Title: Mass synthesis for multiple balancing criteria of complex, planar mechanisms /
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Title: Mass synthesis for multiple balancing criteria of complex, planar mechanisms /
Physical Description: ix, 144 leaves : ill. ; 28 cm.
Language: English
Creator: Elliott, John Lane, 1946-
Publication Date: 1980
Copyright Date: 1980
 Subjects
Subject: Balancing of machinery   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 141-143.
Statement of Responsibility: by John L. Elliott.
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000098336
oclc - 06723627
notis - AAL3782

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MASS SYNTHESIS FOR MULTIPLE BALANCING CRITERIA OF
COMPLEX, PLANAR MECHANISMS














by

JOHN L. ELLIOTT


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA


1980















ACKNOWLEDGEMENTS


Firstly, the author would like to express his appreciation for the

guidance and support of Professor Delbert Tesar throughout his graduate

career. Secondly, he would like to thank the members of his committee

for the assistance that they have rendered in his pursuit of an education.

He would also like to thank Dr. Dennis Riddle and Professor Gary Matthew

and their wives for the encouragement in this undertaking. Lastly, he

would like to express his appreciation to Keilah Matthew for her kind

persistence in the preparation of this dissertation.

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS

LIST OF TABLES

LIST OF FIGURES

ABSTRACT

CHAPTER

1 INTRODUCTION
1.1 Purpose
1.2 Dynamic Properties
1.3 Balancing
1.4 Recent History

2 DERIVATION
2.1 Purpose
2.2 Coordinate Systems
2.3 Basic Transformations
2.4 Linear Momentum and Shaking Force
2.5 Angular Momentum and Shaking Moment
2.6 Kinetic Energy, Inertia Driving Torque and Power
2.7 Reaction Moment Equation

3 METHODS
3.1 Purpose
3.2 Linear Dependence
3.3 Notation
3.4 The Method
3.5 The Ternary
3.6 Linear Momentum and its Derivatives
3.7 Total Momentum and its Derivatives
3.8 Kinetic Energy and its Derivatives
3.9 Reaction Moment
3.10 Theorems for Balancing Mechanisms
3.11 Mixed Criteria and Balancing Options
3.12 Calculation of Counterweights
3.13 Approximate Balancing


PAGE

ii












4 EXAMPLES
4.1 Purpose
4.2 An Eight-Bar Linkage
4.3 A Cam Driven Five-Bar
4.4 Rules of Thumb

5 CONCLUSIONS
5.1 The Problem
5.2 Derivations and Methods
5.3 Restrictions and Limitation
5.4 Further Research

APPENDIX

A GROUNDED LINK ZERO TERMS

B COMMON TERMS ACROSS PIN-JOINTS

C A GENERAL NEGATIVE INERTIA

D GENERAL COMPUTER PROGRAMS

E COMPUTER PROGRAMS FOR SECTION 4.2

REFERENCES

BIOGRAPHICAL SKETCH


iv


PAGE

63
63
-64
75
113

115
115
115
116
118
















LIST OF TABLES


TABLE PAGE

3.5.1 Ternary Links 41

4.2.1 Mass Parameters for the Links of the Eight-Bar 66

4.3.1 Mass Parameters and Link Dimensions of the Original 77
Mechanism

4.3.2 Mass Parameters of Completely Shaking Force Balanced 82
Mechanism

4.3.3 Mass Parameters of Counterweights for Completely Force 82
Balanced Linkage

4.3.4 Mass Parameters of Completely Shaking Moment Balanced 97
Mechanism

4.3.5 Mass Parameters of Counterweights for Completely Shaking 97
Moment Balanced Linkage

4.3.6 Mass Parameters of Mechanism Balanced for Non-Zero 106
Shaking Moment

4.3.7 Mass Parameters of Counterweights for Non-Zero Shaking 106
Mnment Balanced Mechanism

















LIST OF FIGURES


FIGURE PAGE

2.2.1 General Link 8

2.4.1 A General Link with Mass Content 11

2.7.1 Illustration of the Relation Between Dynamic 24
Properties

3.2.1 Typical Four-Bar Linkage 27

3.3.1 Typical Links 30

3.4.1 A General Four-Bar with Mass Content 34

3.5.1 Stephenson 2 Six-Bar Linkage 37

3.5.2 Possible Ternaries 39

3.7.1 A Four-Bar with Two Negative Inertia Gear Pairs 47

3.10.1 Three Links Joined Only by Sliding Joints 52

3.12.1 Counterweight Mass Parameters 60

4.2.1 Eight-Bar Example 65

4.2.2 Plot of Kinetic Energy of Eight-Bar 70

4.2.3 Plot of Inertia Driving Torque of Eight-Bar 71

4.2.4 Plot of D134 vs D223 74

4.3.1 A Cam Driven Five-Bar 76

4.3.2 Forces of Cranks of Unbalanced Five-Bar 83

4.3.3 Forces in Moving Pin-Joints of Unbalanced Five-Bar 84

4.3.4 Inertia Driving Torque and Rocking Moment of 85
Unbalanced Five-Bar

4.3.5 Shaking Moment of Unbalanced Five-Bar 86

4.3.6 Shaking Force of Unbalanced Five-Bar 87










FIGURE PAGE

4.3.7 Crank Reactions of Force Balanced Five-Bar 88

4.3.8 Forces in Moving Pin-Joints of Force Balanced 89
Five-Bar

4.3.9 Inertia Driving Torque and Rocking Moment of Force 90
Balanced Five-Bar

4.3.10 Shaking Moment of Force Balanced Five-Bar 91

4.3.11 Forces on Cranks of Moment Balanced Five-Bar 98

4.3.12 Forces on Moving Pin-Joints of Moment Balanced 99
Five-Bar

4.3.13 Forces on Gear 9c of Moment Balanced Linkage 100

4.3.14 Inertia Driving Torque and Rocking Moment of Moment 101
Balanced Five-Bar

4.3.15 Shaking Force Balanced Five-Bar 103

4.3.16 Moment Balanced Five-Bar 104

4.3.17 Crank Reactions of Non-Zero Moment Balanced Five-Bar 107

4.3.18 Forces in Moving Pin-Joints of Non-Zero Moment 108
Balanced Five-Bar

4.3.19 Forces at Gear 9c of Non-Zero Moment Balanced 109
Five-Bar

4.3.20 Inertia Driving Torque and Rocking Moment of Non-Zero 110
Moment Balanced Five-Bar

4.3.21 Shaking Moment of Non-Zero Moment Balanced Five-Bar 111

4.3.22 Total Shaking Force of Non-Zero Moment Balanced Five- 112
Bar

A.I Links Grounded at the Moving Origin 121

A.2 Links Not Grounded at the Moving Origin 122

B.1 Equalities of D.pq About Common Joints 124

C.1 General Negative Inertia 126










Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy

MASS SYNTHESIS FOR MULTIPLE BALANCING CRITERIA OF
COMPLEX, PLANAR MECHANISMS

By

John L. Elliott

June, 1980

Chairman: Delbert Tesar
Major Department: Mechanical Engineering

This dissertation explores the general area of the balancing of

complex, planar mechanisms. Methods are developed for the theoretical

balancing for the dynamic properties of any balanceable mechanism. The

dynamic properties directly covered are kinetic energy, inertia driving

torque, inertia power, linear momentum, shaking force, total angular

momentum, shaking moment, and rocking moment of the mechanism.

The objective of this work is to develop a method for the closed

form determination of the mass parameters and mass content of a mechanism

to satisfy some specified balancing condition, either zero or non-zero.

The development of such a method for balancing mechanisms would possibly

lead to the improved performance of mechanisms as machine components

through improvement of their non-linear dynamic properties. The speci-

fic problem addressed is the development of an expression for each of

the dynamic properties in a linearly independent form. Once this is

accomplished, then the components of this expression could be used for

the closed form balancing of a mechanism.

This work builds on the method of linearly independent vectors for

shaking force balancing as developed by Lowen et al., and previous work

by the author. A matrix formulation of the dynamic properties of the


viii









planar mechanism is developed and is used to remove the linear depen-

dencies of the expressions for the dynamic properties of the general

planar linkage. Once this has been done, the balancing conditions for

the mechanism become apparent and balancing may be carried out in a

straight-forward manner. These linear dependencies are eliminated

through the use of algebra and simple planar geometry.

This work provides a simple method of developing the equations for

the dynamic properties of planar mechanisms by simple algebraic substi-

tution. The balancing conditions for the mechanism are derived from this

equation in its reduced, linearly independent form. Predictors for the

number of terms to be expected in this reduced form of the equation are

presented. A theorem which definitely eliminates certain mechanisms

from the possibility of complete balancing is included.

The method of balancing developed is applicable to any planar mech-

anism including pin-joints, sliding pairs and gear pairs. The under-

lying assumptions are that the kinematic description of the linkage exists

and that some method for the dynamic analysis of the mechanism is avail-

able to a user attempting to balance for a specific set of non-zero

values for the dynamic property.

Two examples are included. The first is an eight-bar which includes

a ternary. The balancing equation for kinetic energy and driving torque

due to inertia is developed. The second is a five-bar linkage. This

mechanism is balanced for shaking force and shaking moment. The mech-

anism is analyzed before and after balancing to determine the effect of

balancing for one property on the other dynamic properties of the mech-

anism. Computer programs for use and balancing mechanisms are contained

in an appendix.

















CHAPTER ONE

INTRODUCTION


1.1 Purpose

Mechanisms are non-linear devices. As such they exhibit non-linear

dynamic properties. The energy content and momentum of mechanisms vary

not only with their speed of operation but also with their position. This

means that mechanisms exert varying forces and moments on their surround-

ings, which makes it difficult to predict the dynamic response of a mech-

anism and to size the bearings and prime movers to be used with mechanisms

as machine components. If mechanisms could be designed to operate more

smoothly, they would be more acceptable as machine components. It is the

purpose of this dissertation to present a general method for the balancing

of planar mechanisms by mass addition or redistribution to assure smooth-

ness of operation.


1.2 Dynamic Properties

Principal dynamic properties of mechanisms include their kinetic

energy, linear momentum, total momentum, the rocking moment exerted by the

machine on its foundation, and the associated derivatives of these pro-

perties. Direct control of these properties would allow better design of

machines and their components. Control of the energy content of a mech-

anism would allow firstly the reduction in fluctuation in order to put

fewer demands on commonly available prime movers and secondly the adjust-

ment of the shape of the input energy or torque curve requirement to suit










an available non-standard prime mover such as a spring. Since the shaking

force of a mechanism is the first time derivative of the linear momentum

of a mechanism, the direct control of linear momentum would make feasible

a reduction in the shaking force which a mechanism exerted on its founda-

tion for control of vibration. The control of the rocking moment that a

mechanism exerts on its foundation would allow control of vibration and

noise for the same reasons of smoothness of operation of the whole system.


1.3 Balancing

Balancing of mechanisms in this work will be defined as the ability

to distribute or redistribute the mass parameters of the links of a mech-

anism to satisfy certain prescribed conditions. The mass parameters of a

mechanism are the mass of each link, the moment of inertia of each link

about its center of gravity and the location of the center of gravity of

each link in a reference frame attached to the link. Thus there are four

mass parameters associated with each moving link of a mechanism.


1.4 Recent History

Since the author's thesis [9] was finished in 1976, there have been

several researchers active in the field of study which is the subject of

this dissertation. Most of the research that has been carried on has

been of an iterative nature only. There have, however, been contributions

to the field of closed form balancing of mechanisms during that time.

Bagci [1] derived the complete balancing conditions for the shaking force

of the slider-crank and there is good agreement between his work and the

work in [9]. He and Balasubramanian combined [2] to derive the complete

shaking force balancing conditions for the common six-bar revolute link-

ages and the six-bars with one slider at ground.









