6

CHAPTER 2

LITERATURE REVIEW

The review of literature is related to the sy~

thesis of four bar mechanism and the effect of tolerances

on link length and clearances in the joints. Each paper

is reviewed and discussed there itself. The outline of

the present work and its requirement is also briefed at

the end.

2.1 SYNTHESIS IN NINETEENTH CENrURY [6)

Watt (1736-1819) was the first to be concerned

with the synthesis movement. Generally speaking, mecha­

nism designers, before Watt, had confined their attentions

to the motions of the links attached to the frame, i.e.,

the first and the last links; one having been given an

input motion, the other produced a transformed motion,

called the output. It was Watt, who focussed on the

motion of a point on the intermediate link (Coupler)

of a four bar mechanism.

Willies (1800-1875) made a substantial contri­

bution to the caue e of synthesis. He considered that

7

,. all the simple combinations of mechanism i.e. from

input to output may be distributed into three classes.•'

Cl.ass .A.- directional relation and velocity

ratio both being constant

Class B - directional relation constant and

velocity ratio varying

Class C - directional relation changing perio­

dically velocity ratio either con­

stant or varying.

He counted five ways of achieving the relative motion

between input and output links - rolling contact, sliding

contact, wrapping contact (belts and chains), link work

and reduplication (tackle of all sorts).

Reuleaux•s (1829-1905) comprehensive and orderly

views marked a high point in the development of kinema­

tics. It was pointed out that the constrained motion of

a mechanism or the kind of relative motion between parts

was controlled by the form of the surfaces of contact of

the adjacent parts, called as working surfaces of the

connection. Each of the working surfaces was named an

element. Reuleau:z: identified synthesis as a concept

which would help and guide the designers.

8

Samuel Robert {1827-191') showed that a coupler

point described a curve of the sixth order. A theorem

important to synthesis and bearing the name of Robert,

states that there exist three different four bar linkages

capable of drawing identical coupler curves. They are

called cognate linkages.

Aronhold in Germany and Kennedy in Englani

greatly extended the use of instantaneous centres of

velocity by recognizing the theorem of in line three

centres. The 'graphical relative velocity method' of

Smith was a real break through. The great usefUlness

of the velocity and acceleration polygons lay in the

fact that even complicated mechanisms had been studied

by simple graphical means. Burmester in Germany deve­

loped geometric methods that furthered analysis and

showed the way to synthesis. The Russian mathematician

Chebyshev developed the theorem of the triple genera­

tion of coupler curves same as that of Samuel Robert.

Until nearly the end of the nineteenth century,

1 t was oba erved t bat • five basic meahanical powers -

Lever, winch, screw, wedge and pulley' were the build­

ing blocks from which all more complex assemblages were

constructed.

9

2.2 SYNTHESIS AFTER NINETEENTH CENTURY

In the beginning of 20th century, machines were

made from six basic mechanical components.

1. Crank

4. Screw

2. Wheel including gears '3. Cam

5. Intermittent motion devices i.e. ratchets

6. The tension compression parts including one way rigid

parts like belts, chains and hydraulic lines.

The whole problem of synthesis now consists of

three facets namely type synthesis, number synthesis and

dimensional synthesis. For the purpose of discussion they

are separated even though they are having va:cying degrees

of interrelation.

In the type synthesis, the type of mechanism

like, cam linkage, elliptical gear train and so on - is

decided upon. It is with this face of synthesis that the

name of Reuleaux is particularly associated.

In the number synthesis the reqUisite number of

links am lower or higher pairs yielding constrained motion

are established. This is a case yielding a mechanism

having a minimum number of pairs properly connected to

enilure that it will produce the desired motion. Criteria

of mobility were presented by Griib1er (6] who dealt with

planar situations in 1883 - 1885 and the spatial chuns

in 1917.

10

The dimensional synthesis detemines the propor­

tions (lengths} of the links needed to accomplish the

specified motion tranef'omation.

In 1951, J .A.Hrones and. G.L.Nelson [2] presen­

ted a fundamental work on synthesis. This book contained

approximately 7,300 different curves generated by the

coupler of a four bar linkage, Fig.2.1. These curves

helped to get the solutions of most of the practical

contemporary problems. Still it is one of the most useful

reference for mechanism synthesis.

