Dynamic response analysis of complex mechanisms with multiple inputs

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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 co-chairman 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 Input-Output 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 FIRST-ORDER 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

2-1 Influence Coefficients . . . . . . . ... 27


3-1 Equivalent System Forces and Torques . . . . .. 37


3-2 Equivalent System Formulation . . . . . . .. 50


B-1 Five-Bar Parameters . . . . . . . ... 80















LIST OF FIGURES

Figure Page

2-1 Binary Groups . . . . . . . . . . 7

2-2 Assur Groups . . . . . . . . ... . . 9

2-3 General System Point . . . . . . . . .. 10

2-4 Point Paths of Assur Groups . . . . . . ... .14

2-5 Seven-Link System Group . . . . . . . ... .20

2-6 Sliding Pair Constraint . . . . . . . ... .28

3-1 Equivalent System Elements . . . . . . ... 31

3-2 Complex Multiple Input Linkage System . . . . .. 39

3-3 Differential Gear System . . . . . . . .. 42

3-4 Angular Relationship to Input #1 . . . . . .. 43

3-5 Angular Relationship to Input #2 . . . . . . 43

3-6 Complex Two Input Linkage System . . . . . .. .49

4-1 Two Input System . . . . . . . . ... . 54

5-1 Complex Multiple Input Mechanism and Its Coupled

Equivalent Mass System . . . . . . . ... 61

5-2 Linkage System with Elastic Coupler Link . . . .. .64

5-3 Linkage Models with Deformable Bearings . . . .. .65

5-4 Optimal Open Loop Control Example . . . . . .. 67

B-l Two Degrees of Freedom Five-Bar Example . . . ... .76

B-2 Kinematic Position Equations . . . . . . ... .77

B-3 Polar Phase Plane: 1 vs pl .. . . . ...... 78

B-4 Polar Phase Plane: p2 vs P 2 . . .... 79

B-5 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 x-coordinate of system input

X Denotes x-coordinate of system output

y Denotes y-coordinate of system input

Y Denotes y-coordinate 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

Right-hand Cartesian coordinate system

Angles measured positive ccw from positive x-axis













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
Co-Chairman: 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,

second-order differential equations, requiring additional algebraic

equations of constraint, occur as a result of these constraints.

Double iteration algorithms, which are both time-consuming 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 four-link 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 higher-order-complexity

(one which cannot be analyzed as a system of four-link mechanisms con-

nected in series) could be determined, using velocity and acceleration

influence coefficients obtained through a "zero-relax" 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 input-output position information of six- and eight-bar

mechanisms which do not contain a basic four-bar at ground but do contain

an internal four-link 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 four-link 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 time-consuming.










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 [11-14] 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

X-method 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, rigid-link 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 four-link mechanisms and cams

in [1,21-25]. 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 2-1 illustrates its use in constructing the

four-, five-, and six-bar 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 five-bar mechan-

isms and the dyad attached to the intermediate input of the six-bar

mechanism form dipoles while the dyad in Figure 2-1(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 2-1. Binary Groups











The next Assur group of higher complexity is the tetrad (see

Figure 2-2(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 2-2(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 2-3(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 2-2. 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 2-3. 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 2-4(a))


dS
ex.
x xi =-p. i'

(2-1)
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


(2-2)


This gives


(2-3)


Sf
gy -
-x f
gx TSy











The expressions for me and g are given by


S= (2-4)
e


and


-2 .-2
ge = +y (2-5)


Since g is known, then
e


o
g = (2-6)



and

gy = me g (2-7)



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 (2-8)
h = a for 9. = 1.0, (p = 0.
y y 1 i

Define T as follows (see Figure 2-3(b))


S= tan- { } (2-9)
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


(2-10)


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 2-4(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 2-4(a) and (b)) as


ge.
gx
S = 1

i


g = m g
Yi ei xi


(2-13)


h = h cos (0 ) h = h sin () .
x e i Yi ei


and


(2-11)


(2-12)














(x, Y2)


(x ,yl)


(a)


(x2,y2)


(XI ,y1)


(x3 ,Y)


(x,,y,)


(x3, y3)


( b)


Figure 2-4. 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 2-4(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,


(2-14)


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 (2-14) may

the vector loop equation for the


(2-15)


fx(1, x2,X,ylY2, Y) = 0,




fy(X1'2,X,Yy12,Y) = 0.


(2-16)


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


(2-17)


VF X Y










where
2
2 ax ax
VX x. Vy. ,
S1 1 1 1 1
(2-18)
2
r f Ay ay
vV =+V i + T yi VV
1=1 1 1 1 1

The partial derivatives in Equations (2-18) 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

(2-19)
A 3y
Yigy =


Substituting these into Equations (2-18) gives
2
vx= x xg xi + i vy i
i= 1 1 1 1

2
V = v + Yy v } (2-20)
i= 1i1 1 1 y

Equations (2-20) 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 (2-22)
v
yi

where

ax x
xGx i x2


G --(2-23)




Ivxl xl

Lx2


Acceleration Influence Coefficients

Differentiating Equations (2-18) 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 (2-24)
1n 1 1 1 1

and





2 2
a2Y + axi +t aY '
+ 3-Vy Vyj a a (2-25)
y 3 y/ j=1 yi Y I







Equations (2-24) and (2-25) 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'^ l-n


(2-26)


(2-27)


(2-28)


(2-29)


(2-30)


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 input-output 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 input-output relationships developed.



Total Input-Output 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 input-output transformation is now carried out.


Velocity Influence Coefficients

Consider the system group in Figure 2-5. The expression for

the velocity components of point F with respect to points E1 and E2 are




G G 1 .2xI
----- ---- --- (2-31)

_Y Yx YG x_
i : iy -~












F
I (x,y)


E2(x2,y2)


Figure 2-5. Seven-Link 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, (2-32)

LIV .I- ..Lyi,59q

where

8x1 bx1
ax 6


[Xi q] x2= x2



and







Substituting the right-hand side of Equation (2-32) into Equation (2-31)

yields



Vy xi Yi Li c q
= ----;---- --- Jwl (2-33)

[ Y x Y yi i q_

which reduces to




--- ||, q = 1,2. (2-34)

v- YGq










Notu that when q=l, Equation (2-34) becomes


Vx = Xg '1

(2-35)
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 2-5) in terms

of the intermediate inputs are









r -
3X]i] ;x
= -------0----V - i














+ .- (2-36)
x O x


Yx x j 0 HL






2 ------------ IV I +2-36 I

Si I i 0 7


The first and third terms of Equation (2-36) are recognized as real

quadratic forms while the second is real bilinear.










Substitution of Equations (2-26), (2-27), and (2-32) into


Equation (2-36) 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 j-dy ]
YRX. X


(2-37)









The complexity of Equation (2-37) 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, (2-38)

Y q Yr q L


where 11wll is a compound diagonal matrix given by


u 0
= (2-39)
0 jo|

and |)l is the transpose of the column vector IwI.

The expressions (2-34) and (2-38) 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, (2-40)
v G
1 1 (q



V YG
m m q









t
a X 1 q r X G






~m / Xm qr Xm
= ml ----|--- + -- (2-41)
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 (2-40) and

(2-41) 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,k-l (2-42)
agik = 2Ap.

2

Lhijk =) '


i,k+l i,k+l i,k-1 i,k-1
Iij k+1 j,k-1 j1,k+1 + z j,k-1 (2-43)
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)
(2-44)

hij ,k i(i j kjk


Table 2-1 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 (2-41)). 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 2-6.

The influence coefficients in Table 2-1 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 2-6. 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)


(2-45)













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 3-1(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 (3-1)














T41.


(a)


K~.


(b)


C ..


(c)


Figure 3-1. 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 (3-2)
T/ik i- 1


Similarly, for an external system force Fek


T/ik (egik Fek (3-3)


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 (3-4)


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 (3-2) and (3-3) 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 3-2(b)). This

yields the following relation


2 (3-5)
i2 /ik (ik = (ik (3-5)

Solving for K /ik and taking the limit as p ik 0 yields



(At 2k (3-6)
K/ik = K lim = gk3-6
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 quasi-cyclic 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 determined-from its past history

in the form


a/ik /i,k-1 + K/ik ik (3-7)

where

2 2
K* gik-i+ gik (3-8)
K/ik ik 2 3-8








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


(3-9)


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 (3-2). The external torque is given by

Tk = (K f ( .k)= *k (3-10)

where

I,, = free length of spring measured from the reference axis,

A k = deformation of system spring K from free length.

