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This is to certify that the

thesis entitled
An Analytical and experimental investigation
of the dynamic response of a four bar mechanism

with clearance in the ooupler—rocker bearing
presented by

Mehrnam Sharif—Bakhtiar

has been accepted towards fulfillment
of the requirements for

M.S. degree in Mica]. Engineering

 

Major professor

Date I) Mf / 7J9?

0-7639 MS U is an Afimative Action/Equal Opportunity Institution

 

 

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AN ANALYTICAL AND EXPERIMENTAL
INVESTIGATION OF IRE DYNANIC RESPONSE
OF A FOUR BAR NECEANISII WITH (LEARANCE

IN THE mUPLER-ROCIER BEARING

By

Mehrnan Sharif-Bakhtiar

A EEESIS

Snbnitted to
Michigan State university
in partial fulfillment of the requirement

for the degree of

“ASTER OF SCIENCE

Department of Nechanical Engineeing

1984

j

5’77¢~

Iva 4/42 “

v...’

ABSTRACT
AN ANALYTICAL AND EXPEEIIENTAL
INVBTIGATION 0F IEE DYNAIIC RESPONSE
OF A FOUR BAR lECEANISI IIIE CLEARANCE

IN TEE.COUPLERPEOCIER.BEAEING

By

lehrnan Sharif-Bakhtiar

Design problems of mechanisms in industry which were considered
of secondary importance until about a decade ago. are now receiving
acre extensive attention due to the ever-increasing demand for
highrspeed machinery. One of the major problems associated with the
operation of such machinery is the inevitable existence of clearance
in one or more of the bearings of the system. which results in a
number of undesirable effects such as excessive vibrations and defor-
mations in the parts. the generation of large forces in the bearings
with clearance_due to the vibro-impaet phenomenon uhich can create
high levels of acoustic radiation. overloading of the driving motor.

and also prenature failure of the bearings.

The work presented here deals with this specific problem. namely.
the existence of clearance in the bearings of a mechanism. A four bar
linkage with a finite clearance in the coupler-rocker bearing has been
chosen as the mechanism to be studied. Governing equations of action
for one full revolution of the crank are developed which consider the
occurence of contact-loss in the bearing with clearance along with
subsequent impacts. To perform this task. a specific theoretical
model based on some simplifying assumptions has been deve10ped and
simulated. An attempt has been made to compare and correlate the
theoretical results to experimental data. The principal objective of
this study is to investigate the sole effect of bearing clearance on
the kinetic and kinematic behavior of the system. whose links are
assumed to be rigid and hence free of deflection. This project is by
no means intended to be a thorough analysis of the subject of bearing
clearance in high-speed machinery. but rather as a means of deve10ping
guidelines and criteria for the design of mechanisms, and to help pro-
vide a better insight into this relatively new phenomenological

problem.

This work is dedicated to my parents, who taught me that there is

no failure except in nolonger trying.

ACKNOWLEDGEMENTS

The author wishes to express his sincere thanks to Dr. Brian S.
Thompson for his continual suggestions and guidance throughout the
course of this investigation. Sincere appreciation is extended to Dr.
Suhada Jayasuria and. Dr. Behroox Fallahi of the Department of

Mechanical Engineering for serving on the guidance committee.

The frequent assisstance cf the consulting personnel of the
Albert E. Case center for Computer-Aided Design of the Engineering

College is sincerely appreciated.

Special credit must be given to the author's family for their
support and encouragement. morally and financially. that helped comr

plete this work.

LIST OF TABLES

LIST OF FIGURES

TABLE OF CONTENTS

Chapter 1. Introduction

Chapter 2. Assumptions and governing modes

Chapter 3.

Chapter 4.

2.1-

2.2-

Assumptions

lodes of behavior

The following mode

3.1

3.2

3.3

3e4-

3.5

Total kinetic energy of the system
3.1.1- Crank kinetic energy. Ta
3.1.2- Coupler kinetic energy. T,
3.1.3- Rocker kinetic energy. T‘
Generalised forces

3.2.1- Total potential energy. V
Resulting equations of motion
Initial conditions

Tbrmination of the following mode

The free-flight mode

4e1-
4.2-
4.3-

4.4-

Compound pendulum (crank and coupler)
Simple pendulum (rocker)
Initial conditions

Tbrmination of the free-flight mode

Chapter 5. The impact mode

5.1-

Governing equations of motion

iv

vii

iix

14
17
20
21
21
25
25
29
30
35
36
42
43
53
53
55
59

60

Chapter 6.

Chapter 7.

Chapter 8.

Chapter 9.

APPENDIX

5.1.1- Equations of motion for crank
5.1.2- Equations of motion for the coupler
5.1.3- Equations of motion fefr the rocker
5.2- Coefficient of restitution. e
5.3- Nature of contact
5.3.1- The smooth ease
5.3.2- The rough case
5.3.3- The stick-slip case
5.4- Assemblage of the equations of motion
Correlation and numerical solution of the modes of
behavior
6.1- Initial conditions
6.2- Iethod of solution
6.3- Numerical results
Experimental instrumentation and results
7.1- Experimental apparatus
7.2- Instrumentation
7.2.1- Angular acceleration and velocity of
the coupler
7.2.2- Angular acceleration and velocity of
the rocker
7.3- Experimental results
Comparison of analytical and experimental results
8.1- Iodificaticn of the analytical results

Conclusions

65
66
70
74
82
83
84
85

87

92
92
94
101
134
134

138

141

147
152
134
194
213

215

vi

LIST OF WCES 223

vii

LIST OF TABLES

Table 6.1- Specifications of the simulation mechanism 102

Table 7.1- Specifications of the experimental mechanism 153

FIGURE 2.1.
FIGURE 2.2.
FIGURE 2.3.
FIGURE 2.4.
FIGURE 2.5.
FIGURE 3.1.
FIGURE 3.2.
FIGURE 3.3.
FIGURE 3.4.
FIGURE 4.1.
FIGURE 4.2.
FIGURE 4.3.
FIGURE 4.4.
FIGURE 4.5.
FIGURE 4.6.
FIGURE 5.1.
FIGURE 5.2.
FIGURE 5.3.
FIGURE 5.5.
FIGURE 5.5.
FIGURE 5.6.
FIGURE 5.7.

FIGURE 6.1.

iix

LIST OF FIGURES

Schematic view of the nominal mechanism
Angular displacement

Angular velocity

Angular acceleration

Nodes of behavior of the mechanism

mechanism in the following mode

velocity direction of the Coupler's mass center
Vector representation of V,

Coupler-rocker bearing

mechanism in the free-flight mode

Compound pendulum (crank and coupler)

Crank free-body diagram

Coupler free-body diagram

Rocker free-body diagram

Vector representation of the free-flight mode
mechanism in the impact mode

Crank and coupler links at impact

Rocker link at impact

Crank and coupler prior to impact

Vector representation of velocity Vb,

Rocker prior to impact n
Vector representation of Vb

4

Coupler angular displacement

10
ll
12
15
22
24
24
39
42
44
46
48
54
57
61
63
71
75
76
79
79
104

FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE

FIGURE

ix

6.2. Rocker angular displacement

6.3. Coupler angular velocity

6.4. Rocker angular velocity

6.5. Coupler angular acceleration

6.6. Rocker angular acceleration

6.7. C-R bearing reaction; X-eompcnent

6.8. C-R bearing reaction; T-eomponent

6.9. Crank torque

6.10s.
6.10b.
6.11s.
6.11b.
6.12s.
6.12b.
6.13s.
6.13b.
6.14s.
6.14b.
6.15s.
6.15b.
6.16s.
6.16b.
6.17s.
6.17b.

6.18s.

FIGURE 6 .18b .

Coupler angular displacement (e-1.0)
Coupler angular displacement (e-0.50)
Rocker angular displacement (e-1.0)

Rocker angular displacement (e-0.50)
Coupler angular velocity (e-l.0)

Coupler angular velocity (e-0.50)

Rocker angular velocity (e-1.0)

Rocker angular velocity (e-0.50)

Coupler angular acceleration (e-1.0)
Coupler angular acceleration (e-0.50)
Rocker angular acceleration (e-1.0)

Rocker angular acceleration (e-0.50)

C-R bearing reaction; X-eomponent (e-l.0)
C-R bearing reaction; X-eomponent (e-0.50)
C-R bearing reaction; T-eomponent (e-l.0)
C-R bearing reaction; Y-component (e-0.50)
Crank torque (e-1.0)

Crank torque (e80.50)

105
106
107
108
109
110
111
112
113
114
115
116
117
118
121
122
123
124
125
126
127
128
129
130
131

132

FIGURE 7.1a.
FIGURE 7.1b.
FIGURE 7.2

FIGURE 7.3.

Photograph of the experimental rig
Photograph of the instrumentation
Coupler-rocker bearing

Schematic of the instrumentation

FIGURE 7.4. Coupler-rocker bearing mounted on the mechanism

FIGURE 7.5. 2-D vice of the experimental rig

FIGURE 7.6. Graphic representation of equation (7.2.1-1)

FIGURE 7.7. 2-D view of the mechanism

FIGURE 7.8. A typical power spectrum

FIGURE 7.9a.
FIGURE 7.9b.
FIGURE 7.9c.
FIGURE 7.9d.
FIGURE 7.10.
FIGURE 7.11.
FIGURE 7.12s.
FIGURE 7.12b.
FIGURE 7.21e.
FIGURE 7.12d.
FIGURE 7.13s.
FIGURE 7.13b.
FIGURE 7.13e.
FIGURE 7.13d.
FIGURE 7.14a.
FIGURE 7.14b.

FIGURE 7.14e.

Coupler angular acceleration
Coupler angular acceleration
Coupler angular acceleration
Coupler angular acceleration
Effect of variation of t alone
Effect of variation of the RP! alone
Rocker angular acceleration
Rocker angular acceleration
Rocker angular acceleration
Rocker angular acceleration
Coupler angular acceleration
Coupler angular acceleration
Coupler angular acceleration
Coupler angular acceleration
Coupler angular acceleration
Coupler angular acceleration

Coupler angular acceleration

135
135
137
139
140
142
145
148
151
155
156
157
158
160
160
162
163
164
165
166
167
168
169
170
171

172

xi

FIGURE 7.14d. Coupler angular acceleration

FIGURE 7.15s. Coupler angular acceleration

FIGURE 7.15b. Coupler angular acceleration

FIGURE 7.15c. Coupler angular acceleration

FIGURE 7.16e. Coupler angular acceleration

FIGURE 7.16b. Coupler angular acceleration

FIGURE 7.16e. Coupler angular acceleration

FIGURE 7.17s. Coupler angular acceleration

FIGURE 7.17b. Coupler angular acceleration

FIGURE 7.17c. Coupler angular acceleration

FIGURE 8.1a.
FIGURE 8 .11).
FIGURE 8 .2a.
FIGURE 8.2b.
FIGURE 8.3.
FIGURE 8.4.
FIGURE 8.5.
FIGURE 8.6.
FIGURE 8.7.
FIGURE 8.8.
FIGURE 8.9.

FIGURE 8 .10.

FIGURE 8 .11.

FIGURE 8 .12.

Coupler angular velocity

Rocker angular velocity

Coupler angular acceleration

Rocker angular acceleration

Coupler angular acceleration (e-0.50)
Coupler angular acceleration
Rocker angular acceleration

Coupler angular velocity
Rocker angular velocity

C-R bearing reaction; X-component
C-R bearing reaction; T-eomponent
Crank torque

Impulse directions on the bearing components

Directions of the dominant impacts for one cycle

173
174
175
176
177
178
179
180
181
182
186
187
188
189
190
200
201
202
203
205
206
207
210

210

CHAPTER ONE

INTRODUCTION

The presence of clearances in the joints of mechanisms is inevit-
able if plain journal bearings are incorporated into the system.
Owing to the high-speed operation of machinery. more interest is cur-
rently drawn to the study of this problem due to the undesirable
effects of existence of bearing clearance such as excessive noise
radiation. links' deflection. force generation in the bearings and so
forth. The complex effects associated with the existence of clearance
in a bearing cannot be confined to the bearing alone. rather it influ-

ences the behavior of the system as a whole.

During the high-speed operation of a mechanism. intermittent
motions are generated in the bearing(s) with clearance. This. in
'turn. causes high levels of impactive forces in the bearing(s). Large

pulses of forces are transmitted throughout the mechanism causing

extensive vibrations and deflections in the parts. which can partly
reduce the longevity of the links. These pulses are also a major
cause of premature failure of the bearings vith clearance due to the
generation of large impulses and subsequent ovality of the bearings.
and also frequent overheating of the driving motor primarily due to
the inability of the motor to withstand high levels of torque-pulses

transmitted to its driving shaft.

The induced vibrations of the links of the mechanism due to the
intermittent motions of the system. can become a primary source of
excessive acoustic radiation. In addition. these vibratory motions of
the links tend to decrease the accuracy of the mechanism in tracing a
prescribed path. which is a crucial factor in precision instrumenta-

tion and some areas related to robotics.

Numerous researchers have studied the effects of bearing
clearances from different points of views. The vast amount of litera-
ture available ranges from studies wherein the primary focus of the
research is on the on the behavior of the pin within the bushing of
the bearing with the links of the mechanism assumed to be free of
deflection [1-11]. to ones which concentrate on mechanisms whose links
are elastic [12-20]. that investigate the response of the system to
the presence of bearing clearances. Mechanisms with only one of the
bearings with clearance [l.2.5.6.21-24] or with multiple clearances in
the bearings [16.25-29] have been taken under study to investigate the

difference in the behavior of the mechanism based on the number of

clearances and also. the dependency of the response of the system on

the sire of the clearance(s).

In more specialized studies. some related investigations have
been performed on the nature and mechanics of contact between the sur-
faces of the two components of the bearing. the pin and the bushing
[30-34]. Some attempt has been made to mathematically model the bear-
ing components [35-39] in order to observe the effect of such physical
and geometrical parameters as damping and the hydrodynamic phenomenon

associated vith the bearing lubrication.

As previously mentioned. the mechanics of the intermittent motion
of a system is primarily due to the impacts generated in the
bearing(s) as a consequence of trajectory motion of the pin within the
bushing after contact between the two components has been lost
[36.40-62]. This in turn. brings about the problem of determining the
extent of acoustic radiation transmitted to the surroundings [63-71]
caused by the links' resonant vibrations. The intensity of such noise
radiation can become high enough that the employees adjacent to the
machine are exposed to doses of acoustic radiation in excess of that

specified by federal regulations.

Due to the nature of the problem. no one literature can contain
all the aforementioned issues associated with the subject of bearing
clearances in a comprehensive form. for the subject is too broad and

diverse. Thus. one has to acquire a good deal of expertise in several

areas of scientific endeavor in order to be able to analyse and com-
prehend the thorough behavior of all the issues. However. any attempt
in this field provides a basis for further investigation. and helps
provide a better insight into the problem as a whole. For instance.
the results of an investigation which has focused on the magnitude of
the forces generated in the bearing of a rigid-linked mechanism due to
the vibro-impact behavior. can serve as an intuitive guideline for the
prediction of the size of the deflections occuring in the links of the

mechanism. if the links were elastic.

Although the primary effort in all of the investigations has been
to model and predict the behavior. in various conditions. of a mechan-
ism with intermittent motions. there has been some work concentrating
on the feasibility of adaption of different methods to prevent these
vibro-impact motions. For instance. Perera et al. [40] have suggest-
ed the use of 'properly sized torsional springs' fitted into the
bearing of a mechanism that has clearance to prevent the occurence of
the intermittent motion. The results that are presented in this work
are plausible to some extent. But. the optimum solution to this prob-
lem is a well-founded and indepth understanding of its nature. for
once this task has been accomplished. then sound and optimum preven-
tive methods can be developed. which are more likely to be effective.
Some recent work [82] has also been carried out on the effect of using
Idifferent types of materials. such as composites. in order to reduce
the acoustic radiation emitted from the links of mechanism vith bear-

ing clearances. which show some promising results.

For the interested reader. there is a vast number of publications
and investigations devoted to the subject of bearing clearances. which
focus on different aspects of the problem. and one can select to con-
centrate on any one of the aspects of this issue that suits one's

interest. These may be reviewed in a survey paper by Eaines [83].

CHAPTER TIO

ASSUMPTIONS AND GOVERNING MODES

In order to study and develop the governing equations of motion
of a planar four bar linkage with clearance in the coupler-rocker
bearing. some simplifying assumptions need to be imposed upon the sys-
tem. to ease the way for modeling the system. The primary use of
these assumptions is to eliminate some of the uncertainties associated
with the system which could significantly influence the outcome of the
simulation. Prior to listing these assumptions. the term 'nominal

mechanism' should be explained; this phrase refers to a mechanism with

no clearance in any of its bearings.

