—'J-'—.'4_

TO ANALYTIC. SYNTHESIS OF A

PRACTICAL LIMITATIONS
FUNCTION GENERATOR

FOUR-LINK MEQHANISM

Ph. D.
ERSITY

Thesis for the Degree of

MICHIGAN STATE UNIV

Han-GIGS I... DoweII, .Ir.
I965

THE-IS“:

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Michigan Scam
University

 

This is to certify that the

thesis entitled
PRACTICAL LIMITATIONS TO ANALYTIC SYNTHESIS

OF A FOUR-LINK MECHANISM FUNCTION GENERATOR

presented by

Harold L. Dowell, Jr.

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in MeCh. Engr.

 

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(/12flé{/;4xuy// / jZK 214/14 {C t C }

Major professor

Date November 29, 1965

 

 

ABSTRACT

PRACTICAL LIMITATIONS TO.ANALYTIC SYNTHESIS
OF A POLE-LINK MECHANISM FUNCTION CENERATCR

by Harold L. Dowell, Jr.

The problem was to investigate certain practical limitations to synthesis

of a four-bar mechanism function generator. The limitations investigated

were:

8)

Branching: Some of the precision points may lie on one branch of
closure and some on the other, meaning that a joint must be
separated and reconnected in a different configuration of the

mechanism to complete the range.

b) Closure: Moving the links beyond the first or last precision

position, but still within the range may require the mechanism to

enter a region beyond its limits of motion.

c) Change Points: The link lengths may be such that as the links

d)

pass through a folded configuration the follower is able to move

in either direction.

Transmission Angle: During operation within the range of motion,
the acute angle between centerlines of coupler and follower link
may become so small as to preclude effective force transmission.
Excessive Link Length Ratio: Two of the links may be much longer
than the other two making the operation of the mechanism impossible
because of deflection of the longer links or because of space

requirements.

The method of investigation was experimental. Several commonly used

functions and their ranges were chosen; for each, a series of four-bar

.Ila

Harold L. Dowell, Jr.
mechanisms to generate the function were synthesized. When one of the
limitations was encountered, the particulars were recorded and the next
mechanism was synthesized. The syntheses and analyses were performed
with the aid of a digital computer; 4050 function generators were
synthesized for each of eight functions. Since attention was concen-
trated on the design failures attributable to any one of the above

limitations, the sucessful designs were discarded.

For this work, the analytic synthesis was performed with three un—
knowns (the three link length ratios) and four arbitrary parameters

to give the greatest variety of solutions consistent with reasonably
accurate function approximation. The Freudenstein Equation with three
precision points was used. The arbitrary parameters were the starting
angles (corresponding to the first precision point) and range angles
for input and output links. The aim was to find a pattern for each
limitation in terms of these four arbitrary parameters; if such a
pattern is known, then the parameters can be chosen so as to avoid or

minimize encountering the limitations.

The results of the investigation are based on choices of 40 degrees for
minimum transmission angle in the range of operation and six for maxi-
mum link length ratio. On this basis, about 92 percent of the attempts
at synthesis failed because of one of the limitations. Choice of
ranges from 30 degrees to 150 degrees had little influence on whether

or not one of the practical limitations appeared.

Making the starting angles differ will reduce the probability of

Harold L. Dowell, Jr.
branching for input and output links rotating in the same direction if
the function has positive slope or for input and output links rotating
in opposite directions if the function has negative slope. Avoiding
the third quadrant for the starting angle of output link will reduce
the probability of branching for input and output links rotating in
opposite directions if the function has positive slope or for input and
output links rotating in the same direction if the function has negative
slope. These same measures will also reduce the likelihood of problems

with closure.

Patterns for the transmission angle limitation were found and are given
in the thesis. These patterns depend on whether input and output links
rotate in the same or opposite directions and on whether the first and

second derivatives of the function have like or unlike signs.

As long as the starting angles of input and output links were not exactly

the same, change points were not a practical limitation.

No significant generalization can be formed from this work which will aid
the linkage designer in avoiding the limitation of excessive link length

ratio.

PRACTICAL LIMITATIONS TO ANALYTIC SYNTHESIS

OF A FOUR-LINK MECHANISM FUNCTION GENERATCR

BY

Harold Li?bowell, Jr.

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DCIJTCR OF PHILCBOPHY
Department of Mechanical Engineering

1965

COpyright by
HAROLD LEE DOYIELL, JR.

1966‘

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ACKNOWLEDGMENTS

During the academic year 1964-65, I was a Science Faculty Fellow of the

National Science Foundation; the support and confidence signified by

that award have been important to my work.

Professor Palmer J. Reiten, Chairman of Mechanical Engineering, University

of North Dakota, relieved me of teaching duties for two consecutive years,

thereby expressing his confidence in me.

While a student at Michigan State University, I have frequently received

inspiration and encouragement -- exenplary and direct - from my professors.

Many of my hours were spent in the Computer Laboratory of Michigan State
University: the Director, Professor Lawrence von Tersch, made special

arrangements essential to the completion of this task. Mrs. Elizabeth

Unger, with her staff of Programing Consultants, frequently helped me out
of bewilderment; herself an engineer, she showed genuine interest in my

Problem. Mr. Robert Sparbel wrote (and named) the tape handling function,

KLUNKER, and advised me on other programing matters.

At the time I was writing computer programs, I shared an office with

Professor Mahlon C. Smith, who applied his talents and experience in my

1behalf, often and generously.

Mr. Michael Hooth made half of the seventy-two plots, a tedious chore

which he executed with speed and care.

The members of my Graduate Committee gave good counsel — technical and

administrative. They are professors William A. Bradley, Rolland 1'. Hinkle,

ii

iii

George H. Martin, Edward Nordhaus and Howard L. Womochel. Professor

Hinkle served as chairman of this committee.

My interest in kinematics of mechanisms was kindled by Professor Hinkle's

excellent textbook on the subject several years ago; my personal associa-

tion with him has been rewarding.

By her tactful and constant encouragement, my wife, Johnnie, has shared

with me the burdens of the task; she has helped to make it pleasant work.

With care and devoted persistence, in spite of my interruptions,

Mrs. Joyce Medalen typed the manuscript.

TABLE OF CONTENTS

ACIGNOEJLEM'ZENT O O O O O O O O O O O O O 0
LIST OF FIGLV'Rg O O O O O O O O O O O O O 0
INTRODUCTION 0 O I O O O O O O O O O O O O

The Four-Bar Planar Mechanism
A5 a FUUCtion Generator 0 o o o o o 0

DESCRIPTION O O O O O O O O O O O

FUMZTIOAS WHICH CAN BE IJECHANIZED
APPROXIIMITELY o e o o o o o o o o

APPLICATIOBB O O O O O O O C O O

A REVIEW OF THE DEVELCPNENI' OF
DESIGN TECHNIQUES o o o o o o o o

THEPROBLEi-i ...............
Statement of Prgblem . . . . . . . . .

THE BRAICHING LIMITATION . . . .

THE CLCBURE LIMITATION . . . . .

THE CHANGE POINTS LIMITATION . .

THE TRAIEMISSION ANGLE LIMITATION

THE LINK LENGTH RATIO LIMITATION

THE name) or INVESTIGATION . . . . . . . .
Statement of Method . . . . . . . . .

THE FUJR ARBITRARY PARAMETERS . .

FUIIZTIOhB AND RANGES . . . . . .

(BDERING THE LIMITATIONS . . . .
CCTIIPU‘I‘EROU'I'PUT.........

A Preliminary Study . . . . . . . . .

The COWQULGI Programs 0 o o o o o o 0

iv

Page
ii

vii

13
14
16
l7
l7
19
19
21
24
24
2o
26

29

Page

MAINPROGRAMFCR FINALSTUDY............... 29
SUBROUTINES.......................30

MAIN PROGRAM FIR PRELII-IINARY STUDY . . . . . . . . . . . 34
Disglaxing the Results . . . . . . . . . . . . . . . . . . . . 34
THEINDEXI.......................35
THEPLOT........................37
THERESULTS............................40
Number of Design Failures by Kind . . . . . . . . . . . . . . 4O
Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4O
Summaries of Plots . . . . . . . . . . . . . . . . . . . . . . 40
Sample Output from Preliminary Study . . . . . . . . . . . . . 59
INTERPRETATIONOFRESULTS.....................74
Branching . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Transmission Angle . . . . . . . . . . . . . . . . . . . . . . 75

FUIIZTIONS HAVING LIKE SIGNS PCB
FIRST AND SECOND DERIVATIVES . . . . . . . . . . . . . . 75

FUACTIOAB HAVING UNLIKE SIGbS FCR
FIRST AND SECOND DERIVATIVES . . . . . . . . . . . . . . 75

Link Length Ratio . . . . . . . . . . . . . . . . . . . . . . 75
COIxCLUSIOIS............................76
A~PPENDIXI............................78

Derivation of Eguations . . . . . . . . . . . . . . . . . . . 78

THE SYNTHESIS EQUATION . . . . . . . . . . . . . . . . . 78

TI—E AMLYS IS EQUATI OI‘J . C O O O O O O C O O O O C C O O O 80

vi

APPEBDIX II 0 O O O O O O O O O O O O O O

Commgter'Proqrams in Fortran . . . .

MAIN PROGRAM PCR FINAL STUDY .

small-11m O O O O O O O O O 0

MAIN PROCRAM FCR PRELIMINARY STUDY

LIST OF REFEREDCES

Page
82

83
86
99

104

10 .
ll .
12 o
13 .
14 .

l5 .

12
13
13
15
15
18
21

21

25
28
32

32

LIST OF FIGURES

£222.22 P3192 2.99.2
16...36 31.
17...38 32.
18 . . . 41 33 .
l9 . . . 42 34 .
2O . . . 43 35 .
21 . . . 44 36 .
22 . . . 45 37 .
23 . . . 46 38 .
24 . . . 47 39 .
25 . . . 48 4O .
26 . . . 49 41 .
27 . . . 5O 42 .
28 . . . 51 43 .
29 . . . 52 44 .
30 . . . 53 45 .

vii

57
58
6O
61
62

64
65
66
67
68

79

INTR CDLCTI ON

The Four-Bar Planar Mechanism As a Function Generator

DESCRIPTION

A function generator is a passive device which has a prescribed
functional relationship between its inputs and its output. The four-
bar mechanism with pinned joints when used as a function generator has
only a single input. The functional relationship may be accomplished
by coordinating rotations of input and output shafts or by coordinating
derivatives of these rotations. Use of a rack and pinion transforms

these rotations to linear motion.

Thus, if the function be designated f, the relationship is: (Output) =
f(Input). The input and output for the mechanical function generators
considered here will be restricted to rotations. The constraint of the
four-bar mechanism (Fig. 1) provides the necessary unique relationship
between rotations of input (driver) and output (follower), these bars
being joined by the frame and coupler. All bars are considered to be
I‘igid bodies and the pinned joints assure coplanar motion. So if the
IUtations of the input and output, each measured by an angle from some
radial datum fixed to the frame, are called P and 8, respectively,
then the desired function if exactly or ideally mechanized would be

characterized by: S = H?)-

Shaffer and Cochin (5)1 have given a means to determine whether a
particular function can be exactly mechanized by a four—bar mechanism.
\

NUmbers in parentheses indicate References.

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POINT POINT

 

 

 

Very few of the commonly required functions can be exactly mechanized

by'a four—bar mechanism, but many of them can be approximately mechanized

satisfactorily.