Substituting Equation (3-10) into (3-2) gives


T/ik = K (gik) ik (3-11)

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


(3-12)


, 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 3-1(c)). The torque generated by the

dashpot is proportional to the angular velocity of the link attached to

the dashpot, or






Using Equation (3-2), TVk is transferred to the i system input as



TA/ik (ik) C (gj CW j (3-14)
j=l k

The total torque acting on the ith input link resulting from all system

dashpots is

n

Td/ik = C (3-15)

where

N
C j,k= gi gJ)k (3-16)


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 (3-17)

Cnl *
T C C ** C
d/n nl n2 nn k
-'k 1 k











Table 3-1 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 2-1, i.e.,


brick = ngik mgik (3-18)

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 two-force members attached to links m and n at points B and
m

B n. The relative influence coefficient rk is defined in Table 2-1,
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 (3-19)
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 3-2.

Consider the kinetic energy of a complex, multiple input system.


I2 2 + M 2 } (3-20)
(KE)k =k 2 {2 (3-20)


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 (3-20) in terms of the system input velocities yields

N n 2 n 2
(KE)k = ikk + (eik)Wik
=1 i=l 1=1


(3-21)






































n=3
N=13


Figure 3-2. Complex Multiple Input Linkage System









Rewriting Equation (3-21) in a more compact form gives

n n N
(KE)k Z [ )( ) + M i)( j) wj
i=l j=1 2=i

(3-22)

where it is noted that

1 2=i,j; i=j
( gXi)( g ) = (3-23)
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 (3-24)

(nxn)
k



where
N
ij,k = i i(gj) 2+ ei e j(3-25)
&=1k


Matrix Equation (3-24) 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 (3-25).











Two Degrees of Freedom Example

For the purpose of illustration, consider the differential

gear (Figure 3-3), 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
(3-26)
X3 d1
32 = x2 dl+d2


and

3 = 3 1 v1 + 3 2 v2 (3-27)


The angular velocity influence coefficients of 4 are given

by (see Figures 3-4 and 3-5)

a
4 1
4g 1 dl+d2
(3-28)
4 1
4 2 dx2 + dl+d2


and

4 = 4 1 v1 + 42 v2 (3-29)


The kinetic energy of the two input differential gear is


1 2 -2 2 2
KE = 2 Iv + M2v2 + (M3 +M14)v3+ I4w4 (3-30)
1 2v2 ~3 3 44










CL x
0
4-t


C _
x-


Q.

X




















dx, = (d,+ d2)(-d34)

Figure 3-4. Angular Relationship to Input #1


dx2= (d,+ d2)d 4

Figure 3-5. Angular Relationship to Input #2










Substituting Equations (3-27) and (3-29) into (3-30) gives

1 2 2+ 2
KE = (MV1 + M22 + (M3+M4)(3gv1 3g2v2)


+ 14(4g1 1 + 4g2v2) (3-31)

where

v3 = v4

Collecting like terms and rewriting Equation (3-31) 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 (3-32)

or
1 2 2 *
KE = IlV + 1222 + 21121v2} (3-33)

where
2 -2
I = M11 4 4 1 + (M3 + M4)(31)




2 2
12 = 4(4g)(4 g2) + (M3 + M) (3gl)(3g2) (3-34)


2 2 2 + 4(42)2 + (M3 +M432

For the system in Figure 3-3, Equation (3-24) becomes



11 12 v1

KE=[V1 2] (3-35)

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 3-2), the I 's

are not constant but a function of the system input parameters. Note

that whenever v2=0, then

1 2
KE = vI (3-36)


and whenever v1 = O, then
1"2
KE = 22 v2 (3-37)


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) (3-38)

where


KE = W I*j Il i,j = 1,2,...,n (3-24)


Differentiating Equation (3-24) with respect to time gives


P i- j* + JI + i I 1. (3-39)
2 '' dtI ij


The second product on the right side of Equation (3-39) 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 (3-40)

Differentiating Equation (3-40) with respect to time gives


d n N
dij = [(Zg) ( r) + ,
d k r=1 1 I =g


+WNIkt 9 (3-41)
+ i(ei jr+ (oj)( ei k rk', (-41

where


(ii)(hjr) = pr (3-42)









Equation (3-41) can be written as the product of a row and column

vector as


0W


n


(3-43)


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 (3-43) into Equation (3-39) 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.


(3-44)







(3-45)


nnxn)

II
I * I
-- r--
0




--1 r---
-I

S I k
k


(3-46)



ces are 1 Xn











Two Input Example

Consider the system in Figure 3-6 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 (3-47) 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


(3-48)


W k
2- k


(3-49)


(3-50)


Table 3-2 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.


(3-47)























































Figure 3-6. Complex Two Input Linkage System





50





n < Q -I
0P O ,. .,
>
Sxrn Z: Fr
F- C










0 0
c-I 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 [11-14] 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 second-order 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 first-order 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 second-order differential equations is derived

from Lagrange's equations as


d (iL aL* *
dt Ti ( = 1,2...n) (4-1)

where

ci generalized coordinates,

pi generalized velocities,

L Lagrangian of the coupled equivalent mass system,

T. equivalent nonpotential forces and torques.
1

The n second-order differential equations resulting from

Equation (4-1) are nonlinear, coupled, nonhomogeneous, and contain

variable coefficients. It is necessary to reduce this set of equations

to 2n first-order 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 five-bar mechanism in Figure 4-1 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), (4-2)

and the equivalent nonconservative torques are


T = -C34 brl [brl 1 + br2 2] + Tl'

(4-3)
T2 =-34 br 2 brl br2 + T2

where
2 2
KE = j L i j* (4-4)
i=l j=1l

1 2
pE -- (34(f ) (4-5)
PE 2 34 34f 34 (4


Substituting Equations (4-3), (4-4), and (4-5) into

Equation (4-1) and performing the designated differentiation with

respect to the generalized coordinates and velocities yields





























C34


Figure 4-1. Two Input System








dt { 1 l + 112 + 34 (34f 34)(brl)

=T1 34 [br)l + (brl) br2 (4-6)


d -
d I12 91 + 22 ~2 + K34 (P34f 34)(br2)

2 -34 (b br2 1 (b 2] (4-7)

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
(4-8)
*2 .2 * _
21P 1 + (21P2 + 22P112 22 2 2 12 1 122 2-

+ K34(C34f P34)(br2) = 3 (bl) (br2 + br2
(4-9)

Equations (4-8) and (4-9) are nonlinear in the velocities and

are unwieldy in their present form. In addition, they should be reduced

to four first-order equations before they are numerically integrated.

The ..P terms as defined by Equations (3-44) and (3-45) 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 first-order

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 (4-10)
ik j=l ik


as N 8f..
+ .. = 0 (4-11)
2ik j=1J aik


fik= -ik ik At = 0, (i= 1,2,...,n; k= 1,2,...,N), (4-12)

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. (4-13)
k=1

The 3nN first-order difference equations resulting from Equa-

tions (4-10) (4-12) 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 4-1. 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 (4-14)
S1 2 + 2112 2 22 2 (4-14)

1 ( 0 + TI, (4-15)

S=- C 9 + C12 2 + Ts/lk-l + Kb/ik lk (4-15)


S- C21 + C2 2 + T/2,k-1 + Kb/2k 2k +T2, (4-16)


ik = ik yik-l' (4-17)

where the C and K k are defined in Table 3-1.
ij,k b/ik
Substitution of Equation (4-17) into Equation (4-12) reduces

Equations (4-10) and (4-11) to the following forms


i = (iN + (F)k (4-18)
ik+l =

and

as
ki,k = (4-19)



Let ki,k+1 be given by the finite difference relation


i,k+l = ,k+ ,k)/At. (4-20)









Then Equation (4-18) becomes

sk+1
+.i ^(Fi)kk] At
i, k+l = i,k k\~o /


Substituting Equations (4-14) (4-17) into Equations (4-12), (4-19),


and (4-21), 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, (4-22)

2 k 21 22k 2 s/2] k-1 b/2 N2


= + At ,
2k+ 2 k 2 k

and

-1

*1 11 12 k1


? k+1 L- -2k+l 2- k+l


(4-23)


(4-24)


(4-21)










The initial values of the X's are obtained by solving Equation (4-19),

using the initial values for the p's and t's. Hence



I 1 11 1 2 I1
(4-25)

X2 121 1220 -20


The matrix Equations (4-22) (4-24) 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 (4-8) and (4-9) 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 5-1(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, second-order 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 5-1. Complex Multiple Input Mechanism and
Its Coupled Equivalent Mass System











second-order differential equations resulting from their methods has

been sufficient, though unwieldy, time-consuming, 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 high-order-complexity 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 higher-order-

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 higher-order-complexity 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 5-1(a)) to coupled,











equivalent mass systems (Figure 5-1(b); rcquirinr only n g-nci1i-: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.