2.1- Assumptions

1. The nominal mechanism has one degree of freedom associated with a

crank shaft drive. running at constant angular velocity. In other
words. the motor driving the crank is theoretically capable of supply-

ing any magnitude of torque required to keep the crank speed constant.

2. The mechanism. be it nominal or otherwise. is perfectly planar.
with co-planar dynamic and static loads. This assumption eliminates
the forces and bending moments acting in the direction perpendicular
to the plane of the mechanism. thus reducing the problem to that of a

two-dimensional one.

3. All of the link and the bearing surfaces are rigid. implying that
the elastodynamic behavior of the system can be ignored. Eence. prob-
lems such as deflections of the links and/or nature of contact of the

bearing surfaces can be bypassed.

4. Friction is light. and can be neglected. In other words. there is
no energy dissipation in any of the bearings. Also any hydrodynamic
phenomena due to the lubrication of the bearings is assumed to have

negligible effect on the results.

5. The radial clearance in the coupler-rocker bearing is very small
compared to any dimensionally similar expression associated with the
nominal mechanism. The magnitude of the clearance is at least of the
order of 10" times that of any other longitudinal dimension in the

mechanism.

Upon imposition of these assumptions on the system. a theoretical

model can be constructed whose governing equations of motion and their

behavior will be investigated in the subsequent chapters. Since the
nominal mechanism plays a significant role in this study. some of its
governing equations. which will be referred to later. are braught here
(for a thorough derivation the reader is referred to the paper by
Smith and launder [72] ) with a typical plot of their behavior over
one full crank revolution. The ordinates of the plots are insignifi-
cant and hence are omitted. Figure (2.1) is a schematic view of the
nominal mechanism with its corresponding notation. Figures (2.2) to
(2.4) show the angular displacement. velocity. and acceleration of the

coupler link. respectively.

The angular displacements of the coupler and rocker. respective-

ly. can be expressed as follows

cos0.-A-R/o‘t[(AoB/n’)‘-(B‘-c‘)In’]"’ (2.1-la)
where

Ae2l,(l,cosG,-l,)

R-I.‘-1,’-I,'-1,‘+2111,co.e,

C-2l,l,sin0,

D-(A'+c’)"'

cos0.-(l,cos0,+l,cosG,-l1)ll. (2.1-lb)

 

FIGURE 2.1.

 

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The angular velocities of the coupler. and rocker. respectively

RIO

~.=-u.[1.sin(o.—e.)1/[1,.1n(e,-e.)1 (2.1—2.)

u.=-u,[l,sin(0,-0,)llllgsin(0,-0.)] (2.1-2b)

and. the angular accelerations of the coupler. and rocker. respective-

l’e Rt.

a,=(a,/u,)u,-[u,’1,cos(93-0.)
+u’,1.COO(O’-o"u‘zlgl/lp'in(°’.04)

(2.1-3a)

a‘=(u‘/m,)u,-[u,'l,cos(03-0,)
-g‘al‘co.(9,_o.)+.,’1,111..In(e,-o.)

(2.1-3b)

2.2- Nodes of Behavior

Figure (2.5) shows a four bar linkage with clearance in the

coupler-rocker bearing. Following considerable deliberation. cogita-

14

tion and. consideration of the physics of the problem. it is
postulated that the dynamic behavior of such a system can be primarily
contained within three modes at all times. This section is concerned

with explaining this hypothesis.

The first mode. called the "following" mode (Fig. 2.5). refers
to a particular configuration. in which the pin and the bushing of the
bearing under study are in contact at all times. The question of
whether the pin and bushing slip against each other or stick together
in any given time in this mode. is a matter that will be dealt with
later in its proper place. The second mode. referred to as the
"free-flight" mode. describes the system at times when contact has
been lost between the two components of the coupler-rocker bearing.
the pin and the bushing. In this mode. the pin moves along a particu-
lar trajectory path inside the bushing. Figure (2.6) shows the system
in the free-flight mode. The third mode. called the "impact" mode
(Fig. 2.7). which always occurs in succession with the free-flight
mode. primarily describes the system at the instant of time when the
pin collides with the bushing at the termination of the free-flight

.Od. e

Once the equations of motion for each mode are derived. they can
be related together using appropriate boundary and initial conditions.
to yield a set of equations which describe th complete behavior of
the system. The next three chapters are devoted to the derivation of

tile governing equations of each mode. and chapter six deals with

15

         

\\§1\\E\\\3\\=\\\- ‘u' .\u\\\\-\.\ 23“};

(a)- Following mode

- \ t “E

 

 

(c)- Impact mode

FIGURE 2.5. IODES 0F BEHAVIOR OF THE IECHANISI

16

relating these modes and their numerical solution along with the cor-
responding results. Chapter seven describes the experimental
instrumentation along with some digitised results. In chapter eight
the numerical and experimental results have been correlated. Finally.

chapter nine will contain the closing remarks.

CHAPTER THREE

THE.FOLLOIING IODE

In order to study the four bar linkage in the following mode. the
clearance in the coupler-rocker bearing can be represented as a mass-
less fifth link. whose length remains constant (as proposed by Earles
and In [ 3 ] and Grant and Fawcett [ 8 ]. Figure (3.1) shows such a
linkage with the notation that will be used throughout this chapter.
By examining this diagram. it is observed that such a system has a
total of seven degrees of freedom associated with.its dynamic motion.
These are the four angular displacements of the crank. coupler. rock-
er. and the 'clearance link'. along with the crank torque. and the X-
and I-components of the coupler-rocker bearing reaction. The latter

force acts in order to sustain the mechanism in a prescribed configu-

ration.

As suggested by several authors [3.5.10.29.79]. The Lagrangian

18

approach is employed here in order to develop the governing equations

of motions. i.e..

d/dt[dT/dqi]-dT/dq1 - 01+§(lj‘dfj/dqi) (3-1)

where

T : Total kinetic energy of the system
qi : Generalised coordinates

01 : Generalised forces

AJ : Lagrange multipliers

fj : Constraint equations

The following notes regarding equation (3.1) should be

considered;

1. The generalized coordinates. as previously mentioned. are 9,, 9,,

0‘, and 9c, .

2. The Lagrange multipliers include the remaining three unknowns.

namely.
1, : crank torque

i, : I-component of the coupler-rocker bearing reaction

19
A, : I-component of the coupler-rocker bearing reaction

3. The summation on the right hand side of the equation is over
parameter j. for j-l.....n .where n is the number of Lagrange multi-

pliers.

4. The number of constraint equations should be equal to the number
of Langrange multipliers. The first constraint equation can be
deduced from the first assumption stated in section (2.1) that. the
crank has constant angular velocity. This can be expressed in

mathematical form as

where the term t denotes time. In what follows an overdot on a param-

eter indicates the derivative of that variable with respect to time.

The other two constraint equations can be obtained by writing the

vector loop-closure equation of the mechanism (fig. 3.1). namely.

1,.I,.‘c..':.1" (3-3)

where the overbar indicates vector quantity.

However. any vector. L1, can be expressed in complex form as

20

[isLoeje-Lcos8+jLsin0 (3’4)

Thus. decomposing equation (3-3) according to (3-4) and seperating the
real and imaginary parts yields the two constraint equations. f3 and

f,. i.e..

f3-l,sin0,+l,sinG,+C,sin0c.-lgsin0‘ (3-6)

f,=l,cos0,+l,cos0,+C,cos0c.-11-1.cos0‘ (3-7)

5. The generalized forces. which will be discussed in detail later in
section 3.2. generally include both conservative and nonconservative

forces.

3.1-m1ximufiurnefmm

The total kinetic energy. T. of the mechnism (fig. 3.1) can be

expressed as

T3T3+T.+1“ (3 . 1'1)

where the subscripts 2. 3. and 4 denote the crank. coupler. and rocker

21

links. respectively. Note that there is no kinetic energy associated
with the clearance link for it is assumed to be massless. The objec-
tive is to express the total kinetic energy in terms of the angular
velocities of the links to be suitable for substitution into the

Lagrange's equation (3-1).

3.1.1- Crank kinetic Energy. T3

The kinetic energy of the crank can be formulated as (fig. 3.2)

Ta'(1/2’-.S.’-.’+(1’2)Ie,».’ (3.1.1-1>

which is readily in the desired form. and where IGi denotes the mass

moment of inertia of link i about its mass center.

3.1.2- Coupler Kinetic Energy. T,

Refering to Figure (3.2). the kinetic energy of the coupler may

be written as

T.a-(1/2).,v,‘+(Inna...3 (3.1.2-1)

However. using the law of relative velocities

22

 

 

FIGURE 3.1. IECHANISI IN THE FOLLOIING IODE

23

‘V,£VB+Vc/3 (3.1.2—2)

where V, is the absolute velocity of the mass center of the coupler

link. But.

v,=1,u, (3.1.2-3)
and

Vclrs.fl. (3.1.2-4)

Construction of the vector diagram of equation (3.1.2-2) is shown
in Figure (3.3). Refering to this Figure and using the law of cosines

V, can be written as

v,‘-1,’.,’+s,'.,'-21,s,.,.,co.(9,-9.) (3.1.2-5)

Substitution of equation (3.1.2-5) into (3.1.2-1) yields the

expression for the kinetic enegy of the coupler as

Ts‘(1/2).s[1s"s‘+ssa's3'21sss“s“s
cos(0,-G,)]+(l/2)IG'U,’

(3.1.2-6)

24

  

 

 

 

I 8
4L '
\ \Enrai (K '\ =\).\ 3 m:

FIGURE 3.2. VEOCITT DIRECTION OF THE (DUPLER'S IASS (ENTER

 

FIGURE 3.3. VECTOR REPRESENTATION OF V,

25

3.1.3- Rocker Kinetic Energy. T,

Observing Figure (3.2). the kinetic energy of the rocker can be

written as

T,=(1/2).,s,.,’+(1/2)IG‘u,' (3.1.3-1)

Substitution of equations (3.1.1-1). (3.1.2-6). and (3.1.3-1)
into equation (3.1-l) results in the final expression for the total

kinetic energy of the system in the desired form as

T%(1/2)m,s,3.18+(1/2)IG:uas+(1/2).’[lzsm‘s
+S,'m,’-21,S,m,m,cos(O,-—O,)]+(1/2)IG,u,a
+(1/2)-,s,.,‘+(112)16‘.,‘

(3.1-2)

3.2- Generalised 221221

The generalized 10:60.. Q,. are obtained using the Hamiltonian
method and principle of virtual work [80]. The reader can refer to

any advanced calculus or mechanics book for a thorough description of

26

these methods. Thus. for breviety. only the principal features of the

derivation of the generalized forces will be indicated here.

If p forces act upon a system. then the virtual work can be stat-

ed as

GFEJ F106;.i ngseeeaP (302-1)

Since :1. the vector position of the point of action of any one

of the forces. such as F,, cgn be written as

Ej=;j(q,.q,.....qn.t) (3.2-2)

where q, indicates the ith generalised coordinate and t represents the

time. Hence the variation of :1 can be expressed as

szfldzjldqflbmi-(d-rj/dq,)6q,+...+(d;jldqn)8qn (3.2-3)
Eence. equation (3.2-1) can be written as
eh}, (i, - (3?,qu1)6q.+F-jo (3?,Iaq.)sq,+. . .+§, . (aijlaqnmqnl

j'l....pp

(3.2-4)

27

On the other hand. one can express the virtual work as the pro-
duct of n 'generalizod forces'. Qk' acting over n generalized virtual
displacements 59k . The directions of these generalised forces coin-
cide with the corresponding directions of the generalized

displacements. so one writes

sv=o,sq,+q,sq,+. . wound-2, kaqk k=l. . . . .n

(3.2-5)

The generalized forces. Qk' take the place of the single forces
acting upon each particle. These forces form the components of an
n-dimensicnal vector in the configuration space. The configuration
space of a system is a point in an n-dimensional space. corresponding
to values of n generalised coordinates defining the instantaneous con-
figuration of the system. In other words. the configuration of the
system at any instant of time can be represented as only one point in
an n-dimensional reference frame whose axes are the n generalized

coordinates.

Note that the generalized force. Qk' does not necessarily
represent a force. Its units. however. must be such that Qk'qu has

units of work. For instance. 0* could be a moment. in which case bqk

represents an angular displacement.

Comparing equations (3.2-4) and (3.2-5) one concludes that

28

Qt‘Z Ej'(a;j/BQk) jsleeeesp (3.2-6)

The generalized force. Qk' in general contains both conservative
and nonconservative force fields. One can distinguish between the two
by noting that the conservative force. ch, can be derived from the
potential energy expression V. and the nonconservative force. anc'

may include internal dissipative forces or any external forces which

are not derivable from a potential function. so that

ok=oic+oknc (3.2-1)

The system under study (fig. 3.1) is assumed to be acted upon by
a conservative field. since the friction (dissipative) forces are
assumed to be negligible. and also no external forces (path indepen-
dent) act on the system. Note that the crank torque and the x- and
I-components of the coupler-rocker bearing reaction have been account-

ed for as Lagrange multipliers. For a conservative field

tenth-m.) (3.2-3)

because in such a field. the work is the negative of the change in the

potential energy. But

"4.9.2k chbqk (3.2-9)

29

and

6'°(qk)--svtqk)=-2, (dV/qu)6qk (3.2-10)

Comparison of the equations (3.2-9) and (3.2-10) yields

ch"°v’3qk (3.2-11)

3.2.1- Total Potential Energy. V

Refering to Figure (3.2). the potential energy of the crank can

be written as

V,=m,gS,sin0, (3.2.1-1)
and that of the coupler is

V,=m,g(l,sin0,+S,sinO,) (3.2.1-2)
and for the rocker

V,-m,gS,sinO, (3.2.1-3)

Renee. the total potential energy of the mechanism can be

_expressed as

‘.
x
\ \
\‘

Vim,gS,sin0,+m,g(l,sin0,+S,sin9,)+m..gSgsin04 (3.2.1-4)

30

Application of equation (3.2-11) to (3.2.1-4) yields the expres-

sions for the generalized conservative forces as

0.0"(-.S.+-.1.)sc«0. (3.2-12.)

Qac"-.s8.c°89. (3 .2-12b)

Q0, =0 (clearance link) (3.2-12c)
c

Qgc=-m,gS,cos0, (3.2-12d)

where g in the above equations represents the component of the

gravitational acceleration in the plane of the mechanism.

3.3- Resulting Egpgtigpp 2f lotion

Raving all the parameters expressed in the desired form. the
Lagrange's equation (equation 3-1) can now be written for each gener-
alised coordinate. The explicit expression of equation (3-1) on the

first generalised coordinate. 0,, is

31
‘11-(l,cos0,)13+(l,sin0,)k,+[m,l,S,cos(0,-0.)la,
=-m,gS,cos0,-m,gl,cosG,-m,l,S,sin(0,-G,)u,’
(3.3-1)
Note that e, is zero.

For 0, we have

“(1,cos9,)x,+(l,sin0,)A,+(IG.+m,S,')a,
=—-,;s,co.e,+-,1,s,.1n(e,-e,)m,‘
(3.3-2)

and for Oe we have
s

-(C,cosOc.)Ag+(C,sin0°’)A,=0 (3.3-3)

and finally. for 9, one our write

(l,cos0g)A,-(l,sin0,)h,+(IG‘+m,S,’)u,=
-m,gS,cos0,

(3.3-4)

Assuming that. at the time of start the angular displacements of

all the links are known. along with the angular velocities of the

32

links (section 3.4). the only remaining unknowns in equations (3.3-1)
to (3.3-4) are the angular accelerations of the links ( oi, i-2.3.4.c,
) and also the Lagrange multipliers ( 1,, j-1.2.3 ). Hence. at ini-
tial condition (section 3.4). there are six unknowns embedded in these
four nonlinear equations. The other two necessary equations to solve
for the unknowns can be derived from the constraint equations f, and
f, (equations 3-6 and 3-7). which represent the real and imaginary
components of the loop closure equation of the mechanism. However. in
order to have these two latter equations be compatible with equations
(3.3-1) to (3.3-4) in terms of the linearity of the unknowns. the
second time derivative of these constraint equations shall be used.

Thus. equation (3-6) yields

a’t,/at’=o
or
-(1,sin0,)u.+(l,sin0,)u,+(C,sin0°’)u°’
‘lgcos0,a,'-l,cos0,m,’-C,cos0c.00.’-l,cos0,e,’
(3.3-5)
and equation (3-7) yields

a‘r,/at.'°

33
or
-(1,cos0.)u.+(l,cos0.)u,+(C,coch.)u,
I=1,sin0,m,’+l,sin0,u,'+C,sin0¢.cun¢.’-l‘,sin0.m.3
(3.3-6)
Note that. the constraint equation f, (equation 3-2) which states

that the crank has a constant angular velocity. has not been ignored.

rather its first time derivative .i.e.
af1/3t30,=0

has been substituted into the other six equations to eliminate a, .