'The usual procedure is to obtain point-approximation for several positions
of input and output links or some combination of point- and order-approx-

imation. In point-approximation the mechanized function will match the

ithended (ideal) function at three, four or five different positions in
tine range while at other positions the mechanized function will deviate

frwam the ideal function. These points of agreement have been called

"precision points" (Fig. 2). Order—approximation consists of matching

fiEIst, second, third, etc. derivatives of the function at each of several

Precision points.

A<3<:ordingly, an error E usually exists between the ideal function f and

tfiee mechanized function f' : E(P) = f'(P)—f(P). Through a certain range

‘325’ motion of the mechanism, satisfactory approximation (meaning acceptably

Srhall error) is often possible.

FUNCTIONS WHICH CAN BE MECHANIZED APPROXIMATELY

7r11<e kinds of functions which can thus be satisfactorily mechanized by a

IEIDIJr-bar mechanism, and their ranges, are limited. Freudenstein (7,10)

has discussed these limitations.

frifimey are:
Linear functions should be avoided.
The function should be monotonic.
The function should not be symmetric about the mid-point of the range.
The slope of the function should change gradually.

Ranges of input and output should correspond to less than 120 degrees
of shaft rotation.

-"

APPLICATIOIB

Four-bar mechanisms have been used as function generators in mechanical
analog computers and control devices (1.10.11). Their use would be

more common if it were easier to design them since linkage mechanisms
possess advantages that other mechanical function generators do not have.
Cams, non-circular spur gears, wires used as belts with variable pitch-
radius pulleys and sliders guided in curved grooves have been used for
function generation. But these devices are relatively expensive to
manufacture and their performance may be unfavorably affected by large
friction forces or non-positive driving. They can be replaced by

four-link mechanisms designed to have good force transmission character-

is tics.

A REVIEW OF TI-E DEVELCPMENT OF DESIGN TEHNIQUES

The first successful efforts to design linkage mechanisms as function
generators were graphical. Until electronic digital computers became
available graphical methods were faster than analytic techniques;
judging from published literature, graphical synthesis is being replaced
by wholly analytic methods. Not only is the computer solution faster
than the graphical, but the use of a digital plotter for some of the

computer output retains the advantage of graphic procedures.

During World War II, concentrated linkage design work resulted in
Practical mechanical analog comuters. Svoboda's report on this work (1)
has been cited frequently since its appearance in 1948. He presented
graphical design methods using nomograms and overlays and discussed

Various means for refinement of the design to reduce error.

 

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In Germany, Rain (3) used graphical construction (i.e., successive

inversion of the mechanism) with his "point-pasition-reduction" to

synthesize four-bar and six-bar function generators. Point—position-

reduction is the choice of the starting angle for input or output link

so as to obtain angular symmetry with respect to the frame for pairs of

the specified positions. In the United States, Nickson (4) elaborated

on the overlay method discussed by Svoboda, a direct graphical approach

to function generatOr design.

In 1954, Shaffer and Cochin (5) obtained a differential equation to test

a function for exact synthesis by a four—bar mechanism and, if it can be

exactly mechanized, to determine for what range of values such mechan-

ization is possible. Freudenstein (6, 7) presented an analytic approach

to the design of four-bar function generator mechanisms and developed
sYnthesis equations for three, four, and five positions of input—output

as well as equations for specified derivatives of displacement up to

Seventh order at a single position. His equation for three positions

r1<>vv appears in many textbooks for undergraduate engineering students.

LIsing complex numbers to represent vectors in a plane and combining
them to form the quadrilaterals of the four-bar linkage for three and

four specified positions ("loop equations"), Sieker (8) obtained sets

of simultaneous equations. He noted that these are equations for the

"Circle—point-curve" and the "center—point-curve". He obtained
Synthesis equations for a function generator for three and four

Positions as well as for fewer positions with their derivatives. Several

numerical examples were presented.

 

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AJJen (9) applied Hain's point-position—reduction to the graphical
design of four—bar mechanisms as function generators. Freudenstein (10)
tabulated dimensions of numerous four—bar mechanisms to generate various
functions using his equations for 5, 6, and 7 precision points. Each
mechanism is a minimum error approximation; he solved his equations (7)
*with a digital computer. Also, he mentioned some applications of mech—
anical function generators and discussed the suitability of functions

for mechanization.

Michalec (11) gave a summary of types of elements used in mechanical
analog computers based on his experience with a United States manufac—

turer of these computers.

Starting with the graphical construction employing point-position-
l‘E!¢:!uction, Worthley and Hinkle (12) derived analytic equations for design
of four-link function generators. And, to reduce the error of the
approximation, Hahn (13) proposed four-bar mechanisms in series; the
Ifi.rst mechanism is designed using one of the analytic schemes and its
error curve is plotted. Then, from a collection of error curves which
T16! prepared, one chooses a second four-bar mechanism such that the errors

of both when superposed will result in reduced error for the combination.

Taking the four-bar mechanisms used in preparation of Hrones and Nelson's
atlas of coupler curves (2), Vidosic and Johnson (14) developed a set of
146 pairs of curves as suggested earlier by Nickson (4):
l) The trace curve; normalized output vs. normalized input.
2) The deviation curve; (normalized output-normalized input) vs.
normalized input.

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mechanized and comparing them with the trace and deviation curves
of their collection, one can select a suitable mechanism. Their

collection of curves was prepared with the aid of an electronic

digital computer.

Also using the digital computer, Dunk and Hanson (l5) wrote a
Program to analyze the performance of a six-bar mechanism as a

function generator. By iteration with the aim of reducing error,

design is possible.

Hinkle (16) suggested using his angular velocity ratio theorem in
selecting a four-bar mechanism configuration; by studying a plot

of the function the orientation of the coupler with respect to the
flame at any station in the range can be predicted. The electronic
analog computer was used by Keller (22) as a design aid for four-link

Inechanisms; he obtained graphical display with the cathode ray oscillo-

8 cope .

Equations for synthesis of four—bar mechanism function generators
(loop equations) were presented by McLarnan (17, 19); he gave equations
for three, four, and five conditions and Fortran programs to perform
sYnthesis and analysis. Also, he gave (19) a complete discussion of
theory and limitations, including coments about ”branching”, to be
Considered below. McLarnan and Hagan (18) proved that no more than
flure constraints are allowed if scale factors for input and output are
to be arbitrarily chosen. They noted that seven different combinations
°f (precision points and derivatives are possible, and that the linkage
hairing two links very long relative to the other two links can be

replaced by an inversion of the slider-crank. Their development

resulted in a sixth degree equation with three trivial solutions and

either one or three real, nontrivial solutions.

Finally, two modern textbooks contain rather complete discussions,
to date, of design techniques for function generators. One is by

Hirschhorn (20), the other by Hartenberg and Denavit (21).

This review of the development of design techniques has been selective
rather than complete; the intention was to indicate the variety of
approaches to the problem of function generation using a four-bar
mechanism, and to show the movement away from graphical methods toward

analytic methods augmented by the electronic digital computer.

{1

‘1

THE PROBLEM

Statement of the Problem

 

The problem confronted in this thesis is to investigate the practical
limitations to analytic synthesis of a four-link mechanism function
generator with the aim of finding some pattern for their occurrence

and thus a means to avoid them.

In spite of the amount of work and the diversity of attack on the
Problem of designing a four-link mechanism as a function generator,
the solutions obtained are often not usable. While the solutions

do satisfy the theoretical requirements at the precision points, they
do not always satisfy the mechanical requirements.

a) Some of the precision points may lie on one branch of closure
and some on the other, meaning that a joint must be separated and
reconnected in a different configuration to complete the range of
Operation. This limitation is discussed as "branching” below.

b) Moving the links beyond the first or last precision point, but
Still within the range, may cause the mechanism to enter a region beyond
its limits of motion where it is impossible for the links to remain
cOnnected. This is treated below as ”closure".

c) The synthesized link-lengths may be such that as the links pass
' through a folded configuration the follower is able to move in either
direction: it may continue to move in the same direction that brought
it into the folded configuration or it may move in the opposite direc-
tiOn. This is discussed under "change points”, below.

d) During operation within the range of motion, the acute angle

be'tween centerlines of coupler and follower link may become so small

10

as to preclude effective force transmission. This limitation is

discussed below as ”transmission angle”.
e) Two of the links may be relatively much longer than the other
two making the Operation of the mechanism physically impossible because

of deflection of the longer links, or because of space requirements.

This limitation is referred to as ”link length ratio".

Except for the transmission angle and link length ratio, these limita-

tions are not matters of quality or degree: they are more in the nature

Of dichotomies.‘ They are decisive. Graphical synthesis reveals these

limitations as they arise; analytic synthesis can obscure their

Presence until after the solution is completed and a sketch or model

Of the linkage has been made. When such a limitation appears, new values

for the arbitrary parameters must be chosen and another solution begun.

It is impossible to predict such failures of solution.

Concerning branching, Hartenberg and Denavit declare (21), p. 233,
”No systematic way is known to the authors for avoiding the trouble
that was encountered.” Again, McLarnan (19) has this to say about

selecting arbitrary parameters, p. 14, "Normally the designer would

Want to make a number of choices of U2 and U3 [rotations of coupler]

in order to have a number of alternative solutions to examine before
selecting the best one for use. The proper selection of values u2 and
'43 has not been studied extensively enough with regard to Optimizing

the solution to permit any guide lines to be laid down at this time."

Of course, the error or the deviation between the mechanized function

and the ideal function may also be regarded as a practical limitation

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to design but the recent literature is replete with error-reducing

techniques to be applied after a generally usable configuration has been

determined. The five practical limitations mentioned above must be

satisfied prior to such refinement.

The problem for this thesis is an investigation of these practical

limitations to the design of four—bar mechanism function generators.

THE BRAICHING LIMITATION
Given four bars identified by their lengths A, B, C, and P, where these

are input link, coupler, output link and fixed link, respectively, let

the position of the input link be specified. Then the coupler and out-

put link can close the loop of four links in either of two different ways

or branches as shown in Fig. 3.

The output angle $81 corresponds to branch 1; output angle $82 to branch

2- Each branch may be thought of as a reflection of the other about the

diagonal D-D.

AS an example of the branching difficulty, suppose that Fig. 4 represents

a three precision—point approximation to some function. Use of the

synthesis equation with P2 = 45 deg., S2 = 60 deg., range of P 90 deg. and
range of S 60 deg. gives the linkage A, B, C, and frame shown in heavy

s011d lines in Fig. 5. For the first position, the four links are con-

heciao so that Link A is inclined at angle 22 and Link c at angle 52.
As Link A is moved from angle P2 counterclockwise to angle P3, Link C

mo‘Ves to occupy position b1; when A is moved to P4 then C moves to Cl.
But the desired second and third locations for C are b2 and c2, on the

other branch. If after being connected at position a1, Link A is moved

BRANCH.

 

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clockwise to the folded position, Fig. 5, then moved counterclockwise
again with an open rather than a crossed configuration, positions b2
and c2 will be satisfied when A is at P3 and P4, respectively, but then

position a1 will not be satisfied with A at P2.

THE CLCBURE LIMITATION

The input link and output link of a four-bar mechanism may not be able
to rotate through 360 deg. (i.e., turn continuously) but, instead, one
or both may be restricted to oscillate within some angle. A link which
can turn continuously is referred to as a crank; one which can only
oscillate is called a lever. This matter has been thoroughly investi-
gated and the particulars are well-known. One good discussion is in
Hartenberg and Denavit's book (21); part of their discussion is followed
be low to establish some terminology that is needed. Let the length of
the shortest link be calledxb; the length of the longest link, 2 3 the
lengths of the other two links can be called 10 and 32.