Second-order differential equations describing the dynamical

behavior of the equivalent mass system have been derived by the clas-

sical Lagrangian method, while first-order difference equations were

derived by the direct application of Hamilton's principle. This method

[18], yields a set of first-order 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 5-2.

The effect of bearing deformation on the input dynamic response could

be investigated, based on the system models in Figure 5-3. 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 5-2. 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 5-4 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 third-order 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 FIRST-ORDER DIFFERENCE
EQUATIONS FOR DYNAMICAL SYSTEMS



The derivation of the set of first-order 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 First-Order

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, (A-l)
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, (A-2)

where the functions S and S2 are sums given by

N
S1= E n (qln'q2n..A ',qkn'4ln'2n'. qkn)Lt (A-3)
n=1

N k
S2 = FinqinAt. (A-4)
n=l i=l

Equation (A-2) requires that the variation of the function

S1 equal the negative S2. Since the displacements and veloc-

ities are to be related by some finite-difference 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 (A-5)
in At

or

fin = Aqin int = 0, (A-6)

where Aqin is any desired expression for the first-order dif-

ference of qin.










Taking the variation of the constraint function (A-6)

gives

N "of. of" )
6fin = (ln 6q + 6i.)= 0. (A-7)
j=1 1j ij1

Equation (A-2) can now be written as

N k
S + infi + S2 = 0. (A-8)
1 Zn f+ 2=
n=l i=l

Substitution of Equation (A-7) into Equation (A-8) and

rearrangement of terms gives

N k as N fa ]




N k SS N af..
++ 6q. = 0. (A-9)
+-' 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 (A-9) 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, (A-10)
in j=l in in


S+ kij x 6 = O, (A-ll)
in j=l in


f. = Aqin in. At = 0. (A-12)


When the first-order difference form Aq. is substituted
in

into Equation (A-12), the summations in Equations (A-10) and

(A-ll) are reduced to only a few terms. For example, if

qin = qin qin-l

3f.. 6f..
3- = 0, j n, n+ l; and J = 0, if j j n. (A-13)
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 (A-12) 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, Addison-Wesley Publishing
Co., Atlanta, 1965, p. 227.











The derivation just presented is not tied to any particular

inite-difference 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 five-bar mechanism in Chapter IV (see Figure (B-l(a))).

The procedure for obtaining the variable coefficients to

Equations (4-22) (4-25), describing the dynamical behavior of the

system in Figure B-l(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 B-2.

(2) Substitute this position information into the appropriate

blocks of Table 2-1 to determine the velocity and acceller-

ation influence coefficients igi and hij. For example,


3 l,k+l 3 l,k-l
3gi 2p (B-l)


(3) Once the 1g.'s and Lhij's are determined, substitute them

into the appropriate blocks of Tables 3-1 and 3-2. This

furnishes the following expressions for the coefficients

to the differential equations:









2 2
IEgik-I Igik
K/ik = a gik K ik-12 ik (B-2)


N
S(k C ) (B-3)
ij,k = k'-' 9


Tik = (gik)T (B-4)


I and P are determined from the expressions in Table 3-2,
ij,k ij r,k
blocks 1.1 and 1.2. The right-hand side of Equations (4-22) (4-25) 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 (4-22) -

(4-25) represent the equations of motion for the coupled equivalent

mass system shown in Figure (B-l(b)).

The solutions to the dynamical equations of motion are shown

in Figure (B-3) and (B-4). Polar plots of c. vs p1 and c2 vs c2 are

shown for the parameters listed in Table B-l.

For added clarity, Figure (B-5) 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 360-65. 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 B-1. Two Degrees of Freedom Five-Bar Example










34


- k34


-


=- +


- 2/3/4,OS34


/324 (05 COSc -ACOS S-r-2COSS2+ (CSIN 5+ NSIN(+- 1SININ0

3 -04 34
C34= C os 2- 2- 4 -4
2-e3 ,^


C<3 SIN -12-SINO5+


C4 SIN I_ r5SIN0B +


PSINO, -
134

, SINN, -
-34


,4SIN 2 .+ 4SIN03

S 1E .,r 1 .1034
--SIN-2- +SIN SIN
SSINS 1 34


Figure B-2. 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 B-1

FIVE-BAR 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
(B-3) and (B-4).


"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














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29. Quinn, B. E., "Energy Method for Determining Synamic Character-
istics of Mechanisms," Journal of Applied Mechanics, Vol. 16,
September, 1949, pp. 283-288.

30. Sitchin, A., "Problems in Attitude Stability of Dual-Spin 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 one-half 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, Co-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.





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

PAGE 1

DYNAMIC RESPONSE ANALYSIS OF COMPLEX MECHANISMS 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 1971

PAGE 2

UNIVERSITY OF FLORIDA 3 1262 08552 5698

PAGE 3

To Patricia

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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, particularly in the preparation of this dissertation. Through Dr. Tesar' s efforts and support the author was afforded an opportunity to present a major part of this research to the leading researchers in this area at the XSF Advanced Training Workshop in Mechanisms held at Oklahoma State University. Sincere appreciation is expressed to Dr. Calvin C. Oliver for his assistance as co-chairman of the supervisory committee. Appreciation is also expressed to the other committee members for their guidance and support: Dr. W. II. Boykin, Jr. , Dr. T. E. Bullock, Dr. J. Mahig, Dr. J. II. 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.

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii:L LIST OF TABLES vi LIST OF FIGURES vii NOMENCLATURE viii ABSTRACT .' xi CBAPTER I LITERATURE SURVEY 1 Dynamic State Analysis 1 Displacement Analysis 2 Dynamic Response Analysis 3 II KINEMATIC INFLUENCE COEFFICIENTS OF COMPLEX MECHANISMS CONSTRUCTED FROM ASSUR GROUPS 5 Kinematic Influence Coefficients of Primary Input Poles 8 Velocity Influence Coefficients 8 Acceleration Influence Coefficients 12 Kinematic Influence Coefficients of Intermediate Input Poles 15 Velocity Influence Coefficients 15 Acceleration Influence Coefficients 17 Total Input-Output Kinematic Influence Coefficients ... 19 Velocity Influence Coefficients 19 Acceleration Influence Coefficients 22 III EQUIVALENT SYSTEM FORMULATION IN TERMS OF KINEMATIC INFLUENCE COEFFICIENTS 30 Equivalent System Torques 30

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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 FIRST-ORDER 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

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LIST OF TABLES Table Pa S e 2-1 Influence Coefficients 27 3-1 Equivalent System Forces and Torques 37 3-2 Equivalent System Formulation 50 B-l Five-Bar Parameters 80

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LIST OF FIGURES Figure Page 2-1 Binary Groups 7 2-2 Assur Groups 9 2-3 General System Point 10 2-4 Point Paths of Assur Groups 14 2-5 Seven-Link System Group 20 2-6 Sliding Pair Constraint 28 3-1 Equivalent System Elements 31 3-2 Complex Multiple Input Linkage System 39 3-3 Differential G-ar System 42 3-4 Angular Relationship to Input #1 43 3-5 Angular Relationship to Input #2 43 3-6 Complex Two Input Linkage System 49 4-1 Two Input System 54 5-1 Complex Multiple Input Mechanism and Its Coupled Equivalent Mass System 61 5-2 Linkage System with Elastic Coupler Link 64 5-3 Linkage Models with Deformable Bearings 65 5-4 Optimal Open Loop Control Example 67 B-l Two Degrees of Freedom Five-Bar Example 76 B-2 Kinematic Position Equations 77 B-3 Polar Phase Plane: Cf> vs cp 78 B-4 Polar Phase Plane: cp vs cp 79 B-5 Equivalent Inertias vs Time 81 vii