In England, Walker and Oldham [27] developed from "the method of

linearly independent vectors" of Berkof and Lowen [5], the shaking force

balancing conditions for an open chain and showed that under the conditions

that the free end is fixed to ground the force balancing of various mech-

anisms is obtained. This method is applicable to the general complex

mechanism. It is possible to derive the balancing conditions for any of

the dynamic properties of linkages using the method that Walker and Oldham

used, but this appears to be more tedious than the approach used in this

work. In a later paper [28], these authors again collaborated to deter-

mine whether a linkage could be fully force balanced, using the theorem

of Tepper and Lowen [26], to determine the minimum number of counter-

weights necessary to balance a given linkage and the optimum placement

of the counterweights in the linkage, the selection of the "best" link

for the placement of the counterweights.

In 1978, Elliott, Tesar, and Matthew [11] explained a method for

the partial balancing of any mechanism. That paper was restricted to the

balancing (redistribution of mass) of a single coupler link since no

attempt was made to eliminate the linear dependence of the vector des-

cription of the dynamic properties of the mechanism. This work, as well

as the previous works by the author [9] and [10], is limited in that there

has been no development of the required and allowable balancing conditions.

Hence, the designer is restricted in a sense to the iterative application

of the balancing conditions followed by analysis to determine if other

properties of the mechanism have been negatively affected. The reader is

referred to the works [6] and [15] by Berkof and Lowen for what may be an

appropriate predictor technique as an aid to the designer. This work was

originally done for use in prediction of the allowable and desirable










balancing conditions for the shaking moment of the four-bar with constant

speed input that had been previously shaking force balanced. That these

conditions are required for the complete shaking moment balancing of any

four-bar has been amply demonstrated in [9]. The prediction graphs that

have been developed [6]and[15] can be used since one of the other inter-

esting results of the previous work by the author was the demonstration

(as is pointed out by Berkof [4]) that the torque balancing conditions

are satisfied if the shaking moment conditions are first satisfied (with

the unfortunate requirement that negative inertia be provided). The

important point to note here is that the inertia driving torque of the

mechanism will be greatly reduced if one constructs the mechanism so that

the centers of mass of the links of the mechanism lie on the center-lines

of the links. Then it becomes obvious that the prediction technique

developed by Berkof and Lowen for shaking moment may be extended to the

balancing of more complex mechanisms even though it was originally done

for a special class of mechanism.

In 1968, Ogawa and Funabashi [19] balanced a spatial mechanism for

inertia driving torque. Their paper was a combination of theoretical

work and experimental analysis to substantiate the theory. Two of the

methods that were used in the balancing of the mechanism were reasonably

well known: the additions of balancing dyads (auxiliary mechanism) and

harmonic balancing using planetary gears. It should be noted that bal-

ancing using planetary gears had been attempted previously in order to

control the shaking force and shaking moment of the mechanism rather than

the inertia driving torque.

Carson and Stephens [7] present optimization criteria for the bal-

ancing of in-line four-bar linkages. These criteria define usable links










in that the radii of gyration of the links are related to the lengths of

the links of the mechanism. Equations, graphs and monographs are pre-

sented so that the designer may determine if "real" links can be expected

from a mechanism which has been shaking force balanced and root-mean-square

shaking moment balanced.

Paul [20] presents a good summary of the balancing techniques avail-

able until 1978. These include balancing for harmonics, the method of

shaking force balancing used by Berkof and Lowen [5] and an explanation

and extension of a method of sizing a flywheel that was put forth by

Wittenbauer in 1923. Paul's text deals mainly with the analysis and

dynamics of mechanisms. A good description of Lagrangian mechanics is

presented on a basic level.
















CHAPTER TWO

DERIVATION


2.1 Purpose

As was explained in Chapter One, most of the balancing methods that

have been used in the past have been either methods of approximation

(mathematical or graphical) or methods of total balancing as applied to

special configurations of mechanisms. These methods have required a com-

plete understanding of the mechanism to be balanced. By contrast, a com-

pletely general method of balancing planar mechanisms will be presented

in this dissertation. The general form of the equations for the balance-

able dynamic properties of mechanisms will be derived in this chapter.

The work that will follow presumes that the description of the mech-

anism exists. That is, that the lengths and the orientations of the links

of the mechanism are known. These may be from an existing mechanism or

be the result of some synthesis on the part of the designer (see Ref.

[12], [24] and [25]). It is possible to balance a mechanism if the mass

parameters of the system are known, but this is not necessary. It is

also presumed that a relatively efficient program for kinematic or dynamic

analysis is available, such as that in Ref. [21]. Before any balancing

for non-zero dynamic properties is attempted, the mechanism must first

be analyzed and the data made available for use in the dynamic equations

for balancing to be presented in this chapter.









2.2 Coordinate Systems

In the derivations to follow, a special notation and set of coordi-

nate systems will be used. A fixed coordinate system (see Fig. 2.2.1)

will be used to trace the motion of a point, p. This point will be desig-

nated with the letter pair (Url V ). Each moving link will have attached

to it a moving coordinate system. All of the dimensions of points located

in the moving reference system will be presumed to be constant. A point

fixed to the moving reference system will be designated with the letter

pair (ur, vr). An attempt will be made always to fix the origin of the

moving reference system to some point in a link whose motion (Up, Vp) in

the fixed reference system is known. These special reference systems

will be used in order to continually remind the reader that the object

of the work presented here is to synthesize the mass parameters in the

moving reference system.


2.3 Basic Transformations

The work to follow will consist of the transformation of the classic

equation for some dynamic property of a link to two unique forms. In

order to accomplish this, the motion transformations for the position

and velocity of a point will be needed (as well as the representation of

the rotation of a link) in terms of other known motion parameters. These

transformations will be given here for compactness of presentation.

Consider the representation of a link undergoing coplanar motion

(Fig. 2.2.1). Points p and q are two points in the link whose motion,

position and velocity, are known (Up, Vp, Up, Vp, Uq, Vq, Uq, Vq). The

angular motion of the link is also known as Ypqi, pq' A third point, r,

is located in the moving coordinate system attached to point p with the

fixed dimensions ur and vr.
































































FIG. 2.2.1 General Link









Paint q can be located relative to p with the following transform:


Uq = Up + apqcos Ypq, (2.3.1)

Vq = Vp + apqSin pq.


The derivatives with respect to time of these functions yield


Uq = Up apqsin YpqYpq, (2.3.2)

Vq = Vp + apqcos YpqYpq.


The first pair of equations can be solved for cos ypq and sin ypq to yield


cos Ypq = (Uq Up)/apq (2.3.3)

sin y = (V V )/a
pq q P Pq

with their time derivatives as


-sin ppq = (U U)/apq (2.3.4)

cos Ypqpq = (Vq Vp)/aq


Equations (2.3.4) can be squared and added to give


Yp = Y (cos2y + sin2y) = [(Uq p)2 + (q Vp)2]/a2. (2.3.5)
ppqg pg p q p pq=

If Eqs. (2.3.4) are multiplied by -sin y and cos ypq and added the result

is


pq = pq(cs2ypq + sin2 ) = [-sin ypq(U Up) + cos Yp(V V)]/apq.

This equation can be made more useful if Eqs. (2.3.3) are substituted for

Sit ypq and cos Ypq


Ypq = [(Uq U)(Vq V (V V )(Uq p)]aq. (2.3.6)








The position of r in the fixed coordinate system is given as


Ur = Up + Urcos Ypq vrsin Ypq, (2.3.7)

Vr = Vp + ursin ypq + Vrcos Ypq*


The time derivatives of these equations are


br = Up (ursin Ypq + Vrcos Ypq)Ypq, (2.3.8)

Vr = Vp + (urcos Ypq vrsin Ypq)Ypq-


Substitution of Eqs. (2.3.3) into Eqs. (2.3.7) yields


Ur = Up + [ur(Uq Up) Vr(Vq Vp)]/apq, (2.3.9)

Vr = Vp + [ur(Vq Vp) + Vr(Uq Up)]/apq


with their time derivatives


U = Up + [u (Uq Up) Vrq p)]/apq. (2.3.10)

Vr = p + [ur(q V ) + r (q p )]/apq

These are all of the transformations necessary in the derivations to

follow. In Eq. (2.3.9) and Eq. (2.3.10), the transcendental functions of

the link angle, y have been eliminated.


2.4 Linear Momentum and Shaking Force

In [5], Berkof demonstrated that the shaking forces of a mechanism

could be found as the time derivative of the linear momentum of the mech-

anism. The linear momentum of a link such as that shown in Fig. 2.4.1 can

be written as


S= m(UG + iG) (2.4.1)

where UG and VG are the real and imaginary translational velocities of the



































































FIG. 2.4.1 A General Link with Mass Content









center of mass of the link. Substitution of Eqs. (2.3.8) into this equa-

tion yields


= m[Up uGsin ypq + VGCOS ypq)ypq

+ ~Vp + i(UGCOS Ypq vGsin Ypq)Ypq].


If this equation is expanded and the constant coefficients are collected

on the time-dependent variables, the result is


S= m(U + iVp) + muG(icos Ypq sin ypq)ypq

+ mvG(-cos Ypq isin Ypq)Ypq


which may be written as

4
L = YD1 + YD + Y1D1 + YD = Z Y1D1 (2.4.2)
Y2 2 Y33 4 4 i= l

where


Y = m,

Y = muG,

Y= mvG,

Y1 (a term to be defined in the next section),

D = U + i
I p %P'

DI = (cos ypq sin Ypq)Ypq,

D = (-cos ypq isin Ypq)ypq,

D = 0.

Here Y4 and D4 have been defined simply for notational convenience as will

be seen later. This is the formulation of the linear momentum of a link

expressed in terms of the motion of a single point in the link and the

rotation of the link. Alternatively, Eqs. (2.3.10) may be substituted for

the velocities of the center of gravity of the link in Eq. (2.4.1) to yield








the new form of the linear momentum equation


L = m[Up + {uG(Uq Up) VG(Vq Vp)}/apq

+ u{UG(Vq p) + vG(Uq Up)}/apq].


This result may be expanded and rewritten as


L = m (1 uG/apq)(Up + iVp) + m(vG/apq)(Vp Up)

+ m(-vG/apq)(iVq Uq) + m(uG/apq)(Uq + Vq)


which has the reduced form of


L = YD2 + 2D2 + Y2D2 + y2D2 = y2D2 (2.4.3)
f= 1 D 2 22 3D i4 l i3i
i=l

where


Y = m(l uG/apq),

Y2 = m(vG/apq),

Y = m(-vG/apq),
Y2 = m(ug/apq),





D2 = q i, and

D2 = q + 1 q


This is the equation for the linear momentum of a link written in terms

of the motion of two points, p and q, in the link. Both Eqs. (2.4.2) and

(2.4.3) are written in terms of constant terms (defined in terms of some

of the mass parameters of the link and the length of the link) which are

coefficients of time-dependent terms. The total linear momentum of a mech-

anism may be found as the sum of the moment of the individual links.









As was stated at the beginning of this section, the shaking force of

a mechanism may be found as the time derivative of the linear momentum of

the mechanism. The time derivative of Eq. (2.4.2) is

4
Fs = I YiLD (2.4.4)
% i=1

where


F= the vector sum of forces exerted on the link by its surroundings

51 =U + iV

Di (icos pq sin Ypq)Ypq + (icos Ypq + sin Ypq)Ypq'
2 + si p 2,

S= (-cos ypq isin Ypq)Ypq + (sin ypq icos Ypq)ypq,

S= 0.


A similar treatment of Eq. (2.4.3) yields

4 2*2
Fs = YiD (2.4.5)
. i=l

where


D = U + iVp,

D= Vp ip,






So that the shaking force of the mechanism has been found as the sum of a

series of terms, each of which is composed of a constant term, which are

coefficients of time-dependent variables.


2.5 Angular Momentum and Shaking Moment

It was demonstrated by Elliott and Tesar [10], and elsewhere; [1] and

[3], that the shaking moment of a linkage can be found as the derivative









with respect to time of the total angular momentum of a mechanism. The

angular momentum of a link, such as that shown in Fig. 2.4.1; is'given as


Ho = m(UGVG VGUG) + mk2Ypq (2.5.1)

where the first term on the right-hand-side of the equation is recognized

as the moment of momentum of the link about the origin of the fixed refer-

ence system and the second term is the angular momentum of the link due to

its angular velocity. Substitution of Eqs (2.3.7) and (2.3.8) for UG, VG,

UG, VG in the above equation results in


Ho = m[(Up + UGcos ypq vGsin pq)(Vp + {UGCos ypq vGsin ypq}Ypq)

(Vp + uGsin ypq + VGCOS Ypq)(Up {uGsin Ypq + VGCOS ypq}Ypq)]

+ mk2 pq.


