Bar Linkage Computers

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BAR LINKAGE COMPUTERS Thesis Submitted in partial fulfillment of the requirements for the degree of


Warren Hundley May 1950

Approved Adv

ProQuest Number: 27591439

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uest ProQuest 27591439 Published by ProQuest LLO (2019). C opyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C o d e M icroform Edition © ProQuest LLO. ProQuest LLO. 789 East Eisenhower Parkway P.Q. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346


To Prof* E. L. Midgette whose patience and helpful suggestions were invaluable in the progress of this work*

DEDICATION To my wife, without whose untiring help this thesis would not have materialized*


CHAPTER I INTRODUCTION .............................





BAR L I N K A G E S .............................


LINKAGE CLASSIFICATION ...................









CHAPTER VIII CONCLUSIONS................................ 121 APPENDIX A ............................................ 126 APPENDIX B .........................................


APPENDIX C . .......................................... 136


Since the beginning of the second World War the scientific and engineering talent, the time, and the money devoted to the development of computing equipment has changed from a trickle to a torrent.

Great advances have

occurred and although not so well publicized as similar war stimulated advances in such fields as nuclear physics and aircraft propulsion they bear the greatest importance* While the results of the great bulk of this effort have to be channelled to meet military needs, a generous portion which is ever increasing, has been alloted to nonmilitary needs#

Witness the theoretical contributions of

such men as Shannon, Wiener, and Aiken and the problems of scientific, commercial, and industrial nature which the large digital computers are handling.

Witness also the belief

held by many that the forthcoming years will show as great an expansion in the application of analog computers to the problems of industrial process control as occurred in their application to military problems during the war years* The subject of bar linkages has been selected from the multitude of other possible topics relating to computer design for two principal reasons:

— I -


The application of bar linkages for computing

purposes represents a phase of the computer field which today is just beginning to be explored*

Contrary to many

other intensively studied and well-publicized aspects of computer problems the acquired knowledge of bar linkages is sparse and the dissemination of this knowledge is exceptionally meager* 2*

Bar linkages show great promise for industrial

process control.

This is because of their ability to mechanize

in a simple manner tabulated data in addition to functional data plus their freedom from the highly developed electrical computing techniques which often do not represent the optimum design in industrial process control. The objectives of this thesis may be summed up under the following three headings: 1.

Collect the available design information from

the few and scattered sources*

Study this information in

order to reject what is incorrect, irrelevant, or repetitious; then simplify where possible and amplify and develop where needed that which remains in order to present under one cover a comprehensive report of available practical design methods* 2.

From the above, develop a design technique

which an ordinary designer rather than a mathematician can execute. 5*

Make an analysis of the future value of bar

linkages as computing devices and outline possible lines of further investigation*


CHAPTER 11 C O M P U T E R S Since the object of this thesis is to investigate a specific phase of the broad field of computer design, it is advisable that the investigation be proceeded by a brief discussion of computers in general.

This is necessary in

order to appreciate the worthwhileness of an Investigation into any basic mechanism, such as bar linkages, which shows promise for computer applications# An important part of the technology that man has been developing during the past centuries has been the application of mathematical relationships to many and various physical phenomena#

The past few decades have seen the rapid

development of machines, which directly apply mathematical principles to specific problems#

One very prominent class

of such machines is known as "computers”, also referred to as "calculating machines", "calculators", etc# A computer may be defined as a device which performs mathematical operations on input data to yield new and more useful results#

Computers, today, range in complexity from

the dollar slide rule to machines, such as the famed ENIAC, which fills large rooms and contains many thousands of intricate parts# The computing equipment available for carrying out numerical calculations falls into two distinct classes#

— 3-


the one hand there are the arithmetical or "digital" computing machines, which accept inputs directly in digital form and with these numbers perform the simple arithmetical operations of addition, subtraction, multiplication and division, usually by the iteration of addition and subtraction in counting devices#

In their simplest forms as exemplified

by ordinary desk calculators produced by Marchant and Monroe, these machines have the virtue of acceptability in a wide variety of computations, including those requiring very high accuracy.

