DYNAMIC RESPONSE ANALYSIS OF COMPLEX
:.MEClH S.IS WITH MULTIPLE INPUTS
By
CHARLES EDWARD BENEDICT
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
To Patricia
ACKNOWLEDGMENTS
The author expresses his appreciation to Dr. Delbert Tesar
for his interest, encouragement, and assistance in all phases of his
doctoral program as chairman of the supervisory committee, particu
larly in the preparation of this dissertation. Through Dr. Tesar's
efforts and support the author was afforded an opportunity to pre
sent a major part of this research to the leading researchers in this
area at the NSF Advanced Training Workshop in Mechanisms held at
Oklahoma State University.
Sincere appreciation is expressed to Dr. Calvin C. Oliver
for his assistance as cochairman of the supervisory committee.
Appreciation is also expressed to the other committee members
for their guidance and support:
Dr. W. H1. Boykin, Jr.,
Dr. T. E. Bullock,
Dr. J. Mahig,
Dr. J. M. Vance.
The author is indebted to the Graduate Faculty for their support
in making it possible for the author to receive financial assistance
from a NDEA Title IV Fellowship.
Deep gratitude goes to the author's wife, Patricia, and
daughter, Sharla, for their patience and understanding.
TABLE OF CONTENTS
ACNOWLEDGMENTS . . . . . . .
LIST OF TABLES . . . . . . .
LIST OF FIGURES . . . . . . .
NOMENCLATURE . . . . . . . .
ABSTRACT . . . . . . . .
CHAPTER
I LITERATURE SURVEY . . . . . .
Dynamic State Analysis . . .
Displacement Analysis . . . .
Dynamic Response Analysis . . .
II CINEMATIC INFLUENCE COEFFICIENTS OF
CONSTRUCTED FROM ASSUR GROUPS . .
Kinematic Influence Coefficients of
Input Poles . . . . . .
Velocity Influence Coefficients
COMPLEX MECHANISM
Primary
. . . ..
Is
Acceleration Influence Coefficients . . . . .
Kinematic Influence Coefficients of Intermediate
Input Poles . . . . . . . . . . . .
Velocity Influence Coefficients . . . . . .
Acceleration Influence Coefficients . . . . .
Total InputOutput Kinematic Influence Coefficients . .
Velocity Influence Coefficients . . . . . .
Acceleration Influence Coefficients . . . . .
III EQUIVALENT SYSTEM FORMULATION IN TERMS OF KINEMATIC
INFLUENCE COEFFICIENTS . . . . . . . . .
Equivalent System Torques . . . . . . . .
Page
111
. . . vi
. . . . vii
. . . viii
xi
1
2
3
TABLE OF CONTENTS (Continued)
Chapter Page
III (Continued)
External Forces and Torques . . . . ... 30
Internal Springs . . . . . . . ... 32
Internal Viscous Dampers . . . . . . .. 35
Total Equivalent Torque . . . . . ... 38
Equivalent System Inertias . . . . . ... 38
Two Degrees of Freedom Example . . . . ... 41
Equivalent System Inertia Power . . . . ... 45
Two Input Example . . . . . . . ... 48
IV TIME RESPONSE OF EQUIVALENT MASS SYSTEMS . . .. 51
Lagrange's Method . . . . . . . . ... 52
Example . . . . . . . . ... . . 53
Hamilton's Principle . . . . . . ... 56
Example . . . . . . . .... . .. 57
V SUMMARY AND CONCLUSIONS . . . . . . . .. 60
APPENDICES
A DIRECT DERIVATION OF FIRSTORDER DIFFERENCE
EQUATIONS FOR DYNAMICAL SYSTEMS . . . . .. 69
Derivation of Method . . . . . . . ... 69
B NUMERICAL SOLUTION TO A TWO DEGREES
OF FREEDOM EXAMPLE . . . . . . . ... 74
BIBLIOGRAPHY . . . . . . . . . . . . . 82
BIOGRAPHICAL SKETCH . . . . . . . . ... . . 85
LIST OF TABLES
Table Page
21 Influence Coefficients . . . . . . . ... 27
31 Equivalent System Forces and Torques . . . . .. 37
32 Equivalent System Formulation . . . . . . .. 50
B1 FiveBar Parameters . . . . . . . ... 80
LIST OF FIGURES
Figure Page
21 Binary Groups . . . . . . . . . . 7
22 Assur Groups . . . . . . . . ... . . 9
23 General System Point . . . . . . . . .. 10
24 Point Paths of Assur Groups . . . . . . ... .14
25 SevenLink System Group . . . . . . . ... .20
26 Sliding Pair Constraint . . . . . . . ... .28
31 Equivalent System Elements . . . . . . ... 31
32 Complex Multiple Input Linkage System . . . . .. 39
33 Differential Gear System . . . . . . . .. 42
34 Angular Relationship to Input #1 . . . . . .. 43
35 Angular Relationship to Input #2 . . . . . . 43
36 Complex Two Input Linkage System . . . . . .. .49
41 Two Input System . . . . . . . . ... . 54
51 Complex Multiple Input Mechanism and Its Coupled
Equivalent Mass System . . . . . . . ... 61
52 Linkage System with Elastic Coupler Link . . . .. .64
53 Linkage Models with Deformable Bearings . . . .. .65
54 Optimal Open Loop Control Example . . . . . .. 67
Bl Two Degrees of Freedom FiveBar Example . . . ... .76
B2 Kinematic Position Equations . . . . . . ... .77
B3 Polar Phase Plane: 1 vs pl .. . . . ...... 78
B4 Polar Phase Plane: p2 vs P 2 . . .... 79
B5 Equivalent Inertias vs Time . . . . . . ... .81
vii
NOMENCLATURE
B General linkage pin joint or points in links m,n
m,n
C Viscous damping coefficient of dashpot attached to link i
and ground
*
C.. Equivalent viscous damping coefficient associated with the
13
th th
i input link due to a unit angular velocity of j input
E Coordinate point denoting center of gravity of system link
F General output coordinate point
F External force acting through general system point E
e
* th
F Equivalent force acting on ith system input due to a unit
e/1
external force at point E
g.i Velocity influence coefficient of link I with respect to
input link i
Gci Row vector of velocity influence coefficients
h.ij Acceleration influence coefficient of link a with respect
S .th
to the ith and jth input links
Hij Square matrix of acceleration influence coefficients
i Denotes input link or generalized coordinate
I Effective moment of inertia of link I taken about its
center of gravity
I.. Equivalent moment of inertia term
13
viii
j Denotes input link counter
k Corresponding time counter
k Corresponding position counter
K Effective spring constant associated with a spring attached
between link a and ground
K Equivalent spring constant of K with respect to system
input i
2 Denotes general system link
L Lagrangian of the equivalent mass system
m Total number of general system outputs
M, Effective mass of link i at center of gravity
n Denotes total number of system inputs
N Denotes total number of system links
P Total system inertia power
th
..P ij equivalent inertia power coefficient with respect
ij3 r
th
to the r input
q Denotes input link counter
r Denotes input link counter
S1 Summation representing the time integral of the Lagrangian
S2 Summation representing the time integral of the virtual
work of the nonconservative forces
T General system external torque acting on link i
* th
T,e,d,s/i Equivalent torque acting at i input link resulting from
unit torques on link I, unit forces through system point E,
unit velocities on equivalent viscous damper, and unit
displacements on equivalent springs
ix
T Total equivalent torques acting on ith system input link
1
v Linear velocity of general system point E
e
WV Weight of link A acting through center of gravity
x Denotes xcoordinate of system input
X Denotes xcoordinate of system output
y Denotes ycoordinate of system input
Y Denotes ycoordinate of system output
a. Angular acceleration of link i
1
6 Variation of some parameter
A Finite increment
1 Denotes angles
\ Undetermined Lagrangian multiplier
Y. Angle of ith system input link
Angle of 2th system output link
W. Angular velocity of link i
Denotes column vector
Denotes differentiation with respect to time
Denotes transpose of matrix
S Denotes partial differentiation
Sign Convention
Righthand Cartesian coordinate system
Angles measured positive ccw from positive xaxis
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
DYNAMIC RESPONSE ANALYSIS OF COMPLEX
MECHANISMS WITH MULTIPLE INPUTS
By
Charles Edward Benedict
December, 1971
Chairman: Dr. Delbert Tesar
CoChairman: Dr. Calvin C. Oliver
Major Department: Mechanical Engineering
The holonomic constraints associated with complex, multiple
input linkage systems complicate the procedures and methods used in
determining their dynamic response. Large systems of nonlinear,
secondorder differential equations, requiring additional algebraic
equations of constraint, occur as a result of these constraints.
Double iteration algorithms, which are both timeconsuming and subject
to error, are necessary to integrate numerically these differential
equations of motion.
In this dissertation the concepts of kinematic influence coef
ficients of complex, planar, rigid link mechanisms with multiple inputs
are developed and utilized to eliminate the holonomic constraints asso
ciated with such systems. Kinematic influence coefficients associated
with series and parallel linkage combinations are developed, based on
the addition of Assur groups dyadss, tetrads and more complex groups)
to the basic system group.
