80102 6df SO 12
df
Proof of (v): Referring to Figure 513d, it follows
from path cognate property (iii) that pivot j in fourbar
II and pivot k in fourbar III will pass through change
points simultaneously. In this position O^J and O^g are
collinear, as are O^k and O^h. Due to the parallelograms
and similar triangles, it follows:
ge and eh are collinear and
O^d and eh J J C>2f
ld II 2f
ge
225
Program Chart 11
Problem #11 Coplannr Motion
Coordinated With a Crank.
Linkage Solution Stephenson 2
Mode of Cyclicity #4
STEPHENSON 1
Synthesis Procedure
1) CURVATURE TRANSFORM to
find 5.
2) INVERSION to 3.
3) CURVATURE TRANSFORM to
find 1,6.
Sample Program
VPR0B11STEPH2MODE4
[1] * INITIAL
[2] * 31 SYNINP
[3] 32 34 SYNINP
[4] * 36 3 5 SCRVTRN 31
[5] * 40 SINVERT 32 31 1
[6] * 3 7 3 3 38 39 SCRVTRN 4 0
[7] * SYNOUT 3 6 3 5
[8] > SYNOUT 37 33 38 39
[9] * NEC INP
[10]* 41 NDYDTYP 36 35 34 2
[11]* 42 NCPH 37 33 38 39
[12]* 43 NECDYAD 37 33 38 39 36
35 34 40 2 1 7
[13]* NECOUT 43
Necessary Condition Procedure
1)
Necessary Condition
Type (25).
Dyad
2)
Necessary Condition
cedure H for 1346.
Pro
3)
Verify Connection of
25.
4)
Verify nongfunction
Reversal (5/2).
Files
31)
4MSP (4 /2 )
32)
4 MSP (3/2)
33)
Pivot F
34)
pivot A
35)
pivot B
36)
pivot C
37)
pivot D
38)
pivot G
39)
pivot H
40)
4MSP (4/3)
41)
NC Grid (5)
42)
NC Grid (1,6)
43)
NC Grid (1,6,5)
44)
DC Grid (1,6,5)
[14] 44 DESINP 43
[15] 44 DPVTLCA 44 34 35 36 37 33 38 39
[16] 44 DLKLNRT 44 36 38 38 39 39 34
[17]* 44 DLKSIZE 44 37 36 33 34
[18]* 44 DESDYAD 44 37 33 38 39 36 35
34 40 2 i 6
[19]* 44 DESOUT 44 V
110
This approach has the advantage that n can be chosen to
assign performance scores in a nonlinear fashion within
the acceptable region. Note that 6.3 is equivalent to 6.2
when n=l and a factor of 10 is included.
The approach used in this work is the result of a
combination of these concepts and was introduced in the
author's thesis [23]. The designer is asked to specify
for each desirable condition an "ideal", "preferable" and
"acceptable" performance, to which scores of 0, 1 and 10
are attached respectively. Any linkage which falls out
side of the "acceptable" bounds is discarded. A smooth
exponential can be fitted to these performance specifica
tions and linkage scores evaluated on the basis of the ex
ponential :
S, =
Log (P3/P1)
where = difference between preferable
and ideal performance
?2 = difference between actual and
ideal performance
P^ = difference between acceptable
and ideal performance
Figure 61 shows scoring as a function of relative
placement of ideal, preferable and acceptable regions.
169
a) Necessary Condition Pro
cedure H for 1346.
b) Verify Disconnection of 25.
(2 short) .
a) Necessary Condition Dyad
Type (25).
b) Necessary Condition Pro
cedure A for 1346 .
c) Verify Connection of 25.
d) Verify Nonencirclement (AC).
a) Necessary Condition Dyad
Type (25 ) .
b) Necessary Condition Pro
cedure A for 1346.
c) Verify Connection of 25.
d) Verify Encirclement (AC) .
8)
a) Necessary Condition Dyad
Type (25 ) .
b) Necessary Condition Pro
cedure B for 1346 .
c) Verify Connection of 25 .
d) Verify Nonencirclement (AC) .
a) Necessary Condition Dyad
Type (25 ) .
b) Necessary Condition Pro
cedure B for 1346.
c) Verify Connection of 25.
d) Verify Encirclement (AC).
63
of appropriate combinations of the four tools of syn
thesis just presented, and are generally not unique for
any given problem type and linkage solution.
A Stephenson 2 solution to problem number twelve is
shown in Figure 47. The synthesis procedure utilizes
the tools of inversion, path cognates, and (as always)
the curvature transform. The procedure begins with a
combination of the translational motion specifications
of point e and the rotational motion specifications of
crank K to create a new motion specification set describ
ing general coplanar motion (Figure 47a). Operating on
these specifications with the curvature transform will
produce constraint links as shown in Figure 47b (actu
ally a set of constraint links is produced). Operating on
this preliminary linkage solution with path cognate analyt
ics will produce constraint links 7' and 5. These constraint
links will move the coupler, link 4, in a way such that
coupler point e will move on its specified path in coordi
nation with the specified angular motion of crank link 5.
Now, the motion specifications exist for links 4', 5 and 3
relative to fixed link 2. Changing the reference to link 3,
it is possible through inversion analytics to define a new
motion specification set for plane 4 relative to 3 (Figure
47d). Operating on these motion specifications with the cur
vature transform produces constraint links 1 and 6, with
the circle point cubic defining pivots D and G and the
172
a) Necessary Condition Pro
cedure M for 2456.
b) Necessary Condition Pro
cedure H for 1234.
a) Necessary Condition Pro
cedure M for 2456.
b) Necessary Condition Pro
cedure G for 1234.
a) Necessary Condition Pro
cedure N for 2456.
Cond'
4)
43
All of the previously mentioned work involving ISP
and MSP involved geometric derivatives, where the indepen
dent parameter is the angular orientation of the body.
The specifiable higherorder motions are consequently
geometric (slope, curvature, inflection, etc.). In 1974
Myklebust [115] presented a unified theory making dynamic
(i.e., time dependent) higher order properties specifiable.
This is termed "time state synthesis." For the fourbar
problems the dynamic properties are equivalent to the geo
metric properties, but for a sixbar linkage the use of
an equivalent independent parameter for the decoupled
subsections of a linkage solution is a necessity. Use
of time state synthesis allows true "kinematic synthesis"
to be carried out, as conditions can be prescribed for
position and its time derivatives (velocity, acceleration,
etc.) or combination of inertia and momenta effects, up
to the fourth order.
Formulation of the curvature transform
Consider the moving plane in Figure 43 where motion
is described by the coordinates a, 3 and y. E is the fixed
reference with orthogonal coordinates U and V attached to
it, and E is the moving reference with orthogonal coordi
nates u and v attached. They are related by the "coordi
nate transform":
U = u cos y v sin y + a
(4.1)
V = u cos Y v cos Y + B
95
Properties (ii), (iii), (iv) and (v) are represented
pictorially by Figure 511. With this theory, necessary
condition procedures for path cognate problems can be
created. Waldron and Stevenson [119] have performed an
analysis similar to that used in the first and third of
these procedures.
Necessary Condition Procedure I.
Follow necessary condition procedure C for
fourbar I (tests for a Grashof doublerocker) .
Path cognates II and III will be usable crank
rockers with cranks at O^g.
Necessary Condition Procedure J.
Follow necessary condition procedure A for
fourbar I (tests for a crankrocker) If C^f
is the crank in I, fourbar II will be a usable
crankrocker with a crank at O3j: if Opd is the
crank in I, fourbar III will be a usable crank
rocker with a crank at O^k.
Necessary Condition Procedure K.
Follow necessary condition procedure B for four
bar I (tests for a draglink) Path cognates II
and III will be usable draglinks.
Necessary Condition Procedure L.
1) Perform necessary condition procedure D on
fourbar I to indicate change points at d (and
g in cognate III).
2) Perform necessary condition procedure D on
fourbar I to indicate change points at f (and h
in cognate III).
Necessary Condition Procedure M.
