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This is to certify that the

dissertation entitled

NONLINEAR VIBRATIONS OF A FLEXIBLE
CONNECTING ROD

presented by

Shang-Rou Hsieh

has been accepted towards fulﬁllment
of the requirements for

Ph.D. Mechanical Engineering

degree in

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MSU Is An Atﬂrmdlve AotioNEquel Opportunity Institmion

 

NONLINEAR VIBRATIONS OF A FLEXIBLE
CONNECTING ROD

By
Shang-Rou Hsieh

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1991

’ ‘

l” ’1“ \J h

// L

/

ABSTRACT

NONLINEAR VIBRATIONS OF A FLEXIBLE
CONNECTING ROD

By

Shang—Rou Hsieh

An analytical and computer simulation investigation of the dynamic behavior associated
with the ﬂexible, uniform connecting rod of an otherwise rigid, in line, planar slider-crank
mechanism is presented. The main goal is to analyze the ﬂexural response of the elastic
connecting rod and to study how this response depends on the system parameters. More-
over, this work emphasizes nonlinear analyses of the dynamic response and its associated
stability.

Two different approaches are used in modeling the ﬂexural vibration of the elastic con-
necting rod. In the first approach, the model used is general and includes the effects of
finite axial and transverse deformations, internal material damping, bearing friction, slider
friction, shear deformation and rotary inertia. The axial and the ﬂexural vibrations are
described by two nonlinear coupled partial differential equations. In the second approach,
the ﬂexural vibration of the connecting rod is described by a single nonlinear ordinary dif-

ferential equation in the second approach which is obtained from the more general model

by a single mode Galerkin truncation. In deriving this equation, the effects of the shear
deformation and rotary inertia are neglected. Moreover, the contribution of axial inertia is
neglected in analyzing the acceleration components.

In studying the dynamic response of the connecting rod, the method of multiple scales
is employed to solve the equations of motion. The slenderness ratio of the connecting rod is
selected as the small scaling parameter. With the method of multiple scales, the solutions of
the axial and the transverse deformations are solved as asymptotic series expansions in terms
of this scale. In both of the approaches, several resonances, including the primary resonance,
the principal parametric resonance, and various super- and sub- harmonic resonances, are
investigated in detail. The analytical results are confirmed by the numerical solutions.

The analytical results show that the solutions and their associated stabilities depend
in a. nontrivial manner on several parameters. They are the ratios of the crank radius to
the length of the connecting rod, the ratio of the connecting rod mass to the slider mass,
the ratio of the crank speed to the fundamental frequency of ﬂexural vibration, internal
material damping, bearing friction and slider friction. The effects of these parameters on

the ﬂexural vibration of the connecting rod are investigated analytically and numerically.

To the members of my family.

iv

ACKNOWLEDGEMENTS

I would like to take this opportunity to express my sincere gratitude to my thesis advisor
and teacher, Steven W. Shaw, for his inspiring encouragement, advice and patience.

I would also like to thank the other committee members, Alan G. Haddow, C.Y. Wang
and Joseph Whitsell, for their comments and help with this dissertation; and Ki-Yin Chang
who helped me in installing the software package AUTO. I am indebted to Rong-Wong
Chen for inspiring me to pursue Ph.D. degree in Michigan State University.

Finally, I thank my mother, brother, and my wife Shu-Mei for their support and under-

standing throughout my graduate study and research.

Contents

1 Introduction

1.1 Description of the Problem ...........................
1.2 Literature Survey .................................
1.3 Scope and Purpose ................................
1.4 Organization of Dissertation ...........................

2 Mathematical Modeling

2.1 Basic Assumptions ................................
2.2 Equations of Motion ...............................
2.2.1 Deformation ...............................
2.2.2 Acceleration Components ........................
2.2.3 Stress-Strain Relation ..........................
2.2.4 Equation of Motion ...........................
2.2.5 Boundary Conditions ..........................
2.3 Comparisons ...................................
2.4 Nondimensionalization ..............................

3 Analysis of the Distributed Parameter Model

3.1 Application of the Method of Multiple Scales (MMS) .............
3.2 Principal Parametric Resonance .........................
3.3 Primary Resonance (Q as wn) ..........................

vi

10

ll

11

12

13

15

17

18

21

22

25

29

30

41

51

3.4 Superharmonic Resonance (Q a: 9f) ...................... 59

3.5 Superharmonic Resonance (52 z 9311) ...................... 64
3.6 Subharmonic Resonance ((2 z 3%,) ....................... 68
3.7 Summary and Conclusion ............................ 72
3.7.1 Effect of the Length Ratio 5 ....................... 72
3.7.2 Effect of Mass Ratio S .......................... 74
3.7.3 Effect of Damping Parameters #2, p3 and p4 ............. 77
3.7.4 Effect of Shear Deformation and Rotary Inertia ............ 83

4 Analysis of the Lumped Parameter Model 86
4.1 Equation of Motion ................................ 87
4.2 Application of the Method of Multiple Scales ................. 94
4.3 Principal Parametric Resonance (0 z 2) .................... 100
4.4 Primary Resonance ((2 z 1) ........................... 109
4.5 Superharmonic Resonance (9 z %) ....................... 116
4.6 Superharmonic Resonance (Q m %) ....................... 119
4.7 Subharmonic Resonance (0 z 3) ........................ 124
4.8 Summary and Discussion ............................ 127
4.8.1 Effect of the Length Ratio 5 ....................... 127
4.8.2 Effect of the Frequency Ratio (2 ..................... 128
4.8.3 Effect of the Mass Ratio S ........................ 131
4.8.4 Effect of the Damping Parameters p2 and p4 ............. 134

5 Comparisons between Distributed and Lumped Parameter Models 141
5.1 Primary Resonance (9 z 1) ........................... 141

vii

5.2 Principal Parametric Resonance (0 z 2) .................... 143
5.3 Superharmonic Resonance (Q is %) ....................... 145
5.4 Superharmonic Resonance (Q a: :15) ....................... 145
5.5 Subharmonic Resonance ((2 z 3) ........................ 147
5.6 Summary ..................................... 148
Numerical Solutions and Comparisons 150
6.1 Inﬂuence of the Length Ratio 5 ......................... 151
6.2 Inﬂuence of the Mass Ratio 8 .......................... 160
6.3 Inﬂuence of the Friction Parameters p2 and p4 ................ 164

6.3.1 Effect of the Damping Parameter p2 .................. 164

6.3.2 Effect of the Friction Parameter #4 ................... 168
6.4 Conclusions .................................... 172
Summary and Conclusions 177
Construction of the Linear Operator 182
Superharmonic and Principal Parametric Resonances for E = 651 190
Solution Procedure for the Case 6 = 6252 194
Solution Procedure for the Case 6 = 6363 199
Coefficients f7 and ﬁg 204

viii

List of Tables

6.1 Identiﬁcation of Combinations of Parameters .................

6.2 Comparison of Bifurcation Data ........................

6.3 Limits of MMS and AUTO

ix

2.1

2.2

2.3

2.4

2.5

2.6

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

List of Figures

Displacement of the beam element ....................... 13
Geometry of the deﬂection curve ........................ 15
Slider crank mechanism with ﬂexible connecting rod (undeformed state) . . 15
Slider crank mechanism with ﬂexible connecting rod (deformed state) . . . 16
Viscoelastic model (Kelvin-Voigt model) .................... 18
Free body diagram for beam element ...................... 19
Frequency response equation for the principal parametric resonance ..... 44
Frequency response equation for the principal parametric resonance ..... 48
Frequency response equation for the primary resonance ........... 53
Frequency response equation for the primary resonance ........... 57
Frequency response equation for superharmonic resonance ((2 z 3211) . . . . 62

Inﬂuence of mass ratio S on the amplitude of superharmonic resonance (0 z 2211) 64
Frequency response equation for superharmonic resonance ((2 a: 331-1) . . . . 68

Inﬂuence of S on the peak amplitude of the superharmonic resonance ((2 z 131) 69

Inﬂuence of f on the primary resonance .................... 73
Inﬂuence of E on the main nose of instability ................. 73
Inﬂuence of S on the primary resonance .................... 75
Inﬂuence of S on the main nose of instability ................. 76
Inﬂuence of S on the principal parametric resonance ............. 76
Inﬂuence of p; on the primary resonance .................... 78

X

3.15 Inﬂuence of p; on the main nose of instability .................
3.16 Inﬂuence of [12 on the principal parametric resonance .............
3.17 Inﬂuence of [13 on the principal parametric resonance .............
3.18 Inﬂuence of #3 on the primary resonance ....................
3.19 Inﬂuence of #4 on the superharmonic resonance (9 z 3%) ..........
3.20 Inﬂuence of #4 on the main nose of instability (fl z 21.01,.) ..........
3.21 Inﬂuence of m on the principal parametric resonance (9 :3 2m”) ......
3.22 Inﬂuence of [14 on primary resonance (9 z wln) ................
3.23 Inﬂuence of shear deformation and rotary inertia on the primary resonance
3.24 Inﬂuence of shear deformation and rotary inertia on the principal parametric

resonance .....................................

4.1 Frequency response equation for the principal parametric resonance (equa-

tion [4.451) ....................................
4.2 Frequency response equation for the principal parametric resonance (equa-

tion [461]) ....................................
4.3 Frequency response equation for the primary resonance (equation [4.92]) . .
4.4 Frequency response equation for superharmonic resonance (0 z %) .....
4.5 Frequency response equation for the superharmonic resonance (9 z %) . . .
4.6 Inﬂuence of f on the primary resonance ....................
4.7 Inﬂuence of E on the principal parametric resonance .............
4.8 Inﬂuence of S on the primary resonance ....................
4.9 Inﬂuence of S on the main nose of instability .................
4.10 Inﬂuence of S on the principal parametric resonance . . ............
4.11 Inﬂuence of S on superharmonic resonance (9 z %) ..............

xi

78

79

80

81

82

82

83

84

85

114

118

121

131

133

4.12 Inﬂuence of S on superharmonic resonance (9 z :13) .............. 135

4.13 Inﬂuence of [1.2 on the primary resonance (equation [4.92]) .......... 136
4.14 Inﬂuence of m on the main nose of instability ................. 136
4.15 Inﬂuence of p; on the principal parametric resonance ............. 137
4.16 Inﬂuence of m on the primary resonance ((2 z 1) ............... 138
4.17 Inﬂuence of #4 on superharmonic resonance (9 a: %) ............. 139
4.18 Inﬂuence of #4 on the main nose of instability (Q 2 2) ............ 139
4.19 Inﬂuence of m on the principal parametric resonance ((2 z 2) ....... 140
6.1 Frequency response curves from MMS, LSODE and AUTO ........ 151
6.2 Comparison between Analytical result with AUTO ( case 5) ........ 153
6.3 Comparison between Analytical result with AUTO ( case 6) ........ 153
6.4 Frequency response for cases 1, 4 and 7 .................... 154
6.5 Frequency response for cases 2, 5 and 8 .................... 156
6.6 Frequency response for cases 3, 6 and 9 .................... 156
6.7 Location of points PDl, PDz, LP; and LP; .................. 157
6.8 Comparison between simulation and analytical approximation for case 8 . . 157
6.9 Comparison between simulation and analytical approximation for case 5 . . 159
6.10 Comparison between simulation and analytical approximation for case 9 . . 159
6.11 Comparison between simulation and analytical approximation for case 6 . . 160
6.12 Frequency response curves from AUTO (cases 4, 5 and 6) .......... 161
6.13 Frequency response curves from AUTO (cases 7, 8 and 9) .......... 161
6.14 Simulation results for cases 7 and 8 ....................... 163

6.15 Nontrivial responses associated with the principal parametric resonance (cases

5 and 6) ...................................... 163

xii

6.16

6.17

6.18

6.19

6.20

6.21

6.22

6.23

6.24

6.25

6.26

6.27

6.28

6.29

6.30

6.31

Amplitude of the response at Q = % ...................... 164
Frequency response curves from AUTO (case 5 and 14) ........... 165
Frequency response curves from AUTO (case 8 and 17) ........... 165
Simulation results from LSODE (case 8 and 17) ............... 166
Nontrivial responses associated with the principal parametric resonance (case

5 and 14) ..................................... 167
Simulation results for cases 5 and 14 ...................... 167
Frequency response curves from AUTO (cases 6 and 21) .......... 168
Frequency response curves from AUTO (cases 8 and 23) .......... 169
Nontrivial responses associated with the principal parametric resonance from

AUTO (cases 5 and 20) ............................. 169
N ontrivial responses associated with the principal parametric resonance from

AUTO (cases 6 and 21) ............................. 170
Frequency response for cases 7 and 22 from LSODE ............. 171
Analytical approximation and simulation results for case 7 .......... 171
Analytical approximation and simulation results for case 22 ......... 172
Frequency response curve for case 10 ...................... 173
Nontrivial responses associated with the principal parametric resonance (case

4) ......................................... 174
Nontrivial responses associated with the principal parametric resonance (case

Emngr‘

T1. T2
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A3

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7

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5,51,62,53

LIST OF SYMBOLS

cross-sectional area of the connecting rod

the amplitude function of the nth mode of the ﬂexural response
absolute acceleration of the element on the moving frame
shear-deﬂection coefficient

the length of the connecting rod

resultant bending moment

resultant axial force

resultant shear force

mass ratio (= slider mass/mass of connecting rod)

new time independent variables

the axial and transverse displacements, respectively

angle between the median line and the the x-axis
dimensionless parameter related to the slenderness ratio
dimensionless parameter related to the effect of shear deformation
dimensionless parameter related to the effect of rotary inertia
dimensionless parameter related to the quadratic nonlinearity
dimensionless parameter related to the cubic nonlinearity
dimensionless parameter related to shear deformation and rotary inertia
a small, constant parameter

shear angle

the slenderness ratio of the connecting rod

amplitude of the particular solution of v1"

amplitude of the particular solution of 122,,

bearing damping

internal Viscoelastic material damping

friction coefficient

frequency ratio (:53?)

operating frequency of the crank shaft

ﬁrst ﬂexural natural frequency of the connecting rod

the natural frequency of the n-th mode ﬂexural vibration
constant (=3.1415926...)

mass density (mass/ volume) of the connecting rod

detuning parameter of the frequency

normal, shear stresses, respectively

bending angle

length ratio

Chapter 1

Introduction

1.1 Description of the Problem

In this dissertation, we study the ﬂexural vibration associated with the ﬂexible con-
necting rod of an otherwise rigid, in~1ine, planar slide crank mechanism. This problem is
equivalent to ﬁnding the ﬂexural response associated with a simply supported beam sub-
jected to (1) the motion of foundation which arises from the motion of the crank shaft, (2)
axial load arising due to the friction force applied to the slider end and the inertial force of
the slider mass, (3) a distributed load which arises from the inertial loading and is applied
along the span of the beam, and (4) concentrated frictional moments arising due to the
presence of bearing friction.

One of the traditional approaches to the dynamic analysis of mechanisms is based on the
assumption that the system is composed of rigid bodies. In fact, a mechanism is deﬁned as
”an assembly of rigid bodies, connected by movable joints, to form a closed kinematic chain
with one link ﬁxed and having the purpose of transforming motion” [32]. Based on this
deﬁnition, the early works in the dynamics of mechanisms concerned themselves with deriv-
ing the displacements, velocities, accelerations, etc, of the mechanism. A basic concept
of kinematic analysis is to determine the maximum acceleration of the machine member

as a function of the input characteristics. The dynamic analysis includes the derivation of

1

2

inertial forces resulting from the rigid-body accelerations found from the kinematic analysis.
However, the rigid body assumption may not always hold. As required by modern industry,
machinery often needs to operate at high speed while maintaining accurate performance.
Due to their inherent ﬂexibility, machine elements deﬂect when subjected to external loads
and/ or internal forces. An increase of the operating speed results in increased inertial forces;
this increase of the the inertial forces can cause sever deformation of the machine members
even without the action of external loads. In other words, inertial loading arising due to
the high operating speed can be responsible for link deformation and associated vibrational
behavior. Also, link members are being designed to be of smaller mass in order to reduce in-
ertial loads, but this often leads to increased ﬂexibility and thus ﬂexural vibrations. Hence,
the traditional rigid linkage analysis is insufﬁcient to provide a satisfactory prediction and
description of the function of these mechanisms. Recognizing these shortcomings of rigid
body analysis, the idea of kineto-elastodynamics has been introduced[17, 27], which is the
study of the motion of mechanisms consisting of elements that deform due to external loads
or internal forces. Basically, kineto-elastodyanmics is a ﬁeld combining kinematics, dynam-
ics and elasticity together. A study of high-speed mechanisms with ﬂexible links requires
knowledge of this ﬁeld. Among the mechanisms commonly used, the slider-crank mecha-
nism may be the simplest. The simplicity of of the structure of the slider-crank makes it
useful as a beginning example, especially when only one linkage, the connecting rod, of the
mechanism is assumed to be ﬂexible. This mechanism is also of practical interest to the

automotive, manufacturing, and agricultural industries.

1 .2 Literature Survey

The slider-crank mechanism, in which the connecting rod is considered to be a ﬂexible

3

member, has been investigated by several authors. In the following, we would like to review
some of the fundamental works by previous investigators. For more complete information,
please refer to the review articles[27, 17, 26, 45]. The methods employed in studying the
elastic mechanism problem can be grouped into the following categories: the ﬁnite element
method, experimentation and analytical methods.

The basic idea in studying this problem with the ﬁnite element method is to model each
member of the ﬂexible mechanism as discrete system with a ﬁnite number of degrees-of-
freedom. Therefore, each member can be divided into several small discrete elements. The
motion associated with each individual element is described by a set of ordinary differen-
tial equations. Finally, each element equation is assembled together and form the global
equations of motion describing the motion of the whole mechanism. The advantage of ﬁnite
element method is that it provides a systematic and easy-to-apply procedure in modeling
the mechanism. Winfrey[51, 52] was among the ﬁrst investigators who applied the ﬁnite
element methods to study elastic mechanism problem. The application of the ﬁnite element
method in studying elastic mechanisms was limited by lack of computational powers, in the
early 19708, because of the vast manipulation of the matrices associated with each element
and the assembling procedure. With the advent of the high-speed computers, the ﬁnite ele-
ment method has become more popular recently. In order that the ﬁnite element method be
applied to solve the elastic motion of the mechanisms, the continuous motion of the system
is modeled as a sequence of structure conﬁgurations at discrete crank angles upon which the
inertial loading is imposed. In order to solve these structures with ﬁnite element methods,
the rigid body motion associated with these instantaneous structures must be removed from
the model to avoid singular matrices. Winfrey[51, 52] accomplished this by directly apply-

ing the principle of conservation of momentum to the complete mechanism. Imam et al. [25],

4

Midha et al.[l] and Gandhi et al.[18] considered the crank as a cantilever beam to avoid
this complication. Nath and Ghosh[28] removed the rigid-body degree of freedom from the
global matrices using a matrix decomposition approach. Concerning the formulation of the
equations of motion, two different approaches are employed: the stiffness and the ﬂexibility
methods. Winfrey[51, 52] employed the displacement ﬁnite element method (the stiffness
technique of structural analyses) to study the elastic motion of mechanisms. This approach
yields directly the nodal displacements and requires displacement compatibility on inter-
element boundaries. Midha et al.[2] applied this approach to study the dynamic response
of a ﬂexible member of a four-bar linkage. Erdman et al.[4, 3] employed ﬂexibility method
of structural analysis to study ﬂexible mechanisms. With this approach, adjacent elements
have equilibrating stress distributions on inter-element boundaries and the global degrees of
freedom are the stress components. Turcic et al.[48, 47, 13] studied the elastic mechanism
problem by using a general procedure in formulating the equations of motion, including
the effects of rigid-body motion and elastic coupling terms. Instead of a single type of
ﬁnite element, they generalized the procedure so that different types of ﬁnite elements can
be included in the same mechanism. The axial force is solved for by using a quasi-static
analysis, and is then included in the transverse component of the equation of motion by
the introduction of the geometrical stiffness matrix. The effects of elastic coupling terms
are included by adding the geometric stiffness matrix to the element stiffness matrix. This
approach is equivalent to modeling the deformation associated with the connecting rod by
von Karman’s ﬁnite deformation theory. Cronin and Liu[12] studied the linear vibration of
a connecting rod in a planar slider-crank mechanism by using the ﬁnite element analysis
software N ASTRAN . The steady state responses associated with the connecting rod are

investigated with different combinations of parameters. Their results show that superhar-

5

monic resonances can be observed when the operating frequency is near to one third or one
half of the ﬂexural natural frequencies associated with the connecting rod.

Unlike the ﬁnite element method, analytical methods treat each individual ﬂexible mem-
ber associated with the mechanism as a single continuous system. Assuming that only one
of the members is ﬂexible, the motion of the ﬂexible member in an otherwise rigid mecha-
nism is described by a set of partial differential equations which, under some circumstances,
can be converted into ordinary differential equations by modal truncation. The major
analytical methods applied in these investigations are: Newtonian dynamics[15], Hamilto-
nian methods[49], variational methods[44, 43, 42] and lumped mass approaches[35, 36, 34].
We note that the last two methodologies, especially the variational principle approach, are
equivalent to the ﬁnite element method. Some of the related fundamental works are outlined
below.

Neubauer, Cohen and Hall[5] examined the transverse deﬂection of the elastic connecting
rod of the slider-crank mechanism by neglecting the longitudinal deformation, the Corio-
lis, relative tangential, and relative normal components of acceleration. They showed that
the amplitude of the transverse vibration increases when the the dynamical axial load ap-
proaches the Euler criterion for static buckling of long, slender columns. They also pointed
out that a more thorough analysis of the response requires a more complete study which
includes the following dimensionless parameters: the ratio of crank speed to natural fre-
quency of ﬂexural vibration, the ratio of connecting rod mass to slider mass, and the ratio
of crank radius to connecting rod length. By the Method of Averaging, and assuming a
small length ratio, Jasinski, Lee and Sandor[30, 31] investigated the dynamic stability of
a ﬂexible connecting rod in a slider-crack mechanism. In the ﬁrst paper, they studied the

slider-crank mechanism for which a concentrated mass is assumed to be attached to one end

6

of the connecting rod. In the second paper, the connecting rod is assumed to be hinged at
each end, an unrealistic assumption. Both longitudinal and transverse deﬂections are con-
sidered, and the coupled linear equations describing the motion are obtained from the force
and moment balance of a differential element. Linearization of these equations was accom-
plished by neglecting the nonlinear coupling terms. Assuming a one-term product solution,
they obtain two coupled, nonhomogeneous Mathieu equations. Finally, they use the Routh-
Hurwitz criterion investigate the stable regions, and the associated steady-state solutions
of the transverse and axial deﬂections are given. Their results showed that the amplitude
of the transverse responses depends upon the frequency ratio quadratically, while the axial
response is not so sensitive to this parameter. Moreover, the vibration amplitudes increases
as the mass ratio increases. They pointed that future work in this area should include
elastic stability analyses using similar asymptotic methods and include the effects of the
nonlinear coupling terms. Viscomi and Ayre[49] examined the nonlinear bending response
of the connecting rod. Assuming that the inertia force from axial deﬂection is negligible,
the axial dynamic force is approximated by solving the so—called reduced equation. With
this axial force, the transverse equation is reduced to a. single ordinary differential equation
by the Galerkin method. The resultant equation is then studied numerically. They showed
that, within the frequency range of interest, the second mode is relatively unimportant
and the response is closely approximated by the ﬁrst mode dynamics. By comparing the
responses from the linear and nonlinear equations, it was concluded that the system is not
adequately deﬁned by the linear form of the equation. Moreover, the existence of multiple
steady state solutions was also observed in some parameter regions. They pointed out that
the response depends on the following dimensionless parameters: the frequency ratio, the

mass ratio, the length ratio, transverse damping and the external piston force. With the

7

same assumptions, Chu and Pan[11] carried out a similar study by use of the Kantorovich
method and the method of weighted residual. The ﬁnal governing equations are studied
numerically. Their results show the the effect of small viscous damping on longitudinal
vibration is small and can be neglected. Moreover, a sudden increase in the amplitude of
the steady state response accompanied with a small increase of the operating frequency, i.e.
the jump phenomenon, has also been observed. Using variational principles, Thompson and
Ashworth[43] studied a planar slider-crank mechanism mounted on a foundation subjected
to a motion normal to the mechanism plane. They found that the combination resonances
can be observed when the sum or the difference of the operating and foundation frequencies
are approximately equal to the ﬂexural natural frequencies associated with the connecting
rod. By applying regular perturbation method to Euler—Bernoulli and Timoshenko beam
models, Badlani and Kleinhenz[8] considered the dynamic stability of the undamped elastic
connecting rod of an in-line slider-crank, and compared the results from each approach.
Their results indicated that new regions of instability exist when both the rotary inertia
and the shear deformation are included in the analysis. Tadjbakhsh[41] introduces in a
general method for obtaining a single partial differential equation describing the transverse
vibration of an undamped elastic link of a mechanism which contains evolutes only, using a
two-parameter perturbation approach. The ﬁnal dimensionless Hill’s equation is obtained
by assuming a sinusoidal shape function and using Galerkin’s method. His result shows that
contributions from the second mode and higher mode amount to less than one percent. Zhu
and Chen[54] studied the stability of the response of the connecting rod by using a regular
perturbation technique. A linear, undamped, nonhomogeneous partial differential equation
is used to described the transverse response based on Euler—Bernoulli beam theory. Using

this equation and Galerkin’s method, they obtain a set of decoupled homogeneous Hill’s

8

equations. After reducing these to Mathieu equations, a ﬁrst-order Floquet approach was
used to obtain the two highest p-periodic and 2p-periodic instability regions. Badlani and
Midha[9] studied the effect of internal material damping on the dynamic response behavior
of a slider-crank mechanism by using the regular perturbation method. Both steadystate
and transient solutions were determined and compared to those results obtained from the
undamped connecting rod. They concluded that the viscous internal material damping
may have signiﬁcant inﬂuence, both favorable and adverse, on the steady-state and tran-
sient transverse responses. Sandor and Zhuang[37] studied a planar four-bar mechanism
by using a linearized lumped parameter approach. The effects of rotary inertia and end
mass are included in their investigation. A set of nonlinear ordinary differential equations
describing the dynamic behavior of the mechanism is derived based on the lumped mass
approach. These equations are then linearized and decoupled by using the concept of kine-
matic inﬂuence coefﬁcients. The resultant equations are solved by using a ﬁnite difference
approach. Their investigation shows that the linearized approach is sufficient in represent-
ing the dynamic response, and that the rotary inertia and end mass of the moving link have
considerable effect on the elastic response of the mechanisms.

In addition to the analytical and computational works describe above, some experimen-
tal works has been carried out [6, 7, 40, 19]. Golebiewski and Sadler[19] determined the
bending stress at the midpoint of the elastic connecting rod analytically and experimen-
tally for a model derived by a lumped parameter approach using d’Alembert’s principle
and Euler—Bernoulli beam theory. They examined the effects of the crank speed, crank
length, and slider offset on the maximum bending stress at the midpoint of the connecting
rod. Superharmonic responses appeared in both the analytical and experimental results.

Sutherland[40] studied a fully elastic planar four-bar mechanism analytically and experi-

9

mentally. By using Euler-Lagrange formulation and modal analysis with only one mode,
the dynamic behavior associated with the mechanism is described by a set of ordinary
differential equations with two holonomic constraints. The resultant equations are then
simpliﬁed by neglecting the shear deformation, rotary inertia, joint frictions, bearing mass
and all the external forces except the torque required to keep the mechanism operating at
a constant speed. These equations were linearized based on the assumption that the deﬂec-
tions associated with each elastic member are small. Consequently, the resultant equations
describing the dynamic behavior associated with the mechanism are three coupled, linear,
nonhomogeneous ordinary differential equations with time-varying coefﬁcients. The Method
of Harmonic Balance is then applied to solve these equations. He pointed out that length
ratios and frequency ratios are the essential dimensionless parameters for this investiga-
tion. An experimental study was carried out to support the analytical work. High order
superharmonic resonances were observed in the experiments. His results indicated that the
modal analysis using only a single mode is sufﬁcient to represent the dynamic response of

the mechanism.

1.3 Scope and Purpose

All previous investigations lead to a nonhomogeneous Mathieu equation to describe
the transverse vibration, and only linear stability analyses were provided. The existence
of superharmonic, subharmonic and combination resonances have been observed both in
simulations and experiments [49, 11, 19, 40]. An analytical study of the overall nonlinear
response, especially dealing with the superharmonic, subharmonic responses, and their as-
sociated stability, is lacking. It is the purpose of this study to provide such an investigation

of the vibration associated with the ﬂexible connecting rod of a planar slider-crank mech-

10

anism. Moreover, this work will focus on the nonlinear analysis of the dynamic response,

e.g., primary, principal parametric, superharmonic resonances and their stability.

1.4 Organization of Dissertation

This dissertation is organized as follows. Chapter 2 contains the basic assumptions and
the derivation of the partial differential equations used to describe the system. Chapter
3 provides the analysis of the partial differential equations obtained in chapter 2 by using
the method of multiple scales. In chapter 4, we simplify the model by using von Karmon’s
ﬁnite deformation theory and obtain an ordinary differential equation describing the ﬂexural
vibration associated with the ﬁrst ﬂexural mode of the connecting rod. We then analyze this
ordinary differential equation by using the method of multiple scales. Chapter 5 contains
a detailed comparison between the results obtained in chapters 3 and 4. In chapter 6, we
compare the results obtained in chapter 4 with numerical simulations. In chapter 7, we
close this dissertation with some conclusions and some suggestions for future investigations.

In the course of this study, extensive use was made of the computer assisted symbolic
manipulation program M athematicaTM . This was essential in the lengthy calculations

involved in the perturbation procedures.

Chapter 2

Mathematical Modeling

In this chapter, equations describing the rotary, axial and transverse dynamic responses
associated with a ﬂexible connecting of an otherwise rigid slider crank mechanism are de-
rived. The basic assumptions which are used in deriving these equations of motion are
given in section 2.1. Based on these assumptions in section 2.2, we then proceed to de-
rive the equations describing the axial as well as the transverse deﬂections associated with
the connecting rod, and the boundary conditions associated with these equations. In sec-
tion 2.3, we compare the resultant equations to the models which were used by previous
investigators. We then convert the equations of motion and their associated boundary con-
ditions into dimensionless form in order to minimize the number of the system parameters

in section 2.4.