These complex, multiple input linkage systems are then reduced
to coupled equivalent mass systems acted upon by variable rate springs,
variable coefficient viscous dampers, and equivalent external forces
and torques. The holonomic constraints associated with the original
system are eliminated, thus leaving the equivalent mass system free of
all such constraints. The number of generalized coordinates required
to describe the motion of the equivalent system now equals the number
of independent system inputs.
The differential equations of motion describing the system's
dynamical behavior can then be determined by established methods and
put in a suitable form for numerical integration.
CHAPTER I
LITERATURE SURVEY
A substantial survey of the literature representing the state
of the art in dynamic response analysis of constrained mechanisms
appears in Il]. Virtually all systems treated in the literature possess
a single degree of freedom. Furthermore, these systems are analyzed for
their output dynamic state. Few researchers have attempted to deter
mine the input dynamic state of complex mechanisms with multiple inputs
(degrees of freedom) to forces and torques. Those who have, invariably
give examples which are constrained fourlink mechanisms, thus leaving
their methods essentially untested. This emphasizes the need for a
formalized procedure to model complex systems mathematically, utilizing
direct methods for determining their dynamic response to known forces,
torques and energy crossing the system boundaries. This will eliminate
iterative procedures necessary in solving the differential equations
of motion.
Dynamic State Analysis
As stated in [1] "dynamic state" implies that the velocity and
acceleration of each point of every link of the system are piecewise
continuous, differentiable functions of the input characteristics. In
1957 Modrey [2) developed a graphical method whereby the velocities
and accelerations of the links of a system of higherordercomplexity
(one which cannot be analyzed as a system of fourlink mechanisms con
nected in series) could be determined, using velocity and acceleration
influence coefficients obtained through a "zerorelax" procedure. In
a discussion of this paper, T. P. Goodman showed the analytical equiv
alent to Modrey's method and proved the linearity property necessary
for superposition with respect to single degree of freedom systems.
Other graphical methods existed for solving the dynamic state question
of complex mechanisms; namely, Hall and Alt's method, Carter's method
and the method of normal accelerations. All of these methods assume
a known geometric configuration. That is, the point path is known, and
the dynamic state is determined based on this known configuration.
Displacement Analysis
Various methods have been developed for the displacement analysis
of mechanisms containing more than one vector loop. Hain [3] shows how
to determine the inputoutput position information of six and eightbar
mechanisms which do not contain a basic fourbar at ground but do contain
an internal fourlink loop. This is accomplished by inverting the mechan
ism, solving the position information and then reinverting. However,
most methods rely on harmonic analysis, including those by Meyer zur
Capellen [4], Denavit and Hasson [5], Flory and Wolford [6], Markus and
Tomas [7], and the Romanian school [8,9]. They obtain the harmonics by
approximate numerical means within a prescribed error criterion.
Crossley and Seshachar [10] analyze the displacement of planar Assur
groups by an iterative algebraic method which allows half the unknowns
from the matrix of complex numbers to be found first with the other
half found later. All of these methods rely on iteration or approx
imations because of the complex loop equations resulting from the
geometry. However, they yield results which can be made as accurate
as desired. Digital computation capability increases the desirability
of these methods.
Dynamic Response Analysis
Time response of mechanical systems, spatial as well as planar,
by use of Lagrangian mechanics has been advanced during the most recent
years. Chace [11] uses relative coordinates in determining the dynamic
response of multiple degree of freedom spatial mechanisms. He utilizes
Lagrangian multipliers to account for the physical (geometric) con
straints. Smith [12] employs this same technique in analyzing the
reaction forces in generalized machine systems. In both [11] and [12]
the examples are planar single degree of freedom fourlink mechanisms,
leaving the general nature of the method untested.
Uicker [13] and Carson and Trummel [14] apply matrix notation
and methods to the kinematics problem. By applying Lagrangian techniques,
they develop the system differential equations based on the vector loop
displacement equations. This method has been the most useful and power
ful for analyzing the dynamic response of complex multiple degree of
freedom systems to date. As with other methods, it has its weaknesses.
A dual iteration algorithm results, which is ideal for digital computa
tion but is timeconsuming.
The works of Wittenbauer [15], Federhofer [16], and Beyer [17]
are significant in the development of equivalent mass systems and
appear to be the most direct antecedents of the concepts developed in
this dissertation. Wittenbauer and Federhofer introduce the concepts
of reduced mass in terms of general mass content and velocity ratios.
Beyer effectively summarized this work for the modern reader in terms
of the time response problem. The authors in references [1114] ignore
this property, preferring to treat these systems as ones with large
numbers of generalized coordinates, coupled by algebraic equations of
constraint. This property necessitates the use of the Lagrangian
Xmethod in treating the holonomic auxiliary conditions, replacing the
kinematical constraints by the forces necessary in maintaining them.
The X's, then, indicate the degree to which the constraints are violated.
The need for a more concise system formulation, allowing one to
determine the dynamic response in a more precise and direct manner, is
paramount. This will be accomplished in the following chapters by
developing the concepts of kinematic influence coefficients of complex,
planar, rigidlink mechanisms with multiple inputs (degrees of freedom).
These, in effect, account for the holonomic constraints on the system,
resulting in coupled, equivalent mass systems requiring only as many
generalized coordinates as its degrees of freedom. These equivalent
systems can then be uncoupled by standard methods, allowing their time
response to be determined by direct methods such as that developed in
[18].
CHAPTER II
KINEMATIC INFLUENCE COEFFICIENTS OF COMPLEX MECHANISMS
CONSTRUCTED FROM ASSUR GROUPS
Pelecudi [19,20] defines kinematic multipoles as kinematic
chains interpreted as lines for transmitting motion information (posi
tion, velocity, acceleration) of rigid links from an input to an output
or, more generally, from one point in the chain to another. This is
accomplished by motion transforms called kinematic ratios, velocity and
acceleration ratios, or, more precisely, kinematic influence coefficients.
These are well established for the case of fourlink mechanisms and cams
in [1,2125]. Kinematic chains receive information from input poles and
transmit it to output poles. The generality of this concept enables
one to construct highly complex mechanisms consisting of groups of
links known as Assur groups. The degree of mobility of the final sys
tem can therefore be controlled or determined by the mobility associated
with each group added to the basic kinematic chain. Two types of chains
are most important in the construction of complex mechanisms.
(1) Structural groups having a degree of mobility (freedom)
f 0, forming pa dive multipoles. A group may be
overconstrained (f < 0) to form a mechanism with a
degree of mobility which is less than the basic chain.
(2) Mechanisms with a fixedlink having a degree of freedom
f 1, formingactive multipoles.
Thus,one is free to construct mechanisms from basic chains
whose degree of freedom remains invariant under the addition of active
and padLive multipoles. This points out the importance of the struc
tural group in the construction of mechanisms of high complexity.
For the purpose of analysis it is desirable to reduce these
systems to their basic kinematic groups in order to derive the input
output transformation of position, velocity, and acceleration. Systems
possessing these transformations can be reduced to coupled equivalent
mass systems. This coupling will be quadratic and will depend on
velocity influence coefficients.
It is necessary to discuss the types of kinematic groups avail
able for constructing complex mechanisms of desired mobility.
Assur groups [26] are defined in terms of kinematic groups
which form structures when the outer joints are fixed after being
separated from the input links. The binary group is the simplest of
the Assur groups. Figure 21 illustrates its use in constructing the
four, five, and sixbar mechanisms. Each binary group forms a struc
ture when it is separated from its primary or intermediate input and
has its outer joints fixed. The dyad in the four and fivebar mechan
isms and the dyad attached to the intermediate input of the sixbar
mechanism form dipoles while the dyad in Figure 21(c) attached to the
primary input forms a triple, capable of transmitting information from
the system input through the output pole to the dipole.
A B
00
0 1
(a)
B
A
00i O
(b)
(c)
Figure 21. Binary Groups
The next Assur group of higher complexity is the tetrad (see
Figure 22(a)). When the outer joints A, Oc, and E are fixed, the
group becomes a structure with f = 0. This group requires two vector
loop equations to determine the angular relationships between the links.
More complex Assur groups are illustrated in Figures 22(b) and (c).
These and other groups are classified and analyzed for their displace
ments by Crossley and Seshachar [10].
Markus and Tomas [7] define a system group as a group of links
which has zero degrees of freedom when the input links are fixed. The
system groups contain Assur groups but are more general, enabling the
authors to solve for displacement information of any system group by
a single harmonic analysis algorithm. Curtis and Tomas [27] point out
the importance of the binary group in determining position information
of mechanisms of varying complexity. They utilize the law of cosines
in developing a closed form algorithm to obtain the position information
of a large class of mechanisms consisting of various binary group con
figurations.