1) Test the distance of f and 0? relative to O.d
for the specified positions. If the "altitude" of
f is consistently lesser or greater, no change
points will occur at j in cognate II or k in
cognate III.
9
an analog computer. NechL [30] also used an analog simula
tion, using a combined relaxation and gradient modifica
tion of the conjugate gradient approach to circumvent the
problem of instability near the optimum. Sixbar func
tion generators were optimized by Chen and Dalsania [31]
by applying a leastsquares gradient method.
Newton's method in its unmodified form is not often
used for linkage problems, largely due to difficulties as
sociated with the selection of initial guesses. One appli
cation for this method was found by Rose and Sandor [32],
who minimize structural error of fourbar function genera
tors by equalizing error between precision points and at
the ends of the specified range. The resulting formulation
produces ten nonlinear differential equations to which
Newton's method is applied to find a solution.
The socalled "quasiNewton" method, also called the
"variable metric" method, is basically a gradient tech
nique which involves the formulation of a differential ma
trix which approximates the Hessian as the minimum is
reached. Davidson, Fletcher and Powell [33] introduced
this method in 1963, and MclaineCross [34] used it in
1969 to optimize a crankrocker fourbar to generate solar
declination. This formulation, which involved five non
linear differential equations, allowed for some error
tolerance at the precision points. Optimization of mech
anisms with flexible links was discussed by Sevak and
201
Module Chart 24
Name Necessary Condition Output
Label NECOUT
OUTPUT
Description
Displays grids generated by
previous necessary condition
modules. Collects expansion
input for the selected portion
of the grid.
APL Call
"+NECOUT (I ),(I ),
where:
(Ij^) (I2) .... = grids to be displayed.
91
a. Prescribed Path,and Input Crank
m
Figure 510 Use of Path Cognates to Solve the Path
Crank Coordination Problem
160
a) Necessary Condition Pro
cedure H for 1234 .
b) Necessary Condition Pro
cedure H for 2456.
a) Necessary Condition Pro
cedure A for 1234.
b) Necessary Condition Pro
cedure D for 2456.
c) Verify connection of 56.
a) Necessary Condition Pro
cedure B for 1234 .
b) Necessary Condition Pro
cedure D for 2456.
c) Verify connection of 56.
a) Necessary Condition Pro
cedure B for 1234.
b) Necessary Condition Pro
cedure D for 2456.
c) Verify connection of 56.
a) Necessary Condition Pro
cedure A for 1234.
b) Necessary Condition Pro
cedure B for 2456.
a) Necessary Condition Pro
cedure A for 1234.
b) Necessary Condition Pro
cedure A for 2456.
128
Problem #1: PATH SYNTHESIS
The coplanar motion
of point e is con
trol led.
SOLUTION LINKAGE
FOURBAR
Moving body/Linkage Component Match
FourBar
Synthesis Procedure
Link 1 is grounded
Point e * Link 3.
*t CURVATURE TRANSFORM to
find 4,2.
f 4MSP Grid: 20]
14
Lindholm [87] optimized sixbar Watt1 function generators.
Hobson and Torfson [88] introduced a theoretical method
which involves analysis of the centrodes of specified
motion.
Graphic synthesis and analysis packages have been de
veloped and bear some consideration in any discussion of
linkage optimization due to their high level of human/
computer interaction, which allows convenient adjustment
of representations of "good" designs to achieve "better"
designs. Reed and Garrett [89] and Smith and Reed [90]
have introduced IMAGE (Interactive Mechanism Analysis
through Graphic Exchange) and Ricci [91] has introduced
SPACEBAR. Perhaps the most powerful is KINSYN III, intro
duced by Rubel and Kaufman [92], All of these employ
highlevel graphics coupled with linkage synthesis and
analysis routines.
The linkage optimization problem has been discussed
on a philosophical level by Johnson [93], who discusses
stimulants and aids for creativity, such as a building
block approach, and systematics of linkages and circuit
diagrams. Srivastava and Newcombe [94] introduce a multi
factor objective function including such things as pivot
location, bearing load limitations, displacement, velocity
or acceleration constraints and Grashof constraints.
Sutherland and Siddall [95] also use a multifactor objec
tive function with inverse utilics for spherical function
130
Problem #3: ANGULAR COORDINATION OF A TWO LINK CHAIN
The rotation of crank
K is coordinated with
the coplanar motion
of attached link J.
SOLUTION LINKAGE
FOURBAR
FourBar
Synthesis Procedure
Link 1 is
Plane J 
Point e =
Crank K =
Point f =
grounded.
Link 3 and
Pivot B.
Link 2 and
Pivot A.
*t CURVATURE TRANSFORM to
find 4.
[4MSP Grid: 1D]
35
C. Motion Specification Types
The types of coplanar problems which are solvable
and optimizable using the procedures developed in this
work are shown in Figure 39. The first and second prob
lems, coplanar path and motion syntheses, are the subject
of most linkage optimization efforts.
1) Path Synthesis. The motion of a point is guided
by attaching the point to the coupler of an appropriate
fourbar linkage.
2) Motion Synthesis. The motion of a body is guided
by attaching the body to the coupler of an appropriate
fourbar linkage.
3) Angular Coordination of a TwoLink Chain. The
angular motion of two connected bodies, one of which is
connected to ground, is controlled by attaching these
bodies to an appropriate fourbar linkage.
4) Angular Coordination of TwoCranks. The angular
motions of two unconnected bodies, both of which are con
nected to a common ground, are coordinated by attaching
these bodies to an appropriate fourbar linkage.
5) PathCrank Coordination. The motion of a point
is coordinated with the rotation of a body connected to
ground by attaching the point and body to an appropriate
fourbar linkage.
6) Angular Coordination of ThreeLink Chain. The
85
Y
or
Y
02'
Y
03
y
10'
Y
12'
Y
13
y
20'
Y
21'
Y
23
Y ,
30'
Y
31'
Y
33
have the same sign.
ii.) Determination of "compatible" links. A "com
patible" constraint link is one which can be used in com
bination with a permissible constraint link so that the
4MSP will be satisfied on the same branch. The previous
analysis is used to eliminate portions of the cubic which
represent constraint links which will definitely pass
through a change point at the moving pivot while satisfy
ing the 4MSP (regardless of the other constraint link
chosen). A permissible constraint link which has been
selected from a portion of the permissible circle point
cubic might still pass through a change point if coupled
with an "incompatible" constraint link.
Let 0 A be a selected permissible constraint link.
cl
Figure 58a shows one in the four positions in which the
motion specifications are satisfied. defines the rela
m
tive coupler rotation in reaching position j from position
0. Figure 59b shows 0 A in the zeroth position with
angles Y^, "^02 an<^ ^03 drawn through point A. In
this example, Y^ and Y ^ represent the extremes of the
angular range. Now, if a circle point representing a pos
sible second constraint link lies in region Z, it must cross
60
Figure 4
a. Prescribed Path,and Input Crank
5 Use of Path Cognates to Solve the Path
Crank Coordination Problem
107
of operation is an indicator of system precision and in
ternal bearing loads. It is desirable to minimize the
overall rootmeansquare of the hfunction.
7.) Torque
The functional product of the first and secondorder
geometric derivatives throughout a cycle (gh) is indicative
of the driving torque needed to maintain a constant input.
The peaktopeak value is predictive of speed fluctuation
or dynamic windup at the input.
8.) Input Shock Level
A combination of first, second and thirdorder geo
2
metric derivatives given by the relationship T=gh'+(h)
is an index of shocklevel felt at the input of a system.
Higher values will be indicative of increasing significance
of dynamic properties related to vibration, noise, wear
and backlash.
Desirable conditions 3+8 must rely upon a dyadbased
analysis, and are evaluated simultaneously for each solu
tion linkage. Higherorder geometric properties are eval
uated using a finitedifference differentiation of the
zeroth order motion analysis. Taking advantage of the
dyadanalysis is a final desirable condition unrelated to
system dynamics:,
61
pathcrank coordination (Figure 45a). The synthesis
procedure is: 1) specify the motion of coupler def by
using the translation of point e and the rotation 0,
use the curvature transform to synthesize fourbar O^dfC^,
and find its path cognate O^gjO^ or C^hkO^. Either cognate
will coordinate the rotation of a crank parameter 0
(attached to link O^d or C^h) with the motion of point e
(attached to coupler link gej or hek, respectively).