Elaborations, such as the introduction of punched

tape control have greatly increased their possibilities for automatic operation.

Digital machines are, at present, restricted

to processes of arithmetic, and in particular, integration has to be replaced by summation in these machines#


they can be built to work to any finite degree of accuracy. Characteristic of their operation, however, is their computing of numerical results by discrete steps involving finite time delays which may be very large for complex calculations.


strides have been made in the reduction of this calculating time by the introduction of electronic components as computing elements#

A most prominent example of this is the ENIAC which

carries out an addition in a fifth of a millisecond, and a multiplication in less than three milliseconds or at the rate of about a million and a quarter multiplications an hour# In addition, they can handle, if properly designed, any calculation which can be reduced to a sequence of arithmetical operations, although a high premium in bulk and complexity

— 4


must be paid for this flexibility.

These characteristics

of speed, flexibility and bulk have pointed their immediate future development toward the field of large general-purpose machines designed to carry out automatically extended computations according to operating programs which can be altered from one problem to the next.

Problems in such fields

as fluid mechanics, astronomy, pure mathematics, applied statistics, etc., idiich were hitherto incapable of solution because of the impractical amount of computational labor involved in obtaining answers to such matters as 50 or 100 simultaneous non-linear differential equations, etc. will be capable of rapid solution. The second main class of computing equipment is known as analog computers.

These deal with magnitudes

expressed as physical quantities (lengths, angular rotations, voltages, etc.) as opposed to digital numbers.


range from slide rules to differential analyzers and anti­ aircraft computers.

They can handle continuous variables

and in particular can perform integration as a continuous operation.

Possessing less flexibility and less potential

accuracy than digital computers, their great practical advan­ tage is their ability to deliver an instantaneous and continuous solution to specific problems.

Since this type of computer is

particularly adaptable to the solution of specific problems, it is possible to design a very compact and light machine as compared to a digital type even for the solution of a quite


complex problem.

Analog computers in the past and present

have received wide adoption and undergone intense development in various aspects of military science, particularly navigation and fire control. Their immediate future importance in this field and in such fields as commercial navigation, industrial process control, air traffic control, automatic piloting, etc., is assured.

In general, they have possibilities of application

in any field which requires specialized human operations that may be mechanized with the aid of servo systems. Greenwood, Holdam, and McRae in their book "Electronic Instruments" have the following to say regarding the more distant future of computing equipment, "Without exceeding a reasonable extrapolation of known techniques, one may speculate on the possibilities of desk size machines containing the equivalent of whole libraries and capable of high speed selection and cross indexing, machines that perform simple associative reasoning, machines that type spoken words, machines, that translate one language into another, etc.

Even that favorite

of the cartoonists, the 'mechanical man' that can beat its human master in a chess game cannot be said to be an imposs­ ibility."

It is remarkable fact that since 19ii8 when these

words were published the "distant future" has already arrived in the person of the chess playing machine developed by Claude Shannon and his associates at Bell Laboratories, which is capable of anticipating three moves ahead of its human opponent.


6 -



The reasons for the author's restriction of this investigation to the bar linkage aspect of the computer problem have been summed up in Chapter 1 and need not be repeated here#

It may be noted here that these reasons are

in addition to the obvious one that it would require a series of volumes to adequately discuss all the phases of computer design. The term "bar linkage" as used from here on will denote any mechanism consisting of rigid bodies moving in essentially the same plane and pivoted to each other, to a fixed base, or to slides capable of rotation or translation* This is opposed to the classical concept of bar linkages which does not admit slides.

However, this admission is not a real

extension, since bar linkages - in the classical sense, can be designed to apply the same constraints. Historically, engineers and mathematicians have considered bar linkages principally as curve tracers, serving to constrain some point on the bar linkage to move along a given curve.

The classic problem in this field which was

first solved by Peaucellier has been that of finding a bar linkage which would constrain some point to move on a straight line.