Kinematic Influence Coefficients
of Primary Input Poles
Velocity Influence Coefficients
An implicit functional relationship, f(x,y) = 0, describing the
path of some general system point E (see Figure 23(a)) associated with
the outer joint of a binary group, can be written based on known posi
tion information. In general, f(x,y), ge, he, me, and (+P will be
known as a function of some system input parameter such as p9. It is
1
(b)
(c)
Figure 22. Assur Groups
Y h,
e f (x,y) 0
ge
1+ I
x
(a)
y
he
E
I 'I
I jf (x,y)= 0
L_ 9e
(b)
Figure 23. General System Point
noted that g and h are defined in [1) as
e e
dS
e
e dcp
d2S
e2
h
i
and
m = slope of path tangent of point S
e e
In order to proceed through the chain and determine the influence
coefficients associated with F(X,Y) the values for gx, gy hx, h
must be determined. It is noted that (see Figure 24(a))
dS
ex.
x xi =p. i'
(21)
dS
y =y Ci =dp. i 'i
and
d y f v f
S[(f(x,y)j = x + s v
dt x x + y
=f af 
=T gx + 3 g = 0
(22)
This gives
(23)
Sf
gy 
x f
gx TSy
The expressions for me and g are given by
S= (24)
e
and
2 .2
ge = +y (25)
Since g is known, then
e
o
g = (26)
and
gy = me g (27)
Acceleration Influence Coefficients
Now consider h and h in terms of m g and h .
x y e e e
It is noted that
t
h = a
x x (28)
h = a for 9. = 1.0, (p = 0.
y y 1 i
Define T as follows (see Figure 23(b))
S= tan { } (29)
gx
For f(x,y) = 0, a closed path of point E in a general system link,
g is always positive. Then h is positive if directed in the same
sense as g, negative if directed in the opposite direction.
sense as ge negative if directed in the opposite direction.
Therefore,
h = h cos (71) ,
x e
h = h sin () .
y e
(210)
Now consider a dyad as a dipole or binary group in the Assurian
sense with points El and E2 having general point paths defined by
implicit functions fl(x1,Y1) = 0 and f2(x2, 2) = 0 (see Figure 24(a)).
Then,
ge
gx
g =
x1 /1+ i
ge2
e 2
h = cos (6 ,
x1 el 1
h =h cos ( 2) ,
x2 e2 2

g el x1
1 1
g = m g
Y2 e2 gx2
h = h sin (71
y1 e 1
= h sin (2)
Y2 e2
This can be generalized immediately to all classes of Assur groups
(see Figures 24(a) and (b)) as
ge.
gx
S = 1
i
g = m g
Yi ei xi
(213)
h = h cos (0 ) h = h sin () .
x e i Yi ei
and
(211)
(212)
(x, Y2)
(x ,yl)
(a)
(x2,y2)
(XI ,y1)
(x3 ,Y)
(x,,y,)
(x3, y3)
( b)
Figure 24. Point Paths of Assur Groups
Kinematic Influence Coefficients
of Intermediate Input Poles
Let E1 and E2 be the inputs and F be the output of the binary
group in Figure 24(a). Point F is an internal joint to links I and
S+ 1, Hlowover, the following derivation holds for output polo as
general points in either of the links 2 or + 1. The point path of
F can be expressed as an implicit function
f(X,Y) = 0,
(214)
where X and Y are functions of xl,
be expressed by the projections of
dyad,
E1F + FE2 + E E1 = 0,
y1, x2' and y2. Relation (214) may
the vector loop equation for the
(215)
fx(1, x2,X,ylY2, Y) = 0,
fy(X1'2,X,Yy12,Y) = 0.
(216)
Velocity Influence Coefficients
Since X,Y are functions of xl, y1, x2, and y2, then they are
differentiable with respect to them. The velocity of point F is given
as
(217)
VF X Y
where
2
2 ax ax
VX x. Vy. ,
S1 1 1 1 1
(218)
2
r f Ay ay
vV =+V i + T yi VV
1=1 1 1 1 1
The partial derivatives in Equations (218) represent the velocity
influence coefficients for multiple input linkage systems. They are
defined as follows in order to remain consistent with previous literature.
Let
A ax
i TX
(219)
A 3y
Yigy =
Substituting these into Equations (218) gives
2
vx= x xg xi + i vy i
i= 1 1 1 1
2
V = v + Yy v } (220)
i= 1i1 1 1 y
Equations (220) can be put into compound matrix form as
I 1
VX XGx XG i i xI
] [ ] [ Ivy!
SImIlrJl\ for angular odULpa information
GL a G i G I 1i (222)
v
yi
where
ax x
xGx i x2
G (223)
Ivxl xl
Lx2
Acceleration Influence Coefficients
Differentiating Equations (218) with respect to time yields
2 2 2 2x
ax2 ax x Vxi y
i= j=l 1 j
2 2
ax + y fax t ax t 1 24)
Y i yj i1 x + ai (224)
1n 1 1 1 1
and
2 2
a2Y + axi +t aY '
+ 3Vy Vyj a a (225)
y 3 y/ j=1 yi Y I
Equations (224) and (225) can be written in compound matrix form and
generalized to n inputs as
a = vxI I v + 2v vx x ] Ivy
+ vi'  iY j] i'Viy + I Xi K f
y IV y Ii X i ] t
11
i,j = 1,2,...,n,
Xhxxx Xxx x Xhx x
1 1 1 2 n
ix ] ~~xh2x1 X2x2 tx22n
Xhx x X x nx2. Xh x nxn
n 1 X x2 X XnXn
82X
Xhx x. axx
i G g j
x x ... X g y1 X n]
[  i 1'^ ln
(226)
(227)
(228)
(229)
(230)
where
Corresponding expressions relating angular input to angular or
linear output can be derived. Since most kinematic chains consist of
lower pair connections, the remaining derivations will be based on
angular inputs to both linear and angular outputs. Complex kinematic
chnins constructed from basic chains with invarinnt mobility are developed
by adding Assur groups to output poles of the established chain of pro
determined mobility. II the mobility does not remain invariant, then
the basic kinematic chain with its known inputoutput position relation
ships is destroyed and the point paths are no longer predetermined with
respect to the original system group inputs. A new system group must be
established and new inputoutput relationships developed.
Total InputOutput Kinematic
Influence Coefficients
If the inputs to a linkage system group are angular, then the
intermediate linear inputs (xi,Yi) and their dynamic states (vxi'vyi)
t t
and (axi,ayi) can be expressed as functions of the input position param
eters c 's and their dynamic states W 's and a 's. The derivation of
q q q
the total inputoutput transformation is now carried out.
Velocity Influence Coefficients
Consider the system group in Figure 25. The expression for
the velocity components of point F with respect to points E1 and E2 are
G G 1 .2xI
   (231)
_Y Yx YG x_
i : iy ~
F
I (x,y)
E2(x2,y2)
Figure 25. SevenLink System Group
The intermediate input data,in terms of the primary input information,
are given by
v x i x q
  i,q = 1,2, (232)
LIV .I ..Lyi,59q
where
8x1 bx1
ax 6
[Xi q] x2= x2
and
Substituting the righthand side of Equation (232) into Equation (231)
yields
Vy xi Yi Li c q
= ;  Jwl (233)
[ Y x Y yi i q_
which reduces to
 , q = 1,2. (234)
v YGq
Notu that when q=l, Equation (234) becomes
Vx = Xg '1
(235)
V = Y= I1 '
which are recognized as the velocity influence coefficients developed
in reference [1] for single input systems.
Acceleration Influence Coefficients
The component accelerations of point F (Figure 25) in terms
of the intermediate inputs are
r 
3X]i] ;x
= 0V  i
+ . (236)
x O x
Yx x j 0 HL
2  IV I +236 I
Si I i 0 7
The first and third terms of Equation (236) are recognized as real
quadratic forms while the second is real bilinear.
Substitution of Equations (226), (227), and (232) into
Equation (236) gives
(2YX1) (2N X 2n)
t   
[ 0
t I o
ayj I
+2  
0 uw
(2n X 2n)
Y xy
1i3
$ [_
    
Y Yi I i
_ \ N,n= 2.
G
+ I q I 
Yi q
(2n xn) (n xn) (n xl)
XH x x. I
ji jdy ]
YRX. X
(237)
The complexity of Equation (237) is reduced by collecting like terms
and writing it in a more concise and familiar form as
aH G
i = j .. I + l a, q,r= 1,2, (238)
Y q Yr q L
where 11wll is a compound diagonal matrix given by
u 0
= (239)
0 jo
and )l is the transpose of the column vector IwI.
The expressions (234) and (238) for the velocity and acceler
ation can be generalized to treat systems with n inputs and m outputs.
These expressions take the general form
X1 XGlpq
X X G
m m 'q
.  q = 1,2,...,n, (240)
v G
1 1 (q
V YG
m m q
t
a X 1 q r X G
~m / Xm qr Xm
= ml  +  (241)
tly
1 1 q'r 1 q
t
a H G G
m Ym q r m q
q,r = 1,. ,n.
It is clear that expressions similar to Equations (240) and
(241) can be written for the angular properties of the system links.
For the purpose of illustration, both forms will be used where appro
priate throughout the remainder of this dissertation.