The four bar linkages considered, were crank

rocker type i.e. having continuous rotation of the crank

with oscillating follower. The link lengths satisfied

the condition that A, B, C > crank, fig. 2.1.

The new era of mechanism synthesis started in

1955, with the development of displacement equation of

four bar function generating linkage by F.Freudenstein [J).

He developed an analytical method of synthesis. In this

technique the function was represented by three discrete

points taken in sequence within the range. A displacement

equation was formUlated which synthesized the four bar

linkage for three such discrete points, later on called as

accuracy points. The function generated by the linkage

11

-

,... _.(

CRANK 9"" B FOLLOWER

f. c ~ ~

FIG. 2·1. NOTATIONS USED IN HRONES AND NELSON'S FOUR BAR COUPLER CURVE ATLAS-

!

12

was exactly same as the desired one at these accuracy

pointe, At other pointe called the design pointe, within

the same range, there was difference (error) in the gene­

rated function and desired function.

In 1959, F.Freudentein [7] selected five accuracy

points in the given range with Chebyshev spacing criterion

which minimized the structural error, the difference bet­

ween the generated function am desired function at an;y

design point, within the given range.

In 1964, R.S.Hartenberg and J .Denavits [6] dis­

cussed the effect of tolerances on link lengths. The dif­

ference ir. the output due to tolerances is named • 'Mecha­

nical error• • •

2. 2.1 Optimization in Synthesis

In 1966, Dr.Hen Chiyeh [8] developed a general

method for finding the Optimum set of physical parameters

of a mechanism in such a way that a certain physical :··

characteristic of the system would best fit the desired

physical characteristic in the operating range of the

mechanism. This physical characteristic to be fitted,

might be an angular displacement, a linear displacement

in two or three dimensions, a spatial force or a spatial

moment of force, etc. The physical parameters to be

optimised, could be angular or linear dimensions, masses,

moments of inertia of the elements or physical properties

of the elastic members of the mechanism.

Example given in the paper was the design of a

four bar linkage for best match to the output motion of

l~

a point on the extended parts of the connecting rod. The

output motion at a point on the extended part of the con­

necting rod was given by

xexp = 0.4 +sin 21t(t-O.l6); yexp = 2 + 0.9 sin 2nt

Total number of design points were ten. The result was

checked for {i) Positive link lengths {ii) The driving

{link} crank must be able to make complete revolution

about its axis of rotation. The result iS shown in

Fig.2.2.

In 1967, D.W.Lewis and C.K.Gyory (9] suggested a

method for solving for the parameters that defined a mecha­

nism. The approach was essentially the 'Damped Least

Squares • method. The plane curve was specified by a

series of paired coordinates (~,y1 ). Each of these

co-ordinate pairs were related to the input parameter

of a particular mechanism (such as the angUlar position of

one link of a four bar linkage). This parameter was iden-

tified as g, then the set of paired coordinates were written

as

v

0 1

, _, / /' ---·- IAIIT/AL GCI«IS

,... ... ,(, -- OPTJNJZ£0 lfti:SCJI,.T I / .... ,

I I (I"';,TI I I y I' ••

I I s I ' ' .. /

(J,y}:­

(x.yJn,.-a 1 = o·g,gA6

0.2 c 5·SSR'­o., = S·IS09

0.-4 = 3·9134 O.:s ~ 1·9960

o., " 0·5759

2 )( 3

FIG.2·2. OPTIMIZED DESIGN OF COUPLER MOTICXII OF A FOUR· BAR LINK AGE.

J(

X

D

ex .. , y.,., s .. J o._----------~~--~~

__ ..., _,,..,. '

{lco,~,t.J )C X X X X )( ~-

(x,,y,.~,J

I I \ I I

' I ' ' \

\ ' \ ._,.r,-,., 7" .,. ' ,. ,- - ---...,.. ,,//;;/;'

FIG. 2·3. DESIRED Fl.f\ICTION AND ITS SOLUTION •

•

14

I I '

~ = ~ (ei) ;

11 = h2 <ei >

15

To emphasize the association of 9 with the co­

ordinates, a point was represented by the symbol (lj_, Y 1 ,

e1 ). The actual position of a coupler point on a speci­

fic mechanism was written in terms of the parameters that

defined the mechanism including the angular position of

a link, Q •

x1 = H:J. (e1 , Ll' t 2, Lm);

yi = H2 {ei, Ll' L2' Lm)•

The error or mismatch for a particular value of 9 was writ­

ten. A sum of the squares of these errors was the measure

of the degree of success of matching the function genera­

ted by the mechanism with the desired function. The

example problem dealt therein and its solution is shown

in Fig. 2.3.