2.1 Basic Assumptions

In deriving the equations which describe the axial and the transverse deﬂections associ-

ated with the ﬂexible connecting rod, the following assumption are employed:
0 Concerning the connecting rod AB

1. is an elastic member with uniform cross section

11

12

2. all quantities are assumed to be independent of z, the out-of-plane coordinate
3. plane cross section remain plane after deformation
4. made of a Viscoelastic material

5. simply-supported at both ends, with rotary damping at the pivot ends (to model

bearing effects)
6. point C on rod AB moves to point C’, after deformation,

7. shear deformation and rotatory inertia are included
0 Concerning the crank element OA:

1 . perfectly rigid

2. constant crank speed, to
o Concerning the slider mass D:

1. no clearance
2. no offset
3. velocity-dependent friction force acting between the slider and its contact surface

4. no external forces applied to the piston.

2.2 Equations of Motion

In the present section, equations describing the axial and transverse deﬂections as-
sociated with the ﬂexible connecting rod are derived based on the assumptions given in
section 2.1. This problem is equivalent to a simply supported beam subjected to the inertia

forces associated with its motion as speciﬁed by the kinematic constraints. These inertia

13

forces are caused by the acceleration associated with the connecting rod. We ﬁrst analyze
the acceleration components of a point located on the ﬂexible connecting rod. We then
proceed to derive the equations of motion describing the axial and transverse deformations
of the connecting rod following the work of Chow[10]. By combining these results, we obtain
the equations describing the axial and transverse deﬂections associated with the connecting

rod subjected to the inertia forces.

2.2.1 Deformation

We now consider a segment of the beam with length As: taken from the connecting rod

as shown in ﬁgure 2.1. After deformation, the angle between the median line and the x-axis

 

--------------->

 

 

 

Figure 2.1: Displacement of the beam element

is given as

029+? (2.1)

where 0 = the bending angle and 7 = the shear angle. Let a material point located at (x,y)

14

in the undeformed state move to (:c’ , 3”) after deformation. Therefore, we have

x' = x+u—ysin0 (2.2)

v + yc050 (2.3)

t:
II

where u(a:, t) and v(:r,t) are the displacements of the median line along the x and y direc-
tions, respectively. The length of an undeformed line element (dx,0) in the deformed state

is given by

ds = «(12:2 + dy2

\/[ 1),, + (y cos 0);]? + [1+ us - (y sin 0);]2 (1.7: (2.4)

 

where a subscript :1: stands for the partial derivative of the corresponding function with

respect to the spatial variable 1:. The relative elongation in the x-direction is deﬁned as

 

d3 ‘
e=E—1=\/v3+(1+ux)2-1-y9,=eo—y0x (2.5)

where so is the relative elongation of the median line. From the geometry of the deﬂection

curve as shown in Figure 2.2, we also have

”a:

 

 

 

t = , 2.
an a 1 + at ( 6)
sin a = 1 :20 , (2.7)

1 + um
cos a _. 1 + 60 (2-8)

where so is implicitly deﬁned in equation (2.5).

15

 

 

 

 

 

 

 

nu+(du/dx)dxi
[ I‘— '_.l
X I
k.
y t
NP_ x _,
#— dx—VI

Figure 2.2: Geometry of the deﬂection curve
2.2.2 Acceleration Components
With the assumptions given in section 2.1, we now proceed to derive equations which

describe the axial and transverse vibrations associated with the connecting rod AB. Con-

sider a slider-crank mechanism as shown in Figure 2.3. The reference frames OXY and

l:
>
o

p \
$3“ n--. Y

e
is.
is

Figure 2.3: Slider crank mechanism with ﬂexible connecting rod (undeformed state)

oxy represent the ﬁxed and moving coordinate systems associated with this slider-crank

mechanism, respectively. Let u(:r, t) and v(z, t) represent the axial and transverse displace-

16

l
x: 4y
. I
I I
I
I 9
I A ‘ C
I ’~ ~-
I I ~~
I ’ ‘s ‘s
I ~ ~ ~
: , C ‘~~:~~:~ -
I
I

 

I N‘ “ S
I ‘s“s‘
I s‘ s‘
I ‘~ ‘~ 5....
~ ‘
‘ N
‘
b
\ ~
- - -------------------------------- m-T‘--.’ Y

m / //}//

Figure 2.4: Slider crank mechanism with ﬂexible connecting rod (deformed state)

 

 

 

ments of the connecting rod measured in the moving coordinate system, respectively. The
moving coordinate system (i.e., oxy) is shown in Figure 2.3 with its origin located at the
connecting point on the rigid crank and its x-axis always passing through the ends of the
elastic coupler. We now consider a point C located with (22,0) in the moving frame, which
moves to point C’ after defamation (Figure 2.4). Modeled by the embedded geometric

constraint method[33, 53], the position vector i- of point C’ can be expressed as

r- = (z + u)i+ vj (2.9)

where i andj are the unit vectors along the x- and y-axes, respectively. The deviation of the

acceleration components begins with the following well- known result from dynamics[20, 23]:

5=A05,+wx(oxr)+atxr+i~u+2wxn=axi+ayj (2.10)

which allows the absolute acceleration of a point to be written in terms of a moving frame
of reference and where

a = absolute acceleration component seen from a ﬁxed coordinate system,

17

A05, = absolute acceleration of the moving frame,

r = position vector for a particle as reviewed from the moving frame, and is deﬁned in
equation (2.9),

II) = angular velocity of the moving frame.

Substituting equation (2.9) into equation (2.10), we obtain

at = —rw2 cos (wt — (p) + u“ — ¢tv¢ — (but) — 453(9: + u) (2.11)

0,, = —rw2 sin (wt — ¢) + 43?,(3 + u) + 2¢>tut + v“ - 45??) (2.12)

where a subscripts t presents the partial derivative of the corresponding function with
respect to the time variable, 1' is the length of the crank shaft, ¢ represents the angle between
OA and OB, measured counter-clockwise, at and a, are the acceleration components along

the x— and y— axes, respectively.

2.2.3 Stress-Strain Relation

In this section, we consider the relationship between stress and strain. We assume that
the connecting rod is made of a Viscoelastic material which can be described by models built
from discrete elastic and viscous elements. In this work, we assume that the connecting
rod is made of a the Kelvin-Voigt material which is modeled as shown in Figure 2.2.3.

Therefore, the stress-strain-time relationship for this material is given by

a = Ee + poet (2.13)

where 0‘ is the normal stress, and e is the strain, E is the elastic spring constant, and ,uc is

the dashpot coefﬁcient. Based on this, we obtain the following equations for normal stress

18

MN—

 

 

 

 

 

 

 

 

Figure 2.5: Viscoelastic model (Kelvin-Voigt model)

an and shear stress a”:

02:; = E6 ‘l' Poet : E‘30 —' E3102: + #630. " #cyoxt I (2-14)

on = G7. (2.15)

Now, we express the axial force N, bending moment M and shear force Q according to

A

M : / OzzydA = —E103 — #clart (2.17)
A

Q = / and/l = k'GA'y (2,18)
A

where A is the cross-sectional area, I is the area moment of the inertia of cross section, k’

is the shear-deﬂection coefﬁcient, a modifying factor introduced by Timoshenko[46].

2.2.4 Equation of Motion

Consider the free body diagram shown in Figure 2.6, from which the equations of motion

are obtained by considering the balances of forces and bending moments. From this, we

19

 

 

 

 

 

 

 

u A
Y
Figure 2.6: Free body diagram for beam element
obtain the following equations:
(N cos a), - (Q sin 0),, = pAaz , (2.19)
(N sin a), + (Q cos 0),, = pAay , (2.20)
M1, - Q(1 + co) cos (a —— 0) = —J0tt (2.21)

where the acceleration components 0., and ay are deﬁned by equation (2.11) and (2.12),
p stands for the density (mass per unit volume) of the connecting rod, and J is the mass
moment of inertia per unit length. We note the these equations are expressed in terms
of eight unknowns, namely N, M, Q, a, 0, u, v and so. We now proceed to transform
these equations into a form in which only the displacements and forces are involved. With

equations (2.18), (2.6), (2.7) and (2.8), equation (2.21) reduces to

_ J0“ "" Elgar-#cloxzt
— vxsin9+(l+ux)c080'

 

(2.22)

20

In deriving this equation, we assume that I , E,and ye are all independent of 2:. From

equation (2.1) and (2.18), we have

 

 

_ _ Q
Equations (2.16) and (2.8) yield
1 + II; 1+ u,
N EAeo +11 .460, EA( cosa )+p ( cosa )t ( )

Substituting equations (2.24) and (2.22) into equation (2.19), we obtain

 

 

 

 

 

1+ 11,, 1+ 11,;
[( +11.) \/v§+(1+ur)2] +11 [( +11.) \/v§+(l+ur)2lt
J0“ -' E1017: " ”claxxt
_ .— = A x, 2.2
[ vz+(1+ux)cot0 ] p a ( 5)
Equations (2.16) and (2.7) yield
1) ‘U
=EA CA =EA I —1 cA ,3 —1. .
N 60 + It 60. (sina )+ It (Sln a )z (2 26)

Substituting equations (2.26) and (2.22) into equation (2.20), we obtain

”’ 1+ AIv
¢s+a+mr$ W ‘
+ [Jott‘Elazx-Pconz-t
v,tan0+(1+ux)

v3 ]
Jug + (1 + ux)2 It

lat = pAay - (2.27)

 

 

EA[v;,7

 

Substituting equations (2.24), (2.22) and (2.23) into equation (2.6), we obtain

J0“ "' Eloxz - I‘chxrt

0
tan( + k’AG[vac sin0 + (1 + 111-) c080

 

21

Equations (2.25), (2.27) and (2.28) describe the rotational, axial and transverse defections

associated with the ﬂexible connecting rod.

2.2.5 Boundary Conditions

In the present section, we consider the the boundary conditions associated with equa-
tions (2.25), (2.27) and (2.28). We ﬁrst consider the boundary conditions applied to the
connection point on of the crank shaft. Since the crank shaft is assumed to be perfectly
rigid and there is no relative displacement of the connection between the connecting rod

and the crank shaft, we must have

u(0,t)

II
o

(2.29)

v(0,t)

II
o

(2.30)

Due to the presence of the frictional moment at point A from the bearing, we have

Ero.(o,t) + p.10,.(o,t) = _M,, = ”5(0) — a. — v3¢(0, t) ) (2.31)

where [15 represents the friction coefficient, and the moment applied by the bearing is the
product of m, and the relative rotational speed. We now consider the boundary conditions
applied to the sliding end to which a mass m4, the piston, is attached. At point B, the

piston motion is constrained to move along the X direction. Therefore, we have

v(L, t) = —u(L,t) tan cl). (2.32)

Because of the presence of the bearing at point B, there exists a frictional moment at this

22

point. Therefore, at point B, we have

E10x(L,t)+ prOn(L,t) = —M3 = pb(—¢t + vx¢(L,t)) , (2.33)

where the bearing friction law from point A is again employed. We now apply Newton’s

second law to the slider mass 172., and obtain the following condition:

(N cosa — Q sin 0) cos(—¢) + (N sin a + Q cos 0) sin (—¢)
+ m4ar cos (—¢) + m4ay sin (—¢)

+ p,Z¢ + mu, cos(—¢) + #51); sin (—¢) = 0 (2.34)

where p, is the slider friction coefficient and Z, stands for the rigid motion of the slider
mass. We note that there are two slider friction forces in equation (2.34). The (p,Z¢)
term represents the friction force caused by rigid body motion, while [p,u¢cos(-¢) +

pm, sin (—¢)] stands for the friction force due to the elastic deformation.

2.3 Comparisons

In the present section, we compare the equations of motion obtained in section 2.2 to the
models used by other investigators in studying the dynamic response associated with the
ﬂexible connecting rod of a slider crank mechanism. Regarding the acceleration components,
there is no difference between our approach and other’s. Therefore, we shall focus on the
beam theory used to model the connecting rod.

If the effect of shear deformation and rotary inertia terms are neglected in our formula-

23

tion, we have a = 0 and

Q = “E10131: — I-‘clgxxt - (2.35)

Moreover, if we assume that the relative elongation of the median lines is very small com-

pared to unity, then equation (2.19) becomes

EA[(1+ ux) — cos 0]: + Apc[(1+ ux) — cos 9],. — (Q sin 9),, = pAa, , (2.36)

and equation (2.20) becomes

EA[(1 + ux) tan0 — sin 0],; + (Q cos 0),, = pAay . (2.37)

Since the relative elongation of the median line is assumed to be very small compared with
unity, we also have

sin 0 = I); (2.38)

from equation (2.7). Moreover, we assume that 0 is very small so that the last equation can
be reduced to

a z 221. (2.39)

Based on this, equation (2.35) becomes

Q = ’Elvxxz: "’ II-‘cvxxxt . (2.40)

Substituting equations (2.39) and (2.40) into equations (2.36) and (2.37), we obtain the

24

following equations:

pAa, = EA(ux + gulf), + Apc(ux + $152,), , (2.41)

1
pAay = EIvmu + yclvxmzt — (E — pc)A[ (at + 5123a,]; . (2.42)

This derivation is mainly based on the von Karmons ﬁnite deformation theory. These
two equations were used by previous investigators[9] in studying the dynamic behavior
associated with the connecting rod. Viscomi[49] applied this model without considering
the effects of internal material damping pa. In chapter 4, we will use equations (2.41) and
(2.42), together with the work by Viscomi to study the ﬂexural response associated with
the connecting rod. We now consider the model used by Badlani et al.[8]. Linearizing

equation (2.1) and neglecting the effects of internal material damping no, we obtain

Q

0 + m = vl‘ (2.43)
or
Q = k'AG'(v1c — 0). (2.44)
Linearizing equation (2.16), we obtain
N = EA at. (2.45)

Substituting equations (2.44) and (2.45) into equations (2.19) to (2.21) and linearizing the

resultant equations, we obtain

EAun = pAax, (2.46)

25

Q.

pAay, (2.47)

where the mass moment of inertia J is expressed as J = {11. Substituting equation (2.44)

into equations (2.46) and (2.47), we obtain

E10131: + k’AG(vx — 9) 3 p19“ , (2.49)

k’AG(vz — 0),; = pAv“. (2.50)

These two equations were used by Badlani et al.[8] in studying the dynamic behavior asso-
ciated with the ﬂexible connecting rod. This derivation is mainly based on Timoshenko’s

beam theory.

2.4 Nondimensionalization

We now convert the equations and their associated boundary conditions into dimen-
sionless form in order to minimize the number of system parameters. To achieve this, we

introduce the following dimensionless parameters:

 

- _ 3: -_3 -_£ -1 __m..4 2- .1
u _ L9v“L7z—L1£—L’S—pAL,n —AL2,
EIIr" w - pw
2 _ 0-:— = t = C It
wlt pAL4, wu,t wlts”? 2E I
_ m4 _ #3 0 __§_ 0 __{_
”3 ’ 2pAL3wn’ 2pALw1t’ ’ k’G’ 3” pI'

Before we proceed to convert equations (2.25), (2.27) and (2.28) into dimensionless form,

let us explain the physical meanings of these parameters. The parameter 5 is the ratio of

26

the throw length of the crank shaft to the length of the connecting rod, which is referred to
hereafter as the length ratio. The parameter S is the ratio of the mass of the connecting rod
to the mass of the slider, and is speciﬁed as the mass ratio. The parameter n represents the
slenderness ratio of the connecting rod. The parameter (12 is associated with the effects of
shear deformation. The parameter 03 is associated with the effects of rotary inertia. The
parameter Q is the ratio of the operating frequency to the ﬁrst ﬁexural natural frequency
of the connecting rod, to“, and is referred to as the frequency ratio. The parameter In
represents the damping effects of the internal material damping acting on the connecting
rod. The parameter #3 represents the damping coefﬁcients of the bearing friction applied
to the joints at point B and C. The parameter #4 is used to model the friction force
acting between the slider and its contact surface. With these dimensionless parameters,

equation (2.25) becomes, after dropping the overbars for notational convenience,

 

 

 

 

(1+ux) 1+ux
1 1: - I 2 1 17 _ x
[( +11) \/(1+u1:)2+v3.] + #2[( +11) «(1+u3)2+vg]t
0 0

_ 4 4 it 2 1:1:

03” 1r [(1+u,,)cotl9-+-vx]x+1,7 [(1+ux)cot0+v,]‘r
0317
+ 2I‘2772[ lxt

 

(1+ ux)cot0 + v:
: n21r4[ -ff22 cos (Qt — ¢) + u“ - 2¢¢v¢ — 45:20 - (15%,(1‘ + “ll . (2-51)

equation (2.27) becomes

 

 

 

 

 

 

Iv... —- ”I I. + 2.1222. - ”’ I...
\/(1+"z)2+v}.- \/(1+ux)2+v%
0 0

4 4 it _ 2 2:1:

+ nxa3[(1+u3)+vztan0]3 n[(1+u3)+vxtan0]x

0

_ 2 1:2:

2M" [(1+ u,) + v, tan0 L”

= 772772 ‘502 sin Qt - ¢ + ¢tt(-’D + u) + 2¢Iut — ‘Utt — 45%)] , (2.52)

27

and equation (2.28) becomes

We now apply the same procedure to the boundary conditions. We obtain the following

tan( 0 + (1217

2 037727r40tt - 01:1: - 2”2prt
(1 + at) cos0 + v;c sin 0

”1:

1+ux°

 

 

):

conditions at point A:

u(0,t) = 0, (2.54)
v(0,t) = 0, (2.55)
03(0,t) = 2p3[Q-¢¢—vxt(0,t)]. (2.56)

While, we have the following conditions at point B:

“1+ “3)

v(1,t)

0,,(1,t)

1+‘u,r
\/(1+ 11:92 + I)?
031‘
1 + u$)cot0+ v3

«(1%: 132).]. 122] + 2f‘2l (1 + um) _.

4 9t:

[(1 + uz)cot0+ v3
02:1:

(1 + 11,) cot0 + 11,, h

S7727f4[ -£Q2 COS (Qt - ¢) + U“ — 2¢¢vt — $117) — ¢?(.’B + 11.)]

lt
l

 

 

 

(131747r l + 772l (

 

2712 ﬂzl

v1- v3

 

 

 

 

[v \/(1+ux)2+vg] an¢ pal” ¢(1+u$)2+vg lt an¢
0 0
4 4 t1 2 1:1:
t
’7 1r a3[(1+u3)+vxtan0] an¢+17 [(1+ux)+vxtan0]tan¢
01717
2p2n2[ ]¢tan¢

 

(1+ ux) — vxtanﬂ

(2.53)

—u(1,t) tan(;b , (2.57)

2p3i ‘¢t ‘l' ”11(190] I (258)

28

5772772 ‘692 sin Qt - 43 + ¢tt($ + 10+ 2W4: - vtt — $31)] tan¢

Z
costqb + 2n21r4p4[u¢ — v, tan¢] = 0 . (2.59)

 

+ 2027”!“

Equations (2.51) to (2.59) form the basis of the mathematical analysis in the thesis.

Chapter 3

Analysis of the Distributed Parameter Model

In the present chapter, we study equations (2.51), (2.52) and (2.53) by applying the
Method of Multiple Scales[29]. To apply this method, we must ﬁrst locate a small, con-
stant parameter which will be used as a basic unit in scaling the response and the other
parameters. For the current problem, the slenderness ratio 1] of the ﬂexible connecting rod
is chosen to be the basic measurement unit. The axial and transverse displacements u(a:, t)
and v(:r,t) are expressed in asymptotic series in terms of n. We then expand the partial
differential equations describing the dynamical behavior associated with the connecting rod
with these assumed solution sequences and obtain a sequence of linear partial differential
equations. We then study the dynamic response associated with the elastic connecting rod
by solving these equations sequentially. In section 3.1, the Method of Multiple Scales is used
to outline each individual resonance case. In section 3.2, we study the case of the principal
parametric resonance. In section 3.3, we study the case of primary resonance. Superhar-
monic resonances of order 1/2 and 1 / 3 are studied in sections 3.4, and 3.5, respectively.
Section 3.6 contains the analysis for the case of the order two subharmonic resonance. We

summarize the results and provide a detailed parameter study in section 3.7.

29

30
3.1 Application of the Method of Multiple Scales (MMS)

To apply MMS, we need to introduce a set of new independent time variables Tn ac-
cording to

Tn = ("t (3.1)

where 6 is a small constant parameter which is related to the slenderness ratio of the
connecting rod. Because the connecting rod is assumed to made of a slender beam, its
slenderness ratio 7] = 3%, is small. Based on this, we choose this quantity as a basic

measurement unit of the parameters that are used to model the problem. Therefore, we

scale the slenderness ratio of the connecting rod by letting

172 = 0162 . (3.2)

Moreover, the length ratio E is scaled according to

E = 661 . (3.3)

It follows that the derivatives with respect to t become expansions in terms of the partial

derivatives with respect to Tn according to

d _ a a 2 a _ 2
d2
— = 002 + 2.00 D1 + 62(012 + 200 D2) + (3.5)

where D,- stands for the derivatives with respect to the independent time variable Tj. We

next assume that the solutions u(:c,t) and v(:I:,t) can be represented by expansions having

31

the forms:

u(a:,t) = cu1(z,To ,T1 ,T2) + £2u2(:c,To ,T1 ,T2) + c3u3(a:,To ,T1 ,T2) + , (3.6)

12(3), t) = €01($,T0 ,T1 ,T2) + €202($,T0,T1,T2)+ €303($,T0 ,T1 ,T2) + . (3.7)

Note that the number of independent time scales TJ- needed depends upon the order to
which the expansion are carried out. For the present problem, it is sufﬁcient to expand the
solutions up to C(63). Therefore, only the ﬁrst three time scales are used in the expansions.
There are three damping parameters #2, p3 and #4 involved in equations (2.51), (2.52) and
(2.53). Basically, these damping parameters are rescaled in such a way that they will show
up in the ﬁnal resonant condition together with the detuning parameter which is related to
the nearness of the frequency ratio 0 to a resonance condition. A trial and error approach
is used to show that the damping parameter in must be of order 0(3), #3 of order C(62)
and #4 is of order 0(1). This leads to the following ordering of these damping coefﬁcients

in terms of e:

#2 : 62’122, (3.8)
#3 = 62/132, (3.9)
#4 = #40. (3-10)

Substituting equations (3.2) to (3.10) into equation (2.5), expanding and equating the co-
efficients of the like-power terms in c, we obtain the following expansions for the elongation

e0 of the median line along the connecting rod:

co = (Co, + 62602 + 63603 + 64(80‘ — 46021)?» + 0(65)

32

2
v
= an, + 62012: + ‘5‘") + 63(u3, + mum)

 

 

4 v? v3 ”f 2 5
+e[(u4,+ 8=+ 2'+v1,va,)-4(u2,+—2—’-)vl,1+0(c) (3.11)

where e01. represents the j-th order term in the expansion of 60. Applying the same procedure

to equations (2.51), (2.52) and (2.53), we obtain the following equations:

 

Order 6
“'11: = (801)17 : 0 , (3.12)

Order 62

v?

(“2: 'l' 21):: = (602): = 0 9 (3.13)

Order 63
(113, + 121,122, )1. = ((303 )1 = —al7r4£102 cos 9T0 , (3.14)
011r4D02v1 + 01111,,” = (602121,); + 017r4§192(1 — 1:) sin 0T0 , (3.15)

Order 6“

(60, )1: + 2H22(Do€og )1: - (6’02”th + 01(vlzviuxlx = 0117f4lD02U2 — £19201 sin QTo

—§1f22v1 sin 0T0 + 2619(00121) cos 9T0 + {1292 sin,2 9T0 — fﬂzx cos2 0T0] , (3.16)

017T4D0202 + 01112.2“; = (603011;)1: + (€02v2x)x - 2al7r4(DODlvl)

+ 011 «4892 cos 0T0 sin 9T0 , (3.17)

33

Order (5

017F4D02‘U3 + alv3xxra‘ = ’aga2lexxzzx " 2l‘220100v1rxrx + 01(802 v1xxx):c
2 1 3
"’ 01(vlrlezlr + (€02v3r)x + (603 v2x)r " §(802v11:)1‘
+ 2’12; 00(802 ”13):: + (8041,1101: — (€32v1xlx + 02120371'4D02v1x1:
— al1r4(2DoD1v2 + 2Dngv1 + D12v1)+ alw4§f§22v1 cos2 9T0
+ 2al7r4519(Dou2) cos 0T0 — 0:17r45192u2 sin 0T0

— a11r4fi’ﬂzgl—z—x—2 sin3 9T0 + 011r4£i35222 cos2 9T0 sin (Mb. (3.18)

Applying the same procedure to the boundary conditions, we obtain the following boundary

conditions:

Order 6
u1=v1=v1u=0,atx=0, (3.19)
u13=v1=v1u=0, atx= 1, (3.20)

Order 62
u2=v2=v2m=0,atx=0, (3.21)

viz

(”.21- + 7) = v; = 0 ,vgn = ~2n329 , at x = 1, (3.22)

Order 63

“3 = ”3:09

”31$

v3
( ”'31:

”31:1:

Order 64

604

34

—alagvlmm — ”$31123 — 2p32Dov1x — {M32519 cos 9T0 ,at x = O , (3.23)
-‘U.2€1 SlIl 9T0 ,
01302;) = 501145102 cos 9T0 + 2011r4y4o§1§2 sin 0T0 ,

v a:
—alagvlnn — vh¥ — 2p32Dov13 + $13,619 cos 9T0 , at x = 1, (3.24)

+ allelexz = -2#22 D060, + 0177451291140 sin (2QT0)

+ anr‘éfﬂzﬂ cos2 9T0 — sin2 0T0) — al7r4S(D02u2)

+ al7r4£1925v1 sin 9T0 — 2al1r4£1SlS(Dovl) cos 0T0

— aI1r4flS(D02v1) sin QTO — (60,121,051 sin 9T0 + (602 12123:)

+ 01511113” sin (2T0 - 201n4p4(Dou2) , at x = 1 . (3.25)

In the following, we show that u1(:r, T0, T1, T2) = 0 and hence we drop all the terms involving

products with u1(:c,To,T1,T2). To show that u1(a:,To,T1,T2) = 0, we solve equation (3.12)

to obtain the general solution of u1(:r,To,T1, T2), which takes the following form:

“103,710, T1,T2) = b1(To,T1,T2)$ + b2(To,T1,T2) (326)

where 61(T0,T1,T2) and b2(To,T1,T2) can be determined by the boundary conditions asso-

ciated with ul. By applying equations (3.19) and (3.20), we determine that bl = b; = 0.

