|Table of Contents|
Table of Contents
Chapter 1. Introduction
Chapter 2. A philosophy for linkage design and optimization
Chapter 3. Solvable problems
Chapter 4. Synthesis of linkage solutions
Chapter 5. The necessary conditions
Chapter 6. The desirable conditions
Chapter 7. Programming for four-bar and six-bar optimization
Chapter 8. Recommendations and conclusions
Appendix A. Problem types and solution linkages
Appendix B. Modes of cyclicity and necessary condition procedures
Appendix C. Program modules
Appendix D. Sample programs
MULTIPARAMETRIC OPTIMIZATION OF
FOUR-BAR AND SIX-BAR LINKAGES
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
Kim Loring Spitznagel
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
ACKNOWLEDGI4ENTS . . . . . . . . iii
ABSTRACT . . . . . . . . . . vi
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. Four-Bar Necessary Conditions . 71 B. Six-Bar Necessary Conditions . 99 VI THE DESIRABLE CONDITIONS . . . 104
A. The Desirable Conditions . . 104 B. The Objective Function . . . 108
vii PROGRAMMING FOR FOUR-BAR AND SIX-BAR
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
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
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 FOUR-BAR
AND SIX-BAR LINKAGES
Kim Loring Spitznayel
Chairman: Delbert Tesar
Major Department: Mechanical Engineering
An approach for four-bar and six-bar 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,
appendices include synthesis procedures, necessary condition evaluations, program module tables and sample programs.
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 (1452-1519) and Newton (1642-1727), 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  involvement in straight-line linkage design led him to develop his well known "Chebyshev polynomials"and optimality
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 four-bar optimization with respect to the transmission angle criterion was presented by Alt [31 in
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  introduced his computational synthesis method of "successive approximations," and in 1954 Levitskii and Shakvazian  introduced a method to optimize spatial slider-cranks and crank-rockers on the basis of a set of linear equations from a finite'number of precision points.
The work of Freudenstein [7-11] 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 path-generating mechanisms, and Freudenstein  developed a
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
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.
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 1-1 shows such a representation.
Esthetically, a closed-form approach is preferred for any problem, but the inherently complex and nonlinear
ALGEBRAIC RANDOM UNCONSTRAINED CONSTRAINED DIRECT
METHODS SEARCH MINIMIZATION MINIMIZATION METHODS
FOigur I I-. VLinaEe OIptiizi MIt.h
JCN.GAIET JIT PEAT GADETPO.
Figure 1-1. Linkage Optimization Methods
linkage problem makes the optimization analytics somewhat difficult to implement. Algebraic approaches, generally limited to small-scale problems, are rarely attempted. Sutherland  has developed mixed exact-approximate position synthesis in which some points are matched exactly using Burmester theory, and the objective function contains the least-squares 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 , Bagci [141, and Bhatia and Bagci[151 have developed and expanded on a linear superposition technique in which the error of the loop-closure 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 four-bar 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
1968 Tomas  discussed the treatment of linkage synthesis as a nonlinear programming problem, and employed the optimization random-search method of Garrett and Hall . Eschenbach and Tesar  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  approached the optimization of two-degree-of-freedom function generators by generating a small number of "good" designs, and then generating linkages around these to find a "best" design. Spherical four-bar path generators are handled by Sridher and Torfason , 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  and the author and Tesar  have "grid refinement" technique for four-bars which involves generating two-dimensional 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 1-1 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 non-differentiable objective functions. Suh and Mecklenburg [271 have used Powell's method to operate on matrix-derived 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  employed this method for motion analysis using
an analog computer. Nect-i [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. Six-bar function generators were optimized by Chen and Dalsania [31) by applying a least-squares 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 four-bar 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 so-called "quasi-Newton" 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 Mclaine-Cross [341 used it in 1969 to optimize a crank-rocker four-bar 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
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
Newton-Raphson 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 [38-40] has received quite a lot of attention recently. It is