If the indicated multiplication is carried out and terms collected in

terms of constant coefficients, the result is


Ho = m(UpVp VpUp) + muG[(Upypq + Vp)Cos ypq + (Vpypq Up)sin pq]

+ mvG[(Vppq Up)Cos pq (Uppq + Vp)sin ypq] + m (k2 + + vY)


which can be written as


H = Y D + YD3 + Y + YD3 (2.5.2)
1 1 22 33 4 4

where


Y' = m(k2 + u2 + Vg),

D3 = Up Vp,

D = (Upypq + Vp)os ypq + (Vppq Up)sin pq,

Dj = (Vpypq Up)cos ypq (Upypq + Vp)sin Ypq,

D = ypq.









The rest of the Y! are the same as those defined in Eq. (2.4.2). This
1
is the equation for the total angular momentum of a link expressed in

terms of a set of constants ( the Y') multiplied by a set of time-dependent

variables (the Di). The total angular momentum of a mechanism can be found

as the sum of the angular moment of the links of the mechanism. An alter-

native form of the angular momentum of a link may be found by substituting

Eqs. (2.3.9) and (2.3.10) for UG, VG, UG, VG and Eq. (2.4.6) for { in Eq.

(2.5.1) to give


Ho = m[{uG(Uq Up) + vG(Vq Vp) {uG(Vq Vp) + vG(Uq Up)}/q

uG(Vq Vp) + vG(Uq Up)}{uIq Up) VG(Vq Vp)}/aq]

+ mk2[(U Up)(Vq Vp) (V Vp)(Uq p)]/aq.


If the indicated multiplication is carried out and the collection of terms

is done in the previous manner, the equation reduces to


Ho = YJD4 + Y3D4 + Y3D3 + Y3D4 (2.5.3)


where


Y3 = m(l-2uG/apq) + m(k2 + u1 + vd)/aq,

Y23 = m(uG/apq) m(k2 + u2 + v2)/a2

Y3 = m(vG/ap),
3 G pq
3 = m(k2 + u2 + vG)/a,

D4 = U Vp VUp,
1 pp p p
D4 = Uq+ U VU V p
2 pq qp pq qp,
D4 = U q Uq U + VV Vp
p q q p p q q p
D = U V u
qq q q









This is the equation for the total momentum of a link expressed in terms

of the motion of two points, p and q, in the link. It is a collection

of products of constant coefficients and time-dependent variables. The

total momentum of a mechanism may be found as the sum of the moment of

the individual links of the mechanism.

As was stated at the beginning of this section, the shaking moment

of a mechanism may be found as the derivative with respect to time of

the total momentum of the mechanism. If the time derivative of Eq. (2.5.2)

is taken, the result is

4
Mo = Z Y13 (2.5.4)
S i=l i- i

where


O = the shaking moment with respect to the origin of the fixed coordinate

system,

= U Vp- VpUp
pP P P,
2 + q p p U ) q'
D2 = (Upq + V p + Vp)cos + (Vyp Up Up)sin

S= (V U "2 U )cos y (U + + V )sin ypq and
3 p pq p pq pq p pq p pq + pq

4 Ypqp


For Eq. (2.5.3), the differentiation yields


M y= YD + Y2D4 + 3D + y3D4 (2.5.5)


where


D = pVp VpUp

S= Upq + Up VpUq VqUp

4 = UpUq UqUp + VpVq VqVp, and

4 = UqVq VqUq









This provides two formulations of the equation for the shaking moment of

a mechanism; the first, Eq. (2.5.4), expressed in terms of the motion of

a point, p, in a link and the rotation, ypq, of the link and the second,

Eq. (2.5.5), expressed in terms of the motion of two points, p,q, in the

link. Both of these equations, though algebraically different, will

yield the same value for the shaking moment of a mechanism.


2.6 Kinetic Energy, Inertia Driving Torque and Power

The kinetic energy of a link (see Fig. 2.4.1) is given as


Ei =m(U2 + V) +mk22 (2.6.1)
C C G) 2 pq

The first term of this equation is the kinetic energy due to the linear

velocity of the center of gravity of the link and the second term is the

kinetic energy due to the angular velocity of the link. Substitution of

Eq. (2.3.8) for UG and VG into the equation yields


Ei = 21m[p (uGsin Tpq + VGCOs ypq)Ypq}2


+ {fp + (uGcos y pq- vGsin Ypq)pq}2] + mk2-yq.


If this form is expanded and the appropriate collection of terms performed,

the resulting equation is


S= m[l(Up + V2)] + muG(-Upsin Ypq + VpcoS ypq)ypq
2 +p V2p 112

mV(Upcos ypq VGsin ypq)pq + m(k2 + u + V 2pq


which can be written as


Ei Y1D5 (2.6.2)
i=l i i









where the variable terms are


D = (U2 + V2),

D = (-UpSin y + VpCO pq)pq,

D = (Upcos Ypq Vpsin Ypq)Ypq, and

D5 1. Z-
D4 = pq

This equation expresses the kinetic energy of a link in terms of the

translation of a point in the link and the rotation of the link. Alter-

nately, Eq. (2.3.10) may be substituted for UG and VG and Eq. (2.3.5)

may be substituted for Ypq in Eq. (2.6.1) to yield


Ei = [{Up + (uG(Iq p) V q -p

+ {Vp + (uG(Vq Vp) + v(Uq Up)}]/apq

+ mk2[(q p)2 + (Vq V)2]/a.
q p q p pq

When the terms are squared as indicated and the equation simplified by

collecting terms, the result is

m(k2 + u2 + v2) G
Ei = m(l 2u/a ) +( -- )[(2 + 2)]
2 pq a2 2 p p
2pq
G m(k2 + u + G)
+ ma -)a2 UpU + V Vq)
pq pq

+ m(G)(VpUq UpVq) + m(k2 + u2 + v2) [(Up2 + 2)]
apq

which can be rewritten in the economical form


Ei = YD6 + 3D6 + y3D6 + Y3D6
11 2D2 3-3 4D4


(2.6.3)








where


Ds = 1(U2 + V2)
1 2 P p

D| = Upq + Vpq,

D = Uqp Vq, and
q p p q
D6 (2 + 42).
4 j2q q


This is the equation for the kinetic energy of a link experiencing co-

planar motion expressed in terms of the motion of two points in the link.

The total kinetic energy of a mechanism may be found as the sum of

the kinetic energies of the links of the system. The inertia driving

torque, Td, for a system may be found as the positional or geometric

derivative of the kinetic energy of the system. For a single degree of

freedom system, the geometric derivative of the kinetic energy for a

single link with 6i (the input position parameter) as reference may be

found for Eq. (2.6.2) as


Td d -) = E Y( -) = Y (D )' (2.6.4)
1 d i=l i 1di i=l

where


(Dc)' = (U U + Vp4),
1 pp pp
( (Vppq UpYpqYpq + pq)COS ypq
"- ( + VV y' + U q)siny
( pq ppqypq pq Ypq'
((DU)ypq p pqq + Y )pq)CS pq
("' + y + +ppq)sin and
(pYpq ppqyq pq pq
(D5),' = (ijqpq).









Also, for Eq. (2.6.3)


T Z4
T = Z Yi(D)' (2.6.5)
d i=l

where


(D) (ip + ),

(D2)' = (0pq + Up6q + pq + Vp% ),

(D')' = (Ciqp + UqVp Up#q Op44), and

(D) (Cq0' + Vq4).


The power required to drive the mass of a mechanism may be found as

the time derivative of the total kinetic energy of the system. For the

individual links of the system, the time derivative of Eq. (2.6.2) is

4
pi = E YiD" (2.6.6)
i=l

where


" = "pip + qVq
'S2 + V2Cos
2 = (Vp pq p-pq pq pq)co pq
( pq + + U pq)sin pq
ppq p pq p pq pq
5 .- ( 2 + U Yi )Cos
3 = ppq ppq p pq) Ypq

(Vpq + Y2 + V )cos pq, and
p pq p pq pYpq po
D5


Also, for Eq. (2.6.3)

4
p = E Y3Dh (2.6.7)
i=l









where


D51 = UpUp + VpVp,

D = UpUq + UpUq + VpVq + VpVq,

D = UqVp + UqVp VpUq VpUq' and

4D = qqiq + qVqq


Thus, in this section, the algebraic equations for each of the pro-

perties, kinetic energy, inertia driving torque, and power, have been

formulated in two distinct forms. All of these equations are expressed

in terms of a set of constant coefficients, the Yi in terms of the link

dimensions and mass parameters, multiplied by a set of time-dependent

variables, the Di in terms of the motion phenomena of the linkage.


2.7 Reaction Moment Equation

It is well known that the shaking moment, Mo, and the inertia driving

torque, Td, are related (see Ref. [4]) by the equation


oM = T + rix F


where


ri = the vector locating the fixed pivots of a mechanism, and

Fi = the reaction forces to ground of the mechanism.


This equation can be solved to yield

Mr r- x F. T (2.7.1)
= r x Fi = M T (2.7.1)


where


Mr = the reaction moment of the mechanism.
'LP










Figure 2.7.1 is a graphical representation of this equation. Earlier in

this chapter, the equations for the shaking moment and inertia driving

torque were found in two forms. If the definitions of shaking moment

and inertia driving torque are substituted into Eq. (2.7.1), two forms

of the reaction moment equation can be found. Using Eqs. (2.5.4) and

(2.6.3) gives

4 4
M= E (YiD3 Y(D5)') = E Y1( (D5)') (2.7.2)
i=l i=l I

or, if Eqs. (2.5.5) and (2.6.4) are used, the formula takes the form


Mr = 3D (D)'). (2.7.3)
,o i=l (i (

The result is that the reaction moment of the link is expressed in two

similar forms, both of which consist of constant coefficients multiplied

by time-dependent variables. The total reaction moment of the mechanism

may be found as the sum of the contributions of each of the links in

the mechanism.






































































FIG. 2.7.1 Illustration of the Relation
Between Dynamic Properties
















CHAPTER THREE

METHODS


3.1 Purpose

Balancing, as it is defined for this dissertation, is the adjust-

ment of the mass parameters of the links of a mechanism to suit prescribed

conditions in one or more of the dynamic properties. The equations that

were developed in Chapter Two can and have been separated into two parts:

the first part is the collection of the terms that are constants, Yi, and

that are made up of the mass parameters of the mechanism and the kinematic

parameters of the mechanism; the second part is a series of terms that are

time-dependent variables, Di. For any given mechanism, the time-dependent

terms are fixed when the dimensions of the links of the mechanism are

selected and the input state defined. Through control of the constant

premultipliers, Yi, of the time-dependent terms, one can control the

dynamic properties of the mechanism. The methods to be developed in this

chapter will allow one to have a closed form solution for the mass para-

meters that will satisfy the prescribed conditions and will show that the

form of the equations that have been developed lends itself well to various

schemes of optimization. The methods will be developed through the use

of non-numerical examples. In Chapter Four, numerical examples of bal-

anced mechanisms will be given.









3.2 Linear Dependence

In Ref. [5], it was demonstrated that conditions for shaking force

balancing of simple linkages could be derived from the equation which

locates the center of mass of the linkage if that equation were expressed

in terms of a set of linearly independent vectors. This concept was

extended to the shaking moment and inertia driving torque balancing of

four-link mechanisms in [10]. Here it will be shown that the formula-

tions of the equations for the dynamic properties of general planar link-

ages, as derived in Chapter Two, are expressed in terms of a set of

linearly independent vectors. Therefore, it will be possible to derive

a set of balancing conditions for any particular mechanism.