Then, if 4+4 < 10+}, the linkage is termed a Grashof linkage (Fig. 6)
for which three cases exist:

a) If link AD is the input, the input link is a crank and the output
link is a lever. The mechanism is called a crank-lever. Two of
these are possible, since either link adjacent to 4b can be fixed.

b) If link xv is fixed (i.e., if xi is the frame) both input and
output links are cranks; the mechanism is called a drag-link or
a double-crank mechanism.

c) (Otherwise, input and output links are levers. This is called a
double-lever mechanism and there are three possibilities. Note
that when xv is the output link of a Grashof linkage A/ could
turn continuously if the output were the driver rather than

driven, but this is precisely the converse meaning of ”output".

 

 

 

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7.
(“DOUBLE LEVER d. DOUBLE LEvEK

If}? dAE ‘ 2M .5. __E3 9 "1 .TtiE. -___§_Bfl.$_,*i9_ F. .6 . .-‘:L.’:‘__!S_e 6.;

 

ELC: U RE 6

———.__ —._._ —__. ._._ m“

/ J 47

r IIIIII/li/l/

 

 

 

d.PARALLELOGRAr/\ b, KlTE SHAPED

 

 

 

 

16

On the other hand, if A 4-K > 70+; , only double-lever mechanisms are
possible. Or, ifo + Z = 767 + 3, , change points occur; this is con-

sidered below.

bmte in Fig. 6-a, for example, that the output lever is restricted to

nmve within its angle of oscillation and that it occupies the extreme
values of this angle when input link and coupler centerlines become
colinear. In a synthesized linkage, the precision points will lie within
this angle of oscillation but it often happens that the range angle speci—
fied in the synthesis problem must lie partly outside one of the extremes
of the angle of oscillation. Where this happens the mechanism is not a
solution to the synthesis problem. Closure is not possible. Similar

considerations apply to the input link of a double lever mechanism.

THE CHANGE POINTS LIMITATION

In the discussion of closure, one of the possibilities was shown to be:
3, +2 = 70 + 7’ . For this case, if ¢=7a or if sea-j, , the link—
age is a parallelogram or kite-shaped as the equal—length links are
opposite or adjacent to each other, respectively (Fig. 7). But even if
A; is equal neither to 70 nor to 3,, as long as ,d,+’£ = 70+} , change
Points occur when the linkage is folded flat (i.e., when the centerlines
of all links coincide). While the indecisiveness of change points can

he corrected by inertia forces or by mechanical stops to prevent an
undesired change, and even though the range of operation of the function
Emnerator*may never include a change point, the possibility of change

Fmints is regarded herein as a design failure.

17

THE TRAISMISSION ANGLE LIMITATION

Uflmther the coupler pushes or pulls the output link during Operation of
1mm mechanism, the Optimum angle between centerlines of the coupler and
the output link for force transmission is 90 degrees. The transmission
angle, TA, is defined as the angle between centerlines of the coupler
and follower link or its supplement, whichever is smaller. See Fig. 8.

lfldle the optimum transmission angle is well-defined, the minimum value

is a matter of judgement.

A quotation from Hirschhorn (20), p. 257, is pertinent:
”Alt, who first suggested the use of this parameter as a
quality index for the transmission of motion, recommended
a minimum value of 40 deg. for low-speed and 50 deg. for
high-speed applications. Other kinematicians have reit-
erated these values. However, it is extremely doubtful
whether such generalizations should be accepted uncrit-
icaIIYQ'
Fozrthe purposes of this study, 40 degrees was taken as a minimum in
accord with recommendations in most references. 50, if at any position

within the range the transmission angle becomes less than 40 degrees,

the design is regarded as a failure.

THE LINK-LENGTH RATIO LIMITATION
Choosing a maximum link-length ratio, RATIO = 2% , is again a judgement.

er'this work, the maximum link-length ratio was arbitrarily chosen to

be 6.

18

(NPUT

    

IN PulT

 

 

T7/If/If/777

INPUT

 

 

TA

 

TA
‘NPuT—ei

 

11277/ril’ 'TA

INPUT

'- A
THE. TRAN$M\E>$\QN ANGLE-.7 T

 

THE NETHOD OF INVESTIGATION

Statement of the Method

The investigation was basically experimental: several commonly used
functions and their ranges were chosen; for each, a number of mechanisms
to generate the functions were synthesized using a well-known analytic
technique. When a practical limitation was encountered, the particulars
were recorded and a different synthesis was performed. The syntheses
and analyses were done with the aid of a digital computer; in this way
4050 function generators were synthesized and analyzed for each of eight

functions.

For the analytic synthesis of a four-link mechanism, seven variables are
involved: three link-length ratios, the starting positions for the input
link and for the output link, and the range angles for the input link and
for the output link. Theoretically, all seven of these variables may be
treated as unknowns; actually, the mathematical difficulties are such
that Ordinarily only three, four, or five unknowns are attempted, values

for the remaining variables being arbitrarily assumed as parameters.

For this work, the analytic synthesis was performed with three unknowns
and four arbitrarily chosen parameters to give the greatest variety of
solutions consistent with reasonably accurate approximation. With three
unknowns, three precision points may be prescribed; obviously, with more
precision points the error of the approximation can be reduced. But even
with only three precision points, errors of less than one percent of the
range of output are possible. Furthermore, as stated previously, error

reduction methods are plentiful.

19

20

The Freudenstein Equation (derived in the Appendix) was used for synthesis
at three precision points. Here the arbitrary parameters were taken as
the positions of input link and output link for the first precision point
and the range angles of input link and output link. The unknowns were
taken as the three link—length ratios, based on the fourth link which was
assumed to have a length of unity. Hence the following notation (See
Figs. 9 and 10.):

A Length of input link

8 Length of coupler link

C = Length of output link
F = Length of fixed link (frame) = 1

P2 = Angular position of input link at first precision point
configuration.

82 = Angular position of output link at first precision point
configuration.

RP = Range angle for input link

R5 = Range angle for output link

THE FOUR ARBITRARY PARAMETERS

For a function with positive slope, the input and output links can rotate
in the same direction. But Hinkle (16), p.286 , points out that a dif-
ferent mechanism results when the input and output links are assumed to
rotate in opposite directions; he calls it the alternate solution.
Similarly, if the function has negative slope, the input and output links
can rotate in opposite directions or - for the alternate solution - in
the same direction. In this work, the alternate solution is accomplished
by using negative values of one of the range angles: R8 was given both

positive and negative values.

21

    

 

Y .
HUI/ll l/l/I/l/l/l77 \
J i
F__
[(5—77 ,._, .,_..,r,,,_,fi'_
FIGURE 9_
4 3 z 5 4 3

  
    

RP

'NPUT OUTPUT

NOTE: PRECISION
POINT POSITIONS
ARE 2,3 44”

 

[III/III/IIIIII/ll

22

Hence, the schedule for selection of arbitrary parameters was as follows;

Value of Parameter, Degrees

 

P2 10 50 90 130 170 210 250 290 330
$2 20 60 100 140 180 220 260 300 340
RP 30 6O 90 120 150

RS 30 ~30 60 ~60 9O -9O 1 20 --120 150 —150

An91es P2 and 82, corresponding to the first precision point, are always
measured counterclockwise from a datum ray; range angles RP and R5 are

magnitudes. Two hypothetical functions, having a positive slope and the
other having negative slope, are shown in Fig. 11, where the meaning of
the four arbitrary parameters is illustrated. Parameters P2 and S2 fix
a POint in the P-S plane; parameters RP and RS act as scale factors, the

Sign of RS having its further significance mentioned above.

Also, note in Fig. 11 that the first and last precision points (P2 and P4)
are distinct from the ends of the range RP (P1 and p5). This distinction
is necessary for re—spacing the precision points to Optimize the error.
HOVIlever, of course, P2 and Pl, for example, might have the same value.

In the programs, precision points P2, P3, and P4 have been calculated
from the values of x2, x3, and x4, taken as input data. And x2, x3, and
X4 were determined by Chebyshev spacing in the range of X; see Hartenberg

and Denavit (21), p. 140, or Hinkle (16). p. 284.

Whi1e parameters P2 and S2 were chosen in steps of 40 degrees, the data
Shows steps of 20 degrees for them. In solving the synthesis equation,
link lengths A and C may have emerged as negative values whereupon they
were changed to positive values and angles P (if A was negative) and S

(if C was negative) were advanced 180 degrees. Since 180 degrees consists

 

 

 

 

23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fuua‘lou Hume 73°“:"5 .3593? LML'YIMN HAVING NEGATIVE SLOPE
U ’I 9 I
I
gasot)
‘RY 5‘1“) RY
I l I I
l I l f I I l \
XI X} x;‘ xzxs x x. X1.1 ’3' N": 7
Rx RX
Foo. ‘RS Posmvs: Foo. Rs Peon-we:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RS i
$5__ ' I
‘ I
l 1
PI P17 I";Y 7: p
113
rs 5—4
Fee RS NEQATNE: 'Foe RS thA‘erZ
SII Si
524»— —
l 1
R5 I RS 521’— l I
I I . I I
I . | I
l l l = l I 1 _
1’. h‘ 15’ 1'37, P P, PJ P? P43, 7:
RP fl RP

 

 

 

 

 

 

24

of four—and-a-half 40 degree intervals, the 20 degree steps were introduced.

The difference of 10 degrees (or integral multiples of 10 degrees)
between P2 and S2 was introduced to prevent starting a solution with
all four links having colinear centerlines. Incidentally, this precluded

parallelogram linkages.

FUNCTION AND RANGES
The functions used for this study are listed in Fig. 12, which also gives
a function identification number, the range for the approximation and

the precision point spacing: X2, X3, X4.

CRDERING THE LIMITATIONS

Unavoidably, the order of apprehending the practical limitations had the
effect of assigning priority to them. Since it is not known that these
limitations are exclusive, failures attributed to one kind of practical
limitation might have been attributed to some other kind had the order
of testing been different. From the viewpoint of the mechanism designer
this is not important but it does impede developing a theory of failure

for a particular kind of limitation.

Even so, certain ordering criteria prevail:
a) Over-riding influence of certain limitations.
(For example, if branching occurs then the maximum link length
ratio and minimum transmission angle are immaterial.)
b) Consquence of the limitations.
(For example, the change points limitation can be corrected by
other means, so it has low priority.)

c) Time of computation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PRECISION Pawns
N9- FuNcTIoN RANGE. X7- X3 X4
- ""—“‘“—'“‘_" __.:_"..‘—::-:‘.'.::‘_“:(L.,.__ .* “‘77::- '::_—- —:__ _.______.___ __ --- _.____.-_.___
I",
I [4:31— I \ Iéxe’: 7. 1067 1.5 1.933
x I
I I
2 15:9," ‘1 éXé1 l‘.8(:-G O .8(='é
-— i
.,____. I. _- fl __ -._-___ _ 'Y _._1Li. -. -2 ..._... L _._-.__.L_...._._. ._ -
l I ‘I I
‘3 ' we" OéXélO l .467 570 9'33;
i I
I_ ,_ x I J
I 3
4 I WM x ”I I éxéIO [.603 5.5 9.397 ’
- IO
xi
! .
I t4
I 5 gzsguux O éXé 2. . '134. [.0 £866»
--_-_-g_ _- _- -I
g.
9 i / 566
'_ Oéxéz ,:3:4' [.0 .
6 I5- cosn x I
X I
9 A I 5
4- o
7 ! q=TA~x O’Xn .10 .75 /.4
r___- _.---__- _- '_-_ -_._--_.. __ x __-r__rr -
8 ‘1 4 e 10
gum,“ o _. X .ee7 5.0 9.33
51,2 ___.r___._.___.__~.-_... -x _ __ -.__
I I
9 g“(2 Oé—Xé'o .467 570 9.33
L‘-.I-__---_-_-__,_-- _--.,.-x .............. -
FuNm‘LI'ONS AND RANGES
G U RE I2.