The discrete form [28] of the kinematic influence coefficients
of velocity and acceleration are expressed as
1 ik = 1)
g i,k+l Ai,kl (242)
agik = 2Ap.
2
Lhijk =) '
i,k+l i,k+l i,k1 i,k1
Iij k+1 j,k1 j1,k+1 + z j,k1 (243)
AhiJ,k = 4(A )2
where the input type (i.e., angular or linear) is considered understood
and its designation is dropped for simplification of notation as follows:
Agi,k ( i)
(244)
hij ,k i(i j kjk
Table 21 is a compact collection of the influence coefficients
derived thus far, expressed in discrete form. Each column is headed by
a characteristic set of input conditions and each row is designated by
a particular system model. Rows and columns intersect in blocks con
taining the influence coefficients for which these properties hold.
Blocks 1.1 1.3 and 3.1 3.3 have been derived and are expressed in
discrete form.
The coefficients in blocks 2.1 2.3 pertain to the relative
angular motion between any two links m,n in the system (where m,n are
not to be confused with m,n denoting the number of inputs and outputs
as in Equation (241)). The links are shown with coincident points at
B for purposes of simplicity of notation. The influence coefficients
in blocks 4.1 4.3 apply to the relative motion between any two points
B ,B in different links m,n in the system. The system model has been
m n
represented as a sliding pair connecting these two points. It may be
helpful to consider this pair as massless when it is not required for
constraint in the system. An example of when it would be required for
constraint is that shown in Figure 26.
The influence coefficients in Table 21 are a precise defin
ition of the system geometry up to the second order. As such they
27
C\?I jr r) ,
3 fT ~ x? 9 
N E : 3
,, g 
;^ '. < Cr 1^_ .<< :
j7^ C^ . ^ ^ ^
~P I
c^ 
^i [^j cLj ch
It I II
II (\j ro 
1. C7? 
3 3/)
II
tI (J* A
cW CVJEI
' . II I
OIo
Figure 26. Sliding Pair Constraint
represent a powerful definition of the meaning of multiple input link
age systems.
These influence coefficients enable one to determine the
dynamic state of every link in a general linkage system, given the
dynamic state of the system inputs. They will be used in the follow
ing chapter to reduce complex, multiple input linkage systems acted
upon by external forces and torques, internal springs, and viscous
dampers, to equivalent mass systems acted upon by equivalent external
torques, variable rate springs and variable coefficient viscous dampers.
The remaining coefficients will be treated on a discerte time
basis, since some of them will depend on their past history. This
means that expressions such as that for Sk imply that
k
S = f(t)
(245)
CHAPTER III
EQUIVALENT SYSTEM FORMULATION IN TERMS
OF KINEMATIC INFLUENCE COEFFICIENTS
The coefficients developed in Chapter II will now be utilized
to eliminate the holonomic constraints associated with complex linkage
systems, reducing them to coupled, equivalent mass systems. These
same coefficients will be used to reduce the generalized internal and
external force generators acting on the system to equivalent general
ized torques acting at the system inputs.
The Equivalent System Torques
The force related influence coefficients are derived by replac
ing the effect of external forces on the system (i.e., TV, Fe) and inter
nal forces generated by system elements, such as springs and viscous
dampers, with equivalent torques acting on the n system inputs.
External Forces and Torques
If a linkage system (see Figure 31(a)) is given a virtual
displacement by each separate input from some given system position k,
th
then the virtual work done by the equivalent torque at the i input
must be equal to the virtual work done by the system external torque, or
(T1/i iAk = l )k (31)
T41.
(a)
K~.
(b)
C ..
(c)
Figure 31. Equivalent System Elements
CK
C..
solving for Tz/ik and taking the limit as 5 ik 0 gives
.A
T/ik = ln T )k TLk = ik) Tk (32)
T/ik i 1
Similarly, for an external system force Fek
T/ik (egik Fek (33)
The total equivalent torque seen at each input link i, resulting from
all system torques T k, is given by
Te 1 g 2 1 Ngl G T
= 12 2g2 N' 2 (34)
T* G T
n I 1 n 2 n L Ng n k k N k
i = 1,2,...,n,
N = number of system links.
Equations (32) and (33) show that Agik and egik are the influence of
a unit torque or force applied to system link A or point E on the ith
system input link. The gik is the mechanical advantage of link Z with
respect to the ith input link and vice versa.
Internal Springs
Perhaps the most difficult system parameter to develop in terms
*
of an equivalent coefficient is the equivalent spring constant KA/ik.
Suppose a torque TLk is generated by a linear spring K between link I
and the ground link. The question arises: What is the equivalent
spring constant K /ik of the system spring K,? By giving the system
a virtual displacement, the potential energy change in the system
spring can be equated with the potential energy in the equivalent
spring associated with the ith input link (see Figure 32(b)). This
yields the following relation
2 (35)
i2 /ik (ik = (ik (35)
Solving for K /ik and taking the limit as p ik 0 yields
(At 2k (36)
K/ik = K lim = gk36
tipik 0 i k
It is noted that S cannot continuously increase with increas
ing pi otherwise the spring would be destroyed. Rather, is cyclic
or quasicyclic with increasing cp.i Therefore, for part of the cycle
A, (i.e., 2gi) must be negative. This implies the need for the signum
function a = sgn (,gi) which changes sign whenever the gi.'s go
through zero. The torque T/ik is then determinedfrom its past history
in the form
a/ik /i,k1 + K/ik ik (37)
where
2 2
K* giki+ gik (38)
K/ik ik 2 38
The equivalent torques at each system input resulting from all system
springs are given in matrix form as
*
Ts/l
T
s/2
T
s/n
*
Ts/1
T
s/2
Ts
s/n
N
2=1
K/l &1
K /2 2
K/n Mn
2/n n
(39)
An alternative approach to determine the equivalent torque
T2/ik generated by a system spring between link A and ground is to
treat the torque acting on link I due to spring Ke as an external
torque as in Equation (32). The external torque is given by
Tk = (K f ( .k)= *k (310)
where
I,, = free length of spring measured from the reference axis,
A k = deformation of system spring K from free length.
Substituting Equation (310) into (32) gives
T/ik = K (gik) ik (311)
The equivalent torques at each system input resulting from all
system springs K are now given in matrix form as
Ts/1 1Gi 1K
s/n N N N
< k k k
(312)
, i = 1,2,.ber of system lines.
N = number of system links.
Internal Viscous Dampers
Let the torque TIk be generated by a system dashpot C between
link ; and ground (see Figure 31(c)). The torque generated by the
dashpot is proportional to the angular velocity of the link attached to
the dashpot, or
Using Equation (32), TVk is transferred to the i system input as
TA/ik (ik) C (gj CW j (314)
j=l k
The total torque acting on the ith input link resulting from all system
dashpots is
n
Td/ik = C (315)
where
N
C j,k= gi gJ)k (316)
The equivalent torques at all system inputs resulting from all system
dashpots aregiven in matrix form as
I ~
d/l C11 C12 Cln
d/2 C21 C22 2n
S WI k (317)
Cnl *
T C C ** C
d/n nl n2 nn k
'k 1 k
Table 31 is a concise collection of the preceding equivalent
torque concepts. Row 2 corresponds directly with row 1. Those in
row 2 are more general, since the generated torques occur between two
moving links m and n in the system. The relative influence coefficients
brik correspond to those in Table 21, i.e.,
brick = ngik mgik (318)
The equivalent torques in row 3 refer to a force Fek applied
to a particular point or points in the system. Since the direction
angle Tek of this force relative to the path tangent is known as a
function of the input angles i then the appropriate equivalent torque
can be determined. The equivalent torque, T e/ik, is obtained by first
taking its component F along the path tangent and transferring this
ek
th
force to the i input link by the influence coefficient egik. Here,
the product egik cos ( ek) is the effective influence of a unit force
.th
applied in the direction F at point E on the i input link. As such,
ek
it corresponds directly with gik in row 1. Blocks 3.2 and 3.3 follow
directly as a result of this correspondence. Here, the system spring
K and dashpot C are anchored at some fixed point 0 .
e e e
Finally, the equivalent torques in row 4 apply to forces in
extensible twoforce members attached to links m and n at points B and
m
B n. The relative influence coefficient rk is defined in Table 21,
n b ik
row 4. This implies the treatment of very general force generators in
the system.
3z
4::
K)
*o *1I
F ~F
U
uL4~
 ~
+
II
I I
Total Equivalent Torque
The total equivalent torque acting upon the ith input link
is given as
19)
T = T +T + T +T (319)
ik A/ik e/ik + Ts/ik +d/ik
If one or more of the input elements is actually a sliding
pair, then sik represents that input's reference position parameter
and Fik would be the equivalent force acting on that system input.
Equivalent System Inertias
The objective of this section is to transfer the effective
mass of link a to each of the n system inputs in order to obtain equiv
alent mass systems for a linkage system such as that shown in Figure 32.
Consider the kinetic energy of a complex, multiple input system.