Figure 3.2.1 is a line representation of a four-bar linkage. The

well-known vector loop equation for this linkage can be written for

this mechanism as


alei01 + a2eiO2 a3ei03 aei 4 = 0 (3.2.1)


where


ai (i = 1, 2, 3, 4) are the constant link lengths of the linkage, and

ei1i (i = 1, 2, 3, 4) are unit vectors which are determined by the

positions of the linkage.


From the definition of linear independence, as given in Ref. [29], the

unit vectors e will be linearly independent only if all the coeffi-

cients are zero to satisfy the controlling equation such as that of Eq.

(3.2.1). If this is not the case, it may be concluded that the unit

vectors, the e ii, are linearly dependent. In Refs. [10] and [5], Eq.

(3.2.1) was used to eliminate one of the time-dependent variables, i1,


























\c
\


--"









i2, or 03, from an equation for a given dynamic property of a linkage.

This equation was then expressed in terms of two of the vectors, ei1l (i

= 1, 2, 3) and ei4 which is a constant. Then, this equation was found

to be expressed in terms of a set of linearly independent vectors, i.e.,


ale'0l + a3e 3 + aei4 = 0


which can only be satisfied, in general, if all the ai are exactly equal

to zero.

All of the equations for the dynamic properties of linkages, suLh

as that shown in Fig. 3.2.1, which were derived in Chapter Two, are ex-

pressed in terms of the motion of the two pin-joints, 2 and 3, of the

linkage. It is immediately obvious that


S = f(X2, Y2, X3, Y3) = f(,l2) = f(l'i,3)


where


S = a given dynamic property,

f = the resulting function, and

(X2, Y2, X3, Y3) = the motion of pin-joints 2 and 3.


From this observation, it is recognized that the equation for a dynamic

property of a four-link mechanism, which is written in terms of the motion

of the pin-joints, is expressed in terms of a set of linearly independent

vectors. Since this was done in Chapter Two for all of the dynamic pro-

perties of general planar linkages, it is apparent that any of these

equations, Eqs. (2.4.3), (2.5.3) and (2.6.3), will yield a set of bal-

ancing conditions if properly rearranged. In the following sections of

this chapter, that manipulation will be explained.










3.3 Notation

A convenient system of notation is adopted as depicted in Fig. 3.3.1.

A link with two pin-joints is represented by the letters p and q or the

ordered pair, pq, which is representative of the two endpoints of the

link. A reference system is fixed in the moving link with its origin

attached at the p end of the link and the u-axis aligned along the center-

line, pq, of the link. The importance of this orientation will be demon-

strated later. A link that is a part of a sliding pair is designated with

a similar pair of letters, rs. Where r is fixed to a pin-joint in the

link, if one is available; otherwise, it may be any point in the link.

In this work, the direction of rs is taken in the same direction as

relative sliding between the associated sliding links. A moving coordi-

nate system is fixed in the link with the origin attached to r. The u-

axis of the moving coordinate system is aligned along rs. The use of

this notation will result in the designation of each link in the system

by a pair of numbers or letters.

The object of the synthesis procedures, that will be developed here,

is to define the mass parameters of the link. Since this is required,

the location of the center of the mass of the link will be defined in

the moving system with the pair (upq, Vpq) for the pin-jointed link and

similarly for the links of the sliding pair. The mass of the link will

be identified as mpq. The moment of inertia of the link about its center

of gravity will be designated as Ipq = mpqkpq.


3.4 The Method

In Chapter Two, it was demonstrated that any of the dynamic properties

of a mechanism could be defined as the sum of that particular property

for all of the links of the mechanism. Further, it was shown that each

































































FIG. 3.3.1 Typical Links










of the dynamic properties of the individual links could be expressed as

the sum of four terms where each of the terms is the product of a con-

stant, Yi, and a variable, Di. If S represents any dynamic property,

then

4 j m 1 n
S = E YipqDipq YirsDirs (3.4.1)
pq i=1 rs i=l

where


E stands for the sum over all of the links with pin-joints at each
Pq
end,

Y. are the Yq from Chapter Two with the added subscripts to count

over all the pinned links of the system,

Dp are the DT from Chapter Two with the added subscripts to count

over all the pinned links of the system,

E stands for the sum over all of the sliding links,
rs
Ys are the Y! from Chapter Two with the added subscripts, rs, to
irs1

count over all the sliding links of the system,

Dirs are the D from Chapter Two with the added subscripts, rs, to

count over all of the sliding links of the system,

m = 2, 4, or 6,

n = 1, 3, or 5, and

j = 2 or 3.


From this point on it is assumed that Yipq stands for any of the Ypq or

Yirs and that Dipq stands similarly for the Dipq or Dirs. The equation

for any dynamic property can be written as

4
S = E E Y. D. (3.4.2)
pq i=1 ipq ipq









Now, it is presumed that the kinematic representation of the mechanism

exists and that a kinematic analysis of the mechanism has been performed.

If this is the case, then the Dipq may be considered as knowns. Their

functional form will not change so long as the kinematic dimensions of

the mechanism and the input states) are not altered. If the Yipq of

the mechanism are known, then a dynamic property of the mechanism can be

evaluated for each position of the mechanism using Eq. (3.4.2). If this

evaluation is performed for several positions of the mechanism, then the

dynamic property could be evaluated in several positions and the results

tabulated in matrix form as


[S] = [D][X]


where


[S] is a single column containing the values of a dynamic property for

each position of the mechanism,

[X] is a single column made up of the various Yipq of the mechanism, and

[D] is a matrix of the variable Dipq terms, each row of this matrix

corresponds to a single position of the mechanism.


On the other hand, if the dynamic property in each position is known and

it is desired to balance the mechanism by determining the Yipq' a simple

process of matrix manipulation yields


[D]-'[S] = [D]-1[D][X] = [X] (3.4.3)


where [D]-1 is the inverse of the matrix [D]. The inverse of a matrix

will exist if, and only if, the matrix is non-singular. This requires that

[D] be a linearly independent matrix.









It was shown in Section 3.2 that the equations developed in Chapter

Two for the dynamic properties of mechanisms are expressed in terms of a

set of linearly independent vectors. However, a definition of linear

independence from matrix algebra requires that the columns and rows of

the matrix [D] be linearly independent. This means that no columns (rows)

may be equal to any other columns (rows) of the matrix and that no

columns (rows) of the matrix be made up of a linear combination of other

columns (rows) of the matrix. For the mechanism shown in Fig. 3.4.1,

the general equation for any dynamic property can be written as


S = Y112112 + Y212D212 Y312D312 + 412D412 (3.4.4)

+ Y3 +2323 Y223D223 + 323D323 + Y423D423

+ Y134D34 + Y234D234 + Y334D334 + Y424D44'


From Eqs. (2.4.2) and (2.7.3), it is possible to recognize special values

for certain of the Dipq for all dynamic properties (see Appendix A)


D112 = D434= 0


and (see Appendix B)


D412 = D123; D423 = D134-


The substitution of these definitions into Eq. (3.4.4) yields


S = Y1120 + Y212D212 + Y312D312 + Y412D123

+ Y123D123 Y223223 + Y323D323 + Y423D134

+ Y134D134 + Y234D234 + 334D334 + 34


where, in matrix form, each of the Dipq would represent a column of the

matrix [D]. In order for [D] to be nonsingular this form must be




























E -


> E = E










rearranged. The columns of zeros must be eliminated along with the cor-

responding constants, Y112 and Y434, and the number of columns of the

matrix must be reduced since, in two cases, adjacent columns will be

equal to one another. If both of these requirements are fulfilled, the

equation becomes


S = Y212212 + 312D312 + Y223D223 + 323D323 + Y234D234
(3.4.5)
+ Y334D334 + [Y412 + Y123D123 + [Y423 + Y34D134


or


S = X1D212 + X2D312 + X3D223 + X4D323(3.4.6)
(3.4.6)
+ XgD234 + X6D334 + X7D123 + X D134


where


X1 212'

X2 = Y312

X3= Y223,

X4 = Y323'

X5 = 234'

X6 = Y334,

X7 = Y412 + 123, and

Xg = Y423 + Y134*


This is the most compact representation of the general equation for dynamic

properties of the simple four-link mechanism shown in Fig. 3.4.1. It is

expressed in terms of a set of linearly independent vectors (the D terms)

and all of the linear dependencies of the matrix form have been eliminated.

The equation may be used in the matrix manipulation of Eq. (3.4.3) to find









the values of the components of [X] to satisfy a set of specified values

of dynamic properties [S] to balance a mechanism. The equation for the

dynamic property of any mechanism must be reduced in a similar manner to

its linear independent form in order that it may be used to balance the

mechanism. Other examples of the elimination of linear dependence will

be illustrated in the next section so that the extension to more complex

mechanisms will be apparent.


3.5 The Ternary

It was demonstrated in Section 3.2 that the formulation of the equa-

tions for dynamic properties, as given in Chapter Two, eliminates linear

dependence for grounded loops of links. It is further necessary to

eliminate linear dependence which is introduced by any closed loops in

a system which is not grounded. A mechanism containing one of these

loops is shown in Fig. 3.5.1. Observe that the loop 2367 is connected

directly to ground only at pin-joint 1 and that a vector expression may

be written for this loop in the form


a26ei26 + a67ei'67 a23ei923 a37ei'37 = 0


which is of the same form as Eq. (3.2.1). Note that the values for all

of the constant apq are non-zero. This means that the vectors, ei26,

eit67, ei'23 and ei37, are linearly dependent. Therefore, at least one

of these variables must be eliminated from any expression for a dynamic

property of a mechanism in order to use that equation to arrive at a set

of balancing conditions for the mechanism.

A second requirement (definition) of linear independence can be

found in the field of linear algebra as: A square matrix is nonsingular

(possesses an inverse) if, and only if, its columns are linearly independent



























3















FIG. 3.5.1 Stephenson 2 Six-Bar Linkage









[18]. The columns of a matrix will be linearly dependent if any column

can be formed as a linear combination of any other columns, i.e., if any

column can be formed by multiplying one or more of the other columns by

constants and adding the results. This requires that any column of a

matrix which can be decomposed into a linear combination of other columns

of the matrix must be so decomposed and the rank of the matrix reduced

by distribution of the dependent column among its constituents.

In this dissertation, a matrix form of the dynamic equations will

be used to balance mechanisms and therefore all linear dependence must be

eliminated. The possible physical forms of a ternary link using pin-

joints and sliding joints are shown in Fig. 3.5.2. Figure 3.5.2(a) is

a ternary with three pin-joints. The linear dependence for the pin

jointed ternary will be eliminated here for the condition of linear

momentum for a general link.

The time dependent terms of the equation for linear momentum were

defined in Eqs. (2.4.2) and (2.4.3). For the adjacent link, rs, the

first term is

1 2
Dlrs = Dlrs = Ur + Vr"


If the transformations of Eq. (2.3.10) are substituted here, the result

is


Dirs = Drs = "p + [ur(Oq Up) vr(q Vp)]/apq

+ Vp + [ur(Vq Vp) + vr(Uq Up)]/apq.


From the definitions of the D! in Eq. (2.4.3), it is evident that
1pq


I rs = rs = (-ur/apq)Dpq + (vr/apq)D2pq

+ (-vr/apq)Dpq + (ur/apq)D2pq.




















(a) 3R TERNARY


(b)2R-P TERNARY


p --in
"-'^n p


(c) R-2P TERNARY


(d) 3P TERNARY


FIG. 3.5.2 Possible Ternaries


r ---As


P "-in


--4n









Thus, it is demonstrated that D1rs D2s is a linear combination of the

Dpq, i = 1, 2, 3, 4. Because of the definition of linear dependence,

this type of decomposition must be accomplished for all such terms in

order to arrive at a linearly independent matrix formulation of the dy-

namic properties of a mechanism. Table 3.5.1 is a listing of decomposi-

tion for the third point of all four possible ternary links. The ter-

naries are those shown in Fig. 3.5.2. The sub-cases for each ternary

correspond to the various ways that the three joints of the ternary can

be ordered. Case I.1 is the ordering used in the derivation above with

pq as the "base" of the ternary and r as the third point. Case 1.2 is

for the use of pr as the base and q as the third point, while Case 1.3

uses qr as the base and p as the third point. In all cases, the ordering

of the designation may be reversed, i.e., pq and qp are both legitimate

bases for the ternary. All of the cases for the three pin-joint ternary

use the same decomposition if the subscripts p, q, and r are suitably

rearranged.