sometimes called the Sequence of Unconstrained Minimization Technique (SUMT). Fox and Willmert  used this technique to optimize a four-bar path-generating linkage using a least-squares error approach. Tranquilla  used this method to optimize a four-bar under the geometric couplerpoint constraints. Spatial mechanisms were considered by Gupta  with constraints on closure, mobility and transmissability. Bakthavachala and Kimbrell  considered four-bar path generator optimization under clearance, tolerance, transmission angle and link size considerations. Conte, George, Mayne and Sadler  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 two-degree-of-freedom spatial function generators and offset slider-crank path generators. Alizade, Novruzbekov and Sandor  then went on to four-bar function generators with consideration of link length and transmission angle inequality constraints. Kramer and Sandor  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
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  have used the modified simplex to optimize multi-loop 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  which later received attention from McLarnan , who attempts to minimize structural error by requiring the derivatives at the precision positions to be zero. Huang, Sebesta and Soni  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  and Datseris and Freudenstein , who develop the application of Lin's heuristic method (originally developed for the communications field) to mechanism optimization. Moradi and Pappas  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
another criterion is to be considered, it is generally related to transmission angle. Lin , Hamid and Soni [62,63], Shoup, Staffer and Weatherford , Freudenstein and Primrose , Sutherland and Roth , Bagci [671, and Savage and Suchora  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 , Berkof and Lowen [71,72], Conte, George, Mayne and Sadler , Tepper and Lowen , Sadler , Elliot and Tesar , and Sandler . 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  have treated optimization of transmission gears. Optimization using the least-squares method of Gauss has been done by Chi-Yeh  and Mansour and Osman . In 1971 Bagci and Prasad  and Bagci and Parekh  dealt with optimization schemes for coplanar and spherical four-bar and six-bars 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  and stress and deformation . Rao and Ambekar  optimized spherical RRRR function generators, and Sallam and
Lindholm  optimized six-bar Watt-i function generators. Hobson and Torf son  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  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 . All of these employ high-level 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  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  also use a multifactor objective function with inverse utilics for spherical function
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 , Starr  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 , Seireg  Fox and Gupta [1021, Chen [1031 and Root and Ragsdell .
The "philosophical" papers [93-99] 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.
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
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 four-bar linkage or one of the six-bar 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/
analysis/resynthesis loop to be used for this work is shown in Figure 2-1.
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 (1-dimensional), rectangular (2-dimensional) or cubic (3-dimensional) 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.
S Y n thesis (D I (D
Figure 2-1 The computational steps of a Synthesis/Analysis/
Resynthesis Linkage optimization
The third step, evaluation of "desirable conditions," is where the "quality" of the surviving solution linkages is determined by the user-defined 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 re-evaluated. 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 crank-rocker four-bars 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
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 2-1). 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 four-bar 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
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.
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 non-linear mechanism is the four-bar
linkage, as shown in Figure 3-1a. 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 four-bar provides eight free parameters.
Two types of single-degree of freedom six-bar chain exist, the Stephenson and the Watt, as shown in Figure 3-lbc. They are both composed of two ternary links and four binary links, and have different conceptual inversions, as shown in Figures 3-2 and 3-3. Each one provides
B 3 C
a. The Four-Bar Chain
D 4 C
b. The Stephenson Six-Bar Chain
C 4 H
c. The Watt Six-Bar Chain
Figure 3-1 Four-Bar and Six-Bar Chains
D 4 C
F 3 A
a. The Stephenson 1
D 4 C
F 3 A
b. The Stephenson 2
D 4 C
c. The Stephenscn 3
Figure 3-2 Inversions of the Stephenson Six-Bar Chain
C 4 H
A 2 G
a The Watt 1
C 4 H
b. The Watt 2
Figure 3-3 Inversions of the Watt Chain
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 "Watt-l" 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 3-1 will be adhered to.