I2 2 + M 2 } (320)
(KE)k =k 2 {2 (320)
where
1) = effective moment of inertia of link I about its center
of gravity E,
M, = effective mass of link 2 located at its center of gravity E.
Writing Equation (320) in terms of the system input velocities yields
N n 2 n 2
(KE)k = ikk + (eik)Wik
=1 i=l 1=1
(321)
n=3
N=13
Figure 32. Complex Multiple Input Linkage System
Rewriting Equation (321) in a more compact form gives
n n N
(KE)k Z [ )( ) + M i)( j) wj
i=l j=1 2=i
(322)
where it is noted that
1 2=i,j; i=j
( gXi)( g ) = (323)
0 o =i,j; i j,
a direct consequence of the meaning of independent system inputs.
The kinetic energy can be written in matrix form as
(K E)k k ij IWk (324)
(nxn)
k
where
N
ij,k = i i(gj) 2+ ei e j(325)
&=1k
Matrix Equation (324) is recognized as being real quadratic form
since the [I..] matrix is real symmetric. Thus it is seen that any
complex multiple input linkage system can be transformed into a system
of coupled equivalent masses with n degrees of freedom whose equiva
lent inertias are given by Equation (325).
Two Degrees of Freedom Example
For the purpose of illustration, consider the differential
gear (Figure 33), where the translating links are constrained to move
in the horizontal direction and rotation of link 4 is positive
counterclockwise.
The linear velocity influence coefficients of 3 are
3X3 d2
63 2
3gl x dl+d2
1 12
(326)
X3 d1
32 = x2 dl+d2
and
3 = 3 1 v1 + 3 2 v2 (327)
The angular velocity influence coefficients of 4 are given
by (see Figures 34 and 35)
a
4 1
4g 1 dl+d2
(328)
4 1
4 2 dx2 + dl+d2
and
4 = 4 1 v1 + 42 v2 (329)
The kinetic energy of the two input differential gear is
1 2 2 2 2
KE = 2 Iv + M2v2 + (M3 +M14)v3+ I4w4 (330)
1 2v2 ~3 3 44
CL x
0
4t
C _
x
Q.
X
dx, = (d,+ d2)(d34)
Figure 34. Angular Relationship to Input #1
dx2= (d,+ d2)d 4
Figure 35. Angular Relationship to Input #2
Substituting Equations (327) and (329) into (330) gives
1 2 2+ 2
KE = (MV1 + M22 + (M3+M4)(3gv1 3g2v2)
+ 14(4g1 1 + 4g2v2) (331)
where
v3 = v4
Collecting like terms and rewriting Equation (331) gives
1 2 + + )2 2
KE = 2 {I + 4(4gi) + (3 + M(31 1]
+ [M2 + T4(4g)2 + (M13 + M)(3) ]v2
+ 2[4I4g ) (4g2) + (M3 + M4) 3g1) (392)]Vlv2 (332)
or
1 2 2 *
KE = IlV + 1222 + 21121v2} (333)
where
2 2
I = M11 4 4 1 + (M3 + M4)(31)
2 2
12 = 4(4g)(4 g2) + (M3 + M) (3gl)(3g2) (334)
2 2 2 + 4(42)2 + (M3 +M432
For the system in Figure 33, Equation (324) becomes
11 12 v1
KE=[V1 2] (335)
21 122_ 
which is recognized as a quadratic form.
In the case of a differential gear the equivalent inertias are
constant. However, for a general linkage system (Figure 32), the I 's
are not constant but a function of the system input parameters. Note
that whenever v2=0, then
1 2
KE = vI (336)
and whenever v1 = O, then
1"2
KE = 22 v2 (337)
which are simply the expressions for the kinetic energy of a single
input equivalent system,
Inertia coupling terms, I.. i f j, appear for systems with
*
two or more inputs, indicating a relationship between the I..'s
iJ
through the velocity product terms w.iW.. Quinn's energy method [29]
Ij
becomes invalid for these types of systems, since the distribution of
kinetic energy between the links of the system is no longer invariant
with respect to the input velocities. The equivalent mass system
described above remains valid provided certain modifications are made
with respect to the solution technique.
Equivalent System Inertia Power
The final equivalent system parameter to be considered is the
equivalent system inertia power. The coefficients derived in this
section are necessary when deriving the differential equations of
*
motion of complex linkage systems. If the I..'s were constant, as in
the preceding example, these terms would be zero. Although they appear
unwieldy in size and computation for systems with more than two inputs,
they readily lend themselves to computer computation.
The power necessary to drive a linkage system against its
inertias is given as
i d
P = t (KE) (338)
where
KE = W I*j Il i,j = 1,2,...,n (324)
Differentiating Equation (324) with respect to time gives
P i j* + JI + i I 1. (339)
2 '' dtI ij
The second product on the right side of Equation (339) is self
explanatory. The time derivative of the equivalent inertia matrix,
however, is not as trivial. Consider a typical element of [I..]
1J
given by
N
ij = g ) + M(egie j (340)
Differentiating Equation (340) with respect to time gives
d n N
dij = [(Zg) ( r) + ,
d k r=1 1 I =g
+WNIkt 9 (341)
+ i(ei jr+ (oj)( ei k rk', (41
where
(ii)(hjr) = pr (342)
Equation (341) can be written as the product of a row and column
vector as
0W
n
(343)
where
N
ij r T=1 g Zg hj fj (Ah ir
+ M[ z egi)( ejr)+ (egj)(hir k
The subscript notation is defined by
'I
ij
ij r = '
r
Substituting Equation (343) into Equation (339) gives
(IXn) (nxnn) (
I ,7
p* p i w o
11 r 12 r n r
 i  
..I I *1
p p . p .
21 r 22 r 2 2nr 0 o
^JK  r 
I *I
P P ... P 0 0
nl r n2 r nn r
k+ LI jk I
where ijP is a compound nxnn matrix whose submatri
row vectors.
(344)
(345)
nnxn)
II
I * I
 r
0
1 r
I
S I k
k
(346)
ces are 1 Xn
Two Input Example
Consider the system in Figure 36 whose kinetic energy is
given by
I
(KE)k = 1 [2 k
21
I
12 Wl
I* [2
22 k k
where
ij
i k e gi Zgj + i k]
Differentiating Equation (347) with respect to time gives
1
P P P P r1
11 1 11 2 12 1 12 2 W2 0
i 1
=   
k 2 1 2k
k 1 1 0 W
21 1 21 2 22 1 22 2 1
Ik
+ [Yl 2 k
i21 22
where
8
12 1 [I(gi h11+ g2 2h21)+ M(eg1 eh11+ e2 eh21)]k
(348)
W k
2 k
(349)
(350)
Table 32 represents the concepts of equivalent system
inertias and equivalent inertia power for complex, multiple input
linkage systems. With this formulation the holonomic constraints are
eliminated and the equations of motion derived, utilizing only those
generalized coordinates associated with the independent inputs.
(347)
Figure 36. Complex Two Input Linkage System
50
n < Q I
0P O ,. .,
>
Sxrn Z: Fr
F C
0 0
cI i
+.I . I ,
I
I..
' i lil::^~' ' t
ZII "
.7s
+
c^ Ir r~
^^^ \r<\
'.< s . < \ 
^< O
`Q' << f.M 
_ L^ *
^x << >^
,Xi5~ ? i /
tt 5< ^ 
CHAPTER IV
TIME RESPONSE OF EQUIVALENT MASS SYSTEMS
Complex, multiple input linkage systems present a unique problem
with respect to determining their dynamic response. Methods for deter
mining the time response of single input systems are well established,
either in terms of Lagrangian mechanics [1114] or equivalent mass sys
tems. Each method has its strengths and weaknesses. The method chosen
to solve any given problem should be based on need and solution form.
It is possible to obtain phase plane solutions for single input systems
by using techniques in [1] if the input velocity is never less than
zero. With slight modifications in the predictor equation, negative
velocities can also be treated. Here the independent variable is the
input angle, i rather than time, a result of the cyclic nature of most
mechanisms. However, a large class of problems does not meet this
requirement. Hence, a more general method must be employed.
As pointed out earlier, the energy distribution method is
invalid for multiple input systems. Similarly, the kinetic energy
and power concepts utilizing the equivalent mass and force system
developed in [1] fail because they yield only one secondorder differ
ential equation in n independent variables. These failings are a
result of the geometric constraints imposed on the system.
Two methods will be discussed in this chapter. The differ
ential equations of motion for a two input system will be derived,
using both methods. The complete set of firstorder difference equa
tions will be derived for only the second method for reasons which
will be explained later. The methods used to derive the differential
equations and to solve them are not new. The kinematic influence
coefficients and coupled equivalent systems (inertias) described in
Chapters II and III are new and unique, enabling established methods
to be used in deriving and solving the differential equations of motion.