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NOMENCLATURE B General linkage pin joint or points in links m,n m,n C„ Viscous damping coefficient of dashpot attached to link Z Z and ground , * C Equivalent viscous damping coefficient associated with the ij 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 * r e/i * th F Equivalent force acting on i system input due to a unit external force at point E .g Velocity influence coefficient of link I with respect to X 1 input link i -G. Row vector of velocity influence coefficients X i .h Acceleration influence coefficient of link Z with respect Z ij th th to the i and j input links „H . Square matrix of acceleration influence coefficients I ij i Denotes input link or generalized coordinate I . Effective moment of inertia of link Z taken about its center of gravity * I. . Equivalent moment of inertia term viii

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j Denotes input link counter k Corresponding time counter k Corresponding position counter K« Effective spring constant associated with a spring attached between link Z and ground K,.. Equivalent spring constant of K. with respect to system input i Z Denotes general system link L Lagrangian of the equivalent mass system m Total number of general system outputs M« Effective mass of link Z 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 -id equivalent inertia power coefficient with respect th to the r input q Denotes input link counter r Denotes input link counter S Summation representing the time integral of the Lagrangian S„ Summation representing the time integral of the virtual work of the nonconservative forces TGeneral system external torque acting on link Z * th T, .. Equivalent torque acting at i input link resulting from *,e,d,s/i unit torques on link Z, unit forces through system point E, unit velocities on equivalent viscous damper, and unit displacements on equivalent springs ix

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T Total equivalent torques acting on 1 system input link i v Linear velocity of general system point E e w Weight of link I acting through center of gravity x Denotes x-coordinate of system input X Denotes x-coordinate of system output y Denotes y-coordinate of system input Y Denotes y-coordinate of system output a. Angular acceleration of link i 6 Variation of some parameter A Finite increment T) Denotes angles X Undetermined Lagrangian multiplier cp. Angle of i system input link .th § Angle of * system output link 0). Angular velocity of link i J J Denotes column vector Denotes differentiation with respect to time Denotes transpose of matrix S Denotes partial differentiation Sign Convention Right-hand Cartesian coordinate system Angles measured positive ccw from positive x-axis

PAGE 12

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 Co-Chairman: 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, second-order differential equations, requiring additional algebraic equations of constraint, occur as a result of these constraints. Double iteration algorithms, which are both time-consuming and subject to error, are necessary to integrate numerically these differential equations of motion. In this dissertation the concepts of kinematic influence coefficients of complex, planar, rigid link mechanisms with multiple inputs are developed and utilized to eliminate the holonomic constraints associated with such systems. Kinematic influence coefficients associated with series and parallel linkage combinations are developed, based on the addition of Assur groups (dyads, tetrads and more complex groups) to the basic system group. xi

PAGE 13

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.

PAGE 14

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 ri]. 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 determine 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 four-link 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, diff erentiable functions of the input characteristics. In 1957 Modrey [2] developed a graphical method whereby the velocities

PAGE 15

and accelerations of the links of a system of higher-order-complexity (one which cannot be analyzed as a system of four-link mechanisms connected in series) could be determined, using velocity and acceleration influence coefficients obtained through a "zero-relax" procedure. In a discussion of this paper, T. P. Goodman showed the analytical equivalent 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 input-output position information of sixand eight-bar mechanisms which do not contain a basic four-bar at ground but do contain an internal four-link loop. This is accomplished by inverting the mechanism, 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

PAGE 16

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 approximations 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) constraints. 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 four-link 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 powerful 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 computation but is time-consuming.

PAGE 17

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 [11-14] 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 X-method in treating the holonomic auxiliary conditions, replacing the kinematical constraints by the forces necessary in maintaining them. The \' 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, rigid-link 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].

PAGE 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 (position, 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 four-link mechanisms and cams in [1,21-25]. 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 system 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^dLve. multipoles. A group may be overconstrained (f < 0) to form a mechanism with a degree of mobility which is less than the basic chain.

PAGE 19

(2) Mechanisms with a fixed link having a degree of freedom f £ 1, forming active, multipoles. Thus, one is free to construct mechanisms from basic chains whose degree of freedom remains invariant under the addition of aotcve. and paAdJ-Ve. multipoles. This points out the importance of the structural 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 inputoutput 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 available 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 2-1 illustrates its use in constructing the four-, five-, and six-bar mechanisms. Each binary group forms a structure when it is separated from its primary or intermediate input and has its outer joints fixed. The dyad in the fourand five-bar mechanisms and the dyad attached to the intermediate input of the six-bar mechanism form dipoles while the dyad in Figure 2-l(c) attached to the primary input forms a tripole, capable of transmitting information from the system input through the output pole to the dipole.

PAGE 20

B jg (a) (b) (c) Figure 2-1. Binary Groups

PAGE 21

The next Assur group of higher complexity is the tetrad (see Figure 2-2 (a)). When the outer joints A, , 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 2-2(b) and (c) . These and other groups are classified and analyzed for their displacements 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 configurations. 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 2-3(a)) associated with the outer joint of a binary group, can be written based on known position information. In general, f(x,y), g , h , m , cp „ , and cp , „ will be e e el jUI known as a function of some system input parameter such as cp . . It is

PAGE 22

7777 7777 (0 Figure 2-2. Assur Groups

PAGE 23

10 Figure 2-3. General System Point

PAGE 24

11 noted that g and h are defined in [1] as e e dS _ e 'e dcp. ' d 2 S r e 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 g , g , h , h must be determined. It is noted that (see Figure 2-4(a)) dS v = g cp = — — cp , x B x Y i dcp. Y i ' (2-1) dS ey • V y " g y 9 i " dcp. 9 i ' and d r^, si df df dt <*<*.*>} = d^ V x + d? V 3 This gives

PAGE 25

12 The expressions for m and g are given by m = — -£, (2-4) e — and = y? + g 2 . (2-5) y Since g is known, then e g e g v = , (2-6) and ST l = m g (2-7) y e x Acceleration Influence Coefficients Now consider h and h in terms of m , g and h . x y e e e It is noted that h = a* , x x — t h = a , for cp. = 1.0, Cp. = 0. y y i i Define 7] as follows (see Figure 2-3(b)) £ i (2-8) T] = tan" 1 { -£ } . (2-9) 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 e e sense as g , negative if directed in the opposite direction.

PAGE 26

13 Therefore, h = ii cos (7]) , ^ X 6 (2-10) h = h sin (T)) . y e Now consider a dyad as a dipole or binary group in the Assurian sense with points E and E having general point paths defined by implicit functions f (x ,y ) = and f_(x ,y ) = (see Figure 2-4(a)). Then, C l /l + m-i % *> J S„ = m g^ (2-11) 1 + m^ y 2 e 2 X 2 e„ and h = h cos (7] ) , h = h sin (T] ) , x, e, 1 y. e.. 1 h = h cos (71) , h = h sin (7) ) X 2 e 2 2 y 2 6 2 * (2-12) This can be generalized immediately to all classes of Assur groups (see Figures 2-4(a) and (b)) as g„ i i *' / i T~ ~y. e. x. 1 + m z i i (2-13) h = h cos (71) , h = h sin (7]) x e, i ye. i i i ii

PAGE 27

14 (x,,y,) (x 2 ,y 2 ) (a) (x-.y.) i"i (b) Figure 2-4. Point Paths of Assur Groups

PAGE 28

15 Kinematic Influence Coefficients of Intermediate Input Poles Let E and E be the inputs and F be the output of the binary group in Figure 2-4(a). Point F is an internal joint to links i and l+l. However, the following derivation holdi for output poles as general points in either of the links i or X+l. The point path of F can be expressed as an implicit function f(X,Y) = 0, (2-14) where X and Y are functions of x , y , x , and y . Relation (2-14) may 1 1 A A be expressed by the projections of the vector loop equation for the dyad, E.F + FE + E E = 0, (2-15) 1 £t £ \ f x ( W X ' y i' y 2' Y) = °' and (2-16) f y (X l'V X ' y i' y 2' Y) = °Velocity Influence Coefficients Since X,Y are functions of x y , x and y , then they are differentiable with respect to them. The velocity of point F is given V F = v x + v y (2-17)

PAGE 29

16 where ** £ { dx -v J , . dx x . oy . y i=l i 1 'i J i dX d7~ (2-18) Ef dY Sy t^ v x. + 3y~V 1=1 1 1 11 The partial derivatives in Equations (2-18) 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 dx ^x. ~ ox" ' yV A dY (2-19) Substituting these into Equations (2-18) gives V X = t {x^x. V x. + X i y V yJ. ' i=l 2 V y = E {A. V x . + yi y . V y } 1=1 ii 'i J i (2-20) Equations (2-20) can be put into compound matrix form as X G x. Y G x. x G y, yV (2-21)

PAGE 30

Similarly for angular output information 17 M" A. 1 *V II J 1 (2-22) where X G x. " Bx a x" dx dx„ A. "as, sa; 3x i ^ (2-23) Acceleration Influence Coefficients Differentiating Equations (2-18) with respect to time yields 2 2 = y y { sx_ v v + 2 ? .*-* h*. Ldx . ox . x . x . ox d'x "X (-> Cu idx dx x. x. dx.oyT x. y. i=l j=l ijij iJiJ z 1 y f SX t dX t \ V J A* Idx x dy y.J i *j 'i 'j 1=1 li 'i i d'x + 5 S v (2-24) and 2 2 s y y {, 3 , Y v v + 2 « ** .*-*, ldx . dx . x . x . dx a 2 v — T V V dy x . y . i=l j = l ijij i J i J ^2 d Y v v h + Y* "W — a + -r — a r . (2-25) y. y.J , Wx. x. dy. y.J f dy dy y. y.J /-» ldx. x. dy . y.