.. .-.————

 

26

According to these criteria, the following order was used:
1. Maximum link length ratio exceeding 15.
2. Branching.
3. Closure.
4. Minimum transmission angle less than 40 degrees.
5. Maximum link length ratio exceeding 6.

6. Change points.

CCMPUTER OUTPUT
For each mechanism synthesis which failed to be usable, the kind of

practical limitation responsible for the failure and the corresponding
values of the four arbitrary parameters were recorded on magnetic
computer tape. Later, a sorting program read this tape, consolidated
the data as explained below in Displaying the Results and printed the

results in a form convenient for plotting.

A Preliminary Study

Prior to extensive examination of the practical limitations to analytic
synthesis, an intensive preliminary study was made; in addition to
recording the failures, certain performance features of each usable
mechanism were determined and recorded. In the preliminary study, the
practical limitations were discovered using the same methods as for the
final study except that 10 degrees rather than 40 degrees was taken for

the minimum value of the transmission angle.

For each usable mechanism found in this preliminary study, the error was
optimized by re-spacing the precision points in the manner described by

Hinkle (l6) and by Hartenberg and Denavit (21) and the mechanism was

27

classified as crank-lever, double-lever, double-crank, etc. according to

the index NGRAS in Fig. 13. Then all results of interest were printed in

the following order:

Identification number of usable mechanism . . . NPLOT

Cla$$1fication number a o o o o o o o o o o o o NGRAS

Error, percent 0 o o o o o o o o o o o o o o 0 Ep

Minimum transmission angle . . . . . . . . . . TAM

Value of angle P corresponding to TAM . . . . . PTM

Maximum link length ratio 0 o o o o o o o o o RATIO

Number of passes in SPACE subroutine . . . . . NSPAC

Length Of links 0 o o o o o o o o o o o o o o o A,B,C

Indicator for negative A or C . . . . . . . . . NEGP, NEGS

Precision points as re-spaced . . . . . . . . . X2, X3, X4

Values of the four arbitrary parameters . . . . P2,52,RP,RS

For each non—usable mechanism found in this preliminary study, appropriate

results were printed. A portion of the printed output is given in The

Results, pp.70 and 71. The interpretation of this printed output will be

aided by the following notes:

1.

2.

3.

In the line beginning "RATIO EXCEEDS 15.”, the statement
LARGE = 1 appears. This indicates that the maximum link
length ratio, while greater than 15, did not exceed the
numerical capacity of the computer. If it ever did, the
statement LARGE = 2 would appear.

"UNABLE TO CLCSE LOOP" means that a failure due to the
closure limitation occurred.

"NSPAC = 100" means that after 100 passes in the SPACE

subroutine the test for optimizing the error had not

 

NGRAS CLASS

--.._-_——.______ —__———

 

 

-4 —.—. ---——~— -...___-

I CRANK - LEVEK

 

 

 

 

 

 

 

 

’3. DOUIBLEcCKANK
I———-~—-—+—--

DOuBLE-LEVER,’ BuT OuTPwT

3 LOULD ROTATE rimuousI-Y
IF IT WERE. MADE THE DRIVEIL.

4. DOuBLa-LEVEK

5 CHANGE ‘POINTS

6 CHANGE POINTS , KITE- semen
CHANc—le 'Pown‘s,

7 PARALLELOGRAM

 

 

 

 

~

 

THE INDEX NGRAS

 

29

been satisfied, so the effort was abandoned for the mechanism.

The function used for the preliminary study was the reciprocal function,
y = l/x, in the range l‘é x é.2. Sketches of each usable mechanism in
the configuration for the first precision point were drawn by the digital
plotter. A few of the 922 sketches obtained are shown in The Results,

pp. 72 and 73.

The preliminary study confirmed certain aspects of the procedure and
sharpened the pursuit. Most of the subroutines developed for the prelim—
inary study were used for the final study. Furthermore, the sketches of
mechanisms made by the plotter provided the writer much satisfaction and

encouragement.

The Computer Programs
Although a full listing of all digital computer programs and subroutines
appears as Appendix II, this section contains concise descriptions of

these programs to give the reader a comprehensive summary.

MAIN PROGRAM FCR FINAL STUDY

Nine passes in this main program were required to collect the data for
the eight functions (including two different ranges of the exponential
function) listed in Fig. 12. Automatic advance through the nine passes
was planned but an unknown total run—time and limited availability of the
computer required a provision for orderly interruption before the time
required exceeded the time allowed. Such an interruption should occur
between passes and should leave the tape on which the data was being
recorded in proper condition for a re-start. The indexes 11, I2, 13,

I4, 15, and NPLOT and subroutine COPY were intended to be used for such

a re-start. However, as it happened, the total run-time was within the

time allowed, so this feature was not used.

Index NP was used to identify each pass. For each pass, the data input
was the five values of X (i.e., X1, X2, X3, X4, and X5). Proper choice
of each function in subroutines FERD3 and ERROR was also regulated by

index NP 0

Next, the four arbitrary parameters were set for one of the 4050 combina-
tions. For this combination, a mechanism was synthesized by subroutine
FERD3. Then this mechanism was checked for maximum link length ratio by
subroutine MAM, for branching by subroutine BENCH, for minimum trans-
mission angle by subroutine ERRfiR, again for maximum link length ratio

and finally for change points by subroutine GRAS. Following this, certain
parameter restorations were made, if necessary, and the next one of the
4050 parameter combinations was set. At several points during the above

procedure, a failure may have been recorded.

SUBROUTINES
Subroutine FERD3
Input: X(J), J21, 2, ...5, 2(2), 3(2), RP, RS
Output: Y(J), le, 2, ...5, 2(1), 9(3), 2(4), P(5),
5(1), 3(3), 3(4), 5(5), RX, RY, A, B, c,
NEGP, NEGS, LARGE
(The index NP was also used as an input parameter during the final
study to permit changing the functions for successive runs.)
Calculate: A, B, C using the Synthesis Equation. (See Appendix I
for a derivation of this equation.) If A is negative,

rotate input link 180 degrees and make A positive.

31

Set NEGP = 2. If C is negative, rotate output link
180 degrees and make C positive. Set NEGS = 2.
Initially, LARGE = 1. If A or'C is quite large,
set LARGE = 2. If loop-closure is impossible, set

LARGE = 3. (This never happened in FERD3.)

This subroutine calculates the link lengths for the mechanism using the

Synthesis Equation.

Subroutine MAMI

Input: A, B, C

Output: LS, LL, LE1, LE2, LE3

This subroutine determines which of A, B, C and F = l is maximum and

which of these is minimum, finds the maximum link length ratio, RATIO,

and sets indexes according to the schedules shown in Fig. 14.

Subroutine BRANCH

Input: P(J), J22, 3, 4, SL1), J22, 3, 4’ A, B, C,

Output: NS, NSD

Calculate:

Calculate:

Compare:

ssp(3), 3:2, 3, 4 as the output angle for the mechanism
when the positive square root is used in the Analysis

Equation.
ssm(3), 3:2, 3, 4 as the output angle for the mechanism

when the negative square root is used in the Analysis

Equation.

532(3) with 5(3), 3:2, 3, 4. If all three agree (within

1:3 deg.) set NS = 1 and NSD = 1 and return to main program.

If “at, o o 0

Compare: ssm(3) with 5(3), 3:2, 3, 4. If all three agree (within

32

 

 

 

LLleDEx FOR

 

 

 

 

LsleDEx Fog

SHOKTEST LINK,A LONGEST LINK,(
4 A a c. 1 l A a C 1
L5 I 2 3 4 LL I '2. 3 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

INDEXES FOIL EQUAL LINK LENGTHS

(INITIALLY LEI = LE2 = LE3 = I)

A=B A'C A=l‘B=C 3'1 C=l
LE\ 2. 3_4

I———-—-—-4p~~.~-—— — ~-<>——-- —-.—.—._4
2 3

L E 2
I. —"'—".II—"*
2

—-.‘— __..>_____-__-_ _

 

 

 

 

 

 

 

 

 

 

 

 

LE3

 

 

 

INDEXES FOR SUBROUTINE MAMI

.“__--__

E00?

 

XI X2

 

 

Rx -

-—.—-—...-___—.

 

 

 

THE EKEPL-EU NCTION

 

f- I.-C_:LU-_KE--‘--5_

33

i 3 deg.) set as = 2 and Nan = 1 and return to main

program. If not, set NSD = 2 and return to main program.
This subroutine determines if branching occurs (NSD = 2) and, if not, it
also determines whether the positive or the negative square root should

be used in the Analysis Equation.

Subroutine ERROR
Input: P1, P2, S2, RP, RS, X1, Y2, RX, RY, NS, A, B, C
Output: 5(3), 3:1,2,. . . 101, EMAX, xmxx, rap, TAM, PTM, NE.
(The index NP was also used as an input parameter during the final
study to permit changing the functions for successive runs.)
This subroutine builds the error function, E(x), consisting of 101
ordinates (Fig. 15), determines the maximum error, EMAX, the correspond-
ing value of X, XMAX, and the maximum percent error, EP, based on the
range of y. This subroutine also finds the minimum value of the trans-
mission angle in the range of operation, TAM, and the corresponding
value of P, PTM. Finally, it checks for closure throughout the range
of operation. If closure is a limitation, the index, NE, is set to 3

and control is returned to the main program.

Subroutine (IRAS
Input: A, B, c, 15, LL, LE1, LE2, LE3
Output: NGRAS
This subroutine determines to which of the classifications in Fig. 13,

p. 28, the mechanism belongs.

Subroutine ATQ
Input: Y, X

Output: V

34

This subroutine determines the angle V: Arctan Y/X, in the correct

quadrant.

Subroutine SPACE (Not used during the final study.)

Input: x(3), 3:1, 2, ...5, RX, RY, xrmx,

E(K), K=1, 2, ... 101.

Output: x(3), 3:2, 3, 4, NOPT, name
This subroutine re-spaces the precision points X(2), X(3) and X(4) until
the extrema of the error function agree within a tolerance ofI:(.OOl)(RY).
A count of passes through this subroutine, using index NSPAC, is kept.
NOPT = 1 means: Error function not yet optimized. NOPT = 2 means: Error

function is optimized.

Subroutine SCALE (Not used during the final study.)
Input A, B, c, 15, P(2), G
(G is the angle of inclination of the coupler link for the
configuration of the first precision point, measured from the
datum ray. See Fig. 45.)
Output: X2, X3, X4, Y2, Y3
This subroutine scales link lengths preparatory to plotting so that the
shortest link has a length of unity. Also, horizontal and vertical

components of the pinned joints are determined for input to the plotter.

MAIN PROGRAM FOR PRELIMINARY STUDY

Only one range of one function was handled by this program but the
interrupt and re-start feature was needed often because of the long run-
time (more than six hours of computer time). But here, the interruption

had to be between any two parameter combinations among the 4050. Much

35

printed output was obtained; in addition, certain output data was recorded

on tapes 5 and 6, although the data on tape 6 was never used.