This was accomplished by the application of the Peaucellier

inversor to the conversion of the circular motion of a crank

- 7

into a rectilinear motion*

Svoboda points out that Watt

also considered the problem when designing his steam engine and found a sufficiently accurate solution of the problem but the cost and space required forced the adoption of slides in his original design, which represented a corruption of the classical concept of bar linkages* Today, bar linkages are extensively used In mechanical design principally because of their rigidity and simplicity, and their ability to transmit various types of motion with an extremely high order of efficiency*


tions vary all the way from sensitive control linkages for valves and microscope slides to locomotive transmission linkages and quick return devices for automatic meat cutting machines*

It is with these fields of special motion and

power transmission devices that most engineers today associate linkage mechanisms*

Very little is known even among engineers

working in the computer flexd of the usefulness of bar linkages for computing purposes*

It is one of the purposes of

this paper to help make this information available* The amount of published material available to the public dealing with the application of bar linkages to computing purposes is particularly scarce* Appendix B) ,

(See Bibliography -

Svoboda*s publication (1 % 8 ) for the Radiation

Laboratory Series like many another volume in that series is a pioneering work and the cornerstone of the field*


and Silverberg make several important contributions in their -




The remaining information is tied up in unpublished

reports and papers and in the files of companies engaged in this work.

Some of the recently granted patents of Svoboda

are another published source of information.

Svoboda himself

mentions that he knows of only one published work (a Russian treatise) other than his own dealing with the synthetic approach to bar linkage computer design - and this in a more restricted field than he covers*

However, all the work

which is being done in this field, should bring forth further contributions in the near future* There are a few corrolary fields, which are more adequately referenced*

Since these are supplementary fields,

only representative works are included in the Bibiiograpny* Because a knowledge of these fields Is helpful to an engineer designing a computer using bar linkages, a brief discussion of these works and what they cover is warranted* One of these fields is the formal study of kinematic synthesis.

An obstacle encountered here is that very little

has been published in English though a great deal has been published abroad.

However, the interesting though contra-

versial papers of A. E. R. DeJonge, the work of A. W. Klein and the thesis of S. Hersfeld are available*

Also available,

of course. Is the work of Franz Reuleaux (translated by A. B. W. Kennedy) the father of both modern kinematic analysis and synthesis*

Reuleaux*s works consist of some 9^ books

culminating in "Theoretische Kinematik" published In lb75 ^


were followed on the continent by many other learned contributions* It is worth mentioning here that DeJonge speaks ef kinematic synthesis as being divided into three parts as follows: Kinematic Type Synthesis - concerns itself with the establishment ef all existing types and forms of mechanism. Kinematic Number Synthesis - concerns itself with establishment of relationships between the number of element pairs, the number of members, and the constrained movability of kinematic chains* Kinematic Size Synthesis - concerns Itself with creating mechanisms (both the form and required dimensions) to suit given conditions* DeJonge and Herzfeld both feel that the training in kinematics given to most American engineers is very weak in that it consists in the main of a detailed study of a group of mechanisms along with an analytical treatment of certain linkages and omits entirely the methods of kinematic synthesis* It is the opinion of the author, however, that the field of kinematic synthesis is too detailed and much too specialized to warrant the attention of the average engineer­ ing student*

It does seem though that training In formal

synthetic kinematics would be most valuable to engineers actively engaged in kinematic design.

Today when so many

fields of scientific endeavor have advanced so far, it is -f o-

still a far too common practice to design intricate mechanisms in a haphazard trial and error manner*

The solution of the

computer problem will not yield to such methods and a synthetic approach has been found to be the most fruitful.


one should not jump to the conclusion that the methods of kinematic synthesis far reaching though they may be will yield ready made formula-type answers.

This fact will become

apparent in the later chapters* A phase of link work which is sometimes useful in computer design is the venerable study that deals with link­ ages that trace various curves, including the well known mathematical curves*

The amount of literature available in

the English language dealing with these linkages is extensive* A limited list of representative works is included in the Bibliography.

Prominent contributions have been made by E. Hart,

A. Cayley and A. E. Kempe.

Other works on linkages, such

as Michael Goldberg’s, "New Five Bar and Six Bar Linkages in Three Dimensions", which are primarily theoretical in nature are listed here.