Hence, ul must be zero. This indicates that, for simply-supported slender beam, the axial

deformation is much smaller than the transverse one. Next, we consider equation (3.13)

35

and its associated boundary condition (3.22). Together, they reveal that
”13:2
(U2;- + 7 ) = €02 = 0 . (3.27)
Consequently, equation (3.15) reduces to
01x4(1)02v1)+ (11011-333; = 011F4€192(1 -- 2) sin 9T0 (3.28)
which can be converted into the following equations:
2 ~ 2 ~ 2 2 -
(D0 01,.) +wnv1n =2 5610 8111 9T0 (3.29)

by Galerkin’s procedure with 121 = 131,,(sin mrx), where w}, = n4 for n : 1,2,3,.... In
analyzing the particular solution of ﬁlm we need to consider the following cases separately :
(1) Q is near to can and (2) Q is away from w". The case in which 0 is near to can corresponds
to the primary resonance and will be investigated in section 3.3. At the present time, we

assume that Q is away from can and continue our analysis to consider the equation of higher

order. When (2 is away from can, the general solution of 171,, is given as
131,. = An(T1, T2)e:cp(jwnTo) + A1 sin 9T0 + c.c. (3.30)

where c.c. stands for the complex conjugate of the preceding term, and

A. = (ﬂag-25;) (331)

represents the amplitude of the particular solution. The function An(T1,T2) in equa-

36

tion (3.30) is a complex function of time scales T1 and T2, which represents the amplitude
of the homogeneous solution. The second term of equation (3.30) represents the particular
solution. Since the frequency ratio (2 is assumed to be away from can, the magnitude of the
particular solution is ﬁnite (equation [3.31]). Note that, according to the linear theory, the
homogeneous solution will decay to zero due to the presence of the damping parameters.
However, as we will ﬁnd out later on, due to nonlinear effects, this is not always true. When
certain conditions are met, the homogeneous solution may not decay to zero and hence must

be included. Equation (3.14), together with its boundary condition (3.23), leads to
603 '3 (U31; + levgx) ‘2 (117F4(1 — 27 + S)€192 COS ”TO + 2017r4flﬂp40 sin 0T0 (3.32)

which represents the third order expansion of the axial elongation of the median line along

the connecting rod. Substituting equation (3.27) and (3.32) into equation (3.17), we obtain

011r4002v2 + alvgmm = alrr4flﬂz[ (1 -— a: + S)v1$]x cos 9T0 — 2al1r4DoDlv1

+ a11r4fff22 cos 9T0 sin 9T0 + 201 «4;;40 £19121” sin 9T0 (3.33)

which can be converted into the following equation:

002132,. + with” 2: —2jwn(D1An)e:cp(jwnTo) + (WEEK)? sin (29%)

rm
1 . . - . .
7(1 + 2S)(mr)2£.n2[ A..exp(2w.To + Mo) + Anew-M7?) + 19%)]

2
‘l‘40519l(n7r)2/lnexp(jQTo _ jWTo _ j1r/2) _ (mr)

 

Alexp(2jﬂTo)]

— i0 + 2S)(mr)21\1£192 sin (29%) (3-34)

with v2 = 62n(sin mrz), where an overbar stands for the complex conjugate of the corre-

37

sponding term. Now, in analyzing the solution of this equation, we need to distinguish
between the following cases: ( 1) 29 is near to can, (2) Q is near to 2t.)n and (3) Q is away
from 2w... and 923. The ﬁrst case corresponds a superharmonic resonance, and will be in-
vestigated in section 3.4. The second case corresponds the principal parametric resonance
which will be investigated in section 3.2. At the present time, we assume that Q is away

from 2a)., and “—31. Under this condition, we must have

01.4,. = o (3.35)

in order to remove the secular term from the particular solution of {22”. This implies that
the amplitude An of the free oscillation term in the general solution of v1 (equation [3.30])

must be independent of the time scale T1 = ct. The particular solution of 132,, is then given

 

 

 

as
'52,; = A21 sin 2QTO + A22€$p(jQTo + jwnTo) + A23ezp(jQTo — jwnTo)
+A24ezp(2jQTo) + A25ezp(jQTo — jwnTo — j7r/2) + c.c. (3.36)
where
_ 1 — cos mr 892 1 2 {102
A21 — 12W (0): _ 402) -- 1(1+ 2S)(n7r) AIW , (3.37)
A22 - 4 (71”) {In [wg _ (wn + “)2] 7 (3'38)
_ 1 2 2 [an
A23 _ 4(1+ 25)(mr) {In [.33 _ (w. _ 9),] , (3.39)
A 2
A24 — ‘ (m) (3.40)

_2-w,2, — 402 ’

38

and

An
A25 = —p40£19(n1r)w2 _ (Q _- w )2 . (3.41)

 

2
From the expression uh + ”‘7’ = 0, we solve u2(:r, T0, T1, T2) and obtain

$n2nwx
2n1r

u. = -;‘,-(mr)%¥.(x + ) (3.42)

where 1317. is given by equation (3.30). With this information, equation (3.16) reduces to

80. + 01(v1xlexr)x = 017r4[ D02‘U2 - £19201 Sin QTo

+ 2519(D0v1) cos 9T0 + {(92 sin2 9T0 — ($923: cos2 9T0 ] (3.43)
subjected to the following boundary conditions:

e0, + 01(v13v13m) = —.S'al1r4[ Do2u2 + {1292 sin2 9T0 — {(522 cos2 9T0]
- 011 ”4’140510 Sll'l 2QTO + (116101311- sin ”TO

+ a11r4p40(n1r)2(00131n)231n ,at x = 1 . (3.44)

With these two equations, we solve for 4304 and obtain

4 2

v I v
804 = (“42.- + —;r + ~¥ + lev3x)

2
= -a1v1zlen + aln‘Ef92(-:- - £2- + 5) cos2 9T0 + aw‘ffﬂzﬁ - 1 - 5') sin2 9T0

COS 117112 - COS 7MP

 

+a11r4£1( )[ 92131,, sin 9T0 — 20(D01‘21n) cos 0T0]

mr
400253. 2 2 - 2 4 2 -
+al7r —-—8—[(mr) (1+ 25' — a: )— srn mrx] + alvr #40519 sm 2QT0

—011£1(n1r)3 cos mr sin 9T0 + al1r4p4o(n1r)2(Dof)1n)131n . (3.45)

39

With these results, equation (3.18) reduces to

air‘Do2v3 + alv3xxxa: = —a§02v1mm — 2011M; Dolexxr
+ (803122;); - 01(levfu)z + (60.01:);- + a§a31r4D02v1n
— a11r4(2DoD2v1 + 2D0D1v2 + Dlzvl) + al1r4£ff22v1 cos 9T0
+ 201 «4519(00112) cos 9T0 — (11 «45192112 sin 9T0
— alw4f?92£1—:—xl sin3 9T0 + 012461023: cos2 9T0 sin 0T0

+ a11r4p40(n1r)2(D061n)f21nv1u (3.46)

subjected to boundary conditions (3.23) and (3.24). We note that equation (3.46), together
with its associated boundary conditions (3.23) and (3.24), represent a simply supported
beam subjected to (1) motion of its foundation and (2) external moments applied to its both
ends. It is not a straightforward procedure to reduce this partial differential equation into
ordinary differential equation. Here, we use the variational approach[16, 46] to compensate
for the effects of the nonhomogeneous boundary conditions and obtain an equivalent system
to which the Galerkin’s method can be applied directly. Appendix A contains the detailed
approach regarding this procedure. At this moment, we only present the ﬁnal result as

below:

D0203n + “9721531; = ’2jwn(D2An + I121n4An)ezp(jwnT0)

. . 2 .
+ (01103 + a1a2)An(mr)2n4ezp(anTo) + Jpn4p32AneprwnTo)

8 1 - . .
(mr)2n4[-1-§ — (-3- + S)(mr)2]A?,Ane:cp(]wnTo) + flffAneszwnTo)

+ jf2£?€$p(3jQTo) + jf3£1 ﬁgempUQTo — 2jwnTo)

40
. - . .1 .
- anm.(mr)‘A?. AnerchwnTo) -— J 5m. An(n7r)“wnA¥exp(anTo)
1 . - n . .
+ §(n1r)4p4o A?9€$P(3J9To) + (nr)4/‘40A1A121(wn — Elexl’UQTo - 2anTo)

+ N.S.T. + c.c. (3.47)

where N .S .T. represents those terms which have no effect when we consider the secular terms
in the particular solution of 63... In analyzing the particular solution of equation (3.47),
there are three cases which need to be considered separately: (1) Q is near to 3 can, (2) Q
is near to 9311, and (3) Q is away from 3 can and 951. The case in which Q is near to 3 can
corresponds to a subharmonic resonance and will be investigated in section 3.6. The case in
which 9 is near to 33“, corresponds to a superharmonic resonance and will be investigated
in section 3.5.

At this point, we pause and discuss some general features of this pattern of analysis.
After a small, constant parameter c has been located, the solutions are expressed in uniform
expansions based on this parameter. By using these assumed solution sequences, the equa-
tions describing the transverse and the axial deﬂection can be converted into a sequence
of partial differential equations which can be solved easily, since they are linearized by the
nature of the perturbation expansions. We note that in solving every sequential order for
the ﬂexible vibration associated with the ﬂexible connecting rod, we are considering a sim-
ply supported beam subjected to the following generalized forces: ( 1) a distributed load,
which comes form the inertia force associated with the gross beam motion, applied along
the span of the beam, (2) concentrated moments, which are caused by the presences of the
bearing damping p1,, shear deformation and rotary inertia, (3) motion of the foundation
which comes from the motion of the crank, and (4) axial loading which comes from the

inertia force associated with the slider mass and its friction force. In order to make the

41

analysis as clear as possible, we provide a general procedure for solving this kind of prob-
lem in Appendix A. In solving these equations for the ﬂexural vibration associated with
the connecting rod, the primary resonance arises ﬁrst and then the principal parametric
resonance. After these two resonances, several secondary resonances arise. In principle, it
is possible to extend this analysis to study even higher order resonances, for instance the
subharmonic resonance which occurs when (2 is near to Sta... However, this involves an
unreasonable amount of computational work. As a matter of fact, the present results are
accomplished with a great deal help from a Macintosh version of the computer symbolic
manipulation program M athematicaTM . Although the primary resonance occurs ﬁrst, in
the analysis, we investigate the principle parametric resonance ﬁrst rather than the primary
one, since this pattern will simplify the analysis and make it more easily understood. After

these two cases, the secondary resonances will be investigated case by case.

3.2 Principal Parametric Resonance

In analyzing the particular solution of equation (3.34), when the frequency ratio 9 is
near to 2w“, the principal parametric resonance takes place. To describe the nearness of Q
to 2w“, we express (2 as

Q = 2a).. + 2601 (3.48)

where 01 is the detuning parameter. At the same time, the damping parameter [1.2 is rescaled

according to

”'2 = 6”'21 a (3°49)

so that it will show up in the ﬁnal resonance equation, together with detuning parameter 0;.

We ﬁrst expand equations (2.51), (2.52) and (2.53) with these new ordering relations, and

42

then carry out the same procedure as described in the last section. Since the computational
work is routine, we only present a few essential results in the following analysis. For a
detailed solution, refer to Appendix B. Following the same procedure as described in the
last section, the ﬁrst order ﬂexural vibration associated with the connecting rod is described
by equation (3.29). Consequently, the general solution of 61,, is given in equation (3.30).
When we try to solve the second order equation for the ﬂexural vibration, we obtain the

following equation:

Dozﬁzn + 013,132,, = —2jwn(D1An)exp(jwnTo) — 2jp21Ann4wnerp(jwnTo)
- (1 + 23)("7r)2w3./ln€18$1’(jwnTo + 2j01T1)

_ 2p4o(n7r)2wn/lnélexp(jwnTo + 2j01T1 — j%) + N.S.T. + c.c. (3.50)

where N .S.T. stands for those terms which have no effects in considering the secular term

in the particular solution of 132,1. Therefore, we must have

’2jwn(DlAn) - 2jl‘21wnAﬂn4 - (1+ 2S)(n7r)2w3,/ln£1exp(2jolT1)

- . .7l'
- 2p40(n1r)2wnAnfle:cp(2]alT1 — )5) = 0, (3.51)

in order to remove the secular term from the particular solution of 132,1. In solving this

equation, the unknown complex function A1(T1 ,Tz) will be expressed in polar form
a .
An(T1 T2) = §€$p(]\1’) . (3.52)

Expanding equation (3.50) with this expression, and separating the resultant equation into

43

the real and imaginary parts, we obtain

a' = —p2, n4a — 2a§1A1 sin (2<I>1) + a(n1r)2§1p40 cos (2<I>1) (3.53)

a<I>’1 = 01a - 2a£1A1 cos (2&1) — a(n7r)2£1p4o sin (2‘15) (3.54)
where the phase angle {>1 and A1 are deﬁned by

Q1 2 0'1T1 — ‘1’ , (3.55)

A1 = @(nnfnz (3.56)

and the prime indicate derivatives with respect to the time scale T1. Periodic steady state

conditions can be achieved whenever a’ = a<I>’1 = 0, these yields the following conditions:

n‘pgla —2a£1A1 sin (2<I>1) + a(n7r)2£1p4o cos (2‘1>1) ,

01a 2 2a§1A1 cos (2<I>1) + a(mr)2£1,u4° sin (2<I>1) . (3.57)

Squaring these equations and adding them together, we obtain the frequency response

equation which takes the form:

[of + n33. —- (251m)” 4121022318 = 0. (3.58)

From this frequency response equation, we note that the trivial solution a = 0 is always
the steady state response for this resonance. To determine the stability associated with
this trivial solution, we transform equation (3.50) into Cartesian coordinate by substituting

An = (BR + jBI) ezp(j01T1 + 7T1) , and then separating the resultant equation into the

44

 

    
 

 

 

 

0.3 I I r I I
0.25 r- .4
0.2 - ﬂ
{1 0.15 - (unstable region) _
S = 0.00
0.1 t" #2 = 0.02 —1
[13 = 0.00
0.05 - *‘4 = 0-00 ..
n =
0 1 1 1 1 1
1.85 1.9 1.95 2 2.05 2.1 2.15

frequency ratio 0

Figure 3.1: Frequency response equation for the principal parametric resonance

real and imaginary part to obtain two equations. From these two equations, we solve for

the nontrivial solutions of BR and BI. From this, we determine that when

01" + 118113, > {fl 4A? - (1120211402 ] (3-59)

the trivial solution is unstable. Otherwise, it is stable. Figure 3.2 shows an example of a
stability boundary in the {—0 plane.

In order to capture the effects of the nonlinearities, we need to rescale either the solutions
or the parameters, so that the nonlinearity will then be included in the ﬁnal equation
describing the resonant condition. This problem is equivalent to a simply supported beam
subjected to both the transverse and the axial force coming from the same source. We
note that this force source arises because of the inertial force acting on the the connecting
rod. Moreover, these inertial forces are proportional to the length ratio 5. Because the

magnitude of this force source is proportional to the length ratio 6, we rescale the length

45

ratio in order to extend our analysis to include the nonlinearity. Therefore, we restart our
analysis by letting

E = 6262 (3.60)

and keeping the orders of damping parameter #2, p3 and [14 be the same as given in equa~
tions (3.8), (3.9) and (3.10). Expanding equations (2.51), (2.52) and (2.53) with this new
length ratio, and equating the coefﬁcient of like-power terms, we will obtain a new sequence
of linear partial differential equations. Carrying the same procedure for the boundary con-
ditions, we obtain a sequence of boundary conditions associated with these linear partial
differential equations. We then follow the same procedure as that described in the last sec-
tion. Because the computational work is tedious and routine, we only present a few results
which are essential to our work in the following analysis. For a complete solution procedure,

refer to Appendix C. The equation describing the ﬁrst order ﬁexural vibration becomes
00261.. + 613,6... = 0 (3.61)
which admits the following solution
61,, = Anezp(jwnTo) + c.c. . (3.62)

We then obtain the following ordinary differential equation describing the second order

ﬂexural vibration:
130262., + 13,362,. = $5202 sin 0T0 — 2jwn(D1An)e:cp(jwnTo) + c.c. . (3.63)

In analyzing the particular solution of 62m we need to distinguish the following two cases:

46

(1) fl is near to can and (2) Q is away from w... The ﬁrst case is referred to as the the primary
resonance which will be investigated in section 3.3. At the present time, we continue our
analysis to consider the third order ﬂexural vibration with the assumption that (2 is away
from w". Under this condition, we must have D1Aﬂ = 0 in order to remove the secular term
from the particular solution of equation (3.63). This implies that the amplitude An must
be independent of the time scale T1. Based on this, we assume that all the higher order
ﬁexural response 131,, also independent of T1. Therefore, we obtain the following equation

describing the third order ﬂexural vibration:

002037. + 0123,63,, = —2jwn(D2An)exp(jwnTo) — 2J'<..2nn4pg2 Anezp(jwnTo)
+ [01012 + ala3 ](n1r)2n4Anezp(jwnTo) + 2j0.),,pg2 gAneszwnTo)
+ 52("7rlzﬂ409rineprﬂTo - jwnTo - jW/2)
" £93 + ”(erzézﬂzflneipUQTo - jwnTol
- [185- - (é + sxnaz1(n.)2n4131,..puwﬂb)

— jmo(n1r)4A3,/lnwnezp(jwnTo) + N.S.T. + c.c. . (3.64)
To describe the nearness of Q to 20.2", we express (2 as
n = 2 + 2.202 . (3.65)

Hence, we must have

. . .n4 . —
— 2an(D2An) — 2ann4p22An + 2.7;r—5Anpg2 — .7(n7r)4,u40A3410.)n

+ {2(n1r)2p4°Q/1nezp(2j02Tl — j7r/2) + An(n1r)2n4[alag + 0103]

47
_ in + 23)(n1r)2£292/1n€$P(2j02T1l

— [18.5. -— (:13 + S)(n1r)2](n7r)2n4A,2,/1n = 0 (3.66)

in order to remove the secular term from the particular solution of i13n. Expanding this
equation with An = geszW), and separating the resultant equation into the real and

imaginary parts, we obtain

1 n2 ,
a = —n4p2,a + —§(n7r)4p4on203 + ﬁpgza - 2a§2A1 sm (2&2) ,
+ap40 {2(7z1r)2 cos (2&2) , (3.67)

a<I>’2 = (720 + A30 — A203 — 2aA1§2 cos (2<I>2) — (1114062020)2 sin (2‘32) (3.68)

where

A. = 10210211213 -(1+ S)(mr)"’]. (3.69)
8 8 3
“n2 2
A3 2 ?(mr) (011012 + 0103) , (3.70)
T2 = 02T2 - ‘1’ (3.71)

and primes indicate derivatives with respect to the time scale T2. The steady state condition

can be obtained by letting a’ = a<I>’2 = 0, which leads to

—a§2 [2A1 sin (2‘52) — u40(mr)2 cos (2<I>2)]

an2[;12,n2 + E§—°(n7r)4a2 — %],(3.72)

a[ 2A1£2 COS (2&2) + p4o§2(n1r)2 SlIl (2Q2) ]

a[ 0'2 + A3 — A202 ] . (3.73)

48

Squaring these equations and adding them together, we obtain

1 n2 .
a2[ (114,122 + §("7r)41u40wna2 - 311302 + (02 - A2012 + A3)2

— (252m? - (mr)“£%#io] = o (3.74)

which is the frequency response equation for this parametric resonance. An example of this

frequency response equation is plotted in Figure 3.2. From equation (3.74), we solve for the

a, = the magnitude of the nontrivial solution

 

 

 

 

 

3 I I I I I
{1 = 0.01
S = 0.00 _
[1.2 = 0.02
a [13 = 0.00 .4
s [1.4 = 0.00
01 = 1.00
02 = 0.00 ‘
03 = 0.00
n = 1 _
0 l 1 m2 1 m1 1
1.85 1.9 1.95 2 2.05 2.1 2.15

frequency ratio 9

Figure 3.2: Frequency response equation for the principal parametric resonance

nontrivial solution of a2 and obtain

(a2)? = (ﬁn—1+ «12-1....1, (3.75)
(03)2 = (bl—I-x/ﬂ—kml (3.76)

49

where

k = [(f‘gaxmrr]? + Ag, (3.77)
I = (%)(mr>"tn‘m. - $2.13.] — (a. — A3)A2. (3.78)
m = [11%. - (2)2113. l2 + (02 — A3)2 - 463A? - (M)"£§H§o- (3-79)

For the existence of these real amplitudes a1, a2 and a3, the term inside the radical sign
must be positive or zero. This implies that, in order to produce a sustained nontrivial
steady state response, the magnitude of the forcing must be large enough to overcome the

effect of the energy dissipated. Moreover, we must have

 

0. + A. < Joe/:1)? + (mrwguz. — [#22714 — ($2.13.? (3.80)

for the existence of (a2)2, and

 

02 + As < -\/(2€2A1)2 + (Ml453l130 - (#22714 - (%)2(‘32)2 (3-81)

for the existence of (a3)2. In order to determine the stability associated with these steady

state responses, we compute the the Jacobian matrix associated with equations (3.68) and

(3.68) and obtain

—(9;-)n2(mr)4p40 -2(02 — A202 + A3)a
(3.82)

‘2A20 ’2n4l‘22 _ #4o(n7r)4n2(a4_2) + (5921132

After we compute the eigenvalues of this matrix, we obtain the following conclusions:

 

(1) when 02 > —A3 + \/(2§2A1)2 + (n1r)4§§p§o — (p22n4 - (%)2p32)2, there is only one

50

response: the trivial one, and it is stable,

 

(2) when 02 < —A3 + |\/(2§2A1)2 + (n1r)4§§p§0 - (”22124 — (%)2p32 )2I, the trivial solution

becomes unstable, while a2 exists and is stable,

 

(3) when a; < —/_\.3 — \/(2€2A1)2 + (n1r)4£%p§o — (”2,114 — (%)2pg2 )2, a3 exists and is un-

stable, while the trivial solution and a; are stable.

Remark: Before leaving this section, there are few points to make. We ﬁrst compare the
linear and nonlinear results. If we neglect the effects of shear deformation 0:2, rotary inertia
03, the friction force introduced by elastic deformation (p4a3 term) and bearing damping

#3, then equations (3.68) and (3.68) become

a' = —n4p22a — 2a§2A1 sin (202) + a,u40§2(n7r)2 cos (24);») , (3.83)

a<I>’2 2 02a — A2a3 — 2aA1£2 cos (2<I>2) - ap4o£2(n1r)2 sin (2<I>2) . (3.84)
In consequence, the frequency response equation becomes
a2l("4#22)2 + (02 - A202)? — (252A1)2 — (”704531130 l = 0 - (3-85)

It is clear that equations (3.58) and (3.85) coincide with each other in their linear parts. To
show this, we substitute M, = (#2, , and £2 = ((1, into equation (3.85). Then equation (3.85)
reduces to equation (3.58) when on approaches zero. Moreover, the equation describing the
variation of the amplitude of the response (equations [3.53] and [3.84] ) are in the same
form. The difference between the linear and nonlinear analyses appears in the equation
describing the variation of the phase angle {>2 (equations [3.54] and [384]). This implies

that the direct effect of the nonlinearity is to affect the phase angle. From another point of

51

view, the nonlinearity affects the rate at which energy is pumped into the system. In the
absence of nonlinearity, the response is unstable when the parameters are located inside the
unstable regions. According the linear theory, the amplitude of response will grow without
limit. However, this increasing response amplitude will be accompanied with a change of
phase angle because of the presence of nonlinearity. A new steady state condition will
then be achieved when the energy put into the system is balanced by the energy dissipated
by the damping. Furthermore, let us consider the main nose of instability, a region in
which the trivial solution becomes unstable. If we substitute the trivial solution a = 0 into
equation (3.85) then we will obtain equation (3.58). From the conclusion obtained from the
nonlinear analysis, we obtain the same region of instability regarding the trivial solution.
We next consider the effects of the shear deformation and the rotary inertia which are
included in the parameter A3 in this work. It is clear that these effects can only be found
in the nonlinear analysis. These nonlinearities affect the response by shifting the stability
region along the frequency axis to the left. In other words, the main nose of instability in
which the trivial response becomes unstable will be centered to the left of Q = 24.2". We
note that the effect of A3 is of higher order. Therefore, its effect is included in the analysis
by using rescaling relation (3.60). In other words, its effect on the response may not be

signiﬁcant.

3.3 Primary Resonance (9 x can)

In this section, we consider the case in which the frequency 0 is near to w". This is
referred to as the primary resonance. To begin our analysis, let us reconsider equation (3.63)
in section 3.2, and assume that Q is near to can. Moreover, the damping parameters [12 and

m are rescaled by equations (3.49) and (3.10). In order to make the analysis clear, we

52

redeﬁne 01 to quantitatively represent the nearness of Q to can. Hence, we let
9 = I.)n + 601 (3.86)
in equation (3.63). Under this condition, we must have
- 2jwn(D1An) — ﬁﬂzexmjmﬂ — jg) — 2jp21n4Anwn = 0 (3.87)

in order to remove the secular term from the particular solution of €22“. Expanding this equa-
tion with An = §ezp(j\ll), and separating the resultant equation into real and imaginary

parts, we obtain

a’ = —p2,n4a — 73-6—2 cos <I>1 , (3.88)
1r

aQ’l 2 01a + 35—2 sin {>1 (3.89)

where <I>1 is given in equation (3.55). The steady state condition can be achieved when

a’ =: a¢I>’1 = 0. Therefore, we obtain

"52

n4p21a = -7 cos <I’1 , (3.90)
010. = —nT€2 sin Q1 . (3.91)

Squaring these two equations and adding them together, we obtain the following frequency

response equation:

(nap; +a¥)a2 = (if—2):. (3.92)

53

Therefore, solution of the ﬂexural vibration is approximated by

v(a:,t) cv1(z,t) + 0(62)

6] a cos (9T0 -— <I>) ] sin mm: + 0(3) (3.93)

where a and <I> are described by equations (3.88) and (3.89), respectively. The steady state

value of the phase angle 45 is described by

tan <I>, = (2) . (3.94)
#21

Equation (3.92) is the frequency response of the primary resonance. We note that it does

a, = magnitude of '01,. at 9 z an
I

 

0.16

0.14

0.12

0.1

0.08

0.06

0.04

 

 

 

 

0.02 L 1 1
0.9 0.95 1 1.05 1.1

frequency ratio 9

Figure 3.3: Frequency response equation for the primary resonance

not contain any nonlinearity, and hence corresponds to the frequency response condition
for a linear oscillator. Figure 3.3 provides an example for this frequency response equation.
All these imply that the ordering relation (equation [3.60]) fails to capture the nonlinearity

in the analysis. In order to extend our analysis to includes the nonlinearity, we rescale the

54

length ratio 5 by letting

g = 6353 . (3.95)

Moreover, we rescale the damping parameters #2, p3 and #4 according to equation (3.8),
(3.9) and (3.10), respectively. Expanding equations (2.51), (2.52) and (2.53) with these new
ordering relations, and equating the coefﬁcients of the like-power terms, we will obtain a
sequence of partial differential equations. Applying the same procedure to the boundary
conditions, we will obtain a sequence of boundary condition associated with these equations.
We then solve these equations by following the same procedure as described in the previous
sections. Because this procedure is routine and tedious, we only present a few essential
results in the following analysis. For a complete solution procedure, refer to Appendix D.
The equation describing the ﬁrst order ﬂexural vibration is given in equation (3.61). In
consequence, the solution of the ﬁrst order ﬂexural vibration is given by equation (3.62).
The equation describing the second order ﬂexural vibration of the connecting rod is given
as

. . 2
D0262“ + 013,132,. 2 —2]wn(D1An)ezp(]wnTo) — Emma cos mr + c.c. (3.96)

where (—-%p3,f2 cos mr) comes form the nonhomogeneous boundary conditions. The elim-
ination of the secular terms requires that (DlAn) = 0. This implies that the amplitude
function of the transverse response is independent of the time scale T1. The third order

term for the ﬂexural vibration of the connecting rod is described by the following equation:

4
. . . . .n .
0027131. + wivan = -2.7wn(DlAn)€$P(anTo) + 2.7 jx—zllsz Anexp(anT0)
15 1 — .
- l‘s - (5 + Ser)”1(n«)2n‘A3A.ezp(Jw.To)

+ An(n1r)2n6(alag + ala3)ezp(jwnTo) — j(n7r)4p4ownA?,Ane$p(jwnTo)

55

— 2jwnn4p22 Anexp(jwnTo) — j%6$p(jQT2) + N.S.T. + c.c. . (3.97)
To describe the nearness of the frequency ratio 9 to wn, we express 9 as
Q = w, + 6202. (3.98)
Therefore, we must have

- 2jwn(D1An) + An(n7r)2n6(alag + 01023) — 2jwnn4p22 An — j(n7r)“cunn40 A211,,

.124 15 1 - . .
+ 2jﬁp32An — [§- — (3 + S)(n7r)2](mr)2n“A,2,An — J-TE—jreszong) = 0 (3.99)

in order to remove the secular term from the particular solution of 133m. Expanding this equa-
tion with An = §ezp(j\ll), and separating the resultant equation into real and imaginary

parts, we obtain

1 n2 53
a’ = —n4ﬂ220 — -8-(n7r)4)u40wna3 + Fpgza — 5 cos (1)2 , (3.100)
a<I>'2 = 02a — A2a3 + A30 + %92 sin <I>2 (3.101)

where <I>2 is deﬁned by equation (3.71), A2 is deﬁned by equation (3.69) and A3 is deﬁned
by equation (3.70). The steady state condition for these equations can be achieved when

a’ = a<I>’2 = 0. This leads to

n 1 n2
-(;)fa cos <I>2 = n‘nga + §(mr)4u40wna3 - Eusza . (3-102)

—(%)£3 sin Q2 = 020 - A203 + A30. . (3.103)

56

Squaring these equations and adding them together, we obtain the following frequency

response equation:
1 n
02[(02 - A2612 + A3)2 + (714/122 + §(n7r)4#4own02-22H32)21‘ ( {if - (3-104)

Consequently, the frequency response curves take the following form when solved for the

detuning parameter:

 

n 1 n2
02- _ [A202 — A3] :t \/(71-1-r- £3 )2 — [ n4)”, + -8-(n7r)4p40<...vna2 - Fug, ]2 . (3.105)

This equation indicates that the steady-state response can reach its maximum amplitude

described by
n
( £3 )2

1 n2
,, ail n‘uz. + §(mr)4u4.wna§ - 5%: 12 = 0 (3.106)

which occurs at the frequency
(2 = w. + co, = w, + 52(A2ag — A3) . (3.107)

Since A2 < 0 and A3 > 0 for each individual mode, we have 0,, < 0. Therefore, the
frequency response curves bend to the left. Figure 3.4 provides an example for this frequency
response equation. To determine the stability of these steady state responses, we compute
the J acobian matrix associated with equations (3.100) and (3.101), which takes the form:

—n“;122 — 3(n1r)4wnp4oa2 + 51132 —(02 — A202 + A3)a
(3.108)

(02 + 3132“2 + A3)/0 414/122 + $20132 - %(n7r rum/14002

57

a, = magnitude of 131,, at O z w]

 

 
 
 
 
 
 
 
 
   

0.16 _ cl I 7 l l 1 I
£1 = 0.01
0'14 ” s = 0.50
_. [12 = 0.02 g
0.1 - #4 = 0-00 —
a 01 = 1.00
8 0.08 P a; = 0.00 ‘
013 =3 0.00 _
0.06 — n = 1
0.04 —
0.02 P
0 l l l l l l l

 

 

 

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

frequency ratio 9

Figure 3.4: Frequency response equation for the primary resonance

By computing the eigenvalues of this matrix, we determine that when
3 n2 1 n2
["0122 + '8'0"? Won/14002 — 1731132 “"4112: + §(mr)4wn#4oa2 - pﬂszl

+ (0'2 — A202 + A3)(02 — 3A202 + A3) < 0 (3.109)

these steady state responses are unstable. Otherwise, they are stable. The solution v(a:,t)

of the ﬂexural vibration of the connecting rod can be approximated by

v(a:, t)

601(x,t) + 0(62)

e[acos(QTo — <I>)] sin mm: + 0(62) (3.110)

where a and (I) are described by equations (3.100) and (3.101), respectively. The steady

state value of the phase angle is described by

0'2 - A202 + A3

tan (P, = ( 4 1 4 2 n2
#22” + §(n7r) wnﬂ‘ioa — 377/132

 

). (3.111)

58

From Figure 3.4, it is clear that, when 9 is located between point I) and 0, there exists more
than one steady state response. Those steady state responses located along the be portion
of the response curve are unstable, and hence will not be observed either in simulations
or experiments. Moreover, for frequences between point I) and c in Figure 3.4, the initial
conditions will decide the steady state response. In other words, the initial conditions will
decide whether the response shall appear along the ab curve or the cd portion. To be more
precise, there is a saddle node bifurcation[50, 21] associated with points 0 and c in Figure 3.4.
For each stable steady state response, there is one corresponding basin of attraction. The

response is then determined by the location of the initial conditions.

Remark: Before proceeding to the next section, there are few points which need to be
made. We ﬁrst compare the results from the linear and nonlinear analyses. By neglecting the
effects of shear deformation, rotary inertia, the friction parameters #32 and #40 associated

with equations (3.100) and (3.101), we obtain

a' = —n4p22a — 1:3 cos (P2 , (3.112)
a<I>'2 = 0'20. + A203 + 12% sin <I> . (3.113)

The resultant frequency response equation takes the form:
n
aZI (a. — A323)? + (12472021 = ($I". (3.114)

Basically, the equations describing the variation of the amplitude of the response (equa-
tions [3.88] and [3.100]) are exactly the same. The only difference between linear and

nonlinear analyses shows up in the equations describing the variation of the phase angle

59

(equations [3.89] and [3.101]). Moreover, equations (3.92) and (3.114) coincide with each
other in their linear parts. To show this, we substitute M, = cm, , {3 = 662 and 02 = 601
into equation (3.114). Then equation (3.114) will reduce to equation (3.92) as a approaches
zero. This implies that when the amplitude of response is small, linear theory works well.
However, as the amplitude of response increases, the nonlinearity affects the system by
changing the rate at which energy is pumped into the system. As a consequence, a new
”equilibrium” will be reached when the rate at which energy put into the system is balanced
by the rate of energy dissipated.