Other simple linkages exist, such as the slider-crank and inverted slider-crank four-bars, the geared five-bars and six-bars with sliders and oscillating blocks. They have less general applicability and except for the geared five-bar 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 four-bar chain in Figure 3-4, and for the six-bar chains in Figures 3-5 and 3-6. The information in these figures is central to this work. This unfortunate
2 4 2 4
2 4 2 4
1 i 1
24 4 2 4
2 4 2 4
1 1 4
3 2 4 3 2 4
I4 2 4
3 2 4
1 / Driven
2 4 /
1 Complete Incomplete
Figure 3-4 Modes of Rotational Cyclicity of the Four-Bar
Stephenson Six-Bars With No Cyclic Links
K5 A& Any pair of links
+ may be driven (15
1 6 t distinct possi2 bilities)
Stephenson Six-Bars With CyclIic Links
Figure 3-5 Modes of Rotational Cyclicity of the
Stephenson Six-Bar Chain
Watt Six-Bars With No Cyclic Links
5 Any pair of links
+may be driven (15 distinct possi1 6 bilities)
Watt Six-Bars With Cyclic Links
Figure 3-6 Modes of Rotational Cyclicity of the
Watt Six-Bar Chain
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 rocker-driven crank-rocker with complete crank rotation, and the Grashof double-rocker 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 four-bar or six-bar problem will have a
subset of the possible modes of cyclicity, depending upon the grounded link and the conceptual equivalence or
non-equivalence of links (a function of motion specification type). As an example, the four-bar solution to the two-link chain problem (Figure 3-7) 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 3-8 shows some examples of modes of rotational cyclicity which are unacceptable or geometrically impossible. Figures 3-8a and 3-8b might represent the cyclic coupler-driven Grashof double rocker and the cyclic rockerdriven crank-rocker, respectively, which have already been discussed. In Figure 3-8c, control of motion through the change points of pivots r and s is impossible. The motion shown for the six-bar linkages of Figures 3-8d and 3-8f simply cannot exist, and the motion shown for the Watt six-bar linkages of Figure 3-8e would require a linkage with very unusual geometry and an external control of motion for both four-bar chains.
Applicable Nodes of Rotational Cyclicity
Figure 3-7 Applicable Modes of Rotational Cyclicity of the
Four-Bar Solution to the Two-Link Chain Problem
B 3 C
B 3 C
A 1 D
A 6 B
(c) D 4 C
1 G6 B
F C4k 3 C 4 H (d)
(e) c 4
B D F
Figure 3-8 Some Unacceptable Modes of Rotational Cyclicity
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 3-9. 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 four-bar linkage.
2) Motion Synthesis. The motion of a body is guided by attaching the body to the coupler of an appropriate four-bar linkage.
3) Angular Coordination of a Two-Link Chain. The angular motion oil two connected bodies, one of which is connected to ground, is controlled by attaching these bodies to an appropriate four-bar linkage.
4) Angular Coordination of Two-Cranks. 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 four-bar linkage.
5) Path-Crank 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 four-bar linkage.
6) Angular Coordination of Three-Link Chain. The
I.) 2.) 3.) Angular Coordination
Path Synthesis Coplanor Motion of Two- Link
4.) Angular Coordin- Path Crank Angular Coordination
otion of Two Coordination of Three Link Chain
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 3-9 Tes of Motion SCrecifications ns
Figure 3-9 Types of Motion Specifications
motion of three bodies connected in a chain (one connected to ground), is guided by attaching these bodies to the links of an appropriate six-bar linkage.
7) Angular Coordination of a Crank and a Two-Link
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 six-bar linkage.
8) Angular Coordination of Three-Cranks. The rotation of three non-adjacent bodies, all connected to ground, is coordinated by attaching these bodies to the links of an appropriate six-bar linkage.
9) Angular Coordination of Three Adjacent Planes. The motion of two non-adjacent 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 six-bar 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 six-bar 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 six-bar linkage.
12) Point Path Coordinated with Two Cranks. The motion of a point and the rotation of two non-adjacent bodies which are connected to a common ground are coordinated by attaching the point and two bodies to the links of an appropriate six-bar 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.
