Optimal synthesis of a geared four-link mechanism

J nl. MechanismsVolume 4, pp. 291-302/Pergamon Press1969/Printed in Great Britain

Optimal Synthesis of a Geared Four-Link Mechanism* George N. Sandor~ and Donald R. Wilt§ Received 1 2 August 1 968 Abstract Geared linages often present better transmission characteristics and greater ranges of input and output rotation than pivoted linkages. The method of complex numbers (see reference [4]) is well suited for the analysis and synthesis of geared linkages, and is therefore applied in the present paper to develop the equations of synthesis for a geared four-link function generator consisting of a frame, input crank, coupler and output gear, where the coupler is formed as a rack to engage the output gear. Solutions are derived for 3, 4 and 5 point approximations of function generation, including optimization to minimize structural error. The mechanism permits large input and output rotations with good transmission characteristics. The solutions have been programmed for automatic digital computation and numerical examples are given which demonstrate the favorable characteristics of this mechanism. Zusammenfassung--G/~nstigste Synthese eines viergliedrigen Zahnradgetriebes : Prof. G. Sandor und Ing. D. Wilt. Zahnrad-Kurbelgetriebe ergeben 5fters bessere 0bertragungs-Eigenschaften und gr6ssere Bereiche der Antriebs- und Abtriebs Drehungen als Gelenkgetriebe. Die Methode der komplexen Zahlen (siehe Ref. [4]) ist gut geeignet fur die Analyse und Synthese von Zahnrad-Kurbelgetrieben und ist deshalb in der vorliegenden Arbeit angewandt zur Ableitung der Gleichungen fur die Synthese eines viergliedrigen Zahnrad-Funktionsgetriebes, bestehend aus einem Gestell, Antriebskurbel, Koppel und einem Abtriebs-Zahnrad. Die Koppel besteht aus einer Zahnstange verzahnt mit dem Abtriebs-Zahnrad, L6sungen sind abgeleitet for 3, 4 und 5 Punkte zur Erzeugung der Funktion, einschliessend die Optimisierung zur Verminderung von Fehlern. Das Getriebe erlaubt grosse Antriebs- und Abtriebs-Drehungen mit guten Ubertragungs Eigenschaften. Die L6sungen sind programmiert fur automatische elektronische Rechenmaschinen und Zahlenbeispiele sind angef~hrt, die die g/Jnstigen Eigenschaften dieses Getriebes veranschaulischen. Pe3mMe---HaMnbIro~Mel~mH~ CHHX¢33y6,mTO-CTepw,Heeoro qeTbxpex3se~aro .,~exaNH3Ma: 1-Ipodp. I'. CaH~op H Ha~¢. ~. BH.rlbT. 3y6qaTo-cTepw, Hesb[e MexaHH3MM qac'ro ~amT nymziHe nepe~zaxo,m~[¢ ceogc'r~ H 60/IbmHe npc~enb[ ecnyutaro H Be~10Maro BpaRIeHH~I HC~CYn4 mapHHpHb[e MexaHH3Mbl. M C T O A KOMnMeKCHbIX ~Hccn (CM. pcdp. [4]) x o p o m o n p W M e H H M Z aHa.qH3y H CHHTC3y 3y6qaTo-c-rep~esbzx MeXaHH3MOa HO3TOMy ITpHMeHSCTC~ B HaCTO~lLtleltpa60Te ~U~ BbmoIza ypasHeHafl CHHTc3a 3y6qaTaro tICTblpCX3BCHHaro MCXaHH3Ma cocxoau~aro H3 C x o ~ , Be~lynlaro gpm~ommla, LuaTyHa H BelzoMaro 3y6qaTaro Koneca. IIIaTyH COCTOHTH3 3 ~ q a T O ~ ]~I~KH cHenJIeHH01~ C RenO,~,IM 3y~q~.TMM Ko.rICCOM. Pemem~ BMBe~,eHM ~qS 3, 4 H 5 TotleK BOCnpoH3eO~]JaMOl~ ( ~ BKJno=la~ O n T H M H 3 a u m o .a~ y M e R b m e H H ~ OmH6OZ. M e X a H H 3 M ]~onyczaeT 6ombtuHe Bc]IyHIHe H BC~XoMMe epamem~s C x o p o m e M e nepe~laTo~bz~n~ CBOIICTSaMH. Petuemu~ nporpaMMaposaH~[ ~na a]rroMavmieczoR 3negTpoaHofl c~eTHoi~ MaBIH]-gM H npHeo~U~TC~ qacnemm=e npeMepb[, KOTOpble noaTeep:~,cna~oz BblFO~[~'ible CBO~;~L-~'Ba