It is clear that the shear deformation and the rotary inertia do not affect the linear
response, because they are not included in the resonance condition. In consequence, they
will not affect the peak amplitude of the primary resonance. The shear deformation and
the rotary inertia affect the primary resonance by shifting the location of the frequency at
which the primary resonance reaches its peak amplitude to the left by a small amount of
order C(62). In other words, the backbone curve, which is the locus of the peak amplitude
associated with the frequency response curves, will not originate from Q = 02... However,
the effect of A3 is independent of the response and is of order C(62). Hence, we may expect

that A3 affects the linear approximation by a very small amount.

3.4 Superharmonic Resonance (Q a: 92“)

Recalling that in analyzing the particular solution of 132,, in section 3.1, we assumed that
the frequency ratio 9 is away from 20.7,, and 94;, and then proceeded to analyze the equations
of higher order. The case in which (2 is near to 20.7,, corresponds to the principal parametric
resonance and has been studied in section 3.2. In the current section, we investigate a

superharmonic resonance case in which the frequency ratio 9 is near to 9211. To consider

60

this case, let us express the frequency ratio 9 as
1
Q = 5(0),, + 601). (3.115)

The damping parameters [12 and #4 are rescaled according to equations (3.49) and (3.10),
so that they will show up, together with detuning parameter 01, in the ﬁnal resonant condi-
tion. We then expand equations (2.51), (2.52) and (2.53) with these ordering relations, and
follow the same procedure as described in section 3.2. Again, we only present a few results
which are essential to our work. For a completed analysis, please refer to Appendix B.
Following the same procedure as described in the last section, the ﬁrst order ﬂexural vi-
bration associated with the connecting rod is described by equation (3.29). Consequently,
the general solution of 771,, is given in equation (3.30). The equation describing the second

order ﬂexural vibration becomes

13027721; ‘1’ “321327; = ’2jwn(D1An)ezp(jwnTO) "' 2jl‘21wnn4Anemp(jwnT0)

 

mr C Sﬂ‘ll'
+ ( 2) p4oflflAlezp(2jQTo) + -—2—€192€$P(219T0 — j— ’2’)
_ 3(1 + 25)(mr)2A151I22exp(2jQTo - j%) + N.S.T. + c.c. . (3.116)

Therefore, in order to remove the secular terms from the particular solution of 7‘22”, we must

have

— 2jwn(D1A—,.) 2jn4p21wnAn+—— f(n7r)wne:cp(jalT1)

—COS mr

+ (f—LX—H— —$0+2S)(mr)21exp(ja.T.—j-;5).—.o. (3.117)

Expanding this equation with An = %e:cp( j W), and then separating the resultant into the

61

real and imaginary parts, we obtain the following equations:

éiwn

 

a = 424/1210 + -#—gﬂ(n1r)£f sin <I>1— ”[3(1 — cos mr) — 2—] cos (1)1, (3.118)

 

a€I>’ 01a + E33(n77)§2 cos <I>1 + £120.72 [3(1 — cos mr) - 2—1]sin in (3.119)
1 6 1 24n1r n2

where in is given in equation (3.55), and A1 is deﬁned by equation (3.56). The steady state

condition of these equations can be obtained by letting a’ = agb’l = 0, which yields

 

114/1.210. Eg—°(mr)£f sin <I>1 — 3:" [3(1 - cos mr) — 2—) cos (1)1, (3.120)

 

-010 = [13° —(mr)£1 cos <I>1 +2 61:" W[3(1— cos mr) -— 2%)sinQ1. (3.121)

Squaring these steady state responses and adding them together, we obtain the following

equation:

 

A
c.2078”; + 012) = £14[%(n7r)]2+ (24nn7r)2[3(1— cos mr) - 212—21]2 (3.122)

which is the frequency response equation for this superharmonic resonance. Figure 3.4
provides an example for this frequency response equation. Therefore, when 0 is near to 231,

the response of the ﬂexural vibration can be approximated by

v(:c, t) €v1($,To, T1,T2) + 0(62)

6[ A1 sin 0T0 + a cos (2QT0 — <I>) ] sin mra: + 0(62) (3.123)

where A1 is given in equation (3.31), while a and the phase angle <I> are described by

62

a, = magnitude of the homogeneous solution

0.06

 

I T I l

0.05

0.04

as 0.03

0.02

 

0.01

 

 

 

 

0 1 1 1 I 1 1 1 1 1
0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

frequency ratio 9

O . C 0 N w
Figure 3.5. Frequency response equation for superharmonic resonance (9 ~ 31*)

equations (3.119) and (3.119). The steady state value of the phase angle <I>1 is given as

404,712, (W)2 + 01wn[3(1 - COS 7”) - 2%]
41140010”) + wn[3(1— cos mr) - 2%]

 

tan <I>, = (3.124)

Remark: Equation (3.123) is the ﬁrst order approximate solution to the ﬂexural vibration
associated with the connecting rod. Note that this solution consists of two parts: the
homogeneous and the particular solutions. A simple calculation of the eigenvalues associated
with these steady state responses indicates that the homogeneous solution is stable for
#21 > 0. Hence, a steady state superharmonic resonance exists for all conditions and the
solution is composed of two periodic functions. The particular solution is, of course, of the
same frequency as the the operating frequency. While the homogeneous solution oscillates
with twice the frequency of the operating frequency. Therefore, for this superharmonic
resonance, the homogeneous solution of 131,. interacts in 132,” and hence does not decay to

zero. This indicates that the presence of this non-vanishing homogeneous solution is the

63

character of the nonlinearity possessed by the system. Moreover, the magnitude of this
homogeneous solution is proportional to the square of the length ratio {1. This implies that
the homogeneous solution vanishes more rapidly than the particular solution as the length

ratio 6 approaches zero.

Remark: The mass ratio S has a very interesting effect on the overall flexural response.
To analyze this point, we consider the frequency response equation (3.122). We note that
the right hand side of this equation represents the magnitude of the external force and is
always positive. Therefore, a local minimum will imply a local minimum value of magnitude
of the force, and hence a local minimum value of the amplitude of the homogeneous solution.
Figure 3.6 shows the inﬂuence of S on the magnitude of the homogeneous solution. We note
that 0, decreases when S increases from zero to about 0.11, and then 0, increases as S

increases beyond 0.11. Based on this, we ﬁnd that 0, reaches a minimum when

2A1 — 3132(1— cos n7r) = 0 . (3.125)

This implies that a proper choice of the mass ratio will help in suppressing (but not totally
eliminating) the peak amplitude associated with this superharmonic resonance (Figure 3.6).
Figure 3.6 shows the variation of the peak amplitude a, with respect to the mass ratio S.
As shown in this ﬁgure, the peak amplitude a, decreases as S increases from 0 to 0.11, and
then it increases as S increases beyond 0.11. We note that the presence of the damping
parameter #4, in the right hand side of the frequency response equation (3.122) indicates
that the an increase in #40 will increase the amplitude of response. It can be interpreted
as follows. The presence of p40 implies the presence of the friction force acting on the

slider end, and hence increasing [140 corresponding to increasing the axial load acting on

64

a, = magnitude of the homogeneous solution at Q = 021/2
0.06 I —l I I I I I I I

 

0.05

0.04

as 0.03

0.02

0.01

 

 

 

0 1 l l l l l 1 l

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

mass ratio S

 

Figure 3.6: Inﬂuence of mass ratio S on the amplitude of superharmonic resonance ((2 z 2;)

the connecting rod. As a consequence, the amplitude of response will increase.

3.5 Superharmonic Resonance (0 z 9’3“)

In analyzing the particular solution of equation (3.47), a superharmonic resonance takes

place when (2 is near to 3311. Under this condition, equation (3.47) reduces to

130203.. + 0:303, = -2jwn(D2A,, + n4p22An)ezp(jwnTo)
4
+ 2j%An:132ezp(jwnTo) + An(0102 + ala3)(mr)2n4e:cp(jwnTo)
_ 1_5_ _ S 2 6A2A T (11W)
[ 8 (g + )(nﬂ') 1n Aneprwn 0) + 1536wnﬂ40616zp(jwﬂTo +j02T2)
— jwn(n7r) ”40/42/17: ezp(jw,,To)- j(‘3‘*12)/140(NW)2Anwn€1€$P(jwnTo)

+ fllffAneprwnTo) + jf2,{?exp(jwnTo + jang) + N.S.T. + c.c. (3.126)

65

where f1, and f2, are given in Appendix A. Therefore, we must have

4
. . ,n , _
- 2an(D1An) — 2ann4p22An + 2JFAn/132 — an(n1r)4p4o AiAn

l5 1 _
+ (0102 + alag)(n7r)2n4An —- [—8—- + (5 + S)(n7r)2]n6A,2,An + fllAnéf

(mr ——)-1rwnp40€1 + jf2,£1 ]e:rp(jc72T2)- — 0 (3.127)

. 1

in order to remove the secular term from the particular solution. Since the leading coefﬁcient
of p4oezp(j02T2) is very small, we neglect this term in the following analysis. Expanding
equation (3.127) with An = gezp(j‘I'), and separating the resultant equation into real and

imaginary parts, we obtain

a’ = —anz[pgzn2 + “4 8°(n1r)4a 2 +p—°-p4o(n7r)2£1—64#—3—22]+ —£1f2‘3 cos (P2, (3.128)

a<I>'1 a[02 +2 fl—‘—2 7135, + A3 - A2a2] — f2___1_3 -——£1 sin <I>2 (3.129)

where A; is deﬁned in equation (3.69) and A3 is deﬁned in equation (3.70). The steady

state conditions can be achieved by letting a’ = 04>!) : 0. This yields

2
n
%€f€08‘1’2 = alpha“+%(mr)"n2a2+(%f)(mr)2n2€f-77/13.]. (3.130)

[$675111 T2 = [02+ {112 $61 + A3 — A202 ]. (3.131)
Squaring these equations and adding them together, we obtain the following equation:

n2
[(#m‘ + %(..)4.2.2+(E§)(n.)2.2g-5132):

+ (02 +2 -f—“ $5? + A3 .— A202)? ]a= (ii; 5;")2 (3.132)

66

which is the frequency response equation for this superharmonic resonance. We then solve

the values of 02 and obtain the following frequency response curves:

- — -f——‘;£¥+A2a —A11

 

(/(f21 053)” n4lﬂ22n2 + (”7040ng +(lu‘i°)(mr)2€1:3";2 12 . (3-133)

Thus, despite the presence of the damping parameters, the homogeneous solution does not

decay to zero. Furthermore, the steady state response takes the form:

v(:r,t) = €v1(m,t)+0(62)

= 6[ A1 sin 9T0 + acos (3QT0 — <I>)] sin mm: + 0(52) (3.134)

where a and the phase angle <I> are described by equations (3.128) and (3.129). The steady

state value of the phase angle <I> is described by

02 + £55? + A3 - A2a2

ta Q,=
11 1,. n4+(mxno4n2a2+(igrxnoznzé1—(:—2 m3.

 

]. (3.135)

By computing the eigenvalues of the J acobian matrix associated with equations (3.128) and

(3.129), we determine that when

2
n
(m. + -"—g£(mr)‘n2a2 + ($01,021.25: - FM»)
3712
(m. + _"4_O.(111)411212 (”4° )(11)2 251- 7:11.11)

+(0’2 +2 f_1_12 —-El- A202 + A3)(0’2 +2f-1—127ﬂfl - 3A2€12+ A3) < 0 (3.136)

these steady state responses are unstable. Otherwise they are stable.

67

Remark: According to this analysis, the free oscillation term does not decay to zero,
despite the presence of the damping parameter. Therefore, the steady state response is
composed of two periodic functions. The particular solution oscillates with the same fre-
quency as that of the external frequency, while homogeneous solution oscillates with the
frequency that is three times the operating frequency 9. We note that the amplitude of the
particular solution is of order 0(5), and the homogeneous solution is of order C(53). This
indicates that thefree oscillation term vanishes more rapidly than the forcing oscillation as
5 approaches zero. From the frequency response curves and the stability analysis, we know
that within a certain frequency region, there exists more than one steady state response.

We note that the homogeneous solution possesses a peak amplitude (1,, described as

5? 2 2 4 2 4 af, “i 2 2 "2 2
(f21_2_) '— “pl #22" + n (nn) [‘40— + n ("'7") ”'40 '— 7/‘32 ] = 0 (3'13?)
11 8 64 71'
which occurs at
a - —-f—1—’—£2 + A a2 A (3138

We note that A2 < O in the last equation. This indicates that the frequency response
curves bends to the left corresponding a superharmonic resonance with softening type of
nonlinearity. Figure 3.7 provides an example for the frequency response equation for this
superharmonic resonance. Note that, in Figure 3.7, there is almost no bending associated
with the response curve. This implies that, to observe higher order superharmonic reso-
nances, the bending effect may require a larger length ratio as well as a larger mass ratio.
In other words, it is necessary to have strong nonlinearity and a large response amplitude of
response in order to achieve a signiﬁcant bending in the response curve. Another interesting

phenomenon accompanying this superharmonic resonance is the inﬂuence of the mass ratio

68

on the peak amplitude up. The numerator of equation (3.137) is the force applied to this
superharmonic resonance, and it is a quadratic function in terms of S. A proper choice of
the mass ratio S will minimize the magnitude of the force and hence minimize the amplitude
of the response. Figure 3.8 shows the inﬂuence of the mass ratio on the peak amplitude
associated with the superharmonic resonance. From this ﬁgure, we see that the amplitude
of the response will decrease when S is increases from zero to a critical value; while the peak

amplitude will increase when S is beyond this critical value.

a, = amplitude of homogeneous response

 

 

 

 

0.03 l I I
{=0%
0.025 - S = 1.0 r
[12 = 0.02
002 ,_ p3 = 0.00 T
01 = 1.00
as 0.015 '— 02 = 000 1
a3 = 0.00
0.01 - n = 1 '1
0.005 - a
——_—— / \\.1
0 l l J
0.2 0.25 0.3 0.35 0.4

Frequency ratio 9

Figure 3.7: Frequency response equation for superharmonic resonance (9 z 2311)

3.6 Subharmonic Resonance ((2 z 302,.)

In this section, we consider the case in which 9 is near to 3w”. Under this condition,
the system is potentially in a state of subharmonic resonance. To analyze this subharmonic

resonance, we express 0 as

n = 3(0),, + 6202). (3.139)

69

a, = amplitude of homogeneous response

 

 

 

 

 

0-04 I I I I I
0.
0.035 - 0
[£2 = 0.02 '
0.03 - #3 = 0.02
[14 = 0.00
0.025 '— a1 = 1.00
02 = 0.00
as 0.02 L" a3 = 000
0.015 L n = 1
0.01 -
0.005 *-
0 i
0 0.5 1 1.5 2 2.5 3

Mass ratio S

Figure 3.8: Inﬂuence of S on the peak amplitude of the superharmonic resonance ((2 z 9311)

Then equation (3.47) reduces to

D0263" + 013,233,, = —2jw,,[m,n4 + DgAn + jp40(nr)4A:/ln]exp(jwnTo)
+ figfiAneprwnTO) + [37315133. + (g)ﬂ4o(n’r)3wn€11§§l€$P(3j02T2 + jwnTO)
+ (0102 + :1a3)(n1r)2n4Ane:cp(jwnTo) + j(%)p4o£§(n1r)2Anexp(jwnTo)
- "700185 - 4% + 5) (nﬁ)2ln4A§3ne$P(jwnTo)

+ 2j:—:p3, Anezp(jwnTo) + N.S.T. + c.c. (3.140)

where c.c. represents the complex conjugate term of the proceeding terms, and an overbar

stands for the complex conjugate term of the corresponding function. Hence, we must have

-— 2jwn(p2,n +D2An )+2.7':-—:72p;3,.4,,-{~(c11(3z2+crla3)(n7r)2 4A,1
l5
— (mrﬂ— —(- +S)(mr)"’]n“A3.A +f12£¥ A.ezp(jw..To)

- jwnu4o[(nW)‘A§A n+(8— —:¥er)2€ An]

70

+ [1.1-3161A: +(§)#4o(n7r)3wn51/131l€93P(3j02T2) = 0 (3-141)

in order to remove the secular term from the particular solution. Expanding equation (3.141)
with An = gexp(j\Il), and separating the resultant equation into real and imaginary parts,

we obtain

 

a’ = «1124112. - ($121.3, “Ea—5054a? +(2— immow
(1.3161 )a2 cos (3<I>2), (3.142)
a<I>’2 =+éf—1—221ﬂ€la—4€1:2 f31a2 SlIl (3Q2) + A30 - A203 . (3.143)

The steady state conditions can be obtained by letting a’ = a<I>"2 = 0, these yield

(f3161)a C05(3‘1’2) = aln4p22 - :—:#32 +Il4onz(mr)4(:3 2'2‘“)+( :erlgnzahllio 144)

if; a2 sin (3<I>2) = 020 +2 f__1_221ﬂ£la + A30 - A203- 3-145
413'2 1

We obtain the frequency response equation which takes the form:

2 4 "2 2 4 “2 81 2 2 2 2
a [0‘22" " ﬁlls: + #40" (“77) ('gl‘l’ (5;)(mr) n “51)

+(ag+¢f-—122£1+A3—A222a) (fl—15‘)2a2]=0 (3.146)

by squaring equations (3.144) and (3.145), and adding them together. From equation (3.146),
we know that the trivial solution for a is always a steady state response. Due to the com-

plexity of this equation, we neglect the effect of p40 in the following analysis. We solve

71

equation (3.146) for the nontrivial values of a2 and obtain

(02,3)2 = (52711;) 1W — 4g] (3.147)

where coefficients p and q are

p = 2A2(0’2 +2 f-1-22 71261 + A3) + (-—- f3] T61? (3.148)

q = (#2271 -—#3.)2+(0'2+f1"’€1+A)2 ~ (3.149)

We note that q is always positive. Therefore, these nontrivial free oscillations exist only
if p 2 0 and p2 — 4q 2 0. When these conditions hold, it is possible for the system to
respond in such a way that the homogeneous oscillation does not decay to zero, in spite
of the presence of damping parameters. In consequence, the steady state response can be

approximated by
v(a:, t)- — £01 + 0(62)_ — 6(0, 005(— To — —) + A1 sin 0T0) + 0(6 2) (3.150)

where a, is described by equation (3.147), A1 is deﬁned by equation (3.31), and the phase

angle <I>, is given by
0’2 + $612 ‘1’ A3

n‘iuzz- $1132

 

tan Q. =( ) (3-151)

Remark: We note that the response of the system is described as follows. When the initial
conditions are selected in such a way that the resulting steady state corresponds to a stable
nontrivial solution of a,, then the magnitude of the free oscillation term does not equal to
zero. Under this condition, the ﬁrst order ﬂexural response associated with the connecting

rod contains frequencies of Q and %. Otherwise, the ﬁrst term of equation (3.150) will be

72

zero, only particular solution will exist, and the response contains only the frequency (2.
Friction parameter m, which is used ot model the friction force acting between the slider
mass and its contact surface, has a favorable effect in reducing the amplitude of response.
This is caused by the friction force introduced by the elastic deformation, which behaves as

a resistance to prevent the connecting rod from further deformation.

3.7 Summary and Conclusion

In this section, we summarize the results obtained in previous sections. Based on the
observations from previous investigators[ll, 49] and the analytical results obtained in the
previous sections, we select in turn the length ratio 5, the frequency ratio 9, the mass
ratio S, and the damping parameters ”2, p3 and p4 and then provide a detailed discussion

regarding the effect of each individual parameter on the overall response.

3.7.1 Effect of the Length Ratio 5

As we mentioned in the introductory section, the increase of the inertial force is pri-
marily responsible for the failure of the traditional dynamic analysis of mechanisms. In
this problem, the acceleration components 0,, and (1,, are proportional to the length ratio
5 (equations [2.11] and [2.12]). Consequently, the inertial force is pr0portional to this pa—
rameter. The general solution of the first order ﬂexural vibration is given as a combination
of the particular and homogeneous solutions. The magnitude of the particular solution
is proportional to f. If no resonance occurs, the free oscillation term will decay to zero,
and hence steady state response contains the particular solution only. In consequence, the
amplitude of the response will be proportional to the length ratio. In the case of primary

resonance, the amplitude of the resonance is proportional to 6 (equation[3.106]). Figure 3.9

73

shows the effect of f on the amplitude of the primary resonance. Moreover, the frequency
at which this peak amplitude is observed depends upon the square of the magnitude of the

particular solution, which in turn depends on {2.

a, = magnitude of 01,,

 

 

 

 

 

 

 

0.35 I I I I I I I
0.3
0.25
0.2
as
0.15
0.1
0.05
0 1 1 l 1 1 1 1
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
frequency ratio 9
Figure 3.9: Inﬂuence of f on the primary resonance
0, = nontrivial response
0.3 I I I I I I I
f = 0.01 —
0.25 6 = 0.02 —— ..
0.2 S = 0.50 _.
[12 = 0.02
a. 0.15 [1.4 = 0.00 *
01 = 1.00
0.1 02 = 0.00 '4
03 = 0.00
0% "=1 —
O 1 l q 1 Q l l

 

 

 

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency ratio 9

Figure 3.10: Inﬂuence of f on the main nose of instability

In the case of the principal parametric resonance, the width of the frequency region in

which the trivial response becomes unstable also depends on f (equation[3.59]). In other

74

words, the width of main nose of instability is proportional to 6. Figure 3.6 shows the
effect of the length ratio on the main nose of instability. We note that, in Figure 3.6,
the width between the transition curves increases when 6 increases. Figure 3.10 shows the
effect of the length ratio on the amplitude of the nontrivial responses. We note that, in
Figure 3.10, the region between points a and b indicates the region in which the trivial
response becomes unstable. In the case of superharmonic resonances, the magnitude of the
non-zero homogeneous solution also depends on 6. To be more precise, when 0 is near to
can / 2, the magnitude of the non-zero free oscillation term is of order C(52); when 9 is near
to can / 3, while the magnitude of the non-zero free oscillation term is of order C(53). Hence,
the contribution of the non-zero homogeneous solution depends on the order of 5. All these

imply that the length ratio is the primary parameter for the current problem.

3.7 .2 Effect of Mass Ratio S

The effect of this parameter is not immediately clear. Generally speaking, it acts as the
nonlinearity associated with the system. In order to make our discussion as clear as possible,
we need to consider each resonance case separately. In the case of primary resonance case, a
slight increase of S will cause a significant increase of the value of Up, the value of detuning
at which the peak amplitude occurs. To make this point more clear, let us consider the
following cases: S = 0 and S = 1. When there is no mass attached to the slider end, the
value of 0,, is given as

II2 15 1r2

(0p)s=o = (3 - g 73— (3.152)

for the first mode (n=1). In consequence, the frequency at which the peak amplitude
associated with the primary resonance can be observed will shift to the left by an amount

(20,, with the value of 0,, being given by equation (3.152). Now, we consider the case in

75

which S = 1. When S = 1, the value of 0,, becomes

1r2 15 x2 W4 1r“
(Uplsu = (3' - -8—)—8_ - ‘8- = (0105.0 — 3 (3-153)
A rough estimate of the difference between these two values of 0,, shows that (01,)58l is
about eight times the value of (0,) 5.,. Moreover, this ratio will increase as the value of
S increases. This implies that the value of 0p is very sensitive to the value of S. A slight
change of S will cause a signiﬁcant change of 01,. Figure 3.11 shows the inﬂuence of the

mass ratio S on the primary resonance.

a, = the magnitude of 01,.
0.2 r I I I I T I

 

S = 1.00 S = 0.00

0.15 _

as 0.1 -

0.05

1

 

 

 

 

0 1 l l l l l i

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

frequency ratio 0

 

Figure 3.11: Inﬂuence of S on the primary resonance

The mass ratio S affects the principal parametric resonance by increasing the width
of the main noise of instability. Figure 3.12 shows the inﬂuence of S to the main nose of
instability. From this ﬁgure, it is very clear that the distance between two transition curves
increases when the mass ratio S increases. Figure 3.13 shows the amplitude of the nontrivial
homogeneous response associated with the principal parametric resonance.

In all the superharmonic resonances, S has a very interesting effect on the response. We

76

 

0.3 I I I I I I

0.25 r

0.2 -

    
 

 

 

 

 

 

 

 

 

f 0.15 - (unstable region) —
0.1 - 01 = 1.0 [12 = 0.02 f
a; = 0.00 #3 = 0.00
0 05 _ 013 = 0.00 [14 = 0.00 _

. n = 1
0 l l l l I 1 l
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio 9
Figure 3.12: Inﬂuence of S on the main nose of instability
a, :2 nontrivial response
0.3 I I I I n I n
S = 0.50 -—
0.25 S = 1.00 — —
0.2 f = 0.01 a
[£2 = 0.02
as 0.15 [1.4 = 0.00 .-
01 = 1.0
0.1 02 = 0.00 “
a3 = 0.00
em "=1 ~
0 l l 1
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency ratio 9

Figure 3.13: Inﬂuence of S on the principal parametric resonance

77

ﬁrst consider the case in which (I is near to 53h. When this resonance occurs, the steady
state response is composed of particular and homogeneous solutions. The magnitude of
the free oscillation component depends on the value of S. It is interesting to find that the
magnitude of the free oscillation term decreases as S increases from zero up to a critical
value, while the magnitude increases as S is increased beyond this critical value (which is
the root of equation (3.125)). Hence, the magnitude of the overall response follows the same
trend as, the homogeneous solution (Figure 3.6). This implies that a proper choice of the
mass ratio 8 can aid in decreasing the amplitude of superharmonic resonances. This may

have important practical implication for the design of slider crack mechanisms.

3.7.3 Effect of Damping Parameters pg, #3 and #4

The internal material damping #2 has a favorable effect on the overall response of the
system. In the primary resonance case, the presence of m in the frequency response equa-
tions (3.104) and (3.92) will reduce the peak amplitude of the system. The amplitude of
this resonance will decrease as p22 is increased (Figure 3.14). In the principal parametric
resonance, the width of the main nose of instability will decrease as we increase the value
of pg, in equation (3.58). Moreover, the amplitude of the response decreases as we increase
the value of p22 in equation (3.74). Figure 3.15 shows the effect of the damping parameter
#2 on the width of the main nose of instability. We note that, in Figure 3.15, the presence
of #2 will cause the upward vertical movement of the transition curve. Therefore, the width
of the main nose of instability will decrease. As a matter of fact, the shrinking accompanied
with the presence of m may be so large that the width of the main nose of instability re-
duces to zero for a speciﬁc value of 5. Figure 3.16 shows the inﬂuence of m on the branches

of the nontrivial responses associated with the principal parametric resonance. The am-

78

a, = magnitude of 01,,

 

 

 

 

 

 

 

 

 

 

0.2 I F I
f = 0.01
#2 = 0-02 p3 = 0.00
0.15
as 0.1
0.05
0 l l I
0.8 0.9 1 1.1 1.2
frequency ratio 9
Figure 3.14: Inﬂuence of p; on the primary resonance
0.3
0.25 *-
0.2 —
E 0.15 -
0.1 -
0.05 ~
' n = 1
0 J l l l l
1.7 1.8 1.9 2 2.1 2.2 2.3

Frequency ratio (1

Figure 3.15: Inﬂuence of M on the main nose of instability

79

a, = nontrivial response

 

0.3 I I I I I I I
[12 = 0.02 ——
0.25 [12 = 0.04 —- -.
0.2 5 = 0.01 _.
S : 0.50
= 0.00
as 0.15 S: ___ 000 ""
01 = 1.0
0.1 02 = 0.00 "‘
a3 = 0.00
n = 1

0.05

 

 

 

0 l l l l l l

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency ratio 9

 

Figure 3.16: Inﬂuence of #2 on the principal parametric resonance

plitude of the nontrivial responses associated with the parametric resonance will decrease
as [12 increases. The parameter [12 is a measure of the system’s ability to dissipate energy
through material damping. Hence, in order to produce a sustained nontrivial steady-state
response, the magnitude of the external force must be large enough to overcome the energy
dissipated. Therefore, for a given system, the width of the main nose of instability shrinks
as the value of p; is increased. In both of the superharmonic resonances, the amplitude
of the free oscillation term will decrease as we increase the value of #2, (equations [3.122],
[3.132]). These observations show that the presence of m has a favorable effect on the
overall response.

The bearing friction #3 has an adverse effect on the overall response of the system. To
show this, let us consider the sign associated with parameter #3 in the frequency response
equation of each resonance case. The negative sign associated with parameter p3 in every
frequency response equation indicates that the presence of this parameter has an adverse

effect on the system. From another point of view, the presence of this parameter implies a

80

constant positive energy ﬂux and hence increase the amplitude of the response. Figure 3.17
shows the inﬂuence of #3 on the amplitude of the nontrivial response associated with the
principal parametric resonance. We note that [13 does not affect the main nose of instability
associated with the principal parametric resonance. From this ﬁgure, it is seen that the effect
of [13 on the nontrivial response is not very signiﬁcant. Figure 3.18 shows the inﬂuence of
#3 on the primary resonance. From these two ﬁgures, it is very clear that the inﬂuence of
#3 on each resonance case is not very signiﬁcant, although #3 has an adverse effect. The
reason for this is that the relative angular motion associated with each joint point (A and
B) is constant in the ﬁrst order. The presence of p;, will cause a constant friction moment

acting on each joint. These friction moments will enlarge the transverse deformation.