The case system and corresponding ordering of points that were used

for Ternary I will be used for each of the other ternaries. For Ternary

II, Case II.1 is unique and Cases 11.2 and 11.3 use the same transforma-

tion with reordering of subscripts. For Ternary III, Case 111.2 is unique

and II.1 and 111.3 use the same transformation if the subscripts are

changed accordingly. Ternary IV is similar to Ternary I in that the decom-

position is the same for each of the cases with reordering of subscripts.

Table 3.5.1 has been constructed so that the linear dependence in-

cluded with a ternary may be readily eliminated by simple substitution and

rearrangement of terms. Any link, with more than three joints, will be

considered as if it were a series of ternaries, all using the same base.









TABLE 3.5.1 Ternary Links

SUBSTITUTE FDR THESE PROPERTIES AND THEIR DERIVATIVES


TERNARY AND CASE ANGULAR MOMENTUM i
LINEAR MOMENTUM
KINETIC ENERGY -

1.1 pq(r) Drs rs 2 = D
irs lrs ITr rs irs A sr
1.2 qr ) D 1 p(a Lur) 2 D )p v r a ) I D (p(aq ur 2 + 2 2I
1.3 priq) P Dir pqapq 'r P

+ D q(-vr pq) 04p q(r -pq) apq ( v)) a

r r2+ D pq r pq)
D2rs Dlpq(-r + apq) Opq(apq- r) + a + 'p(u + )
Pq

P 'n p(ur apq + D4 p('r 'pq)



2.1 pA(r) N D Dl (pac ) a pq+ (sin ) fl ;sr = p zpq +

r9 q + Dp q = p(1sU) a D pq,() p













-
r -n i ir ) ap + S2R (c p
3 D) (-ls) ap + Dp (-sire) +a


2.3 pr(q) I- I I
0/n Dlqr + qr( ) + Dqr(p) Dpn qr + 02qr(p)

D2p D+pn + Dqrp D4qr(U + v p











P Dq () + 3qrp
3.1 p(r) s p ( ) + n0 4r
3.3 pr(q) r DADp(-sasS)














D 2ps = 1np + rsip' Aqr pu p
q0


4 1 pq(r)
4.2 qr(p)
4.3 pr(q)


/ /


-t
p


ir D pqlcos6) + D pq(sne)

IDsr D (-s'"e)' a I (c"se)


Dnsr ;p
'4 r 'pq










3.6 Linear Momentum and its Derivatives

The equation for the general dynamic property of the four-link mech-

anism was shown to be Eq. (3.4.5). To balance a mechanism for a specific

property, it is necessary only to substitute the definitions of the Yipq

and the Dipq from Chapter Two. For linear momentum, these definitions

are found in Eq. (2.4.3). For the four-bar mechanism, the following are

true


U1 = V1 = Ui = V4 = 0 (the fixed pivots do not move)


so that (see Appendix A)


D12 = D2 = 0


and


D2 = D23; D323 = D23 (the moving pin-joints have common velocities in

neighboring links (see Appendix B)).


After substitution of these values in Eq. (3.4.5), the equation for the

linear momentum of the four-bar is found to be


L = [Y2 + y2]D2 [y212 + y23 D223
L 12 + Y23]DI23 + [Y2 + 2 ]D223

+ [y2 + 2 + ]D2
[Y223 Y34]D34 323 34 234'


The time derivative of this equation is the equation for the shaking force,

F of the mechanism. Complete shaking force balancing has been defined

(see Refs. [10] and [5]) as forcing the total shaking force of a mech-

anism to be zero. This was accomplished in Ref. [5] by making the cen-

ter of mass of the mechanism stationary. The complete balancing of the

mechanism may be accomplished by forcing the four constant coefficients









oftheDipq in the above equation to be equal to zero. If this is done and

the definitions of the Yipq from Eq. (2.4.3) are substituted, the balancing

conditions for the four-bar are


Y212 + Y23] = [(m12u12/a12) + m23(1 u23/a23)] = 0, (3.6.1)

[Y12 + 23] = [(-m12v12/a12) + m23v23/a23] = 0, (3.6.2)

423 + 34] = [(m23u23/a23) + m34(1 u34/a34)] = 0, and (3.6.3)

[Y23 + Y34 = [(-23v23/a23) + m34v34/a34] = 0- (3.6.4)


If it is presumed, as in Refs. [5] and [9], that the mass parameters of

link 23 are known, then the location of the center of mass of link 12 is

given from Eqs. (3.6.1) and (3.6.2)


u12 = m23(1 u23/a23)(al2/m12), (3.6.5)
v12 m v a
V = 22LJ3 (3.6.6)
a23m12


and, for link 34, from Eqs. (3.6.3) and (3.6.4)



34 = (1 + 2323a34, and (3.6.7)
a23m3 4

Sm23v23a34 (3.6.8)
34 a23m34


These conditions are identical to the balancing conditions found in Refs.

[5] and [10]. Hence, it is demonstrated that this new method agrees for

the shaking force balancing of mechanisms as found previously by the

author and others, Refs. [2], [5] and [10].


3.7 Total Momentum and its Derivatives

Again, Eq. (3.4.5) is the equation for the general dynamic property

of a mechanism. If substitution of the definitions of the Y3 and the
ipq
iDtpq is made, it can be shown that Eq. (3.4.5) is also of the same form









as the equation for total momentum of the mechanism. From the field of

dynamics, it is known that the time derivative of the total angular

momentum is equal to the sum of the moments exerted on the mechanism.

This time derivative is recognized to be the shaking moment of the mech-

anism, or


Mo = Ti + rm x Fm (3.7.1)


where


M is the shaking moment,

T is the inertial driving torque or torques supplied to the inputs

of the mechanism,

S is the vector locating the mth fixed pivot,

F is the force exerted on the mechanism by the mth pivot, and

,m x Fm is the moment about the origin exerted by the forces.


If it is desired to completely balance the shaking moment of the

four-bar (achieve MO = 0 for the entire cycle), it is necessary only to

force each of the constant terms of Eq. (3.4.5) to be zero. In order to

accomplish this, each of the constant terms of Eq. (3.4.5) are separately

set to be zero and the definitions of the Y3 from Eq. (2.5.3) are
ipq
substituted


Y12 = (m12u12/a12) m12(k 2 + u2 + v12)/a12 = 0, (3.7.2)

Y312 = m12v12/a12 = 0, (3.7.3)

12 + 123 = m12(k2 + +2 + v2)/a12 + m23( u23/a23)

+ m23(k 3 + u3 + v3)/a3 = 0, (3.7.4)
2 2
Y -23 = (m23u23/a23) m23(k 3 + U23 + 3)/a23 = 0, (3.7.5)

Y323 = m23v23/a23 = 0, (3.7.6)









2 2
23 + 34 =23(k23 + u3 + 3)/a23 + 34(1 u34/a34)

+ m34(k(4 + u24 + v2 )/a 4 = 0, (3.7.7)
33 /4 34),4 (3.7.7)

234 = (m34u34/a34) m34(k24 + u4 + v24)/a 4 = 0, and (3.7.8)

Y334 = m34v34/a34 = 0. (3.7.9)


If each of Eqs. (3.7.3), (3.7.6) and (3.7.9) must be zero and if each of

the links are physically real, then the only possible choice is to make

each of the v12, v23, and v34 equal to zero. If this is compared with

Eqs. (3.6.6) and (3.6.8) from the shaking force balancing, it is apparent

that, with the v-coordinate zero, the shaking force and shaking moment

locations are the same for all three links. Further, if Eq. (3.7.5) is

solved for (m23u23/a23), the result substituted into Eqs. (3.7.4) and

(3.7.7), these equations solved for m12(k2 + u2 + 2)/a12 and m34(k4

+ u24 + v2 ), these results substituted into Eqs. (3.7.2) and (3.7.8)

respectively, then the resulting equations may be solved for u12 and u34

as


u12 = -m23(1 u23/a23)a12/m12, and
m u
34 = (1 + 23 )a34
u23m34


These results are exactly equal to Eqs. (3.6.5) and (3.6.6), the criteria

for the shaking force balancing of the four-bar. Thus, it is demonstrated

that complete shaking moment balancing of a mechanism ensures complete

shaking force balancing of the mechanism. Of the three remaining bal-

ancing conditions, Eq. (3.7.5) is relatively easy to accomplish since

this is the requirement that link 23 is a physical'pendulum. This requires

that the link have the same total moment of inertia about either of the

pivots, 2 or 3.









The remaining two balancing conditions, Eqs. (3.7.4) and (3.7.7),

are the most difficult to achieve. They can be used to determine (sup-

posing that the mass parameters of link 23 have been fixed) the moments

of inertia of links 12 and 34 about fixed pivots 1 and 4, respectively.

It appears that these two conditions require that the sum of two positive

numbers be zero. Because of this, it becomes necessary to introduce the

concept of "negative" inertia. For shaking moment balancing, negative

inertia can be simulated by adding a body which counter-rotates with some

existing body. In Ref. [3], this was achieved by adding a gear pair to

the chain for exact balancing and in Ref. [10], by adding a dyad (pair of

links) which simulated a gear pair over a small range of motion for approx-

imate balancing. If it is presumed that this negative inertia will be

used as shown in Fig. (3.7.1), then Eqs. (3.7.4) and (3.7.7) must be

modified by the addition of a balancing inertia to satisfy


m12(k22 + u 12 + v2)/a2 + m23(l u2323)

+ m23(k + 2 + v23)/a3 I = 0, and

23(k + u3 + v23)/a3 + m34(1 u34/a34)
+ m3(k2 + u 4 + v2 )/a2 6 2 = 0,
34 3 34 34 34 34 0'


where 15 and 16 are the rotary inertias of a pair of gears, as shown in

Fig. (3.7.1). So, at the cost of the addition of two pairs of gears, it

is possible to completely eliminate the shaking moment and shaking force

of a four-bar linkage. In general, it will be necessary to add negative

inertia gear pairs to any mechanism which is to be balanced in order to

completely eliminate shaking moment.

Note that the last six of the eight equations, Eqs. (2.7.2) through

(3.7.9), are exactly those balancing conditions for complete moment













balancing of the four-bar as found in Ref. [9].. It is now understood

that the reason that only six balancing conditions were found in Ref. [9]

is that a special reference was taken at the center of the input link to

derive the balancing conditions found in that work.

Thus far, it has been shown that it is possible to completely bal-

ance a four-bar mechanism for shaking moment and that this balancing

includes the complete shaking force balancing of the mechanism. This is

equivalent to making the specification of the column [S] of Eq. (3.4.3)

as a column of eight zeros. If it is decided that this complete bal-

ancing is not desirable, perhaps because of the negative inertia require-

ments or other unattractive link configurations, it is possible to

specify [S] as eight non-zero values and to solve for the required values

of the constants of Eq. (3.4.5). This may result in more attractive

links and will satisfy exactly the specified values of [S].


3.8 Kinetic Energy and its Derivatives

The substitution of the definitions of the Yipq and the Dipq from

Eq. (2.6.3) into Eq. (3.4.5) yields the equation for the total kinetic

energy of the four-bar. If it is recognized that, for the four-bar

=6 D6 -D6 D- -0
U1 V1 = U4 = V4 = D212 = D312 D34 = D34 = 0


(see Appendix B) then the controlling equation becomes


Ei = [Y3 + y3 ]D6 + y3 D6 + y3 D6 + [y3 + y3 ]D6 (3.8.1)
412 123 123 223 223 323 323 423 134 134. (3.