3 TOrOMexaa~3Ma. * A paper from the Proceedings of the Second International Congress on the Theory of Mechanics and Mechanisms, held in Zakopand, Poland, September 1969. Fellow, ASME; Member, U.S. Committee on Mechanisms; ALCOA Foundation Professor of Mcchardcal Design and Chairman, Machines and Structures Division, Rcnssclaer Polytechnic Institute, Troy, New York 12181. § Associate Member ASME, Graduate Student, R.P.I.. Present address, General Electric Company, U.S.A. 291

292 Introduction THE METHODof complex numbers [4] is welt suited for the analysis and synthesis of geared linkages. Since geared linkages often present better transmission characteristics and greater ranges of input and output rotation than pivoted linkages, the development of geared four-link function generators suggested itself. One of the simplest such mechanisms has a rack and gear connection between the coupler and output. It therefore was chosen as the subject of the present paper.

Description As shown in Fig. 1, the linkage consists of a crank connected to a gear by an offset rack.

\

\

///

t/

i-

\\\

# ,:.,x

;~ /.~///'1

Figure 1. A geared four-link function generator. ~b and ~ are linear analogs of the input and output variables. The anticipated method of operation is to use the crank as the input and the gear as the output. However, it is conceivable that in some applications it would be preferable to have the gear as the input and the crank as the output. The purpose of the linkage is to provide a prescribed relation between input and output displacements. This may be in the form of arbitrary angle relationships or a functional relationship. In either case, there are a limited number of "exact" angle conditions or precision points which may be prescribed. As it turns out, if scale factors are prescribed, there are a maximum of five precision points. However, if we wish to synthesize a function generator and regard the scale factors of the input and output variables as unknowns, then the maximum number of precision points rises to seven.

Synthesis Referring to Fig. 2, the vector equation of closure is: Zz+Z3+Z~.+Z ,=Z~ .

(l)

In developing the mathematical model for synthesis, we observe the following four notes: (1) Since angular relations are all we are interested in, the scale of the linkage and its orientation in space will not affect the results. Thus, we are free to choose the magnitude and the direction of one vector. Therefore, let [ZI[= I, and Zt be the positive x-axis. (2) The rack Z~ is tangent to the gear at all times, thus Z,,.LZ5 and therefore only three quantities are needed to specify the two vectors Z , and Zs.

293 (3) Since the offset Z3 is an iategral part of the rack, the angle which Z3 makes with Z~ will not affect the operation of the linkage, thus for convenience we choose Z3_I_Z4 and again we need only three quantities to specify two vectors. (4) Due to the geometry of the rack-gear connection, the rack and gear, in general, will rotate through different angles y and ~. However, if we introduce the stretch ratio p, where, for the j-th position:

p _lz.I

j-[-~,

(2)

and (in view of note 2) the magnitude ratio 2 5, where:

/5_lZ,tl

(3)

Z3==Z31

ZZ ==ZZi

ZL=Z.

Figure 2. The vector diagram of a geared four-link function generator. Note the displaced vectors (lightweight linkes) and the output angle ~iThen, assuming pj is positive, we obtain:

Iz ,l

,

(4)

or

q,j = v~ + ( ° L- 1).

(5)

A5

Writing the closure equation for the linkage in the fith position, we obtain: Z2j+Z3j+Z41+Z~j=Z~j,

(6)

(a) Zt is fixed, so that Z i j = Z t ,

(7)

(b) Iz.l=lz,,I

(8)

where

and Z2~=Z21e '4J.

294

Here, ei~J=cos ~ j + i s i n ~bj is the complex rotation operator.

(c) Iz l=p lz ,t, Z , j = pjZ,~e ;~'j,

(9) ( 1O)

{ll

(d) Z 3 t = 23Z41e I~/2 ,

(12)

Z3j = ';'3 Z4 t e/t( ~/2 ) + ej],

(13)

(e) IZ..I-- IZ , I Z5, =

:.5Z~,e-i(~::)

(14) (15)

and Z5 i = 2sZ4 zeltyj- (r¢/2)]

(16)

Substituting in the closure equation we obtain:

Zz tei4~J+ 2324.teltt~/2)+Y~] + pjZ4.telrJ + J.~Z4.1eiDJ-(~t/2)J=Zl t .