0, = nontrivial response

 

 

 

 

0.3 I I I I I I I
[‘3 = 0.00 —-—
0.25 [1.3 = 0.08 —- _
0.2 f = 0.01 _
S = 0.50
#2 = 0.02
as 0.15 F #4 = 000 j
01 = 1.00
0.1 b (12 = 0.00 -‘
a3 = 0.00
0.05 — n = 1 ~
0 l l I I I l 1

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio 0

Figure 3.17: Inﬂuence of #3 on the principal parametric resonance

The effect of the friction parameter #4, which is used to model the friction between
the slider mass and its contact surface, is complicated. In order to have a clear picture
regarding the effect of [14 on the response, one must consider the nature of this friction

force. The slider friction force is composed of the actions of the rigid body motion as

81

a, = overall response

0.18 I I I I I I I r I
0.16
0.14
0.12

0.1
0.08
0.06
0.04
0.02

0 l L l l l l l I l

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
Frequency ratio 9

 

as

 

 

 

 

Figure 3.18: Inﬂuence of #3 on the primary resonance

well as the elastic deformation. When the response amplitude is small, slider friction is
mainly composed of the friction introduced by rigid body motion. As the value of #4
increases, slider friction increases. This results an increase in the axial force acting on the
connecting rod. Consequently, the response amplitude increases. From this, #4 has an
adverse effect on the response. This can be observed in the second order superharmonic
(Q m %) resonance (Figure 3.19). Moreover, the width of the main nose of instability
associated with the principal parametric resonance resonance increases as we increase the
value of [14 (Figure 3.20).

The increase of the amplitude of response implies an increase of the foreshortening. This
results an increase in the friction force introduced by elastic deformation. The friction force
introduced by foreshortening acts as resistance which prevents the connecting rod from fur-
ther deformation and hence has a favorable effect. In general, the slider friction is composed
of two friction forces with compatible magnitude, and the system reaches dynamic equilib-

rium. This can be observed in the principal parametric resonance (Figure 3.21). Figure 3.21

82

a, = amplitude of the homogeneous solution

 

 

 

 

 

 

  
 

0025 I I I I I I j I
0.2
0.15
0vs
0.1
0.05
0 l J l l l l L l l
0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
Frequency ratio 9
Figure 3.19: Inﬂuence of #4 on the superharmonic resonance (Q z 221,11)
0.03 I I I T f r I
#4 = 0.00 —
0.025 #4 = 0.40
0.02
as 0.015
0.01 - 01 = 1.00 f = 0.01 '-
02 = 0.00 S = 1.00
n = l [13 = 0.00
0 1 l l l l l 1

 

 

 

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency ratio 9

Figure 3.20: Inﬂuence of [14 on the main nose of instability ((2 z 2w”)

83

shows branches of response originating from the main nose of instability associated with
the principal parametric resonance. From this ﬁgure, we find that ”4 has a favorable effect
on the outgoing response by reducing the response amplitude; while it also has a tendency

to increase the width of the main nose of instability.

 

0.22

 

 

 

 

T j
0.2 [14 = 0.00 — -
0.18 ”4 = 0'40 “" —

0.16

0 14 5 = 0.01

° S = 1.00
0.12 [12 = 0.02 "‘
6 0.1 [13 = 0.00 _I

01 = 1.00

0-08 a2 = 0.00 J
0.06 03 = 0.00 "
0.04 n = ..
0.02 ..

0 I I I I I L I

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio 9

Figure 3.21: Inﬂuence of m on the principal parametric resonance ((2 z 20.21”)

Figure 3.22 shows the frequency response curve for the primary resonance. From this
ﬁgure, it is clear that friction parameter III has a favorable effect on the primary resonance
by reducing the amplitude of response. This is caused by the friction force introduced by

large elastic deformation.

3.7 .4 Effect of Shear Deformation and Rotary Inertia

In this work, the effects of shear deformation and rotary inertia are included in the pa-
rameter A3. In the primary resonance, the presence of shear deformation and rotary inertia
affect the frequency response curve in such a way that the locus of the peak amplitude, i.e.,
the backbone curve, will not originate from can. The origin of the backbone curve is shifted

to the left by an amount of order 0(62). Figure 3.23 shows the effects of shear deformation

84

 

 

 

 

 

 

0.2 I I I I I
[14 = 0.00 —-
[£4 = 0.08 —
0.15 -
as 0.1 b
0.05 '-
0 l l l l I
0.6 0.7 0.8 0.9 1 1.1 1.2

Frequency ratio 9

Figure 3.22: Inﬂuence of #4 on primary resonance (9 a: com)

and rotary inertia to the primary resonance. We note that there is no difference in the
proﬁle of the frequency response equations, while the shift between these curves is caused
by the presence of shear deformation and rotary inertia (02 = 03 = 1). In the principal
parametric resonance, the presence of A3 causes a shift of the center of the main nose of in-
stability (Figure 3.24). In other words, the center of the region in which the trivial solution
becomes unstable will shift to the left by an amount of order C(62), while the width of this
region remains unchanged. Figure 3.24 shows the effect of shear deformation and rotary
inertia on the principal parametric resonance. We note that the proﬁles associated with the
nontrivial responses for these two cases are very similar, while the shift of these nontrivial
responses is caused by the presence of shear deformation and rotary inertia (02 = (13 = 1).

Shear deformation and rotary inertia do not affect the superharmonic resonance in which
0 is near to 9.}. In the superharmonic resonance in which 9 is near to 9131, these two terms
affect the response of the system in a way similar to the primary resonance. In other words,

the backbone curve will not originate from the point 9 = 131. It shifts to the left by an

85
amount of order 0(62).

0, = magnitude of 131,.

 

 

 

 

 

0.2 , I I
curve a: a; = a3 = 0,0
cur e b curve b: a; = a3 = 1,0
0.15 - curve a _
f = 0.01
[12 = 0.02
#3 = 0.00
a. 0.1 l’ #4 = 000 _
a} = 1.00
02 = 0.00
0.05 - a3 = 0.00 _
n = l
0 i L I I
0.5 0.75 1 1.25 1.5

frequency ratio 9

Figure 3.23: Inﬂuence of shear deformation and rotary inertia on the primary resonance

0, = nontrivial response

0.3 I I I I I I I

 

0.25

I

l

0.05

 

 

 

 

0 I l l l l l l

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio 0

Figure 3.24: Inﬂuence of shear deformation and rotary inertia on the principal parametric
resonance

Chapter 4

Analysis of the Lumped Parameter Model

In the present chapter, we consider an alternative approach to the problem under in-
vestigation. Instead of a set of partial differential equations, the dynamic behavior of the
connecting rod is modeled by a single ordinary differential equation. This approach is based
on extending the work by Viscomi and Ayre[49] by the inclusion of internal material damp-
ing, bearing friction and sliding friction. In this chapter, we also demonstrate how an active
piston force can be incorperated in this model. Based on the assumptions given in section
2.1, except that the effects of the shear deformation and rotary inertia are neglected, the
elastic response of the connecting rod is modeled by two coupled nonlinear partial differ-
ential equations. Based on the ﬁrst order approximation, the equation describing the axial
component is ﬁrst solved to obtain the time-varying axial force. Using this axial force and
neglecting the axial displacement, the equation describing the ﬂexural response becomes a
single uncoupled partial differential equation. This equation is then converted into a single
ordinary differential equation by Galerkin’s method with a single mode. We ﬁrst consider
the linearized model and obtain the corresponding response curve. The Method of Multiple
Scales is then employed to locate the conditions in which the linear approximation fails
to provide a satisfactory prediction. Improved approximations are also obtained by the
Method of Multiple Scales. In section 4.2, the equation describing the elastic response as-

sociated with the connecting rod is derived, and converted into dimensionless form. Section

86

87

4.2 contains the main outline of the analysis, and several resonance cases are introduced.
The principal parametric resonance is investigated in section 4.3. Section 4.4 contains the
analysis for the primary resonance. Two superharmonic resonances in which the operating
frequency is near to one half and one third of the fundamental natural ﬂexural frequency
are studied in section 4.5 and section 4.6, respectively. In section 4.7, we consider the sub-
harmonic resonance of order three. In section 4.8, we summarize the results and provide a

detailed parameter study.

4.1 Equation of Motion

In this section, an equation describing the ﬂexural vibration associated with the con-
necting rod of an otherwise rigid slider-crank mechanism is derived. After this equation is
obtained, it is then transformed into dimensionless form in order to minimize the number
of system parameters.

The basic equations describing the motion along the transverse direction are derived
based on the assumptions given in section 2.1, except that the the effects of shear deforma-
tion and rotary inertia are neglected. In other words, we model the connecting rod using
the Euler-Bernoulli beam theory. With these assumptions, we proceed to derive equations
which describe the ﬂexural vibration associated with the connecting rod AB. Consider a
slider-crank mechanism as shown in Figure 2.3. The OXY coordinate system represents a
ﬁxed inertial reference frame with its origin being attached on the crank shaft. The oxy
coordinate system represents the moving reference frame with its origin being attached on
joint A. The ox axis passes through the idea pin joints at the ends of the elastic connecting
rod and makes an angle -¢ with the OX axis. Let u(z,t) and v(:c,t) represent the axial

and transverse displacements of the connecting rod, respectively. The equations of motion

88

in the oxy coordinate system are described by force and moment balance and are given by

95g?) = EA(u. + $223.). + Aﬂoat. +£22.12)“ = PM“ (4-1)

1 1
4):)xe lx — #cAl (”r + 51212,.)va ix = (”my ° (4-2)

EIvzzxz + Iﬂcvxxxxt ‘ EA[ (“1: + 2

where P(:1:, t) = axial force acting on the connecting rod,
p = mass density (mass per unit volume) of the connecting rod,
A = cross-sectional area of the connecting rod, and

03(3, t) and ay(a:, t) are acceleration components given by

0212 045 61) (9—2451)

0,, = —rw2cos (wt — (,b) +0t — — 2552- — 3—t2v —(% ¢x)2( + u), (4.3)
2 2?
0,, = —rwzsin (wt — 45) + 0 1&0: + u) + 2%??3—1;+ g-; -%—(f')-)2 (4.4)

The axial strain 60 at the median line of the connecting rod is approximated by

_3u+ 1 3v 2
co — — WI 7.) (4.5)
From the analysis in chapter 3, we know that the axial displacement is small compared to the
transverse displacement. Therefore, we neglect the contribution of the axial displacement

on the inertia forces pa, and pay. Equation (4.1) then becomes

0P

1 1
5; EA(u,,. + i”:):: + +A#c(ua‘ + '03:)1't

pAl-rw’cos(wt—¢ ¢)— 2(%,— x 8,) %%-(%>2x1. (4.6)

89

and equation (4.2) becomes

1 1
Evaxza: + Iﬂcvzxxzt " EA[(1}; + 51):)‘03 lx — Al‘cl (”1: + 51):)11’1: 1:: =

2 2
= —pA[—rw2 sin (wt - 4)) + gig-a: + 9&7; — (g—va] . (4.7)
Integrating equation (4.1) from 0 to x, we obtain
P(a:,t) = P(0,t) + pA / axdz (4.8)
o

where P(O,t) stands for the force applied to the end to which the end mass m4 attached.
If the axial load acting along the connecting rod is known, then this equation is a partial
differential equation describing the transverse displacement v(:r,t). Using the free body
diagram as shown in Figure 2.4 and summing forces along the x-direction, we can determine

the time-varying axial force P(:c, t) to be of the form

 

P(:I:,t) = pA/L azdz+m45+(MA+MB)ta2¢+/I,5

17(1) — 177.42 - ﬂsz
cos¢

ta

L
21¢ [0 (aye: — axv)d:c +

 

 

+pA (4.9)

where the overdot represents the time derivatives, F (t) is the active piston force, Z rep-
resents the piston acceleration from the rigid body motion of the mechanism, 0,2 is the
friction force acting on the slider due to rigid body motion, 6 is the foreshortening due the
bending deﬂection of the rod, and the p,5 term represents the slider friction due to elastic
deformation.

In order to determine the piston acceleration Z, let us digress and consider the rigid

body kinematic analysis. From Figure 2.4, in order to retain the motion of the piston end

90

along the X-direction, we must have
Lsin(—¢) = rsin wt (4.10)
which can be written in the following dimensionless form:
sin¢ = —§sin wt (4.11)

where 5 represents the ratio of the effective radius of the crank shaft to the length of the
connecting rod and is speciﬁed as the length ratio. With this geometrical relation, the

piston displacement function can be expressed as
Z(t) = rcos Qt + LcosqS = L(£cos Qt + cos 03) . (4.12)
The foreshortening due to the bending deﬂection of the connecting rod can be approximated

L 612 1 L (912
= _.__2 _ z- —2 , .
a /o‘/1+(6$)dx L 21) (63”.; (413)

This approximation of the foreshortening effect is based on the small transversal deforma-

by

tion assumption. Hence, we keep only the ﬁrst order term of the foreshortening effect.

Substituting equation (4.6) into equation (4.7), we obtain the following equation

Elvzrxx + Mcvaxxzt - [1307:9003 la: =

, 62¢ 62v 005
pA[ rw Sln (wt 45) + 3t2 :1: + 6t2 (9t ) v] (4.14)

where P(a:, t) is given by equation (4.9). Equation (4.14) describes the transverse deﬂection

91

of the connecting rod.
We now consider the boundary conditions for the connecting rod. Since the crank shaft
is assumed to be perfectly rigid and there is no relative displacement of the connection

between the connecting rod and the crank shaft, we have

v(0,t) = 0 . (4.15)

Also, at :c = L, the piston motion is constrained to move along the X direction. Thus, we
have

v(L,t) = u(L,t) tan(—¢) . (4.16)

Again, equation (4.5) implies that the axial displacement u(a:,t) is much smaller than the
transverse one. It is shown in [9] that this boundary condition can be approximated by
v(L,t) = 0. Due to the presence of the ﬁctional moments on points A and B due to

bearings, we have the following boundary conditions:

EIvm(0,t) —MA = pb[w — q) — vzt(0,t)] , (4.17)

EIv,,(L,t) —M3 = III”) — v,,,(L,t)] , (4.18)

where III, represents the friction coefﬁcient such that the moment applied by the bearings
is the product of III, and the relative rotational speed. We assume the value of the friction
coefﬁcient [15 is very small compared to the length ratio so that these reaction moments can
be neglected for the ﬁrst order approximation. Therefore, with the ﬁrst order approxima-

tion, the reaction moments acting on points A and B are zero. Based on these observations

92

and assumptions, the solution v(:c,t) can be approximated as

v(:c,t) = $540 sin (32—3) (4.19)
where 27,,(t) represents the amplitude of the displacement at the midpoint of the beam, and
sin a? are the natural modes of transverse vibrations of a uniform beam with pinned ends.
Previous work[49] has shown that higher order modes have insigniﬁcant inﬂuence on the
transverse response of the connecting rod for the region in which the model is valid. Hence,
we assume that the solution of v(:I:, t) can be approximated by a single mode (1)-1(t) sin (ELI—D.
Substituting this assumed solution into equation (4.14) and projecting the equations onto

the ﬁrst mode, we obtain the following equation:

2pALd; 4pAr sin (wt — ¢)w2 + EIII4v 5pA<iSzv + pA1r2052v

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1r 1r L4 4 3
+ Izmvdi sin (1) wzpbvd tan (1) pA1r2v$ tan (1) «2p, rwv sin wt
cos 43L L3 3 L2 cos d)
+ w’pbwv tan 43 + pArwzwzv cos (wt — <15) pAr‘Irzwzv tan <1) sin (wt — 45)
L3 2L 2L
+ «202mm «2m4div sin d) «2m4rvw2 cos wt + «2vF(t)
L L cos ()5 L2 cos 45 L cos 43
pA1rgf> 2pA1'Irw2 tan ¢ cos (wt — 45) 2 pA‘Irzciiv3 tan d)
+ ( L + L2 )” + L2
1r‘I pct: 2pA1r§iSviJ pArrzgis tan 4313122 w‘mwﬁz [1, 4 2 ,
+ L4 +——L + 1.2 2L3 tm“”
prv tan 05 141714122 .. _
+ (pA + L 2L3 )1) — 0 (4.20)

which is a single nonlinear ordinary differential equation describing the ﬂexural vibration
associated with the connecting rod of a slider-crank mechanism under the assumptions
given above. In the following analyses, we only consider the cases in which there is no

external force acting on the piston, i.e., F (t) = 0 in equation (4.20). We note that these

93

simpliﬁcations (equations [4.15] to [4.18]) are essential in truncating equations (4.14) with
the linear mode shape function (equation [4.19]) and obtaining equation (4.20). With the
same dimensionless parameters as those described in section 2.4, equation (4.20) can be

written into the following dimensionless form:

(1+ 7rvtan¢ +g1r4v2)ii + 2m?) + %( cf) — 2592 sin (Qt — ¢))

.. 1 . 1r2 5 1 cosflt
2 - 2 — 2 __ _ 2 2—

l
+ v[ 2p3(Q — 2(1'5)7r2 tan 4) + 2114(43 — (2)7r2 tan (15]
+ v2[1r<;3 + 2x502 tan 43 cos (Qt — d>)] + 21rdn'2v tan d)

+ v3( giditangb) + (1;)sz12 + v21}(1r2d>tan¢ + p47r4) = 0 (4.21)

where all the overdots represent the derivatives with respect to time, 1'51 is replaced by v and
all the overbars have been dropped for notational simpliﬁcation. Equation(4.21) is a second
order differential equation describing the ﬂexural response associated with the connecting
rod. We note that it contains the following features: (1) nonlinear inertial terms, (2) exter-
nal as well as parametric forces arising from inertial forces, (3) dissipation effects, and (4)
time-dependent quadratic and cubic nonlinearities. Despite of its complexities, each term of
equation (4.21) has its own physical source. All the terms involving the parameter S , except
(gr’vzii) and (g-r‘mﬂ), which represent the foreshortening effect, arise because of the iner-
tial force associated with the slider mass m4. The forcing term, [ %(¢— 2512 sin (Qt — ¢)) ],
arises because of the action of the transverse acceleration component (1,, on the connecting
rod. The following terms arise because of the effects of bending moments caused by the
actions of as and “v: xvii tan ¢, grzv tan qt, v43”, gﬂzw’vﬁ, 21(52sz tan¢cos(ﬂt - ¢),

31,31,3(5 tan¢ and w’vzbé tan (b. All the terms involving the parameters [13 and p4 arise from

94

the actions of the bearing friction and friction force on slider mass, respectively.
Now, let us reconsider equation (4.11). Since in most of the applications, the length
ratio is smaller than one, we expand equation (4.11) in terms of the length ratio £ to obtain

the following asymptotic series:

¢ = sin—1(sith)

 

(fsin Qt)3 _ (Esin Qt)5 +

= —(£srn Qt) - 6 120

353

3 5
a: = -( E + T + ...) sin Qt — ( 34— + ..) sin3Qt — ( % + ...)sin 5Qt + ...... (4.22)

This equation indicates that the excitation provided to the connecting rod is a superposition
of harmonic inputs at frequencies which are multiples the crank rotation speed. Note that

the force amplitudes associated with higher harmonics are of small magnitude.

4.2 Application of the Method of Multiple Scales

Before applying MMS (Method of Multiple Scales) to study equation (4.21), let us ﬁrst
consider a solution to the linearized model of equation (4.21). To achieve this end, we
expand the original equation (4.21), with the help of equations (4.11) and (4.22), and retain

all the ﬁrst order terms in f, v and its time derivatives. This results in the equation
2
v(t) + 2p21}(t) + v(t) = ;§Q2 sin Qt (4.23)

which represents a linear oscillator subjected to an external excitation. The response of this
equation is given as

v(t) = Xsin(Qt — (,9) (4.24)

95

 

 

where
X = (3) 592 (4.25)
W ,/(1 — m)2 + 4ng2
and
,0 = tan’l( 125232) . (4.26)

Equation (4.24) represents the response curve of the linearized system for small length
ratios. From equation (4.25), we see that, to ﬁrst order, the amplitude of the steady-state
response is proportional to the length ratio E as well as the square of the frequency ratio
Q. However, this estimate may not be sufficient to capture the true response under certain
conditions. In the current section, we use the Method of Multiple Scales[29] to locate these
conditions, while we will improve our approximation in the subsequent sections for each
individual condition.

To apply MMS, we need to introduce a set of new independent time variables Tn and, in
addition, reorder some parameters, such as the length ratio E and the damping parameters
#2, p3 and [14. The new independent time variables Tn are introduced according to Tn = c" t.
It follows that the derivatives with respect to t become expansions in terms of the partial
derivatives with respect to T,, according to equations (3.4) and (3.5). Moreover, we assume
that the length ratio 5 can be ordered by 5 = 651. Since the amplitude of the response is
proportional to the length ratio £, we assume that the solution v(t) can be represented by

an expansion taking the following form

v(t) : (1)1(T0 ,T1 ,T2) + £2 02(T0 ,T1,T2)+ 63 03(T0 , T1,T2)+ . (4.27)

The damping parameters [12, p3 and #4 are treated in the same way as described in chapter 3.

96

In other words, the ordering of #3 is ﬁxed to of order 0(62), the ordering of #4 is ﬁxed to
be of order 0(1), and p; is reordered according to the order of a, the detuning parameter.
Substituting equations (3.1), (3.4), (3.5), (3.8), (3.9) and (3.10) into equation (4.21),

expanding and equating the coefficients of the like-power terms we obtain the following

equations:
Order 6
2 2 2 -
Do v1 + 121 = (;)£1 Q snn QTo , (4.28)
Order 62
Do2 122 + 222 = —2Do D1 121 - (S + %)£11r2 Q2221 cos QTo + (i) Q2 6? sin 2QT0 ,(4.29)
Order 63
D0203 + 123 = —200D1’02 - 2[122Do’01 - D1201 - 2D0D2v1 — 11'2(S + %)£1Q2v2 COS 0T0
_ f: 2 _ § 2 2 2 l 2 2 - 2
(3 +1 S 4)§1v1Q cosQTo +(3 +S)£1Q 121 stTo

4
— 21r2p4oQ£1v1 SlIl 0T0 + 2£1Q1r(Dov1)v1 COS 0T0 — :2-S(D0‘Ul )2'01

4
1r .
_ _2_S(D02v1)vf + n5,[Do(Dov;~’)] sm 9T0 — p407r4(Dov,)vf . (4.30)

In obtaining the general solution of 121(To ,T1 ,T2) from equation (4.28), we need to distin-
guish between the following special cases: (1) Q is near to unity and (2) Q is away from
unity. The case in which Q is near to one corresponds to the primary resonance. When
this occurs, the linear estimate (equations [4.24]) fails to provide a good prediction of the

response. Section 4.4 contains a complete analysis of the primary resonance. At the present

97

time, let us assume that Q is away from one and continue our analysis. Under this condi-
tion, the general solution of v1(To ,T1 ,T2) is given as the combination of the homogeneous

and particular solutions,
01(T0 , T1 , T2) = A1(T1 , T2)e:cp(j To) + A4 sin 9 To + c.c. (4.31)

where A1 is an unknown complex function which will be determined later, A, is given as

2 £1Q2

#1755 (43?)

4=(

and c.c. stands for the complex conjugate terms of the preceding terms on the right hand
side of equation (4.31). Note that, according to the linear theory, the homogeneous part of
solution (4.31), i.e. Alexp(jTo) term, will decay to zero due the presence of the damping
parameters. Thus, the homogeneous solution is not included in the steady- state response
of the linear system (equation [4.24]). At the present moment, we include the homoge-
neous solution in this general solution and proceed to ﬁnd the conditions in which this
homogeneous solution will not decay to zero. Substituting the solution v1(To ,T1 ,T2) into

equation (4.29), we obtain

. . 1 . .
130202 + 02 = "2J(D1 A1)€$P(JT0) + (;)§f92€3P(2JQTo - 1%)
— (S + §)£192«2’-‘lexp(2jmo — 2'1)
4 2
1 2 2A] . .
- (5 + ‘2')619 1r 76931’097‘0 + 1T0)

1 A . .
— (5 + §)£1Q21r27lezp(JQTo — 3T0) + c.c. . (4.33)

In analyzing the particular solution of equation (4.33), there are three cases which need to

98

be considered separately: (1) Q is near to %, (2) Q is near to 2 and (3) Q is neither near

to % nor to 2. The ﬁrst case corresponds to a superharmonic resonance, while the second

one corresponds to a subharmonic resonance, or the principle parametric resonance. Both

of these cases will be studied in detail and improved approximations will be provided later
1

on. For the present, let us assume that Q is away from 2 and 2, and continue our analysis.

When Q is away from % and 2, we must have

D1A1(T1 ,Tz) 1' 0 (4.34)

in order to remove the secular term from the particular solution of v2. This implies that
the unknown amplitude function A1(T1 ,T2) must be independent of the time variable T1.
As a consequence, all the higher order solutions are assumed to be independent of the
time variable T1. Thus, all the derivatives with respect to time variable T1 are zero. The

particular solution of 122(To ,T2) is then given as

 

 

 

1 1’92 3 1 2 2 . .1r
02 1_ 402[ 1r '— (Z + §)£11f Q A1]exp(2]QTo - J5)
1 S 1 2 2 , .
1_(1+ (2)2(5 + Z)£IQ 7" A1€$p(]QTO +JT0)
1 s 1 - , _
1 _ (1 _ Q)2(§ + Z)£192"2A1€$P(J9T0 - 1T0) + c.c.. (4.35)

Substituting 121(To , T2) and v2(To , T2) into equation(4.30), we obtain

Do2v3 + v3 = —2jp2,A1e:rp(jTo) — jp4o1r4Af/llexp(jTo) — 2j(DgA1)e:cp(jTo)
+ W‘sAiz‘fiezPUTO) + f7€i41€$P(jTo) + jf8€i€$P(3j9To)

+ 2j1rnglﬁl’ezpuntro — 23711,) — j(%)w4A,A3exp(jT0)

99

4 Q _ . .
+ g—Ai’nmoezpwjmo) + uiow‘Am — §)Ai6$P(JQTo — 22%)
+ N.S.T.+c.c. (4.36)

where N .S.T. represents those terms which will not produce secular term in the particular
solution of v3. In analyzing the particular solution of equation (4.36), there are three cases
which need to be considered separately: (1) Q is near to 3, (2) Q is near to %, and (3) Q
is away from 3 and %. The ﬁrst case corresponds to a subharmonic resonance, while the
second case corresponds to a superharmonic resonance. These two cases will be investigated
in sections 4.6 and 4.7, respectively, in order to obtain better approximations.

At this point, we pause and briefly consider some of the general features of this analysis
procedure. By using the assumed solution sequence, the equation describing the transverse
vibration of the connecting rod can be transformed into a sequence of ordinary differential
equations which can be solved quite easily, since they are linearized by the nature of pertur-
bation expansion. In solving these sequential equations, several resonant conditions arise.
The primary resonance appears ﬁrst, and then the principle parametric resonance. After
these two resonances, some secondary resonances of subharmonic and superharmonic types
arise. When these occur, the linear approximation is insufficient to provide a good estimate
of the response of the system.

For simplicity, we follow the same analysis pattern as described in the last chapter. The
principal parametric resonance is examined ﬁrst in section 3.2. The primary resonance is
investigated in section 3.3. After these two resonances, secondary resonances are examined

case by case.

100

4.3 Principal Parametric Resonance (Q m 2)

Recalling that in analyzing the particular solution of equation (4.33) in section 4.2, when
the frequency ratio Q is near to 2, a principle parametric resonance takes place. To describe

the dynamics when the frequency ratio is close to 2, we express Q as

Q = 2 + 2c 01 (4.37)

where 01 is the detuning parameter. The damping parameter #4 is reordered by #4 = [440.
Moreover, the damping parameter #2 is rescaled according to M 2 (#2, so that it will show
up, together with detuning parameter 01, in the ﬁnal resonant condition. After carrying

out the same procedure as that described in the last section, equation (4.29) reduces to

D021); + 122 = —2j(DlA1)exp(jTo) — 2.71121 Alezp(jT0)

S 1 ‘ - . .
(‘2' + z)€1927f2A16$P(JT1 + 2.701%)

_ 5,9,2p404,exp(2ja,rl _ jg) + N.S.T. + c.c. (4.38)

where N .S .T. represent those terms which will not produce the secular terms in the partic-

ular solution of v2(To ,T1 ,T2 ). In order to eliminate the secular terms, the following must

hold

. . S 1 - .
- 21(D1 A1) — 2Jp2,A1 — 4(5 + Z)El7r2A1ea:p(2_70'1To)

— . .1r
— 212p40£1Alexp(210‘1T1 — 35) = 0 . (4.39)

Expanding equation (4.39) with A1 = (%)ezp(j\II), and separating the resultant equation

101

into real and imaginary parts, we obtain

a = —u2 a — 2a§1A4 sin 2<I>1 + a7r2y4 {1 cos 24>] , 4.40
l 0

a<I>'1 = (no — 2a£1A4 cos 2<I>1 — a7r2p40£1 sin 2<I>1 (4.41)
where {>1 = alTl — \II and A4 are deﬁned by
7r2
A4 = (25' + 1)(-4—) (4.42)

and the primes indicate derivatives with respect to T1. The steady state response conditions

are obtained by letting a’ = 0 and <I>’ = 0, which yield

p2,a = —2a{1A4sin 2<I>1 + a7r2£1p4ocos 2<I>1 , (4.43)

ala = 2a£1A4cos 2(121 + a1261u4osin 2<I>1 . (4.44)
Squaring these equations, and adding them together, we obtain
[of + #3, - (2511302 - ("2611140)2102 = 0 - (4-45)

Equation (4.45) is an equation for the amplitude of the response as a function of the
detuning parameter, the damping parameters and the length ratio. From this equation,
we see that the trivial solution a = 0 is the unique steady state response for the cur-
rent resonant condition. To determine the stability of this solution, we substitute A1 2
(83 + jBI)ea:p(j01T1 + 7T1) into equation (4.39), in which 33 and B; are two constants,

and separate the resultant equations into real and imaginary parts to obtain two equations.

102

 

    
 

 

 

0.03 I I I I I I I
0.025 "- "
0.02 '_ ._(
E 0 015 _ (unstable region) _
0.01 I’ S = 1.00 "
#2 = 0.02
_ #3 = 0-00 ..
0.005 #4 = 0.00
0 l l l I 1 I l

 

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio Q

Figure 4.1: Frequency response equation for the principal parametric resonance (equa-
tion [4.45])

We then solve the nontrivial constant solutions of BR and B; from these two equations.