SYNTHESIS OF LINKAGE SOLUTIONS
In the last chapter the types of problems under consideration were presented (Figure 3-9). 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 3-9 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 4-1.
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 maybe-operated upon by the other tools of
CURVATURE TRANSFORM PATH COGNATES
ANGULAR COGNATES INVERSION
Figure 4-1 The Four Tools of Synthesis
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 4-2a.
Finitely separated position synthesis of four-bars
was first done by Burmester  in 1888, using graphical techniques for five positions. In 1964 the problem was solved analytically by Bottema  and Primrose, Freudenstein and Sandor .
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 4-2b. Mueller 11091, Allievi , Krause , Wolford  and Veldkamp  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 4-2c). The unified theory for MSP was developed by Tesar  in 1967.
b, Infinitesimally Separated Positions (ISP)
a. Finitely Separated Positions c. Multiply Separated Positions
Figure 4-2 Finitely, Infinitesimally and Multiply Separated Positions
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 higher-order motions are consequently geometric (slope, curvature, inflection, etc.). In 1974 Myklebust  presented a unified theory making dynamic (i.e., time dependent) higher order properties specifiable. This is termed "time state synthesis." For the four-bar problems the dynamic properties are equivalent to the geometric properties, but for a six-bar 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 4-3 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
V = u CosY -v Cos Y+~
Fixed Plane U
Figure 4-3 Parameters Relating Fixed and Moving Planes
and the "inverse coordinate transform":
u = (U-a) cos y + (V-3) sin y
v = (V-a) cos y (U-a) 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:
F d [f(u,v,CC,,y,Q0,Q1Q2* .. )] = 0
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)
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
+ 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 cosy-2asiny)
and the generalized circular constraint function (divided by 2) takes the form: Gz = dk Q0 (X 2 2)+ u(ccosY+6sinY) + u(-asinY+cosY))
+ Q1(u(cosY-1) usinY +0) + Q2(usinY
+ u(cosY-1) + 8)] = 0
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 4-1. They are a function of the motion specifications only.
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)
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
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)
for k. = 0,1,2
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 4-2.
In time-state 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 time-state 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 4-4) 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,
I m \
n j I r Ir
nj I r n
Figure 4-4 Inversion Notation
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 4-3.
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
_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 = ISP counter ( for example, for k=2, a = a ).
so A r r0 mj nj
T B r r
T0 = mj nj
= ,r r
S =A + BC
T1 = -AC + B
S2 = -2AC AC + B BC
T = A AC + 2BC + BC
.... *2 + * "3 **
S = A 3ACC 3AC2 + 3BC + 3BC BC + BC
T = 1B 3BCC 3BC 3AC 3AC + A 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 R0 reference ).
Independent parameter for given MSP: yr
Independent parameter for derived MSP: y
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.
r ,. n
Let a n an; an dyr ;etc. (same for 13n and y~ n
a *r a *
an a~m i etc. (same for1 andy )
A'/ m n f
A" =n 2' n 3 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 )
-y"(y' 1) + 2
C "" = m
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
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.
r / n
Let an =X n a =n ; etc. ( same for n and y
am a m da y etc. ( same for and y
M m m dr m m
dyr 2 r
Ynm r Ynm r2 ;etc.
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
C, = nm
'(1 y/ ) + 3y"2
Gnm nm 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.
(k) r (k)
Smj = S cos C T sin C + a
-m] k k nj
(k) r (k)
S= Sk sin C + Tk cos C + nj
(k) r (k) (k)
r m Y m + Y nj
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
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
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 (').
Let a a ; n; etc. ( same for and y )
n Ln n r n n
am a ; a d ; etc. ( same for 8m and y
m m m rmm
Now, dan a /
n m m
m dy 1 + y
a'(l + ym) a'y
/'n m- m m
am( + ym)
(1 + 2y' + 2) a"(y' + 2y" + 2y"y' + y2)
a =m m m m m m mm
+ a'y"/(2y' 1)
(1 + Y)5
6 h and are similar ( substitute B's for a's).
m m m
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
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 (').
r / n
Let a = n a -- ; etc. ( same for 8and
n n n dy rnn dan
a dy ; etc. ( same for am dyM dyr y r
yn Y/ dyn-dn etc.