A few texts dealing with related mathematical

fields, particularly statistics are also listed* As for the mechanical design of bar linkages it can be said that this aspect of bar linkage design has received thorough treatment even in elementary texts, a few of which are listed, and is well understood throughout the industry* This is primarily due to the fact that it is possible to

treat analytically the kinematic properties of a given linkage such as the distribution of velocities of its parts, its accellerations, forces, and inertias. However, these fields are only supplementary to our problem of designing bar linkages for computing purposes and it is here that we find the great scarcity of published material that we have mentioned* Bar Linkages can be characterized by a considerable number of dimensional parameters, and the field of functions which they are capable of generating is correspondingly large* The core of the difficulty in bar linkage design is the fact that given a well behaved function of one variable, one should be able to select from the field of functions generated by bar linkages with one degree of freedom at least one and probably more than one function that differs from the given function by a relatively small amount.

The characteristic

problem then is one of selecting a curve generated by a linkage from among numerous curves generated by other linkages one that agrees with a given function within specified tolerances*

Svoboda states that these curves are too numerous

and varied for effective cataloguing. The author has seen fit to take exception to this statement and has developed (See Apendix B and Chapter VI) what is believed to be an effective method for cataloging bar linkage functions of one variable.


At present no linkage catalog



There is no question of the value of such a

catalog to the designer particularly one who is a novice as regards bar linkage functions* Svoboda also states that there are a minimum of standard bar linkage function generators and that in the present state of the art, it is nearly always necessary to design a particular bar linkage for any given function. While in general this is correct, particularly with respect to tabular functions, it is expected that anyone working in the field will soon compile a standard set of linkages for such common operations as multiplication, aadition, and the generation of the simple trigonometric and logarithmic functions. Indeed, several linkages are already available# It must also be noted that, in general, it is impossible to design a linkage which is simple and yet match a given function exactly.

Prom a practical point of view,

however, even the simpler bar linkages offer enough flexibility to permit solution of the design problem with an acceptably small error. This error, which in general is not zero, is represented by the difference between the curve selected, with its corresponding linkage and the given function.


presence of this mathematical error which we shall call

(a) One company that I know of and possibly others in the field have done some effective cataloging and consider it extremely useful. However, this work is not available to the public and in all probability so constructed to fit the company’s particular needs as to lose its general value*


"residual error" sets bar linkages apart from other computing mechanisms where the error arose purely from their construction as physical mechanisms with unavoidable mechanical imperfec­ tions*

These mechanical imperfections are capable of reduction

by more careful design, more precise construction, enlargement of entire computer, etc* No amount of careful construction can reduce the "residual error" of bar linkages since it is a function of the mathematical design of the linkage and must be carried over into the finished product*

Since the residual error

does not depend on the mechanical properties or overall size of the linkage it is necessary to alter its structural nature to affect a reduction in this error* A method adopted in the past for the reduction of this residual error has been the addition of links which has the effect of increasing the number of adjustable constants thereby extending the field of functions that the linkage can generate.

From this extended field one can theoretically

select a better match to the given function.

This addition

of parts inevitably increases the design difficulties. Since bar linkages can attain extensive use as computer elements only as efficient and comprehensible design methods are established the notion of reducing the residual error by the introduction of additional links will not be pursued here.

Instead, systematic design procedures involving

Only simple basic linkages and combinations thereof will be




With this approach it is usually easy enough

to design a linkage with a residual error that does not exceed 0*5 per cent of the whole range of motion of the computer.

It becomes relatively laborious to reduce the

structural error below 0.1 per cent, which can be considered in general a high accuracy for analog computers.

When the

tolerances are below this figure, as a typical value, alternatives to the use of bar linkages should be explored. Experience in computer design has shown these alternatives to be few in number and costly both in price and complexity. One of these alternatives which is very powerful, relatively simple, and still permits the use of bar linkages, is their combination with cams.

If, for example, a given

function of one independent variable were to be mechanized with an error of not more than 0.01 per cent, it might be worthwhile to mechanize the function by a simple bar linkage with a maximum error of about 1 per cent and use a cam to introduce the required correction term.

This procedure is

discussed in detail in Chapter VI. Bar linkages cannot be used to serve as differen­ tiators and integrators.