From this, we determine that when

0? + #3, < {H 4A} + ”4143, ) (4.46)

this trivial solution become unstable. It is clear that of + p31 = {ﬂ 4A2 + «4&0 ) represents
the transition curve which is the solid curve shown in Figure 4.1. Along this curve, the
eigenvalues associated with the responses are either +1 or -1. In the unstable region, any
disturbance, no matter how small, applied to the steady-state response results an unbounded
growth in the response amplitude of the linear model. However, as the amplitude of the
response becomes large, the nonlinear terms can no longer be neglected. In the remaining
part of this section, we extend our analysis to investigate these nonlinear effects. In stable

regions, equation (4.24) can be used to predict the response, since the trivial response is

103

stable and will remain zero. Thus, the steady state response can be approximated by

v(t) = £v1(T0,T1 ,T2 ) + 0(62) = 6A] sin QTo + 0(62) . (4.47)

When the response is unstable, any small disturbance applied to the trivial response will
grow unbounded, and then dominate the response curve. This is an example of parametric
resonance.

In order to capture the effects of nonlinearities, it is necessary to reorder either the solu-
tion or the parameters so that the nonlinearities will be included in the equation describing
the resonant condition. For the present problem, the amplitude of the response is shown to
be proportional to the length ratio (equation [4.24] in section 4.2). Therefore, a reordering
of the length ratio is equivalent to the reordering of the solution. To carry out the nonlinear
analysis, we reorder the length ratio 5 according to £ = 6252. Expanding equation (4.21)

with this new length ratio, and equating the coefficients of the like-power terms in e, we

obtain
Order 6
D02 ’01 + 01 = 0 , (4.48)
Order 62
D02 122 + 02 = -2Do D1 121 + (%)£2stin 9T0 , (4.49)
Order 63

D02 ‘03 + 1);; = —2Do ngl — 2ﬂ22 D001 — W4S(D02’U:13) — 2y40 #262901. sin ”To

1
— 012121 — 2D0D1v2 — «2(5 + §)£2Q2 v1 cos QTo — «4p4o(Dov1)v¥ . (4.50)

104

Equation (4.48) admits the solution
v1(To ,T1 ,T2) = A(T1 ,T2)e:cp(j To) + c.c. . (4.51)
Substituting this solution into equation (4.49), we obtain
Do2 122 + 122 = -2j(D1 A1)e:rp(jTo ) + (3.-)6ng sin QTo + c.c. . (4.52)

In analyzing the solution of this equation, we need to distinguish between these two cases:
(1) Q is near to one and (2) Q is away from one. The ﬁrst case is referred to as the primary
resonance and will be investigated in next section. At this moment, let us consider case (2).
When Q is away from one, we have D1 A1 = 0 which implies the independence of A1 to the
time variable T1. Therefore, all the higher order solutions are assumed to be independent

of T1. With this assumption, equation (4.50) reduces to

002 Us + 123 = -2jM2241€$P(jT0) - 2j(DlA1)€$P(jTo) - 5114407r 4Ali/ii61131437110)
1 A . . _ . .
— (S + §)£2Q21r2(—21)e:cp(]QTo — 1T0) — 2p407r2£2Alexp(JQTo — 312:)

+ w‘SAfzfleszTo) + N.S.T. + c.c. (4.53)

which describes the third order term in the ﬂexural response associated with the connecting

rod. To describe the nearness of Q to 2, we express Q as

a = 2 + 26202 . (4.54)

105

Hence, we must have

-2j[1.22A1 - 2j(D2 A1) — 462A414-1827p(2j0’2T2) + W4SA¥41

- . .1r . _
- 2p401r252Alexp(2]a'2T2 — .75) — Jp4o7f4A¥A1 = 0 (4.55)

in order to remove the secular terms from the particular solution of 121. Substituting
A1 = (%)ezp( j W) into equation (4.55) and separating the resultant equation into real and

imaginary parts, we obtain

3
a’ = —p2,a — maﬁa? — 262A” sin 2<I>2 + £21r2u40a cos 2<I>2 , (4.56)
a<I>’2 = 02a — 2£2A4a cos 2<I>2 — £21r2p40a sin 2(1); + A5a3 (4.57)

where <I>2 = ang - \II and A5 is deﬁned by

A5 = lﬁ‘S (4.58)

and the primes represent the derivatives with respect to the time scale T2. The steady state

conditions for these equations are given as

3

#220 + [1407(4‘95' = —2€20A4 sin 2T2 + £27f2ﬂ4oa COS 2T2 , (4.59)
02a + A5a3 = 2fgaA4 cos 2<I>2 + 52731140“ sin 2<I>2 . (4.60)

The frequency response equation takes the following form

2
02l (#22 + Morlaglz + (02 + A50%?)2 - (252402 - (W2€2#4o)2l = 0 - (4-61)

106

From this equation, we solve for the nontrivial solution of a2 and obtain

(62)2 = (in —l + \/12 — km 1 , (4.62)
(as)? = (71,-) —I — W2 - km 1, (4.63)
where
k = [(94.44 12 + A: . (4.64)
l = (%)#4o#2,7r4 + A502 1 (4-55)
m = #32 + 0.3—(252134)? — (W2€2#4o)2 - (4-66)

Figure 4.2 shows an example for these frequency response curves. From equations (4.62)

and (4.63), we know that a2 and a3 exist only when

I2 — km > 0. (4.67)

This implies that the magnitude of the forcing term must be large enough in order to

produce a sustained nontrivial steady-state response. Furthermore, we need

 

02 < ./(2:2A.)2 + («264.52 — #4. (4.68)

for the existence of a2, and

 

02 < —\/(242A4)2 + («242mm — #4, (4.69)

for the existence of a3.

107

a, = magnitude of homogeneous solution

 

 

 

 

 

0.25 I I I I I
0.2 :\\\\ 4
0.15 T
as
0.1 a
0.05 ._
0
1.6 1.7 1.8 1.9 2 2.1 2.2

Frequency ratio Q

Figure 4.2: Frequency response equation for the principal parametric resonance (equa-
tion [4.61])

In order to determine the stability of these steady-state responses, we compute the

Jacobian matrix associated with equations (4.56) and (4.57), which takes the form,

—(5§‘1)7r2a2 —2(02 + A502)a ( )
4.70
2A5a —2;422 — (i‘iﬂyr'za2

The eigenvalues associated with this matrix are

 

41.2 = —(/42., + ”—iﬂwzaz) :t \/(”—8431r2a2)(p22 + Eg—Wrza?) + A5a2(02 + A5a2) . (4.71)

Thus, the stability of each individual steady-state response is decided by the sign of the
term inside the radical sign. When this quantity is negative, we have a stable steady state

response. Otherwise, it is unstable. Thus, the following conclusion is obtained:

 

(1) when 02 > \/(2£2A4)2 + ("2521‘4ol2 — #32, only the trivial solution is possible, and

it is stable.

108

 

(2) when 02 < I \/ (2521\4)2 + («252,440 )2 — pg: I, the trivial solution becomes unstable,

while a2 exists and is stable, and a3 does not exist.

 

(3)when 02 < —\/(2£2A4)2 + (1r252p40)2 - ”32, a3 exists and is unstable, and a2 and al

are stable.

Remark: Before leaving this section, a few points need to be made. First, let us compare
linear resonant equations (equations [4.40] and [4.41]) with the nonlinear version (equa-
tions [4.56] and [4.57]). Basically, they coincide with each other in the linear part. The dif-
ference in equations describing the variation of the amplitude is the nonlinear term (@1440 a3.
We note that ($114003 arises because of the slider friction introduced by elastic deforma-
tion. The difference in equations describing the variation of phase angle is the A5a3 term in
equation (4.57). Next, let us compare the frequency response equations obtained from the
linear and nonlinear analyses. Again, they coincide with each other in the linear part. The
inﬂuence of the nonlinearity on the response can be determined by a direct comparison. In
absence of the nonlinearity, the response is unstable when the parameters located inside the
unstable regions. According to linear theory, the amplitude of the response grows without
limit. Due to the nonlinear effects, this increasing response amplitude will be accompanied
with an increase of the resistance force and a change of the phase angle. The resistance
force arises because of the foreshortening introduced by elastic deformation. As the am-
plitude of the response increases, this resistance force whose magnitude is proportional to
the foreshortening also increases. The amplitude of response reaches its equilibrium state
when this resistance compensates the effect of the slider friction introduced by the rigid
body motion. We now consider the effect of the nonlinear terms in the equation describing

the variation of phase angle. Due the effects of nonlinearity, the increase of the amplitude

109

of response will accompany with a change of the phase angle. This implies a change in the
rate at which energy is pumped into the system. When the energy put into the system is

balanced by the energy dissipated by the system, the system reaches its steady state.

4.4 Primary Resonance (Q m 1)

From equations (4.31) and (4.32) in section 4.2, it is clear that the amplitude of the
particular solution approaches inﬁnity as Q approaches one. This phenomenon is referred
to as the primary resonance. Let us reconsider equation (4.49), and now assume that Q is

near to one. To make the analysis more clear, let us express Q as
Q = 1 + £01 . (4.72)

In addition, the damping parameter #2 is rescaled according to equation (3.49). With these

arrangements, equation (4.49) becomes
2 .
D0202 + ’02 2’ —2DoDlv1 + (;)£202 S111 9T0 — 2p21 D001 (4.73)

by following the same procedure as that described in section 4.3. To retain the periodicity

of the solution, the following must hold
. . . ,1!‘
— 23(0141) _ 23,421.41 — (%)Qzexp(]alTo _ )5) = o (4.74)

after substituting v1 = Alexp(jTo) into equation (4.73). Expanding the last equation with

A1 = (§)ezp( j '1!) and separating the the resultant equation into the real and imaginary

110

parts, we obtain

a’ = —p2,a —(§1r3)cos‘1>1 , (4.75)

a<I>'1 = 01a+(%)sin<l>1 (4.76)

where in is given in equation (3.55). After solving these equations, solution v(t) of the

original equation is approximated by
v(t) = (121 + 0(62) 2 ca cos(QT0 — <I>) + 0(8) (4.77)

where the magnitude of the steady state response a, is described by

a, = 54—1—— (4.78)

I (M. + a?

and the steady state value of the phase angle (D, is given by

tan <1>, = (ﬂ . (4.79)
I121

It is interesting, and not surprising, to ﬁnd that equation (4.77) is the ﬁrst order ap-
proximation of equation (4.24) which is the response for linearized model when Q is near

to one. The local equivalence between equations (4.77) and (4.24) can be shown as below:

2 £02 2 £262

) .4 (- = ‘5‘
7r \/(1 — Q2)? + 4p§Q2 1r \/4€20§ + 462p§1

2 2' = 6as a
\/01 +le

i (4.80)

111

while angles w and 45 are related by

2ﬂgﬂ 269
t ‘p 1 — Q2 -2£01 co (1)1 ( )

 

Hence, equations (4.77) and (4.24) are within 0(62) in an e-neighborhood of Q = 1. This
shows that the reordering relation (3.60) can not help us in including the nonlinearity
in our analysis. Based on this fact, let us rescale the length ratio 5 according to 5 =
(3&3. Substituting this reordering relation into equation (4.21), expanding and equating the

coefﬁcients of the like-power terms, we obtain

Order 6
0021)] + ’01 = 0 , (4.82)

Order 62
D0202 + 02 = —2Do DI v1 , (4.83)

Order 63

002123 + 123 = -QDo Dz vi - 2D001122 - 012111 - 2H22-DO vi
2

— «45(00272?) + (:r-){3Q2 sin (QTo) - p4o7r4vi"(Dov1) . (4.84)

The solution of equation (4.82) is given by equation (4.51). Also, from equation (4.83), we
know that Al must be independent of the time scale T1 in order to eliminate the secular

term from the particular solution of 122. With this, equation (4.84) can be reduced to

00203 + ”3 = -2j(DzA1)€$P(jT0) - 2jﬂ22A1€$P(jTo)

1 . . . - .
+ (;)£39’ewp(JﬂTo — 24/2) — JMM‘AiAiexPUTo)

112

+ 174A¥41ezp( jTo) + N.S.T. + c.c. . (4.85)
Therefore, we must have
. . . - 1 , , -
- 2J(D2AI) — 23I122A1 '7 .7/1'407r4AfA1 + (;)5392€93P(J02T2 - JW/Q) + W4AfAl = 0 (4-85)

in order to eliminate the secular terms from the particular solution of 123. Expanding
equation (4.86) with A1 = (%)ezp(j\lf), and separating the resultant equation into real and

imaginary parts, we obtain

I _ 403 £3 Q
a — —p22a - p403 —— — —cos 2 , (4.87)
8 1r
a<I>'2 = 02a + A5a3 + s—f—sin <I>2 (4.88)

where <I>2 is given in equation (3.71), A5 is given in equation (4.58) and the frequency ratio
Q is expressed as

Q = 1 + 6202 . (4.89)

The steady-state response can be obtained by letting a’ = ’2 = 0 in equations (4.87) and

(4.88), which gives

3
p2,a + #40240? = —£—:cos {>2 , (4.90)
02a + A5a3 = —%—sin <I>2 . (4.91)

The frequency response equation is then given by

a2[( 4“_22 22 _ Q2
m.+u4.« 8) +(02+A50) 1—(1r) . (4.92)

113

and the frequency response curve as

 

= —A5 02 :l: E)? - ([122 + ”40146;? - (4°93)

Figure 4.3 provides an example of the frequency response equation associated with the
primary resonance. To ﬁnd the peak amplitude tip of the steady-state response, we need to

solve the following equation:

(6; 3)2 - “pl/‘22 +(- 8)/‘4o7r 0,2512: 0 (494)

which is a cubic equation in terms of a3. Moreover, the steady-state response achieves its
maximum amplitude at frequency Q = 1 + 6202? where

0'2p = —A5a: (4.95)

where (1,, represents the solution of equation (4.94). The negative sign in the last equation
implies that the response curve reaches this peak amplitude at a frequency of less than one.
Moreover, over a region of Q values, there exist more than one steady—state response (Fig-
ure 4.3). The frequency response curve bends to the left, corresponding to a softening type
of nonlinearity. The stability of these steady-state responses can be decided by computing
the eigenvalues of the Jacobian matrix associated with equations (4.87) and (4.88) which
takes the form
—m. - (ﬂ‘gww —(a2 + 45.2).

(4.96)
(02 + 3A5a2)/a -#22 - (97‘3“)??402

114

From its eigenvalues, we determine that when

1 3
[I122 + (‘8')11407'402 ll #22 + (§)#4o7r402 ] + (02 + Asazxaz + 3A502) < 0 (4-97)

the steady-state responses are unstable. Otherwise, they are stable. The solution v(t) can

approximated by
v(t) = 6v1(To ,T1 ,T2 ) + 0(62) = ca cos (9T0 —- Q) + 0(62) (4.98)

where a and <I> are described by equations (4.87) and (4.88). The steady state value of the

phase angle <I> is given by
02 + A502

<I> = tan’1( .
8 ”'22 + %#4o7r4a2

 

(4.99)

a, = magnitude of v1(t)
0.2 r I T r I

 

0.15

 

 

 

 

0.05
0 ‘ 1 1 1 1 1
0.6 0.7 0.8 0.9 1 1.1 1.2

frequency ratio 0

Figure 4.3: Frequency response equation for the primary resonance (equation [4.92])

115

Remark: Before leaving this section, a few points need to be mentioned. First, let us
neglect the effect of p40 and then compare the ﬁnal equations describing the resonant con-
ditions between linear and nonlinear analyses. We note that the amplitude function a is
described by equations of the same type (equations [4.87] and [4.75]), while the equations
describing the phase angles (1)1 and (P2 (equations [4.76] and [488]) are coincident in the
linear part. The only difference between these two analyses is the existence of the nonlinear
term A5a2 in the equation describing {>2 (equation [4.88]). This implies that the nonlinear
affects the response of the system by changing the phase angle Q2. Therefore, there is a
change in the rate at which energy is pumped into the system accompanying this phase
change as a varies. The system achieves its steady state amplitude when the energy dissi-
pated by the system is balanced by the energy pumped into the system. This implies that
the magnitude of the steady state response may be remain unchanged but a considerable
shift of the phase angle of the steady state response can be expected. We now compare
the maximum amplitude of the response obtained from the linear and nonlinear analyses.
Again, the effect of slider friction is neglected. A prudent observation shows that these two
values are exactly the same. This implies that linear theory is sufﬁcient to provide a good
approximation of the amplitude of the steady—state response. However, as we compare the
frequency at which the steady state response of the system reaches its maximum ampli-
tude, there exists a considerable difference between the results from linear and nonlinear
analyses. Linear analysis indicates that this peak amplitude can be observed when 9 = 1.
The nonlinear analysis show that the response achieves this amplitude when 9 is at the
value given by equations (4.92) and (4.93). This indicates that linear theory may offer a
poor prediction of the frequency near which large amplitudes occur. Finally, let us consider

the effect of #40 which represents the effect of slider friction. According the analysis, #40

116

has a favorable effect in reducing the amplitude of the response. This is caused by the
foreshortening accompanying with the transverse deformation of the connecting rod. The
existence of the slider friction will resist further deformation and hence has a tendency in

reducing the amplitude of the response.

4.5 Superharmonic Resonance (Q m %)

In analyzing the particular solution of v; in section 4.2, we assumed that Q is away from
2 and -§— and proceeded to analyze the solution of 223. When (2 is near to %, a superharmonic

resonance occurs. For (2 near to %, the frequency ratio 9 is expressed as
1
Q 2 5(14- 601). (4.100)

At the same time, the damping parameter p; is rescaled according to #2 = mg, so that it will
appear together with detuning parameter 0; in the final resonant condition. Consequently,

equation (4.29) reduces to

D02 v2 + '02 = —2j(D1 A1)ea:p(jTo) — 2jp21Alezp(jTo)
A .
+ marzﬁﬂéeszJQTo)
1 2 2 A4 . ,7l'
- (5 + §)€19 7' (7)83142197‘0 - .75)

+ iﬁfﬂzezpﬂjﬂﬂ) — j%) + N.S.T. + c.c. . (4.101)

The elimination of the secular terms of the particular solution of the last equation requires

117

that
”40 2 - - - {f - '7’ _
_6_£l W3$P(JUIT1) “ 21(D1 A1) - 2J#21A1 + T2—1r(3 - A4)exp(]alT1 — 15) _ 0 (4.102)

where 01 is given in equation (4.100) and A4 is deﬁned by equation (4.42). Substituting
A1 = §ezp( j \I!) into equation (4.102), expanding and separating the resultant equation into

real and imaginary parts, we obtain

2 2
a’ = —p21 a + E—‘iiél—E sin <I>1 - 1%?(3 — A4) cos <I>1 , (4.103)

2 2
a<I>’l = 01 a + $06—51: cos (1)] + 1%“ - A4) sin in (4.104)

where in is given in equation (3.55). Therefore, when S2 is near to %, the response equation

takes the form

v(t) = €‘Ul(To ,T1 ,T2 ) + 0(62)

= 6[ A4 sin 9T0 + a cos (2QT0 — <I>)] + 0(62) (4.105)

where A; is given in equation (4.32), a and (I) are given by equations (4.103) and (4.104),

respectively. The steady state value of phase angle <I> is given as

-1 01(3 - A4) - 2W2I121ﬂ4o

 

(I), = tan 4.106
2772011140 + (3 - A0112: ( )
and the amplitude of the steady state response a, is described by
1r 3 - A
(9%. + aha: = €1‘[(E‘—g-)2 + (Er—4?]. (4.107)

118

Equation (4.107) represents the frequency response equation of this superharmonic reso-
nance. Figure 4.4 provide an example of the frequency response equation associated with

this superharmonic resonance. Equation (4.105) represents the ﬁrst order approximation to

a, = magnitude of homogeneous solution

0.25 I I l

0.2 s \ 4

\ gznm
0.15 ~ \ 3 =1.00

 

l

as

0.05 -

 

 

 

O l 4 1

0.4 0.45 0.5 0.55 0.6

frequency ratio Q

 

. . o 0 ~ 1
F rgure 4.4. Frequency response equation for superharmonic resonance ((2 ~ 5)

the steady state response. Note that v1 consists of two parts, the first term is the particular
solution and the second term is from the free oscillation term. Hence, the homogeneous
solution does not decay to zero. This is caused by the last term in equation (4.29). Because
of the existence of this oscillating term, resonance arises when (2 z %. The amplitude of
the particular solution is proportional to the length ratio 5, while the second term is pro-
portional to the square of length ratio. This implies that the second term vanishes more
rapidly than the ﬁrst one as the length ratio 5 approaches to zero. A simple calculation of
the eigenvalues of the Jacobian matrix associated with equations (4.103) and (4.104) shows
that this homogeneous solution is stable everywhere for m, > 0. Hence, a steady state su-

perharmonic resonance exists for all conditions and the response is composed by two terms

with different frequencies. Certainly, the particular solution possesses the same frequency

119

as that of the external excitation, while, the homogeneous solution, which is included to im-
prove the estimate, oscillates with twice the frequency of the external excitation. It should

be noted that this result is simply an expansion of the linear response in terms of 5.

Remark: An interesting observation about the inﬂuence of the mass ratio S on the am-
plitude of the homogeneous solution needs to be pointed out. Since the amplitude of the
homogeneous solution is described by equation (4.107), a minimum value of this equation
implies a minimum amplitude of the homogeneous solution. Based on this fact, we ﬁnd that
a, reaches a minimum value when A; = 3. This implies that a proper choice of the mass
ratio can be used to suppress, but not eliminate, this superharmonic resonance. Further-
more, the existence of the damping parameter [:4 , from the slider end, prevents us from
having a homogeneous solution with a zero amplitude for this superharmonic resonance.
This indicates that the damping parameter p40 has an adverse effect on the response of the

system for this superharmonic resonance.

4.6 Superharmonic Resonance (Q m %)

In the current section, we consider the superharmonic resonance case where Q is near

to G). In this case, equation (4.29) reduces to

002 ”3 + v3 = -2j(Dz A1)€$P(jTo) - 2jll22A1€$P(jTo) - jmgr‘AfAleprTo)

+ r45A¥4163p(jT0)+ f7£fAlezp(jTo) — j(%52£)1r2A1€fe:cp(jTo)

ﬁt;

1536)7r€f€-TP(J'T0 + 10273)

+ jf8§f€3P(jTo + j02T2) +(

+ N.S.T. + c.c. (4.108)

120

where the coefﬁcients f7 and [3 are given in Appendix E, and the detuning parameter 02

is deﬁned by

= 21:“, + 602) . (4'109)

Therefore, in order to eliminate the secular term of 123, we must have

-2j#22A1 - 21(DzA1)— .7Il4o7r 7l"1142/11 -J(M° )71’2/1 {1 +W4SA2A1

 

+ (IZE’GMrEfepragTz) + fag/ll + jfglfi’eprang) = 0 . (4.110)

Substituting A1 = 92—exp( j ‘1!) into equation (4.110), expanding and separating the resultant

equation into the real and imaginary parts, we obtain

a' —m.a-("—g°)vr‘a 3 <"‘°)vr2£.a+f.,s. cos<1>2

+(1—57r36)£1p4osin¢1>2, (4.111)

f712

a<I>'2 02 a + -—£la + A5a3 - f3,£3 sin <I>2 + ()81140 cos (P2 (4.112)

1536

where <I>2 is given in equation (3.71), A5 is deﬁned by equation (4.58), coefficients f71 and
f8, are given in Appendix E. Because the coefﬁcient of 11406? is small compared with the
rest of terms in equations (4.111) and (4.112), we neglect the inﬂuence of this term in the
following analysis. From these equations, we ﬁnd that the frequency response equation
takes the form,

a2[ (#22 + Lgo’ra “2 +L10 251 )2 + (‘72 +— f712 ‘2—51 + A 502)2l — 1.8161 (4113)

121

and the frequency response curves are given by

 

02 = —-f——7‘£2 a2 5JI—l-i‘f6i —[;1p22 + ”4° --—-1r“a2 +lg‘2’1r252]2. (4.114)

Figure 4.5 provides an example of the frequency response curve associated with this super-

harmonic resonance.

a, = amplitude of the homogeneous solution

 

 

 

 

0.16 I I l
0.14 - a
0.12 *- ..
6 = 0.25
on] S=Lm ‘
' [1.2 = 0.02
as 0.08 " // p4 = 0.00 d
0.06 - _
0.04 - 4
0.02 - _.
/
0 1 1 1 \
0.2 0.25 0.3 0.35 0.4

Frequency ratio 0

Figure 4.5: Frequency response equation for the superharmonic resonance ((2 z %)

Despite the presence of the damping parameters, the homogeneous solution of 121 does

not decay to zero. Moreover, the steady state response of the system can be approximated

by

v(t) 6121(T0,T1,T2)+0(€2)

6[ A4 sin 9T0 + acos (3QT0 — <I>)] + 0(62) (4.115)

where a and <I> are described by equations (4.111) and (4.112), respectively. The steady

122

state value Q, of the phase angle is deﬁned by

l 02 + (f71/2)€f+ A502

tan Q, =
#22 + (131)4492 + (%{L)”2§f

] . (4.116)

 

Note that equation (4.113) indicates that the homogeneous solution reaches its own maxi-

mum amplitude which is described by
(41414114 - «114.1 (141. +(”_LO)1411;+(4;_:;.)1253 14 = o (4.117)
when the frequency ratio is Q = (1 + cap) / 3 with 0,, being deﬁned by
a, = —(f7, /2)§;~’ — Asag. (4.118)

To determine the stability of each individual steady state response, we compute the J acobian

matrix associated with equations (4.111) and (4.112), which takes the form,

—#22 - (%)p4o1r4az-—(Ei°-)1r2£1 "(02 + (f71/2)€12 + A563)“

(02 + (f71/2)€f + 3A502)/a "f‘22 " (%)#407f4 —(%)”2£1

(4.119)

By computing the eigenvalues of this J acobian matrix, we determine that these steady state

responses are unstable when

[1121 +(§)u111r4a4 + (’;—4,° >444? 1192. + (— §)u1.«4a4 + (”—4—° 144414 1

+102 + €i(f71/2) + As“2 H02 + £f(f71/2) + 3A5a2] < 0- (4-120)

Otherwise, they are stable.

123

Remarks: First, the homogeneous solution does not decay to zero in spite of the exis-
tence of the damping parameters. Therefore, the steady state response is composed of two
functions with two different frequencies. The particular solution oscillates with the same
frequency as that of the external excitation, the homogeneous solution oscillates with a fre-
quency that is three times the external frequency. Note that the amplitude of the particular
solution given in equation (4.32), is proportional to the length ratio {1, while the amplitude
of the homogeneous solution (equation [4.113]) is proportional to the cube of the length
ratio {1. This implies that the homogeneous solution vanishes more rapidly than the partic-
ular solution as f approaches zero. Note that the linear approximation may be applicable
when length ratio 5 is small, since the homogeneous solution may be insigniﬁcant.

Next, let us consider the frequency response equation (4.113), which is cubic in a3. There
exists more than one steady state responses in some parameter regions (Figure 4.5). Note
that the value of A5 is always positive, this implies that the backbone curve, which is the
locus of the frequency responses, bends to the left. Moreover, the existence of f7, implies a
shift of the origin of the backbone curve. In other words, the backbone curve associated with
this superharmonic resonance does not originate from Q = Sign. Together with the stability
results, we know that the frequency response curves bends to the left corresponding to
a superharmonic resonance with a softening type of nonlinearity. However, the inﬂuence
of Am}, may be insigniﬁcant so that the no bending of the backbone is observed. This
indicates that, for small values of the length ratio E, there is no jump phenomenon.

Finally, let us discuss the inﬂuence of the mass ratio S on the amplitude of this su-
perharmonic resonance. Since the amplitude of the free oscillation solution is governed by
equation (4.113), it is possible to reduce the amplitude of this solution by reducing the value

of coefﬁcient f81 - In other words, we can suppress the contribution of the homogeneous so-

124

lution on the overall response by choosing a proper mass ratio. From a more physical point
of view, we can select a mass ratio so that the external force is diminished. When such a
mass ratio, which is about 1.40, is selected, equation (4.113) admits only the trivial solution,

and hence the contribution of the free oscillation term is eliminated.

4.7 Subharmonic Resonance (9 z 3)

In order to analyze a subharmonic resonance for 9 near to 3, we introduce the detuning
parameter 0 according to

Q: 3+3€202. (4.121)
Therefore, equation (4.36) reduces to
00203 + vs = “-ZjllngIG-TPUTO) - jH4OW4AfAI€$P(jTo) - 2j(DzA1)€$P(jTo)

81 . — . .
_. 52—)ﬂ40W2£¥A1€$P(JT0) + 1r4SA¥A1€$P(JT0) + f7£fA133P(JTO)

. — 9 - - -
+ [2J1rQ£1A¥+(-8—)p4°£17r3A¥]exp(JTo + 3102T2)+ N.S.T. + c.c. (4.122)

and we must have

— 2jp22Alexp(jTo) - 2j(D2A1)ezp(jTo) + 7r4SA¥14183p(jT0)
81 2 2 o . 4 2 " a 2 .
- (5)1141?r flAieszTo) - #140“ AlAlexPUTO) + fvélAleprTo)

. — 9 - . .
+ [231r051/ﬁ + (§)p40£11raA¥ ]€$p(]T0 + 3302B) = 0 (4.123)

in order to eliminate the secular terms from the particular solution of 121 and retain the

periodicity of the solution. Expanding equation (4.123) with A1 = gezp( j ‘11), and separating

125

the resultant equation into real and imaginary parts, we obtain

a, = ”220 ”go ”4‘13 _ gﬂ4o “W251
+39—2p401r3a261 sin (3<I>2) + —£11ra2 cos (302), (4.124)
a<I>52 = 02a +— —£1af7’2 + A5a3 +—9—p4°1r3a2£1 cos 3<I>2 — Efnraz sin 3<I>2 (4.125)

32

where <I>2 is given in equation (3.71), f7, can be found in Appendix E, and A5 is deﬁned by
equation (4.58). From the last two equations, we obtain the frequency response equation of

the form,

a+11%».+"—,‘;°7r“a2 + 3:... 263)2+(02+-f—;’€i+4502)2
- [(%£1«)2+(:—21rp403£1)2 1021—0 (4-126)

From equation (4.126), we see that the trivial solution is always a steady-state response of
this speciﬁc resonance. Due to the complexity of equation (4.126), we only consider the
case in which #40 is zero in the following analysis. After this simpliﬁcation, equation (4.126)

reduces to

a2lﬂ§2 + (02 + _;_2£1f + 135—02)2 (351702021: 0- (4-127)

Now, besides the trivial response, we obtain the following nontrivial responses,

(antes)? = 5,1,? -pi \/p2 - q] (4.128)

126

where coefﬁcients p and q are given by

p = (202+f7.£¥)A5—(%£11r)2, (4.129)

a = A§[(u22)2+(02+f—;’-£¥)21- (4-130)

We note that q is always positive. Therefore, these nontrivial free oscillations exist only if

p2 2 4g and p < 0. These conditions imply that
3 4 3 2 2 2 2
(5517‘) " 2(55170 (202 + f7251)A5 — 4A5H22 .>_ 0 , (4-131)

and

(3m)2 > 4(2a2 + f7,£f)A5 . (4.132)

From equation (4.131), we solve for the values of A5 for given values of 02, p2, and £1, and

obtain the following inequality:

1 3
2115,... s — —(-1r£1)’(202+f72i)2
#22 2

 

+ (/(—1-)2(§«:1)4(2a2 + fvzéi) + «21704. (4.133)
#22 2 2

This inequality provides the criterion regarding the existence of the nontrivial subharmonic
response.