In this equation, the kinetic energy of a four-link mechanism is determined

by the sum of four terms. The time derivative of this equation is the

inertia power required to drive the mechanism. An ideally balanced mech-

anism should appear as a flywheel to its prime mover; this would mean









that a mechanism operating at constant speed would require no energy in-

put (in the absence of friction) to maintain its speed. If the deriva-

tive of Eq. (3.8.1) is taken the result is

Pi = L6 + Y33232 + [23 + Y334 y 34 (3.8.2)
p yL 223 323 25 423 34 34


since


123 2 2 2 + 2 = 12 112 =

for a mechanism operating at constant input crank speed. In order for

this mechanism to have zero power input, it is sufficient to force the

three constant coefficients of Eq. (3.8.2) to be equal to zero. If this

is done and the definitions of the Y3pq substituted from Eq. (2.5.3), the

resulting conditions are


223 = (m23u23/a23) m (k + u2 + v )/a =0, (3.8.3)

Y23 = m23v23/a23 = 0, and (3.8.4)
y3 + y3 (k2 + u2 + v2 )/a2 + m (- u /a
423 134 2323 + 23 23)/a23 + m34 34
+ m (k2 + u3 + vq)/a = 0. (3.8.5)


Observe that these three equations are exactly the same as Eqs. (3.7.5),

(3.7.6), and (3.7.7). However, any attempt to satisfy Eq. (3.8.5) with

negative inertia results in an increase in the power required to drive

the mechanism. This is best illustrated by writing the power equation

of the mechanism with a gear pair added to provide negative inertia.

Consider the mechanism in Fig. (3.7.1), the equation for the power of

this device is


pi = y3 3 233 +3 + [23 + 34 + I /a34]34
223223 323 323 42









If this equation is compared with Eq. (3.8.2), the power equation for the

unbalanced mechanism, it is seen that the difference is the addition of

the positive number, T6/a4 The addition of the balancing gear pair can

only increase the power required to drive the mechanism. The above argu-

ment holds equally well for the inertia driving torque of the device

since the inertia power and the inertia driving torque are related by


p = Tda i = input speed.


At times, it will appear to be advisable to attempt to balance for

non-zero driving torque or power. When this is the case, it will be pos-

sible to balance for up to four specified values of the dynamic property

if the mechanism has an accelerating input crank. The dynamic property

which the system is to satisfy can be kinetic energy or any of its

derivatives.


3.9 Reaction Moment

It was demonstrated in Section 2.7 that the shaking moment, the

inertia driving torque, and reaction moment are related by Eq. (3.7.1).

This leads to the conclusion that the equation for the reaction moment

may be found as


Mr =rmx F= MO Tx.


The equation for a general dynamic property, Eq. (3.4.5), is still applie-

able in this instance if two new Dipq's are defined as


D7 = D (D )
ipq ipq ipq

and


D8pq = D (D )
ip ipq (Dipq










where D n = 3, 4, 5, 6, are defined in Eqs. (2.5.4), (2.5.5), (2.6.4),
ipq
and (2.6.5). This allows the writing of the equation for reaction moment

for the four-bar in the form


S= Y2323D223 + Y323323 8 23 + 34]D34 + Y233D (3.9.1)

+ Y334D 34 + [Y412 + 323]D8 + Y312D 12 + 3 2
2334 2 1 312


3.10 Theorems For Balancing Mechanisms

In Chapter One, reference was made to the theorem on shaking force

balancing of mechanisms as stated by Tepper and Lowen [26]. In this sec-

tion, it is proposed that the theorem be revised or that a new theorem

be advanced. This theorem is a result of the form of the equations for

the dynamic properties of mechanisms. The theorem as previously stated

in the literature deals only with the shaking force balancing of mech-

anisms. It is proposed that the theorem be changed to read:

THEOREM

A planar mechanism without axisymmetric link groupings can be
fully balanced for any dynamic property by internal mass
redistribution or the addition of "negative inertia" if, and
only if, from each link there is a contour to the ground by
way of revolute joints only.

The phrase "fully balanced" has the same meaning as that for completely

balanced which has been used throughout this work; i.e., to force the

value of some dynamic property or combination of properties to be zero

for the complete cycle of the mechanism regardless of position or dynamic

input state.

As proof of this theorem, consider Fig. 3.10.1 which is a group of

three links considered to be part of some mechanism which is connected at

p and r to the rest of the mechanism. The generalized equation for a

dynamic property of the mechanism containing these links will be:



































































FIG. 3.10.1 Three Links Joined Only by Sliding Joints









S = YlpqDlpq + Y2pqD2pq + Y3pqD3pq + Y4pqD4pq (link pq)

+ YlqsDlqs + Y2qsD2qs + Y3qsD3qs + Y4qsD4qs (link qs) (3.10.1)

+ YlrsDlrs + Y2rsD2rs + Y3rsD3rs + YrsDrs (link rs)

+ other terms for other links in the mechanism.


Since Dpq and Dlrs will combine with elements due to components from

other links, they will be lumped here and ignored. Also in this case, by

definition


D4pq = D4qs = D4rs = f(Ypq = Yrs = Yqs)"


This is true regardless of the dynamic property in question. After

these observations, Eq. (3.10.1) reduces to


S = 2pqD2pq + 2rs 2rs + 2qsD2qs

+ Y3pqD3pq + YrsD3rs + 3qsD3qs
+ [Y pq + Y4rs + Y4qs]D4pq + YqsDqs + other terms.


This is the appropriate equation for the balancing of the triad of links

of the mechanism shown. If the property in question is angular momentum,

kinetic energy or any of their derivatives, the mechanism may be fully

balanced by making all of the constant coefficients go to zero. All of

the Y2pq and Ypq may be made to be zero by choosing upq and vpq equal to

zero. The constant coefficient of D4pq may be made to be zero for angular

momentum if some form of negative inertia (even though it is unattractive)

can be used. It cannot be made zero for kinetic energy. Therefore, this

mechanism cannot be fully balanced for kinetic energy or its derivatives.

In either case, note that Ylqs appears alone in this equation; it is de-

fined as Y1 = m .Clearly to make Y1 = 0 would require that a
Iqs qs Iqs
physical link be constructed with zero mass. Therefore, a mechanism









containing this link triad cannot be fully balanced for angular momentum

or its derivatives. This requirement for zero mass links holds also for

balancing for kinetic energy and further precludes balancing for that

property.

If the dynamic property in question is linear momentum, further com-

bining of terms is necessary. For the orientation of the three moving

reference systems as shown in Fig. 3.10.1, the angles of the three links

are related as follows:



Ypq = qs' and

y =y +6 = y +
rs pq qs


and their time derivatives



Ypq = Yrs = qs


are all the same. From this information and the definitions of the D1
ipq
of Eq. (2.4.2), it is possible to determine that (see Appendix B)


D1 = D1 ;D1 = D1 ; D1 = D1 D1 0,
2pq 2qs 3pq 3qs' 4pq 4qs 4rs

and that (see Appendices A and B)


Drs = cos Dl + sin D1 and
2rs 2pq 2pq'

Dfrs = sin Dlpq + cos Dpq
rs pq 3pq


Substitution of the above into Eq. (3.10.2) yields


F = pq + qs + cos Y1rs sin Y3rs]Dpq
\,S 2 qs 2rs 3 2pq
Spq +Yrs + sin Yrs + cos Yrs]Dpq
pq+ [ s]Dlqs other terms for other links in the mechanism.
+ [Y' qa]DD + other terms for other links in the mechanism.
I iqa









Again, to fully force balance the mechanism, it is necessary only to

force the constant coefficients (in brackets) of this equation to be zero.

The coefficients of Dipq and D4pq can easily be forced to zero. However,

Y1pq appears alone again; making it zero would require that a physical

link be constructed with zero mass. Since this is true, it is impossible

to balance the given mechanism for linear momentum or its derivatives.

Shaking force is the time derivative of the linear momentum of the mech-

anism. The above conclusion for linear momentum was proved by Tepper and

Lowen [26] and is a special case of the above theorem. It is readily

apparent, then, that the above theorem, in its revised form, holds true

for all of the dynamic properties of a mechanism that contains link

series (i.e., the link triad) which makes reaching ground through re-

volutes from all sliding pairs impossible.


3.11 Mixed Criteria and Balancing Options

Since the shaking force criterion is a subset of the shaking moment

criteria, it follows that one cannot balance for specified non-zero values

of shaking moment and then for specified non-zero values of shaking force,

or vice versa. It is also obvious that, since the balancing conditions

for driving torque are a subset of those for shaking moment, that it is

not possible to balance for either torque or shaking moment and then to

balance for the other. It is possible, however, to balance for non-zero

specification of inertia driving torque and then to balance for specified

shaking force for the same positions and to exactly satisfy both sets of

specifications. Since both of the cases mentioned above are encompassed

in the balancing for reaction moment, it is clear that a mechanism cannot

be balanced for reaction moment and any other of the dynamic properties.










In Ref. [10], Elliott and Tesar have defined the concept of balancing

for multiply-separated non-zero conditions for shaking moment and inertia

driving torque. If this concept is extended to balancing for general

dynamic properties, it becomes obvious that one can balance, say, for

the kinetic energy (and the inertia driving torque) of a mechanism. The

specification of the values of energy (and torque) may be made at the

same position of the mechanism or at different positions. It is possible

to balance only for the same number of conditions that could be balanced

for if one were balancing in either property alone since the number of

positions or specifications which can be made is the same as the number

of unknowns in the dynamic equation which remains the same whether deri-

vatives are taken or not. This type of balancing of a mechanism would

allow the control of both energy content of the mechanism or the tailor-

ing of the mass content of the mechanism to suit some available energy

source. All of the above analytical methods allow the development of a

few rules of thumb or predictors.

It was shown in Chapter Two that there are four mass parameters

(m, u, v, k) in each moving link of a linkage system. For balancing, it

becomes desirable to know or to be able to predict the number of mass

parameters in the system, the number of specifications of dynamic property

which can be made, and the number of mass parameters remaining for optimi-

zation. It is possible to formulate rules or equations to provide this

information. If n is taken as the number of links in a given kinematic

chain, the number of mass parameters available for balancing is.found

to be


Q = 4(n 1)


(3.11.1)










where Q is the total number of mass parameters in the system. If j is

taken as the number of lower-pair connectors (pin-joints or sliders) in

a given chain, then the number of positions or values of the various

dynamic properties is found as.,


S =Q -j,

S2 = Q 2j, and

3 = Q j 2f


(3.11.2)

(3.11.3)

(3.11.4)


where


S is the number of specifications possible in total momentum and its

derivatives,

S2 is the number of specifications possible in linear momentum and its

derivatives,

S3 is the number of specifications possible in kinetic (inertial) energy

and its derivatives, and

f is the number of fixed pivots in the mechanism, both pin-joints and

sliders-


These Si are the maximum number of specifications which can be made if a

closed form exact solution to the non-zero balancing specifications is

desired. If this number of specifications has been made then the number

of design parameters available for optimization are found with the fol-

lowing equation:


Pi = Q Si Pq i = 1, 2, 3


where Pq is the number of grounded sliders. The Pi can be found more

specifically as









P1 = J Pq, (3.11.5)

P2 = 2j Pq, and (3.11.6)

P3 = j + 2f Pq. (3.11.7)


The last bit of information which can be gleaned from the kinematic

chains is the maximum number of prismatic or sliding pairs that can be

contained in a kinematic chain to be completely balanced for shaking

force or shaking moment. This maximum number of sliding pairs is found

by inspection to be


PM = (j + 1) n. (3.11.8)


This is the maximum number of sliding pairs that can be contained in the

kinematic chain without violating the theorem of Section 3.10, for all of

the mechanisms derived from the given chain. This is the maximum num-

ber tolerable; it is still necessary to examine individually each mechan-

ism with more than one slider to determine that it has not violated the

theorem by isolating a slider or sliders from ground.

When using the above results, it should be noted that it is possible

to balance for kinetic energy (or its derivatives) and then for linear

momentum (or its derivatives). If this dual balancing is done, it has

the desirable effects of reducing the number of design parameters avail-

able to the designer to optimize the system. In some mechanisms, this

dual balancing will be more restrictive than the balancing for shaking

moment alone as can be seen from the following equation which will pre-

dict the number of parameters remaining for optimization for the dual

balancing case


(3.11.9)


P23 = Q (S2 + S3) = 3j + 2f Q Pq.