In 1967, R.L.Fox and K.D.Willmert [10] solved

the problems of path generating mechanisms by nwnerical

methods using Newton-Raphson technique. The objective

function which seemed desirable for expressing the error

between the generated curve and given curve was the inte­

gral of the square of the difference between the two curves

i.e.

error=

where, y g - Y co ordinate of generated curve,

Yd - Y coordinate of given curve.

They defined the part of the objective function as :

where Yig and Yid were the discrete values of Yg and Yd

respectively and a1 was an arbitrary positive constant.

16

An expression representing the crank angle error was written

as

where yi was the crank angle, 6 y1 was the desired rota­

tion of crank and a2 was arbitra:cy positive constant.

The objective function F(U) was then defined as

For minimizing the objective function, the equality con­

straints and inequality constraints were specified. Example

problems of straight line linkage, linkage generating the

circular arc and linkage generating figure-S were dealt

therein. The trial linkage was obtained from the atlas

[ 2]. Maximum structural error was calculated in each case.

The finaJ. straight line linkage is shown in Fig.2.4.

In 1968, J .Tomas [11] presented non linear

programming technique for optimizing the synthesis of

the mechanis-m,

Mathematically; any problem was formulated as

f(x) = f(Xl, ••••• , xn)

and the domain,..n..., was determined by inequalities

~j(X) = fllj (:z::t••• .. ~) ~ 0 for j = 1, •••• , m

where X was the unknown n-dimeneioned vector with com­

ponents (Xi••••••• xn)•

A point x* was searched w1 thin the domain,..n.., for

which

f(X ) = min f(X), X£~.

17

The given function was denoted by F(fll) atxi the generated

function by G(JII,X), where J11 was a variable parameter. The

mean square error and maximum error of both the functions

were expressed, Example problems dealt therein were

(i) four bar linkage for function generation for the

c otxi it ions

a) feasible for the crank to rotate 360°

v

FIG.2·4. FINAL STRAIGHT LINE LINKAGE

A

FIG. 2·5. NOMENClATURE FOR THE FOUR-BAR LINKAGE.

•

18

b) values of transmission angles to stay within

a prescribed interval

c) dimension of each of the four members to hold

to the prescribed limits, and

(ii) four bar linkage for path generation for the same

conditione as in (i) • Least mean square error

and least maximal error were determined. The

results were compared with the results obtained

by Han [8].

No relationship was mentioned between least

mean square error and least maximal error.

19

2.2.2 Effect of Tolerances on Link Lengths

The tolerances on link lengths deviate the output

of the linkages. This deviation, in four bar function

generator, called the mechanical error, was first consi­

dered by Hartenberg and Denavit [6]. In 1969, R.E.Garrett

and Allen s.Rall Jr. [5] presented the effects of tole­

rances arrl clearances in the form of mobility bands for

the linkages. This band was of varying width around the

desired function within which a given percentage of

mechani.S ms would operate. The limits o:f this mobility

band were fixed from the results of a particular-combina­

tion of tolerances and clearances. The tolerances on the

20

link lengths were assumed to follow any statistical pattern

i.e. Gaussian, exponential or rectangular distribution

Fig.(2.5) shows the nomenclature for the four

bar linkage. The mechanical error, f1 was represented by

where ~ represented the position of the input link, R the

link length parameter, and C the radial clearances in pin

jOints.

Evaluation of mobility band was done by sampling

technique and Delta method. In the example problem dealt

therein was solved by sampl.ing technique and the variation

in mechanical error was determined for 300 sample mechanisms

and input angle e = 15° for various combinations of clea­

rances and tolerances. In another example problem, the 3a

mobility band was determined by Delta method for a mecha­

nism to generate functions like - (i) y = x2, (11) y = log x,

(iii) y = sin x. The paper ie still being referred for com­

parison of results.

In 1970, S.A.Kolhatkar and K. S.Yajnik (12) also

dealt in their paper, the effect of tolerances am clearan­

ces in a plane function generating linkages. In a four

2l

bar linkage for function generation, first, they tried to

determine the maximum error in clockwise and anticlockwise

direction for a given value of the input variable am second

one was to find the maximum error when the input variable

was in a given range.