Ymn n ;n mn dyn 2 ;ec
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,
= (ly 3
(I + Y/)
"' (1 + y' )- 3y, 2
Ym n Ymn ) mn
Yn +Y 5
3.) Synthesis Tool #3. Path Cognates
The Roberts- Chebyshev theorem states that three different coplanar four-bar mechanisms will trace identical coupler-curves. An example is shown in Figure 4-5b, where coupler point e has the same motion whether carried by four-bar O1dfO2, four-bar OlgjO3 or four-bar 02 hkO3. An additional property is that the angular parameters of those links with identical hash marks in Figure 4-5b are identical for any reachable position or position derivative of point e.
These properties can be used to solve the problem of
a. Prescribed Path,and Input Crank 03
b. Robert's Cognates
Figure 4-5 Use of Path Cognates to Solve the PathCrank Coordination Problem
path-crank coordination (Figure 4-5a). 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 four-bar 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 A-6c. 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 4-6 outlines the use of the concept of angular cognates to solve a crank-coordination 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
a. Input Output Crank3
b. Curvature Transform
C' e C
c. Angular Cognate
Figure 4-6 Use of Angular Cognates to Solve the
Function Generation Problem
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 4-7. 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 4-7a). Operating on these specifications with the curvature transform will produce constraint links as shown in Figure 4-7b (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 4-7d). 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
D 4 C
35 5 F 3 A
\(a) (b) f3
+ CURVATURE e
e C (d) (c)
4' B 5
2 INVERSION + 7' '2
e (e) (f) e
D 4 C D 4 C
SB (+ REINVERSION)i B
F A F A
Figure 4-7 A Stephenson-2 Solution to Point/Path
Coordination with Two Cranks
center point cubic defining pivots F and H. A conceptual reinversion (which requires no computational activity) produces a Stephenson 2 six-bar 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 four-bar (which occupies a place on a two-dimensional grid) has two path cognates, a "left" cognate and a "right" cognate.
The example discussed in the last section produces two (2) two-dimensional grids representing link 5 and 71 and another two-dimensional grid representing links 1 and
6. The total, if these grids are combined, is two (2) four-dimensional 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 one-dimensional grids. The nature of the component grids is a
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 six-bar synthesis procedures
involve dependent syntheses. In fact, all of the problems
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 4-1. 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. ($!~-a.)-Cos yj sin yj Cos y. ax!
+(r3!-ct!) sin y. -(Ox!+. sin yj 2 (Ux6) +N%) (X "+otO
0 0~2x -l00x
3 3cx6co+3F13" Ot 1 -Ncx 1 -3%-30t I 0 -1 a i 1
For k > 2, the position j=0, is assumed as the initial position. When k. j
a 0 =30 =Y0 = 0. The primes denote derivatives with respect to y.