PAGE 31

18 Equations (2-24) and (2-25) can be written in compound matrix form and generalized to n inputs as a X " ' V x' i

PAGE 32

19 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 ehaini constructed from basic chains with invariant mobility arc dcvclopod by adding Assur groups Lo output, polos of the established chain of predetermined mobility. If the mobility does not remain invariant, then the basic kinematic chain with its known input-output position relationships 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 input-output relationships developed. Total Input-Output Kinematic Influence Coefficients If the inputs to a linkage system group are angular, then the intermediate linear inputs (x. ,y.) and their dynamic states (v x . ,Vy.) t t and (a x . , a y . ) can be expressed as functions of the input position parameters CD 's and their dynamic states uu 's and a 's. The derivation of q q q the total input-output transformation is now carried out. Velocity Influence Coefficients Consider the system group in Figure 2-5. The expression for the velocity components of point F with respect to points E and E are 1 £ X G x. Y G x. x G y v G y 1 ' X 1 (2-31)

PAGE 33

20 E 2 (x 2 ,y 2 ) Figure 2-5. Seven-Link System Group

PAGE 34

21 The intermediate input data, in terms of the primary input information, are given by x. C 9 i q y. G 9 i q CD , i,q 1,2 ; where G X. CD i Y q

PAGE 35

22 Note that when q=l, Equation (2-34) becomes '1 1 (2-35) V Y = Y g l W l ' 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 2-5) in terms of the intermediate inputs are LI, i j Y x. x . i J X»x i y j A i y j V + y 1 L 1 ! Xy . y . I, i J ; v JL*Vj X G x. Y°x. xV yV 1 x'

PAGE 36

23 Substitution of Equations (2-26) , (2-27) , and (2-32) into Equation (2-36) gives (2.Nxl) (2N X 2n) (2n x 2n) (2n xn) (n xn) (n xD

PAGE 37

24 The complexity of Equation (2-37) is reduced by collecting like terms and writing it in a more concise and familiar form as t"I X H cp 9 q r Y H 9 9 q l x G 9 A \a\ , q,r = 1,2 (2-38) where || U)|J is a compound diagonal matrix given by (2-39) and j co I is the transpose of the column vector |u)|. The expressions (2-34) and (2-38) for the velocity and acceleration can be generalized to treat systems with n inputs and m outputs. These expressions take the general form 1 T q x 9 m q Y l\ Y 9 m q hi f Q = 1,2,. . . ,n, (2-40)

PAGE 38

25 t Y „ H X i ^ 1 q r m q r 1 q r Y 9 9 m q T r U) + *l\ X G 9 m q h\ G Y 9 m q !« , (2-41) q,r = 1, . . . ,n. It is clear that expressions similar to Equations (2-40) and (2-41) can be written for the angular properties of the system links. For the purpose of illustration, both forms will be used where appropriate throughout the remainder of this dissertation. The discrete form [28] of the kinematic influence coefficients of velocity and acceleration are expressed as ,8i, X g ik = rik i k 6 _ 5 li,k+l I i,k-l 2A9. (2-42) d 2 $ * h ij,k = br4:\ • i,k+l i,k+l i,k-l i,k-l /ij.k S _ 6 _ v + i 4j,k+l Xj,k-1 X,j,k+1 Xj,k-1 4(A9-) 2 ). (2-43)

PAGE 39

26 where the input type (i.e., angular or linear) is considered understood and its designation is dropped for simplification of notation as follows; (2-44) Table 2-1 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 containing 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 (2-41)). 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 • j j 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 2-6. The influence coefficients in Table 2-1 are a precise definition of the system geometry up to the second order. As such they

PAGE 40

27

PAGE 41

28 Figure 2-6. Sliding Pair Constraint

PAGE 42

29 represent a powerful definition of the meaning of multiple input linkage 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 following 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 S imply that S = f(t) . (2-45)

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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 generalized torques acting at the system inputs. The Equivalent System Torques The force related influence coefficients are derived by replacing the effect of external forces on the system (i.e., T ., F ) and internal 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 3-l(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 1 input must be equal to the virtual work done by the system external torque, or T Vi *Jk l T i A§ Jk' . (3 1} 30

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31 (a) %%*' K Vi (b) * (c) Figure 3-1. Equivalent System Elements

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32 solving for T^ /iR and taking the limit as ^? ik "*0 gives J '4k Similarly, for an external system force F , T X/ik U g ikj F ek (3-2) (3-3) The total equivalent torque seen at each input link i, resulting from all system torques T . , is given by ,Si 1 & 1 2 g l ' ' ^1 1 S 2 2 g 2 ' ' 1^2 l g n 2 g n ' ' ^n i_ _tk u

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33 and the ground link. The question arises: What is the equivalent spring constant K, . 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 th spring associated with the i input link (see Figure 3-2(b)). This yields the following relation I K I/ik GO § e j KJ • (3 5) Solving for K,. and taking the limit as #p -• yields *vikr j . lim ~ (*r). = At*i (3_6) ^ik It is noted that §» cannot continuously increase with increasing cp. , otherwise the spring would be destroyed. Rather, §. is cyclic i * or quasi-cyclic with increasing 9.. Therefore, for part of the cycle A§. (i.e. , .g.) must be negative. This implies the need for the signum function a = sgn („g.) which changes sign whenever the fl g.'s go fg. Ill * 1 through zero. The torque T«. is then determined from its past history in the form T */-, = T a/1 , + K,,,. tip., , (3-7) £/ik £/i,k-l Vik lk where •U Vu r * (^4^) •

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34 The equivalent torques at each system input resulting from all system springs are given in matrix form as s/1 s/2 s/n 4 L s/1 s/2 * * s/n 11 k-l K £/l *i K X/2 ^2 K ly A? */n n (3-9) An alternative approach to determine the equivalent torque T.. generated by a system spring between link JL and ground is to treat the torque acting on link i, due to spring K. as an external torque as in Equation (3-2). The external torque is given by * T Xk = K i ( § Xf " $ Xk) = K X A§ if ^£kj "X " Xk ' where $,_ = free length of spring measured from the reference axis, A$. = deformation of system spring Kfrom free length. Substituting Equation (3-10) into (3-2) gives (3-10) T £/ik " K JL Ce g iJ A$* . lk (3-11) The equivalent torques at each system input resulting from all system springs K^ are now given in matrix form as s/1 s/n -•k L l G i N°i k L K l M l K A$ N N , i = 1,2,. . . ,n, (3-12) N = number of system links.

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35 Inter, Ml Viscous Dampers Let the torque T. be generated by a system dashpot C. between link I and ground (see Figure 3-l(c)). The torque generated by the dashpot is proportional to the angular velocity of the link attached to the dashpot , or j=l J J 'k (3-13) — th Using Equation (3-2) , T. is transferred to the i system input as T 4/ik = U s ilJ ~^l £ \#i w j ) • (3-14) th The total torque acting on the i input link resulting from all system dashpots is ( * d/ik 3=1 J J (3-15) where
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36 Table 3-1 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 r correspond to those in Table 2-1, i.e. , b ik r., = g g . (3-18) b ik n ik m ik The equivalent torques in row 3 refer to a force F applied to a particular point or points in the system. Since the direction angle T) of this force relative to the path tangent is known as a function of the input angles cp . , then the appropriate equivalent torque can be determined. The equivalent torque, T ... , is obtained by first e/ik taking its component F along the path tangent and transferring this force to the i input link by the influence coefficient Sk Here, the product g cos (71 ) is the effective influence of a unit force e ik ek applied in the direction F at point E on the i input link. As such, it corresponds directly with .g„ in row 1. Blocks 3.2 and 3.3 follow a ik directly as a result of this correspondence. Here, the system spring K and dashpot C are anchored at some fixed point . Finally, the equivalent torques in row 4 apply to forces in extensible two-force members attached to links m and n at points B and m B . The relative influence coefficient r is defined in Table 2-1, n b ik row 4. This implies the treatment of very general force generators in the system.