The effect of clearance in a revolute jOint was

taken by imaginery equivalent clearance link and then lin­

kage had only ideal revolute joints. As shown in Fig.2.6,

.6. a is the imaginary link connecting points A.j and ~·

This link was called the equivalent clearance link. Its

mass was taken to be zero in dynamical analysis. A four

bar function generator was theoretically analysed for

maximum error in output for a given position of input

crank.

No example problem was solved for explaining

the theory in their paper. They did not considerable

effect of tolerances on the link lengths.

In 1971, K.H.Hunt and H.Nolle [13] presented

a very important paper. The paper contained the Optimum

synthesis of planar linkage coupler curves represented by

position coordinates of points on the curve and corres­

ponding input link angular intervals. All independent

(0) (b)

FIG.2·6. TYPICAL REVOLUTE JOINTS WITH CLEARANCES EXAGGERATED.

FIG.2·7. LOOP CLOSURE AND POINT PATH ERRORS FOR THE PLANAR FOUR BAR.

22

23

variables were determined from a set of ten simultaneous

non-homogeneous linear equations through the use of the

least square criteriOn in the minimization of error func­

tion and of a stated selection procedure for certain lin­

kage parameters.

In the Fig.2.7. the given curve is to be gene­

rated by the coupler point E on the four bar OABG. The

ideal conditione would prevail when point E (on coupler)

and point E* (on given curve) coincided for all values of

input angle Q • In practice coincidence could not be

achieved. since it occured at discrete points only (not

necessarily at the desired values of e). The link loop

was opened by disconnecting the revolute pair at A and

moving E to E* • The gap AA.' could now be used as a

measure of error, instead of the distance EE* which was

made zero. For optimization, both loop closure and cor­

rect coupler point position were included as simultaneous

requirements of the error function.

For 'n' discrete positions of the input link,

corresponding to 'n' given points E* on the curve,

and minimizing AA' and EE* , the error fUnction was

expressed as

24

B = tw1[(e~)i + Cey>f) + w2[Ce;• >f + Cey.• )~]

where the mUl. tipliers w1 am w2 are weighting factors.

Example problems of straight l.ine synthesis and

circular arc synthesis were explained. Precision pointe

taken were 7 and 6 respectively. There are ten partial

differential equations which are to be solved for a pro­

blem. Lot of data are to be assumed which may lead to

numerous solutions.

In 1972, N.L.Levitski et al [14) in their paper

presented the optimal synthesis of function generating

four bar mechanism as a non-linear programming problem.

The optimal synthesis was done for given range of input

and output for the conditions of

( i) required type of four bar linkage, and

(ii) favourable transmission angles.

The pin joint C was replaced by a slider and

thus converted the four bar mechanism in a give bar mecha­

nism with two degrees of freedom as shown in Fig. 2.8.

Then the error, 6b, was represented by

6b = b - bf' when exact function was generated.

FIG.2·8. FOUR BAR MECHANISM WITH SLIDER.

R~.tl I I

FIG.2·9. LINE DIAGRAM OF FOUR BAR LINKAGE WITH TOLERANCE AND CLEARANCE.

25

where, b = coupler length

bf = slider length

A weighted difference was expressed as

The objective function for optimization was derived from

the weighted difference in form of linkage parameters.

The objective function was

m

s = l: i=O

[ oq( ¢1 )) 2

4 b2c 2sin2,

One example problem of generating log x was solved for

crank rocker type mechanism for a given range of trans­

mission angles.

26

There were no guide lines for ini tiel solution

and many assumptions in derivation of objective function

were made.

In 1973, Dhande and Chakraborty [15] construc­

ted a stochastic model of the plane four bar function

generator considering tolerances on link lengths and

clearances on the hinges. The comparison was made bet­

ween the mechanical error :for the three sigma band and

the structural error. A dynamic programming model wae

then presented for allocation of tolerances and clearan­

ces to different members such that mechanical error did

not exceed the s:p.ecified. value. The :procedure adopted

27

was analysis oriented and could suitably be appHed both

for analysis and synthesis of planar linkages. Fig. 2. 9

shows the effect of tolerances on link lengths and

clearances in joints. The equivalent mechanism which

takes the effects of tolerances and clearance into account

is shown by dot ted link.