The plotting was done with the aid of an NSU Computer Center library
subroutine, PLOT. Another library subroutine, CHAR, was used to plot the
small circles indicating the pinned joints and to plot the identification
number, NPLOT, for each sketch. Otherwise, this program was similar to

the main program for the final study.

Displaying the Results
Since the purpose of the work was to investigate the practical limitations

 

to analytic synthesis of linkage function generators, the occurrences of
these limitations had to be displayed in such a way as to show a pattern,
if such a pattern exists. For each limitation studied, it was to be
determined if that limitation was dependent on the function being gen-
erated and if so, in what way. Also, the influence of the four arbitrary

parameters on each of these limitations had to be revealed by the display.

In view of the thousands of items of data stored on tape,the anticipated
manner of display affected the way in which this data was to be read

from the tape and printed. Graphical display on transparent paper was
indicated because of the possibility of stacking sheets of paper together

and looking through them, somewhat in the manner of a contour map.

THE INDEX I

In order to reduce the number of variables involved and still preserve
distinction, two of the four arbitrary parameters, the range angles RP
and RS, were combined into a single index, 1. Referring to Fig. 16, it

is seen that any pair of values for RP and RS gives a distinct value of

Fore RS (-31

I: “RP-3O + )RS)

 

 

 

 

 

 

G 30
up
Rs so so so I20 I50
30 I 6 II I6 2|
(,0 2 7 I2 I7 22
90 3 8 I3 I8 23
no.4 9 I4 9 24
I50 5 IO I5 20 25

 

 

 

 

 

 

 

 

FOR R5 (+) 2

 

RP
'RS

30

60

90

I20

I50

 

3O

26

3|

3C.

4|

4e

 

60

27

32

37

42

47

 

90

28

33

38

43

48

 

I20

29

34

39

44

49

 

 

 

ISO

30

 

 

35

40

 

 

45

 

50

 

QHE

INDEX I

 

FIGURE

I G

 

 

37

the index I, from 1 through 50. For example, for'RP = 60, R5 = -l20,

I = 9 and for'RP = 90, R3 = +60, I = 37.

THE PLOT

Then, for a single function and its range, using P2 as abscissa and 82
as ordinate, the event of a failure due to branching, say, could be
indicated by placing the number I in the cell given by coordinates (92,
52). It would then always be possible, though perhaps tedious, to
recapture all values of the four parameters for which branching was a
limitation for that function by reading the number I and its associated

coordinates from the plot. See Fig. 17 for some examples of the plot.

This procedure has the disadvantage of subduing the effect of changes
in the range angles RP and RS. But some initial plots indicated that
the limitations being studied were not sensitive to variation of RP and

RS, so the procedure was adopted.

Initial plots also revealed similarities in pattern for plots belonging
to RS(positive) and similarities in pattern for plots belonging to
RS(negative) but no similarities when plots belonging to RS(positive)
were compared with plots belonging to RS(negative). Thus, two families
of patterns were quickly identified: the RS(+) family and the RS(-)
family, or the family for the solutions and the family for the alternate

solutions.

This knowledge simplified the plot procedure since it reduced from 50
to 25 the maximum number of entries I to be plotted in each cell. At
each point (P2, S2), as many as eight entries I were placed inside the

square cell; additional entries I were placed outside the plot and

38

P2 = 330, 52 a I20 RP=90IIzs=-eo
l\—__Y—__—/
I =I'L
P2: 330, sz-Izo, Rinse, RS = —eo

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I17.
I
P2-uo, $2:zzo,121>=co,Rs=—I20 I I
\——-—‘,———/ ; 3 1
i=9 V, t ‘.
/ I I
P2=|I0,51=220,RP=COJRS=—30 / '. I
- P I I
1‘; / / E I
1>2=uo, sz= 220, RP-30,RS=—GO , 1,4/ I I
r—v——-—/7 ; , a .
1‘2 / ‘ I
34 — -' / 7
O / l, / I
/ ' 3
/ , ‘ ’
/ /" I
300- ‘1. . q
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2 7 I2
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~ALPHABETIL ChAKAC’TEK ”D" USEb FOL '3 [8

CONV|MuAYIoN o: auTlias I __-_-_-_‘_/ 22 23 D

 

 

 

TYPICAL “Du-mus Fonz Tue: ”PLOT

3ng I7

39

referenced with an alphabetic character inside the cell.

Finally then, each plot on transparent paper was to be for one of the five
practical limitations, for one function and its range, and for one family,
either'RS(+) or RS(-). By stacking transparencies and peering through

the stack with a strong light in back, the plots could be compared on the

basis of function sensitivity or family sensitivity. This, in fact, was

the way the Summaries of Plots were made.

With this plan in mind, the sorting program to read the data tape was
written to tabulate the data, function—by-function, in sets of five
columns, each column corresponding to a single practical limitation. The
tape was read once for the RS(+) family, obtaining 25 such sets of five
columns for each function; then the tape was read again for the RS(-)

family, obtaining another 25 sets of five columns for each function.

THE RESULTS

Number of Design Failures by Kind

By subtracting the total number of design failures from 4050, the number
of usable mechanisms obtained for each function is as shown in Fig. 18.
In Fig. 19 are given numbers of failures attributed to the practical
limitations of each kind, listed by function. Notice that not a single

failure for any function was attributed to the change points limitation.

Plots
In all, seventy-two plots were made, as tabulated below:

Function Branching Closure Transmission Link Length Total
No. Ratio

Angle
RS(+)RS(-) RS(+)RS(-) R$(+) RS(-) RS(+)RS(-)

\OCDQO‘U‘AOOMH
HHI—‘i—‘l—‘Hl—‘l—‘H
I-‘l-‘D-‘I-‘Hl—‘l-‘I-‘H
HHI—n—H—H—HHH
I-‘l-‘I-‘Hl-‘Hl-‘HH
I—‘l-‘D-‘l-‘HHI—‘l-‘H
I—‘HHI—‘Hl—‘Hl—‘H
HHHHHHI—‘HH
HI—‘l-‘I-‘l—‘I-‘t-‘HH
cocooooooooooooooo

Total 72

Several of these appear as Figs. 20 to 35, inclusive.

Summaries of Plots
By superimposing transparencies of the plots and comparing their patterns

by groups, certain similarities of pattern were observed.

40

41

._______.__ -. .__. ..---- --- _ - -1

‘ '" Isolate” oe'“‘u’s;'s‘ s’" '
NCT I o N “
FU __--__-___----._-.._-- ,MECHANIEMS OBTAINED

 

N9. KIND AND RANGE. FROM 4050 svumescs

 

 

.__._.--_ .... -.---_-.__ __ _ __ ..___. ray--. s1.

 

 

 

Iéxél . 333

 

 

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NUMBERS (_DF _EAMILUKES ArftzIBuTED _To CHE TDRAQTICAL

___.

 

LIMITATIONS: OF EACH: KIND

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FUNCTION BRANCMNQ CLOSURE Tamsmssuon L|NK Lemma CHANGE
N9 ANGLE RATIO “Pom-rs
RS(+) 34c, 429 364 454 o
I RSI-I 435 44I 344 475
TOTAL 78I 870 708 959 o
Rs(+) 585 .394 4Ie 389 O
2 R3(—) 530” 437 552 - 340 0
TOTAL. IIISM I531_I_.998 L729 0
Ragga} 939 I252 1_ 339 I 379 o
3 IT_{__S(:) 44-742; _I_285 I 53! I4I2 0
,TOTAL 1 |§_42 J 537 870 79I O
IEEfit) I-..5..?-§ I 4' 6 _4 53 ..2- 399 o
4 R$_<-_-_)_ I _4_7__5__ I. 407 - 42 I__~__~_L_;Ia2 o
ITOTAL I003 I 823N074 L7‘3I L 0
£5<+>I 3.48 I 45I 5.5.5..--I-.405 I O
5 15<+>I 45.5-_I -445 -454 = 5.55 I o
’IOTALL 786 I934:rII60 763 O
‘KSCHI 807 I £834.33] _, 369 0
e 3.55:2 I 6. 5.5.1 -5..8.2_-I_.E_i-_0 344.-- o
pomLLIZIéo Legal 907 7/3 0
IRSH) _I_ _6 26 I _BHLQH I-fi.§_I__.__ @465 I" o
,7 Bit.) 34.3.2.7.-- 5L4 ___5Z-5---__§0I o
ITOTAL1023I629 IIOI age 0
£5._<fi__I_ -99 55.12.58. 5.5.7 - “.552 I- 0
5 RE“) I 62.27 ’93? -5 -E?--..I 347 0
E'I‘ZLIIsso Igol 324, I727 o
55 (+2 6 55 L 55.4 5.5 7.. ---;I__-.5-.5_2... . _ 0'
9 Rs(—) I610 4I4 556 334» 0
"TOTAL. Im7— 6 5 ~ ”.72.? 5.5”.3 “HI-7.1.3 O
F IGURE I9

 

 

 

43

 

44

 

45

 

46

 

47

 

49

 

50

 

52

 

54

 

55

 

56

 

57

58

59

An attempt to represent these likenesses of pattern resulted in the
Summaries of Plots, as listed below:

Functions

Branching . . . . . RS(+) l to 5, incl. Fig. 36

Branching . . . . . . . . RS(-) 1 to 9, incl. Fig. 37
Closure o o o o o o o o o RS(+) 1 to 9, inCIO Fig. 38
Closure o o o o o o o o 0 RS(-) 1 t0 9, incl. Fig. 39
Transmission Angle. . . . RS(+) 2,3,5,6,7,9 Fig. 40
Transmission Angle. . . . RS(+) 1,4,8 Fig. 41
Transmission Angle. . . . RS(—) 2,3,5,6,7,9 Fig. 42
Transmission Angle. . . . RS(-) 1,4,8 Fig. 43
Link Length 0 o o o o 0 RS(") 1 to 9, inClo Fig. 44

For all Summaries of Plots, the shaded area represents design failures
due to that kind of limitation; the clear area represents freedom from

failure due to that kind of limitation.

A certain judgement was employed in making the Summaries of Plots:
where no more than two cells having one entry apiece coincided in a
stack of six or more functions, this was regarded as clear area. Also,
where only one cell having no more than three entries appeared in a

stack of six or more functions, this was regarded as clear area.

A Summary of Plots was not made for Link Length.Ratio, RS(+), because

no pattern was diScernible.

For the Summaries of Plots of the branching limitation, the dense regions
caused by a concentration of entries in these regions were judged worthy

of special notice.

Sample Output from Preliminary Study
Two pages of printed output and two pages of corresponding sketches of

mechanisms are given as a sample of the output from the preliminary study.

52 DEGREES

60

 

220 ~

 

[00“

m4

 

 

20 T “‘\

 

 

l I 1 I l ' f I I F
IO 50 90 :30 I70 ’llO 750 290 330
P2 DEGREES

NOTE-:2 SHADED AREA REPRESENTs oesmn FMLUKE.
SUMMARY OF PLOTS
FUNCTlONS’. ITO 9, mcLuswE,‘RS H")

BRANCHKNG

HeuRE 36

$2 DEGREES

61

 

340 4

 

 

 

 

 

 

 

 

 

20 -L

t ‘ l I 1 I I l I
i0 50 90 I30 I70 ZIO 250 290 330
‘ P7. DEGREES

 

 

 

 

. >»._
NOTE; SHADED AREA REPRESENTS DEEMGN FAILURE.