Table 4-2. Motion Coefficients, D mt
k D D 3 P4 D 5 D 6
0 cosf 4- S i. nY P -cosy --,A Cos Y C' i'( sinY .-sinY
2 0 0 j- j
0 c 0 SY 0 0 S i rl"f 0 0 CDSYO-kl 0 S in'f 0 L (-,-tj0-y) siny Y) silly -YS. Ln ;cosy
+ + cosy
1*6*) Giny -25y) siri,( 3 i n y Sin'(
+ 2 +2 2 COSY
Cosy +(-.,ty-2uy- y +13 2 COS-,, 4COS y
:-,LL t 3 + 01 c ;% 3,*,'Y* 3 3,, y) z; i- n,( (-3yy)sin),
3 Y S I n Y -6y-3al -36y) s in-,, +(-3y'f)C0SY +(-y ty)cosy
+ .y 3 4 y
+ 0 3 3,x+3 6-j 11,T 3 3,; 3 1!' 6)cosy
Related Motion Specification Sets for Two Planes M and N (2ISP)
(Notation: Figure 4-4)
Inversion Formulation #1 Inversion Formulation #4
(Independent parameter t) (Independent parameter t)
Plane N: Plane M: Plane N: Plane M:
r r r r r r r r r n n n
an n Yn a Yi n n'n n am' m m
dar d r dy r dar d r dy d r d fr dyr dan d n dn
n n n dm m m n n m minm
dt' dt' dt dt' dt' dt dt' dt' dt dt' dt' dt
Inversion Formulation #2 Inversion Formulation #5
(Independent parameter r (Independent parameter r
coupler angle y) coupler angle y)
Plane N: Plane M: Plane N: Plane M:
r r r r r r r r r n n n
an' n n am' m m an n Y n m' m m
r r r r r r r n dam dy
da dB da dr dy da dB da d$ 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))
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
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. Fuur-Bar Necessary Conditions
It is entirely possible that a linkage solution will
satisfy the specified positions in an improper sequence,
-as displayed in Figure 5-1 for a four-bar 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 four-bar chain may have either zero pivots with complete rotation, or two adjacent pivots with complete
a. Improper Position Sequence
b. Order Analysis Crank Rocker)
Figure 5-1 Order
rotation. The latter situation occurs if the linkage satisfies the Grashof condition:
kZ+ s < p+ q
Z= length of longest link
s = length of shortest link p,q = length of other two links Grounded four-bar chains which satisfy this relationship are known as "Grashof four-bars," and take the forms shown in Figure 5-2 a,b,c. They are simply different inversions of one another, and are generally more useful than non-Grashof linkages. The crank-rocker, for example, can be used to generate an arbitrary shape with a coupler point while its crank is driven continuously, and the drag-link is often used for continuous function generation. In a Grashof four-bar only those pivots which are cyclic will ever enter change points.
The non-Grashof four-bars take only one form (regardless of inversion), the non-Grashof double-rocker. 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.
(c) GRASHOF DOUBLE -ROCKER
(d) NON- GRASHOF DOUBLE- ROCKER
Figure 5-2 Four-Bar Grashof Types
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 six-bar indicate simultaneous disconnection (or a limit in motion of the rest of the linkage, Figure 5-3b). At this point the dyad could pass from one dyad "type" (relative angular orientation, Figure 5-2c) 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 dyad-driven six-bar linkage (Figure 5-3d).
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 three-link 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 dyad-type must be checked during necessary condition analysis.
Figure 5-3 Dyad Change Points
b.) Change points in four-bar chains
All of the solution linkages can be considered as
single four-bar chains, a combination of four-bar chains and dyads, or a combination of two four-bar chains. A detailed analysis of change points within four-bar chains is therefore very worthwhile. It involves considerations of both synthesis procedure and mode of cyclicity.
If a four-bar chain (or any Grashof type) is driven as shown in Figure 5-4a 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, four-bar chains driven through opposite links (Figure 5-4b) 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 four-bar chains) on the basis of change-point 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
Figure 5-4 Change Point Considerations in a
"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 5-5). For example, the follower of a crank-rocker four-bar is both a trailer and a rocker, but a constraint link of a drag-link four bar is a trailer only. Trailers
def ined by circle- and center-point 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
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
circle point A bisects the angle if ij
(see Figure 5-6a).
This property applies to any T ik' where k is another position, so the angle T jk can be found from:
Limited relative rotation at the moving pivot a. Trailer
Limited relative rotation at the fixed pivot b. "Rocker
Figure 5-5 Trailer and Rocker
p Image pole for positions i,j SkOa
-y A jk
jk2 2 1Ik Ij
(Vij Yik) '
Figure 5-6 Determination of Relative Moving Pivot
Rotation for 2FSP
T jk = -(T 13 T ik) (see Figure 5-6b)
Now the location of circle points with positive and negative values of the T in terms of P!. and P! (Figure 5-7c) 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 5-7a). On the opposite side of P! P! the signs will be reversed. Figure 5-7b
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
+ Circle Point
Figure 5-7 Determination of Permissible Trailers for 2FSP
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 5-8a 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 5-9b 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
IQ~~2 A01,- Position of
=0 -point A when coupler is in
00 specified pos.