Two example -problems, viz. y = sin x and y -

log x were explained. The method. was very lengthy. The

tolerances and clearances determined by dynamic program­

ming procedure for the position of linkage giving maximum

specified mechanical error, might :::1ot be economical. All

the links and hinges were given same tolerances and clea­

rances respectively, which would not be the usual practice.

In 1974, Bhakthavachalam and. Kimbrell [16] con­

sidered the problem of synthesis of path generating four

bar mechanism as an optimization problem with ine·qtlality

and. equality constraint. The penalty function approach

was used. The effects of clearances and tolerances in

manufacturing were considered in order to make sure that

28

the inequality constraints were within the acceptable to­

lerance limit during the required motion. The Fletcher­

Powell-Devideon minimization technique had been modified

slightly so that one could use it in a non-convex program­

ming problem effectively.

In 1975, R.I.Alizade et al [17) also applied

penalty function approach for solving the problem of opti­

mal mechanism design. A new algorithm was developed for

finding the parameters of a • first mechanism, • satisfying

all inequality constraints, to serve as an initial. appro­

ximation. Example problem of four bar mechanism genera­

ting given function y = f(x) = log x was demonstrated.

Eleven discrete points were taken. Optimization process

was very much similar to Levi tskii [14).

K.M.Ragsdell [18] formulated the synthesis of

mechanism as a nonlinear programming problem. Three

methode were discussed for solving non-linear programming

problem

( i) Penalty functions,

(ii) The Grifish-Steward method,

(iii) The Reduced Gradient method

The optimization of synthesis was done by the Generalized

Reduced Gradient method. Example problem of Tomas [11]

was solved and compared. It was pointed out that the

Reduced Gradient Method was more reliable and efficient

non-linear optimization algorithm.

29

A.H.Youssef et a1 [19) utilised Simplex method

of optimization for minimization of the difference between

the actual and desired motion. The four bar path gene­

rat or had nine design parameters as shown in Fig. 2.10.

One set of initial values of all the parameters were

assumed. The desired path was defined by the coordinates

of 19 points evenly spaced around the curve. .All the

designed parameters were determined. The programme gene­

rated the nine other sets of starting values and error

was compared in all ten sets of values and best one was

selected. Maximum error, however, was not mentioned.

J .Chakraborty [ 20) attempted to formulate a

general procedure to allocate tolerances on link lengths

and clearances in the y hinges for a given set of nominal

dimensions of the link lengths and a specified maximum

mechanical error. The equivalent linkage, which takes into

consideration the effect of clearances ani tolerances is

shown by dotted line in Fig.2.11. The objective function

was formed with maximum permissible tolerances and clea­

rances and total mechanical error as the constraints. The

P(r,y)

FIG.2·IO. SINGLE LOOP LINKAGE FOR THE PATH GENERATrON.

v ... - Ru_

I R41f

I I

FIG-2·11. FOUR BAR LINKAGE SHOWING THE EFFECT OF CLEARANCE AND TOLERANCE.

30

form of the objective function was to be eo chosen that

its minimum corresponded to the maximum values of tole­

ranee and clearance and giving equal weight age to each

of them. The objective function was minimized by using

the interior penalty function method for constrained

minimization. The minimization process started with a

feasible set of values of the design variables so that

no constraint was violated. Example problems of func­

tion generation Viz. y = x 2 , y = sin ·x and y = log x

were illustrated. 0 The maximum mechanical error was 1 •

The location of pin centre in the clearance circle was

taken to :f'o110>7 normal distribution. The results showed

equal magnitude of tolerance on link lengths and clea­

rance in joints. The result was impractical because

clearances should have been much less than the toleran-

ces on link lengths for rolling element bearings.

K.C.Gupta and B.Roth [21] discussed the effect

on the actual structural errors of the various approxi-

31

mations made in the measurement of the structural errors.

To measure these errors, three types of approximations

were used - (i) the errors were set to zero at several

positions - precision point method (i~) the root mean-square

of an error sum was minimized-least square·method, (iii)the

maximum error was minimized - Chebyshev spacing method.

32

They studied the effect on the error of the coupler point

motion of a planar four bar linkage. Instead of working

with the entire four bar chain, they designed each link

separately. They, then showed, the effect of applying

the results to a complete linkage design. The linear

error and angular error in the generated curve was

discussed separately. The errors were explained in the

percentage form and not as specific values.