(17)

Factoring out Z4te~V~: Z21 ei*l + ('~3 ei(~/2 ) + Pj + "~'5e - i(~/2 ))Z4.1 e I~'1= Z l t ,

(18)

ei(~/2)= cos rt + i sin-n = i, 2 2

(19)

e-~(~/2~= - i,

(20)

Z11=1,

(21)

where

and therefore 22 eh~' 4- [pj 4- i(,2.3 -- ;.5)]Z4e irJ = 1.

(22)

Noting that this equation contains no ¢ terms, we introduce Cj by way of expressing pj from equation (5): Pi = 1 + (~ki - yi)25.

(23)

Substituting this in equation (22), we obtain our final equation of closure: Z2e~J 4- [1 +(~,j--y j)25 + i('J'3 -'J'5)']Z4 ei':J = i .

(24)

The question now at hand is how many precision points can we synthesize the linkage for? To answer this we must consider the mathematics involved. We are all familiar with the fact that one equation in one unknown can be solved. Similarly two equations in two

295

unknowns can be solved. However, one equation in two unknowns will only yield the ratio of the unknowns, that is, there will be an infinite number of solutions, since specifying one unknown allows solving for the other. On the other hand, two equations in one unknown yield no solution (assuming independent equations). Thus, we can solve for only those numbers of precision points for which the number of unknowns equals or exceeds the number of equations. For ease in determining when the number of equations equals the number of unknowns, we will use the following table

No. of equations

j 1 2 3 4 5

2 real 4 real 6 real 8 real I0 real

Unknowns

No. of real unknowns

No. of solutions

6 7 8 9 l0

0(o¢4) 0(oo3) 0(~2)

Z2,A3,Z4).5 ditto +72 ditto + 73 ditto + 74 ditto + 75

0(oo)

Finite

Scale factors are assumed to be predetermined. The table is compiled on the basis that for ea~t%alue o f j we obtain one closure equation (24), which is a vector (or complex) equation that can be broken into two separate equations, and that the values of ~b~.and ~bs are prescribed by the conditions of synthesis (purpose of the linkage). This leads to ten unknowns and ten equations in case of five precision points. It can be seen that, if the scale factors of t~ and ~k are regarded as additional unknowns, the number of unknowns and equations will balance at seven precision points. Writing the five vector (ten real) equations in matrix form, we obtain: 1

[I + i(23-25)]

eiq~z ei,lla

[ l + (#z 2 --'/2)25 + i(23 - - 2 5 ) ] e i~

-- 1

[1 + (¢3-./3)2, + i().3- 25)]W "~

-1

e~'~4

[1 + (~b4-./4)). 5 + i(). 3 -- 2,)]e ~'4

-- 1

ei4'~

[ l + (~, --./5)).5 + i(23 - ,~.,)]e'"

-i

-

1

=o.

(25)

For a non-trivial solution, the matrix of coefficients must be of rank two, i.e. the determinant of any three rows must be zero (since Z2 S 0 and Z 4 ~ 0 ) . Writing this out we have: I

ei4,2 e~s

[[ + i().3-45)]

- 1 [

[1+(~2-')~2)25-}-/(23-25)]e i;'2

-1 I=0'

[ ]. "I- (l~j -- ]/j)"~'5 Jr- i(23 -- 2s)]e iys

-- 1

j = 3 , 4, 5.

(26)

Expanding this by cofactors of the third column:

( - z)' * 3 ( _ 1){e'*'[z + (q,j - ./s)'% + i(23 - 2 , ) ] # " -

e'*'D + (~02 -./5)2,

+ i(2 s - 2,)e""] } + ( - 1)2 + 3(_ 1){ 1. ['1 + (#Js - 7s)2s ÷ i(23 - 25) levz' - e'*J[1 + i(23 - 4 5 ) ] } + ( -

-e'¢'[1+ i(23-4,)]} ----0.

1)3+3( - 1){1-[1 + ( ¢ 5 -'/2);.5 + i(43 - 25)]e'~'

(27)

296

For a three-point solution, j = 3 and there are four unknowns (5.3, ,;.~, 72, 73) tO be found from two equations. Thus we can specify two of the four unknowns arbitrarily and then solve for the remaining two. Noting that 23 and 23 appear in linear form, we specify 7_, and 73 and then solve for 23 and 25. For four precision points, the solution requires solving for four of five unknowns using four non-linear equations. One approach to the four-point solution is to solve two separate three-point systems and then, using the computer, finding the gradient of 23 and 25 ,and then calculate that change in the assumed unknowns 73 and ~,.~which will bring the two three-point solutions together in a four-point solution. This has been done, and there is a working program which gives reasonably fast results, requiring about ten iterations to equalize the lambdas to eight decimal places. Upon extending the process to obtain a five-point solution, the results were not nearly as good. The convergence is very slow and, unless the initial guess for the gammas is close to the solution, the chances of finding it are very small.