Lagrange's Method
A system of n secondorder differential equations is derived
from Lagrange's equations as
d (iL aL* *
dt Ti ( = 1,2...n) (41)
where
ci generalized coordinates,
pi generalized velocities,
L Lagrangian of the coupled equivalent mass system,
T. equivalent nonpotential forces and torques.
1
The n secondorder differential equations resulting from
Equation (41) are nonlinear, coupled, nonhomogeneous, and contain
variable coefficients. It is necessary to reduce this set of equations
to 2n firstorder differential equations in order to solve them on
a digital computer. Although this reduction is entirely possible, it
is often tedious, yielding unwieldy equations. The variable coefficients
are functions of geometry and are hence implicit functions of time.
Tlii characteristic requires the determination of the equivalent
inertia power coefficients of the system, terms which are very diffi
cult to obtain (see Chapter III) and only adds to the complexity of
the problem.
Example
The fivebar mechanism in Figure 41 is a representative two
input system. The generalized coordinates are 91 and 92; a helical
spring and viscous damper are attached between links 3 and 4 at pin
joint B, while external torques T1 and T2 are applied to links 1 and
2, respectively.
The Lagrangian for this system is given as
L = (KE PE), (42)
and the equivalent nonconservative torques are
T = C34 brl [brl 1 + br2 2] + Tl'
(43)
T2 =34 br 2 brl br2 + T2
where
2 2
KE = j L i j* (44)
i=l j=1l
1 2
pE  (34(f ) (45)
PE 2 34 34f 34 (4
Substituting Equations (43), (44), and (45) into
Equation (41) and performing the designated differentiation with
respect to the generalized coordinates and velocities yields
C34
Figure 41. Two Input System
dt { 1 l + 112 + 34 (34f 34)(brl)
=T1 34 [br)l + (brl) br2 (46)
d 
d I12 91 + 22 ~2 + K34 (P34f 34)(br2)
2 34 (b br2 1 (b 2] (47)
Completing the differentiation with respect to time gives
* .+ .2 
11 '9 + 1 + 12P) l + 12P 2 11 1 + 12 2 
+ K34 34f 34)brl) = 34 r)l + (brl)(br2 2
(48)
*2 .2 * _
21P 1 + (21P2 + 22P112 22 2 2 12 1 122 2
+ K34(C34f P34)(br2) = 3 (bl) (br2 + br2
(49)
Equations (48) and (49) are nonlinear in the velocities and
are unwieldy in their present form. In addition, they should be reduced
to four firstorder equations before they are numerically integrated.
The ..P terms as defined by Equations (344) and (345) are not simple
expressions, and require calculation at each integration step.
expressions, and require calculation at each integration step.
Hamilton's Principle
The method used in this section for deriving firstorder
difference equations by direct application of Hamilton's principle was
developed by Vance and Sitchin [18]. The section on the derivation of
the method from [18] is included in the Appendix for completeness.
The motion of a dynamical system is determined by solving the
3nN equations derived from
as,1 N 6f.
3 Z x ij F Fk At (410)
ik j=l ik
as N 8f..
+ .. = 0 (411)
2ik j=1J aik
fik= ik ik At = 0, (i= 1,2,...,n; k= 1,2,...,N), (412)
where
X.. it undetermined Lagrangian multiplier at j time
13
interval,
th
F i generalized equivalent nonpotential force, at
ik
kth time interval,
N
and S = Lk(9~Ik' 2k ... k' k' CP2k ... nk) At. (413)
k=1
The 3nN firstorder difference equations resulting from Equa
tions (410) (412) in conjunction with an appropriate difference
expression for p ik are obtained from only one differentiation and are
in a form which allows their solution to be marched out with time.
Furthermore, the differentiations (aS /Sp ) and (aS l/i.) produce the
. .P terms and uncouple the equations in one step.
ij r
Example
In order to compare the two methods, the above procedure will
be applied to the system in Figure 41. Since the internal spring K34
produces an internal torque which is transferrable to each input as a
generalized equivalent torque (see Chapter II), it will be included in
the Fik terms, leaving the sum S1 made up of only the kinetic energy of
the system. Therefore,
1 1 212, }* Y (414)
S1 2 + 2112 2 22 2 (414)
1 ( 0 + TI, (415)
S= C 9 + C12 2 + Ts/lkl + Kb/ik lk (415)
S C21 + C2 2 + T/2,k1 + Kb/2k 2k +T2, (416)
ik = ik yikl' (417)
where the C and K k are defined in Table 31.
ij,k b/ik
Substitution of Equation (417) into Equation (412) reduces
Equations (410) and (411) to the following forms
i = (iN + (F)k (418)
ik+l =
and
as
ki,k = (419)
Let ki,k+1 be given by the finite difference relation
i,k+l = ,k+ ,k)/At. (420)
Then Equation (418) becomes
sk+1
+.i ^(Fi)kk] At
i, k+l = i,k k\~o /
Substituting Equations (414) (417) into Equations (412), (419),
and (421), and solving for ci,k+l
equations
p111
1+1 1
 2k+1 L k 11 2
12 1
12 2
yields the following set of six
*
21 1 22 1 0
2
1 2k
21 2 22 2 k 2 k
1 11 C12 1 s/1 Kb/1 1
+ + + At, (422)
2 k 21 22k 2 s/2] k1 b/2 N2
= + At ,
2k+ 2 k 2 k
and
1
*1 11 12 k1
? k+1 L 2k+l 2 k+l
(423)
(424)
(421)
The initial values of the X's are obtained by solving Equation (419),
using the initial values for the p's and t's. Hence
I 1 11 1 2 I1
(425)
X2 121 1220 20
The matrix Equations (422) (424) require one less numerical step of
integration as pointed out in [30]. It is also pointed out in refer
ence [30] that the X's are the moment, usually called pi's, and that
Hamilton's canonical equations are obtained directly without formerly
deriving the generalized moment. These equations may now be marched
out with time to obtain the solution to the dynamical equations of
motion. They require one less integration step as opposed to solving
Equations (48) and (49) numerically. The matrix inversion is only
necessary once per integration and hence any error is only integrated
once as opposed to twice for Lagrange's equations.
The significance of this development is the system formulation
in terms of kinematic influence coefficients developed in Chapters II
and III, allowing complex mechanisms with multiple inputs to be analyzed
for their dynamic response by established numerical methods. This for
mulation provides a way to reduce complex, multiple input mechanisms to
coupled equivalent mass systems, yielding differential equations of
motion possessing variable coefficients. These coefficients are known
in terms of the mechanism geometry through the kinematic influence coef
ficients of velocity and acceleration.
CHAPTER V
SUNVMARY AND CONCLUSIONS
Complex multiple input linkage systems have been difficult to
analyze for their dynamic response because of their nonlinear geometric
character. This characteristic generates holonomic constraints asso
ciated with the generalized coordinates necessary in describing the
motion of the linkage system. Large numbers of generalized coordinates
(see Figure 51(a)) have been required to obtain the systems dynamical
equations of motion. Algebraic equations of constraint are required to
account for the generalized coordinates other than those associated with
independent system inputs. The result is a large number of coupled,
nonlinear, secondorder differential equations together with a set of
algebraic equations in terms of undetermined Lagrangian multipliers
which account for the geometrical constraints on the system. The algo
rithm required to integrate numerically and solve this set of equations
requires a dual iteration scheme, one to solve the differential equa
tions of motion and one to satisfy the geometric constraints on the
linkage system.
Methods such as those developed by Chace [11], Uicker [13],
and Carson and Trummel [14], utilizing relative coordinates and 4 X 4
matrix coordinate transformations, have been the only tools available
to solve the dynamic response question for these systems. The set of
N= 8
n =2
8 Generalized Coordinates
6 Algebraic Equations of Constraint
212
n= 2
(b)
Figure 51. Complex Multiple Input Mechanism and
Its Coupled Equivalent Mass System
secondorder differential equations resulting from their methods has
been sufficient, though unwieldy, timeconsuming, and subject to error,
to describe the dynamical behavior of linkage systems.
The goal of this dissertation has been threefold: to develop a
systematic method whereby linkage systems of highordercomplexity can
be constructed from Assur groups in terms of kinematic influence coef
ficients of velocity and acceleration of the basic system group; to
reduce these highly complex linkage systems to coupled, equivalent mass
systems acted upon by equivalent variable rate springs, variable coeffi
cient viscous dampers, and equivalent external forces and torques; and,
to determine the differential equations of motion for the coupled equiv
alent mass system in terms of the minimum number of generalized coordi
nates (i.e., the number of independent system inputs).
The construction of general linkage systems of higherorder
complexity, as discussed in Chapter II, is seen to be expressed in terms
of series and parallel link connections. The connection types are
defined by multiplication (series) and addition (parallel) of successive
velocity influence coefficients. The use of Assur groups to construct
mechanisms of higherordercomplexity from existing system groups, with
out modifying the mobility of the basic chains, allows the displacement
analysis to be performed by established procedures. This is the basis
for eliminating the holonomic constraints on the system, reducing the
number of generalized coordinates required to describe the motion from
N (number of system links) to n (number of independent system inputs).