K .Laxminarayan and G. Ramaiya [ 22] analysed the

error and clearance effects in mechanisms. The coeffi­

cient of sensitivity of the output position to an infi­

nitesimal error or clearance in a mechanism was repre­

sented by the relevant force in the link or contact force

(or force sum) in a pair. This was derived from the

principle of Virtual work. The error insensitivity was

identified with transmission effectiveness of the mecha­

nism and was defined to be high when the forces in the

mechanism were low for given output force or torque.

The force analysis was done for locating contact points

in the pair and thus clearance was determined. Friction

was also considered and contact forces in the various

types of pairs were analysed and presented in a practi­

cal form. No example was given.

J.R.Baumgarten and J,V.Fixemer [23] analysed

the effect of manufacturing tolerances on four bar path

generating mechanisms using probability theory. The

errors were assumed to follow normal distribution and mean

and standard deviation of vector position of the coupler

point were determined. The mean and stamard deviation

of errors in linkage lengths were thus accumulated to

produce the mean coupler curve and the derivative curves.

The analysis did not optimise the errors.

R.G.~Titchiner and H.H.Maybie [24] presented a

procedure for the case of an approximately straight coup­

ler point path by approximating the zeros of the first

and second derivatives of the radius of the curvature.

Both the first and second derivatives of the radius of

curvature were specified with respect to the loci of

the zeros of the derivatives. The synthesis procedure

was illustrated by an example.

The procedure was extended to the case where

four bar path generator would trace an approximately

circular path. The radius and centre of this circular

coupler curve were prescribed. Through the approximation

of the behaviour of the zeros of the derivatives of the

radius of the curvature of the coupler point paths, a

family of four bar linkages was identified. Each member

of the family was approximately tracing the same coupler

motion. The problem illustrated assumed the initial

design.

The intermediate phases of the optimization

process, beginning from the initial design to the final

optimum were not explained.

A.Ghosh and G.Dittnch [25] presented a graphi­

cal method of synthesis of four bar function generating

linkage satisfying four conditions. The method was based

on the coordination of positions along with the respec­

tive velocities of the input and output links at two posi­

tions. so only four quantities out of total eight could

be chosen. To achieve better accuracy the velocity and

position matching was done at some optimally placed sui­

table in between points and not at the extreme points

of the whole range to be generated. The accuracy points

were determined by Chebyshev's method. An example problem

was solved to illustrate the procedure.

l'I.Discinto et al. [26) presented the problem of

synthesis as a non-linear programming problem. The non­

linear programming problem was reduced to one of uncon­

strained minimization via the introduction of a penalty

35

function. The use of new tyPe of penalty function was

suggested which was neither an interior nor an exterior

penalty function, as claimed, did not require initial

feasible design, and the solution of the problem was

approached from the inside of its feasible region. Appli­

cation to four bar path generator was presented.

A.C.Rao [27] presented an iterative method for

the synthesis of linkages taking into account the com­

bined effect of link deformation, tolerances and joint

clearances. The linkages, which were under dynamic condi­

tions, were brought to a state of static equilibrium by

including the inertia forces and torques in the active

force system acting on the mechanism. The mechanism was

analysed to get the forces am torques acting on each

link. From the free body diagram of the coupler of the

four bar linkage, the axial denection and the lateral.

rlenection were determined. The position of the journal

in the bearing was located on the assumption of coloumb

friction and elasto-hydro-dynamic friction. The linkage

with its deformed links (dotted lines) and displaced

journals iB shown in Fig. 2.12.

The procedure was explained to determine the

link dimensions to get the output with desired accuracy

at various design points.

----------I J I I

I Q• I I I '

FIG.2·12. FOUR BAR LINK AGE WITH DEFORMED LINKS.

y

0 .r,_,J

FIG.2·13 BASIC OPEN CHAIN

•

36

Mikio Horie et al [28] presented a systematic

method of displacement analysis taking into account the

effect of both clearances and tolerances. When a mecha­

nism with a single degree of freedom had clearances,

37

any point on coupler link, for a prescribed crank angle,

could not be determined uniquely and vrould take a certain

value in a range. The conditione, that the point on

coupler link was in the limit values, could be determined

from the forces on each link. Conditions for binary link

or ternary links were different.

Four bar and six bar mechanisms were analysed.