SUMMARY OF PLOTS

 

Fuuc-nous: \To 9,1McLu5NE , RS Q”)

BRANcrH N6

FIGURE 37

$7. DEERE ES

340

300 - ' - ’ [—
260 -‘

220 -

W I:
.00-

w~ D

‘20.

62

 

 

 

 

    

 

 

 

 

 

 

1 ‘ V \ x ‘ I l I I A x. 1

IO 50 90 130 |7O 2\O 250 290 330
'P2 Daakcas

NOTE: SHADE!) AREA REPRESENT: DE$\GN Fm LUKE.

SUMMARY OF PLOTS

FuNCTlousi 1 To 9, mcLuswE ,‘RS (+3

CLOSURE

FiGUKE- 38

5’2. DEGREES

 

340‘ .

300‘

260 d

2.20 ‘

180-1

‘40-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

)001

204

 

 

 

 

 

 

 

 

 

 

 

 

 

r T I

j T i f T I
IO 50 90 I30 no 210 250 290 330
P2. DEGREEs

MOVE: SHADED AREA REPRESENTS DESIGN FA\LuQE.

SUMMARY OF PLOTS

 

FUNCTlONS; | TO 9) mcLusuva,RSC—)

CLOSURE.

 

FIGURE. 39

 

 

 

64

 

 

52 DEGREES

 

 

 

 

l I ' I I I I I
IO 50 90 I30 I70 2|O 7.50 290 330
1:“). DEGREES ‘

NOTE: SHADED AREA REPRESENTE) DESlGN FAILURE.

SUMMARY OF PLOT s

 

FUNCTIONS: 2.,3,S,6,7 AND 9 1 R5 (+7

FIRST 6. secouo 'DEK‘VATIVIE3 WITH
LIKE sIGNs.

TRANSNHSSKDN ANGLE

 

FIGURE 40

52 DEGREES

340

300' - ‘

1
2.60 J

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

220 -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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so?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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I 1 V I l I T I
I0 so 90 I30 no me 250 290 330
P2 DEGREES

NOTE: SHADED AREA REPRESENTS DESIGN FAILuKE.
SUMMARY OF PLOTS

FUNCTION81l74 AND a , RS (+3

FIRST é, sacouo DEENATIVES WITH
UNLIKE SIGNS.

TKANSMI$$\ON ANGLE

FIGURE. 4|

 

 

52 DEGREES

66

 

340

300 d

2601

 

 

 

 

 

 

 

 

 

 

 

 

 

22.0 -

l90~

 

 

 

 

 

 

 

 

 

 

I40—

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I l I I I r I I I
I0 50 90 I30 I70 2|O 250 2.90 330
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NOTE: SHADE!) AKEA REPKESEN‘Ts DESIGN FAILURE.

 

SUMMARY OF PLOTS

FUNCTIONS: 2,3,5,G,7 AND 9 ,KS L“) _
FIRST Q, Sgcouo ‘DEOJVATIVES van-H
LIKE $|C1N s.

TRANSMI5$I O‘N ANGLE.

 

FIGURE. 42

 

 

52. DEGREES

60‘

67

 

340 "

300‘

Zea-I I

210‘,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I80“

 

 

 

 

 

 

 

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r T I I I
I0 50 90 I30 I70 740 9.50- 290" 330

P7. DEGREES

NOTE: $HhDED AREA REPRESENTS ‘DEQIGIII FAILURE.

SUMMARY OF PLOTS

 

FUNCTIONS: I,4 AND 8 , R-S‘C")

FIRST Q sccouo DERIVATIVES WIT“

UNLIKE SIGNS.

TRANQSMISSION ANGLE

 

FIGURE 43

 

 

$2 DEQRE ES

300‘
2604

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j I A V U V I _ l V 1' f ' 1;“ A V 1
IO 50 9o ' I30 I70 ZIO . 250 290 330
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NOTE: SMADID AREA Kafka-azure. DESIGN FAILuILe,

SUMMARY OF PLOTé

 

FUNCTIONS: I To 9, INCLUSIVE. , RSC“)

LINK LENGTH RATIO

1

 

FIGURE 44

 

 

69

Among the 922 usable mechanisms obtained for generating the function l/x
in the range 1 f x f 2, there are 190 mechanisms having less than one
percent error and having the minimum transmission angle greater than 10
degrees. And among these 190 mechanisms, 46 have the transmission angle
in the range of operation always greater than 40 degrees. For mechanisms
with less than one percent error, the greatest value for minimum trans-
mission angle in the range of operation was 68 degrees. See NPLCT : 296,
p. 70. The smallest error for a mechanism with IAMl>>4O degrees was

0.1 percent. See NPLOT : 427, p. 71.

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INTERPRETATION OF RESULTS

Only those results relatively independent of the function being generated
were considered significant. Thus, in the Summary of Plots for branching,

attention was focused on the dense areas.

However, the Summaries of Plots for transmission angle were clearly sepa-
rable into those for Functions 2, 3, 5, 6, 7, and 9 and those for Functions
1, 4, and 8, according to similarities in pattern. Note that Functions 2,
3, 5, 6, 7, and 9 have like sign for their first and second derivatives

while Functions 1, 4, and 8 have unlike signs for them.

Branching
For RS(+), the diagonal dense band in the Summary of Plots, Fig. 36,

shows four—bar mechanisms having P2 = $2 to be quite susceptible to the
branching limitation. The Summary of Plots for RS(-), Fig. 37, predicts
frequent encounter with the branching limitation when 52 is in the third

quadrant (180 to 270 degrees).

Closure

The Summaries of Plots for closure, Fig. 38 and Fig. 39, have enough
resemblance to those for branching to suggest a correspondence. Branching
is concerned with the two different configurations that are possible with
four links after two of the links have been fixed relative to each other,
Closure is concerned with the configurations that are impossible after

two of the links have been fixed relative to each other. The nature of
the correspondence, if there is any, is not clear. But it is clear that
choosing P2 and 82 so as to avoid branching will also avoid closure
problems.

74

75

Transmission Angle

FUN'JTIOBS HAVING LIKE SIGhB FCR FIRST AND SECOND DERIVATIVES

The Summary of Plots for Rs(+), Fig. 40, indicates that 52 should be
chosen in the first two quadrants and P2 anywhere but in the first
quadrant, or $2 in the third quadrant and P2 in the first or third

quadrants to have a good transmission angle.

For'RS(-), the Summary of Plots, Fig. 42, condemns the third and fourth

quadrants for P2 unless S2 is in the first or fourth quadrant.

FUNCTIONS HAVING UNLIKE SIGNS FCR FIRST AND SECOND DERIVATIVES
The Summary of Plots for RS(+), Fig. 41, shows by the shaded diagonal

pattern that equal angles P2 and $2 give a poor transmission angle.

For RS(-), the Summary of Plots, Fig. 43, reveals the second and third
quadrants for'P2 to have relative freedom from the transmission angle

limitation, whatever the value of S2.

Link Length Ratio
For’RS(+), no pattern independent of the functions could be seen, so a
Summary of Plots was not made. For RS(-), the Summary of Plots, Fig. 44,

gives two clear areas and was impossible to ignore.

C OP-CLUS I ons

In designing a four-bar mechanism for a function generator, if a minimum
transmission angle of 40 degrees in the range of Operation and a maximum
link length ratio of six are assumed, only about 8 percent of the solu-
tions obtainable from a three precision point approximation are usable,

exclusive of any consideration of error.

Choice of range angles from 30 degrees to 150 degrees has little influence

on whether or not one of the practical limitations is encountered.

Branching is a practical limitation to such linkage design; making the
starting angles for input and output links differ will reduce the proba-
bility of branching for input and output links rotating in the same
direction if the function has positive slope or for input and output links
rotating in opposite directions if the function has negative slope. Avoid-
ing the third quadrant for the starting angle of the output link will
reduce the probability of branching for input and output links rotating

in Opposite directions if the function has positive slope or for input and
output links rotating in the same direction if the function has negative
slope. These same measures will reduce the likelihood of problems with

closure at the extremes of the range of operation.

To have good force transmission characteristics in four-bar mechanisms
(i.e., to keep the transmission angle greater than 40 degrees in the
range of operation), the starting angles for input and output links
should be chosen from the clear areas of Fig. 40, 41, 42, or 43, pp. 64

to 67, inclusive, as appropriate.

76

77

As long as the starting angles of input and output links are not exactly

the same, change points will not be a practical limitation.

No significant generalization can be formed from this work which will

aid the linkage designer in avoiding the limitation of excessive link

length ratio.

 

 

APPENDIX I

Derivation of Eguations

THE SYNTHESIS EQUATION

This is the well-known Freudenstein Equation; its derivation (21) is
given here primarily for the purpose of defining notation. Refer to
Fig. 45, the four-bar mechanism.

A Sin P

x2 =.A Cos P y2

C Sin 5

x3 = C Cos S + F y3

By the Pythagorean Law:

B2

2
= (x3 -x2)2 + (Y3 -y2)
= (CCosS+F-ACosP)2 + (CSinS-ASinP)2

C2(Sin25 + C0525) + A2(Sin 2P + CosZP) + F2

- 2AC Sin p Sin 3 - 2AF Cos p + 2(CF —AC Cos P)Cos s.
Rearrangement gives
2A0 Sin P Sin 5 + 2(AC Cos P - CF) Cos s =
= A2—BZ+C2+F2-2AFC05P.
And, after division by 2AC,
Sin P Sin 5 + (Cos P - §-)Cos S =

2 2 2 2
_A‘B+C+F-E
_ C CCOSPooooooooooo(Eqnol)

Using the trigonometric identity for the cosine of the difference of two
angles,

Cos (p -s) =Cos p Cos s +Sin p Sin 3,
in (Eqn. 1), it becomes

2-132+CZ+I=2

Cos (P-S) = A 2A0

-§CosP+§ CosS, or

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2 2 2 2
F F A — B“ + C + F _
A Cos S — C Cos P + 2AC - Cos (P — S).
F F
LET. H1 :: 'A' , HQ : E and
___ A2 - 82 + C2 + F2
H3 2AC ‘

Substitution in the above equation gives the Freudenstein Equation, or
the Synthesis Equation:

HICOSS-H2C05P+H3=CO$(P-S)........o Eqnc2)

This equation must hold whatever the values of P and 8, since it was

derived from the geometry of the four-bar mechanism.

For a three precision point synthesis, (Eqn. 2) can be written three
times, once with each pair of (P2, 32), (P3, S3) and (P4, 84). The

result is three equations in three unknowns, H1, H and H3. From H1, H2

2
and H3, the ratios of the lengths of three of the links to the length of

and .g . As used in sub-

the fourth can be determined, e.g., é',

*HK1

routine FERD3, F = 1.

THE AI-JALYS IS EQUATI ON
Once link lengths have been determined, the Analysis Equation is needed
to investigate the performance of the mechanism. That is, an equation is
needed for S as a function of P. This is obtained from (Eqn. 1), above.
To simplify the expression, let

Q1 = Sin P , Q2 = Cos P -‘§’ and

A2-B2+C2+F2

2AC
Then, (Eqn. 1) becomes

— E.
Q3 — C COS p o

QISinS+Q2CosS 2: Q3 ,

81

or, when sine and cosine are expressed in terms of the tangent of the

 

 

half-angle,
2Tanir s 1 Tan2’gs
+Q - = .
1+Tan2é-S 2 1+TanZ%S Q3

Now, this is a quadratic equation in Tan % S giving S as a function of P.
After some rearrangement, the Analysis Equation appears:
+ 2 2 2t
<21 -(Q1 +02 -03)

S=2Arctan ........(Eqn.3)
Q2+Qs

 

The positive square root belongs to one branch of closure, the negative

square root to the other branch.