Analysis Having synthesized the linkage and knowing its displacements at the precision points, it is desirable to also know the displacements between precision points. This is particularly true for a function generator, where the error between precision points is of importance. The link vectors are known for the first precision point, so the analysis requires only some vector geometry and a few constraints. First, a vector is drawn from the tip of Z2 to the great pivot. Also, a line parallel to Z4 is drawn through the tip of Zz. These are shown in Fig. 3. Z4

//

Z3

F i g u r e 3. Vectors used in the analysis of performance of a synthesized mechanism. The vector D can easily be determined for any displacement ~bp of Z2 since: D t --- 1 - Z 2 t

(28)

D p = 1 - - Z 2 l e I~" •

(29)

and

Having found D u, the value of

Iz4pl can be found by trigonometry:

[Z,,p[ = ~ / [ o p l z - EO-5 - ; . a ) [ z , , , I:] 2 •

(30)

297

From this we can calculate pp:

p _iz,,I

(31)

Now only the arguemcnt of Z,p is needed to determine ?p, from which ~p can bc calculated. From Fig. 3 we see that arg Z,p = arg Dp + ~,

(32)

c~= tan- t(2s - x3)l Z4~ ].

(33)

where

Iz.I

Referring to Fig. 2, yp =arg Z4p-- arg Z , t .

(34)

Thus the value of~bp can be found by substituting into the constraint equation $P = VP+ (PP2-~1).

(35)

In programming the performance calculations,there were two logicaldecisions which had to be built into the program. The firstof these was the determination of whether the angle ccwas positiveor negative. By examining Fig. 3, it becomes apparent that the sign of c~will always bc the same for a given linkage as long as p doesn't change sign. Since there would be an instabilityin the linkage for p to change sign, the program rejectssolutions with p negative. Thus the sign of ~,is found for position one and assigned to ¢ in other positions. The second problem encountered was that of finding angles andthe differencesof angics. The computer only gives arctangents in the range of -Ir/2 to + 7r/2,which was readily corrected by use of a two argument function subprogram that gives the arctangent in the range from 0 to 27r. This made it possible to calculate the correct angic of a vector, but it didn't solve the problem of finding the angle between two vectors. For example, having found Z4p ,the displacement yp is given by ~p = arg Z , ~ - arg Z , t .

(30

However, if the arg Z,p=2 ° and the arg Z,t =350 °, then simple subtraction gives ?p= - 348 °,while the desired answer might bc + 12°. To correctthis,the magnitude of the angle ?p was compared to that for the previous displacement and, if they differed by close to 2~, then 27r was added or subtracted from ?p to makc the magnitudes approximately the same.

Computer Program The computer program as shown in Fig. 4 is made up of modules and can bc used to synthesize and analyze for three, four or five precision points for either function or position generators. In addition, a small program was written to generate Chebishcv-spaced data for function generators.

298

Prescribed

con~?ions

I

J -- - I

CHEBI Calculates

Chebishev spaced data

I

I

1 THREE

QUAD

Calculates

Calculates g a m m a s and

linkage for 3 prec. pts.

lambdas for 4 prec. pts.

QUINT Calculates g a m m a s and

lambdos for

5 prec. pfs. I !

ZETA Calculates linkage

PERFP Posi tion

generator performance

PERFF

Function generator performance FRF respaced prec. pts. L--

Figure 4.

Modules of the computer program shown in a cybernetic diagram. Dashed lines indicate data flow. Solid lines represent possible combinations of components, i.e. the module THREE must be combined with PERFP or PERFF to obtain a complete program, whereas QUAD is a complete program in itself. The program is available from the second author upon request.