It may be pointed out that this

should not on the face of it rule out bar linkages in a computer which must mechanize a few differentials or integrals, just as the more familiar computing mechanisms should not be considered non-applicable where a computer consists primarily of linkages.

It is often the skillful use of many different

types of components which means the difference between an outstanding computer design and an ordinary one.


From the above, it may be concluded that while bar linkage computing mechanisms in general possess a residual mathematical error this fact alone should not deter anyone from considering them for computer work for even the simpler linkages can be designed with what are often acceptably small errors.

Neither should the fact that the

design approach appears somewhat strange, being mainly synthetic and approximative rather than analytic and exact (an analytic treatment will give rise to complicated simultaneous equations which are difficult to manipulate) deter anyone.

It is the hope of the author that use of the

systematic design approach of Chapter Vll plus a reasonable amount of experience will bring the "art of designing a linkage function generator" a little more into the realm of science.


CHAPTER IV LINKAGE CLASSIFICATION Linkage computing mechanisms may be conveniently divided into two major classifications as follows:



Linkages with one degree of freedom for the generation of functions of one independent variable*


Bar Linkages

with two degrees of freedom for the generation of functions of two independent variables*

This represents a very practical

and natural breakdown of linkage computing mechanisms as they presently exist* BAR LINKAGES WITH ONE DEGREE OF FREEDOM These form the great bulk of linkages found in present day linkage computers and knowledge of their characteristics and design is essential to the computer designer* There are many well-known linkages of this class in existence today such as the locomotive parallelogram linkage, the quick return mechanism, etc*

The motions of

many of these linkages can often be expressed by simple analytical formulae, but they are generally not useful in the mechanization of given functions. In the last chapter it was stated that in the interests of simplicity, the principle of improving the conformity of a linkage’s output to a desired function through the addition of further links would not be pursued* Instead, only simple basic linkages and combinations thereof


' -

will be used in reproducing various functions.

As a matter

of fact, only two basic linkages with one degree of freedom will be employed#

These are

1# the harmonic transformer

2# the familiar four-bar linkage*

These two linkages and

their combinations will mechanise a tremendous field of functions.

This is a simplification of great importance# HARMONIC TRANSFORMER A typical harmonic transformer is shown in Fig# 1|.-1.

>t F(C>4-(

MOAJIC___ *T R A N 5 f O R M B A

It serves to establish a relation between an angular parameter X and a translational parameter Y# It is theoretically correct to consider as equivalent two geometrically similar mechanisms and to disregard changes in scale of the mechanism# The field of functions

X= { C Y )


may then be considered to depend on two ratios of dimensions: L/R and E/R#

Only rarely in computers is the complete range

of motion of a harmonic transformer used#

' ) 8 '

When the range of

the angular parameter X la limited toXwiN^

X ~



/ and

the functiona defined within theae limits are taken as elements of a functional field there la obtained what may be considered a four dimensional functional field depending on Xh/ix

as well as on L/R and E/k# FOUR-BAR LIRKAGE This la the linkage which Svoboda refers to as

a three-bar linkage but which is better known in this country as a four-bar linkage#

It is shown in Fig* ij.-2#

A four-

bar linkage with given dimensions has two forms in which it may be used, with a different function for each form#


dotted line of Pig. li-2 shows the second of the two possible forms#

The four-bar linkage despite its simple structure

will, like the harmonic transformer, serve to mechanize a wide variety of functions#


Fo ur th


\n r k



As a computing mechanism this linkage sets up a relation between the angular parameterX| and the angular parameterX*i_® The linkage is described by four lengths:

Since geometrically similar mechanisms establish the same functional relation between



the field of functions

can be considered to depend on any three ratios of dimensions which establish the structure of the linkage - for example 0

Limits of motion can be assignedto

Xi o r X ^ though not Independently. X(m*A^

For instance one may fix

This increases the number of independent

parameters by two; the field of functions of a four-bar linkage operating within fixed limits may be considered to be five dimensional* LINKAGE COMBINATIONS

- In many cases it is not

possible to mechanize a given function with sufficient accuracy by a four-bar linkage or a harmonic transformer*

Instead of

attempting to devise entirely new structures a procedure which defies systematic design, various combinations of the two basic linkages in series will be employed*

These combinations

greatly expand the field of mechanizable functions, particularly where the curvature of the function is high* A second very important advantage of the linkage combinations is that they enable us to get desired combina­ tions of rotary and translational inputs and outputs.