4.) Synthesis Tool #4. Angular Cognates
Consider the parallelogram formed by links BCD and
BC'D in Figure 46c. Regardless of the motion of point B
(guided by crank AB for this example) the angular param
eters of links BC and C'D are identical. So too for links
BC' and CD. Figure 46 outlines the use of the concept of
angular cognates to solve a crankcoordination problem.
The second angular parameter is used to define the mo
tion of coupler plane BC, the curvature transform is used
to find a second constraint link CD, and the angular cog
nate of dyad BCD is taken to provide solution linkage
ABC'D.
B. Coordination of the Tools of Synthesis
Appendix A contains tables of linkage solutions and
recommended synthesis procedures for each of the twelve
problem types. These synthesis procedures consist entirely
80
"trailer" signify constraint links with limited rotation
at the moving pivot and the name "rocker" signify con
straint links with limited rotation at the fixed pivot
(Figure 55). For example, the follower of a crankrocker
fourbar is both a trailer and a rocker, but a constraint
link of a draglink four bar is a trailer only. Trailers
defined by circle and centerpoint cubics which have mov
ing pivot rotation less than tt for the 4MSP are termed
"permissible". The first technique of Waldron is used
to determine sections of the circle point cubic which
represent permissible constraint links.
Let be the rotation of the coupler relative to
the constraint link in moving from positions i to j. This
angle will be defined between tt and tt, clockwise positive.
Let n denote the position with the smallest or most nega
tive value of 'h j so that it defines one end of the angular
range. If is ^ 0 for all possible values of j, the
angular range must be less than tt. If it can be estab
lished that for some value k, is always either positive
or negative, the constraint link under consideration must
be permissible.
The following property can be used to find the sign
of y
Jk
A line through the image pole P.. and
circle point A bisects the angle^ T
(see Figure 56a) .
This property applies to any where k is another posi
tion, so the angle 'F .. can be found from:
1K
26
C 4 H
C 4 H
Figure 33 Inversions of the Watt Chain
196
Module Chart 19
Name Necessary Condition Procedure K
Label NCPK
Description
Evaluates synthesized four
bar I on the basis of change
point considerations within
path cognate fourbars II and
III as shown. Generates a
necessary condition grid
(2x2D) indicating acceptable
cognates.
APL Call
where:
(01) NCPK
dx), d2),
d8r
d^, d2)
d3), d4)
(15), d6)
(I7),(Ig)
(ox)
= link dO,
= link fO.
= link gO^ (and h02)
= link jO^ (and KO^)
= Output grid.
3
procedure to optimally approximate a given function using
the precision points as design variables.
In the mid 1960's the mechanical design field began
to feel the real impact of modern optimization theory,
which involved numerical methods initially developed to
solve management science and control system problems.
Initial application to mechanical designs was primarily
in the fields of structural mechanics and aeronautics, but
mechanism design was not far begind. The development of
such high level languages as Fortran, Algol, APL and Basic
have made the use of the computer much simpler, more ef
fective and less system dependent for the engineering com
munity .
A general expression for the linkage optimization
problem is:
Minimize Objective Function F(x)
where x = (x.,x . ., x )
i ^ P
Subject to:
Inequality constraints h^ (x) <_ 0; k1 to m
Equality constraints e^(x) = 0; j=l to n
A condition for the existence of an optimization prob
lem is that the number of design parameters must exceed
the number of constraining equations, in order to ensure
the existence of "free" parameters.
94
To use (iv), we wish to ensure that 0^ in fourbar II
does not pass through its change point by verifying that
link df never becomes parallel to link 0^C>2 furbar 1
while moving in such a way as to satisfy the four specified
positions. Every fourbar chain, regardless of Grashof
type, has two positions in which two opposite links become
parallel. The total motion of the fourbar can be concep
tually divided into two segments delimited by these crit
ical positions. The test to be performed, therefore, is
a verification that all specified positions lie on the same
motion segment.
Since pivots d and f are equidistant from a line
through 0^ and 02 at the critical positions, it follows
that throughout one motion segment d is farther away from
0^C>2 than f, and throughout another motion segment d is
closer. If the "altitude"of d is consistently closer or
farther from 0^C>2 than f in all specified positions, then
fourbar II will not have a change point at 0^ (and four
bar III will not have a change point at C>2) while satisfy
ing the specified positions.
To make use of (v) similar arguments can be made.
If the "altitude" of f (relative to O^d) is consistently
lesser or greater than the "altitude" of C>2 for all speci
fied positions, pivot j in cognate fourbar II and pivot
k in cognate fourbar III will not have change points in
moving through the specified positions.
=0,1,2,3,4
b. Infinitesimally Separated Positions
(ISP)
a. Finitely Separated Positions c. Multiply Separated Positions
(FSP) (MSP)
Figure 42 Finitely, Infinitesimally and Multiply Separated Positions
ro
37
motion of three bodies connected in a chain (one connected
to ground), is guided by attaching these bodies to the links
of an appropriate sixbar linkage.
7) Angular Coordination of a Crank and a TwoLink
Chain. The motion of two connected bodies (one connected
to ground) is coordinated with the rotation of another body
which is connected to ground by attaching these bodies to
the links of an appropriate sixbar linkage.
8) Angular Coordination of ThreeCranks. The rota
tion of three nonadjacent bodies, all connected to ground,
is coordinated by attaching these bodies to the links of
an appropriate sixbar linkage.
9) Angular Coordination of Three Adjacent Planes.
The motion of two nonadjacent bodies and a third body
which is connected to ground and to which the first two
bodies are connected, is controlled by attaching these
bodies to the appropriate links of an appropriate sixbar
linkage.
10) Coplanar Motion Synthesis of Two Adjacent Planes.
The general motion of two connected bodies is controlled
by attaching these bodies to the links of an appropriate
sixbar linkage.
11) Coplanar Motion Crank Coordination. The
motion of a body is coordinated with the rotation of a
body connected to ground by attaching these bodies to the
links of an appropriate sixbar linkage.
105
pivots may be prespecified but the location of others
will vary throughout the solution set. If some latitude
is not available in these locations the designer must re
duce his specification. Otherwise he may use the 4MSP
motion specifications and treat location of the variable
pivots as an optimization criterion.
2.) Link Length Ratio
The link length ratio is simply the length of the
longest specified link divided by the length of the short
est specified link (i.e., LLR>1) and is indicative of dynam
ic characteristics of a linkage. Previous optimization
schemes dealing with fourbar coplanar motion problems
[20,23] have automatically considered all four links in
this ratio. For sixbar problems it becomes important to
allow the designer to specify the links of particular interest.
3.) Link Size
Kinematic and dynamic characteristics may be func
tion of link sizes relative to overall motion parameters.
Again, some of the links may be prespecified, but all
problems have variable links.
The link length ratio and link size criteria are both
simple geometric linkage parameters which can provide in
sight into the dynamic characteristics of a linkage. Al
though they are not particularly reliable, they are use
ful because they are computationally inexpensive and can
179
Module Chart 2
Name Synthesis Input
Label SYNINP
Description
Interactively collects
input for synthesis routines.
APL Call
">(01) (02) SYNINP
where:
(01) = 4MSP set associated with E
(O^) = Point e. Optional, used if E rotates
as a crank and e is a prespecified
pivot.
239
119. Waldron, K.J., and E.N. Stevenson, Jr., "Elimina
tion of branch, Grashof and order defects in path
angle generation and function generation synthesis,"
to be presented at the ASME Design Engrg. Tech. Conf.
Minneapolis, Minn., September 2427, 1978.
120. Tesar, D., and G.K. Matthew, The Dynamic Synthesis,
Analysis and Design of Modeled Cam Systems, Lexing
ton Books, 1976.
121. Taat, M., "The fundamental development of design
procedures for a full spectrum of linkage dwell
mechanisms," PhD. Dissertation, University of
Florida, 1978.