The three-point linkage is the most flexible, although generally of lower accuracy than those synthesized for four or five precision points. By varying the arbitrary unknowns 72 and Y3, the shape and proportions of the linkage can be varied to suit the application. There are two types of linkages which can be designed using the program. Position generators, where arbitrary angular displacements are specified, or function generators, in which there is a desired functional relationship between input and output rotations, specified exactly at the precision points. The method of solution for 23, 25 and 7j is the same for both types. The analysis differs in that the performance of a function generator includes an error print-out. In addition, the error profile is analyzed and the precision points respaced using the Freudenstein respacing formula [I, 12] from which a new linkage with equalized (and thus minimized) errors can be synthesized. Examples

Several linkages generating the function y = x 2 synthesized with the present method are tabulated in Table 1. To illustrate the achievable accuracy under different conditions, the generation of the same function with varying scale factors and numbers of precision points has been presented. The first three examples show how the error can be reduced using the Freudenstein respacing formula [1]. Unfortunately the transmission angles were small and became even smaller making the resulting linkages impractical. However, by adjusting the coupler rotations 72 and 73, the transmission angles and the accuracy were both greatly

299

t m p r o v e d as shown in e x a m p l e (4). T h e effect o f adjusting the c o u p l e r r o t a t i o n s 72 a n d Y3 u p o n transmission angles a n d a c c u r a c y is s h o w n by examples (5)--(7). Here the i n p u t has a 120 ° range a n d the o u t p u t a 180 ° range. T h e p o i n t to be n o t e d is t h a t g o o d t r a n s m i s s i o n angles m a y or m a y n o t a c c o m p a n y g o o d accuracy. T a b l e 1. L i n k a g e s t o g e n e r a t e y = x 2. Example No. 1 2 3 4 5

6 7 8 9 10

11 12 13

Input range

Scale factors Maximum error Minimum (degrees/unit) (% output range) transmission Remarks Input Output angle=tan-l(Z4/Za) l~
* FRF: Freudenstein's Respacing Formula, [1]. E x a m p l e s (8)--(10) indicate the range limitations. I n all three, the o u t p u t oscillates over a 100 ° range. T h e input range varies f r o m 100 ° for e x a m p l e (8) to 200 ° for e x a m p l e (10). N o t e t h a t linkage (10) is m u c h better t h a n i n d i c a t e d by m a x i m u m e r r o r a n d m i n i m u m t r a n s m i s s i o n angle over 90 per cent o f its range (see r e m a r k s in T a b l e 1). O f p a r t i c u l a r n o t e is the fact t h a t for the three linkages the e r r o r varies by a f a c t o r o f 15 while t r a n s m i s s i o n angles r e m a i n a l m o s t c o n s t a n t a n d quite good. E x a m p l e s (12) a n d (13) are i n t r o d u c e d to give s o m e c o m p a r i s o n o f four- a n d five-point solutions. Both o f these linkages have small errors. H o w e v e r , the five-point linkage has less t h a n h a l f the e r r o r o f the f o u r - p o i n t linkage. It thus a p p e a r s to represent a w o r t h w h i l e gain in accuracy. I f very high precision is required, then t h e time a n d effort required to o b t a i n a five-point solution m a y be w a r r a n t e d . O n the other h a n d , the f o u r - p o i n t s o l u t i o n requires only a fraction o f the effort a n d m a n y m o r e linkages can be found, thus a l l o w i n g s o m e choice o f link p r o p o r t i o n s a n d configurations to suit design conditions. N o t e t h a t the f o u r - p o i n t solution in item (12) is at least as a c c u r a t e as a 10-in. slide rule. T a b l e 2 lists sine a n d log generators, a n d Fig. 5 shows scale d r a w i n g s o f s o m e o f the linkages f r o m T a b l e s 1 a n d 2 at their first precision point.

Table 2.

l
l
l
l
17

18

19

20

1~

1~

I~

I~

1.5

2

2

5~

5~

250

250

150

1~

1~

Scale factors (degrees/unit) Input Output

0-1318

1-2634

0"2215

0"2550

0.2307

0"5884

0"9464

Maximum error ( ~ output range)

* F R F : Freudenstein°s Respacing Formula, [I].

--90°
--~°
16

°

15

--~°
Input range

14

Example No.

30"3

22 "4

44 "4

44"2

48"2

39"8

62"5 deg.