Dismembering the four and six bar mechanisms, four open

basic chains were formed. One such basic open chain is

shown in Fig.2.13. Transformation functions for all basic

open chains were determined. The effect of tolerances

and clearances on transformation functions was considered.

The four bar mechanism was dismembered into two basic open

chains as shown in Fig.2.14. For the given clearance,

the upper an:l lower limit of the coupler point for com­

plete rotation of the crank was determined from the trans­

formation function.

In the above analysis, the clearances were the

part of tolerances. For given tolerances, the maximum

y

FIG.2·14. DESCRIPTION OF FOUR BAR MECHANISM BY TRANSFORMATION FUCTIONS.

/ I I

'

/

/ /

/

' .....

A.-­-:..~ - ....._

\ I

I

o~----~~~--------------~x ~XAo _ __,

FrG. 2·15. SCHEMATIC OF PLANAR FOUR BAR PATH GENERATING LINKAGE •

38

39

error in the output could be determined but not vice versa.

The theory was not explained by taking any example.

S.V.Kulkarni and R.A.Khan [29] designed the four

bar linkage for minimum positional velocity and accelera­

tion errors. Modified geometric programming had been

used. A composite objective function was obtained by

adding the positional, velocit.y and acceleration errore

in non-dimensional form. Relative import-ance or weigh­

tage had been given to these errors. The resUlt presen­

ted showp,d that the method could be successfully adopted

for four bar function generator.

M.M.Agrawal [30] presented an improved nonlinear

mathematical programming method for optimal synthesis of

tolerances and clearances in four bar linkage. The metho­

dology was based on transformation of relationship between

variance of random variables and va.riance of output func­

tion into a parametric constant. This constant was handled

in the penalty formulation of the Fiacco-Mc Cormick type.

Example problem illustrated the fomulation and solution

for three different functions each for two permissible

values of the mechanical errors.

M.Choubey and A.C.Rao [31] analysed the planar

mechanisms. The objective function was formulated so

that both structural and mechanical error could be mini­

mised together by summing the squares of the output

errore.

40

As the link dimensions vary at random about

their mean values during production, the optimization

problem was :formulated as a stochastic nonlinear program­

ming problem. The random variables i.e. link dimensions

were taken independently and followed normal distribution.

The modified objective function was then formu­

lated and minimised with respect to link dimensions

yielding nominal dimensi one of the links of the plane

linkages for function generation. The four bar chain

problem was illustrated for the minimization of the mecha­

nical error in a four bar function generating mechanism.

O.M.Sharfi and M.R.Smith [32] presented a method

of calculation of approximate tolerances on the link

lengths and clearances in the joints of linkage mechanisms.

The variation of the sensitivity of the output of the

mechanisms to the mechanism parameters which arises from

the change in the geometry of mechanism throughout i te

cycle of operation vas analysed. This sensitivity analysis

41

allocated the tolerances on link lengths. The problem

of four bar 1inkage was used for illustrating the theory.

In 1983, A.J.Kacachioe and S.J.Tricamo [}3)

presented a nonlinear programming technique for the

minimization of maximum structural error in the dimen­

sional synthesis of four bar linkage. The maximum struc­

tural error was minimized by inequality constraints which

bounded this error to be smaller than a minimization para­

meter at as many discrete mechanism positions as required.

Additional constraints were imposed on the design variables

to ensure linkage mobility, an acceptab1e range of trans­

mission angle and realistic mechanism dimensions. The

methc.'l. employed an autmented Lagrangian penalty function

coupled with a variable metric algorithm. The Fig. 2.15

shows the various dimensions of the four bar path gene­

rator.

The structural error was given by

Ei = Y[(xdi- xpi)2

+ (ydi- Ypi)2

]

An example problem was solved. The structural error was

minimised by this.

Ashok K.Mallik and S.G.Dhande [34] presented,

in their paper, stochastic modele of planar !0111' bar path

42

generating linkage and its associated cognate mechanisms

considering tolerances on the link lengths and clearances

in hinge joints as random variables. The mechanical error

for the three sigma band of confidence l.evel was evaluated.

A synthesis procedure for allocating tol.erances and cl.ea­

ranc es had be en developed by Lagrange 1 s mUl. tipli er approach.

The tolerances and clearances were assigned such that the

maximum errors in the path of a coupl.er point along each

of the two reference directions were within certain spe­

cified limit. One example problem was solved to illus­

trate the theory.