APPENDIX II

82

Computer Programs in Fortran

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__-_-_-______.51-3__,I,.I'_.§_?_i-__'/-'l :53"; 3“ f5 f3 .8 ‘3 L3 8 _a_ 2 0° “___ ~ ________
507 V =‘130. nJF(-Y/X)*L 00 /P1
GQ TU 513A
509 V =' uTAi‘gF' (Y/th 180 ./PI
_“ nglflmél“ 3 - - __________
512' V = 5‘. .
______________ i 13.-.?5LT“.1._'__...w~_.-....-_.._-____-_ _ - -
: 3 2'3
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96

11141-31 _________________________________________________________________________________ 1121-13755 ___________ ‘

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“ \I; =13J *stx
_mGp-T°-!??Gj w 7J‘_5 7:0 3 7o19>_._1

-7” \42}; r1+M

 

m e .s m 1: 711172111 5 >7 111‘: ‘4") :‘2‘ 1*‘c‘1’1‘5‘ffzm‘rrrz‘c‘s‘;
NuPT=1

’171"Q’EYI"E1WTTW3“E"W3 WU. 3 E(1U47:5:73”ETIQ§T§WT

_Z(1)=1. 3 7(5_)=2. 3 .Hfix 1. +(XMAX- X<W1))/RX~

___.....,__,___ __._._._.._..__- -..—....”

 

 

2(2)=1. +<x!2>- (1)379x s z<3>=1.+<x<3>-xc1))/Rx
2(4)=1. +<xr~>~1<1>>qu

-——_-—————..-

 

@.X= ZCJ+1)_- ZtJ)

 

 

 

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Hi} lfiiJR (fr-'14.}?

 

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7015 MSPAC JSPAC+1
,.._-_-___w_ RETURDC _- -- - .................... —— — --—

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”_Zfif11137511g2

-.D?’J--/.-. U51 JFW1-13 ___

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1*(EV(J>' LfillZHEZ-ZAQELZQEE

 

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U TO [00"
L‘(J)~->-'x<))7009 2009,70un

 

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TOL=-1.*TOL

 

ETUL= “(J7-L’(’+3}
IF(T3L+ETUL770121731237004

-JE1I3L:EIDLlZfiLZLZF12LZ0U4

CONTIJUE

 

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IF<2¢AX~ 2(1))7n14.7015.7143
.-LJ \[T 1.31112;

 

 

GO TO 7017
DM=Efl11)15m(2)+EH(3)+EM(41_H_

 

GO TO (7016:701617‘13o70187od

 

IF(CHE K)70?1:7‘ 2417024

 

2(2): 2(2)-~TE~*(Z<») 2(1))

 

 

ZS: Z(J) 3 731=Z(J-1)

PRDP=ZMAK-231

--..‘-.M__fi._-______- ...--.-.1-_‘ -.. -..-E - .-.. ..__._..__.___ -...___..~.-.___._._.--—.-_—__—_—_~-_._.....-__—--——_-_—_-

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—_———.——-__-._-. »._....

__.._..._..___._ .... -....—

 

 

97

ETN413U ,__.07/13/65.,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7(J1= ->-\f *RIQP
-Y(J1= K<11+ti<J1-1.)*RX
PRCP=ZS-Zhéx 7 A __”uwm_ ____________________________________
___________________ 2(3411523145T65IPR3P"”
____________________ 1’ ( J " 111“ 1.2_t£.2.$31_:.1_2:_1_9_118X___._____ _____________
VOPAC=NSDAC*1
_“_________h3m_-J?=TLJ1Nwwm-~wmuflu-uu_ ._._. -n. - -
7017 00-: '1(11+FW(21+E <3)+EM(4)
___ Cir—K-‘MVM41/J'1 -.?5 __________
IF(C%4 K170g l 7 25,7025
_““~m_wm_792_§_ _S_TEP= a1%T( 2113.33 __ 3 _____
2(41‘/(41+“*"3*(Z(51 -Z(411
1(41-A(11+«Z(41-1 )wRX
'" \QDA,-V“DA”+1
........................... 3‘6- __ .193- v ______
7021MJPT=
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98
FT.‘~."4 ..30 ____________________________ .- ______.___-._

 

. 93nr{x> = “v3.14150255/150.

....................................... 417113/65 _-_-_______.-
SaganUTIug snMLE <A.a.c.Ls.P.G.x2.X3,X4.Y2;Y3) '

 

 

GU,” TI! 5 "[1 ’ 9 "’..%3_5_9_3_a-5 4.)) '13- --—-—

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(2.4 = Q/A
>14. = “sh/n.
1'. 11—35253 5 t3 ’3 ”A *
_ 5_9_?_-_,p5.°-_r“-.'__3.._o__ _____________________ -
'59 3 .‘3/45
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‘< 4 -= 3. . / 7?
Cu m as
503 C9 = l.
________________________ I4 r3--_’5__,-\[_C -_____ _-___,_~_______-
5:? .1 v; C
______________________ ‘1‘?” 5.. J. o-/_ (3.- -..... -----."
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_____________ .5. D_‘5__)<..2_-E__59__1C $15.1. RAD F- (-P H .. __
X35 =‘ ){2' 4" N" *CUSFCRADFKGH
Y2 2 AP *SLLLEKRADFIPH
Y3 ‘=~ Y2 *' F3!“ *SINF(RAUF(G3)
___ EEILIRE‘J --..
END ’
__________________________ a ---____*___m___-_____

. ................................................................... — .————._

 

 

MAIN PROCRAM FCR PRELIMINARY STUDY

99

 

 

 

 

 

 

 

 

 

 

     

 

 

 

"’ 100
16130 _m_u__m“fl__ H-.Awn_1"_m_wu..nmm1“1wumfl.u_n_h_m_mmmmElléiiéé __________________
PR'JD ”II-1A1 FUU 0 ° .
01%;:o101 x<b),Y(5)7P<5), 5(5) M<3>gXSAt521Etlos);NAME(10)**—-—“—--~-
5056 FORHAT(13A8) ' ' '

———————— '3300"FUR1-1'AT(59"9'.‘J)‘“ - -~- 1
5051 FURm\T(25HM FUJCTIUN Y = 1/X: X1; 1615. 8:7H X5 3 €15.8l)

“““““ 5054‘”FURI’1 11‘1T'('121’i'i""t3r 11:15 IN FERULS :‘T‘"‘""“"""“““"““""“""”:"“""“""‘-""“"“"""'
5010 FURHAT(2$H UM ULE TO CLOSE LUOP )
§2113“‘f-‘U.'-<.“11\‘I'(ELM—RAT1L3 1-Lxc 61.1.38 1;»: RA!IO‘B‘f’i‘élfil§‘1.10H"“EARGE“‘=T‘o"1‘4')“"‘""‘“"
5n17 FURM1T<12H 821~0n1~9.1 ”' ‘ " ‘ "" "'

--------- 5019' FuRy-z 3C1" ('1 411““(1‘11‘1‘7153 1:HRU‘~’ 1. ) "“ ““"""“"""“"
5052 FURn1T(23m CM€K NEG IV S;°ACE.)

--------- 5023 1'1 041.1r1; 1%‘4'11‘5131‘10 111.1; ) -----__1__ 5 —-*-----~----'
5026 FURn\f(de TM LESS T1AJ nu- ES. TAM = 1614.518H PIN-=~.El4.8)
b029~F521111271 RATIO EXCEEUS 6; HAIIO -'TE15:87 ” -*

9050 FOR:- .ATIISH CriAnSE POINTS. )
----------- :10olr-F1121121'1I 11--’n: 1 11111r;>---~-“----~~—--- --* - -—- - E41
5032 FuRH1T(16H PARALL:LOGRAM.J ‘1

"""""" 5 U 3 4‘”F‘UR1‘IRT(1IIH‘_—~"3".0 T 5 I 4'; .1. "1””?"Lin A 5"”=‘"I’4‘i7H”"EP"“—T1’ €15 ,‘ 3 )‘7'"""""‘"“"""'”""I
bfléb FDRHAT(6U I[\1=ClD: 3: 5H PT34 .L5. 3,7H RATIU- E14. 8:7H NSPAC: I4)
5067’FUHHAIC4U“”2=“‘flooth H: Cl4oaiéH C: t14o 3) 3H NEUP31206H NEGS3 I2)

5038 FORnaIC7U A73 2 251518174 X5 =‘9E3.5 8:7H X4 =5:E15 8)

 

 

 

-------- “5039WFUR1«Ttod‘“V2-r/1214H 14-17.2111 HP=F7-?74H”R§= F7 2” ‘I —

 

50u5 FURH11<26H Cu1HLETED in4UUUH LR=1315H LI_=;325H LRP=1315H LRS=1311 E}

 

1'5'1'1‘51‘13‘3353'1CSTW’MWPLQUTS’I 4 ) __________________________________

RADFCZ)=/ko1-4199265/ILU '

III¢~IWII r13)‘””"*“"”

Ncfifi-L 5 uzuS :,1

7'35" {1‘ 15357:);I1.4431515771531713 )5 —————————
RCA” DOUU:X(1)1X(2)9X(5)JX(4)1X(5)

--------------- PHIRT“5UUU?XIIJTXI2)7XI3)oXIEIJXISI ’—
REAU 5055aIl:I2:I3:I4OI5:NPLD[

-—————5055“FURHWTT6TWI

IF(IigEQolgANUUIZoElegANDoI31EG11§ANOQI41EQUEZAND:I5gEQ:2)50571

--—-___'___—-———..

 

 

--n———-—-— —--——-.——-_--

 

 

-— .—--—-.---————<

 

 

 

_-___—__.-.—.___— .f‘.. a , . -.---.__-._—~————_—_——_-———_._--....—._.—.___—....—.-_._._...._._

5057 1<s=2
---------------- ISO—"'1'J"5U"395"wm _—
5893 K5=l
_— 5 O EJ9I'K3'31‘L-) ‘IKER'CKS‘J'
GU I") (506‘715UUIIoKS
_5USIERRITEWTAPEID:’WAHCCI)oI--1JU7
URITE TAPE a1<1AME(I)1I= 1 1U)
” ““““ 50¢n“P&1UT”5055;11;12:13r11715.UBLUT‘ """"""
PNINT 55:111H1); XCSI
‘ Fi" I1-133F23 12- USERS: I3-15I5F'4-‘3'I4' 1$P59 I53 '1'
P(2)=10. +F3 r43.
________________ I30 55153.~J=i.p5""m-I___———_""~'_I-"I"I"_-'___-I_“"—-“~'—'—-_-_"—"‘__—~__--'5-——__""”_—_'—"-—"
50111 XSACJ)=;’;(J‘
.----.“ DU :33") 11431119
PIZ)-HI2)+4U.~
S(2)=20.+F2*401
F230. 2 5 5
_..___ ______ DO 5120 LIV-”2; 9 IMAM"
5 12:.1
58(2'3;{2)+45.i
RP: I" JadU-..
F3:Uo-

 

 

       

 

 

 

.... __-..___._ _-.-...._——._—-

 

.1_,__....--_.____....._._.________-._..____.___-_“-_._._._~-___.-_--.._._.—-

 

 

 