122. Taat, M., and D. Tesar, "A new interpretation for
the dynamic phenomena associated with Geneva mech
anisms," to be presented at the ASME Design Engrg.
Tech. Conf., Minneapolis, Minn., September 2427,1978
123. Pollock, S.F., "Dynamic model formulation programmed
for dyad based machines," M.S. Thesis, University
of Florida, 1975.
124. Sarkisyan, Y.L., K.C. Gupta, and B. Roth, "Kinematic
geometry associated with the leastsquare approxima
tion of a given motion," Journal of Engineering for
Industry, Trans. ASME, Series B, Vol. 95, No. 2,
May 1973, pp. 503510.
Sutherland, G.H., "Mixed exactapproximate planar
mechanism position synthesis," Journal of Engineer
ing for Industry, Trans. ASME, paper no. 76DET29.
125.
CHAPTER V
THE NECESSARY CONDITIONS
Linkages generated through the use of the kinematic
synthesis techniques of Chapter IV are definitely capable
of satisfying the specified positions. Unfortunately, the
analytics presented there take no consideration of linkage
performance in positions other than those specified. Real
linkages may fall into a number of physical pitfalls, any
one of which is sufficient to eliminate the linkage as a
viable solution to the specified motion problem. The
analytics now presented are for "necessary conditions"
which must be satisfied before a given solution linkage
can be considered for further optimization analysis. The
conditions which apply to a given problem are dependent
upon the solution linkage type, the means by which the
linkage is to be driven and the nature of its motion (i.e.,
mode of cyclicity). It is primarily because of the depen
dence of the necessary conditions upon mode of cyclicity
that the concept of the latter has been developed.
A, FourBar Necessary Conditions
1.) Order
It is entirely possible that a linkage solution will
71
73
Figure 51 Order
117
1) Look through Appendix A, find the problem type
applying to his problem, and select a solution linkage
type.
2) Familiarize himself with the synthesis procedure,
select appropriate synthesis modules from Appendix C,
and outline the synthesis portion of his program:
V PROBLEM
INITIAL,
SYNINP
SCRVTRN
SPTHOXT
Input
Tools of Synthesis
SYNOUT } Output
3)Select from available modes of cyclicity (Appen
dix B and Chapter IV), familiarize himself with the
necessary condition analysis, and through use of Ap
pendix C write the necessary condition portion of the
program:
NECINP} Input
NCPK j
NCPA Necessary Condition Procedures
NECOUT} Output
4)Select from the available desirable condition
modules (or write his own for additional criteria)
and write the desirable condition portion of the
program:
DESINP} Input
DLKLNRT,
DESDYAD^Desirable Conditions
DESOUT} Output
220
Program Chart 6
Problem It 6 Angular Coordination
of a Three Link Chain.
Linkage Solution Watt 1
Mode of Cyclicity K8
Synthesis Procedure
Necessary Condition Procedure
1)
CURVATURE
TRANSFORM
1)
Necessary Condition
Pro
to find 3.
cedure D for 1234.
2)
INVERSION
to 2.
2)
Necessary Condition
Pro
3)
CURVATURE
TRANSFORM
cedure D for 2456.
to find 5,
3)
Verify connection of
56.
Sample Program
Files
VPROB6WATT1MODE8
[1]
INITIAL
[2]
V
31
34 SYNINP
13]
V
32
40 SYNINP
[4]
X
33
37 SYNINP
[5]
b
40
34 38 39 SCRVTRN
32
40
34
[6]
b
41
SINVERT 33
31
1
[7]
b
42
SINVERT 0
31
3
(8]
V
37
40 36 35 SCRVTRN
41
37
40
(9]
>
SYNOUT 38 39
40
34
[10]
b
SYNOUT 36 35
37
40
(11)
V
NECIMP
[12]
b
43
MCPB 38 39
40
34
(13]
4 4
MCPD 36 35
37
40
1
[14]
V
45
NECDYAD 39
34
38
4 0
37
36
35
42
1 1
[15]
V
NECOUT 45
1 16]
4 6
DES INP 4 5
( ) 7]
4 6
DPVTLCA 46
34
3 9
38
40
36
[18]
b
46
DLKI.NRT 4 6
37
36
38
39
[19]
V
46
DLKSJZE 46
38
40
[20]
V
46
DESDYAD 46
39
34
38
40
37
36
35
42
1,
i 6
[211* DESOUT 4 6 V
31) 4MS P (2/1)
32) 4 MSP (4/1)
33) 4 MSP (5/1)
34) pivot A
35) pivot G
36) pivot F
37) pivot H
38) pivot C
39) pivot B
40) pivot D
41) 4MSP (5/2)
42) 4MSP (1/2)
43) NC Grid (3)
44) NC GRID (6)
4 5) NC Grid (3,6)
46) DC Grid (3,6)
127
Necessary Condition Analysis covered in
Appendix B.
Independent syntheses.
"PATH COGNATES"
"find 2"
"INVERSION to 4"
Tools of synthesis are cap
italized.
Motion of Link 2 is derivable
for 4MSP, even though link
shape may not. yet be com
pletely definable (if E is a
ternary link).
4 is the new reference link.
7'
[4MSP Grid: (2)xlD]
The apostrophe indicates that
link 7' is temporary and does
not appear on the linkage
drawing.
The grids produced by a 4MSP
synthesis and necessary con
dition analysis will take the
shape of . (here, (2)
1dimensional grids). For
path cognate problems, "L,R
2D" indicates that 2dimen
sional grids for both "left"
and "right" cognates are gen
erated.
146
Problem #12: POINT PATH COORDINATION WITH TWO CRANKS
The coplanar motion
of point e and the
rotations of cranks
J and K are coordi
nated .
SOLUTION LINKAGES
WATT
Watt 1 Link 1 is grounded.
a.) Crank J * Link 3 and
Point f = Pivot B.
Crank K ~y Link 2.
Point e on Link 5.
Synthesis ProcedureÂ£
CURVATURE TRANSFORM to find
4,6.
INVERSION to 1.
CURVATURE TRANSFORM and PATH
COGNATES to find 3.
I 4MS P Grid:L,R 2D, and
10 J
217
Program Chart 3
Problem if 3 Angular Coordination
of a TwoLink Chain.
Solution Linkage FourBar
Mode of Cyclicity H1
B 3 *
1A
A 777 /// D
1
FOURBAR
Synthesis Procedure
Necessary Condition Procedure
1) CURVATURE TRANSFORM to
find 4.
1) Necessary Condition Pro
cedure D for 1234.
Sample Program
VPR0B3FBM0DE1
[1] * INITIAL
[2] * 31 33 SYNINP
[3] v 32 36 SYNINP
[4] * 36 33 35 34 SCRVTRN 32 36 33
[5] * SYNOUT 35 34 36 33
[6] * NECINP
[7] * 37 NCPD 35 34 36 33 1
18] * NECOUT 3 7
[9] 38 DESINP 37
[10]* 38 DPVTLCA 38 33 3C 35 34
[11]* 38 DLKLNRT 38 35 34 36 33
[12] 1 38 DLKSIZE 38 34 33
[13]* 38 DESDYAD 38 36 33 35 34 ,
0 0 0 31 2, i 6
[14] DESOUT 38V
Files
31) 4MSP(2/1)
32) 4MSP(3/1)
33) pivot A
34) pivot D
35) pivot C
36) pivot B
37) NC Grid (4)
38) DC Grid (4)
59
The f ormulatiorfS for an, 3 and
m' m
equivalent to those of section (2c),
except that y
,n
m
w
y +
'm
Y/ etc,
ran
their derivatives are
under the above notation,
Now,
n
mn
1 + Y
/
mn
Yn =
mn
(1 + Y' )
mn
"
Yn =
y'" (1 + y' )
'mn 'mn
3y" '
' mn
d + Y' )
mn
3.) Synthesis Tool #3. Path Cognates
The Roberts Chebyshev theorem states that three dif
ferent coplanar fourbar mechanisms will trace identical
couplercurves. An example is shown in Figure 45b,
where coupler point e has the same motion whether carried
by fourbar O^dfC^, fourbar C^gjO^ or fourbar C^hkO^.