Minimum transmission angle 1 +sin (x)

1 +sin (x)

y = log I o (x)

y = l o g t o (x)

y = l o g l o (x)

y = l o g l o (x)

y=

y = 1 +sin (x)

y=

Function generated

4 PPT, Chebishev spacing

4 PPT, Chebishev spacing

3 PPT, 1st FRF* respacing of example (17).

3 PPT, Chebishev spacing

4 PPT, Chebishev spacing

4 PPT, Chebishev spacing

4 PPT, Chebishev spacing

Remarks (PPT= Precision Point)

g

301

Ex. 16

Figure 5. Scale drawings of some of the examples in Tables 1 and 2. Examples 4, 5, 8 and 13 generate the function y = x z, examples 4 and 5 in the interval of 1< x < 2 with three precision points, example 8 for - l < x < 1 with four precision points, and example 13 for O < x < 1 with five precision points. Example 16 generates y = 1+sin x for - 9 0 ° < x < O ° with four precision points. Example 20 is a Iog-zo generator for 1< x < 2 with four precision points. Acknowledgements---Support under NSF Research Grant No. GK-1231, sponsored by the Engineering Mechanics Program, Engineering Division of the National Science Foundation at Renssclaer Polytechnic Institute, is gratefully acknowledged. The authors arc grateful to Roger E. Kaufman for his invaluable assistance with the computer programming and Mrs. F. Willson for her help with the typing and would like to express their sincere thanks.

References [1] FREUDENSl"~.NF. Structural error analysis in plane kinematic synthesis. J'. Eng. Ind (Trans. ASME) Series E 81(1), 15-22, (1959). [2] FR£UDENSTEINF. Lea.st-error four-bar function generators. Trans. Fifth Conf. Mechanisms, Purdue UniversiW, West Lafayette, Indiana (1958). [3] HARTENBEROR.. S. and DENAVrrJ. Kinematic Synthesis of Linkages. McGraw-Hill, New York 0964). [4] SANDORG. N. A general complexonumbcr method of plane kinematic synthesis with applications. Doctoral dissertation, Columbia University, New York; University Microfilms, Library of Congress card No. Mic: 59-2596, Ann Arbor, Michigan, 305 pp. (1959). [5] SANDORG. N. and FI~UDENSTEINF. Kinematic Synthesis of Path-Generating Mechanisms by Means of the IBM°650 Computer. Published in the IBM-650 Program Library, #9.5.003, Applied Programming Publications, IBM Corporation, 590 Madison Ave., New York 22, New York (1959). [6] FR~UDE~I~ F. and SANDORG. N. "Synthesis of path-generating mechanisms by means of a programmed digital computer. J. Engng Ind., Trans. ASME 81B(2), 159-168 (1959).

302

[7] FREUDENSTEINF. and SA,NDOR G. N. On the Burmester Points of a plane. J. appl. Mech., Trans. A S M E Series E 28(1), 41-49 (March 1961). Discussions and Authors' Closure (containing additional developments) published in J. appL Mech., Trans. ASME Series E 28(3), 473-475 (September 1961). [8] SA,',,'DORG. N. On the loop equations in kinematics. Trans. Seventh Conf. Mechanisms, Purdue University, pp. 49-56 (October 8-9, 1962). [9] ROTH B., FREUDENSTEINF. and SANDORG. N. Synthesis of four-link path-generating mechanisms with optimum transmission characteristics. Trans. Seventh Con]~ Mechanisms, Purdue University, pp. 4648 (October 8-9, 1962). [10] SANDOR G. N. On Computer-Aided Graphical Kinematic Synthesis, The Techn&al Seminar Series, Report No. 4, Princeton University Department of Graphics and Engineering Drawing, 63 pp (1962). [11] PRIMROSEE. J. F., FREUDENSTEXNF. and SANDOR G. N. Finite Burmester theory in plane kinematics. J. appl. Mech., Trans. A S M E Series E 31(4), 683-693 (1964). [12] MCLAR,~AN C. W. On linkage synthesis with minimum error. J. Mechanisms 3(2), 101-105 (1968).

Appendix : User's Guide to C o m p u t e r Programs Refer to the program flow chart (Fig. 4) for the assembly of decks. CHEBI is a data program for function generation. It has instructions built in in the form of comments. A source listing will provide all the needed information to use it. Q U A D and Q U I N T are the four- and five-point compatibility programs. These also have comment blocks explaining the input and output, thus source listings are required before running them. Q U A D is the most useful, generally converging to a solution and yielding one or two good practical solutions per 5 min of computer time. Q U I N T is much less likely to converge and at best requires some help from the programmer to find the solution. However, it may be used to find points close to a five-point solution and thus obtain higher-precision three- and four-point solutions. ZETA is a partial program and must b¢ combined with one of the performance subroutines to run. Obtain a source listing for detailed running instructions. The punch-card decks for the above programs may be obtained from the second author.