A glance at the equations (3.3-1) to (3.3-6) reveals the fact
that these equations are nonlinear in terms of the angular displace-
ments and velocities. However. they are linear in terms of the angu-
lar accelerations and the Lagrange's multipliers. Hence. they can be

written in a matrix form as

[ A ][ I ]=[ B J (3.3-7)

34

where matrix [ A ] is

 

 

-1 'l,cos0, l,sin0, 0 m,l,S,cos8 0
0 -l,cos0, l,sin0, 0 IG’+m,S,3 0
0 -C,cos0°. C,sin0c’ 0 0 0
0 1,cos0, -l,sin0, IG‘+m,S,' 0 0
0 0 0 -l,sin0, l,sin0, C,sin0c’
0 0 0 'lgcosG, l,cos0, C,cos9c
(3.3-8a)
'here 8=O,-0,. Vector [ X ] can be written as
P—
i A, 1
18
i, (3.3-8b)
“a
“a
“c,
L 41

 

 

35

and vector [ B ] is

r- —,
“‘183.°°‘9.’I.s1.cosO,-m,1,8,sin(0,-0,)e,’

-m,gS,cos0,+m,l,S,sin(0,-0,)e,3
0 (3.3-8c)
-m,gS,cos0,

l,cos8,a,'-l,cos0,a,'-C,cos0c'u '-l,cos0,m,

l,sin0,e,'+l,sin0,a,’+C,sin0°’u ’-l,sin9,u,’

 

——1

 

3.4“ 1311111 921115122!

Since the solution of equation (3.3-7) is an inital value prob-
lem. the state of the system at some time is required. The criteria
for selecting a particular instant of time as the initial time is not
universal and primarily depends on the geometrical aspects of the
mechanism under study. For instance. Earles and In in their paper [3]
use a certain position called the dead-center-point (d.c.p.). which is
referred to the instant of time when the crank and the coupler are in
line. and the clearance (being in the crank-coupler bearing) link has
a translatory motion without any rotational component. Obviously.
this criterion cannot be generalised to be applicable to any typical

mechanism. for this condition requiring the existense of a so-called

36

d.c.p. would only occur in a few mechanisms with similar geometric

dimensions to that of the Earles and Vu's.

However. a rather customary approach is to assume that at the
initial time. the clearance link's direction is coincident with the
direction of the bearing force in a corresponding 'nominal mechanism'.
Such an assumption is relatively well-founded. as investigated by
Haines [1]. In this paper. the expression for the angle of lag. S. of
the clearance link relative to the bearing reaction has been derived.
Then it is shown that with light damping. the system will rapidly
approach a state where the angle of lag is negligibly small. At any

rate. for reasons to be discussed in six. the determination of the

initial condition will be based on a trial and error approach.

3.5- 12121221122 21 £22 221122121 !222

As previously mentioned in this chapter. the fundamental assump-
tion of the following mode is the representation of the clearance as a
massleas fifth link. whose length remains constant. However. at some
time t the contact between the pin and the bushing within the
coupler-rocker bearing is lost and the above assumption no longer pre-
vails. Once this occurs the following node's equations of motion
cease to apply. Honce. the task is now to determine such a point in

time when contact-loss occurs between the bearing components.

37

Again. this problem has had no unique method of approach. and it
is. to some extent. based on intuition and the characteristics of the
mechanism under study. Also. the accuracy and capabilities of the
available digital computer play a significant role. as to how sensi-
tive the method of contact-loss detection should be. Earles and In in
their work [3]. have adapted a criterion for determining the occurence
of contact-loss which is solely dependent on the shape of the polar
force plot of the bearing under study. In other words. contact-loss
is assumed to take place when the force vector plot passes through the
origin with a finite slope. Obviously. such a method seems to be too
specialized and cannot be given a universal status as a method of

determination of contact-loss.

The second criterion proposed by Earles and Wu [4] is based on
monitoring the rate of change of direction of the bearing reaction
under study. It states that. the contact-loss positions are located
at the points where the rate of change of direction of the force in
the bearing is maximum and. simultaneously. the magnitude of this
force is at or near its minimum value. This may be formulated

mathematically as

(ygll-in) ) 1.0 (rad/N/sec)
where

'1. : rate of change of direction of the bearing reaction

38

Ruin : the corresponding minimum value of bearing reaction

This method. although being more general than the preceding cri-
terion. is not sufficiently accurate and sensitive. According to
Haines [83]. impacts (a subsequent of contact-loss) have been recorded
(albeit small) for (70’nmin) as low as 0.32. whereas no impacts have
been recorded (at small clearance) in one case where this term. is as

high as 220.

Grant and Faucett [84]. have interpreted the latter criterion
develcped by Earles and In [4] as indicating that the actual time when
contact is lost between the pin and the bushing of the bearing cor-
responds to the instant of time when (yganin) equals unity
(rad/N/sec). But. this condition would be met in rather specialized
mechanisms and. again. cannot be generalized. In another case [8].
Fawcett et. al. have made use of the fact that the 'clearance linki
can only be in tension. Hence. once in compressive force field. con-
tact-loss occurs. However. it has not been clarified as to how these

two conditions are identified.

The distinct but interrelated methods have been adapted here to
predict the occurence of contact-loss. which are basically modified
forms of those used by Earles and In [3]. and Grant and Fawcett [84].
The first method is based on the angular displacement of the
'clearance link' relative to the reaction force in the bearing.

Figure (3.4a) shows the bearing at a time when there is contact

39

  

(a)- Continuous contact

TANG EJT

(b)- Imminent contact-loss

 

FIGURE 3 .4. (”UPLER-ROCIER BEARING

40

between the pin and the bushing. while Figure (3.4b) illustrates the
imminent occurence of contact-loss in the bearing at some later time.
In a sense. these two figures depict the clearance link as being in
compression (contact) and tension (contact-loss). respectively. In
other words. if the bearing reaction. F. is in a direction such that
the clearance link would acquire an acceleration directed towards the
center of the bearing. then the bearing components would seperate and
lose contact (fig. 3.4b). 0n the other hand. so long as the direc-
tion of the acceleration is inclined towards the 'outside' of the
bearing. it implies that the pin is pressing against the bushing.

indicating a continuous contact between the two (fig. 3.4a).

The second method detects the occurence of the contact-loss by
sensing the magnitude of the bearing reaction. That is. contact is
lost if the magnitude of this force drops to small values in the
neighborhood of zero. indicating that contact-loss is imminent. This
method acts as a complimentary sensor to the first one and has been
developed in order to compensate for the precision errors generated

during the numerical solution of the equations of motion.

Once contact has been lost between the pin and the bushing. the
system no longer abides by the equations of motion developed for the
following mode. At this point the mechanism enters the second mode.

called the free-flight mode. which is discussed in the next chapter.

CHAPTER FOUR

mE FREE-FLIGHT MOE

Once the criterion for contact-loss has been met. the pin in the
coupler-rocker bearing seperates from the bushing. and the mechanism
enters the free-flight mode. Unlike the following mode. the system
does not act as an integrated mechanism in this mode. but rather as
two independent systems (fig. 4.1) with specific boundary conditions
and constraint equations. The crank and coupler represent a compound
pendulum. while the rocker. which is seperated from the coupler.
represents a simple pendulum. As suggested by Hansour et. al.
[5-7]. to find the governing equations of motion of these two pendu-
lums a Newtonian approach appears more attractive than the Lagrangian
method. since there are less number of unknowns in this mode than in
the following mode. Note that the only assumption made at this stage
is that the driving torque acting on the crank is ignored in order to

reduce the degree of indeterminacy of the compound pendulum. and make

42

uni gaming—mm ma 5 Sammy—SUN! .n.v mun—cum

 

 

 

 

i.e..—REE 3..an

 

 

wanna: 298:8

  

43

it possible to find the remaining unknown parameters. However. the
assumption of the crank having constant angular velocity still pre-

vails.

4.1- Cogpgugd ppndplum (pggnk ggd 223213;)

Figure (4.2) shows the crank and coupler in an integrated manner
along with the pertaining notation. Figure (4.3) shows a free-body
diagram of the crank. Iriting the two equations of motion in the X-

and T-direction yields

§Fx=m,ax

where ax is the absolute acceleration of the mass center of the crank

in the X-direction. Hut.

txsansinB-S,m,’sin(9,-n/2)
or

axu-S,e,'cos8, (4.1-2)

where. hereafter. subscripts n and t denote the components in the

44

 

 

 

‘4

O, x
1s

"\ ‘.‘. .3

ER)
(DUPL

R AND

UND PENDULUH (CRAN

NIPO

4.2.

FIGURE

45

positive normal and tangential directions. respectively. Combining

equations (4.1-1) and (4.1-2) yield

£,-R,=-.,s,u,‘co.o, (4.1-3)
also

Erya't'y

Fy—ny-."n.,‘y (4.1-4)
and

ay=-ancos8-S,m,'cos(0,-n/2)
or

.ya-s,.,‘.1ne, (4.1-5)

Combining (4.1-4) and (4.1-5) results in

Ry-Ry=.,(.+s,.,’.1ne,) (4.1-6)

Taking moments about the pivot point A (fig. 4.3)

ElAflA‘n,-0

since u,-O

46

 

 

A
v

FIGURE 4.3. (RANK FREE-BODY DIAGRAM

47

Note that Iij denotes the mass moment of inertia of link j about

point i. Hence.

‘Fxl,sinO,+Fyl,cosO,=m,gS,cos0, (4.1-7)

Equations (4.1-3). and (4.1-6). and (4.1-7) are such that all the
terms on the left hand sides are unknowns. and. the ones on the right

hand side are known.

Figure (4.4) is a free-body diagram of the coupler. Again.

applying the first two Newton's equations results in

2Fx=I,RBx
or
and

§Fy3-..By
or

Fy-.-’(‘8i+‘) (4.1'9)

Applying the moment law about point A for this link yields

48

 

X

 

FIGURE 4.4. COUPLER FREE-BODY DIAGRAM

49

E'E‘IE,“s

or

FyS,cosO,-FxS,sinO,=IE.u, (4.1-10)

However. in equations (4.1-8) to (4.1-10). the x- and

Y-components of the linear acceleration of the mass center. point E.

0f th. coupler, ‘E and ‘E, respectively. and the angular acceleration
x

of the coupler link. u,. along with the forces Fx and Fy are unknown.

The absolute acceleration of the mass center can be written in vector

form using the laws of non-inertial reference frames [73.74]. Thus.
IE‘;R+:E/R (4.1-11)

where :1], denotes the acceleration of point i relative to point j.

Decomposition of (4.1-11) into its 1- and I-eomponents yields

‘R,"E,+'E/Rx (4.1-12.)

'Ey‘wyfln/Ry (4.1-12b)
But

ans-lge,'eos0, (4.1-l3a)

any-1,e,’sine, (4.1-13b)

50

also
‘EIB‘="En°°'°I"Et‘1n°I (4,1—14.)
'E/By"‘En'in°I+‘Et°°'°3 (4.1-l4b)
where
.Enas,.,’ (4.1-15.)
ant-S,a, (4.1-15b)

Combining equations (4.1-12) to (4.1-15) yields

.E 8-l,u,‘cosO,-S,u,’cosO,-S,u,sin0, (4.1-16)
x
and

.E -1,.,‘.ine,-s,.,’.1ne,+s,a,co.e, (4.1-11)
y

Hence. there are six unknowns. namely. Fx' F , Rx' By, 3,, and
0,. The six equations needed to find these unknowns are (4.1-3).
(4.1-6). (4.1-7). (4.1-10). (4.1-8). and (4.1—9). where in the latter
two equations. the acceleration components sax gnd .Ey hgvg been

replaced by equations (4.1-16) and (4.1-17). respectively.

For ease of reference. these six equations are rearranged and

written in the following

51

Fx-Rx=-m,S,m,’cosO, (4.1-18a)
Fy-Rf-Jpswfung) (4.1-18b)
Fxl,sinO,-Fyl,ccs0,=-m,gS,cos0, (4.1-l8c)
-m,S,sin0,u,-m,S,co .e,.,’+R,=-,11,.,’co.e, (4.1-18d)
m,S,cosO,u,-m,S,sin0,e,'+Fy=m,(l,m,’sin0,+g) (4.1-l8e)
IE'a,+FxS,sin0,-FyS,cosO,-0 (4.1-18f)

Equations (4.1-l8c) to (4.1-l8f) can be solved simultaneously and
independent of the first two. and then. (4.1-18a) and (4.1-l8b) can be
solved for R, and Ry seperately. Iriting the last four equations of

(4.1-18) in a matrix form we have

I A ][ X ]={ B ] (4.1-19)

where matrix [ A ] is

 

 

‘FII 0 1,sin0, -l,cosG;-
-m,S,sin0, -m,S,cos0, l 0
m,S,cos0, -m,S,sin0, 0 1
IE, 0 S,sin0, -S,cos0,

L. .1.

52

and. vector [ X ] is

F!

L J

and where vector [ B ] is

1‘ 1'

“I,gS,cos0,

 

 

I,l,m,3cos8,

I,(l,m,'sin0,+g)

 

 

Solving the matrix equation (4.1-19) yields the angular accelera-

tion of the coupler as

a,-(-,s,/IE,)[1,.,’.1n(e,-e,)-....e,] (4.1-20)

The expression for other unknowns in (4.1-19) have been omitted

here. and will be stated later as deemed necessary.

53

4.2- Sigple Pendglgg isostarl

Figure (4.5) illustrates the rocker as a simple pendulum. The
task of finding the angular acceleration of this link is less tedious
than the case of the compound pendulum. and since it only involves

taking the moments about point 0, (fig. 4.5)

81
2-0. on.

02'

u,--(m,S,/Io‘)g-cos8, (4.2-1)

Hence. at this stage. the two required governing equations of
motion. equations (4.1-20) and (4.2-1). of the four bar linkage which
define the behavior of the system in the free-flight mode are
obtained. These equations may be solved and integrated numerically
using a digital computer. and other system parameters can be computed

at any time once these two equations are solved.

4.3- 1111111 2211212122.!

As previously mentioned. the equations of motion of the
free-flight mode are to be integrated numerically. Hence. some ini-
tial values are needed to start the iteration. Suppose at time t-O

the simulation starts with the mechanism assumed to be in the follow-

 

54

 

 

FIGURE 4.5.

ROCKER FREE-BODY DIAGRAI

55

ing ‘040- If 4t til! t‘t, the pin and the bushing seperation phase
has reached (refer to section 3.5). and the system enters the
free-flight mode. then the initial values taken for the equations of
motion of this mode are the corresponding values of the parameters in
the following mode just prior to contact-loss. If the mesh size of
iteration is small enough. the magnitude of the error tends to become
negligible and does not enter the simulation results as a propagation

error.

4.4- Igggination 91 the Free-Flight Node

The interval of time during which the mechanism is in the
free-flight mode is finite and bounded. The lower bound of this
interval is marked by the occurence of the contact-loss in the
coupler-rocker bearing. Then. as time progresses. the pin travels
along a trajectory path within the bushing until it collides with the
bushing at some point in time. which is an indication of the termina-
tion of the free-flight mode. The extent of time-lapse for the
duration of this mode is not fixed but depends on the relative loca-
tion of the pin and the bushing. and in addition. on the kinematics of

the two components of the bearing at the instant of contact-loss.

Once the pin is travelling inside the bearing housing in the
free-flight mode. it is not constrained to move along a path pres-
cribed by the inner surface of the bushing. as is the case in the

following mode. Hence. the distance between the pin and the center of

56

the bushing. which was taken as the length of the clearance link in
the following mode. becomes less than the actual clearance size of the
bearing. Thus. the optimum method to ensure the prompt detection of
the collision of the pin and the bushing is to monitor the distance
between the pin and the center of the bushing. Vhen this distance
equals the clearance size of the bearing. it is an indication of the
collision of the two surfaces. hence the termination of the

free-flight mode.

Such a task can be performed on a digital computer by making use
of the loop closure equation of the mechanimm after each step of
iteration. Figure (4.6) shows the mechanism in the free-flight mode

whose leap closure equation in vector form can be written as

T +1',+" IpI. (4.4-1)

where a is the vector extended from the pin to the center of the bush-
ing. Decomposing (4.4-1) into its real and imaginary components. and

rearranging yields

-0cose-1,cos0,+l,cos0,-l,-l,ccs0, (4.4-2a)

-0sinuPl,sinO,+l,sin0,-l.sin0g (4.4-2b)

where n denotes the angular displacement of the vector '5 . Squaring

57

1.