An additional property is that the angular parameters of
those links with identical hash marks in Figure 45b are
identical for any reachable position or position deriva
tive of point e.
These properties can be used to solve the problem of
CHAPTER I
INTRODUCTION
The level of interest in the application of optimiza
tion methods to mechanism design seems to be rising expo
nentially, with the great majority of effort coming from
the past decade. The ultimate goals are means to achieve
mechanism designs which are globally optimum relative to
all pertinent criteria.
Some early attempts to optimize were those of da Vinci
(14521519) and Newton (16421727), who were both involved
in specific mechanical design problems, and made modifica
tions of previous designs in order to enhance their per
formances. Galileo developed a rational mechanical opti
mization in 1638 by creating an algorithm for the form of
a bent beam for uniform strength.
The advent of the differential calculus was fundamen
tal to the development of many modern optimization tech
niques. The introduction of elegant evaluations of maxima
and minima of differentiable functions eliminated the need
for discrete evaluations of these functions.
The development of variational techniques is largely
credited to Bernoulli, Lagrange and Euler. Chebyshev's [1]
involvement in straightline linkage design led him to de
velop his well known "Chebyshev polynomials" and optimality
1
Copyright 1978
by
Kim Loring Spitznagel
13
another criterion is to be considered, it is generally
related to transmission angle. Lin [61], Hamid and Soni
[62,63], Shoup, Staffer and Weatherford [64], Freudenstein
and Primrose [65], Sutherland and Roth [66], Bagci [67],
and Savage and Suchora [68] have all dealt with optimiza
tion relative to transmission angle. Optimization with
respect to dynamic criteria is also a field of interest,
with contributions coming from Tomas [69], Benedict and
Tesar [70], Berkof and Lowen [71,72], Conte, George, Mayne
and Sadler [36], Tepper and Lowen [73], Sadler [74],
Elliot and Tesar [75], and Sandler [76]. Optimization of
more general mechanical systems such as planetary gear
reducers and a shaping mechanism are discussed by Golinski
[77], and Osman, Sankar and Dukkipati [78] have treated
optimization of transmission gears. Optimization using
the leastsquares method of Gauss has been done by ChiYeh
[79] and Mansour and Osman [80]. In 1971 Bagci and Prasad
[81] and Bagci and Parekh [82] dealt with optimization
schemes for coplanar and spherical fourbar and sixbars
used for rigid body guidance. Speckhart [83] used minimum
weight and the cost criterion as his objective function.
Khan, Thornton and Willmert have developed "optimality
criterion techniques" for minimizing weight in linkages
subject to restrictions in stress and natural frequency [84]
and stress and deformation [85]. Rao and Ambekar [86] op
timized spherical RRRR function generators, and Sallam and
24
B 3 C
a. The FourBar Chain
D 4 C
b. The Stephenson SixBar Chain
C 4 H
c. The Watt SixBar Chain
Figure 31 FourBar and SixBar Chains
CHAPTER III
SOLVABLE PROBLEMS
The previous chapter described an optimization phi
losophy upon which the efforts of this work are based.
In this chapter the range of problems which are solvable
using this philosophy (and subsequently developed analytics)
is discussed. The range of solvable problems is defined by
the three concepts of linkage type, mode of rotational
cyclicity and motion specification type.
A. Linkage Types
The simplest nonlinear mechanism is the fourbar
linkage, as shown in Figure 3la. It possesses one degree
of freedom after one of its links has been grounded, and
is conceptually identical in all of its inversions, i.e.,
regardless of which link is grounded. The fourbar pro
vides eight free parameters.
Two types of singledegree of freedom sixbar chain
exist, the Stephenson and the Watt, as shown in Figure
3lb,c. They are both composed of two ternary links and
four binary links, and have different conceptual inver
sions, as shown in Figures 32 and 33. Each one provides
23
170
2)
*
Necessary Condition Procedures
a)
Necessary Condition
Type (25).
Dyad
b)
Necessary Condition
cedure H for 1346.
Pro
c)
Verify Connection of
25.
a)
Necessary
Condition
Dyad
Type (25)
b)
Necessary
Condition
Pro
cedure
H
for 1346.
c)
Verify
Connection of
25
d)
Verify
Noncifunction Re
versal
(2
relative to 3).
a)
Necessary
Condition
Dyad
Type (2
Â¡5)
b)
Necessary
Condition
Pro
cedure
H
for 1346.
c)
Verify
Connection of
25 .
a)
Nocoss
iry
Condition
Dyad
Typo (2
5)
b)
Necessa
ir y
Condition
Pro
codu re
II
for 1346.
c)
Verify
Connection of
25.
d)
Vorify
Not
icjfunction Re
versa 1
(5
relative to 2) .
4)
138
Stephenson 3 Link 3 is grounded.
Crank
J
Link 1 and
INVERSION
to 1.
Point
e
=
Pivot F.
CURVATURE
TRANSFORM
to
find
4
Crank
K
>
Link 6 and
INVERSION
to 2 .
Point
f
=
Pivot H.
CURVATURE
TRANSFORM
to
find
5
Crank
L
>
Link 2 and
Point
g
=
Pivot A.
[4MSP Grid
: (2)xlD]
173
a) Necessary Condition Pro
cedure L for 2456.
b) Necessary Condition Pro
cedure G for 1234.
6) *
a) Necessary Condition Pro
cedure M for 2456.
b) Necessary Condition Pro
cedure B for 1234.
c) Verify Connection of 56.
a) Necessary Condition Pro
cedure J for 2456.
b) Necessary Condition Pro
cedure G for 1234.
c) Verify Connection of 13.
a) Necessary Condition Pro
cedure K for 2456.
b) Necessary Condition Pro
cedure A for 1234 .
a) Necessary Condition Pro
cedure K for 2456.
b) Necessary Condition Pro
cedure A for 1234 .
f'
o
1
*
a)
Necessary Condition
cedure K for 2456 .
Pro
/*
w*
b)
Necessary Condition
cedure A for 1234 .
Pro
/ / /
215
Program Chart 1
Problem jf 1 Path Synthesis.
Linkage Solution FourBar
Mode of Cyclicity #2
Synthesis Procedure
1) CURVATURE TRANSFORM to
find 2,4.
FOURBAR
Necessary Condition Procedure
1) Necessary Condition Pro
cedure H for 1234.
Sample Program
VPROB1FBMODF2
Files
[1]
V
INITIAL
31)
4MSP(3/1)
32)
pivot A
[2]
>
31 SYNINP
33)
pivot D
[3]
>
35 32 34 33 SCRVTRN
31
34)
pivot C
[ 4 J
>
SYNOUT 35 32
34 33
35)
pivot B
36)
NC Grid(4x2)
[5]
>
NECINP
37)
DC Grid(4x2)
[6]
3 6 NCPII 35 32
34 33
[7]
>
NECOUT 36
[8]
>
37 DESINP 36
[9]
>
37 DPVTLCA 37
32 35
34 33
flO]
V
37 DLKLNRT 37
33 34
34 35 35 32
til]

37 DLKSIZE 37
35 34
[ 12 J

37 DESDYAD 37
35 32
34 33 0 0 0 31
2
, i 6
[13]
V
DESOUT 37V
84
Figure 57 Determination of Permissible Trailers for 2FSP
CHAPTER IV
SYNTHESIS OF LINKAGE SOLUTIONS
In the last chapter the types of problems under con
sideration were presented (Figure 39). The synthesis
solution to each consisted of attaching these bodies with
specified motions to the links of an "appropriate" linkage.
The means by which this "appropriate" linkage is found is
the subject of this chapter.
A. The Four Tools of Synthesis
The synthesis procedure for the problems shown in
Figure 39 depends on the problem type and the solution
linkage type. In each case some combination of four syn
thesis techniques is employed. These techniques are
termed the "four tools of synthesis" and are depicted
graphically in Figure 41.
1.) Synthesis Tool #1. The Curvature Transform
Probably the single most important concept in the
field of kinematic synthesis is the curvature transform.