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37 «0 3 o * ^ P a -o ii K * A O" •O fO p * 3 3^ ii * ; P m u < CVJ c\2 * V P ISC, ti II * P

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38 Total Equivalent Torque th The total equivalent torque acting upon the i input link is given as T* = T* ., + T* + T* ... + T* .. . (3-19) ik 4/ik e/ik s/ik d/ik If one or more of the input elements is actually a sliding oair then s represents that input's reference position parameter ^ ik * and F would be the equivalent force acting on that system input, ik Equivalent System Inertias The objective of this section is to transfer the effective mass of link I to each of the n system inputs in order to obtain equivalent mass systems for a linkage system such as that shown in Figure 3-2. Consider the kinetic energy of a complex, multiple input system. k 1 £ { h -4 + 5 * 4 } • (3 20) (KE) where I . = effective moment of inertia of link I about its center Jw of gravity E, M. = effective mass of link i located at its center of gravity E. Writing Equation (3-20) in terms of the system input velocities yields N n 2 n _2 (KE) k = | Z { T i L E <# ik )ik J «i LE ( e i ik )ik J } &=i 1=1 1=1 (3-21)

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39 n = 3 N = 13 Figure 3-2. Complex Multiple Input Linkage System

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40 Rewriting Equation (3-21) in a more compact form gives n n jj CKE) k = § £ Z t { frtWW + M X ( e g i )( e i j ) ] vl 1=1 J = l *=1 where it is noted that ) = { i 6 i ,v rj 1 Z=i,j; i = j X=i,j; i^j, k (3-22) (3-23) a direct consequence of the meaning of independent system inputs. The kinetic energy can be written in matrix form as (KE) k = \ |o,| k (nxn) (3-24) where * = E £=1 LI /#i )( # j ) + Ve g .)( g.)l 1 ej Jk (3-25) Matrix Equation (3-24) 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 equivalent inertias are given by Equation (3-25).

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41 Two Degrees of Freedom Example For the purpose of illustration, consider the differential gear (Figure 3-3) , where the translating links are constrained to move in the horizontal direction and rotation of link Z is positive countorclockwise. The linear velocity influence coefficients of & are SX 3 _^2_ 3 g l = 3^ = d i+ d 2 ' (3-26) SX 3 _^1_ 3 g 2 " o^ d i+ d 2 and v = s: v + k v (3-27) 3 3 B 1 1 3 & 2 2 v The angular velocity influence coefficients of SL are given by (see Figures 3-4 and 3-5) 4 g l " ^l ~ d 1+ d 2 ' (3-28) S$ 4 _±_ 4 g 2 3x7 + d.+d and 2 12 ^4 = 4 g l V l + 4 g 2 V 2 • (3_29) The kinetic energy of the two input differential gear is 1 ,2 -2 -.2 -2. KE = 2 i'Vl + M 2 V 2 + (M 3 + M 4 )V 3 + W • (3 ' 30)

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42

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43

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Substituting Equations (3-27) and (3-29) into (3-30) gives KE = \ {M lV 2 + U 2 v 2 2 + (M 3 + M 4 )( 3 g 1 v i + 3 g 2 v 2 ) 2 44 + I 4 ( 4 g l V l + 4 g 2 V 2 } ) ' (3-31) where Collecting like terms and rewriting Equation (3-31) gives KE = \ {[\ + T 4 ( 4gl ) 2 + (M 3 + V^i/lv* + [M 2 + ^ 4 ( 4 g 2 ) 2 + (S3 + M 4 )( 3 i 2 ) 2 ]v^ + 2[T 4 ( 4gl )( 4 g 2 ) + (M 3 + M 4 )( 3 i 1 )( 3 i 2 )]v i v 2 } , (3-32) 1 r * 2 * 2 * KE = 2 ^11 V 1 + ^2*2 + 2I 12 V 1 V 2 } (3-33) where hl = \ + hW* + (M 3 + M 4 )( 3 g l )2 ' T 12= I 4 ( 4 B l )( 4 g 2 ) + (M 3 + V ( 3 g l )( 3 g 2 ) ' (3-34) X 22 = M 2 + F 4 ( 4 B 2 )2+ (M 3 + V ( 3 g 2 )2 For the system in Figure 3-3, Equation (3-24) becomes KE = 2 [V 1 V 2 ] 11 "21 *

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45 In the case of a differential gear the equivalent inertias are constant. However, for a general linkage system (Figure 3-2), the I 's are not constant but a function of the system input parameters. Note that whenever v =0, then KE = | I* x v\ , (3-36) and whenever v = , then KE \ i V 2 ' (3 " 37) which are simply the expressions for the kinetic energy of a single input equivalent system. Inertia coupling terms, I. . i ^ j, appear for systems with two or more inputs, indicating a relationship between the I . S through the velocity product terms cu.cu.. Quinn' s energy method [29] 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

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46 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 where P = -£ (KE) , dt K E = f w'[i;Jw . i,j = 1,2,. . . ,n Differentiating Equation (3-24) with respect to time gives „* 1 I I d P 2 H dt ij |u>| + \a\ h*. |u)|. (3-38) (3-24) (3-39) The second product on the right side of Equation (3-39) is selfexplanatory. The time derivative of the equivalent inertia matrix, however, is not as trivial. Consider a typical element of C I ^ 3 given by N Differentiating Equation (3-40) with respect to time gives (3-40) n N &);: ?.{' k r=l W^V + .W ( 4 h ir> where + M, WW + W
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47 Equation (3-41) can be written as the product of a row and col\ vector as dt / [_ij 1 ij'nj (3-43) where ij r is £=1 L ^ l^L^W + WW J ( a gJ(h.J + ( g.)( h. ) 1 } e i ejr ej e lr J J (3-44) The subscript notation is defined by * a Bl • • • -P = -IT 1 (3-45) Substituting Equation (3-43) into Equation (3-39) gives (lxn) (nxnn) (nnxn) K • § K ll"r P 21 r * 1 P 1 12 r ' P ' 22 r ' nl r P n2 r In r „ P 2n r I 1 * ! p nn r -"k L_ tl < fci K (3-46) where P ij r is a compound nxnn matrix whose submatrices are lxn row vectors.

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48 Two Input Example Consider the system in Figure 3-6 whose kinetic energy is given by (KE) k = [o, 1 u, 2 ] k 11 21 12 22 ->k (3-47) where ij.k s = E X=l L. jfii #j + M -e £ i *] k (3-48) P P 11 1 11 2 P P 12 1 12 2 Differentiating Equation (3-47) with respect to time gives lo, , o " 1 I w 9 ! o z i o ! w i i (0„ k 2 L 1 2 J k 21 P 1 P 21 2 P P 22 1 22 2 ID L 2 J k -Ik L 1 2 J k 11 '21 *

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49 Figure 3-6. Complex Two Input Linkage System

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50 V II II J2, H TD n <1^ .Tl X ro # e 2 n CaJ I O c < > m Z H -< CO H m 3 O 20 > H O

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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 determining the time response of single input systems are well established, either in terms of Lagrangian mechanics [11-14] or equivalent mass systems. 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, cp . , 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 second-order differential equation in n independent variables. These failings are a result of the geometric constraints imposed on the system. 51

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52 Two methods will be discussed in this chapter. The differential equations of motion for a two input system will be derived, using both methods. The complete set of first-order difference equations 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 second-order differential equations is derived from Lagrange's equations as d /dL \ dL * St b-)-<5T= V Ci=1.2,...,n) (4-D o^ i T i where cp. generalized coordinates, cp. generalized velocities, L Lagrangian of the coupled equivalent mass system, T. equivalent nonpotential forces and torques. The n second-order differential equations resulting from Equation (4-1) are nonlinear, coupled, nonhomogeneous , and contain variable coefficients. It is necessary to reduce this set of equations to 2n first-order 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