Ag 94

FIGURE 4.6. VECTOR PRESENTATION OF THE FREE-FLIGHT UDE

58

and adding equations (4.4-2) results in

0’=(l,cos0,+l,cos0,-l,-l,cos04)1+(l,sin0,+l,sin0,-l.sin9.)3
(4.4-3)
If the clearance size in the bearing is denoted by C . then the

collision between the pin and the bushing occurs at the instant of

time when

oz: or o'sc‘ (4.4-4)

Hence. from equations (4.4-3) and (4.4-4) it can be deduced that

the constraint eqauticn for the free-flight mode can be expressed as

o‘<§’
or
(l,cos0,+l,cosO,-l,-lgcos0.)1+(1,sin0,+l,sinO,-lgsin6.]a“a

(4.4-5)

In other words. the free-flight mode is terminated once the ine-
quality (4.4-5) is violated. At this point the mechanism enters the
third mode. namely. the impact mode. which is discussed in the next

chapter.

CHAPTER FIVE

THE IIPACT IODE

The termination of the free-flight mode triggers the initiation
of the third mode. namely. the impact mode. As previously mentioned
in chapter one. a great deal of literature is available on the subject
of impacts in the bearings with clearances and some related areas.
According to Nansour [5]. some of the analytical research carried out
in the past [3.44.75-77] considered the idealixation of the contacting
surfaces by linear and nonlinear springs and dashpots. Veluswami.
Crossley. and Horvay [78] showed experimentally that the dynamic beha-
vior. the impact pattern. and the time of contact of impacting
surfaces constitute a complex function of initial conditions.
approaching velocities of the colliding surfaces. and the size of the
clearance in the bearing. Nanscur and Townsend [6.7] developed a dif-
ferent model which did not use the spring-dashpot approach. It was

tnailt around the momentum-exchange principles for impacting pairs.

60

This is the method that will be adapted here. In this method.
instead of following the rather difficult traditional path of trying
to define the force-time history during the impact. the total area
under that curve is evaluated. This area. which is essentially the
impulse associated with the impact. is all that is needed to determine
the subsequent motion of impacting pair. This approach bypasses many
unsettled issues regarding the exact nature of the stiffnesses of the
contacting surfaces. and its dependence on the actual contact time.
It also enables one to account for the actual geometry of the joint as
well as for the surface roughness. These are what constitute the set
of parameters which were difficult to handle by the previous

approaches.

However. the primary difference between the work presented here.
and Iansour's investigation is that. he has modeled the mechanism as
being in only two modes of free-flight and impact mode. and does not
take into account the intervals of time when the pin and the bushing
of the bearing with clearance are in continuous contact. This defi-
ciency is eliminated here by augmenting the third mode. or the
following mode. to the first two modes. hence encompassing all the

possible configurations the mechanism can acquire.

5.1- 2911:1111 222111234 of 111220

Figure (5.1) depicts the four-bar linkage at the instant of

impact. along with its corresponding notation. This notation will be

61

man: HU¢HIH um? 2H

lmHz<uuml

.a.n mmaenh

Va

62
clarified during the course of the derivation of the equations of

motion.

As discussed in the preceding chapter (section 4.4). impact

occurs when the following condition is violated

u.’+u.’ < t' (5.1-1)
where

“1:13°o'03+1’00‘0.-1:-1‘°0.e‘ (5.1-2‘)

u,-l,sin0,+l,sin0,-l,sin0, (5.1-2b)

and where t is the size of the clearance.

Dubowsky and Freudenstein [76] estimated that the duration of the
impact of a typical pin-connection is in the neighborhood of 10"
seconds. This time is associated with about one degree-displacement
for the driving link with operating speed of 2000 RPR or about 209
radians per second. Consequently. one can consider that the configu-
ration of the mechanism is hardly changing during an impact with that

speed.

Contact between the pin and the bushing at impact occurs at a

clearance link's angle of q (fig. 5.2) given by the relation

 

34.
V
{L
E
9:
A A
.
12
‘
U
’ a
\°‘
2. ‘* 1:. =
02

FIGURE 5.2. CRANK AND COUPLER LINKS AT INRACT

64

tann=u,/n, (5.1-3)

where n, and n, are given in equations (5.1-2).

In what follows. the superscript (+) indicates a variable after
impact. The principle of angular impulse-momentum [73] can be stated

I j Emma’s, (5.1.-4)

where the integral in the above equation. which is vector form. is
from t, to t,. which is the duration of the impact. and denotes the
angular momentum of the system. and the right hand side is merely the
change in the angular impulse of the mechanism about a point 0 . The

vector Ho indicates the angular momentum about point 0 . If point 0

is the mass center of the system. then

Ho-Ie (5 .1-5)

where I is the mass moment of inertia of the body about the mass
center. and e is the angular velocity of the body. However. if point

0 is different from the mass center. then

65

Ho-Ia+de (5.1-6)

where m is the mass. V the linear velocity of the mass center. and d

is the distance from point 0 to the mass center.

5.1.1- Equation of Notion for Crank

Figure (5.2) shows a free-body diagram of the crank and coupler
at the instant of impact. Rewriting equation (5.1-1) for pivot O, we

have

I, loadt=(Hoa),-(Ho‘), (5.1.1-1)

where the integration interval on time. t. is from t, to t,, and where
subscripts l and 2 on Ho: denote the variable at times t1 and t,.
respectively. Also. for convenience. the vector notation has been
omitted without loss of correctness since the equation is written for
only one dimension at a time. Keeping in mind that the angular accel-
eration of the crank is zero at all times. the following expression

may be derived

(30’),-(Ia,+m,V,S,), (5.1.1—2)

and

66

(Hog),=(Iu,+m,V,Sz)a ‘ (5.1.1-3)

The linear and angular velocity terms in equations (5.1.1-2) and
(5.1.1-3) are unchanged before and after impact. Thus. the right hand
sides of the above two equations are equal . This implies that the
angular momentum of the crank is conserved and remains constant.
Hence. the magnitude of angular impulse on this link due to the impact
in the coupler-rocker bearing is zero. Consequently. the equations
corresponding to this link can be omitted altogether from the

remainder of the analysis.

5.1.2- Equations of Notion for the Coupler

Application of the angular impulse-momentum equation (5.1-1) to

point A of the coupler (fig.5.2) yields

filAdt-(HA),-(HA), (5.1.2-1)

From here on. the limits of integration on all the integrals are
eliminated. keeping in mind that the limits are for the duration of
the impact. from t, to t,. If Fn anf Ft (not shown on the figure) are
forces in the normal and tangential directions with respect to the
direction of approach. which are exerted on the pin at the end of the
coupler at impact. respectively. then the moment of all forces acting

on the coupler about point A can be written as

67

ElAf-1,Fncose-1,Ftcos(n/2-u)
or

2'A‘“1.Fn'in(n-9. )-1.F.cos (vi-0.) (5.1 .2-2)

Note that (5.1.2-2) is written for the general case. in the sense that
Fn represents the force due to the collision of the approaching sur-
faces in the normal direction. and Ft indicgtsg the force in the

tangential direction due to the generation of friction between the two

contacting surfaces at the time of impact.

Integrating equation (5.1.2-2) with respect to time from t, to

t,. which is the duration of impact. yields

I.ElAdt--I1,Fnsin(n-0,)dt-Il,Ftcos(n-0,)dt

(5.1.2-3)

As discussed in section (5.1). the configuration of the mechanism
hardly changes during impact. This implies that the angular displace-
ments can be assumed constant. Thus. taking the invariant parameters
in equation (5.1.2-3) outside of the integrals on the right hand side

results in

68

JimAdt=-l, sin(n-0, )IFndt-l, co s(1|-O, )thdt

(5.1.2-4)

The integrals on the right hand side of (5.1.2-4) are but expres-
sions for the 'impulses' in the normal and tangential directions of
the bushing. respectively. From here on. the normal impulse will be

denoted by F. and the tangential impulse by f . i.e..

pajrnd. ; r-jrtat (5.1.2-5)

Note that F and f are vector quantities as shown in Figure (5.1).
that have the dimensions of FORCE x TIRE. They will be refered to in
this study as impulsive forces for ease of identification. The impul-
sive reactions at the bearings are also defined as the time integrals
(DI the large reactions at the bearings during impact. Similarly. t
denotes the time integral of the large torque acting on the crank from
time t, to t,. It is referred to as the impulsive torque with the

dimensions of FORCE x LMTH x TIIE.

Substitution of equation (5.1.2-5) into (5.1.2-4) yields the

OXpression for the total impulse on the coupler. i.e..

ElAdt-dfisinm-O,)-l,fcos(n-G,) (5.1 .2-6)

The angular momentum of the coupler link about point A (fig.

69

5.2) can be written as
HA'tln’u,+m,VB’S, (5 .1 .2-7)

where VB, is the linear velocity of the mass center. E. of the coupler

relative to point A. Noting that
VB’=S,m,

equation (5.1.2-7) becomes
nA,'IE,”s+'sss’“a

‘IA.H. (50102-8) '

Hence. the change in angular momentum of the coupler during

impact becomes
AHA,'(HA,)8’(RA,’1 (5.1.2-9)
“As-IA (of-u.) (5.1.2-10)
3

Substitution of equations (5.1.2-6). (5.1.2-9). and (5.1.2-10)

into (5.1.2-1) yields

7O

-l,Fsin(n-O,)-l,fcos(n-G,-IA'(m,+-m,) (5.1.2—11)

which is the desired angular impulse-momentum equation for the

coupler.

5.1.3- Equtions of lotion for the Rocker

Figure (5.3) depicts a diagram of the rocker at the time of
impact. with corresponding normal. F. and tangential. f. impulses act-
ing on the bushing. Using a similar argument as in the derivation of
equation (5.1.2-10). the change in the angular momentum of the rocker

about the pivot 0. can be written as

Eo‘sro‘., (5.1.3-1)'

and the change in angular momentum becomes

An¢,=(no,),-(no‘).

0t

Aflo‘sro‘(a.*--.) (5.1.3-2)

Since the change in the angular momentum is with respect to point

0,. the angular impulse should also be expressed about the same point.

71

 

FIGURE 5.3. ROCKER LINK AT IMPACT

72

Taking moments about point 0, yields

2 N0‘=-Fnl,cos(n/2-0g+n)+Ft[l.cos(Gg-n)+§]

(5.1.3-3)

where. again. Fn and Ft (not shown on the figure). are the impact
forces whose corresponding integrals with respect to time yield the
impulses F and f. respectively.

Integrating (5.1.3-3) with respect to time gives

JENo‘dt=-IFnl,cos(p/2-O,+n)dt+th[l,cos(0.-n)+§]dt

(5.1.3-4)

and since the angular displacements remain constant. equation

(5.1.3-4) becomes

IEMo‘dtt-l.cos(n/2-0g+n)andt+[l.cos(0.-n)+§]IFtdt

(5.1.3-5)

Substituting equation (5.1.2-5) into (5.1.3-5) yields

73

I.Emo‘dt--Fl.cos(n/2-G.+n)+f[l.cos(O.-n)+§]

(5.1.3-6)

Equating (5.1.3-2) and (5.1.3-6) yields

Fl,sin(n-G,)+f[l,cos(n-O.)+§]'Io‘(fl.+"s)

(5.1.3-7)

which is the angular impulse-momentum expression for the rocker link.

A glance at the above equation and equation (5.1.2-11) reveals
that these equations contain four unknown parameters. These are the
normal and tangential impulses during impact. F and f. and also the
angular velocities of the coupler and rocker after impact. m,+ and
e,+. respectively. The remaining variables in these relations are
quantites that are known from the end values of the preceding
free-flight mode. Thus. two more equations are needed in terms of
these parameters in order to find the unknowns. These two will be
obtained by utilising the theoretical arguments presented in the fol-
lowing section concerning the coefficient of restitution. and also.
the effect of the smoothness of the surfaces of the colliding bodies.

namely. the pin and the bushing of the coupler-rocker bearing.

74

5.2- Qgetfigient 91 Rggtitgtion. g

The definition of the coefficient of restitution. e. provides the
third equation by relating 0, and “i+' since the values of these
parameters as well as the impulses depend on the coefficient of resti-
tution between the pin and the bushing. The parameter c formally
relates the normal impulse of the approaching (compression) phase. to
that in the receding (rebound) phase in the direction of the line of
impact. which is essentially the n direction (Figures 5.2 and 5.3).
For an external view of the impacts. an equivalent relation is used in
terms of the normal velocities of approach (Vb. and VB, ) before and

n n
after impact. Thus. the coefficient of restitution is defined such

that

4

vB T-VB +=e(VB -v3 ) (5.2-1)
n 'n 'n ‘n

Figure (5.4) on the following page illustrates the crank and
coupler at a given configuration just prior to impact. and Figure
(5.5) depicts a graphical representation of the velocity vectors of
the two links. As can be seen from Figure (5.5) on the next page. the
absolute velocity of the coupler end. Vb’, is the vector addition of
the crank end absolute linear velocity. VA‘, whose magnitude and
direction is 1,0, perpendicular to the crank at point A, in the direc-
tion of crank rotation. and the linear velocity of point B, gelstive

‘9 IKDint A,. which is directed normal to the coupler at B, in the

direction of rotation of the coupler. and with magnitude of 1,0,.

75

1s

0s

\‘h “:1, 3,? u?“ 7 “tr-:3“;

0s

FIGURE 5.5. (RANK AND (DUPLR PRIOR TO IMPACT

76

 

FIGURE 5.5. VECTOR REPRESENTATION OF VEOCITY VB,

77
Line x-x in Figure (5.5) represents the direction of the normal

approach of the pin and the bushing. Thus

VB, =-l,u,cos8+l,u,cos6 (5.2-2)
n
But.
B=n~c
a=0,-u/2
or
B=nl2+(n-9.) (5.2-3a)
8=n-(y+n)
1-n/2-O,
or
B'n/Z‘F (on) (5 .2-31.)
Substituting equations (5.2-3) into (5.2-2) and rearranging
yields

Vb, -l,u,sin(n-O,)+l,m,sin(n-O,) (5.2-4)
n

78
Equation (5.2-4) is an expression for the normal velocity of

approach of the coupler end.

To find the corresponding equation for the rocker. reference is
made to Figure (5.6) on the next page. which depicts the rocker at a
given position prior to impact. It should be noted that. 1,. which is
assumed as the rocker link with angular displacement 0,, spans between
point 0, and the center of the bushing. point C. and. p which is an
imaginary line with angular displacement 1. extends from the pivot
point 0, to the point of contact of the pin and the bushing. 8,. u is

the difference in the angular displacements of l, and 9.

Figure (5.7) is a vector presentation of the absolute velocity of
point B. at the instant of impact. It is directed perpendicular to p.
and has a aasnitndo of 90.. Line x—x reflects the direction of normal
approach of the pin and the bushing. and line Y-! is the direction of

the imaginary line 9. Hence

VR, -vg,co.e (5.2-5)
n
or
Vh‘ m—pu,cogb (5.2-6)
n

But. from Figure (5.7)

79

 

FIGURE 5.6. ROCKER PRIOR TO IMPACT

 

FIGURE 5.7. VECTOR REPRESENTATION OF VB
4

80

6=nl2+(n-7) (5.2-7a)
and
then
88n/2+(n-O,+e) (5.2-7c)
Substitution of (5.2-7c) into (5.2-6) and rearranging yields
Vh‘ -pe,sin(n-O,+u) (5.2-8)
n
Angle u can be found using the law of cosines in triangle O,CB,.
i.e..
{/sinuPp/sin(0,-n)
or

sinuP(§/p)sin(O,-n) (5.2-9)

The magnitude of p can be found by writing the vector loop clo-

sure equation for the triangle 0,CB. (fig. 5.6). i.e..

p: +2. (5.2-10)

81
Decomposition of (5.2-10) into its real and imaginary parts

yields

P‘inflFlgsinO,+§sinn (5.2-11a)
PcOtu'l.cosO,+§cosn (5.2-11b)
Squaring and adding of the resulting expressions yields
pali,‘+¢‘+2i.§cp.(n-o.)11’2 (5.2-12)
Equation (5.2-6) combined with (5.2-9) and (5.2-12) yields an

expression for the normal velocity of approach of the rocker.

Substitution of equations (5.2-4) and (5.2-6) into (5.2-1) gives

-[l,sin(n-O,)]0,++[psin(n-O,+c)lo,+=
[el,sin(n-O,)]e,-[pesin(n-O.+u)]u.+
[(l+e)l,sin(n-G,)]e,
(5.2.13)

Equation (5.2-13) provides the third relation necessary for find-

ing the unknown parameters in the impact mode.