All of the synthesis procedures used in this work will use
this tool of synthesis at least once to generate constraint
links (which may be operated upon by the other tools of
39
224
Program Chart 10
Problem Â¡(10 Double Coplanar Syn
thesis of Two Adjacent Planes.
Linkage Solution Stephenson 3
Mode of Cyclicity 05
STEPHENSON 3
Synthesis Procedure
Necessary Condition Procedure
1)
CURVATURE
find 1,6.
TRANSFORM
to
1)
Necessary Condition Pro
cedure H for 1346.
2)
CURVATURE
find 2.
TRANSFORM
to
2)
Verify Disconnection of
25 (2 short)
Sample Program
VPROB10STEPH3MODE5
[1] * INITIAL
[2] * 31 SYNINP
[3] * 32 SYNINP
[4]
V
37 33 38 39 SCRVTRN
31
[5]
35 34 SCRVTRN
31
[6]
>
SYNOUT 37
33
38
39
[7]
>
SYNOUT 35
34
[8]
>
NECINP
[9]

40 NCPH 37
33
38
39
[10]
V
41 NECDYAD
37
33
38
39
36
35
34
31
2
2
[11]
y
NECOUT 41
[12]
V
42 DESINP
41
[13]
y
42 DPVTLCA
42
34
35
36
37
[14]
y
42 DLKLNRT
42
3 6
3 5
38
39
[15]
~y
42 DLKSIZE
42
37
38
38
36
[16]
y
42 DESDYAD
42
37
33
38
39
35
34
31
2,
t 6
[17]^ DESOUT 42 V
Files
31)
4 MSP (4/3)
32)
4MSP (5/3)
33)
pivot F
34)
pivot A
35)
pivot B
36)
pivot C
37)
pivot D
38)
pivot G
39)
pivot B
40)
NC Grid (1
41)
NC Grid(If
42)
DC Grid(l/
CHAPTER VII
PROGRAMMING FOR FOURBAR
AND SIXBAR OPTIMIZATION
A. Philosophy of Program Modularity
The three major computational blocks in the synthesis/
analysis/resynthesis approach to linkage optimization and
their execution sequence (Figure 71) are common to the
optimization of all twelve of the motion specification
types. The difference between problems lie within these
blocks.
To gain an understanding of flexibility needed in an
allencompassing fourbar and sixbar optimization program
package, consider that most of the motion specification
types can be solved by several types of mechanism, each
entailing separate synthesis procedures (fourtyeight com
binations of motion specification/solution linkage exist).
Every combination has four to fifteen modes of cyclicity,
each one requiring some procedure for necessary condition
evaluation (the same is true for desirable conditions to
a lesser extent).
The program package providing a computational base for
the analytics presented in this work is also intended to
serve as a foundation for a future mechanisms package
113
114
>
*
Synthesis
"...
I
Necessary
Conditions
f ,.
Desirable
Conditions
Figure 71 The Computational Steps of a Synthesis/Analysis/
Resynthesis Linkage Optimization
Table 41. Motion Coefficients, A
m2.
k
A02
A12
A2 2
A32
A42
A52
A6 2
0
l(aj+8j)
a cos y.
3 3
a. sin
3
Yj
cos Y 1
3
sin y
3
a .
3
B:
+3 sin y.
3 13
+3 cos
3
Yj
1
a a '. + 6 6
3D 3 3
(a + 3 ) cos y
3 3 3
(3j"otj)
cos y
3
 sin y
3
cos y
3
a
3
Bj
+ ( 3 a ) sin y
3 3 3
 (a'. + 3 )
3 3
sin y
3
CT
00
2
(a) 2 + (3) 2
ao+*b
3"o"2a
1
0
a"
a0
60
3
3aao+3eBo
ao 3+3Bq
3^
3a"
Jao
0
1
a"
0
30
For k > 2, the position j = 0, is assumed as the initial position. When 2 = j = ),
a0 = 60 = Y0 =  ^he prices denote derivatives with respect to y.
U
159
Modes of Cyclicity
Necessary Condition Procedures
a) Necessary Condition Pro
cedure D for 1234.
b) Necessary Condition Pro
cedure D for 2456.
2)
a) Necessary Condition Pro
cedure D for 1234.
b) Necessary Condition Pro
cedure D for 2456.
a) Necessary Condition Pro
cedure H for 1234.
b) Necessary Condition Pro
cedure D for 2456.
a) Necessary Condition Pro
cedure II for 12 34 .
b) Necessary Condition Pro
cedure H for 2456.
CURVATURE TRANSFORM
ANGULAR COGNATES
PATH COGNATES
INVERSION
Figure 41 The Four Tools of Synthesis
22
the solution linkage types are presented and discussed.
Subsequent chapters will treat in detail the analytics
behind the steps of synthesis, necessary condition evalu
ation and desirable condition evaluation.
181
Module Chart 4
Name Curvature Transform (Synthesis Tool)
Label SCRVTRN
Description
Performs a curvature trans
form, generates circlepoint
and centerpoint cubics for a
given 4MSP.
APL Call
"^(01) (02) (03) (04) SCRVTRN (I 1) (12 ) (13 ) "
where:
(I1) = MSP set.
= Pivots of predefined constraint
link (moving, fixed), if one
exists.
(0^) (C>2) = Moving, fixed pivots, first con
straint link.
(0 ),(0^) = Moving, fixed pivots, second con
straint link (optional for
Stephenson dyad links).
121
slightly different types of motion specifications to be made.
For example, Sarkisyan, Gupta and Roth [124] have extended
curvature transform theory to satisfy motion specifications
in a leastsquares minimumerror sense. Sutherland [125]
has treated mixed exactapproximate specifications, again
making use of leastsquares approximations. Kramer and
Sandor [48] have introduced "Selective Precision Synthesis
(SPS)," a technique which produces dyadic constructions of
mechanisms which satisfy "accuracy neighborhoods" about
specified pivots.
Any synthesis procedure which 1.) produces linkage
solution sets which can be mapped onto orthogonal grids
and 2.) allows resynthesis within specified regions of the
grid, can be used in the proposed optimization procedure.
The availability of alternative synthesis procedures would
enhance control of the number of optimization parameters.
2.) Alternate Solution Linkages
T
Another way to control the number of optimization
parameters is to allow the use of different types of link
ages. Synthesis procedures are presented for fourbar and
sixbar linkages, which provide eight and fourteen free
parameters respectively. Untreated here but also directly
synthesizable are the slidercrank and inverted slidercrank
(7 parameters), geared fivebar (12 parameters) and numer
ous variations of the sixbar using sliders and oscillating
97
3 ^3
Figure 512 Necessary Condition Procedures for
Cognate FourBar Linkages
REFERENCES
1. Chebyshev, P.L., Oeuvres, Volume I New York: Chelsea
Publishing Company, 1899.
2. Cauchy, A., "Method generale pour la resolution des
systems d'equationes simultanees," Comptes Rendus de
L1Academic des Sciences, Volume 25, 1847, pp. 53638.
3. Alt, H., "Uber die totlagen des gelenkvierecks,"
Zeitshrift fur Augewandte Mathematik und Mechanik,
Volume 5, No. 1, 1925, pp. 34754.
4. Dantzig, G.B., "Programming of independent activities,"
II, Mathematical Model, Econometrica, Volume 17, 1949,
pp. 20011.
5. Svcboda, A., Computing Mechanisms and Linkages New
York: McGrawHill, 1948, pp. 15456.
6. Levitskii, N.J., and K.K. Shakvazian, "Synthesis of
four element spatial mechanisms with lower pairs,"
Trudi Seminara po Teorri Mushin, Mekhanizmov,
Akademiia Nauk, USSR, Volume 54, No. 5, 1954.
7. Freudenstein, F., "Approximate synthesis of four bar
linkage," Trans ASME, Volume 77, August 1955, pp. 85361.
8. Freudenstein, F., "Fourbar function generators,"
Transactions of the Fifth Conference on Mechanisms,
Purdue University, 1958, pp. 10407.