...-.h
v————_ __‘m—F-

A .A. ._ .. ...—2 ..-... ..-_——-——..__._—-_- __-—~——-——————c——~_....————.——.--—--._.—_.—__.........._1_..-

—.~_._-_L-_.-____.L_1 1. 5 ._ ,_ a7- m.——-—-———.—.—.—-..-V_._-—_.____———-——_-———.——_—._-k--_-..._._——_.__-Lr

 

 

f “___—u

._.__-..._.--.. ..1__-_~- _—.-.____—_-.___—_——~—---_._--_._--_.—__1_-

’1.” Elyi'vz 11 . .~ .
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a --——-—-—'0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 101
'N 4 ' ‘5 ° ________________ 0. Z£§E{j‘_§_-____________-3
”U 5548 Lnr31019
- ,-U-Ib= _,_LL__ _ _.. . L_-_ H _-ML_
RP: hP+301~
------------------1<‘t:3= M a: 3 of? ----------------------------- --.. --------------------
F4: 0.
5V+8m LRB31475 ________________ -- _____
14:1
“‘ R5=HS+30. “‘ “ ’ “‘
DO 5J~8 333:112
TEE) "'T 3(50351'531’5 3771 5 ”"5““
5066 GU [0(5007. 5UU.9}1wHS
""""""" 5 065 T5: 2"‘3 TERS=Z""""""" "” . ‘
GO TO 5066
“_“_*~5UU9‘R35:175H3
5007 NSPAC=O A . . , ..
————————— 5UDEWUACCMFERUXAX:YWPVSWRP3351'X75YTA73;‘75:32753637EARGE)
GU TDC50351501215053)1LARG
--------- 531533"?5R7RVT ‘5Cfi34mm-"‘"-!~-—-’-”‘—_Qm"""‘—--~H’
5006 PRINT 5010
"'15IT§ TADE 51 IX1PC2)15(2)1HP;RS.
P11 T DU371A1 1L1NFB1“EGS
GU T7 5015
"““""‘5‘o‘3‘5 ”CALL ..Aii I c A . 971:. LS. LL. L’E‘ATCE‘A'I’CEs‘fA‘AT'fb>
IF(%A T10 15 353111501L 5:12
5U12flpdlzfiT 5U1é1nATIU1LARG “
IX=2
'HRTT?”TAFE*§T*TXTFT2YTS (ZTTRDTPS
“_ ‘G0 TV SulS
________ 5511"IIfCNBRAVCfiT?“S“K"§“575$TXJU) —
GU T0 (53141501615006)1NSD
5315” P.75T 5U17
X=3
“___“: —————— €116:IE'O'XIEWTAb—E—57 ’ X; P ( 2W) 1 S ( 2-37RFP'TRSI

_______ T 50571A13101NEGPIEGS
____________ gOJ'TU 5U'5 “m—
5014 Pl=P(i) 5 223 9(2) 5 82: 3(2) 5 X1: X(1) S Y23YC2’ .. ‘

CALL EQRWQ<F11 213? RP1HS1X11f21HX1RYVQS K1B1C1.E1EMKX1XHWAX 591+ .—

---—...
———.—————-—-———.w... ----.n ....-n- .2 ._ __ W._._.---._...._..——- "_._--—._. —-..—-_....

--_----__-_P_Q}_§_.D__-_I_'_e_T.-_‘5~lJ_1_?_-__
1X34 ' _,
lnlr" __TAPE 51 IX1PC2)1S(2)1RP1RS
(:0 Tr) 501.5 .

5003 CALL_53AAC(7 ,E,XMAX,RX,RY,NOPT1NSPAC)

‘-—-——-_
_—___*—~--.

G? f (5 u;57¢ ‘5035>.~JDT """""""""

 

 

 

 

 

 

'--_-—
---—._-_-.__.—_.-.. -. _ , ~__._._..----‘__._—_————_..—-————->~—__~-__---__._————_—-_

--~Em¥ quT‘ T_Ap51-IX1P(221§12)1RP1RST
GO T615U1:
_M»_HH 5020 IrC‘Tixpf17 35308 5_02215022
502'2 PHI‘T 5323

 

_————-

 

 

..--21. _ -.:-_. _ V-.._A.___—...._-—--——._—————._—-—_——-——.—_—~——._._ __————__—_—_.____

 

 

“M _._IX=b _ 7 _________________________________
AKITE TAPE 51 IX P‘2)1S(2)1RP RS
KLHW‘ GU T7 5015
5021 IF<TA1'13.)9524 502515 25_

--.. _‘
"'—--‘..__

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”N4 '- 3 0 ' ~__________ ____“_____'_ _ ___9 Z[1_3(_6_5
5024 PRINT 5026 TAiuI PTM
1X37
NRITE TAPE 5: IX Pf2) 3(2):RP0RS
“““““““““““““““““ G‘V’T’f 5 u 15‘”"“""""" ‘ """““""
5025 CALL HAHI(A 9 CoLS LL L51. LE2 LE3. RATIO)
""""""""""""""" IF(RATTU- 5375377557327; 5028"""" . ___-_-____..
5023 PRINT 5029, RATIO
1X38 ‘ ' f7 7— .
UNITE TAPE 5:-IX:P(2):S(2)?RP:RS.
GO TD"‘5U15 "-
5027 CALL GRA3(A B:CaL3 LL LE1 LEZ;LE3oNGRAS)
——————————————————————— VONTUT7373737oaiv w100 EUISIJNGRAS
8 PRINT 5030
IK=9 .
WRITE TAPE 5:‘[X0P(2):S(2)0RPJRSl
"GO-T’T'S‘UIE‘
9 PRINT'SOSi
-_ TX31U

WRITE TAPE 53~IX0P(2)35‘2)0RP0RSA
GU TH 5015
10 PRINT 5032
---------------------- TX311. , . _
«RITE TADE-5.:!X;P(2)a5(2)DRP>R5:
______________________ 33—77775U15 _ _
7 TY=CtSINP(PADP(S(2)))PA*SINF(RADF(P(2)))
TX5C§CUSF(QADFTS(217"A‘CUSPfKAfiPCP(Z)33315.
. CALL ATG(TV TX:G)
---------------------- JKCC"S"KT:TXT"CILS{P"G{X2oX3¢X3:Y?:Y3)
NPLUT=NPLOT*1
CALmetUTT577:1177271UW771WUoI
CALL PL0T(0000030p5000509)
CALL PLOT(009110;235077500)
CALL PL0T(D.:0.:D:50.9300 )
--------------------- *rQEmCHAP—;mfi32poou37 1.H0:1 001006310551)
“ CALL PLOT‘CQDOOIZp 5001500)
---------------- PWCAECmPCUTTYQTX77135OTTSEG)
YP= Y2’3032 $ XP=X299032
CALL CqAR(YP“XwF1Hn1119.I:564AQUG4)
’ _CALL PLOT("/ X7 2:509:500 )
______________________ CALL— PLDI "(Y‘S‘IXSZ I 50 g 0 DOT)
YP=Y3'bO32 $ XP=X32.03?
AWWPHu—"CALitu'VCHIA'Q—(YP 1677-1110: 1 a 027.13 547705.41
CALL PLUT(Y5:X312:50opSUo)
CALL PEUTTWTTX1.ITBO.oSU.)
N YP3P.032 5 XP=X4'. 037 _
_____________________ CALL CHAR ( VP: X7 {1140 1: ‘53-'75 5473-0354.)“ ___- “---
_H- CALL PL0T(0o:X49205001500 ) _
————————————————— CALL PLOTC CT; 07;; 1 50 q :30—§~)-"““_““““ “___---
M0P=JPLOT ’
'”DJ”1§UH33133
dA‘J)-NOP-JOP/1U*1n
““100 NUP=JUPI10 “7" PM“ u H“ ----------------------------------------------------
VPLO“(NA(3)*64+NA(2))*64+NA(1))*1073741824
YP=-.5 $ XP=.25 """"""""""""""""""""""""""""""""""""""""""""
CALL CHAR(YP XP NPL003109o0125a.125)
- CALL PL0T14090002050905001

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—~—~____

 

 

 

‘_-_._,., .--r. _ A-.._.--w.u..-.._._,..- A ...- . 11-, .._._-...--_-......__.__.._-._-___..._..___.-.-____-_“.___.__.._-_..__________

.---—

IN4c3D 07/13/65

..—_—_¢-—.-——————..____--.._...._.- ~_._._-__-————.—.—___—-__._—____--———.-_.-—.-—-—_..._

JALL HLDT :0..;.;-1;3.00.:1?0.)

 

 

 

iKKrJ:\(0)av(fi} P‘2)35(2)0RP2R3

 

331 ~ #4034, Pvmvms. a? -------

PR- T 50:53)“;\ 3PTII’RAT10’NSPAC

”TTTTTTA'ETST %OT7NGRAS;EPTTAM. PTMTRKTTUoNSPICo I. BTUTNEGPTNEGSa

 

 

—————————————— fiR‘ut-5557,4.u.u.w uPT ‘EGS“”"‘
03147'5033,A(2),x<3 ) x<4>

 

5015"PRILT"5039;FI2T}$I2)oRQaRS""
TIM2=TIMCFIU}

 

------------------ T r (TT 1:. . uTz-T ”71751-52”;ng 0 2._...___--___-
53-62 IFIITIVZ' TI 1)oGT. 7290000 0)500215004

—————————— 3004 GO IT’DUZbaJL4ljakRS m“_ T ‘
5041 QS=-“.. *nS

 

5043 ,3‘T3\Sj+p,g 4777NEGP
5043 ”(°)‘)Id)-ID”

 

IFIFI ))304455 42 5042

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"“""”“5344 PI2)~T' <23+533 ‘
56-4 '2 '40 T9 (5u459574.6 )aNEGS
5046 SIP.)= 3(233350
'1:85:J
""""""""""" TTISQ )>)o/7,5:45 5045
5047 8(2)=b I2‘+?5no
“"”"”“5045 93 5 49‘sz75
50’0 X(J)=ASA{J§
5348‘CUNTTFU:
Sfififl CO“TI. UE
“““““ 5002 PRIJT 5334""U‘CTS‘ERP"CRS‘YVS 3JPL0T
END FiLE 5
' “EENTT‘TTI ”E'"S
Emu =1LE 6
EMP"PTCETB
___________________ SIR?
END

‘~~H
_————___~—.-~._—~—_—

 

 

LIST 9;; REFEBENZES

1. Svoboda, Antonin, Computing Mechanisms and Linkages, Dover Pub-
lications, New York, 1965. (First published by McGrawbHill Book
Co., Inc., 1948.)

2. Hrones, John A., and George L. Nelson, Analysis gilthg_Four-bar
Linka e, published jointly by Technology Press, M.I.T., Cambridge,
Mass., and John Wiley & Sons, Inc., New York, 1951.

3. Hain, Kurt, Angewandte Getriebelehre, Hermann Schroedel Verlag
K.G., Hannover, 1952.

4. Nickson, Philip T., ”A Simplified Approach to Linkage Design",
Trans. lst ggnf. on Mechanisms, Penton Pub. Co., Cleveland, 1953.

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7. Freudenstein, Ferdinand, "Approximate Synthesis of Four-Bar Link-
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104

105

12. Worthley, W. W. and R. T. Hinkle, "Four-Bar Linkages, Approximate
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l4.‘ Vidosic, J. P. and H. L. Johnson, "Synthesis of Four-Bar Function
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15. Dunk, A. C. and C. L. Hanson, "Six-Bar Linkages", I£§2§;.§£h;gggig
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