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53 are functions of geometry and are hence implicit functions of time. This characteristic requires the determination of the equivalent inertia power coefficients of the system, terms which are very difficult to obtain (see Chapter III) and only adds to the complexity of the problem. Example The five-bar mechanism in Figure 4-1 is a representative two input system. The generalized coordinates are cp and cp • a helical spring and viscous damper are attached between links 3 and 4 at pin joint B, while external torques T and T are applied to links 1 and 1 £ 2, respectively. The Lagrangian for this system is given as L = (KE PE) , (4-2) and the equivalent nonconservative torques are V"S4b r i C b r i *1 + b r 2 ^2 ] +T 1' (4-3) T 2=C 34b r 2 C b r i *1 + b r 2 *2 ] +T 2' where 2 2 KE = T V I*. cp. cp. , (4-4) 1 — 2 PE = K (CD -(D). (4-5) 2 34 y 34f y 34 Substituting Equations (4-3) , (4-4) , and (4-5) into Equation (4-1) and performing the designated differentiation with respect to the generalized coordinates and velocities yields

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54 7777 Figure 4-1. Two Input System

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55 dT { *11 *1 + J 12 »2 ) + K 34 <*34f " »34 )( b r i> = T 1 " C 34 [( b r ?^l + WWV ' (4 " 6) 3? { X 12 *l + T 22 ^J + K 34 ( *34f " C P 3 4 )( b r 2 ) = T 2 ~ C 34 [( b r i )( b r 2^1 + V^V • (4 " ?) Completing the differentiation with respect to time gives * • * +•• * • 2 *•• %•• 11 P 1 *1 + ( 11 P 2 + 12V V 2 + 12 P 2 *2 + X ll »l + X 12 *2 " T l + ? 34 (C P 3 4f " CP 34 )( b r i ) S4 [V^l + ( b r i )( b r 2 ) ^ 2 ] ' (4-8) 21*1 *\ + ( 21 P 2 + 22VV2 + 22 P 2 *2 + J 12 *1 + J 22 ^2 " T 2 + ? 34 (C P 3 4f " C P34 )( b r 2 ) Si W^l + VJ* J ' (4-9) Equations (4-8) and (4-9) are nonlinear in the velocities and are unwieldy in their present form. In addition, they should be reduced to four first-order equations before they are numerically integrated. * The . .P terms as defined by Equations (3-44) and (3—15) are not simple ij r expressions, and require calculation at each integration step.

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56 Hamilton's Principle The method used in this section for deriving first-order 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 oS 2J, df.. ^ sr=+ L X. . -r-^ 2 " F., At , (4-10) ^ J-l 1J °?ik lk as N 5f . . — — + £ X . . — =3. = o , (4-ii) ap. _ j=i 1J aj>. . ik T ik f., = Ap., -cp., At = o, (i=l,2,...,n; k= 1,2,... ,N) , (4-12) ik ik ik where and \ i undetermined Lagrangian multiplier at j time ij interval , * th F i generalized equivalent nonpotential force, at , th * • k time interval, N S = Y L*(c? , cp cp , cp cp , . . .cp ) At . (4-13) 1 M k^ Y lk' Y 2k v nk' Y lk' Y 2k' Y nk' k=l The 3nN first-order difference equations resulting from Equations (4-10) (4-12) in conjunction with an appropriate difference expression for Up. are obtained from only one differentiation and are in a form which allows their solution to be marched out with time. Furthermore, the differentiations (3S /cf9 .) and (dS /<&> ) produce the 11 1 l .P terms and uncouple the equations in one step.

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57 Example In order to compare the two methods, the above procedure will be applied to the system in Figure 4-1. Since the internal spring K„ produces an internal torque which is transf errable to each input as a generalized equivalent torque (see Chapter II) , it will be included in * the F terms, leaving the sum S made up of only the kinetic energy of the system. Therefore, S l = 2 1 Z ll *1 + 2I 12 V2 + X 22 *2 I • (4 " 14) P lk ="( C 11 9 1 + °1*2 O + T s/l,k-l + Vlk *?lk + T l' (4 " 15) K F 2k = " (Si *1 + C 2*2 O + T I/2,k-l + K b/2k *2k + T 2' (4 " 16) k A? =

Sep . k Let X. be given by the finite difference relation X. . , = (X. . , X. . )/At . (4-20) i,k+l i,k+l i,k

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58 Then Equation (4-18) becomes r ,ds K. , , = A . , 1 , k+ 1 l , k w?.; k i kj (4-21) Substituting Equations (4-14) (4-17) into Equations (4-12) , (4-19) , and (4-21), and solving for cp. yields the following set of six equations

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59 The initial values of the X' s are obtained by solving Equation (4-19), using the initial values for the cp's and cp's. Hence L> J0 hi '12 X 21 X 22 9. (4-25) The matrix Equations (4-22) (4-24) require one less numerical step of integration as pointed out in I" 30 ]It is also pointed out in reference [30] that the X' s are the momenta, usually called p.'s, and that Hamilton' s canonical equations are obtained directly without formerly deriving the generalized momenta. 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 (4-8) and (4-9) 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 formulation 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 coefficients of velocity and acceleration.

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CHAPTER V SUMMARY 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 associated with the generalized coordinates necessary in describing the motion of the linkage system. Large numbers of generalized coordinates (see Figure 5-1 (a)) have been required to obtain the system's 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, second-order 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 algorithm required to integrate numerically and solve this set of equations requires a dual iteration scheme, one to solve the differential equations 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 4x4 matrix coordinate transformations, have been the only tools available to solve the dynamic response question for these systems. The set of 60

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N = 8 n = 2 8 Generalized Coordinates 6 Algebraic Equations of Constraint (a) Figure 5-1. Complex Multiple Input Mechanism and Its Coupled Equivalent Mass System

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62 second-order differential equations resulting from their methods has been sufficient, though unwieldy, time-consuming, 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 high-order-complexity can be constructed from Assur groups in terms of kinematic influence coefficients 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 coefficient viscous dampers, and equivalent external forces and torques; and, to determine the differential equations of motion for the coupled equivalent mass system in terms of the minimum number of generalized coordinates (i.e. , the number of independent system inputs). The construction of general linkage systems of higher-ordercomplexity, 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 (scries) and addition (parallel) of successive velocity influence coefficients. The use of Assur groups to construct mechanisms of higher-order-complexity from existing system groups, without 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 5-l(a)) to coupled,

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63 equivalent mass systems (Figure 5-l(b)), requiring only n generalized coordinates to describe its motion. This reduction eliminates the 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. Second-order differential equations describing the dynamical behavior of the equivalent mass system have been derived by the classical Lagrangian method, while first-order difference equations were derived by the direct application of Hamilton's principle. This method [18], yields a set of first-order 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 5-2. The effect of bearing deformation on the input dynamic response could be investigated, based on the system models in Figure 5-3. The equivalent 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 principles of optimal control theory. Optimal open loop control laws may

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64 Figure 5-2. Linkage System with Elastic Coupler Link

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65 s vvvvvs * eg .

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66 be determined which minimize specified performance indexes while satisfying prescribed constraints on the control itself, the states, or both. Figure 5-4 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 s associated with position s over the range s £ S £ S„, subject to the 1 \ & .* inequality constraint on the control, u ^ u . This in essence places an upper bound on the jerk u,thus giving third-order 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.