This is

of great practical Importance in the design of actual computers* Pigs* 4-3 to 4"5 illustrate various possible combina­ tions*

Pig* 4"5# for example shows an harmonic transformer

being driven by a four-bar linkage.

For equal increments of

F 16 4*3


rotation of the input of the four-bar linkage, the input of the harmonic transformer is driven first slowly, then rapidly, accentuating the curvature of the original harmonic transformer trace*

Other typical useful combinations are two four-bar

linkages in series Pig*

and the double harmonic transformer,

which in addition to providing a slide to slide input to output is also very useful for mechanizing functions wnose deviations cross the zero line* COMBINATIONS OF LINKAGES WITH CAMS AND ECCENTRICS When the tolerance on the theoretical error is very small, for example less than 0.1 per cent, it is often wise to consider the combination of a linkage with a cam.

The cam

is combined with a lever system to make up the difference between the desired function and the linkage trace*


the correction motion applied by the cam is only a small fraction of the linkage motion, the cam radius can be very small resulting in a compact overall structure*

The use of

a lever system eliminates the need for precision in cutting the cam.

Fig. 4-6 Illustrates a typical arrangement* Another method for bringing the linkage output into

more exact agreement with the requirea function is the combina­ tion of a linkage with an eccentric. Fig.

The combination

of linkages with cams and eccentrics is considered further in Chapter VI* At this point it is suitable to mention a linkage which finds occasional use as a computing mechanism in combina­ tion with the two basic linkages, that is the one-bar linkage.





16 4-6


L i W K A GE" -






d u t P{J T



O WE ^ B A R -^3



Fig. i^-8.

The combination oi a one-bar linkage with a harmonic

transformer Is sometimes used in mechanizing parabolic functions of high curvature*

However, it is more often found

in a computer in the role of a motion transferring linkage (slides parallel) than as part of a computing mechanism* BAR LINKAGES WITH TWO DEGREES OF FREEDOM These linkages Fig* 14.-9 can be employed for the genera* tion of functions of two variables.

They may conveniently

be divided into two classes, namely, standard linkages which carry out the basic operations of addition and multiplication and non-standard linkages for the generation of more general functions of two variables. Fig* 14.-10 illustrates a standard linkage adder which is adapted for the slide outputs of a harmonic transformer. The adder illustrated is theoretically exact*

If X/ ]

^^ X 3

are three parameters measured along parallel lines from a common zero line, then

(fli- A 2^1 X 3 ~

A 2.X c


It is possible to simplify this mechanism even further with the introduction of only negligible error.

The straight

forward analysis of linkage adders is treated in Chapter VI* In contrast to adders, bar linkage multipliers do not in general perform the operation of multiplication exactly, but careful design has reduced the theoretical error to acceptable limits in nearly all cases. At the time Svoboda was composing his book the idea of a standard multiplier or indeed, any standard linkage

-2 4 -

Fife 4 - Q -

Tw o











6/9 R


was in the realm of future possibilities since at that time, he was designing ab initio his first multiplier*


Svoboda’3 own text and patents on various mechanisms, plus the work of others has shown the author that it is not only possible but extremely useful to develop a standard linkage multiplier or more properly a series of standard multipliers to meet various accuracy requirements, range variations, and extension of one or more of the variables into negative values The second class of two degree of freedom linkages serves to mechanize the more general functions of two variables and can handle almost any well-behaved function of two variables*

By "well behaved" is meant functions without

discontinuities and radical changes in curvature* Practical methods for the design of linkages that generate directly three or more independent variables are not now available and require further development of the synthetic methods of kinematics*

It is possible however,

to mechanize functions of thee or more independent variables when the problem can be mathematically reduced to the mechanization of functions of one or two variables*


represents the primary contribution of Pike and Silverberg*

CHAPTER V TERMINOLOGY AND BASIC CONCEPTS The present chapter will concern itself with a definition of the terminology employed in bar linkage design and the introduction of some basic concepts with wide application in the field* TERMINOLOGY Ideal Functional Mechanism - Any mechanism that establishes definite geometrical relationships between its parts* Terminals

Those elements of the computing

mechanism whose motions represent the variables involved in the computation.

