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 EightBar Linkage
4.3 A Cam Driven FiveBar
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 PINJOINTS
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 EightBar 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 NonZero 106
Shaking Moment
4.3.7 Mass Parameters of Counterweights for NonZero 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 FourBar Linkage 27
3.3.1 Typical Links 30
3.4.1 A General FourBar with Mass Content 34
3.5.1 Stephenson 2 SixBar Linkage 37
3.5.2 Possible Ternaries 39
3.7.1 A FourBar 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 EightBar Example 65
4.2.2 Plot of Kinetic Energy of EightBar 70
4.2.3 Plot of Inertia Driving Torque of EightBar 71
4.2.4 Plot of D134 vs D223 74
4.3.1 A Cam Driven FiveBar 76
4.3.2 Forces of Cranks of Unbalanced FiveBar 83
4.3.3 Forces in Moving PinJoints of Unbalanced FiveBar 84
4.3.4 Inertia Driving Torque and Rocking Moment of 85
Unbalanced FiveBar
4.3.5 Shaking Moment of Unbalanced FiveBar 86
4.3.6 Shaking Force of Unbalanced FiveBar 87
FIGURE PAGE
4.3.7 Crank Reactions of Force Balanced FiveBar 88
4.3.8 Forces in Moving PinJoints of Force Balanced 89
FiveBar
4.3.9 Inertia Driving Torque and Rocking Moment of Force 90
Balanced FiveBar
4.3.10 Shaking Moment of Force Balanced FiveBar 91
4.3.11 Forces on Cranks of Moment Balanced FiveBar 98
4.3.12 Forces on Moving PinJoints of Moment Balanced 99
FiveBar
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 FiveBar
4.3.15 Shaking Force Balanced FiveBar 103
4.3.16 Moment Balanced FiveBar 104
4.3.17 Crank Reactions of NonZero Moment Balanced FiveBar 107
4.3.18 Forces in Moving PinJoints of NonZero Moment 108
Balanced FiveBar
4.3.19 Forces at Gear 9c of NonZero Moment Balanced 109
FiveBar
4.3.20 Inertia Driving Torque and Rocking Moment of NonZero 110
Moment Balanced FiveBar
4.3.21 Shaking Moment of NonZero Moment Balanced FiveBar 111
4.3.22 Total Shaking Force of NonZero 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 nonzero.
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 nonlinear 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
straightforward 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 pinjoints, 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 nonzero
values for the dynamic property.
Two examples are included. The first is an eightbar which includes
a ternary. The balancing equation for kinetic energy and driving torque
due to inertia is developed. The second is a fivebar 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 nonlinear devices. As such they exhibit nonlinear
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 nonstandard 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 slidercrank 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 sixbar revolute link
ages and the sixbars 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 fourbar with constant
speed input that had been previously shaking force balanced. That these
conditions are required for the complete shaking moment balancing of any
fourbar 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 centerlines
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 inline fourbar 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 rootmeansquare
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 nonzero 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 timedependent 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 timedependent 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 timedependent 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 righthandside 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 timedependent
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(l2uG/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 timedependent 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] + mk2yq.
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 33 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 ppq 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 timedependent
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 timedependent 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
timedependent variables, Di. For any given mechanism, the timedependent
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 timedependent 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 nonnumerical 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
fourlink 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 fourbar linkage. The
wellknown 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 timedependent 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 pinjoints, 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 pinjoints 2 and 3.
From this observation, it is recognized that the equation for a dynamic
property of a fourlink mechanism, which is written in terms of the motion
of the pinjoints, 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 pinjoints 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 uaxis 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 pinjoint 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 pinjointed 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 pinjoints 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 nonsingular. 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 fourlink 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 pinjoint 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 nonzero. 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 SixBar 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 pinjoints. 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)2RP TERNARY
p in
"'^n p
(c) R2P 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 subcases 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 pinjoint 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 fourlink 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 fourbar 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 pinjoints 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 fourbar 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 fourbar 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
fourbar (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 vcoordinate 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 fourbar. 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 counterrotates 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 fourbar 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 fourbar 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 fourbar 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 nonzero 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 fourbar. If it is recognized that, for the fourbar
=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 fourlink 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
nonzero 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 fourbar 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 nonzero values
of shaking moment and then for specified nonzero 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 nonzero
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 multiplyseparated nonzero 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 lowerpair connectors (pinjoints 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 pinjoints and
sliders
These Si are the maximum number of specifications which can be made if a
closed form exact solution to the nonzero 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
squaresense. This means that Xm returned by this process will satisfy
the specified values of the dynamic property in a leastsquare 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 nonnumerical 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 EightBar 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 pinjoints, 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 pinjoints 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 pinjoints, j = 10; and the num
ber of fixed or grounded pinjoints, 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 EightBar
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 pinjoints, 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 pinjoints 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 pinjoints. 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 pinjoints. 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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44
0
0
C\11
H F
0 4
0
0s
c'J
0
0
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 fourbar 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 inline
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 fourbars in Ref. [19].
However, in the eightbar mechanism being considered, the required mass
parameters to satisfy this method, when used on the link pairs 1223,
4556 and 7889, caused a tenfold 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,
C4
;
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
inline 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 pinjoint.
4.3 A Cam Driven FiveBar
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 fourbar linkage.
76
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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 pinjoints (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 nonzero 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 fivebar. 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
fivebar
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 fivebar. 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 pinjoints (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
fivebar; 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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