It is clear from the example problem, the to­

lerances and clearances were optimised to minimize the

error in the output. But the output error was not first

specified as claimed by the authors. They concl.uded that

the mean path generated by a cognate mechanism was the

same, still the mechanical errors produced in a given

path generating mechanism and its corresponding cognate

linkages were different, which is quite obvious. The

tolerance on the coupler length and clearances in the

moving hinges had more influence on the output error in

the path of the coupler point.

Medha Dharap and S.Krishnamurthy (35) presented

in their paper signomial geometric programming technique

for synthesis of four bar mechaniems, Instead of trying

to make the zero error at predetermined point or pointe,

the sum of squares of errors for a number of points in

the range of motion was sought to be minimized, Using

the reciprocal. of the input crank length as a design

parameter for any number of joints, the problem was re­

duced to an unconstrained geometric programming problem

of the signomial type with two degrees of difficulty,

43

The computer programme had been developed to solve the

problem in an iterative fashion for different val.ues of

the design parameters (reciprocal of crank length), The

link proportions were obtained for a specified vaJ.ue of

the design parameter at which the sum of squares of struc­

tural. errors was minimum. The resuJ. ts of the problem

were compared with those obtained by Freudenstein [3],

Mruthyunjaya et al. [37] in their paper presented

a comparative study of the techniques of continuous dynamic

programming, discrete dynamic programming and parametric

programming as applied to the solution of the problem of

optimal allocation of tolerances am clearances in four

bar function generating linkages, For this purpose three

different mechanisms generating the functions y = sin x,

y = log x and y = x2 had been considered. The resul. ts

showed that the parametric programming technique yielded

44

strictest tolerance and clearance values and took minimum

computational. time. Discrete dynamic programming techni­

que yielded comparatively liberal. tolerance and clearam e

values and took moderate computational time.

2.3 OUTLINE OF THE PRESEN'l' WORX

The review of the literature reveals that the

synthesis of the four bar linkage and tolerance alloca­

tion on link lengths, constantly attract the research

workers for more exactness of function or path generation.

New techniques of mathematical programming and computations

are being applied for optimization of errors, velocity

and acceleration of links and transmission angle. One

can also conclude that almost all the methods with the

exception of a few deal with the synthesis problems in a

deterministic sense. Since the link lengths and joint

clearances are rand om in nature, it is realized that the

realistic approach must be based on the probability

coroept.

Secondly one can observe that the methods of

analysis and synthesis reported so far, are iterative

in nature and thus making the solution cumbersome. Thus

closed form solutions are mostly desirable.

45

Thirdly, the synthesis and tolerance allocation

for the specified errors are not considered so far. The

literature reports the synthesis and tolerance allocation

by optimizing the errors. With optimization of errQI:'B ,

there is no control on them. The control on the errors

is a must. Therefore, the synthesis and tolerance alloca­

tion for specified error have the advantage over the opti­

mization technique.

In the present work an earnest effort is made to

accomplish all the objectives discussed above.

2.3.1 Approach for the Present Work

The various approaches used to achieve the above

objectives are based on:

( i)

(ii)

(iii)

The Reliability Approach

The Information Theory

The r·fodified Least Square F-1ethod •

(i) The reliability of a system can be defined as

the probability that the system will perform its intended

design function under specified conditions. We may define

the system i.e. linkage completely reliable, when the

linkage generates the path exactly. So the reliability

is the measure of the deviation of the generated path from

the desired path. The Reliability Index is the overall

reliability f'or the mechanism. The Reliability Index is

used for the synthesis of four bar path generator. The

effect of link tolerances and the type synthesis condi­

tions are also considered.

46

(ii) The Information theory based on probability was

developed for application in Communication Engineering.

This theory utilises the concept of 1 entropy' as a measure

of uncertainty. An analogical. model is developed. The

error existing in the output of a path generating mecha­

nism is expressed by the 'entropy'. The 'entropy• func­

tion is then formulated in terms of design parameters.

~1nimization of the maximum entropy leads to the synthesis

of the path generating mechanism. Allocation of toleran­

ces on the link lengths is al.so carried out.

(iii) Modified least square method or Auto Correlation

is a useful method for the correlation of the generated

path with the desired one. This technique is preferable

when the form similarity is of paramount importance. The

method is applied to synthesize the path generating

mechanism.