The motion of all terminals is usually

specified with respect to some common frame of reference* Terminals that are mechanically practical are of two kinds: 1*

Cranks or rotating shaft terminals which represent a

variable by their angular motion*


Slide terminals

which represent a variable by a linear motion* Parameters - A parameter is defined as a geometrical quantity that specifys the position of a terminal with respect to some specified zero position.

Upper case letters

will be used to symbolize parameters* Variable - The term "variable" will denote the variables of the problem which the computing mechanism is trying to solve*

A variable will be associated with each

terminal of the mechanism*

To each value of a variable there -xr

will correspond a definite configuration of the terminal, each variable (symbolized by lower case letters) therefore, will be functionally related to parameter of the mechanism# Z = j



Travel - The range of motion of a terminal is called its "travel"® A X ~ Range



(5 *2 )

of a Variable - As a parameter changes between

its l i m i t X m i a )

» the associated variable varies within

fixed, but not necessarily finite limits#

In the case of a

regular mechanism this may be referred to as the "domain" of the variable whose range is A 9^ ~

~ %’/w*n

(5 *5 )

Scales - The value of the variable corresponding to a given configuration of a terminal can be read associated with

that terminal*

from a scale

The calibrationof this scale

is determined by the functional relation between X and x* If the functional relationship is linear the scale will be linear.

For nearly all cases where the computing mechanism

is part of a more complex device this must be the case® Domain - The parameters of a computing mechanism cannot, in general, assume all values.

This limitation arises

either from the geometrical nature of the mechanism or from the way in which it is employed#

It must be emphasized that

the range or domain of the parameters is not necessarily determined by the structure of the mechanism, but by the task

set for It# A mechanism will be said to be a "regular mechanism" when each input parameter can vary independently of all others, between definite upper and lower limits: XimiA, —



» .. .

which define the domain of the parameters. The output parameters of a "regular mechanism" will vary between definite limits as the input parameters take on all possible values# Mechanization of a Function An ideal functional mechanism establishes definite relations between its parameters# Fk ( X , , X ^ . . O = O


Such a mechanism, together with its associated scales, similarly provides a mechanization of functional relations, ■) = o between the variables (

T - / , z... (5:6) ) within a given domain ©f

the independent variables# The forms of these relations may be derived by eliminating the value of the parameters X

between equation

(5 :5 ), which characterizes the mechanism, and equation (5 :1 ) which characterizes the scales# Operator Notation It is often required to combine mechanisms in series where the output parameter of the first becomes the parameter of the second, and so on.

The first mechanism

determines an output parameter Xi-

a function of the input

parameter X( • X 2, - - f v C X ) ^


The second mechanism determines an output parameter X 3I11 terms of X%.#

X t)


while the third determines an output parameter X ^ in terms of X 3 «

X and so on.


= i 3 C A 3)

(5 :7 0 )

The final output parameter, say/)(^, is then

determined as a function of '

X\ :

i, [ u l u - ^ . % ]

While the conventional notation of Eqs. (5:7) nnd (528) is fully explicit, it becomes very cumbersome when dealing with any but the simplest mechanisms# Svoboda has proposed the much more convenient and suggestive notation illustrated below: Eq. (527a) can be written as follows: (5:9») where ( X%. | Xi ) replaces the function

and denotes an

operator which converts the parameter Xj into the parameter Xz, #

A specific one degree of freedom linkage will be

characterized by a particular operator# Similarly, Eqs# (527%) and (527c) become

^3 " C X 3 l X % _ ^ " X z

( 5: 9%)

X 4 - C X ^ 1 X 3 V X 3

(5:9c )

Using this notation, Eq. (528) becomes

X 4 =

CA4 1 X 3 ) ' ( X ^ l X i V C K z l X i V X i -




This form of notation shows clearly the successive operations carried out on Xi to produce X 4 •

The overall

effect of Eqs. (5:9) can be expressed as follows: X4 -

' X,


On comparing Eqs. (5:10) and (5:11) we obtain the operator equation ( X 4 IX,) = C X ^ l X ^ X X s l X O ' C X z l y , ' )


The form of this equation calls out attention to the fact that in a meaningful product of operators each internal parameter will occur twice in neighboring positions in adjacent operators.