When these conditions hold, it is possible for the system to respond in such a way that
the homogeneous oscillation (the Alexp(jTo) term in equation [4.31]) does not decay to
zero, in spite of the presence of damping. Moreover, in the steady state, the nonlinear term

adjusts the frequency of the free oscillation to one third that of the excitation. In this case,

127

the steady state response can be approximated by

0(1) = 601(T0,T1,T2)+0(62)

c[acos(%To — 4:1) + A1 sin 0T0] + 0(62) (4.134)

where a is described by equations (4.128), where A1 is deﬁned by equation (4.32), a and <I>
are described by equations (4.124) and (4.125), respectively. The steady state value of the

phase angle is given as

02 + fag/2 + A502

tan<I>,=( [12
2

) . (4.135)

 

Note that the response of the system is dictated by the initial conditions as follow: When
the initial conditions are selected in such a way that the nontrivial subharmonic resonance
exists, then the response of the system can be approximated by equation (4.134). Otherwise,

the system settles onto a steady state given by equation (4.134) with a = O.

4.8 Summary and Discussion

In this section, we summarize the results obtained in the previous sections. Based on the
observations of previous investigators[49], and the results from previous section, we select
important system parameters and provide detailed discussions regarding their individual

inﬂuence on the steady state response.

4.8.1 Effect of the Length Ratio 5

This problem is equivalent to a simply-supported beam subjected to both vertical and
horizontal excitations coming from the same source. The magnitude of this excitation is

proportional to the length ratio 5. At the beginning of section 4.2, we solved a linearized

128

model to obtain a response whose amplitude is proportional to this parameter. In the
primary resonance (section 4.4), although the amplitude of the steady state response can
be predicted by using linear theory, the frequency at which the steady state response reaches
its maximum amplitude also depends on the length ratio. Figure 4.6 shows the effect of
the length ratio 6 on the amplitude of the primary resonance. From this ﬁgure, it is
very clear that the amplitude of the primary resonance increases when 6 increases. In the
parametric resonant case, the width of the frequency region in which the trivial solution
becomes unstable is also proportional to the length ratio. Reconsidering Figure 4.1, the
width between the two transition curves increases as 5 increases. Moreover, the amplitude
of the nontrivial response depends upon 5 also. Figure 4.7 shows the effect of varying E on
the nontrivial solutions for the principal parametric resonance. We note that, in Figure 4.7,
the amplitude of the nontrivial solution increases as 5 increases. Moreover, points m1 and
mg in Figure 4.7, which are used to mark the origins of the branches of the nontrivial
responses, represent the same points as those in Figure 4.1. Furthermore, the frequency
region between points 112; and m2 corresponds to the region in which the trivial response is
unstable. In all the superharmonic resonances, there exists an additional solution, besides
the particular solution. The magnitude of these homogeneous solutions are proportional to
a higher order power of the length ratio. To be speciﬁc, this non-zero homogeneous solution
is of order C(52) in the case where Q is near to (%), and, is of order C(53) when 0 is near
to (:13). These facts imply that the length ratio is the primary parameter, or in other words,

the most important parameter, for the current problem.

4.8.2 Effect of the Frequency Ratio 9

The inﬂuence of this parameter on the response is not immediately obvious. In order

129

a, = amplitude of v1(t)

 

0.35
0.3 -
0.25 -
0.2 L
a.
0.15 r
0.1 b

0.05 -

I l I I I I I I

 

 

 

 

 

'L i r l l l l l

 

0.3

0.25

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Frequency ratio (2

Figure 4.6: Inﬂuence of 5 on the primary resonance

a, = amplitude of homogeneous solution

 

0.2

0.15

0.1

0.05

 

 
 
 
 
 
  

f I I I I I I
g: 0.01—
g: 0.02 —

szrm
p2 = 0.02 J
#4 = 0.00

 

 

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio 9

Figure 4.7: Inﬂuence of 5 on the principal parametric resonance

130

to understand the inﬂuence of Q on the response of the system, we need to separate our
discussion in three categories: low frequency ratios, non-resonant frequency ratios and
resonant frequency ratios.

First, let us begin our analysis in the low frequency ratio region. Let us consider the
inertial force applied to the system. The accelerations ax and 0,, depend upon the square
of $2. This implies that the inertial force is also proportional to the square of the frequency
ratio, although this does not imply that the amplitude of the steady state response is overall
proportional to the square of (2. However, in the region of low frequency ratio, the amplitude
of the steady state response is proportional to the square of the frequency ratio. To show

this, let us consider the amplitude of the linear response, i.e., equation (4.25),

2
X = 3 in z $522 (4.136)

1r \/(1_ 92y + 4%92

for 9 << 1. This implies that the amplitude of the response depends upon the length ratio as
well as the frequency ratio. This point had been pointed out by previous investigators[30].

Next, let us consider the non-resonant case. When resonance is excited, the free oscilla-
tion term will decay to zero due to the presence of the damping. Consequently, the linear
approximation will be sufficient to provide a good prediction. When a resonance is possible
at a given operating frequency, we have to include an additional term in order to capture
the effect of the homogeneous solution. Therefore, we may ﬁnd that some additional peaks
exist along the smooth proﬁle of the linear response curve.

However, let us consider the asymptotic expansion of the geometrical relation, i.e. equa-
tion (4.22). From this expansion, it is clear that the external applied force is composed of

multiple frequencies. Generally speaking, when 0 is near to (%), we can expect a super-

131

harmonic resonance arising from the response of the system. It will be .very difﬁth to
verify these higher order(n > 3) superharmonic resonances, since the magnitude of the
homogeneous solution is proportional to the corresponding order of the length ratio. Un-
less the length ratio is large enough, the contribution of the homogeneous solution on the
steady state response will be very small. This explains the reason why these homogeneous

resonances do not appear in the simulation results.

4.8.3 Effect of the Mass Ratio S

In the primary resonance case, the mass ratio S appears in the nonlinearity of the system
and affects the response of the system by bending the backbone of the frequency response
curve to the left. Figure 4.8 shows the effect of S on the frequency response equation

(equation [4.92]). From this ﬁgure, it is very clear that S plays the role of nonlinearity in

a, = magnitude of v1(t)
0.2 1 I 1 I l

 

0.15

I

as 0.1 -

0.05

T

 

 

 

 

 

0.6 0.7 0.8 0 9 1 1.1 1.2

frequency ratio 9

Figure 4.8: Inﬂuence of S on the primary resonance

the primary resonance. When S = 0, there is no nonlinearity associated with the primary
resonance, and hence no bending of the frequency response curve. When 3 7i 0 there exists

some nonlinearity associated with the frequency response equation and the backbone curve

132

of the frequency response curve bend to the left.

In the principal parametric resonance, the inﬂuence of S is included in the terms A,
and A5. According to the linear analysis, the main nose of instability, or in other words,
the unstable regions on the 5 - 01 domain, will be enlarged if the value of S is increased. To
show this point, let us consider equations (4.62) to (4.69). From these equation we are aware
of the following facts: (1) the width of the region in which the trivial solution is unstable
increases when the mass ratio increases, (2) the amplitude of the nontrivial solutions a2 and
a3 change slightly when the mass ratio increases. With these two observations, we know
that this parameter has both favorable and adverse effects on the principal parametric
resonance. An increment of S will enlarge the unstable region, while this increment will
also help to reduce the amplitude of the nontrivial solution. Figure 4.9 shows the effects
of S on the width of the main nose of instability. From this ﬁgure, it is very clear that the
distance between the two transition curves increases as S increases. Figure 4.10 shows the
effect of S on the branches of the nontrivial responses of the principal parametric resonance
(equation [4.92]). We use points m1, m2, m3 and m4 to mark the origins of different
branches of nontrivial responses. We note that those two branches which originate from m1
and ma correspond to the unstable responses, while those branches which originate from
points m2 and 1124 represent the stable responses.

In both of the superharmonic resonant cases, this parameter has a very interesting
inﬂuence on the response curves. When the frequency (2 is near to (%), the amplitude of the
homogeneous solution depends upon the length ratio and the mass ratio. Equation (4.107)
shows that the amplitude of the homogeneous solution is a linear function of the mass ratio.
Figure 4.11 shows the effect of S on the amplitude of the non-zero homogeneous solution

for the superharmonic resonance (0 z %). The overall amplitude of this superharmonic

133

a, = amplitude of the homogeneous solution

0.03 I I I I I I I

 

I

0.025

I

0.02

  

I

 
 
 
 

as 0.015

(unstable region)

0.01 ’-
E = 0.01
#2 = 0.02 __
[£4 = 0.00

I

0.005

 

 

 

0 J l L l l l l

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Frequency ratio 0

Figure 4.9: Inﬂuence of S on the main nose of instability

a, = amplitude of the homogeneous solution

03 I I I I I I
S = 1.00 —

 

  
  

 

 

 

0.25 S = 0.50 —" -

0.2 _

as 0.15 i2==06032 _

[14 = 0.00
0.1 -‘
0.05 _
0 "‘21
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

Frequency ratio 9

Figure 4.10: Inﬂuence of S on the principal parametric resonance

134

response ﬁrst decreases as the mass ratio increases from 0 to a critical value which is about
0.11, while this amplitude will increase as the mass ratio increases beyond this critical value.
We note that the existence of this resonance is independent of the mass ratio. Hence, it is
possible to suppress the contribution of this resonance by choosing the mass ratio properly,

i.e., near 0.11. The mass ratio has the same effect on the other superharmonic resonance,

a, = magnitude of the homogeneous solution at Q = %

0.06 I I I I I I I I I

 

 
     

0.05 i— 61 = 0.10 d
[12 = 0.02

as 0.03

I

0.02

I

I

0.01

 

         

l l l l l l l l

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mass ratio S

 

0

 

Figure 4.11: Inﬂuence of S on superharmonic resonance (9 z %)

i.e, when 9 is near to (15). Figure 4.12 shows the effect of S on the superharmonic resonance
in which (2 z :13. From this ﬁgure, we can observe the trace of the peak amplitude ap
associated with this superharmonic resonance. From this it is seen that a proper choice of
the mass ratio will help us in reducing the amplitude of this resonance. A simple calculation
of the roots of coefficient f3, show that the optimal value of S is about 1.44. Therefore, for

this superharmonic resonance, we can diminish the contribution of the free oscillation term

by letting S be near to 1.44.

4.8.4 Effect of the Damping Parameters p2 and #4

The internal material damping parameter 112 has favorable effect on the response of the

135

a, = peak amplitude associated with superharmonic resonance (Q m 31;)

0.14 r 1 1 r 1

 

0.12

0.1

0.08

0.06

0.04

0.02

 

 

 

 

0 0.5 1 1.5 2 2.5 3

Mass ratio S

Figure 4.12: Inﬂuence of S on superharmonic resonance ((2 z %)

system. For the region in which the linear approximation is valid, the amplitude of the
particular solution decreases as [12 increases (equation [4.25]). As shown by the analysis,
the peak amplitude associated with the primary resonance can be predicted by the linear
approximation. Hence, it; is expected to have the same effect on the primary resonance.
Figure 4.13 shows the effect of M on the the primary resonance. From this ﬁgure, it is very
clear that the amplitude of the primary resonance decreases as 11; increases. Moreover, the
shift of the frequency associated with the primary resonance also decreases as 11; increases.

In the principal parametric resonance, this damping parameter decreases the region of
the instability by lifting it from the Q-axis and narrowing its boundaries in the 552 - plane.
Figure 4.14 shows the effect of m on the region in which the trivial response becomes
unstable. From this ﬁgure, it is very clear that the main nose of instability associated with
the principal parametric resonance shrinks as )1; increases. Figure 4.15 shows the effect of
112 on the branches of the nontrivial responses associated with the parametric resonance.

Note that we use m1, m2, m5 and me, to mark the origins of the nontrivial solutions. Those

136

branches originating from points m1 and m5 correspond to the unstable responses, while

branches originating from points m2 and ms represents the stable responses.

0.2

0.15

as 0.1

0.05

0

0.6

a, = magnitude of 121(1)

 

[12 = 0.02

I

 

 

 

l

I I I I

 

l l l l

 

0.7

0.8 0.9 1 1.2

frequency ratio 9

1.1

Figure 4.13: Inﬂuence of #2 on the primary resonance (equation [4.92])

 

 

 

 

0.03 I I I I I I
[12 = 0.02 —

0.025 "' [£2 = 0.08 — _1

0.02 '" l

5 0.015 " ..
m1 m5 m6 2

0.01 - _

S = 1.00

0.005 — ”4 = 0'00 -

0 1 l 1 1 1 1
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

Frequency ratio Q

Figure 4.14: Inﬂuence of m on the main nose of instability

In deriving the equation of motion, we assumed that the bearing damping is very small

compared to the transverse displacement. Based on this assumption, the bearing damping

113 is assumed to be of order 0(5)). The analyses in previous sections show that this damping

137

a, = amplitude of the homogeneous solution

 

 

 

 

 

0.25 I I I I I I
#2 = 0.02 ——
s = 0.08 ——
0.2 —\ M2 _‘
g=dm
0.15 S = 1.00 ~
as #4 = 0.00
0.1 _,
0.05 _
0 I715 1 1” m2 1
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

Frequency ratio 9

Figure 4.15: Inﬂuence of 112 on the principal parametric resonance

parameter has no inﬂuence on the response within the frequency region under investigation.
This does not imply that 113 can be ignored. If we extend the path of this work, it will be
very clear that this parameter can have inﬂuence on the higher-order (n > 3) superharmonic
resonances.

The effect of the damping parameter #4, which is used to model the friction between
the slider mass and its contact surface, is complicated. This slider friction force is made up
of two parts. The ﬁrst part arises from the action of rigid body motion, while the second
one is caused by foreshortening. The friction force introduced by rigid body motion can be
treated as linear and has an adverse effect on the response. The slider friction introduced
by foreshortening is nonlinear and has a favorable effect on the response of the system.
When the response amplitude is small, the foreshortening introduced is small. In this case,
the slider friction is mainly composed of friction introduced by rigid body motion. Hence,
an increase of the value of 114 will be accompanied with an increase of the slider friction

force. This corresponds to an increase of the axial force acting on the connecting rod and

138

hence an increase of the amplitude of the response. However, as the amplitude of the
response increases, the friction force introduced by the elastic deformation increases also.
This friction force acts as a resistance to prevent the connecting rod from further elastic
deformation. Hence, the friction introduced by foreshortening has a favorable eﬂect on the
response of the system. Based on these observations, we now proceed to consider the effect

of [14 in each individual resonance case.

 

0.2

0.15 ~

0.05 *-

 

 

 

 

0 1 l l l 1

0.6 0.7 0.8 0.9 1 1.1 1.2

Frequency ratio 9

 

Figure 4.16: Inﬂuence of 114 on the primary reéonance ((2 z 1)

In the cases of primary resonance ((2 z 1), third order superharmonic (Q m %) and
subharmonic ((2 z 3) resonances, in, has a favorable effect. Figure 4.16 shows the effect of
114 on the primary resonance. As shown in this ﬁgure, it is very clear that [14 has a favorable
effect on the response by reducing the amplitude of the response. In the superharmonic
resonance, we can not reduce the amplitude of the free oscillation term to zero because of
its existence. Figure 4.17 shows the effect of [14 on the superharmonic resonance in which
9 z %. From this ﬁgure, it is very clear that the presence of 114 will increase the amplitude of

the non-zero homogeneous solution and hence has an adverse effect. Figures 4.18 and 4.19

139

a, = amplitude of the homogeneous solution

0-25 I T I I I I I I

 

0.2

0.15

0.1

0.05

 

 

 

0 l P m A l l I 4 l

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

Frequency ratio Q

 

Figure 4.17: Inﬂuence of 114 on superharmonic resonance (9 a:

NIH

)

 

0.03 I . I I I 1 1
0.025
0.02

g 0.015
0.01

0.005

 

 

 

 

0 J l l l l l l

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency ratio Q

Figure 4.18: Inﬂuence of ”4 on the main nose of instability (9 z 2)

140

 

0.22

 

I I
0.2 [‘4 = 0.00 — -1
3:: 11. = 0.40 — “
0'14 5 = 0.01
° S = 1.00
a 0.12 ,1, = 0.02 -
’ 0.1 1
0.08 h .1
0.06 - a
0.04 a
0.02 -
0 l l l l l L L

 

 

 

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency ratio 9

Figure 4.19: Inﬂuence of 114 on the principal parametric resonance (0 z 2)

show the effect of [14 on the the principal parametric resonance. From Figure 4.18, we note
that the distance between the transition curves increases as #4 increases. In other words,
the presence of this damping parameter will increase the possibility for the existence of the
nontrivial response and hence has an adverse effect. However, Figure 4.18 only reveals the
effect of the friction force introduced by the rigid body motion. Hence, we conclude that
114 has an adverse effect of the principal parametric resonance by increasing the width of
the main nose of instability associated with the principal parametric resonance. We now
consider Figure 4.19 which shows the effect of [14 on the principal parametric resonance.
From this ﬁgure, it is very clear that 114 has a favorable effect regarding response amplitudes.
This implies that the friction force introduced by the elastic deformation can help in reducing

the amplitude of the response.

Chapter 5

Comparisons between Distributed and Lumped

Parameter Models

In the present chapter, we compare the analytical results obtained by using the dis-
tributed and lumped parameter models. Instead of examining the effects of each parameter
on the overall response of the ﬂexural vibration, this comparison proceeds by considering
the resultant equations describing the resonance conditions. In order to make this com-
parison, the analytical results obtained in chapter 3 need to be modiﬁed. Since the effects
of shear deformation and rotary inertia are not included in deriving equation (4.21), we
delete these two effects by setting a; = a3 = 0 from the results obtained in chapter 3.
Moreover, we substitute the value of n = 1 into the results from chapter 3, because only the
first normal mode was used to convert equation (4.20) into a discrete system. After these
modiﬁcations have been completed, we ﬁrst consider the primary resonance case, and then
the principal parametric resonance. After these resonances, we compare the results for the
superharmonic resonances, and then the subharmonic resonance. We close this chapter by

providing some summary points in the ﬁnal section.

5.1 Primary Resonance ((2 z 1)

In this section, we consider the case in which the operating frequency is near to the ﬁrst

141

142

ﬂexural natural frequency. We ﬁrst consider the linear results obtained in sections 3.3 and

4.4. By letting n = 1 in equations (3.88 ) and (3.89)

a = -p2,a—%cos<1>1, (5.1)

a0; = 01a + $8111 in (5.2)

which are exactly the same as equations (4.75) and (4.76) in section 4.4. This implies
that the continuous and discrete approaches will give us exactly the same result. Next, we
consider the nonlinear results. After the effects of shear deformation, rotary inertia and

bearing damping have been removed, we obtain the following equations:

a' = —p2,a — (ﬁg—‘1)7r4a3 -— €r§cos (1)2 , (5.3)
a<I>’2 2 02a + A’za3 + $1 sin <I>2 (5.4)
where
2 2
I _ 1L 2 _ E L
2 — 3 ‘1' 1r 5 8 8 (5.5)

from equations (3.112) and (3.113) by letting n = 1. From the discrete analysis, we obtain

a’ = —p2, -— (ﬁg—Urr‘a3 — grams in , (5.6)
a<I>§ 2 ago + A5a3 + 5—1:_3sin <I>2 , (5.7)
where
4
A5 = 7%5 (5.8)

in describing the resonance condition when 9 z 1, which is repeated for convenience.

143

Basically, these two sets of equations are of the same form. The equations describing the
variation of the amplitude function a are exactly the same. The only difference between
these two sets of equations is the strength of nonlinear terms, A’2 and A5, appearing in the
equations describing the variation of the phase angle. As we compare the values of [3’2 and
A5, we ﬁnd that the difference between them is strongly dependent on the mass ratio S.
Moreover, the distributed parameter approach will provide a slightly stronger nonlinearity

than the lumped parameter one.

5.2 Principal Parametric Resonance (Q a: 2)

In this section, we consider the principal parametric resonance. We ﬁrst compare the
linear results obtained in the ﬁrst parts of section 3.2 and 4.3. After we set 12 = 1 and
delete the effects of shear deformation, rotary inertia and bearing damping, we obtain the

following equations:

a’ = —p21 a — 2a51 ’1 sin (201) + a7r2p4051 cos (2<I>1) , (5.9)
at}; = ala — 2a51 '1 cos (2%) — a1r251p40 sin (2%) , (5.10)
where
1,2
i= (1+25)-4- (5.11)

from equations (3.53) and (3.54). From the discrete approach, we obtained

a’ = —p2,a —- 2a51A4 sin 2<I>1 + a1r2p4051 cos (2%) , (5.12)

a<I>’1 = 01a — 2a51A4 cos 2<I>1 - a7r2p4051 sin (2%) , (5.13)

144

where

2
A4 = (1 + 25):}. (5.14)

When we compare these two sets of equations, it is interesting to ﬁnd that they are exactly
the same. This implies that these two approaches will lead to the same linear results. In
consequence, the main nose of instability for the principal parametric resonance will be the
same. Next, we compare the nonlinear results obtained from the second part of sections 3.2

and 4.3. We obtain the following equations:

a
ll

’ —p2,a - (E§Q)7r4a3 — 2a52A'1 sin (2<I>2) + ap40527r2 cos (2<I>2) , (5.15)

04>; = 02a + A'2a3 — 2aA’152 cos (2&2) — a1140521r2 sin (2<I>2) (5.16)

from equations (3.84) and (3.84) by letting n = 1, 02 = 013 = 0 and deleting the effect of

bearing damping, 113, = 0. From the discrete analysis, we obtained the following equations:

a' '— —[.l.220 — (”—§Q)1r4a3 — 2052A4 sin (2&2) + 5211211400. COS (2Q2) , (5.17)

a<I>’2 - 020 + A5a3 — 2a52A4 cos (2<I>2) —521r2p4oa sin (2<I>2) . (5.18)

We ﬁnd that the equations describing the variation of the amplitude function are exactly
the same. The difference between these two approaches is the strength of the nonlinearities
(A; and A5) appearing in the equations describing the variation of the phase angle. From
these, we recognize that the distributed parameter and the lumped parameter models will
provide the same result in determining the main nose of instability. Since the effects of
the nonlinearities are again described by the parameters, A; and A5, we obtain the same

conclusion as that obtained in the last section. Brieﬂy, the effect of nonlinearity is very

145

sensitive to the mass ratio, and the distributed parameter approach will provide a stronger

nonlinearity than the lumped parameter one.

5.3 Superharmonic Resonance (Q m %)

In this section, we consider the superharmonic resonance in which the operating angular
speed is near to one half of the ﬁrst ﬂexural natural frequency. After we delete the effects
of shear deformation, rotary inertia and the bearing damping, we obtain the following

equations:

2
a’ = -p21 a + gig-13in <I>1 - €131F—(3 A' 1)cos<I>1 , (5.19)
a¢I _ U I‘ll—2051 I
1 — 1 a +— 6 cos <I>1 +1%(3- A) $111 <I>1 (5.20)

from equation (3.119) and (3.119) in section 3.4, where A’l is given in equation (5.11). From

the lumped parameter model, we obtain

2 2
a' = —p2, a + 3%{1— sin <I>1 — 62%(3 — A4) cos <I>1 , (5.21)
a<I>'l = a1 a +— #4051 —cos <I>1 +1%(3— A4) sin <I>1 (5.22)

from equations (4.103) and (4.104), where A4 is given in equation (4.42). As we compare

these two sets of equations, we see that A4 = A’1 and thus they are identical.

5.4 Superharmonic Resonance ((2 z §)

In this section, we consider the superharmonic resonance in which the the operating

angular speed is near to one third of the ﬁrst natural ﬂexural frequency associated with the

146

connecting rod. We ﬁrst compare the results obtained in sections 3.5 and 4.6. After we
delete the effects of shear deformation, rotary inertia and bearing damping, we obtain the

following equations:

 

a' = —p22a — (”—gﬂ)a31r4— (ﬁg—2° ){f 11'2 a — (1—-—536)€1;1.1o sin <I>2 + f§1£13 cos (D2 ,(5. 23)
a<I>'2 = 02a + [14:15, + A’a 3 +(1536)£f1440 cos (1)2 - £1513 sin (1’2 (5.24)
where
2 I2
fi1 : _2_51_+_1_1L+_5_.A’1_9_11 (5.25)

2304 1728 144 63
1559 17:: 9111'1 A12
+ + -————- -
36864:: 27648 11520:: 360::

 

 

f5, (5.26)

from equations (3.128) and (3.129). From the lumped parameter analysis, we obtained

a' = —p22a — (E19)a37r4— (”Z—J )6211' )511140 sin <I>2 + f31§1 cos <I>2 ,(5. 27)

(_.__ 1536

a<I>'2 02a +—— f7 7’ ——£1a + A5a3 + (—— —)£1p4o cos Q2 — f81£1 sin <I>2 (5.28)

1536

from equations (4.111) and ( 4.112), where

17 1r4 «25' «45' W452

 

 

 

 

f“ = ‘72“ 1008 32 " 252 ' 252

_ 17 3 2 3 A3

‘ ’72 " 641r +16A“ 63’ (5'29)
f _ __5_ 1711' 1r3 +46171'S 1r3S W352
8‘ ‘ 64:: 2160 5760 23040 1440 1440

__ 5 591: 461A4 A:
‘ —641r ' 27648 + 11520:: ’ 3601r ' (5'30)

 

For this superharmonic resonance, the results obtained from two approaches agree with each

147

other in a qualitative sense. However, the numerical values of each coefﬁcient are different.
Generally speaking, the magnitude of coefﬁcient f81 is less than the magnitude of f1’,1 for
0.40 < S < 1.00. Otherwise, the magnitude of f2, is larger than that of f31. This implies
that the lumped parameter model predicts a larger response amplitude than the distributed
parameter model for 0.4 < S < 1.0, while the lumped parameter model predicts a smaller

response amplitude for either 5 > 1.0 or 0 < S < 0.4.

5.5 Subharmonic Resonance (Q m 3)

In this section, we consider the subharmonic resonance in which the frequency ratio is
near to three times of the ﬁrst natural ﬂexural frequency associated with the connecting
rod. After we neglect the effects of shear deformation, rotary inertia and bearing damping,
we obtain the following equations:

It: )7'403 ‘:(—:)€17r2#4oa

+(§§)§1p40a2 sin (3<I>2)+ f31§1 —a2 cos (3<I>2) , (5.31)

a : —p220 _(—

I
a<I>'2 2 02a + A'za3 + LZI-z-ﬁfa + (21%)611140a2 cos (3<I>2) - Elf-f5.1 a2 sin (3‘I’2) (5.32)

where

f1? ‘ 256 + 64’r + 16 A 5 LA‘ ’ (5'33)
45 9 27 3A’

I _ _ 3 _ 3 l

f31 - “6‘4” 7+ 811' + 8 7r S + —81l’ (5.34)

from equations (3.142) and (3.143) in section (3.6). While, for the lumped parameter model,

148

we obtain the following equations:

11 81
a' = —:4.a - (€21.43 - (53:32:43
9 2 . 3 2
+ E)E1p4oa sm (3<I>2) + -2—£17ra cos 3‘32 , (5.35)
a<I>’2 = 020 + 45153:: + A5a3 + (395)511140112 cos(3<I>2) — gémaz sin (34%;) (5.36)

from equations (4.124) and (4.125), where

 

 

387 271r4 + 243135 _ 271r‘S 2711'452

 

f7” : Ta" 80 32 20 20
387 243::2 243A, 2741:
’3" 64 16 ' 5 ' (5'37)

From the results obtained by these two different approaches, we observe that they give the
same qualitative behavior, although the numerical values of the coefficients are different.
Generally speaking, the magnitude of coefficient f1: is less than that of coefficient f72. The
magnitude of coefficient fél is larger than 1.5. These implies that the distributed parameter

model predicts a larger nontrivial homogeneous solution.

5.6 Summary

In this chapter, we compared the results obtained by using two different models in
investigating the ﬂexural vibration associated with the ﬂexible connecting rod. Instead of
discussing the effects of each parameter on the overall response, we proceeded by considering
the resulting equations which describe the resonance conditions. The following conclusions
are drawn: (1) these two models provide the same results in the linear part, and (2) the

distributed parameter model will provide a stronger nonlinearity than the lumped parameter

149

model. It can be seen from the value of A'2 that in the distributed parameter model the mass
ratio is not the only source of nonlinearity, and that the geometric nonlinearity (associated
with ﬁnite deformations) is more signiﬁcant in the distributed parameter model than in the
lumped parameter model.

The absence of the bearing damping parameter 143 in the lumped parameter model is
caused by assuming that it is of order C(62). Although the same assumed order is applied
in the distributed parameter analysis, the effect of this parameter has been recovered from
the final equation describing the resonance conditions, since the expansions included term
beyond C(62). However, because of the nature of the lumped parameter model, the effect of
bearing damping was neglected from the very beginning of the analysis, since its presence
would require modiﬁcation of the assumed mode shape employed. Actually, this parameter
can be included in the lumped parameter analysis by following the analysis pattern outlined

in Appendix A.