82

5.3- Ngture 21 Qgptggt

The fourth amending equation which is required to find the
unknowns is furnished by the nature of contact of the colliding sur-

faces. Three distinct types of contact can occur

i- The SMOOTH case. in which the contacting surfaces offer negligible

frictional forces in the tangential direction.

ii- The ROUGH case. where the surfaces are badly worn. and no relative

slip can occur in the tangential direction during impact.

iii- The STICK-SLIP case. in which the surfaces offer a Coulomb-type
friction in the tangential direction. The coefficient of dry fric-
tion. 2.. is assumed to be constant in this case. It is obvious that.
the smooth and rough cases are special instances of the stick-slip
case. In other words. when u, is set to zero. the result is the
smooth case. and when u, is assumed extremely large. it becomes ident-

ical to the rough case.

The following notation will be employed to simplify the notations

u,-l,sin(n-O,) (5.3-1a)
u,=1,sin(n-O,) (5.3-11.)
u,=l,sin(n-O,) (5.3-1c)

u.=psin(n-Og+u) (5.3-ld)

83

N.=l,cos(n-O,) (5.3-1e)
n.=l,cos(n-O.+t) (5.3-lf)
p,=l,cos(n-O,) (5.3-1g)
u,,-pcos(n-O,+u) (5.3-1h)

Just before impact. all the parameters in equations (5.3-1) are
known quantities. In terms of these notations. equations (5.1.2-11).

(5.1.3-7). and (5.2-13) can be written in matrix form as following

 

 

 

 

 

 

 

 

‘FIA. o p, uflrufl IA, 0 o M «J
0 IO, ”"4 ’Ps “4+ 3 0 IO, 0 “a
eru, u, 0 0 F en, -ep. (l+e)u, w,
J (L ~L d
L.‘ .J
(5.3-2)

At this stage. the related equations of the types of contact will

be investigated

5.3.1- The Smooth Case

Since in this type of contact frictional forces are negligible.

the tangential £0300 Ft' at the point of contact due to friction van-

84

ishes. Hence. equation (5.1.2-5) becomes

f'IFtdt=0 (5.1.3-1)
Therefore. f can be eliminated from equation (5.3-2) and the
resulting system of equations can be solved for the remaining unk-

nowns. Note that. equation (5.3.1-1) implies that the tangential

velocities remain unchanged in this case. i.e..

and

5.3.2- The Rough Case

Since due to friction. there is no slipping or relative motion in

the tangential direction. then

VR +'VB + (5.3.2-1)

By referring to Figure (5.5). one can write

85

VB. 31303008 (“—03)+1,.’co. (“—03,

t
(5.3.2-2a)
and using Figure (5.7)
Vb, =pu,cos(n-0,-e) (5.3.2-2b)
t
Combining equations (5.3.2-1) and (5.3.2-2) yields
[l,cos(n-0,)]e,+-[pcos(n-O,-u)]o,+=
-[1,cos(n-0,)]u, (5.3.2-3)
and using the notations of equations (5.3-1)
(u,)u,+-(u,.)mg+-(u,)m, (5.3.2-4)

The above equation is the fourth relation necessary to solve

equation (5.3-2) for this type of contact

5.3.3- The Stick-Slip Case

In this case. due to Coulomb friction. sticking occurs if the

tangential force. Ft' during impact is less than the frictional force.

86

i.e..
F: 3 qun (5.3.3-1)

where u, is the Coulomb coefficient of friction. Integrating both

sides of inequality (5.3.3-1) with respect to time yields

It... 5 u.IFndt
or
r S u,F (5.3.3-2)

The system then behaves in an identical manner to that of the

rough case. which was discussed in section 5.3.2.

Slipping is initiated once the inequality (5.3.3-2) is violated.

At this stage. Ft. the tangential force assumes a constant value given

by

where

e=sign(Vh. -Vb‘ ) (5.3.3-4)

87
Again. integration of equation (5.3.3-3) with respect to time

will result in

f=ep,F (5.3.3-5)

5.4- Aggemblage 2f the Equations of Motion

The matrix equation (5.3-2) can be combined with equations
(5.3.1-1). (5.3.2-4). and (5.3.3-5) to yield the general equations of

motion of the mechanism in the impact mode as follows

 

 

 

 

 

 

 

 

 

 

 

P I ‘1 1P .1 P T ~ ‘A
IA, 0 (“is “1 ”3+ IA, 0 0 ”3
I
0 I | + I 0 I
O, I'Pss 'l‘ai ”a 0‘ 0 (e,
-"s ”no '0 0 F °Fs ’9Ps (1+°)fls 9's
L r
- I0 0 f 0 0 - L J
L"? “10' i “e
" ' -‘ _ .2

where for smooth or rough (stick) case

”ss‘fls

“ss‘fle

and. for slip case

"11:" .‘Wflgu 1
“ss‘fls+“flefla

Equation (5.4-1) includes all three types of nature of contact.

Iith proper partitioning. it can be solved for each case as follows

1. For the smooth case and the slip case. one considers only the top

3 x 3 submatrix (solid line partitioning).

2. For the rough and stick cases. the entire matrix in equation
(5.4-1) has to be considered. Even then. the solution is easily
obtained by considering the 2 x 2 system of partitioning shown by the
dotted lines. Due to the existence of a sero submatrix in the bottom
right. it is possible to solve explicitly for »,+ and u.+. then. F

and f can be given in terms of these two parameters.

Although in this study the only case considered will be the
smooth type of contact. equation (5.4-1) is solved for each case. and

their explicit forms are shown as follows
For the smooth and the slip cases
4.
”a '1/D1[[1A.flsflss'°lo,flauss]'s+[Ra’s

-u,.,1[(1+.)10‘uul]

(5.4-2a)

89

+
”a ‘1/Ds[[10,Fsflss‘°lg,fleflss]“e+[Pius

+u.e,][(l+e)IA,u1,]]
F=1/D,[IA’IO‘(1+e)(p,o,-u‘u,+u,o,)]

f- 0 (for smooth) 3 op,F (for slip)

DI‘IA,Pe“is+IO,FsFii

For the rough and the stick cases

u,+=l/D,[eu,.[n.m,-p,o,]+[{ngp,

’(1+°)nsuse]“s]

Ug+‘1/D‘[°u[7 [“634-9gmg1'.’ “I'll"(l‘i'ehl‘ul'il 1.3}

F-lID,[IA’u,(u,+-m,)+Io‘u,(u,+-u,)]
fs-IID,[IA'“‘(..+—u,)+lo‘p,(u,+_u,)I
Ds'nsuie'usus

Ds'nefls-"sns

(5.4-2b)

(5.4-2c)

(5.4-2d)

(5.4-2e)

(5.4-3a)

(5.4-3b)

(5.4-3c)

(5.4-3d)

(5.4-3e)

(5.4-3f)

90

The equations of motion developed in chapters three. four. and

five furnish all of the required equations for the three modes of

behavior. namely. the following. the free-flight. and the impact mode.

The next chapter discusses the method of solution of these equations

along with some results.

CHAPTER SIX

CORRELATION AND NUMERICAL SOLUTION OF

THE MODES OF BEHAVIOR

The preceding chapters. taken together. comprise a complete set
of governing equations of motion of the mechanism under study. The
task is now to tie these equations together. using appropriate boun-
dary conditions. and then solve them numerically. which in turn
necessitates the existence of a set of system parameters used as ini-

tial conditions.

The appendix represents the flow chart of a code. called SIMUL.
used for the numerical solution of the equations. It is not practi-
cal. if not redundant. to explain every line of this code. However.
its main framework and objectives will probably become clear in the

remainder of this chapter.

92

6-1- _i___11n tie 9222121222

The choice of initial conditions plays a significant role in the
results of the simulation. For instance. for a given set of parame-
ters and a prescribed upper bound value of the integration error. and
for one full revolution of the crank. the outcome of the numerical
solution is entirely different for the initial condition corresponding
to a crank angular displacement of. say. sero degree than that cor-
responding to a crank angle of 20 degrees. This inconsistency in the
results is due to several reasons. The assumed configuration of the
mechanism corresponding to the value of the crank angle chosen as
being the 'start-up' position might not be the tgpg configuration.
Obviously. at any instant of time. due to the small size of the clear-
ance. the angular displacements of the links of the mechanism (crank.
coupler. and rocker) are practically the same as their counterparts
corresponding to the nominal mechanism. However. as mentioned in sec-
tion (3.4). when selecting apprOpriate initial conditions. the angular
displacement of the 'clearance link' is taken to be that of the cor-
responding bearing reaction in the nominal mechanism. So. now the
question is that how accurately does the direction of the 'nominal
bearing force' approximate that of the clearance link in the mechanism
under study at initial conditions 7 A method of such approximation
has been proposed by Earles and In [3] - explained in section 3.4 -
which relies on the determination of the dead-center-point (d.c.p.).
which is too specialised and cannot be extended to any mechanism such

as the one under investigation. Since there is no universally accept-

93

ed method of selection of the start-up time. the optimum path seems to

be a trial-and-error approach.

Although during the the first few cycles of the simulation. the
influence of the choice of initial conditions on the response of the
system is quite pronounced. this dependency on the start-up position
tends to diminish as the solution is advanced into higher number of
cycles (about 10 or 20 cycles). In other words. as pointed out by
Nansour et. al. [7]. the response of the system during the numerical
solution begins with a 'transient' phase which continues to converge
to a 'quasi-steady' phase after a few simulated revolutions of the
crank. In the transient phase the response does not behave periodic
and has no tendency to follow a consistent path. But. in the
quasi-steady phase. the response tends to become periodic whose funda-

mental wavelength often spans over a number of crank revolutions.

Hence. any angular displacement of the crank can be taken as
initial conditions. so far as the quasi-steady response is concerned.
Once the solution is initiated. it can be carried on to this phase.
and the results can readily be studied. However. more critical than
the choice of the initial conditions is the efficiency and accuracy
lhmits dictated by the type of the digital computer utilised to carry
on the task of numerical solution of the equations. The computer used
in this work was a PRIME 750 which has an accuracy of maximum of 14
significant digits. depending on the case under study. in the double

precision mode. But. as will be apparent. the accuracy needed to

94

solve the equations of motion at hand requires a machine with far
greater accuracy than the PRIME's. Also. due to the round-up and
truncation errors. at some points the degree of accuracy tends to fall
below the specified maximum. thus magnifying the problem. In other
words. the choice of initial conditions was governed by the PRIME sys-
tem to solve the system of equations with reasonable speed and

accuracy.

There were only a few values of the crank angular displacement
that seemed likely for the PRIME to use them as initial conditions and
iterate the solution through one full crank revolution. The other
values would either cause an overflow problem with subsequent halt of

the solution or. totally erroneous and divergent results.

The final note that should be mentioned is that due to length of
time involved. all the numerical solutions shown hereafter are for the
first cycle of solution or. the first crank revolution. Carrying the
solution into higher crank revolutions using the PRIME. if possible at

all. would require unacceptable amount of computer time.

5.3“!22222 21 £212.82!

It would be helpful to take a quick glance at the appendix. which
contains the main flow chart of the code (SIMUL) along with an alpha-
betical listing of all of the related subroutines and functions

developed for the numerical solution of the equations. The primary

95

thrust of the main code is to manipulate and correlate the parameters
computed by the subroutines. and also. perform the task of storing the
data and shifting the 'flow-direction' of the computation to appropri-

ate points and places.

Once the code is compiled and started. all of the desired
parameters corresponding to the nominal mechanism are computed. This
is done prior to the numerical solution of the equations. These
parameters are stored aside for further comparison with the final

results.

After determining the instant of time used as the time of initial
condition. along with a few other auxiliary parameters. the numerical
solution is initiated. This task is performed using a numerical
integration package called HPCG. It is not overemphasised to state
that this package is the heart of the computer code used for the
numerical solution. For in this subroutine all of the required itera-
tions and integrations are carried out. Hence. it is obvious that.
the overall efficiency of the whole program is primarily dictated by
the efficiency of the numerical integration package used. In other
words. the more efficient the method adapted in the package. the more

accurate and reliable are the final results.

Another limiting factor. again. is the capabilities of the com-
puter at hand. This implies that. as the accuracy of the package. and

consequently that of the final result is increased. there is a thres-

96

hold beyond which either there is no further improvement in the
results or. in the worse case. the round-up and truncation errors dom-
inate the results significantly and introduce a substantial level of
'ditgital noise' into the final results. For instance. a numerical
integration package was used first which was quite powerful in terms
of solving the equations occuring in this work. This code was
developed by the Lawrence Livermore Laboratory. California. called
LSODI (The Livermore Solver For Ordinary Differential Equations In
Linearly Implicit Form). Tho methods of integration could be used to
solve any problem depending on its type. The first one was for sets
of equations with stiff characteristics (sparse eigenvalues). which
adapted the backward differentiation formulae to integrate the equa-
tions numerically. The second method which was essentially the
implicit Adams method. was suitable for sets of equations that did not

show stiff characteristics.

Unfortunately. such a package could not be compiled and handled
by the PRIME system due to the occurence of overflow problems caused
by the round-up errors generated in the system. Thus. replacement of
the LSODI package with another one was unavoidable. and LSODI was sub-
stituted by the HPCG to carry out the task. This package (HPCG) is
intended to solve a set of first order ordinary differential equations
with prescribed initial conditions. Integration of the equations is
done by means of the Homming's modified corrector-predictor method.
which uses four preceding points for computation of a new vector of

the dependent variables. A fourth order Runge-Kutta method. as sug-

97

gested by Ralston [85.86]. is used for adjustment of the initial

increment. and for computation of the starting values.

To help give a better insight into the differences between the
former package (LSODI) and the HPCG. it should be stated that. the
example problem (along with answers) illustrated in the commentary
section of the LSODI package was solved using the HPCG subroutine.

The major differences were observed.

1. The time required for the HPCG to solve the problem through the
whole integration interval was computed to be in the neighborhood of
several hours. whereas the same problem presumably would be carried
out in a matter of minutes using LSODI. Obviously. when running the
example problem on the HPCG the execution was halted due to the large

extent of time required.

2. The answers obtained at the specified points of the integration
interval were compared with the ones given by the LSODI. It was noted
that all of the results obtained using HPCG resembled those found

using LSODI only up to the third significant digit.

The apparent conclusion which can be drawn from these observa-
tions is that. the existence of 'digital noise' in the final results
can be attributed primarily to the generation of round-up and trunca-
tion errors in the PRIME system. and partly to the approximation

errors within HPCG during the iteration process.

98

Returning to the set of equations in this study; once the initial
time is specified. the code is written such that it automatically
assumes this instant of time to be in the following mode. This. in
turn. implies that equation (3.3-7) is iterated using the parameters

known at the initial condition. i.e..

[A][X]'{B]

where matrix [ A ] and vectors { X ] and [ B ] are described in equa-

tions (3.3-8).

The iteration in HPCG is advanced until the contact-loss criteria
(section 3.5) is met. At this point. the code exits from the HPCG and
the above equation is replaced by equations (4.1-20) and (4.2-1).
which are the governing equations of motion of the free-flight mode.
The iteration is performed based on initial conditions corresponding
to the values of parameters in the following mode just prior to con-
tact-loss. As soon as the criteria for the termination of the

free-flight mode. inequality (4.4-5). is met. the iteration is paused.

Utilising the values of the parameters at the end of the
free-flight interval. the impact mode is initiated and. equations
(5.4-2) are solved for the unknowns. namely. the angular velocities of
the coupler and rocker after impact. These two new parameters. along
with the other known ones. constitute an 'updated' set of initial con-

ditions for either the following mode or. the free-flight mode.

99

depending on the particular point of impact in the bearing and. other
system parameters such as links' angular displacements. velocities.
etc. The solution. therefore. is carried on in this manner over the

desired time interval.

One of the major difficulties in obtaining the final results of
the numerical solution was the time required to solve the equations
with a prescribed upper bound value of the absolute error of integra-
tion. This problem is distinctly pronounced when high degrees of
accuracy are desired. For instance. if the value of the maximum abo-
lute error was prescribed to be 0.001. it would require an integration
time step as small as one corresponding to a crank angular displace-
ment of about 10" degrees per step to bypass the occurence of
overflow. This would mean that. at least 3.6 x 10‘ iterations are
required to solve the equations over one crank revolution only. As an
experiment. such conditions were generated on the PRIME and. it wis
observed that after 17 continuous hours of CPU time spent in solving
the equations on the system. the iteration had advanced only about 5
degrees of crank angular displacement relative to the initial condi-
tion. Obviously. the execution of the code was terminated since it
required approximately 1200 hours to iterate through one full crank

revolution at the same pace.