One can, without changing the

significance of the operator, strike out such duplicated symbols and condense the notation thus;

C X4lX 3>C X 3lXO -C X ^lX ,)

= C X4lXO -(XzlX,)-C X ^1X.)(5:15») ar

C X-A-l Xj)"CX'3lX'2.X X ^ iXi^ - ( X 41

IX 1) “

(5 :15 %)

Conversely, one can obtain more information about the structure of an operator by an expansion in a direction opposite to that shown in Eqs. (5:15)* Both sides of an operator can be multiplied by the same operator if the resulting operators have meaning, that is the multiplied operators must have neighboring symbols in common. The operator notation is also applied to variables in reference to their relationship to the parameters of a mechanism.

An input scale which determines a parameter X»


as a function

of a variable'Xi can be represented by the

operator X i I'Xi * Xt-CXv(Xi^*Xi

The expression ♦



An output scale would be represented

by the operator

Diaiensionlesa Parameters and Variables The concept of dimensionless or normalized variables and parameters is an extremely useful one in the design of computing linkages, and complex computers# Associated with each variable x having a finite range is a dimenslonless variable (called "homogeneous variable") by Svoboda defined by the following relation: "X -

(5:14) AX

As X varies from its lower to its upper limit,X varies linearly with it from 0 to 1# Another dimensionless variable known as the "complementary dimensionless variable" is defined by X'



M4X ~


or by -



(5 :16)

In the same way there are associated with each parameter X, having a finite travel A x two complementary homogeneous or dimensionless parameters,

X = X-

( 5 : 17)


X' = / -



which change linearly with X between limits 0 to 1,

X - Xmw 4- X



The use of these dimensionlessvariables in linearly mechanizing a given function reduces the complexity of the problem considerably.

For example, if the given function

involves a single independent variable, it may be expressed in terms of a dimensionless input variable

and a

homogeneous output variable X^, 9^2. - f ^


(5 :20)

A linkage with one degree of freedom, operating in a specified domain of the input parameter,Xm^


generate a relation between dimensionless input and output parameters, X ^



f (X.^


It is then necessary to design a mechanism with a domain of operation such that Eq. (5:21) can be transformed into Eq. (5:20) by direct or complementary identification of


with %

, and



Graphical R epre sent at ion of Operators The operator (Y/X) like the function f(X), is conveniently represented by a plot of Y against X* The respresentation takes on considerable value when dimensionless parameters or variables are used.


the respective dimensionless parameters X and Y only vary between 0 and 1 their plot always lies in a unit square. See Fig. 5-1 a. If the deviation (

) of the dimensionless

operator from the diagonal line, is plotted as a function of the input X, the resulting deviation curve has the same

properties as the original curve. maximum value ( Y ~

However, because the

) is usually much less than one, it can

be plotted to a larger vertical scale, a real advantage in many cases.

The deviation curve corresponding to Pig. 5*^1 &

is shown in Pig.


Both of these plots can be used in cataloging various mechanisms, in representing products of operators and in the solution of operator equations through graphical factorization. Estimation This is the process of converting experimental information, the raw material of experience into mathematical form.

Pike and Silverberg have attackea this common problem

from the unique position of the function generator designer; namely, given a table of experimental data, what mathematical form or estimating structure will result in the simplest mechanism?

In the process, they employ a certain terminology

and procedure, the explanation of which is appropriately deferred to Chapter VI.


1 /





j / /

3 a h .2 D Q O Q

/! / / 1/ ' / /

y ,z






I WPUT 5( fj. D I H E W S lOKJLhSS

.6 5