9. Freudenstein, F. "Structural error analysis in plane
kinematic synthesis," Journal of Engineering for In
dustry Trans ASME, Series B, Volume 81, No. 1,
February 1959, pp. 1522.
10. Freudenstein, F., and G.N. Sandor, "Synthesis of path
generating mechanisms by a programmed digital computer,"
Trans ASME, Series B, Volume 81, No. 2, May 1959,
pp. 15968.
11. Sandor, G.N., and F. Freudenstein, "Kinematic synthesis
of path generating mechanisms by means of the IBM 650,"
IBM 650 Program Library, File No. 95300, 1959.
228
46
For multiply separated positions these two expres
sions would be mixed. Now, the circular constraint
equation can be expressed as:
Q0(U2+V2) + 2Q1U + 2Q2V + Q3 = 0
For a point A on plane E whose path describes a circle
for multiply separated positions, this condition will take
the form:
gÂ£(U,V) = dk[QQ(U2+V2) + 2Q1U + 2Q2V + Q3
dYk
= 0
(4.3)
If the zeroth position (9o=0) is subtracted, these
equations will reduce to:
GÂ£(U,V)
%<30 _4tQ(u2+v2"UP"V0) + 2Ql(UU0)
dYk
(4.4)
+ 2Q2(VVq)]
= 0
for 2 = 1,2,3
and coefficient Q3 is eliminated.
Now, if the specifications are normalized so that
a0 = 60 = Y0 = 0/ then Uq=u and Vp=v. Substituting the
coordinate transform into (4.4):
2 2 2 2 2 2,
U +V u v = a +3 +u (2acosy+23siny)+v (23cosy2ctsiny)
88
Necessary
Condition
Procedure
Case
As Synthesized:
NCPA ~NCPB* q22*
7^77 1(3
ra
NCPC
NCPB NCPA
* *
NCPA
NCPD NCPD NCPG
/* / / / ,
NCPG
me md
* *
NCPH NCPH NCPH NCPH
EZa EZb
EZc
EZd
Figure 59 Necessary Condition Procedures for
FourBar Chains
187
Module Chart 10
Name Dyad Type (Necessary Condition)
Label NDYDTYP
Description
Evaluates dyads with a syn
thesized link on the basis of
consistency of dyad type for
the 4MSP. Generates a neces
sary condition grid (1D) in
dicating acceptable dyads.
APL Call
,,^(01) NDYDTYP (I1) (I2) (I3) '(I4) "
where:
(1^),(I^) = Pivots defining synthesized link
(1^) = Pivot defining third dyad pivot.
I. = 1 or 2, case number.
4
(Ox) = Output grid.
226
Program Chart: 12
Problem #12 Path Coordination
with Two Cranks.
Linkage Solution Watt 2
Mode of Cyclicity #6
Synthesis Procedure
Necessary Condition Procedure
1)
CURVATURE
TRANSFORM
and
PATH COGNATES to find
4 6
r
2)
INVERSION
to 1.
3)
CURVATURE
find 3#
TRANSFORM
to
1)
Necessary Condition
cedure M for 2456.
Pro
2)
Necessary Condition
cedure B for 1234.
Pro
3)
Verify Connection of
56 .
Â§ ample Progr am
Files
VPROB7WATT2MODE6
(1)
> INITIAL

[2]
*31 SYNINP
[3]
 32 33 34 35
SCRVTRN
31
[4]
* 36 37 38 39
SPTHCOG
31 32 33 34 35
[5]
y
NECINP
[6]
y
40
NCPM 31+18
[7]
y
NECOUT 40
[8]
41
DESINP 40
[9]
>
41
DPVTLCA 41
37
39
38
36
[10]
V
41
DLKLNRT 41
36
38
36
37
[11]
41
DLKSIZE 41
36
38
[12]
>
41
DESDYAD 41
36
37
38
39
0
0
0
2,
16
[13]
V
DESOUT 41
[14]
 V
EXTRACT 41
[15]
y
45
0 0 ANLZPCG
31
36
37
31
[16]
y
44
49 SYNINP
31)
4 MSP
(4
/2)
32)
pivot
H
1
33)
pivot
D
I
34)
pivot
F
1
35)
pivot
G
1
36)
pivot
H
37)
pivot
D
38)
pivot
F
39)
pivot
G
40)
NC Gr
id
(4,
41)
DC Grid
(4,
42)
43)
44)
4MSP
(1/2)
45)
4MSP
(4/2)
46)
4 MSP
(4/1)
47)
pivot
C
48)
pivot
B
49)
pivot
A
50)
NC Gri
id
(3)
51)
NC Gri
id
(3,
52)
DC Grid
(3,
PAGE 1
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134
c.) Plane J > Link 5 and
Point e = Pivot H.
Plane K = Link 4 and
Point f = Pivot D.
Crank L = Link 2 and
Point g = Pivot A.
*t CURVATURE TRANSFORM to
find 3.
INVERSION to 2.
CURVATURE TRANSFORM to
find 6.
[4MSP Grid: (2)xlD]
Stephenson 1 Link 1 is grounded.
Plane
J
y
Link 5 and
CURVATURE
TRANSFORM
to
Point
e
=
Pivot B.
find 4.
Plane
K
=
Link 2 and
INVERSION
to 3 .
Point
f
=
Pivot A.
CURVATURE
TRANSFORM
to
Crank
L
=
Link 3 and
find 6.
Point
g
=
Pivot F.
[4MSP Grid
l: (2) xlD]
Stephenson '
2
 Link 2 is
grounded.
a.)
Plane
j
V
Link 4 and
t CURVATURE
TRANSFORM
to
Point
e
=
Pivot D.
find 5.
Plane
K
=
Link 1 and
INVERSION
to 3 .
Point
f
=
Pivot F.
CURVATURE
TRANSFORM
to
Crank
L
=
Link 3 and
find 6.
Point
g
=
Pivot A.
[4MSP Grid
l: (2)xlD]
b.)
Plane
j
V
Link 1 and
CURVATURE
TRANSFORM
to
Point
e

Pivot D.
find 3.
Plane
K
=
Link 4 and
INVERSION
to 4 .
Point
f
=
Pivot C.
CURVATURE
TRANSFORM
to
Crank
L
=
Link 5 and
find 6.
Point
g
=
Pivot B.
t 4MSP Grid: (2)xlD]
Stephenson 3 Link 3 is grounded.
a.) Plane
J
Link 5 and
t CURVATURE
TRANSFORM
to
Point
e
=
Pivot C.
find 2 .
Plane
K
=
Link 4 and
CURVATURE
TRANSFORM
to
Point
f
=
Pivot D.
find 6 .
Crank
L
=
Link 1 and
Point
g
=
Pivot F.
f 4MSP Grid
:(2)xlD]
b.) Plane
j
y
Link 4 and
t CURVATURE
TRANSFORM
to
Point
e
' =
Pivot C.
find 1,6.
Plane
K
=
Link 5 and
Point
f
=
Pivot 3.
[4MSP Grid
: 2D]
Crank
L
=
Link 2 and
Point
g
=
Pivot A.
116
package never becomes so large as to become unmanageable.
2.) The user designs his program for his particular
application. The only modules needed in his workspace
are the ones actually executed. Only pertinent I/O is
performed.
3.) The independent nature of the modules permits
easy expansions and substitutions, and greatly simplifies
maintenance.
B. Control of Input/Output
The projected package could be implemented on almost
any interactive system providing sufficient memory, but is
very likely most conveniently implemented in an APL en
vironment. All further discussion will relate to an APL
based package.
Data flow between modules utilizes userdesignated
files, and all user/computer interaction is interactive.
Like a Pollock dyadanalysis routine, each module requires
file numbers as input. Unlike a dyadanalysis routine
(which outputs data into files concatenated onto those
already existing), the designer also specifies the loca
tions in which output files are to be placed. Overwriting
of obsolete data from the previous optimization loop is
now possible and in fact necessary. An example of a typical
program for each motion specification type is presented in
Appendix D. To write such programs, the user would:
144
c.)
Crank
J +
Link 3.