3.12 Calculation of Counterweights

The balancing methods of the previous chapter return the proper

values of the mass parameters in order to satisfy the specified values

of a dynamic property. If the balancing has been undertaken for a mech-

anism that does not exist except as kinematic dimensions, then it appears

that all the designer has to do is to locate the mass of the mechanism

in each of the links to satisfy those requirements. If, however, the

balancing has been done for a mechanism that already exists, whose mass

content is known in advance, it becomes necessary to calculate for each

link the counterweight mass and location to properly balance the mechanism.

This relatively simple procedure has been presented in Ref. [9] and is

repeated here. Let Fig. 3.12.la represent the original unbalanced link

and Fig. 3.12.1b represent the balanced link with its mass content such

that it satisfies the balancing requirements. Then the locations of the

mass content for the counterweights, as shown in Fig. 3.12.1c, may be

calculated using the following:


mc = mb mu, (3.12.1)

uc (mbub mu)/mc, and (3.12.2)

vc = (mbvb muvu)/mc (3.12.3)


and the required radius of gyration of the counterweight is found as



kc Ib v21/2 (3.12.4)
mc c c


where


mb, ub, vb, and kb are the balanced mass parameters,

mu, uu, vu, and ku are the original unbalanced mass parameters, and

mc, uc, vc, and kc are the counterweight mass parameters.










mu
k"


mb
b
k


mu
k"


mec
kc


FIG. 3.12.1 Counterweight Mass Parameters









Note that I = m(u2 + v2 + k2) is referenced to the same pin joint p for

mass parameters such as u, v, k. Fulfillment of these conditions will

achieve the proper placement of the counterweights to balance the mechanism.

If some dynamic property other than the linear momentum or its deriva-

tives is being undertaken, then it will be necessary to calculate a value

for the addition of "negative inertia." A grounded link with its asso-

ciated negative inertia requirement is shown in Fig. 3.5.1. The require-

ment for the inertia of this balancer will be found from

Y. +Y. I /a2 -x

ipq +irs gpq = X


where

Yipq' irs are some of the constant coefficients as found in Chapter Two,

X is the result returned by the solution of the equations [see Eqs.

(3.4.3) or (3.4.6)], and

Ig is the moment of inertia of the counter rotating balancing gear.


This equation may be solved for I as


g = (X + Yipq + Yirs)a2q" (3.12.5)


It remains only to calculate the radius of gyration of the gears where

this parameter is involved in the balancing.


3.13 Approximate Balancing

In Chapter Two, the equations for the dynamic properties of mechan-

isms were developed in several forms. In the preceding sections of this

chapter, a method of exactly balancing any mechanism was described. In

this section, use will be made of the special forms of the equations that

were developed previously to illustrate possible methods of balancing

mechanisms in the approximate sense.









The first of these makes use of a readily available tool, the matrix

inversion capabilities of the APL computer language. This allows one to

overspecify the dynamic property which is being controlled; i.e., the

vector [S] of Eq. (3.4.3) is specified in more positions than that allowed

by Eqs. (3.11.2) (3.11.3) or (3.11.4) and the matrix inversion operation

is carried out. This results in the solution of the equations in a least

square-sense. This means that Xm returned by this process will satisfy

the specified values of the dynamic property in a least-square sense only.

This method was used in [10] and was beneficial in that it allowed con-

siderable smoothing of the shaking moment of a mechanism without the ex-

pected penalty of a 300 percent increase in inertia driving torque. At

times, it appears that this may be a better method to use in the balancing

of mechanisms than the exact method that is described earlier in this

chapter.

It is hoped that further development of various approximation tech-

niques will be carried out by future researchers since the equations pre-

sented in this work are given in their definitive forms. The equation for

each dynamic property is expressed as a sum of a series of terms. Each

term consists of a constant multiplier, the Yipq or XM, and a kinematic

variable, the Dipq. Since this is the case, the dynamics of the mechanism

is separated completely from the kinematics (or geometry) for purposes of

analysis.

















CHAPTER FOUR

EXAMPLES


4.1 Purpose

The purpose of this chapter is to expose the reader to the use of

the methods as developed in the previous chapter. This will be done

through the treatment of a numerical and a non-numerical example. During

the development of these examples, certain special cases and considera-

tions will be pointed out. Towards the end of the chapter, certain rules

of thumb will be developed and listed for the user's convenience. The

examples, wherever possible, are taken from existing literature or from

industrial problems. The main concept that should become clear to the

reader, as progress is made through the chapter, is the ease of applica-

tion of the method and the fact that it can be applied to any problem

which is kinematically analyzable. The restrictions or assumptions for

the method are stated again here:

1. The kinematic representation of the mechanism must exist.

2. A method of analysis of the mechanism exists. This

analysis may be based on the kinematics of the mechanism

assuming rigid links. If an existing mechanism is to

be redesigned, the analysis of the motion may be taken from

the mechanism itself with the appropriate instrumentation.









4.2 An Eight-Bar Linkage

The mechanism shown in Fig. 4.2.1 was designed and built for use in

the textile industry. In the original prototype, all of the links were

made of steel. When this mechanism was run at its design speed of 3500

rpm, the bronze sleeve bearings in the pin-joints, particularly those in

and near the input, failed after a few hours of operation. A new version

of the mechanism was constructed with links of aluminum. This version

appeared to have a longer life. The dimensions of the links and the mass

parameters of the aluminum links are listed in Table 4.2.1.

At the time that the problem became available to this researcher,

the designer of this linkage was still concerned with the life of the

bronze bearings. The observable dynamic property, which was to be con-

trolled in the linkage, was the inertia driving torque, as severe torque

reversals were evident. The designer hypothesized that these would lead

to severe force reversals in the pin-joints of the mechanism which would

lead to early failure of the bearings due to high shock loadings. The

object of the balancing then was to reduce the variation in energy con-

tent of the mechanism in order to reduce the severe torque reversals and

therefore increase the life of the bearings of the mechanism.

Equations (3.11.1) and (3.11.4) can be used to predict the quality

of balancing which may be expected for the mechanism. For the given

mechanism in Fig. 4.2.1, the pertinent parameters are the number of

moving links, n = 8; the total number of pin-joints, j = 10; and the num-

ber of fixed or grounded pin-joints, f = 4. Using this information, Eq.

(3.11.1) indicates that the number of mass parameters in the mechanism is


Q = 4(n 1) = 4(8 1) = 28.














TABLE 4.2.1 Mass Parameters for the Links of the Eight-Bar


LOCATION OF
LINK CENTER OF GRAVITY
pq Upq Vpq


0.187

0.156

0.334

0.625

0.216

2.101

0.216


0

0.025

-0.140

0

0

0.217

0


MASS
mpq


0.00057

0.00011

0.00014

0.000046

0.00036

0.00014

0.00036


CENTROIDAL
MOMENT OF INERTIA
mpqkpq


0.0000032

0.000032

0.000033

0.000011

0.000117

0.00028

0.00012


LINK
LENGTH
apq

0.187

1.25

1.00

1.25

1.05

4.375

1.05









Equation (3.11.4) predicts the number of specifications which may be

made in kinetic energy (or its derivatives) as


S3 = Q j 2f = 28 10 8 = 10.


This means that the energy level of the device can be specified at ten

positions of the input crank or 360 intervals. Because of this result,

it was expected that significant improvement could be made in the dynamics

of the mechanism.

The next step in the balancing of the mechanism was to develop the

specific equation for the kinetic energy of the mechanism. The kinetic

energy of the mechanism is found as

4
Ei = Z E Y6
pq i 1 ipqDipq

and, using the notation of Fig. 4.2.1, yields

Ei = YIDI + YoDo1 + Yo01D00 + y 01 (link 01)

+ Y12D12 2 12 12 + Y12D12 + 12D412 (link 12)

+ Y23D623 + Y221D21 + 23D323 + Y23D423 (link 23)

+ Y 4D4+ Y245D245 45 + Y445445 (link 45) (4.2.1)

+ 56D6 + Y5D56 + Y56D5 + YY56DD56 (link 56)

+ Y78D7 278D27 + 3378D78 + 78D78 (link 78)

+ Y8 89 + 89 289 389D + Y89D9 (link 89).


However, from the definitions of the D0pq In Eq. (2.6.3) and the know-

ledge of the kinematics of the mechanism,


Up = Vp = 0; p = 0, 3, 6, 9


for the fixed pivots, it is known that (see Appendix A)









D6201 = D623 = 56 = D6 = 0
201 223 256 289
D60 = 23 = D6 =D = 0, and
301 323 356 389
D6 = D6 = D6 = = 0
101 123 456 489


Further, from the fact that certain of the pin-joints, 1, 2, 5, and 8, are

shared between links, it is observed that


D6 = D612 612 623; D 56 = D6 ; D678 = D6
401 112' "412 423' 456 -445' 78 189,


Substitution of the above information into Eq. (4.2.1) yields a much re-

duced equation


i = y3 6 + y3 D6 + y3 D6
212 212 245 245 278 278
6 + y3 +
+ 312D612 345D45 +37D78

+ [Y401 + Y12]DB12 + [Y12 + Y23]D23 (4.2.2)
Sy3 + 53 ]D6 + [y3 + y3 ]D6
4 15 56156 478 189 189

145D145 +178 178'


This equation has twelve terms, two more than was predicted by Eq. (3.11.4).

The extra two terms are the last two in Eq. (4.2.2). These terms are con-

cerned with the motion of the pin-joints which are connected to the

quaternary link 2374. They must be combined with the terms from the base

of the quaternary 23 as was shown in Section 3.5. The quaternary is

treated as two ternaries 234 and 237 and substitutions are made using

Table 3.5.1. To use the table, each ternary is treated separately. The

first ternary becomes a case 1.1 ternary with the substitutions p = 2,

q = 3, and = 4 being made for the pin-joints. With this information, the

table yields


D 5 D6 [(a u )2 + v2]/a2
145 123 23 4- 23"


(4.2.3)









The second ternary is also a case 1.1 ternary and the substitutions, p =

2, q = 3, and r = 7, apply for the pin-joints. With this information,

the table yields


D78 = D 23 2) + v/a23. (4.2.4)

After the substitution of the results represented by Eqs. (4.2.3) and

(4.2.4) into Eq. (4.2.2), the final reduced equation for the kinetic

energy of the mechanism is found to be


Ei = y3 + + y3 Y6 + Y31D2 + 3 6
212D212 245D 45 278U278 312 1 345D345
y3 +3 6 [Y5+ Y56]D56
+ Y78D~7 + [Y01 + 12D12 445 + Y56

+ [Y78 + Y+89]D89 + [Y12 + 23 + ({(a23 u4)2 +

v42/a23) 5 + (((a23 u?)2 + v23/a3)Y178]DI23


This equation is expressed in ten terms, the number predicted by Eq.

(3.11.4). It is also expressed in terms of a linearly independent set

of vectors. Hence, this is the equation which may properly be used to

balance the mechanism.

The energy and torque curves for the unbalanced mechanism are shown

in Figs. 4.2.2 and 4.2.3. Notice the changes in the kinetic energy of

the device and the required rapid fluctuations in the torque curve. The

first attempt to use the expected power of the balancing methods of

Chapter Three 'was to specify ten values of kinetic energy which corres-

ponded to the average of the curve in Fig. 4.2.2. This attempt resulted

in the placement of all of the mass of the mechanism in the constant term

associated with D6 2 with all of the rest of the constant terms going to

zero. For the reasons set forth in Section 3.8, it is impossible to force

all of the constant coefficients to be zero. Therefore, this is an







70













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unacceptable mass distribution. Because of this unfortunate result, it

was decided that perhaps the mechanism could be balanced if the term

associated with D62 were ignored and specification made for the remaining

nine terms. This was tried after removing the contribution for the energy

contained in the input crank. The results called for links either too

massive or too large physically to be physically realizable in the

mechanism.