Elimination of the holonomic constraints subsequently reduces
complex, multiple input linkage systems (see Figure 51(a)) to coupled,
equivalent mass systems (Figure 51(b); rcquirinr only n gnci1i:eJ
coordinates to describe its motion. This reduction tl1iminjt,: ther
need for relative coordinates and their matrix transformations required
by the existing methods. Rather, the coefficients of the resulting
differential equations of motion become known variables of the system's
independent input parameters expressed in terms of kinematic influence
coefficients.
Secondorder differential equations describing the dynamical
behavior of the equivalent mass system have been derived by the clas
sical Lagrangian method, while firstorder difference equations were
derived by the direct application of Hamilton's principle. This method
[18], yields a set of firstorder difference equations derived through
only one differentiation. These difference equations can be marched out
with time, requiring only one matrix inversion per integration step.
The equivalent mass system formulation developed here provides
a convenient and unique medium through which many problems concerning
the dynamical behavior of linkage systems can be studied and simulated.
The influence of elastic deformation of system links on the
dynamic response of the linkage system primary input can be studied,
based on a hinged beam model such as the one shown in Figure 52.
The effect of bearing deformation on the input dynamic response could
be investigated, based on the system models in Figure 53. The equiv
alent mass systems are shown below the system models.
The elimination of the holonomic constraints places complex
linkage systems in a convenient form for directly applying the prin
ciples of optimal control theory. Optimal open loop control laws may
C34
Figure 52. Linkage System with Elastic Coupler Link
`
N t
I I I
CM
I I
/
/\ ,
's
e
H
+4
U)
o
N
SI .
be determined which minimize specified performance indexes while satis
fying prescribed constraints on the control itself, the states, or both.
Figure 54 illustrates this concept as applied to complex linkage systems.
The problem could be formulated as follows: Determine the control u
which minimizes the variation of the velocity s from the velocity s1
associated with position s1 over the range s 1 s 5 s2, subject to the
..*
inequality constraint on the control, u u This in essence places an
upper bound on the jerk u,thus giving thirdorder control for the cam
surface designed to produce the required control u.
The preceding problems are not intended to be solved here.
Rather, they are provided to point out possible research areas which
can be pursued, utilizing the system formulation developed in this
dissertation.
(C
0m
APPENDIX
APPENDIX A
DIRECT DERIVATION OF FIRSTORDER DIFFERENCE
EQUATIONS FOR DYNAMICAL SYSTEMS
The derivation of the set of firstorder difference equations
used in Chapter IV to solve the time response of complex linkage systems
is presented in part from the paper [18] "Derivation of FirstOrder
Difference Equations for Dynamical Systems by Direct Application of
Hamilton's Principle" by Vance and Sitchin. The purpose for presenting
this derivation is for completeness and convenience to the reader of
this dissertation, since the method proves to be ideal for treating
complex multiple input linkage systems. The nomenclature in this der
ivation does not correspond directly with that in the main text. It is
therefore listed separately at the end of this appendix.
Derivation of Method
Hamilton's principle for nonconservative systems with k
degrees of freedom is
T T k
61 = 6 S Ldt + Y' ( Fiq.)dt = 0, (Al)
0 0 i=1
where the integrand of the second integral is the virtual
work of the nonconservative forces. This second integral is
zero for conservative systems. After partitioning the
interval 0 to T into N small increments AT, the two integrals
can be approximated by sums and the principle can be rewritten
as
6S1 + S2 = 0, (A2)
where the functions S and S2 are sums given by
N
S1= E n (qln'q2n..A ',qkn'4ln'2n'. qkn)Lt (A3)
n=1
N k
S2 = FinqinAt. (A4)
n=l i=l
Equation (A2) requires that the variation of the function
S1 equal the negative S2. Since the displacements and veloc
ities are to be related by some finitedifference expression,
they are not independent of each other. The problem lends
itself to the use of Lagrangian multipliers in order to achieve
an independent variation of coordinates. The equations of
constraint, defined by the previously mentioned finite
difference relationship, have the general form
Aq.
in
= i (A5)
in At
or
fin = Aqin int = 0, (A6)
where Aqin is any desired expression for the firstorder dif
ference of qin.
Taking the variation of the constraint function (A6)
gives
N "of. of" )
6fin = (ln 6q + 6i.)= 0. (A7)
j=1 1j ij1
Equation (A2) can now be written as
N k
S + infi + S2 = 0. (A8)
1 Zn f+ 2=
n=l i=l
Substitution of Equation (A7) into Equation (A8) and
rearrangement of terms gives
N k as N fa ]
N k SS N af..
++ 6q. = 0. (A9)
+' ij Eqj a in
n=l i=l n j=1 in
The kN Xi are chosen so that the kN bracketed expressions
ij
in the second double summation are all zero. This leaves the
6qin as independent variations. Equation (A9) can then be
satisfied by independently requiring the kN bracketed expres
sions in the first double summation to be zero. Thus the
motion of the dynamical system will be such that the following
3kN equations are satisfied (i = 1,2,...,k; n = 1,2,...,N):
as1 N 6fij
7 ij + =A F. At, (A10)
in j=l in in
S+ kij x 6 = O, (All)
in j=l in
f. = Aqin in. At = 0. (A12)
When the firstorder difference form Aq. is substituted
in
into Equation (A12), the summations in Equations (A10) and
(All) are reduced to only a few terms. For example, if
qin = qin qinl
3f.. 6f..
3 = 0, j n, n+ l; and J = 0, if j j n. (A13)
in in
As is characteristic of the Lagrangian multiplier method,
the convenience of treating dependent variables as if they
were independent has been gained at the expense of an added
set of unknowns, the X's. Unlike many applications of
Lagrangian multipliers, however, the \'s are not in general
constant. In fact the X's represent moment, ., and have
the status of independent coordinates [3] .
The set of Equations (A12) may be considered trivial
(although necessary) in the sense that they are simply the
equations of constraint between the velocities and displace
ments.
The X's will be constant only in the case of ignorable
coordinates.
Goldstein, H., Classical Mechanics, AddisonWesley Publishing
Co., Atlanta, 1965, p. 227.
The derivation just presented is not tied to any particular
initedifference form. A determination of the best form to
use will depend on the application.
The symbols are defined as
6 = "variation of"
A = finite increment
S= dot appearing directly above a variable designates
derivative with respect to time
X = undetermined Lagrangian multiplier constant
i = the corresponding generalized coordinate
n = the corresponding time interval
j = the corresponding time interval
t = time
F. = generalized nonpotential forces
in
APPENDIX B
NUMERICAL SOLUTION TO A TWO DEGREES
OF FREEDOM EXAMPLE
The purpose of this appendix is to illustrate the actual
responses obtained from the dynamical equations of motion derived
for the fivebar mechanism in Chapter IV (see Figure (Bl(a))).
The procedure for obtaining the variable coefficients to
Equations (422) (425), describing the dynamical behavior of the
system in Figure Bl(a), is described below. The necessary coefficients
*
are K C. Tik I. k and P
are K/ik' Cij,k' i,k ij,k' and ij r,k.
(1) For the given initial position of the mechanism defined
by p1 and (p2' calculate the kinematic position informa
tion pertaining to the other links $3' ,4 etc., by
using the equations in Figure B2.
(2) Substitute this position information into the appropriate
blocks of Table 21 to determine the velocity and acceller
ation influence coefficients igi and hij. For example,
3 l,k+l 3 l,kl
3gi 2p (Bl)
(3) Once the 1g.'s and Lhij's are determined, substitute them
into the appropriate blocks of Tables 31 and 32. This
furnishes the following expressions for the coefficients
to the differential equations:
2 2
IEgikI Igik
K/ik = a gik K ik12 ik (B2)
N
S(k C ) (B3)
ij,k = k'' 9
Tik = (gik)T (B4)
I and P are determined from the expressions in Table 32,
ij,k ij r,k
blocks 1.1 and 1.2. The righthand side of Equations (422) (425) are
now completely known, allowing the k+1st 's, p's and p's, to be deter
mined. This procedure is repeated until the equations of motion are
integrated over a predetermined time interval. Equations (422) 
(425) represent the equations of motion for the coupled equivalent
mass system shown in Figure (Bl(b)).
The solutions to the dynamical equations of motion are shown
in Figure (B3) and (B4). Polar plots of c. vs p1 and c2 vs c2 are
shown for the parameters listed in Table Bl.
For added clarity, Figure (B5) illustrates the equivalent
inertias as functions of time. The equivalent inertias show no cyclic
phenomenon due to the noncyclic character of the input links. However,
the inertia coupling term, I12, illustrates the coupling between the
*
positive inertia terms Ill and I22.
The influence coefficients were calculated by finite differences.
The total problem was programmed on the IBM 36065. The program con
sisted of 270 cards and required 0.38 minutes to execute for an integra
tion step size of At = .000025 second. This step size can be increased
considerably without affecting the solution accuracy, thus decreasing
the execution time by that factor.