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67

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APPENDIX

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APPENDIX A DIRECT DERIVATION OF FIRST-ORDER DIFFERENCE EQUATIONS FOR DYNAMICAL SYSTEMS The derivation of the set of first-order 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 First-Order 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 derivation 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 ^ r Ldt + j* '1=1 T T / » . 6l = 6 J* Ldt + J* (£ F.Sqldt = 0, (A-l) 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 69

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70 interval to T into N small increments AT, the two integrals can be approximated by sums and the principle can be rewritten as 6S 1 + S 2 = 0, (A-2) where the functions S and S are sums given by N S l = E L n (q in' q 2n'---' q kn'V'V'"-'\n )At ' (A " 3) n=l N k S 2 E E F in 6q in At ' (A " 4) n=l i=l Equation (A-2) requires that the variation of the function S equal the negative S . Since the displacements and velocities are to be related by some finite-difference 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 finitedifference relationship, have the general form Aq. in . . _« in At f. = Aq. q. At = 0, (A-S) in in m where Aq. is any desired expression for the first-order difference of q. . in

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71 Taking the variation of the constraint function (A-6) gives N /0 f 6f = E in *—> df. 6q. . + -rill 6q. . ) = 0. 3=1 V ^ij " iJ ' 6 ^ij ^ (A-7) Equation (A-2) can now be written as N k 6S , + I! Z *. 6f + s„ = o. 1 *-*, .V, in in 2 n=l i=l (A-8) Substitution of Equation (A-7) into Equation (A-8) and rearrangement of terms gives N

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72 as, af . . ^ + ^ Vj ^ = F. At, (A10) in j = l in in 3S. N Bf . . 557 + ?, x « ** " (A " 11} in j=l in f . = Aq. q. At = 0. (A-12) in in in When the first-order difference form Aq. is substituted in into Equation (A-12) , the summations in Equations (A-10) and (A-ll) are reduced to only a few terms. For example, if Aq . = q . q . ' , , in in in-1 3f . . of . . 3-22 _ o, j 4 n, n + 1; and -^^=0, if j 4 n. (A-13) 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 \'s. Unlike many applications of Lagrangian multipliers, however, the \' s are not in general constant. In fact the X' s represent momenta, . . ., and have ** the status of independent coordinates [3] . The set of Equations (A-12) may be considered trivial (although necessary) in the sense that they are simply the equations of constraint between the velocities and displacements. The \' s will be constant only in the case of ignorable coordinates. Goldstein, H. , Classical Mechanics , Addison-Wesley Publishing Co., Atlanta, 1965, p. 227.

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73 The derivation just presented is not tied to any particular finite-difference 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 . = 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

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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 five-bar mechanism in Chapter IV (see Figure (B-l(a))). The procedure for obtaining the variable coefficients to Equations (4-22) (4-25) , describing the dynamical behavior of the system in Figure B-l(a), is described below. The necessary coefficients * * * * * are K, ... , C. . . , T. , , I. . . , and . .P , . X/ik xj,k i,k' ij,k' ij r,k (1) For the given initial position of the mechanism defined by cp and cp calculate the kinematic position information pertaining to the other links § , $ , etc. , by using the equations in Figure B-2. (2) Substitute this position information into the appropriate blocks of Table 2-1 to determine the velocity and acceleration influence coefficients ..g. and „h... For example, Jfc 1 X ij r , $ _ $ 3 l,k+l 3 l,k-l 3 g i 25jT ' ' CB-1) (3) Once the »g.'s and „h..'s are determined, substitute them * l x IJ into the appropriate blocks of Tables 3-1 and 3-2. This furnishes the following expressions for the coefficients to the differential equations: 74

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75 2 2 * N /_ \ c • „ = I (c,(,g.)(,e.)J . (b-3) 1 J ' jtl 1 j k T* = (,B.„)T, • (B4 ) xk Z lk £ I and P are determined from the expressions in Table 3-2, ij,k ij r,k blocks 1.1 and 1.2. The right-hand side of Equations (4-22) (4-25) are st now completely known, allowing the k+1 \'s, cp's and cp's, to be determined. This procedure is repeated until the equations of motion are integrated over a predetermined time interval. Equations (4-22) (4-25) represent the equations of motion for the coupled equivalent mass system shown in Figure (B-l(b)). The solutions to the dynamical equations of motion are shown in Figure (B-3) and (B-4). Polar plots of

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76 77777 7777 (a) L I2 (b) Figure B-l. Two Degrees of Freedom Five-Bar Example

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77 / 34 = A + A 2/ 3 Acosc£ 34 / 34 = (./ 5 COS(£ 6 -/ | C0Sc/> | +i' 2 C0S<^ 2 +(/ 5 SIN(^ 5 + ^SIN^> r / 2 SIN^ *34= C0S -i •*3 -*4 ^34 $s SIN -I 2 AA ^ 5 SlN(/> 5 + ^SINcjb,/ 2 SIN<^> 2 34 $ 4 * SIN* (" ^ 5 SIN^>3+ -^, SIN , " ^SlNC^j 34 + SIN SIN Figure B-2. Kinematic Position Equations

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78 CO i ffl o u 3 to

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79 UJ < o CO ai o

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TABLE B-l FIVE-BAR PARAMETERS 80 Length (inches) Weight (lbs. ) Moment of inertia (in. lb. sec. 2 ) 4.0

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81 t t t t t (•aVa-3T9NV) (^OaS'BV-NI Vliy3NI lN31VAin03)

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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. 184-188. 3. Hain, K. , Applied Kinematics , McGraw-Hill Book Company, Inc., New York, 1967, pp. 53-56. 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. 171-185. 5. Denavit, J. , and S. Hasson, "On the Harmonic Analysis of the FourBar Linkage," Proceedings of the International Conference for Teachers of Mechanisms , Shoe String Press, New Haven, 1961, pp. 171185. 6. Flory, J. F. , and J. C. Wolford, "Harmonic Analysis of Kinematic Linkages," ASME Mechanisms Conference, Paper No. 64-Mech-38, October, 1964. 7. Markus , L. , and J. Tomas , "Harmonic Analysis of Planar MechanismsKinematics," Journal of Mechanisms, Vol. 5, No. 4-A, 1971, pp. 171185. 8. Bogdan, R. C. , and T. V. Huncher, "General Systematization and Unified Calculation of Fiveand Four-Bar Plane Basic Mechanisms," ASME Mechanisms Conference, Paper No. 66-Mech-ll, 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. 68-Mech-62 , October, 1968. 10. Crossley, F. R. E. , and N. Seshachar, "Analysis of the Displacement of Planar Assur Groups of Computer," Transactions of the International Federation of Theory of Mechanisms and Machines, September, 1971. 82

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83 11. Chace, M. A. , "Analysis of the Time-Dependence of Multi-Freedom Mechanical Systems in Relative Coordinates," ASME Mechanisms Conference, Paper No. 66-Mech-23, October, 1966. 12. Smith, D. A. , "Reaction Force Analysis in Generalized Machine Systems," Ph.D. Dissertation, University of Michigan, 1971. 13. Uicker, 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. 251-265. 14. Carson, W. L. , and J. M. Trummel , "Time Response of Lower Pair Spatial Mechanisms Subjected to General Forces," ASME Mechanisms Conference, Paper No. 68-Mech-57 , October, 1968. 15. Wittenbauer, F. , Graphische Dynamik , Springer-Verlag, Berlin, 1923. 16. Federhofer, K. , Kinetastatik f lachenlauf iger Systeme , S.-B. Akad. Wiss. Wien, Math. -Naturwiss. Kl. , Abt. Ha. Jg. 139, 1930. 17. Beyer, R. , Kinematisch-Getriebeanalytisches Practikum , SpringerVerlag, Berlin, 1960. 18. Vance, J. M. , and A. Sitchin, "Derivation of First-Order Difference Equations for Dynamical Systems by Direct Application of Hamilton's Principle," Journal of Applied Mechanics , Paper No. 70-APM-PP, June, 1970. 19. Pelecudi, CHR. , "Kinematic Multipoles with Rigid Links," Rev . Roum. Sci. Techn. Mec. Appl . , Tome 13, No. 5, pp. 997-1013, 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. 1225-1237, 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 Conference, Paper No. 70-Mech-82, November, 1970. 23. Benedict, C. E. , and D. Tesar, "Dynamic Response Analysis of Quasi-Rigid Mechanical Systems Using Kinematic Influence Coefficients," accepted for publication in 1971 in the Journal of Mechanisms.

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84 24. Benedict, C. E. , and D. Tesar, "Dynamic Response of a Mechanical 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 Sub-Unit Cam Systems," Proceedings of the Applied Mechanisms Conferenc e, 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. 28. Abramowitz, M. , and I. A. Stegun, Handbook of Mathematical Functions Dover Publications, Inc. , New York, 1968, pp. 883-884. 29. Quinn, B. E. , "Energy Method for Determining Synamic Characteristics of Mechanisms," Journal of Applied Mechanics, Vol. 16, September, 1949, pp. 283-288. 30. Sitchin, A. , "Problems in Attitude Stability of Dual-Spin Spacecraft," Ph.D. Dissertation, University of Florida, 1970.

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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 one-half years for Florida Gas Transmission 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. 85

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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. U UU)6S7 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<^-e-^ C. C. Oliver, Co-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. W. H. Bbyfcih/ 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

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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. \l^V^\}^& Associate 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 1/L Vance sistant Professor of Mechanical Engineering This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1971 s^r^ZZfcc^^ ; College of Engineering Dean, Graduate School

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14 6.8 0.