In light of the failure of the exact balancing methods to achieve a

significant reduction in the fluctuations of the kinetic energy of the

mechanism, an attempt was made to use the approximate balancing technique

first suggested by Ogawa and Funabashi [19]. Briefly this method is:

1. Express the inertia driving torque of a four-bar as the

geometric derivative of Eq. (3.8.1) to give


Ti = ([Y3 +Y3 D6 + y3 D6
412 123 123 223 223 (4.2.5)

+ Y3 D6 + [y3 + Y3 ]16 ) i
323D 23 423 134 D134 w


where i is the input speed.

2. If the input is operating at constant speed, then Dg23 = 0,

and it is always possible to make link 23 be an in-line

link by making v23 = 0. This choice of v23 substituted into

the definition of Y323 yields


323 = m23v23/a23 = 0.


These simplifications yield an equation for the inertia

driving torque of the mechanism as a sum of two terms, i.e.,


T = (Y3 D6 + [3 + 3 D6 ) 1
223 223 423 134 134 Wi









3. Multiply this equation for Wi/Y223 to find


T(mi/Y223) = D3+ ([23 + 13 ]/23)34


The driving torque of the mechanism will be zero if the

term on the left is zero.

4. Plot D.34 vs. D223, as is done in Fig. 4.2.4. Approximate

this curve with a straight line. Set the constant multiplier

of D134 equal to the negative of the slope of the approximating

straight line. The constant is made up of the mass parameters

of links 23 and 34.

5. Adjust the mass parameters of link 23 until this ratio is

satisfied. Substitute these mass parameters into Eq.

(4.2.5).

This procedure was used with great success for four-bars in Ref. [19].

However, in the eight-bar mechanism being considered, the required mass

parameters to satisfy this method, when used on the link pairs 12-23,

45-56 and 78-89, caused a ten-fold increase in the kinetic energy of the

mechanism and yielded an increased driving torque. It is hypothesized by

the writer that this mechanism is of such a nature that it is impossible

to balance by mass redistribution to significantly reduce the fluctua-

tions of kinetic energy and their required torque. The possible explanation

is that the input crank is quite small so that all of the system masses

appear to be moving simultaneously with the same sinusoidal motion.

In light of the above negative results, attempts were made to balance

the mechanism for specified values of kinetic energy which were not con-

stant but which, if achieved, would reduce the inertia driving torque.

It was found after several attempts that any departure from the "natural"





























































44
0











C,












C-4
;









kinetic energy curve of the mechanism resulted in the requirement for

mass parameters which were not physically realizable. This natural

kinetic energy curve is the sum of the Dpq or Dq remaining in the re-

duced equation for the kinetic energy of the mechanism. Hence, an indica-

tor has been found for the shape of the kinetic energy curve and, by ex-

trapolation, for the remaining properties of the mechanism for balancing.

Also, it is possible to state that the minimum energy configuration for

this mechanism will be found if all of the Y3p and Y3 are made to be
2pq 3pq
zero and the constants multiplying the Dlpq are made as small as possible.

The Y3pq can be made to be zero by making the links of the mechanism

in-line links, i.e., by choosing Vpq = 0. The Y3 can be satisfied by

making the links in the form of physical pendula, i.e., links having

the same radius of gyration if measured from either pin-joint.


4.3 A Cam Driven Five-Bar

A mechanism similar to that shown in Fig. 4.3.1 was proposed in U.S.

Patent number 3,657,052 and was to be used in the formation of a looped

pile carpet. The object of the mechanism shown was to move point 1 in a

programmed fashion to fold a sheet of yarn into continuous loops. There

would be an opposed pair of the mechanisms alternately folding the yarn

to form a sandwich of yarns between two backing substrates as shown in

Fig. 4.3.1. Such a mechanism, if it could be balanced, would be more

attractive to operate as a component of a machine. The dimensions of the

links and the mass parameters of the mechanism are shown in Table 4.3.1.*



*Since the actual motion of the endpoint, point 1, is not shown in the
patent drawings, the author used, as an approximation of this surve, a
coupler curve which was taken from a four-bar linkage.





76

















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OC


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TABLE 4.3.1
Mass Parameters and Link Dimensions
of the Original Mechanism

LOCATION CENTROIDAL LINK
LINK CENTER OF GRAVITY MASS MOMENT OF INERTIA LENGTH
pq Upq Vpq mpq mpqk2 apq

65 3.12 0 0.0025 0.0095 6.24
45 5.13 0 0.0037 0.0330 10.23
42 3 0 0.0092 0.4776 14.52
32 3.36 0 0.0027 0.0118 6.72



For the mechanism under consideration, it is desirable first to con-

ceptualize the possible modes of balancing for this mechanism; there are

five links (n = 5), five pin-joints (j = 5), and two fixed pivots (f = 2).

Using Eq. (3.11.2), the number of positions possible for balancing for

shaking moment is found to be


Si = Q j = 4(n 1) j = 16 5 = 11.


Using Eq. (3.11.3) for shaking force, similar calculations yield


S2 = Q 2j = 16 2(5) = 6.


Using Eq. (3.11.4), the number of positions for kinetic energy is found

as


S3 = Q j 2f = 16 5 2(2) = 7.


The significance of the above results is that these are the number of

balancing conditions which must be satisfied to completely balance the

mechanism for the designated dynamic properties. They are also the maxi-

mum number of non-zero specifications which may be made in the properties

and still be satisfied exactly. To analytically determine what the bal-

ancing conditions are, it is necessary to derive the equations for the

various properties.









It was shown in Section 3.4 that the equation for the dynamic pro-

perties of a mechanism can be written using Eq. (3.4.2). The general

equation for the dynamic properties of the mechanism of interest is


S = Y165D165 + 265D265 + 365365 + Y465D65 (link 65)

+ Y145DI45 + Y245D245 + Y345D345 + Y445D445 (link 45)

+ Y142D42 + Y242242 + Y3 2 + Y442D442 (link 42)

+ Y132D132 + Y232D232 + Y332D332 + Y432D432. (link 32)


However, because of the fixed pivots 6 and 3, the terms D715 = D132 = 0

for all dynamic properties (see Appendix A). For each of the moving pin-

joints, 5, 4 and 2, Dlpq = Dpq; D4pq = Dqr (see Appendix B). Hence,

the above equation may be reduced in complexity by making these substitu-

tions and collecting in terms of the constant coefficients of identical

variable factors. The resulting equation is


S = Y265D265 + Y365D365

+ Y245D245 + Y345D345 + [Y465 + Y445]D465
(4.3.1)
+ Y242D242 + Y342D342 + Y145 + Y1421]45

+ Y232D232 + Y332D332 + Y442 + Y432]D432*


If the substitutions


Y > Y3 ; Dp =? D-
1pq ipq ipq Ipq

from Eq. (2.5.3) are made in the above equation, it becomes the equation

for the total angular momentum of the mechanism. The time derivative of

this resulting equation is the shaking moment of the mechanism. There are

eleven terms in this equation which was predicted using Eq. (3.11.2).









Using the substitutions


Yipq > Ypq and Dipq = D pq


from Eq. (2.6.3) in Eq. (4.3.1), the resulting equation is the equation

for the kinetic energy of the five-bar. Since, the cranks must rotate

about fixed pivots (see Appendix A), then it is evident from the defini-

tions of Eq. (2.6.3) that


D65 = D6 = D6 = D6 = 0.
265 365 232 332


This reduces Eq. (4.3.1) to the equation for the kinetic energy of the

five-bar


Ei = 53 5 + D6 + [Y5 + Y3 ]D6 + y3 D6
E Y5 5 345 345 365 45 465 242 242 (4.3.2)
(4.3.2)
S43 y3 ] + Y3 6
Y342D342 + [145 4+ 2YD1 + Y442 432+


which is seen to contain seven terms, the number predicted by Eq. (3.11.4).

As a last development, use the definitions from Eq. (2.3.3) in the

form


Y. = Y? and D. =- D2
ipq ipq ipq ipq

and substitute these results into Eq. (4.3.1) to provide the equation for

the total linear momentum of the five-bar. Again, if the appropriate

substitutions from Eq. (2.3.3) and the special nature of the motion of

the cranks of the mechanism are accounted for (see Appendix A), then


D2 = D2 =
265 232

and accounting for the common moving pin-joints (see Appendix B), the

qualities









465 445; D442 D432 145 = D145

3D65 = D345; D245 = D242; D42 = D32

follow. Finally the equation reduces to

L [y2 + Y2 ]D2 + [Y2 + y2 ]D2
3L65 345 345 465 4 45 445

+ 265 242 245 + 45 + 42]D45 (4.3.3)
S[2 + y2 ]D2 + [y2+2 +2 D2
3Y42 332 ]D42 442 432 D442

which clearly involves six constant terms which multiply six variable

terms. There are six balancing conditions that may be specified for this

equation as predicted by Eq. (3.11.3).

The definition of complete balancing as used in this work means that

some dynamic property is identically zero for the complete cycle of the

mechanism. Complete balancing will be illustrated in this case by con-

sidering the above three equations in reverse order. For Eq. (4.3.3),

the shaking force of the mechanism may be forced to be zero by requiring

that the six constant terms of the equation be identically zero. The

definitions of these terms yield


0 = Y65 + Y 45 = "6565/a65 m 45/a45 (4.3.4)

0 = Y65 + Y45 = m65u65/a65 + m5u45/a45, (4.3.5)

0 = 65 242 = m65v65/a65 + m42v42/a42' (4.3.6)
0 = 45 + Y2 = m5( u45/a45) + m42(l u42/a42)' (4.3.7)

2 + 332 = m6565/a65 m32v32/a32' and (4.3.8)

0 = 42 + 2 m2u 42 32u32/a32. (4.3.9)


These six equations are expressed in terms of twelve of the sixteen mass

parameters of the linkage. This means that six of the mass parameters in

the equations are free choices and that the values of the four radii of









gyration of the links have no influence on the shaking force of the mech-

anism. These six equations are the complete balancing conditions for the

five-bar; they are relatively easily satisfied.

For this mechanism, link 42 is geometrically the largest link; it is

also the link which is preforming the useful work of the mechanism. For

these reasons, it is assumed that the configuration of link 42 is fixed.

This means that the mass parameters of this link will be taken as three

of the free choices; i.e., m42, u42, and v42 are given. There are three

remaining arbitrary choices. The author made the decision to pick the

mass content (m32, m45, and m65) of the three other moving links. This

was done and a computer program written (see Appendix E) which calculated

the remaining mass parameters based on the algebraic solution of Eqs.

(4.3.6) through (4.3.9). It was found that the original choices of the

values for the masses of certain of the links were too small and these

were adjusted through several iterations to give both convenient location

of the centers of mass and positive values for the radii of gyration of

the links. A final, but by no means optimum, set of mass parameters for

the completely force balanced mechanism is shown in Table 4.3.2. The

placement of the counterweights was next calculated and these values are

shown in Table 4.3.3. Note that there are calculated values for the

required radii of gyration of the counterweights shown (see Eqs. (3.12.1)

through (3.12.4)).

After the selection of the balanced links and the locations of the

counterweights, an analysis program based on the dyad approach of Pollock

[21] was run to determine the effects of balancing on the mechanism.

Figures 4.3.2 through 4.3.6 illustrate some dynamic properties of interest

in the unbalanced mechanism for comparison. Figures 4.3.7 through 4.3.10









TABLE 4.3.2 Mass Parameters of
Completely Shaking Force Balanced
Mechanism


LOCATION OF
CENTER OF GRAVITY


LINK


pq up
Pq


65 -8. 665 0


45 12.435

42 3.000

32 -0.799


MASS

m
pq


0.0295

0.0337

0.0092

0.0159


CENTROIDAL
MOMENT OF INERTIA


m "k
pq pq


0.6568

0.4045

0.4776

0,0784


TABLE 4.3.3 Mass Parameters of Counterweights for
Completely Shaking Force Balanced
Mechanism


LOCATION OF
CENTER OF GRAVITY


LINK


pq u


MASS

m
Pq


CENTROIDAL
MOMENT OF INERTIA

m k2
Pq Pq


65 -9.737

45 13.349

42 0

32 -1.632


0.0270

0.0300


0 0.1325


0.2745

0.1466

0

0.1155




































































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