Thus. with a few trials. a much coarser mesh in terms of the time
step was used in order to yield the final results in a reasonable

length of time. i.e.. from two to ten hours depending on the size of

100

the mesh and the acurracy required. This. of course. implied a com-
promise on the initially intended degree of accuracy required. It is
noteworthy to mention that. Mansour et. al. [6] solved a similar
problem (all the equations except those of the following mode) on an
IBM 370/165 and. the time required to iterate the equations up to as
high as ten crank revolutions was reported to be less than 15 seconds
including the compilation time. Admitting that. in this work a third
(following) mode is coupled with the other two. and also that all the
dimensions used here are different than those used by Msnsour. this
comparison still gives an idea as to the advantage of using fast and

efficient computers.

Iith all of the aforementioned difficulties involved which could
not be eliminated altogether. the results in this work were obtained
for the first crank revolution. This implies that. only the start-up
cycle was simulated which possesses the severest response in terms of
the vibrations and oscillations in the parameters due to the existence
of the bearing clearance. The previously mentioned limiting factors
made it impossible to study the response of the system in the higher
cycles. Also. as described in chapters two and five. the
coupler-rocker bearing that has a finite clearance is assumed to be
frictionless (Coulomb coefficient of friction is sero) and. the coef-
ficient of restitution at impact equals unity (no energy dissipation
due to impact). Once introduced into the system. these two latter
assumptions imply that. there is no dissipation of energy into the

surroundings from the coupler-rocker bearing at any instant of time

101

during the simulated motion.

6.3- Numeriggi Regults

Table (6.1) is a compilation of all of the relevant information
concerning the physical and dynamic dimensions of the mechanism used
for the simulation. which is a model of the four bar linkage serving
as the experimental rig. As previously mentioned in chapters two and
five. the Coulomb coefficient of friction is assumed to be zero.
implying that. there is no friction between the pin and the bushing of

the coupler-rocker bearing.

Tho complete sets of simulation results are presented here. All
of the dimensional characteristics of these two simulations are the
same except that. the coefficient of restitution at impact is assumed
to be unity in one case and. 0.50 in another. The former implies
that. there is no type of energy dissipation into the surroundings at
the time of impact. while the latter allows for some absorption of the
energy of impact by the colliding surfaces and. yielding it to the
surroundings through heat transfer or acoustic generation or. other

means of system-surroundings interaction.

In order to achieve a better understanding of the results of the
simulation. each curve is superimposed on its nominal counterpart.
However. in some of these plots. which will be presented on subsequent

pages. the scale of the plot is largely dominated by the simulation

102

TABLE (6.1)
SPECIFICATIONS on m 315113103 may

 

mass
center moment

length of of

link (sise) gravity mass inertia
(mm) (mm) (kg) (kg.mm')

Crank 64.0 9.0 0.6095 2645.0

Coupler 309.0 185.0 0.2439 2950.0

Rocker 312.0 134.0 0.2334 3176.0

Ground 387.0 -- --.- _..-
Clearance 0.2540 —-- —-. ....

 

Crank angular velocity : 36.6519 rad/sec (350 RPM)
Crank angular acceleration : 0.0 rad/see‘
Coefficient of restitution : 1.0

Coulomb coefficient of friction : 0.0

Gravitational acceleration : 9814.5 mmlsec'

103

result and. the behavior of the superimposed nominal plot cannot be
readily recognised. For this reason. Figures (6.1) to (6.9) illus-
trate the behavior of all of the studied parameters of the system for
the nominal (clearance free) mechanism. Note that. in chapter eight a
selected number of the simulation results will be compared to the

experimental results which will be brought in chapter seven.

Figures (6.10s) and (6.10b) are the coupler angular displacements
corresponding to coefficients of restitution of 1.0 and 0.50. respec-
tively. Figures (6.11s) and (6.11b) are the analogous plots for the
rocker link of the mechanism. As expected. there are no significant
differences between the angular displacements of the simulation
mechanism (called the 'base' mechanism hereafter) and. those of the
nominal mechanism. This is primarily due to the fact that. the size
of the clearance in the coupler-rocker bearing of the base mechanism
is small enough (0.010 inches or 0.2540 millimeters) to have negligi-

ble effect on the 'geometrical' configuration of the system.

Figures (6.12s) and (6.12b) are the angular velocities of the
couples. with e-1.0 and e-0.50. respectively. Although the effect of
the presence of clearance is quite pronounced on these plots. the
influence of the magnitude of the coefficient of restitution cannot be
clearly sensed. Note that. in Figure (6.12b). there is a rather sev-
ere disturbance in the coupler angular velocity of the base mechanism
occuring at crank angle of about 20 degrees and. spanning over approx-

imately four degrees of crank angular displacement. It is the

104

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suthor's opinion thst two possibilities exist

1. This disturbsnce. which sttenustes sfter s few tine-steps. is the
result of s digitsl noise genersted during the iterstion snd integrs-

tion of the equstions of notion.

2. It is the response of the nechsnisn bssed on the solution of the
equstions of notion. In this csse. the tern 'disturbsnce' could not
be spplied to this portion of the plot since it is now s psrt of the
response. However. ss previously mentioned. sll of these si-ulstions
sre for the stsrt-up cycle with severest response. This could in psrt
be due to the existense of sny insccurscies in the initisl vslues of
the psrsneters selected st the stsrt of the iterstion of equstions.
ns-ely. initisl conditions. In other words. this would imply thst.
such s severe response could occur in the process of nunericslly
'forcing' the results to trsce s psth in the neighborhood of their
noninsl countsrpsrt. The resder should slso note thst. st this point
one of the nsjor ressons for the occurence of the overflow conditions
during the sinulstion becones nore clesr. Thst is. the overflow
occurs prinsrily in this stsge while sttenpting to resch s conpro-ise
between sny insecurscies in the vslues of the initisl conditions snd.
the true response of the systel. Obviously. the precision of the con-
puting nschine csn often dictste the outcone of the sinulstion process
in ten-s of either integrsting the equstions successfully through the
prescribed tine intervsl or. introducing enough error into the results
ss to esuse the whole iterstion process to hslt st some point within

the tine intervsl.

120

Iith the sbove two possibilities in nind. one csn clsin thst.
this s-sll portion of the response (Figure 6.12b) csn be isolsted fro-
the reusinder of the plot with no loss of sccurscy snd generslity in
the results. Such s phenomens csn slso be observed in Figure (6.12s)
which occurs in the neighborhood of crsnh sngle of 10 degrees.
However, in this csse. it is not ss severe ss in the former csse. The
resder should hes: in nind thst. these so-cslled disturbsnces sre pro-
psgsted to the other systel psrsneters. Thus. sny plot presented in
subsequent which besrs this type of behsvior, will be studied with
thst specific portion of the response isolsted fro. the rensinder of

the plot.

Figures (6.13s) snd (6.13b) sre the sngulsr velocities of the
rocker link for e-l.0 snd e-O.50. respectively. Eli-insting the dis-
turbsnces on the plots. there seems to be no significsnt difference
between the two sngulsr velocity curves. Figures (6.14s) snd (6.14b)
depict the sngulsr sccelerstions of the coupler. Note thst. there is
s clesr distinction in the sccelerstion responses for different vslues
of coefficient of restitution. As csn be seen. the coupler sngulsr
sccelerstion for e-l.0 (Figure 6.14s) is. on the sversge. twice thst
of the one for e=0.50 (Figure 6.14b). This csn be sttributed to the
fsct thst, when e-0.50. s portion of the inpsct energy is sbsorbed by
the coupler-rocker besring components. while in the csse when e81.0
(Figure 6.14s). the energy of the inpsct is theoreticslly conserved.
Figures (6.15s) snd (6.15b) illustrste the corresponding plots for the

rocker link.

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133

Figures (6.16) snd (6.17) present the X9 snd Y- components of the
coupler-rocker besring. respectively. Each figure consists of two
plots. One for when e-l.0 end. the other for e-0.50. Figures (6.18s)
snd (6.18b) show the snslogous plots of the crsnk torque for different
vslues of the coefficient of restitution. In sll of the letter six
plots the effect of magnitude of the coefficient of restitution csn be

observed.

The render should note sgsin thst. s portion of these plots will
be studied further in chspter eight in en effort to correlste the sus-
lyticsl snd experimentsl results. Other sets of plots. such ss
describing the effect of clesrsnce size on the genersl response of the
system or. other (preferrsbly snsller) vslues of the coefficient of
restitution could not be obtsined due to the frequent occurence of
overflow conditions, snd/or the extent of time required to sinulste
the system through one full crsnk revolution successfully. This is
primarily due to the fsct thst. for instsnce. es the clesrsnce size is
reduced. more precision is required to solve the equstions of notion
snd. simultsneously. detect the occurence of contsct-loss or impsct

time promptly.

The next chspter describes the experimentsl sppsrstus snd slso
the instrumentstion slong with subsequent experimentsl results. This
provides the bssis for further compsrison of the snnlyticsl snd exper-

imentsl results.

CHAPTER SEVEN

EXPERIIENTAL INSTIDIENTAITON AND RESULTS

Figure (7.1s) is s photogrsph of s four bsr linksge which fur-
nished the expeimentsl date end. Figure (7.1b) shows the relevsnt
instrumentstion for signsl conditioning and recording of the dsts. A
detsiled description of the method of dstn scquisition is slso

presented.

7-1- W W

The experimentsl mechsnism (Figure 7.1s) wss mounted on s lsrge
csst iron tsble which wss bolted to the floor of the lsborstory.
'hile the length of the crsnk link wss fixed. the rig wss designed
such that it could operate with vsrying lengths of the coupler snd

rocker links. The two common link lengths used were 16 cm snd 30 cm.

 

FIGURE 7.1x.

   

FIGURE 7 . 1b .

135

 

PHOTOGRAPH OF'IHE EXPERIENTAL RIG

    

PHOTOGRAPH OF 1113 IN STRUIEITATION

136

The links were prepared to form coupler and rocker links in prac-
tically identical dimensions. At the end of each link two clearance
holes were drilled in order to secure them to the bearings. These
sccomodated socket screws which clamped each link to the bearing hous-
ing and. permitted the experimental four bar linkage to be
constructed. The links were bolted to the aluminum bearing housings

by two socket screws at either end.

The crank-ground. crank-coupler and. rocker-ground joints were
all matched pairs of FAG 23 DB 212 ball bearings of 0.25 inches in
bore. Ehch of these bearings was preloaded using a Dresser torque
limiting screw driver with tl.0 in-lbf preloading. This procedure was
to ensure that bearing clearance was eliminated. Inaccurate preload-
ing of these bearings would introduce additional clearance into the
system or otherwise. produce large frictional forces in these bear-
ings. Either of these conditions would invalidate the assunptions
made in chapter two. The coupler-rocker bearing (Figure 7.2).
however. which was the joint designed to possess a finite clearance.
was a simple pin-and-bushing Journal bearing. The bushing was made of
oil impregnated brass and. the pin was made of steel. The bearing was
designed in a such a manner that the pin could easily be removed. thus
making it possible to incorporate a varying range of clearnce sizes
into the bearing by simply changing the pin with a one having a dif-
ferent diameter. The range of clearances that could be incorporated
into the bearing was from 0.0010 to 0.010 inches with a tolerance of

t0.0005 inches.

137

 

in fingnn-LL . O

(a )- DISASSEIBLED

(b)- ASSEMBLED

 

FIGURE 73.. WUPLER-ROCKER BEARING

138

A 0.75 hp variable speed DC notor (Dayton.ZZB46) powered the
crank through a 0.625 inch diameter shaft supported by a cast iron
pillow box bearing. A 4.0 inch dianeter flywheel was keyed to the
shaft. so as to provide a large inertia to ensure ninimum fluctuations
in the crank angular velocity once a desired crank frequency had been

established.

7.2- unen t o

A schematic of the the instrunentation employed in the experimenr
tal study is shown in Figure (7.3). The rated speed of the electric
motor was measured to three decimal places by an HP 5314A universal
counter which was activated by a digital-magnetic pickup. model
58423 - Electro Corporation. These devices allowed the operator to
adjust the speed controller of the notor in order to achieve the

desired crank angular velocity.

The experimental results. shown in subsequent sections. present
the variation of coupler and rocker angular velocities and accelera-
tions with respect to the crank angular displacement. These
measurements were obtained using a 4371 Bruel and Kjaer accelerometer
with delta-shear piezoelectric type of sensor. Figure (7.4) shows the
coupler-rocker bearing with two accelerometers mounted for obtaining
the data for the coupler link. In order to obtain the same data for
the rocker link. the bearing housing is rotated 180 degrees relative

to the two links.

139

Visual monitoring of

W

HP 5314A
universal counter

r 1

 

 

 

 

 

 

 

 

Blectro Corp.
pickup
( _
60 tooth D.C.
Accelerometers lochgnig.
B and K 4371 ’9“! 80'! Iotor

 

 

 

 

 

 

    
  
 

     

power supply conditioning

amplifier

 

 

: and [ zero crankrangle
Chgg‘. ..p111£. configuration

 

components of

 

 

 

 

joint
’ acceleration
DEC
pdp 11/03
micro-
computer

 

 

post-processed
experimental results

 

 

'FIGURE 7.3 -SCHEMATIC OF THE INSTRUMENTRJION

140

 

FIGURE 7.4. (DUPLER-ROCKER BEARING NUNTEU ON ME NEGANISI

141

The signal from the accelerometer is sent to a conditioning
amplifier of type 2635 Bruel and Kjaer with built-in low-pass filter
of variable cutoff frequency. The amplifier can also be adjusted for
different gains of amplification ranging from 0.1 to 1000. In addi-
tion. the conditioner is capable of simulating a band-pass filter of
variable bandwidths. Finally. it is equipped with an integrator. thus
enabling one to measure the time-integral of the accelerometer's sig-

nal.

The two accelerometers shown in Figure (7.4) measure the normal
and tangential components of the absolute acceleration
(velocity/displacement) of the four bar linkage at the point of mount-
ing (coupler's end in this figure) with respect to the link. To find
the angular acceleration of the links (coupler or rocker). expressions
must be found which relate these components of the absolute accelera-

tion to the angular accelration of the link.

7.2.1- Angular Acceleration and Velocity of the Coupler

Figure (7.5) is a two-dimensional view of the mechanism with an
accelerometer mounted on the coupler-rocker bearing in order to meas-
ure the component of the absolute accelration of the coupler in the
direction normal to the link. Using the law of relative reference

frames. the following expression for the accelerations can be written

142

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IB.:A+;BIA (7.2.1-1)

where the equation is in vector form. and ;R/A is the acceleration of

point B relative to point A.

The terms on the right hand side of equation (7.2.1-1) can be
written in their normal and tangential components with respect to

their corresponding links as follows

I .- +- (7e2e1’2

A IA“ .At .)
where

.An-la“sa

aAt=0
and also

;E/A'(;E/A)n+(;R/A)t (7.2.1-2b)
where

(‘B/A)n'1s“s’

(RB/A) t'l'a,

144

A graphical representation of equation (7.2.1-1) with the help of
equations (7.2.1-2) is shown in Figure (7.6). From this figure the

following expressions can be derived

‘3 =-l,u,‘cosO,-l,a,'cos0,-l,u,sin0, (7.2.1—3a)
x

as =-1,u,’sine,-1,u,’sino,+1,a,coso, (7.2.1-3b)
y

thus the tangential component of equations (7.2.1-3) can be expressed

R8
(.83) t'IBxHAHO. (7.2.1'4‘)
(s ) -a cosG (7.2.1-4b)
By t B, 3
but
( ‘ - .2.1-5
‘3}: (‘3‘)t (Ray): (7 )

Hence. substitution of equations (7.2.1—3) and (7.2.1-4) into
equation (7.2.1-5) and carrying out a few algabraic steps yields the
desired expression of the angular acceleration of the coupler link in
terms of the measured tangential component of the absolute accelera-

tion of the coupler link in the tangential direction. i.e..

a,.1.0/1,[1,.,’s1n(e,-o,)-(a3)t] (7.2.1-6)

145

 

 

 

FIGURE 7.6. GRAPHIC REPRESMATION OF MUATION (7.2.1-1)

146

The angular velocity of the coupler can be found in an analogous
manner to the above procedure. However. due to some difficulties.
which was mainly attributed to the characteristic behavior and
response of the integrating circuit components of the signal condi-
tioner/amplifier used. the computation of the conversion factor needed
to obtain the correct values of the angular velocities was not advan-
tageous due to the extent of time and the laborious calculations
involved. Thus. another method was used to determine the angular
velocity of the coupler link. This alternate method is an adaptation
of the simple trapezoidal rule whose thorough description can be found
in any numerical analysis text such as [88]. Nevertheless. the sal-

ient features of this method will be presented here.

Suppose f(x). a continuous function over interval [a.b]. is to be

integrated. i.e..