B
K 34
4
C34
(a)
(b)
Figure B1. Two Degrees of Freedom FiveBar Example
34
 k34

= +
 2/3/4,OS34
/324 (05 COSc ACOS Sr2COSS2+ (CSIN 5+ NSIN(+ 1SININ0
3 04 34
C34= C os 2 2 4 4
2e3 ,^
C<3 SIN 12SINO5+
C4 SIN I_ r5SIN0B +
PSINO, 
134
, SINN, 
34
,4SIN 2 .+ 4SIN03
S 1E .,r 1 .1034
SIN2 +SIN SIN
SSINS 1 34
Figure B2. Kinematic Position Equations
O/
O
\~ ~ v \ oTOs /
cs
14
.8"
0
'4
1
bo
U,
U,
* U,
'4
0
C.
U,
'4
C3
w
0
00
Ijp
002
(T)4
o o
o .9
4
02
02
No 02
w 4
01
02
12
'I
//i
TABLE B1
FIVEBAR PARAMETERS
Length (inches)
Weight (lbs.)
Moment of inertia
(in. lb. sec.2)
1
4.0
2.97
.0488
34 =
C34=
T =
2 =
2
2.0
1.48
1.14
50.0 in. lb./rad.
5.0 in. lb. sec./rad.
.05 in. lb.
.02 in. lb.
(c1)o
(Yp2)o
(2 o)
(9 o
= 00
= 250 rad./sec.
= 900
= 0
Solution to differential equations for known forcing
functions, constant T1 and T2, are shown on Figures
(B3) and (B4).
"3
8.0
5.52
.1385
4
12.0
4.16
.4062
5
11.18
 0 
 
06 4
*a
I F
Im
EC3
cr
*
w coo o
w
i 3i
3N
I
*I I I I
*
*
I
**
* r, l
*s( tR
*'~l3)~ 358 H~l~~N N~nn3
BIBLIOGRAPHY
1. Benedict, C.E., "Dynamic Response Analysis of Real Mechanical
Systems Using Kinematic Influence Coefficients," Master's
Thesis, University of Florida, December, 1969.
2. Modrey, J., "Analysis of Complex Kinematic Chains with Influence
Coefficients," Journal of Applied Mechanics, Vol. 26, Transactions
of the ASME, Vol. 81, June, 1957, pp. 184188.
3. Hain, K., Applied Kinematics, McGrawHill Book Company, Inc.,
New York, 1967, pp. 5356.
4. Meyer zur Capellen, W., ". . Harmonic Analysis of Periodic
Mechanisms' Proceedings of the International Conference for
Teachers of Mechanisms, Shoe String Press, New Haven, 1961,
pp. 171185.
5. Denavit, J., and S. Hasson, "On the Harmonic Analysis of the Four
Bar Linkage," Proceedings of the International Conference for
Teachers of Mechanisms, Shoe String Press, New Haven, 1961, pp. 171
185.
6. Flory, J. F., and J. C. Wolford, "Harmonic Analysis of Kinematic
Linkages," ASME Mechanisms Conference, Paper No. 64Mech38,
October, 1964.
7. Markus, L., and J. Tomas, "Harmonic Analysis of Planar Mechanisms
Kinematics," Journal of Mechanisms, Vol. 5, No. 4A, 1971, pp. 171
185.
8. Bogdan, R. C., and T. V. Huncher, "General Systematization and
Unified Calculation of Five and FourBar Plane Basic Mechanisms,"
ASME Mechanisms Conference, Paper No. 66Mechll, October, 1966.
9. Bogdan, R. C., D. Larionescu, and I. Carutasu, "Complex Harmonic
Analysis of Plane Mechanisms, Programming on Digital Computers
and Experimental Examples," ASME Mechanisms Conference,
Paper No. 68Mech62, October, 1968.
10. Crossley, F. R. E., and N. Seshachar, "Analysis of the Displace
ment of Planar Assur Groups of Computer," Transactions of the
International Federation of Theory of Mechanisms and Machines,
September, 1971.
11. Chace, M. A., "Analysis of the TimeDependence of M.ultiFreedom
Mechanical Systems in Relative Coordinates," ASME lMechlanisms
Conference, Paper No. 66Mech23, October, 1966.
12. Smith, D. A., "Reaction Force Analysis in Generalized Machine
Systems," Ph.D. Dissertation, University of Michigan, 1971.
13. Quicker, J. J., Jr., "Dynamic Behavior of Spatial Linkages:
Part 1 Exact Equations of Motion; Part 2 Small Oscillations
About Equilibrium," Journal of Engineering for Industry, February,
1969, pp. 251265.
14. Carson, W. L., and J. M. Trummel, "Time Response of Lower Pair
Spatial Mechanisms Subjected to General Forces," ASME Mechanisms
Conference, Paper No. 68Mech57, October, 1968.
15. Wittenbauer, F., Graphische Dynamik, SpringerVerlag, Berlin, 1923.
16. Federhofer, K., Kinetastatik flachenlaufiger Systeme, S.B. Akad.
Wiss. Wien, Math.Naturwiss. Kl., Abt. IIa. Jg. 139, 1930.
17. Beyer, R., KinematischGetriebeanalytisches Practikum, Springer
Verlag, Berlin, 1960.
18. Vance, J. M., and A. Sitchin, "Derivation of FirstOrder Differ
ence Equations for Dynamical Systems by Direct Application of
Hamilton's Principle," Journal of Applied Mechanics, Paper No.
70APMPP, 'June, 1970.
19. Pelecudi, CHR., "Kinematic Multipoles with Rigid Links," Rev.
Roum. Sci. Techn. Mec. Appl., Tome 13, No. 5, pp. 9971013,
Bucarest, 1968.
20. Pelecudi, CHR., "Interpretation of the Dyad as Kinematic Dipole
and Quadripole," Rev. Roum. Sci. Techn. Mec. Appl., Tome 13,
No. 6, pp. 12251237, Bucarest, 1968.
21. Benedict, C. E., and D. Tesar, "Analysis of a Mechanical System
Using Kinematic Influence Coefficients," Proceedings of the
Applied Mechanisms Conference, Paper No. 37, July, 1969.
22. Benedict, C. E., and D. Tesar, "Optimal Torque Balance for a
Complex Stamping and Indexing Mechanism," ASME Mechanisms Con
ference, Paper No. 70Mech82, November, 1970.
23. Benedict, C. E., and D. Tesar, "Dynamic Response Analysis of
QuasiRigid Mechanical Systems Using Kinematic Influence Coef
ficients," accepted for publication in 1971 in the Journal of
Mechanisms.
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System Containing a Coulomb Friction Force," Transactions of the
International Federation of Theory of Mechanisms and Machines,
September, 1971.
25. Benedict, C. E., G. K. Matthew, and D. Tesar, "Torque Balancing of
Machines by SubUnit Cam Systems," Proceedings of the Applied
Mechanisms Conference, Paper No. 15, October, 1971.
26. Assur, L. V., "Issledovanie Ploskikh Sterzhnevykh Mekhanizmov S
Nizshimi Parami," Izdat. Akad. Nauk SSSR, Moscow, 1952.
27. Curtis, M. F., and J. Tomas, "Analytical Solution of Planar Binary
Groups," Proceedings of the Applied Mechanisms Conference,
Paper No. 30, October, 1971.
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Dover Publications, Inc., New York, 1968, pp. 883884.
29. Quinn, B. E., "Energy Method for Determining Synamic Character
istics of Mechanisms," Journal of Applied Mechanics, Vol. 16,
September, 1949, pp. 283288.
30. Sitchin, A., "Problems in Attitude Stability of DualSpin Space
craft," Ph.D. Dissertation, University of Florida, 1970.
BIOGRAPHICAL SKETCH
Charles Edward Benedict was born March 21, 1939, at Tallahassee,
Florida. He graduated from Leon High School in June, 1957. He began
studies at Florida State University in 1958, and graduated with the
degree Bachelor of Science with a major in Mathematics in December,
1963. After working for three and onehalf years for Florida Gas Trans
mission Company, he began studies at the University of Florida in the
field of Mechanical Engineering in April, 1967. He graduated with the
degree Bachelor of Science in Engineering with high honors in August,
1968. He received an Engineering College Fellowship and continued his
advanced education, receiving a Master of Science in Engineering from
the University of Florida in December, 1969. He was awarded a NDEA
Title IV Fellowship and continued his studies toward a degree of Doctor
of Philosophy. This dissertation completes these studies.
Charles Edward Benedict is married to the former Patricia Ann
Casey and has one daughter, age seven. He is a member of Kappa Alpha
Order, Tau Beta Pi, Pi Tau Sigma, Phi Kappa Phi, Florida Engineering
Society, and the American Society of Mechanical Engineers.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
D. Tesar, Chairman
Professor of Mechanical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
C. C. Oliver, CoChairman
Professor of Mechanical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
W. H. Bykin Jr.
Assistant Professor of
Engineering Science & Mechanics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
T. E. Bullock
Associate Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Associ te Professor of MechLnical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
,K1l. Vance
ASsistant Professor of Mechanical Engineering
This dissertation was submitted to the Dean of the College of Engineer
ing and to the Graduate Council, and was accepted as partial fulfillment
of the requirements for the degree of Doctor of Philosophy.
December, 1971
Dea College of Engineering
Dean, Graduate School