CURVATURE
TRANSFORM
to
find
2.
Point
e =
Pivot B.
INVERSION
to 2.
Plane
K *
Link 6.
CURVATURE
TRANSFORM
to
find
4.
INVERSION
to 4.
CURVATURE
TRANSFORM
to
find
5.
[ 4MSP Grid :
(3)xlD]
Watt 2 Link
2 is grounded.
Crank
J +
Link 1.
CURVATURE
TRANSFORM
to
find
Point
e =
Pivot A.
4, 6.
Plane
K *
Link 5.
INVERSION
to 1.
CURVATURE
TRANSFORM
to
find
3.
[4MSP Grid: 2D+lD]
Stephenson 1
 Link 1 is
grounded.
Crank
J *
Link 3.
CURVATURE
TRANSFORM
to
find
4.
Point
e =
Pivot F.
INVERSION
to 3
Plane
K >
Link 5.
CURVATURE
TRANSFORM
to
find
2.
CURVATURE
TRANSFORM
to
find
6.
[4MSP Grid
1: (3) xlD]
Stephenson 2
 Link F is
grounded.
a. )
Crank
J >
Link 3.
*+ CURVATURE
TRANSFORM
to
find
5.
Point
e =
Pivot A.
INVERSION
to 3.
Plane
K 
Link 4.
CURVATURE
TRANSFORM
to
find
1,6.
[4MSP Grid
1: 1D+2D]
b.)
Crank
J +
Link 5.
CURVATURE
TRANSFORM
to
find
3.
Point
e =
Pivot B.
INVERSION
to 5.
Plane
K >
Link 1.
CURVATURE
TRANSFORM
to
find
4 .
INVERSION
to 4 .
CURVATURE
TRANSFORM
to
find
B.
[4MSP Grid
1: ( 3) xlD]
Stephenson
i 3
 Link 3 is
grounded.
a. )
Crank
J >
Link 1.
CURVATURE
TRANSFORM
to
find
2.
Point
e =
Pivot F.
INVERSION
to 1.
Plane
K *
Link 5.
CURVATURE
TRANSFORM
to
find
4 .
INVERSION
to 3 .
CURVATURE
TRANSFORM
to
find
6 .
[4MSP Grid: (3)xlD]
55
be redefined.
Now,
c' =
y /
nm
1 y '
nm
II
*
u
(l y' )3
' nm
c'" =
y (1 y / ) + 3 y ^
'nm 1 nm nm
(1 Ynm)2
nm
d.) Inversion formulation number 4.
Given: Motions of N relative to R, M relative to N
Find: Motion of M relative to R.
Independent parameter for given MSP: time t.
Independent parameter for derived MSP: time t.
00 r
a
 mj
(k) r
= S, cos C T, sin C + a
k k nj
(k)
{k)r
= S, sin C + T, cos C + 3
k k nj
(?\
 mj
(k) (k)
n r
= Y + Y
mj nj
4
The objective function may represent the deviation
between actual and ideal linkage performance, which may be
a measure of prescribed path error ("structural error"),
dynamic imbalances, error due to elasticity, or some
weighted combination of errors relative to pertinent cri
teria.
Equality constraints have the effect of narrowing the
scope of the optimization problem by reducing the number of
free parameters. They commonly take the form of a syn
thesis step in which a linkage solution set is generated
to satisfy prescribed precision points or positions.
Inequality constraints may take the form of design
constraints placed on available linkage characteristics,
such as link length maxima or minima, range of acceptable
transmission angles, or geometric bounds on pivot loca
tions or coupler motions.
The range of possible formulations of the linkage op
timization problem is extremely large due to the highly
variant sets of 1) linkage applications and 2) optimiza
tion criteria. It is highly desirable to organize the
available approaches in order to get a feel for what is
available, help classify past and present efforts, and
perhaps enhance decisions in future efforts. Figure 11
shows such a representation.
Esthetically, a closedform approach is preferred for
any problem, but the inherently complex and nonlinear
193
Modulo Chari: 16
Name Necessary Condition Procedure II
Label NCPH
n
c ^
*
*
a nil}
D
Description
Evaluates fourbar chains
on the basis of changepoint
considerations for the case
shown. Creates a necessary
condition grid indicating ac
ceptable linkages.
APL Call
"*(0.^ NCPH (1^ (I2) (I3) (I4) "
where:
(1^ (I2)
= Pivots of first constraint link
(moving, fixed pivots).
= Pivots of second constraint link
(moving, fixed pivot).
(o1)
= Output grid.
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
MULTIPARAMETRIC OPTIMIZATION OF FOURBAR
AND SIXBAR LINKAGES
By
Kim Loring Spitznagel
December, 1978
Chairman: Delbert Tesar
Major Department: Mechanical Engineering
An approach for fourbar and sixbar linkage optimiza
tion relative to numerous design criteria is presented. A
linkage solution set is synthesized through an analytical
combination of curvature transform theory, inversion,
Robert's cognates and angular cognates under the specifica
tion of one of twelve possible types of motion specification.
Evaluations are made first on the basis of necessary condi
tions, and then on the basis of assigned weighting factors
and design zones relative to the optimization criteria.
Subsequent expansions and refinements through resynthesis
about the best linkages are used to approach an optimum
solution.
The detailed design for an interactive computer program
to implement the analytics is presented and discussed. The
vi
8
By far most of the emphasis in linkage optimization
has been towards the iterative methods. These methods in
volve "mathematical" (or "numerical") programming techniques
which may be (in order of increasing complexity) linear,
quadratic, geometric or dynamic. Many techniques exist,
Figure 11 showing only those used more commonly in linkage
applications.
Constrained minimization techniques involve minimiza
tion of the objective function in the absence of inequality
constraints, and are generally based on a "stepping" itera
tion, where each "new" step is a function of the location
of the "old" step and some formulation of a new direction
and step size.
The methods of Powell [25,26] involve unidirectional
orthogonal searches to find local minima for each itera
tion. It is most applicable for problems with nondif
ferentiable objective functions. Suh and Mecklenburg [27]
have used Powell's method to operate on matrixderived
objective functions and constraints for spatial mechanisms.
The conjugate gradient technique bears some resemblance
to Cauchy's "steepest descent" method but is much more ef
ficient. Unfortunately, convergence degrades as the op
tima are reached or when the objective function has compli
cated features or "ridges." Fletcher and Reeves [28] in
troduced this method in 1964, and later Rees Jones and
Rooney [29] employed this method for motion analysis using
206
Module Chart 29
Name Desirable Condition Dyad Analysis
Label DESDYAD
Description
Performs a dyadbased
analysis, evaluates the solu
tion set of linkages on the
basis of:
 Overall Boundary
 gfunction
 Ah
 hrms
 ATorque
 Shock Level
APL Call
"^(01),(02) ... DESDYAD (I^,^), ..."
where:
(1^) = Previous scoring grid
(I2)(1^) = Crank pivots (moving, fixed)
(I^) (It) = Follower pivots (moving, fixed)
(Ig) (I?) (Ig) = Dyad pivots (= 0 for fourbar)
(Ig) = MSP set, coupler or crank relative to
ground (may be 0 if I^q = 1)
I10 = Mode of drive.
= 1 for cyclic drive.
= 2 from position 0 to 3.
= 3 through range of disconnection.
Ij,+ = Desirable conditions
= 1 for Overall boundary
= 2 for gfunction
 3 for Ahfunction
= 4 for hrmsfunction
= 5 for ATorque
= 6 for Shock Level
for six bars,
Stephenson 3
of Watt 2
inversion.
205
Module Chart 28
Name Link Size (Desirable Condition)
Label DLKSIZE
Description
Evaluates the solution set
of linkages on the basis of
link size. User specifies
links of interest.
"M01),(02) DLKSIZE (I ) (I ) (I3) . .
where:
(1^) = Previous scoring grid.
(I ) (I ) ; (I.) (b) ; . = Constraint link
Pivots (1 pair =
1 link)
(0^) = Output scoring grid, cumulative.
(0^) = Output scoring grid, this condition
only. Optional.