I'If(x)dx

where the limits of integration are from xts to x=b. The area under
the curve f(x) between a and b can be divided into n equal intervals.
The boundaries of the resulting trapezoids are X.. x,. ... . In on one
side and. f(xx). f(xa). ... . f(xn) on the other side. It can be
readily noted that. the above integral. I. can be approximated by the

total area of the n trapezoids. i.e..

147
Isv/a[£(x.)+2£(x,)+z:(x,)+...+2£(xn,,)+£(xn)]
(7.2.1-8)
where w is the length of the subintervals or.

w=(b-a)/n

In our case. the function to be integrated is defined at
equally-spaced. discrete points. thus fixing the number of subinter-
vals. This is due to the fact that the signal is sampled at a certain
rate implying that the accuracy is directly proportional to the sam-

pling rate.

7.2.2- Angular Acceleration and Velocity of the Rocker

Figure (7.7) is similar to Figure (7.5) except that. the
accelerometer is mounted to measure the component of the absolute
acceleration or velocity of the rocker tangent to the link. It can be
observed that the angular acceleration (velocity) of the rocker link
can be obtained by simply dividing the experimentally measured value

of the acceleration (velocity) by the length of the rocker link. i.e..

a‘..Bt’1‘ (7.2.2.1)
and

uj-vht/lj (7.2.2-2)

148

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149

In order to relate the accelerometer signal to the configuration
of the experimental mechanism. another transducer arrangement was
established. A zero velocity digital pickup. an Airpax 14-0001. was
located so as to sense the bolt head mounted at the end of the crank
when the four bar linkage was in the reference position of zero-degree

crank angular displacement.

Thus the mechanism configuration signal and also. the conditioned
and amplified output from the accelerometer were fed to the oscillo-
scope for real-time visual monitoring of the response. In addition to
obtaining photographs using a C-SC camera attachment for the oscillo-
scope. the signal could also be fed to a digital data acquisition

system (DEC PDP-ll/03 microcomputer with 5 lb storage on hard disk).

The ENC cables from the experimental apparatus were connected to
an input-output module. This device has 16 analog-digital conversion
channels. 4 digital-analog channels and. Schmitt triggers one and two.
Using the code developed for acquisition of data. the response of the
mechanism was recorded from the zero crank angle position through 360
degrees. The sampling of the data was initiated by firing of the

Schmitt trigger two.

Once the data was sampled and stored. some further conditioning
was carried out on the results. In almost all of the experiments. the
signal had some degree of noise embedded within it. even after passing

through the conditioning amplifier previously mentioned. There are

150

various sources that could contribute to the generation of the noise.
For instance. if the environment is 'polluted' by electromagnetic
fields generated by nearby radar and satellite stations or fluorescent
lighting. the effect can be picked up at practically any point along
the instrumentation. such as the accelerometer itself. the cables and
connections and. any unshielded section of the circuitry or. poor

grounding. and so forth.

Nevertheless. the severity of the noise in the sampled signal can
be alleviated by simulating a digital band-pass filter. which is mere-
ly a code incorporating the Fast Fourier Transform (FFT) algorithm.
The filtering is done by transforming the data from time-domain into
frequency-domain. thus yielding the power spectrum of the signal.
Figure (7.8) illustrates one such spectrum. At this stage. the signal
can be studied and analyzed at different and discrete frequencies
depending on the frequency resolution of the FFT routine. which in
turn is dictated by the sampling rate. In this manner. the frequen-
cies associated with noise can be identified and eliminated by
'fading' the amplitude of the signal at those particular frequencies.
Once the noise is attenuated. the result. which is in the
frequency-domain. is transformed back into the time-domain using the

same FFT routine in reverse order.

Since the sampling rate is about 2.7 KHz. the maximum frequency
available to work with in the frequency-domain is in the neighborhood

of 1.3 KHz. In other words. in order to avoid aliasing (overlapping

151

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152

of two successive periods of the signal). the Nyquist rate (minimum
allowable sampling rate) should be at least twice the maximum frequen-

cy desired [87].

It should be noted that. the windowing function adapted in the
FFT routine is the simplest type. namely. the rectangular windowing
function. This function has the disadvantage of introducing some
additional degree of noise in the frequency-domain. However. once
transformed back into the time-domain. all of these secondary generat-
ed noises are offset and hence. the net result of the noise associated
with the type of the windowing function is zero. If a more accurate
or. less noisy power spectrum is desired. windowing functions of

higher order such as Kazier's function may be used [87].

7.3- W 31133.:

Table (7.1) is a duplicate of table (6.1) along with some addi-
tional information. such as the working range of the motor driving the
crank link and. so forth. Although the motor was capable of driving
the crank with angular velocities in excess of 2000 RPN. the highest
speed used for the experiments was about 350 RPN. This was merely a
precautionary step in order to avoid damaging the setup due to the
exertion of large magnitudes of forces and impulses acting upon some
parts of the system vith relatively low strength such as the pin of

the coupler-rocker bearing.

153

TABLE (7.1)

§£§QLEL£AIIQ!§.QEII!§.§IZEEL!!!IAL !§£HAEIS!

 

mass
center moment

length of of

link (size) gravity mass inertia
(mm) (mm) (kg) (kg.mm')

Crank 64.0 9.0 0.6095 2645.0

Coupler 309.0 185.0 0.2439 2950.0

Rocker 312.0 134.0 0.2334 3176.0

Ground 387.0 -- ---

 

Clearance size : 0.02540—0.2540 mm (0.0010—0.010 inches)

Crank angular velocity : 0-2500 RP!
Crank angular acceleration : 0.0 rad/sec’

Gravitational acceleration : 9814.5 mm/sec’

154

Two different types of effects will be investigated in this
experimental phase of the work. First. the effect of varying the
angular velocity of the crank for a fixed clearance size will be stu-
died. To perform this task. comparison will be made between the
angular acceleration of the coupler link at 250. 300 and. 350 RPN.
This test is carried out for four different clearance sizes of 0.0010,

0.0040. 0.0070 and. 0.010 inches.

In the second phase of the study. the effect of the variation of
the clearance size is investigated for a fixed crank RPM. This set of
experiments is performed for three different crank angular velocities
of 250. 300 and. 350 RPM. Analogous experiments are carried out for

the rocker link.

Figures (7.9a) to (7.9d) show the coupler angular acceleration at
350 RP! with four different clearance sizes. Note that. superposition
of all of the experimental results are avoided in order to prevent
loss of clarity of individual plots since the responses differ from

each other by relatively small amounts.

A note is in order at this point. Nansour et. al. [7] intro-
duced a design-oriented concept in order to be able to describe the
severity of the response of the system due to the occurence of fre-
quent impacts in the bearing of the mechanism possessing a finite
clearance. Recalling from chapter five that F represents the magni-

tude of the tangential impulse at impact. it would be reasonable to

155

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159

assume that. a useful way of describing the effect of presence of
bearing clearance in a mechanism is to express that in terms of the
summation of all of the tangential impulses for one crank revolution.
namely. 2F. This criterion can be improved further if an index number
is deviced which would be the ratio of the ER over the total number of
impacts. n. for that specific cycle of revolution. If such an index

is called In then by definition

InBEF/n

The advantage of using such terminology is that. the severity and
sensitivity of the response of the system to the presence of clearance
in a bearing of the mechanism can be described with a single number.
It is quite helpful to present two of the plots that Nansour et. al.
[7] develcped in their work. Although these plots are based on a com?
puter simulation of a model which only adapted the free-flight and
impact modes for modeling. they can be correlated to some extent to
the experimental results presented in the following. Such a correla-
tion is merely for a better understanding of the experimental results
by being able to observe the effect of. for instance. variation of the

crank angular velocity on a single plot.

Figure (7.10) depicts the effect of the variation of the clear-
ance size and. Figure (7.11) shows the effect of the variation of the

crank speed. The numerical values on the axes of the plots are of no

 

160

 

 

 

 

 

4.0“
30 r-
N
20 “- Q
as
u.
H
l0*- ‘—"_‘ In 50‘400
——o—~— 2 F
--.o.-- n
l J l J
O 2.5 5.0 7.5 '00 |25
E x l04

FIGURE 7.10.

EFFECT OF VARIATION

OFtALONE

 

 

 

 

 

1 J J 1
600 IZOO IBOO 2400 3000 3600
R P M
FIGURE 7.11. EFFECT OF VARIATION OF 1m: RPI ALONE

161

primary value here and. emphasis should be given mainly to the 'shape'

of the individual plots.

Referring to Figures (7.9). it may be observed that. the severity
of the amplitudes of oscillations of the coupler angular acceleration
diminishes slightly as the clearance is increased from 0.0010 to
0.010 inches. Such a behavior can also be observed on the 5; curve of
Figure (7.10) corresponding to the portion of the plot whose slope is
negative. Similar results of Figures (7.9) are shown for the rocker

link in Figures (7.12).

Figures (7.13) illustrate another case of variation of the clear-
ance size for a crank angular acceleration of 250 RPN. In this case.
as the clearance size increases. so does the severity of the fluctua-
tions or EF. corresponding to the positive-slepe portion of the EF
curve of Figure (7.10). On the other hand. Figures (7.14) show the
same comparison while the crank was running at 300 rpm. As can be
seen. the oscillations amplitudes decrease as the clearance size is
increased from 0.0010 to 0.0040 inches (Figures 7.14s and 7.14b) and
then remain relatively the same for a clearance of 0.0070 inches (Fig-
ure 7.14c) and. then again. it starts to increase for 0.010 inches
(Figure 7.14d). Such a behavior can be related to the EB curve of
Figure (7.10) with cases corresponding to Figures (7.14s) and (7.14b)
being on the negative-slope portion of the curve and. those of Figures

(7.14c) and (7.14d) on the positive-slope portion of the curve.

162

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183

The effect of variation of the crank angular velocity can be
investigated now. For a clearance size of 0.010 inches. Figures
(7.15) illustrate the variation of crank RPM for 250. 300 and.
350 RBI. As the crank angular velocity is increased. the severity of
thO fluctuations (2F or In) also increase. However. the number of
impacts in one revolution is decreased (Figure 7.11). The same beha-
vior can be observed in Figures (7.16) and (7.17) which correspond to

a clearance size of 0.0040 and. 0.0010 inches. respectively.

Based on the experimental results presented in this chapter and.
with reference to Figures (7.10) and (7.11). it can be stated that.
for practical purposes. one can design a mechanism which is 'tuned' in
the sense of having minimal values of EB (In) (Figure 7.10) with

simultaneous minimum number of impacts per revolution.

In the next chapter. attempt has been made to compare and corre-
late the analytical results obtained by the numerical solution
described in the previous chapters with the experimental results

presented in this chapter.

CHAPTER EIGHT

COIPARISON 0F
ANALYTICAL AND EXPERIIENTAL

RESULTS

In chapter six the complete results of the numerical simulations
were studied and. chapter seven dealt with the investigation of the
experimental phase of the work. A seperate chapter has been devoted
to the comparison and correlation of these two types of results since
only the angular velocities and accelerations of the coupler and rockr
er links can be compared and. the reliability of other parameters
obtained by the numerical solution may be deduced from the above. In
other words. it is assumed that. any conclusions or remarks drawn from
the comparison of. say. angular accelerations of coupler and rocker
links will apply to the X? and 1- components of the coupler-rocker

bearing reaction. since these parameters are linearly interrelated.

185

Figures (8.1a) and (8.1b) show the superposition of the numerical
and experimental plots of the angular velocities of the coupler and
the rocker links. respectively. Note that. unless otherwise speci-
fied. in all of the foregoing plots the coefficient of restitution
assumed in the numerical simulation is unity. Figures (8.2a) and
(8.2b) are analogous to Figures (8.1). but for angular accelerations

of the links.

It can be clearly observed that. there is a significant differ-
ence between the magnitudes of numerical and experimental results.
especially for the angular accelerations (Figures 8.2) in which the
numerical results are about 30 times the experimental ones. As previ-
ously mentioned. such disagreement in the magnitude and intensity of
the experimental and simulation results is primarily due to several

reasons such as

1. In the numerical simulation the coefficient of restitution. e. was
assumed to be unity. This is hardly the case from practical stand-
points. The bushing of the coupler-rocker hearing was made of
oil-impregnated brass (chapter seven) which would certainly introduce

a finite degree of damping into the system.

By referring to Figure (8.2a) and comparing it to the coupler
angular acceleration for e80.50 (Figure 8.3). it can be observed that.
as the coefficient of restitution is reduced the angular acceleration
of the coupler becomes less severe and. closer to the experimental

results. although it is still about 15 times the experimental results

186

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191

in scale. Hence. it is conceivable to predict that. as the coeffi-
cient of restitution decreases. so does the severity of the analytical
results. However. the magnitude of the coefficient of restitution can
be reduced down to a realistic value such as 0.10 since e'0.0 implies
the sticking of the bearing parts together after impact.
Consequently. the value of the parameter e cannot be varied indepenr
dently without taking other determining factors into consideration

such as the Coulomb coefficient of friction.

The above assumption is coupled with the fact that. the Coulomb
coefficient of friction in the simulation was assumed to be zero.
implying that. the pin can slide against the bushing of the bearing
without encountering any resistant frictional forces. This also. is
an ideal assumption since in practice there is always some type of
frictional force present when two bodies slide against each other.
Thus. in real life situation. the combined effect of the coefficient
of restitution and. the Coulomb coefficient of friction (or any other
type of friction) on the mechanism is to decrease the severity of the

motion of the system compared to the simulated results obtained here.

2. The existense of digital/numerical noise in the results of the

simulation (chapter six) tends to magnify the aforementioned effects.

Thus. it becomes apparent that. for a realistic simulation of the
mechanism investigation. the 'ninimal' modifications needed to be per-
formed on the original assumptions outlined in chapter two are as

follows

 

192

i)- Development of a means of measuring or estimating the true value

of the coefficient of restitution.

ii)- Ieasurement of the Coulomb coefficient of friction in the

coupler-rocker bearing by proper means and equipment.

iii)- Simulation of the model on an efficient and fast computing

machine with high accuracy.

Up to this point. the primary concern of the discussion on the
correlation of the experimental and analytical results was limited to
the subject of severity and/or smoothness of each response. It is now
appropriate to reflect upon and analyze the reliability of prediction
of the points of occurence of such fluctuations. namely. the points of
occurence of contact-loss and impact during the interval of motion of

the system. The following points are in order

1. Ihen the characteristic parameters in the simulation are as chosen
above (e-l.0 and p-0.0). which are almost certainly different from
their true values corresponding to the experimental mechanism (say
e-et and u8u‘). then the contact-loss and subsequent impact points for
the latter two systems occur at different instants of time except for
a few cases which pertain to the 'geometrical' configuration of the
systems. The reason behind this is that. suppose at time tst1 during
the simulation. a point of occurence of contact-loss is detected by
the code. Then. after a few time steps in the free-flight mode. the
impact mode takes place. Due to the value of e being equal to unity.

there is almost always an immediate further contact-loss and. theoret-

193

ically. it takes infinite time for the system to re-stabilize itself.
But. the round up errors tend to increase this so-called stabilization
process and decrease the severity of the response after a few time
steps. However. in real life situation the coefficient of restitu-
tion. e. is less than unity (and nore likely to be even less than
0.50) and hence. the rate of dissipation of energy is much faster than

in the simulation case.

Another factor is the Coulomb coefficient of friction. u. For
all practical purposes the coefficient u is always greater than zero.
whereas in the simulation the assumption was that “80.0 . The third
factor that dominates the location of the points of occurence of the
variations in the plots is the simulated phase of motion of the
mechanism. That is. if the simulation corresponds to the transient or
the start-up cycles of revolution (which is the case here). then the
results will certainly be different than those obtained for higher

cycles of revolution or the quasi-steady phase.

The combined effect of the above three simulation characteristics
is complex. But. it could be stated that. any disagreement in the
values and phases in the simulation with their true states would cause
frequent leads and delays in the occurence of the points of
contact-loss and impact. with its degree of discrepancy dependent upon
how close the assumed cases are to their real ones. Thus. as noted
above. the optimum way of improving the results is to use the correct

values of the coupler-rocker characteristic parameters (e and u) with

194

the simulation performed on a computing machine that would allow the
study of the responses in the quasi-steady phase of the motion with

optimum accuracy and speed.

Before closing this chapter. a further. simplistic attempt will
be made to modify the available simulation results so that they resem-
ble the realistic case as close as possible. The reader should be
forewarned that. the following so-called 'modification' of the analyt-
ical results simply provides an 'estimation' of the true value of the
system parameters and. they are by no means intended to be accurate
and exact results. In fact. as will be clear. their accuracy is no
greater than the analytical results and. their only advantage is that
they are 're-scaled' in such a manner as to be in the range of the

experimental results.