Fgr -5 C t C
Figure 5-8 Determination of, compatible Constraints
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 four-barchains in sixbar linkages
Displayed in the left-hand column of Figure 5-9 are
the possible forms of four-bar chain which may be needed as all or part (for a six-bar) 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).
FinalForm: As Synthesized:
NCA NOPO NCPA NCPC
NCPB NCPA NCPC NCPA
NPD NCPD NCPG NCPG
___________ lMa Mlb Mlc flld
NCPH NCPH NOPH NOPH
I~ lb WVc
Figure 5-9 Necessary Condition Procedures for
Necessary Condition Procedure A (for cases Ia, Ic, Ib and IId)1) Choose a follower-constraint-link to be a
2) Choose a crank-constraint-link of proper order
which is compatible with the follower.
3) Test Grashof type for a crank-rocker.
Necessary Condition Procedure B (for cases Ib and IIa)1) Choose a constraint link to be a trailer of
2) Choose another constraint link to be a trailer
which is compatible with the first link.
3) Test Grashof type for a drag-link.
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 circle-point and center-point
cubic curves, etc.
2) Follow procedure B.
Necessary Condition Procedure D (for case IIIa)1) Choose a follower-constraint-link to be a
2) Choose a crank-constraint-link 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
3) Check mutual compatibility.
4) Invert to change reference to the coupler link.
5) Repeat 1), 2) and 3) (no need to check order
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 path-cognates
For motion specification types #5 and #12 (Figure 3-9) four-bar chains are produced by performing a curvature transform followed by a generation of path cognate fourbars.
We assume four-bar I in Figure 5-10b ha's been produced by the curvature transform and its path cognates, four-bars II and III, are to be usable as all or part of a linkage solution. What tests can be made on four-bar I. to ensure proper performance relative to change points in four-bars II and III? This question is central to establishing practical necessary condition analyses for path-cognate problems.
A study of four-bar II will now be made in detail. All comments made hereafter regarding "left" cognate four-bar II also apply to "right" cognate four-bar III. The modes of cyclicity which II may take are shown in Figure 5-12. Each of these modes will involve a separate analysis (necessary condition procedures I N) to be performed on four-bar I
-to ensure proper performance in II.
a. Prescribed Path,and Input Crank
b. Robert's Cognates
Figure 5-10 Use of Path Cognates to Solve the PathCrank Coordination Problem
Necessary condition procedures I N take advantage of some unusual properties of the "cognate 10-bar" (Figure 5-l0b):
i.) A cognate 10-bar will be made up of one of three sets of three four-bars: either three drag-links, three non-Grashof double-rockers, or a Grashof double-rocker and two crank-rockers.
ii.) When 0 3 in four-bar II is at a change point, 0 2 in four-bar I is also at a change point.
iii.) When g in four-bar II is at a change point, d in four-bar I is also at a change point.
iv.) When 0 1 in four-bar II is at a change point, links df and 0 1l02 in four-bar I become parallel.
v.) When j in four-bar II is at a change point, links
0 1d and 0 2f in four-bar I become parallel.
Proof of (ii): Referring to Figure 5-13a,
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 four-bar I at 0 2 and in four-bar II at 0 3.
Proof of (iii): Referring to Figure 5-13b,
c(x + + + (D Bx + e 2 7T
A A because 0lged is a parallelogram,
A = 0 at simultaneous change
points for g in four-bar II and d in four-bar I.
Proof of (iv): Referring to Figure 5-13c, it follows from path-cognate property (ii) that change points occur simultaneously at 01in four-bar II and at 0 2 in four-bar 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:
0-102 df 6' 1
Proof of (v): Referring to Figure 5-13d, it follows from path cognate property (iii) that pivot j in four-bar II and pivot k in four-bar 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