EPWQMWW

Ideally. the task of simulating the mechanism under study on a
digital computer effectively requires finding the correct values of
the coefficient of restitution and the Coulomb coefficient of friction
to be used in the numerical simulation of the problem. which in turn
is a tedious and time consuming job. provided that the available comr
puting machine is capable of preventing the obstruction of the system
response by digital noise which is again a problem by itself.
However. an alternate approach can be adapted at this point which. to

some extent. ’modifiea' the final results of the simulation to compen-

195

sate for the inaccuracies of the scaling of the numerical results

obtained due to the difficulties mentioned above.

Reference is made to Figure (8.2a) which depicts the angular
acceleration of the coupler link for the analytical and the experimenr
tal cases. Let us express both of these types of results can be
expressed in terms of their Fourier expansions. In other words. if

the numerical response can be expressed as

a‘=[B1sinmt+B,sinIut+...+B.sinmut]+

[Agein(m+1)ut+A,sin(m+3)ut+...+An.-sinnut]

(8.1.1a)
then the experimental results can be stated as
n.8[81sinmt+3,sin80t+...B-sinmut}+
[C‘sin(m+1)ut+C,sin(m+3)mt+...Cn..sinnut]
(8 01 01b)

Note that. in the above two equations the terms inside the first
set of brackets on the right hand side of the equations corresponds to
a function defining the 'nominal' response. while the remainder are

the superimposed oscillations and fluctuations. Obviously. the nomi-

196

nal part of these two equations are the same and. the difference

occurs in the oscillating parts of the two responses.

In an ideal simulation. both the experimental and analytical
results are made up of ideal combinations of sine waves at different
frequencies with amplitudes related by a magnification/conversion fac-

tor o. i.e.

A1=uC1 3 A,=uC, ; ... ; An_.§ucn_.
(8.1.2)
where. again. for the ideal case. we have

081.0

Assuming that equations (8.1.1) and (8.1.2) hold for the results
shown in Figure (8.2a). The validity of such an assumption will be
discussed shortly. But. for present. it is assumed that it holds
true. Hence. the only remaining parameter to be determined in equa-

tion (8.1.2) is the magnification/conversion factor. a.

Two seperate methods can be adapted to determine the factor a.
The first and more accurate method is to obtain the frequency spectrum
of the two analytical and experimental results based on the fast
fourier transform routine (chapter six) and then. through a simple

comparison between the amplitudes of the spectra at a given frequency.

197

the desired factor a can be determined. Although as mentioned above.
this method is preferred to the second one in terms of accuracy. its
use has been avoided here due to the extent of noise associated with

the rectangular windowing function used in the FFT computer routine.

The second method. which is simpler and less accurate than the
first one. deals with the results in their time-domain. In other
words. the difference in the magnitudes of the numerical and experi-
mental results at a certain instant of time is compared against each
other. After a series of comparisons at several instants of time dur-
ing the operation of the mechanism. an approximate value for the

factor u can be obtained.

From the two methods described above. the latter was used in this
work. However. direct comparisons between the experimental and numer-
ical results is not possible in this case. The reason behind this is
that. each of the experimental and analytical plots is a collection of
several points sampled (or simulated) at different instants of time.
This implies that. if there exists a point in the experimental data
which has been sampled at time. say tttt, the corresponding point on
the numerical results for t-t1 may not be available and. the next clo-
sest point would correspond to a point for t=t1t;, with 3 depending on
the integration mesh size (chapter six) on one hand and. the sampling

rate of the experimental data (chapter seven) on the other.

Hence. an indirect comparison can be made between the experimen-

198

tal and the simulation results. This is performed by comparing each
of these results to a corresponding 'nominal mechanism' response. The
latter is obtained with the aid of the computer routine adapting the
nominal mechanism equations described in chapter two. Such a compari-
son provides enough flexibility that. each of the sets of results can
be compared to the nominal response at their specified instants of
time. The simplest method of finding the factor a would be to compute
the average (or root mean square) of the deviation of each of the
plots from the nominal plot. Then. the factor a would equal the (the
square root of the) ratio of the two average deviations corresponding

to the experimental and nominal mechanisms.

For the responses shown in Figure (8.2a) this factor was found to
be in the neighborhood of 30.0 . Thus. in order to construct the
'modified' analytical response. the deviation of the analytical result
for each point is divided by this factor and the resulting modified
deviation for that point is added algabraically to the 'nominal'
response corresponding to the same instant of time. This latter addi-

tion yields the final result. called the modified analytical result.

The effect of the utilisation of such a conversion/magnification
factor on the plots of Figure (8.2a) can be observed in Figure (8.4)
which. illustrates the modified analytical coupler angular accelera-
tion superimposed on the experimental response. A note is in order at
this point that. the 'disturbance' associated with the original simu-

lated coupler angular acceleration accruing at about zero to 20

199

degrees of crank angular displacement has been.omitted during the pro-
cess of computing the factor n. This step was taken in order to avoid
the offsetting of the value of a due to this localised high-amplitude

disturbance.

As can be seen in Figure (8.4). the simulated response agrees
well with the experimental one. This would be expected. since the
analytical response was in a sense 'forced' to fall within the range
of the experimental response by the use of the factor u. The objec-
tive is now to examine the effect of such a modification on the other
simulated system characteristics with the help of the 11mg

conversion/magnification factor a computed perviously.

Figure (8.5) illustrates the correlation of the experimental and
modified analytical response of the rocker angular acceleration. The
numerical response seems to predict a smoother behavior than the
experimental one. But. there is still a fairly good correlation
between the two curves. Figures (8.6) and (8.7) show the analogous
comparisons for the coupler and. the rocker angular velocities.
respectively. Although in both of the latter figures the modified
analytical results show reasonable correlation to their correponding
experimental responses in terms of smoothness. there exists a small
offsetting of each pair of curves between their end values. This is
primarily attributed to the value of the conversion factor (not to be
mistaken with the factor a) used to convert the experimental results

obtained from the probes in millivolts to meters per second (squared).

200

 

 

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204

The above comparisons indicate that. there is a good correlation
between the experimental and the modified analytical results in terms
of scaling. Hence. the modified analytical response of other system
characteristics for which there is no experimental data available. can
be studied with caution based on the notions and remarks deduced from
the aforementioned correlations. (Figures 8.4 to»8.7). Figures (8.8)
and (8.9) show the modified analytical responses of the x- and the Y-
components of the coupler-rocker bearing reaction. Also. Figure
(8.10) depicts the simulated torque required to drive the crank link

at the specified angular velocity.

Obviously. by now the reader has realized that. although based on
the preceding discussions the X- and Y- components of the
coupler-rocker bearing reaction (Figures 8.8 and 8.9) would predict-
ably correlate reasonably with their experimental counterparts (from
scaling point of view). probably the only case of misrepresentation of
these two latter plots caused by the use of the factor a occurs for
the intervals of time during which the mechanism is in the free-flight
mode. In other words. during the free-flight mode. there should be no
interaction between the two components of the coupler-rocker bearing.
namely. the pin and the bushing. This implies that. for these cases.
the bearing reaction components should necessarily reduce to zero.
Hence. there should exist some points on the responses shown in Fig-
ures (8.7) and (8.8) which indicate such behavior and. as can be seen
such points practically do not occur in these plots. This is due to

the method by which the conversion factor c has been obtained. That

205

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208

is. once a is used to modify the responses. it does not distinguish
between the different modes of motion of the mechanism and. rather it
deals with the average of the response throughout the whole period of

motion of the system.

However. such a flaw due to the rescaling of the plots is not a
serious one since these 'xero's which are not shown on the modified
simulated responses do not. in practice. contribute to the severity of
the response in terms of questions related to design and. strengths of
the mechanism's parts under investigation. Also. as mentioned previ-
ously. it is more likely that. the outcome of the modification of the

analytical results can be improved greatly if.

1. The conversion/magnification factor. a. is obtained using the fast
fourier transform algorithm incorporating an accurate windowing func-

tion.

2. Instead of finding the average deviation of the analytical and
experimental results from the nominal response which. implicitly
assumes a uniform deviation of each response from the nominal one. a
more realistic. if not complicated. type of distribution of deviations

would be used such as normal distribution and. so forth.

Thus. in practical design applications. once the simulation is
carried out using an effective and accurate digital computer. only one
variable needs to be measured from the real-life system and. based on

the conversion/magnification factor obtained by comparing the sampled

209

data with its simulation counterpart other system characteristics
found by the simulation can be readily modified and studied in terms

of their parametric range of values.

Now. for the sake of mentioning a few other points. suppose that
the simulation and the experimental results correlate accurately both
in scaling and points of variations. Assuming that such a task can be
performed easily. other significant plots can be obtained such as the
ones presented by Iansour et. al. [7] (although based on two modes of
free-flight and impact only) and shown here in Figures (8.11) and
(8.12). As can be observed. Figure (8.11) shows the impact spectra
for the pin and the bushing of the coupler—rocker bearing. The six
largest impacts on these system components are marked as Y1 to I. in
the descending order of their magnitudes. It bgin‘ the largest. Note
that. these are locally-oriented spectra. That is. for instance. the
orientation of the impacts shown for the pin of the bearing are those
viewed by an observer moving with the rocker arm. Also. Figure (8.12)
shows the largest three impacts acting on the coupler-rocker bearing
of the mechanism during its motion. Note that. these figures were
obtained from a simulation of the model being in the quasi-steady
phase. Thus. based on the above two figures. one can realize the rea-
son behind the out-of-roundness of the bearings that possess a finite

clearance.

Assuming that. the capabilities are such that spectra and figures

such as the ones in Figures (8.11) and (8.12) can be obtained with

l9:

 

 

 

a) IMPULSES ON PIN b) IMPULSES ON SOCKET

FIGURE 8 .11. IIPULSB DIRECTIONS ON THE BEARING COIPONEQTS

ARROWS INDICATE IMPULSES
ON SOCKET

DEAD-POINT

 

 

3.3
‘3
\:

’31' (b)

(0)
FIGURE 8.12. DIRECTIONS 01? m8 nounmrr IMPACTS FOR ONE CYCLE

211

good reliability. bearings can be tailor-designed to prevent such phe-
nomena. In other words. the bushing of the bearing. for instance. can
be made of composite materials with different stiffness and strength
regions along its inner surface by locally strengthening the surface
of the bushing at regions which are likely to encounter more severe
vibro-impact shocks and impulses. This would certainly prolong the
working lifespan of the bearing along with a significant reduction of

the acoustic noise emitted as a result of these impulses [82].

CHAPTER NINE

CONCLUSIONS

A four bar linkage with rigid links and. with a finite clearance
in the coupler-rocker bearing was modelled and simulated. The
observed disagreement between the experimental and the analytical
results were presumably due to the ideal values assumed for the coefr
ficient of restitution and the Coulomb coefficient of friction. along
with the presence of certain level of round up/truncation error. cou-
pled with the existense of manufacturing errors in building the
experimental apparatus. Also. the simulation was presented for the
start-up or the transient phase of the motion of the mechanism under
study. whereas the experimental results corresponded to the

quasi-steady phase.

The scaling of the simulation data was modified further using a

conversion/magnification factor computed based on the comparison of

213

the experimental and. the original simulation results. The outcome of
such modification suggested reasonable correlation between the analyt-
ical and the experimental responses in terms of the parametric scaling
of the results. The reliability of those system parameters for which
there was only simulation data available. was deduced based on the
aforementioned comparison of the experimental data with their simula-

tion counterparts.

Improvements in the simulation results obtained in this work can

be achieved by

l. DevelOpment of a means of measuring the correct values of the coef-
ficient of restitution and the Coulomb coefficient of friction of the
experimental rig under study. The resulting values can be incorporat-
ed into the simulation in order to test the threshold of reliability
of model assumed in this work and. if possible. enhance the analytical

results.

2. Iodeling the four bar linkage under study as having elastic links
for the coupler and. the rocker links. This. in turn. necessitates
the adaption of different approaches to solve and develcpe the govern-

ing equations of motion.

3. Incorporation of the effect of the bearing lubrication and
hydrodynamic phenomena occuring within the bearing with finite clear-

4. Iodeling a four bar linkage with simultaneous clearances existing

214
in more than one bearing of the mechanism.

5. Development of a model to investigate the acoustic radiation due to

the existense of clearance in the bearing(s) of the mechanism.

6. Nodelling a four bar linkage which encompasses all of the above

four cases.

7. Extension of the model to the other types of mechanisms with higher

number of links and. more complicated configurations.

8. Construction of a three-dimensional analytical model. which would
be able to incorporate the torsional loading on the bearing with

clearance due to possible misalignments of the mechanism links.

APPENDIX

215

APPENDIX

SCHEIATIC ELOICHART OF THE SIIULATION
CODE -STIUL-

(”1”)

Variable type decleration]

A I _
_ Fulani

Primary data input
(known system parameters)

I

Array and vector initialization}

1

Set up vector containing the
values of crank angle at
desired intervals

 

‘4

 

 

 

 

 

 

 

 

 

 

 

 

1

#‘

Compute nominal mechanism
system parameters

Compute the reaction components

of the coupler-rocker bearing
of the nominal mechanism

 

 

*—

216

Input the starting value of the
crankianale

Set up the vector containing the parameter

values corresponding to the initial
conditions of the simulation

1

Input secondary data. i.e.. the integration
mesh size. upper bound error sixe. No. of
crank revolutions ...

1

Input the value of the coeficient of
restitution

I

‘A

 

 

 

 

 

 

 

 

 

 

Start the simulation

 

 

 

<9

      
 

Contact-loss

 

 

 

 

 

 

217

[*Set up new initial conditions

 

 

 

N

Iterate one step—]

 

 

 

 

 

 

 

 

 

 

Compute coupler and rocker angular
velocities after impact

 

- 1

Set up new initial conditions for
the following mode

218

Irite down the result of the
simulation into seperate
output files

7* I

Cm)

—__.

219

SUBROUTINES ‘ND FUNCTIONS USED
IN CONJUNCTION 'ITH THE CODE

-STNUL-

ANGLE : Given the two orthogonal sides of a right triangle. com-

putes the corresponding anales in the range of 0 to 2a.

CLANG : Given the Y- and Y- components of the coupler-rocker
bearing reaction of the nominal mechanism. computes the angular dis-
placement. velocity and. acceleration of the total force at a given
time t‘t1. where t; is the starting time of the integration. Note
that. these angular parameters correspond to those of the 'clearance

link' at initial conditions.

CRKSET : Given an initial starting crank angle. computes the
values of the crank angular displacement at prescribed intervals and.

stores them in a pre-initialixed vector.

DIFF : Given an array containing one of the system parameters
corresponding to equal time intervals. computes the first and second
derivatives of the array with respect to time. The differentiation is

crried out using difference folmulae (backvard. central and. forward).

FCT : Used in conjunction with the numerical integration package

220

HPCG. computes the values of the unknowns of equation 3.3-7 at each

time step. Corresponds to the following mode.

FCTF : Used in conjunction with HPCG. computes the unknowns of
the equations 4.1-20 and 4.2-1 at each time step. Corresponds to the

free-flight mode.

FORCE : Given the angular displacements. velocties and. accelera-
tions of the links of the nominal mechanism. computes the values of
the '- and '- components of the coupler-rocker bearing reaction at
each time step. The result is fed into a predesignated vector forl
further manipulations. This subroutine also computes the crank torque

at each time step.

HPCG : The numerical integration package. Using subroutines FCT
and OUT? (following mode) or FCTF and OUTPF (free-flight mode). along
with the integration mesh size. upper bound of the absolute error of
the integration and. other secondary inputs. this package solves the

given differential equations numerically.

OUTDAT : Generates seperate output files to store the final

results of the simulation obtained from HPCG.

OUT? : Used in conjunction with HPCG. Ihen in the following
mode. it examines the contact-loss criteria and. signals the main code

for such occurence.

221

ODTPF : Used in conjunction with HPCG. Ihen in the free-flight
mode. it examines whether contact has been reestablished. If so. the

main code is signaled accordingly.

OUTPUT : Generates seperate output files to store the system

parameters of the nominal mechanism.

IADDEG : Given an angle in radians. this function convert the

value to degrees.

SIDRl : Given the geometric data of the nominal mechanism along
with the crank angular velocity. it computes the coupler and rocker

angular displacements.

SNDHZ : Given both the input and the output of the subroutine
SlDRl. conputes the angular velocities of the coupler and the rocker

links of the nominal mechanism.

SIDES : Given both the input and the output of the subroutine
SIDRZ. computes the angular accelerations of the coupler and rocker

links of the nominal mechanism.

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I i I l l I II. I I I I I I 1

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