Chapter 6

Numerical Solutions and Comparisons

In the present chapter, we present some numerical simulation studies of equation (4.21).
The main purpose is to verify the analytical work with these numerical solutions. In order to
take a systematic approach and obtain a clear picture of the results, we identify the various
cases considered with the help of a table, Table 6.1. In obtaining the numerical solutions,
we use the programs LSODE[22] and AUTO[14]. The former is a numerical solution
package for solving initial value problems for ordinary differential equations, and the latter
is a software package capable of tracking steady state response curves and, simultaneously
providing stability information. AUTO is used to obtain the global picture for the response
curves, and LSODE to provide time traces for speciﬁc parameter values.

In Figure 6.1 we demonstrate that the general features of the response of the nonlinear
lumped paramter model are well predicted by our asymptotic results. This is done by com-
paring the analytically predicted frequency response curve with that computed by AUTO
and with simulations from LSODE at several frequencies for a speciﬁc set of parameter
values.

In Figure 6.1 we show the frequency response generated by the three methods described
above. We note that the agreement is very good, and that the improtant features are
captured by the analysis. In particular, the bending of the primary resonance near 9 = 1,

and the parametric resonance near (I = 2 all appear in the simulations and in both response

150

151

curves. We now turn to the details of how the individual parameters affect the overall

response, and describe the accuracy of the analysis.

a, = amplitude of the ﬂexural response

 

 

 

 

0.2 r l I 1 1
AUTO —
MMS —
0.15 - LSODE <> -
= 0.01
a‘ 0'1 ‘ 5:100
[£2 = 0.02
[14 = 0.00
0.05 ~ _
0 1 1 1
0 0.5 1 1.5 2 2.5 3

Frequency ratio 0

Figure 6.1: Frequency response curves from MMS, LSODE and AUTO

In section 6.1, we consider the effect of the length ratio 5 on the overall response of
equation (4.21). In section 6.2, the effect of the mass ratio on the overall response is
considered. We examine the effects of the damping parameters 112, p3 and 11.1 on the overall

response in section 6.3. In section 6.4, we summarize the observations to close this chapter.

6.1 Inﬂuence of the Length Ratio :5

In this section, we consider the the effect of the length ratio 6 on the overall response
of the model given by equation (4.21). It is the purpose of this section to verify the predic-
tions from the nonlinear approximation in chapter 4 with the help of the numerical tools
LSODE and AUTO. We use Table 6.1 to identify the various cases considered for different
parameters combinations.

We ﬁrst compare the frequency response curves obtained in chapter 4 with the results

 

 

 

 

 

 

 

 

Case 6 S #2 #3 #4 Case 6 S #2 #3 #4
1 0.005 0.00 0.02 0.00 0.00 2 0.005 0.50 0.02 0.00 0.00
3 0.005 1.00 0.02 0.00 0.00 4 0.010 0.00 0.02 0.00 0.00
5 0.010 0.50 0.02 0.00 0.00 6 0.010 1.00 0.02 0.00 0.00
7 0.050 0.00 0.02 0.00 0.00 8 0.050 0.50 0.02 0.00 0.00
9 0.050 1.00 0.02 0.00 0.00 10 0.100 0.00 0.02 0.00 0.00
11 0.100 0.50 0.02 0.00 0.00 12 0.100 1.00 0.02 0.00 0.00

 

13 0.010 0.00 0.04 0.00 0.00 14 0.010 0.50 0.04 0.00 0.00
15 0.010 1.00 0.04 0.00 0.00 16 0.050 0.00 0.04 0.00 0.00
17 0.050 0.50 0.04 0.00 0.00 18 0.050 1.00 0.04 0.00 0.00
19 0.010 0.00 0.02 0.00 0.40 20 0.010 0.50 0.02 0.00 0.40
21 0.010 1.00 0.02 0.00 0.40 22 0.050 0.00 0.02 0.00 0.40
23 0.050 0.50 0.02 0.00 0.40 24 0.050 1.00 0.02 0.00 0.40
25 0.010 0.00 0.02 0.08 0.00 26 0.010 0.50 0.02 0.08 0.00
27 0.010 1.00 0.02 0.08 0.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 6.1: Identiﬁcation of Combinations of Parameters

from AUTO. After this, we focus our attention on qualitative observations. Figure 6.2
shows the frequency response curves obtained from the analytical work and AUTO for
case 5. Figure 6.3 shows the frequency response curves for case 6. From these two ﬁgures,
it is clear that the nonlinear approximation matches the numerical results quite well when 6
is small. When the length ratio 6 is larger than 0.10, the analytical result can provide only
qualitative consistency with the numerical results. According to the analytical results, when
g is larger than 0.10, the detuning associated with the primary resonance becomes so large
that the peak amplitude occurs at negative values of the frequency ratio. This is physically
impossible. Based on this observation, we would like to use results from AUTO to carry out
a qualitative investigation regarding the effect of the length ratio £ on the overall response
and verify the analytic predictions. Figure 6.4 shows frequency response curves for S =
0.0 with different 6 values (cases 1, 4 and 7). According to the analytical results, S = 0
corresponds to a linear oscillator, because S plays the role of nonlinearity in equation (4.21).
This implies that there is no nonlinearity associated with equation (4.21), and hence there

is no detuning associated with the primary resonance. Moreover, the amplitude of the

153

response is proportional to 5. The main nose of instability does exist, because its existence
is independent of the nonlinearity. Both of these two points are veriﬁed by Figure 6.4. We
note that, in Figure 6.4, the peak amplitude increases from 0.06 to 0.39 when 6 increases

from 0.005 to 0.05.

a, 2 overall response

 

 

 

 

 

 

0.16 I I I I I
0.14 - AUTO -— ..
Analytical —
0.12 - a
f = 0.01
DJ“ S=0w ‘
#2 = 0.02
as 0.08 - #4 = 0.00 _
0.06 - _
0.04 *- _
0.02 - ._
0 1 1 1 1
0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.2: Comparison between Analytical result with AUTO ( case 5)

a, 2 overall response

 

 

 

 

 

 

0-16 T I I I I
0.14 )- AUTO — ..
Analytical —
0.12 r .4
f = 0.05
0.1 - S = 0.50 T
#2 = 0.02
as 0.08 - [14 = 0.00 —.
0.06 )- _.
0.04 r _
0.02 _
0 i— l l 1 I
0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.3: Comparison between Analytical result with AUTO ( case 6)

Next, we consider cases in which S 51E 0. According to the analytical predictions, the

154

a, = overall response

 

0.4 I I I I I
0.35 _ f = 0.005 — _
6 = 0.01 —
0.3 - f = 0.05 -— ~
0-25 7 s = 0.00 ‘
[£2 = 0.02
as 0.2 "’ l ”4 = 000 .1

 

 

 

 

 

 

 

0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.4: Frequency response for cases 1, 4 and 7

cases in which 5' > 0 correspond to systems with a softening type of nonlinearity. Moreover,
the strength of nonlinearity depends on the value of S. The peak amplitude of the response
shall be the same as that given in the linear approximation, while the detuning associated
with the primary resonance (0,) is proportional to the square of 5. Figure 6.5 shows the
numerical results for different 6 values with S = 0.50 (cases 2, 5 and 8), and the results for
S = 1.00 (cases 3, 6 and 9) are given in Figure 6.6. From these ﬁgures, we observe that the
analytical work does provide results which are consistent with the numerical solution. It is
interesting to note that as 5 increases, the superharmonic resonance near (I z % begins to
appear in Figures 6.5 and 6.6. This is as predicted by the analysis of section 4.5. When
6 is smaller than 0.10, the analytical approximation provides very good results. When 6
is larger than 0.10, the analytical prediction again provides only qualitative information.
Table 6.1 contains bifurcation information obtained from analytical results and AUTO. In
this table, we use PD1 and PD; to mark the points at which principal parametric resonance

instabilities occur, while we mark the turning point associated with the primary resonance

155

by LP1 and LP2. In other words, the region between points PD1 and PD; represents the
main nose of instability, and the response curve between points LP1 and LP; represents
the unstable response near the nonlinear primary resonance. The readers should refer to
Figure 6.7 for the locations of these points. For the sake of clearness in the ﬁgures, the
branches of the nontrivial response associated with the parametric resonance are generally
not included. However, the bifurcation points listed in table 6.1 are sufficient to provide
the important information. From Table 6.1, we draw the following conclusions. When the
length ratio .5 increases, (1) the detuning associated with the primary resonance (the shift
of point LPg) increases, and (2) the width of the main nose of instability (the distance

between points PD1 and P02) increases, and (3) the superharmonic resonance near

NIH

appears. We note that, when the length ratio f is larger than 0.10, the analytical results
predict a considerable detuning associated with the primary resonance and the value of LP1
becomes negative. Since the Method of Multiple Scales can only provide local analysis, this
prediction becomes unreliable. Based on this fact, the bifurcation points for large 5 values
are not provided. As a matter of fact, AUTO also fails to provide the overall response
curve when 5 is larger than 0.10. We note that, in some cases, there may exist no limit
points LP1 and LP; associated with the frequency response curves either from the analytical
approximations or from AUTO. This will be explained in section 6.2.

Figures 6.8 to 6.11 show comparisons between analytical approximations and simulation
results for time traces at the (2 = % superharmonic resonance. From these ﬁgures, it is very

clear that the analytical approximations match the simulation results quite well.

as

as

0.45
0.4
0.35
0.3
0.25

0.4
0.35
0.3
0.25

156

a, = overall response

 

r I I T I

5 = 0.005 —
g = 0.01 —
f = 0.05 —

I
l

I
J

I
l

I

S = 0.50 ‘
112 = 0.02 _
[1.4 = 0.00

I

 

 

 

 

 

 

I
0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.5: Frequency response for cases 2, 5 and 8

a, = overall response

 

I I I I

 

 

 

 

 

1 ——_______._—__
0 0.5 l 1.5 2 2.5 3

Frequency ratio 9

Figure 6.6: Frequency response for cases 3, 6 and 9

157

a, = overall response

 

 

 

 

 

0.2 I I I I I
L 1
0.15 r /P -‘
as 0.1 " "
0.05 F L 2 _.
PD1 PD;

0 1 1 1 1

0 0.5 1 1.5 2 2.5 3

Frequency ratio 51

Figure 6.7: Location of points PD1, PD;, LP1 and LP;

a, = overall response

0.02 I I T I I I I
0,015 _ LSODE — -

Analytical —
0.01- _

”M /V\ M /V\

as
-0.005 -
-0.01 _
-0.015 -

—. .2 —
00 s:0.,50§: 0.0,:50 0.:50,p;=0.02,p4=0.00
—0.025 1

0 0.5 1 1.5 2 2.5 3 3.5 4
Crank angle (cycles)

 

 

 

 

 

Figure 6.8: Comparison between simulation and analytical approximation for case 8

158

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Case Analytical AUTO
1 PD1 = 1.9855 PD; = 2.0145 PD1 = 1.9700 PD; = 2.0330
2 LP1 = 0.9711 LP; = 0.999 LP1 = - LP; = —
PD1 = 1.9549 PD; = 2.0451 PD1 = 1.9131 PD; = 2.1013
3 LP1 = 0.9270 LP; = 0.9643 LP1 = 0.9505 LP; = 0.9390
PD1 = 1.9287 PD; = 2.0713 PD1 = 1.8680 PD; = 2.1670
4 PD1 = 1.9549 PD; = 2.0450 PD1 = 1.9153 PD; = 2.0996
5 LP1 = 0.8477 LP; = 0.9296 LP1 : 0.9353 LP; : 0.9104
PD1 = 1.9034 PD; = 2.0966 PD1 = 1.8290 PD; = 2.2312
6 LP1 = 0.6925 LP; = 0.9092 LP1 = 0.8582 LP; = 0.9215
PD1 = 1.8533 PD; = 2.1467 PD1 = 1.7564 PD; = 2.3909
7 PD1 = 1.7541 PD; = 2.2459 PD1 = 1.6354 PD; 2 2.8600
8 LP1 = 0.7830 LP; = —2.8534 LP1 = 0.8430 LP; = 0.7003
PD1 = 1.5067 PD; = 2.4931 PD1 = 1.3593 PD; > 3.00
9 LP1 = - LP; = - LP1 = 0.8133 LP; = 0.5956
PD1 = 1.2600 PD; = 2.7400 PD1 = 1.0936 PD; > 3.00
13 PD1: 1.9711 PD; = 2.0289 PD1: 1.9460 PD; = 2.0664
14 LP1: - LP; = — LP1: - LP; 2 —
PD1: 1.9098 PD; = 2.0902 PD1: 1.8406 PD; 2 2.2169
15 LP1: 0.94222 LP; 2 0.9985 LP1: - LP; 2 —
PD1: 1.8575 PD; = 2.1425 PD1: 1.7637 PD; = 2.3811
16 PD1: 1.7565 PD; = 2.2435 PD1: 1.6382 PD; 2 2.8602
17 LP1: 0.7873 LP; 2 - LP1: 0.8473 LP; : 0.7799
PD1: 1.5081 PD; = 2.4919 PD1: 1.3625 PD; > 3.0
18 LP1: 0.7293 LP; = - LP1: 0.8150 LP; = -
PD]: 1.2609 PD; = 2.7391 PD1: PD; =
19 PD1: 1.9542 PD; = 2.0458 PD1: 1.8882 PD; = 2.1314
20 LP1: 0.8477 LP; = — LP1: 0.9376 LP; = 0.9120
PD1: 1.9030 PD; = 2.0970 PD1: 1.8156 PD; = 2.2493
21 LP1: 0.6925 LP; = - LP1: 0.8612 LP; : 0.9202
PD1: 1.8531 PD; = 2.1469 PD1: 1.7481 PD; = 2.4045
22 PD1: 1.7509 PD; = 2.2491 PD1: 1.5537 PD; = -
23 LP1: 0.7830 LPz = - LP1: - LP; = *-
PD1= 1.5053 PD; = 2.4947 PD1: 1.3400 PD; = -
25 PD1: 1.9549 PD; = 2.0450 PD1: 1.9141 PD; = 2.1009
26 LP1: 0.8477 LP; = 0.9296 LP1: 0.9376 LP; = 0.9088
PD1: 1.9034 PD; = 2.0906 PD1: 1.8283 PD; = 2.2319
27 LP1: 0.6925 LP; = 0.9092 LP1: 0.8554 LP; = 0.9210
PD1: 1.8533 PD; = 2.1467 PD1: 1.7560 PD; = 2.3915

 

 

 

 

Table 6.2: Comparison of Bifurcation Data

 

 

 

159

a , = overall response

0-004 I l T I l I I

.. LSODE — ..
0003 Analytical —
0.002

om /\ /\ /\

—0.001 -
—0.002 - -
—0.003 - ‘

_0'004 __ a o 50 o 02 _
4.005 5.9.01.3T 0.50, . _ . 14.2-1. ,p4l_o.oo .

0 0.5 1 l .5 2 2.5 3 3.5 4
Crank angle (cycles)

 

—1

l

 

as

 

 

 

Figure 6.9: Comparison between simulation and analytical approximation for case 5

a , = overall response

 

 

 

 

 

 

 

 

 

 

0.02 I I . 1 1 1 I I
0.015 V" — 0D —

0.01 -

0.005 -

O
a. -0.005 — V V V V -

—0.01 ~ ..
—0.015 - ﬂ
-0.02 ~ q
-0.025 +— _
_0.03 S = 1.00, f 3'- 0.05, S} = 0.50,}12 = 0.02,n4l= 0.00l

 

 

 

0 0.5 1 l .5 2 2.5 3 3.5 4
Crank angle (cycles)

Figure 6.10: Comparison between simulation and analytical approximation for case 9

160

a, = overall response

 

I I I 1

I I I
0.002 Analytical — LSODE —

mono

as
—0.001
-—0.002 -

—0.003

I

I

 

 

S =11.00,€‘f 0.01, 1? = 0.59,}12 = 9.02,ﬂ41= 0'001

0 0.5 l 1.5 2 2.5 3 3.5 4
Crank angle (cycles)

—0.004

 

Figure 6.11: Comparison between simulation and analytical approximation for case 6

6.2 Inﬂuence of the Mass Ratio S

In this section, we investigate the effect of the mass ratio S on the overall response
predicted by equation (4.21). According to the analytical work, the mass ratio S plays
the role of the nonlinearity in equation (4.21). Moreover, the strength of the nonlinearity
depends in a complicated way on the mass ratio. In section 6.1, it was shown that the
linear approximation matches the nonlinear approximation as well as the numerical solu-
tions for case in which S is zero. For the primary resonance, the detuning associated with
the primary resonance increases linearly as S increases, while the peak amplitude remains
unchanged. There will be no detuning associated with the primary resonance when S =
0. Hence, when S = 0, there is no bifurcation associated with the primary resonance. In
the principal parametric resonance, the width of the main nose of instability increases as
S increases. Moreover, the amplitude of the responses changes as the mass ratio changes.

In the superharmonic resonance in which 0 is near to one half, the mass ratio affects the

161

response in a very interesting way. The amplitude of the response decreases as S increases
from zero to a critical value, while the amplitude increases after S increases beyond this
critical value. Figure 6.12 contains the frequency response curves for cases 4, 5 and 6.

Figure 6.13 shows the frequency response curves associated with cases 7, 8 and 9.

a, = overall response

 

 

 

 

 

 

0.16 I I I I I
S = 0.00 "—
0.14 — S = 0.50 '—
_. S = 1.00 —
0.12 T
0.1 - f = 0.01 "
#2 = 0.02
as 0.08 - [‘4 = 0.00 ..
0.06 - ...
0.04 F’ ..
0.02 - .l
0 l l l l
0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.12: Frequency response curves from AUTO (cases 4, 5 and 6)

a, = overall response

0.45 I I I I
0.4
0.35
0.3
0.25

 

CDUJU)
II II II
HOG "I
OO'IO
coo

(za%
[1; = 0.02 4

0’ [£4 = 0.00

 

 

 

 

0 0.5 1 1.5 2 2.5 3
Frequency ratio 9

Figure 6.13: Frequency response curves from AUTO (cases 7, 8 and 9)

162

We ﬁrst consider the primary resonance. As these ﬁgures show, the peak amplitude of
the response remains essentially unchanged while the detuning 0,, increases as S increases
from zero to one. This in turn enlarges the instability region associated with the primary
resonance. From the numerical results listed in Table 6.1, we know that the width of the
instability region increases from zero (case 4) to 0.0633 (case 6) as 8 increases from zero
to one. Figure 6.14 contains the simulation results for cases 7 and 8 with Q = 1.00. From
this ﬁgure, we observe that the larger the mass ratio is, the smaller the peak amplitude
of the response will be. This is caused by the bending of the frequency response curves
associated with the primary resonance (Figure 6.12 and Figure 6.13). Next, let us consider
the principal parametric resonance. The width of the main nose of instability, as shown
in Table 6.1, increases from 0.2338 (case 4) to 0.6409 (case 6) when S increases from zero
(case 4) to one (case 6). (In order to keep Figures 6.12 and 6.13 as clear as possible, the
branches of the nontrivial solution are not shown in these ﬁgures.) Figure 6.15 shows the
branches of parametric response for cases 5 and 6. From this ﬁgure, it is very clear that
as S is increased these response branches bend more, and the main nose of instability is
enlarged.

In order to consider the effect of the mass ratio S on the amplitude of the superharmonic
resonance, we need to consider the response curves when (I is near to one half. Figure 6.16
shows the amplitude of the response at Q = 0.50. Comparing this figure with Figure 4.11,

it is very clear that the numerical solution agrees the analytical prediction.

0.4

0.2

0.3
0.25

0.2

Figure 6.15:
5 and 6)

163

a, = overall response

 

 

 

 

 

I I I I I I I
__ S = 0.00 -— _
R50 7
Q=1I00 5:005 [12:002 lp4=9.00
0 0.5 1 1.5 2 2.5 3 3.5 4
Crank angle (cycles)
Figure 6.14: Simulation results for cases 7 and 8
a, = overall response

 

 

 

 

 

2.2 2.4

Frequency ratio 9

N ontrivial responses associated with the principal parametric resonance (cases

164

a, = overall response

0.0025 I I I I I
0.0024
0.0023
0.0022

as 0.0021
0.002
0.0019
0.0018
0.0017

 

 

 

 

 

0 0.5 1 1.5 2 2.5 3

Mass ratio

Figure 6.16: Amplitude of the response at Q = %

6.3 Inﬂuence of the Friction Parameters p; and #4

In this section, we consider the effects of the damping parameters [12, p3 and #4 on the
overall response of equation (4.21). The parameter [1; corresponds to the internal material
damping associated with the connecting rod. The parameter #3 represents the damping
effect of the bearings at the joints between members. The parameter #4 is used to model
the friction force acting at the slider end. We focus our attention on the qualitative behavior

of the response as each of these parameters is changed.

6.3.1 Effect of the Damping Parameter ,u;

The analytic results indicate that this parameter has a favorable effect on the overall
response curves. According to the analytical results, this parameter affects the response by
reducing the amplitude of the response and the width of the instability region associated

with each resonance case. Near the primary resonance, the amplitude of the response

165

decreases as p; increases. Therefore, the detuning of the frequency (0,) decreases. This
implies that the unstable region associated with the response curves (region between points

LP1 and LP;) will shrink as )1; increases. In the the principal parametric resonance, the

a, = overall response

 

 

 

 

 

 

 

 

 

 

 

 

 

0.14 m I I I I
[£2 = 0.02 —
0.12 ’— ﬂ; = 0.04 — '1
04' §=am
S = 0.50
0-08 ' p4 = 0.00 ‘
as
0.06 h‘ _I
0.04 L i
0.02 — ..
0 I l J l l
0 0.5 1 1.5 2 2.5 3
Frequency ratio 0
Figure 6.17: Frequency response curves from AUTO (case 5 and 14)
a, = overall response
0.45 I I I I I
.4 __ #2 = 0.02 — _
0 u; = 0.04 —
0.35 - _
0.3 - f: 0.05 _
S = 0.50
a 0.25 " [14 = 0.00 T
0.2 - ~
0.15 - a
0.1 r -
0.05 - 4
0 1 1 L 1
0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.18: Frequency response curves from AUTO (case 8 and 17)

main nose of instability also shrinks as It; increases. Moreover, the amplitude of response

which originates from the main nose of instability also deceases as )1; increases. Figure 6.17

166

shows the response curves for cases 5 and 14. Figure 6.18 shows the response curves for cases

8 and 17. Figure 6.19 shows the simulation results for cases 8 and 17. From these ﬁgures, it

a, = overall response

0.01 I

 

0.005

. W
as -0.005 ..
—0.01 ..
—0.015 a

S=0.50, £=0.,05 p4=0.00, 0:0.50
—0.02 ' 1 ’

0 0.5 1 1 .5 2 2.5 3 3.5 4
Crank angle (cycles)

 

 

 

 

 

Figure 6.19: Simulation results from LSODE (case 8 and 17)

is very clear that It; does have a favorable effect on the response. Moreover, all the trends
predicted by the analytical work are veriﬁed. Figure 6.17 and Figure 6.18 show that the
amplitude of response decreases as p; increases. This in turn reduces the contribution of
the nonlinearity on the overall response. In consequence, the response curve is smoother for
larger values of m. This point is clearly shown from Figure 6.19 which demonstrates how
the second harmonic 18 reduced as [1; is increased near the Q: %superharmonic resonance.
Regarding the instability region associated with the primary resonance, let us consider the
locations of LP1 and LP;. From Table 6.1, we ﬁnd that the width of the instability region
decreases from 0.1427 (case 8) to 0.0674 (case 17) as [1; increases from 0.02 to 0.04.

Next, let us consider the principal parametric resonance. Figure 6.20 shows branches
of response originating from the main nose of instability for cases 5 and 14, that is near

(2 = 2.0 with different p; values. Figure 6.21 shows the simulation results for cases 5 and

167

14 associated with the principal parametric resonance. From Figure 6.20, it is clear that [1;
has a small but favorable effect on the principal parametric resonance. The width of the
main nose of instability associated with the principal parametric resonance decreases from

0.4022 (case 5) to 0.3763 (case 14), as p; increases from 0.02 to 0.04.

a, = overall response

 

0.3 I

I I T T I I
[12 = 0.02 —
0.25 " [1.2 = 0.04 —- _
0.2 £ : 0.01 ..
S = 0.50
= 0.00
as 0.15 #4 "
0.1 -
0.05 _
0 l l l l l l l
1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Figure 6.20: Nontrivial responses associated with the principal parametric resonance (case

5 and 14)

0.2

0.15

0.1

0.05

as 0
—0.05
—0.1
—0.15
—0.2

 

 

 

 

ﬁequency ratio 9

a , = overall response

 

I I

I

I

[1; = 0.02 ——

#2 =0.04 — —

 

 

g = 9.01,!) I: 2.00.5 = 0.511.114 =IO-00

l l

l

 

 

0.5

1 1.5

2

2.5

Crank angle (cycles)

3 3.5

Figure 6.21: Simulation results for cases 5 and 14

168
6.3.2 Effect of the Friction Parameter #4

According to the analytical results, the friction parameter ”4 has a favorable effect on
all the resonances, except the second order superharmonic resonance (0 z %) and the main
nose of instability associated with the principal parametric resonance ((2 z 2). Figure 6.22

shows the overall frequency response curves for cases 6 and 21. Figure 6.23 shows that the

a, = overall response

 

 

 

 

 

 

0.14 I I I I I
[14 = 0.00 —
0.12 h” [14 = 0.40 — -
91" §=am
S = 1.00
a 0.08 " p2 = 0.02 T
0.06 - _.
0.04 - _
0.02 - ..
0 J l l l
0 0.5 1 1.5 2 2.5 3

Frequency ratio 9

Figure 6.22: Frequency response curves from AUTO (cases 6 and 21)

overall frequency response curves for cases 8 and 23. As shown in these ﬁgures, the friction
parameter In, affects the responses by reducing the peak amplitudes of the responses and
hence has a favorable effect. However, it should be noted that relative magnitude of the
superharmonic resonance near 9 = 0.5 is increased due to an increase ion the amplitude of
the second harmonic. Figure 6.24 depicts branches of response originating from the main
nose of instability associated with the principal parametric resonance (9 z 2) for cases 5 and
20. Figure 6.25 shows branches of response originating from the main nose of instability

for cases 6 and 21. From these ﬁgures, it is very clear that the friction parameter #4 has

a favorable on the principal parametric resonance. Form the bifurcation data contained in

 

 

169

a, 2 overall response

 

 

 

 

 

 

 

 

 

 

 

 

0.45 I I I I l
_. [£4 = 0.00 — _
0.4 [1.4 = 0.40 -—
0.35 _ ..
0.3 - E: 0.05 ..,
S = 0.50
0.25 ~ p; = 0.02 ~
as
0.2 - _
0.15 - _
0.1 - d
0.05 -
0 l l l l
0 0.5 1 1.5 2 2.5 3
Frequency ratio Q
Figure 6.23: Frequency response curves from AUTO (cases 8 and 23)
a, = overall response
0.3 I I I I I
[14 = 0.00 —
0.25 #4 = 0.40 —- _.
0.2 g = 0.01 ..
S = 0.50
= 0.02
a. 0.15 "2 —
0.1 -
0.05 1
O l l l l l

1.6 1.8 2 2.2 2.4

Frequency ratio 9

Figure 6.24: Nontrivial responses associated with the principal parametric resonance from

AUTO (cases 5 and 20)

170

a, = overall response

 

0.3 m I I ﬂ I
#4 = 0.00 —
0.25 — #4 = 0.40 — _,
0.2 f = 0.01
S = 1.00
= 0.02
a. 0.15 ”2 —
0.1 "
0.05 q

     

 

 

0 l l l l l

1.6 1.8 2 2.2 2.4

Frequency ratio 51

 

Figure 6.25: Nontrivial responses associated with the principal parametric resonance from
AUTO (cases 6 and 21)

Table 6.1, we ﬁnd that the width of the main nose of instability increases from 0.2934 in case
6 to 0.6564 in case 21. This indicates that #4 has a tendency to increase the width of the
main nose of instability and hence has an adverse effect. Figure 6.26 shows the response for
9 z % for cases 7 and 22. Figure 6.27 and Figure 6.28 show the analytical approximations
and simulation results for case 7 and case 22, respectively. From these ﬁgures, it is clear
that the amplitude near the superharmonic resonance in which 9 is near to -§- increases as
[14 increases, as predicted. This increase of the amplitude is due to the increasing inﬂuence
of the nonlinearity on the overall response. Near this resonance, as #4 is increased, the
amplitude of the second harmonic, which comes from the free oscillation in this case, also
increases. In comparing Figures 6.26 and 6.27, we ﬁnd that the perturbation approach can
provide very accurate results compared with the numerical solutions. This point is shown

from the proﬁle of the responses given in Figures 6.26 and 6.27.

171

a , = overall response

 

 

 

 

 

 

 

0003 I I T r I I I I I
0.025
0.02
as
0.015
0.01
0.005 1 1 L 1 l l 1 1 1
0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
Frequency ratio 9
Figure 6.26: Frequency response for cases 7 and 22 from LSODE
a, = overall response
0.03 I I I I I I T
MMS ——
0.02 - LSODE — —
0.01 -
—0.02 r- ..
S = 0.00, f = 005,11; 2 0.02, 9 = 0.50, [14 = 0.00
_0.03 1 1 1 1 1 1 1

 

 

 

0 0.5 1 1 .5 2 2.5 3 3.5 4
Crank angle (cycles)

Figure 6.27: Analytical approximation and simulation results for case 7

172

a, = overall response

0.03 I I I I I I I

MMS —-
0.02 - LSODE — -<

., °‘°I,/\ A A A;
V v \/ v

—0.02—

 

 

 

 

IS = 0.00, f = 0105,11; f 0.02, (12 = 0.50, 114 = 0.08

0 0.5 1 1.5 2 2.5 3 3.5 4
Crank angle (cycles)

-0.03

 

Figure 6.28: Analytical approximation and simulation results for case 22

6.4 Conclusions

In this chapter, we considered the lumped parameter model with the help of the soft-
ware packages LSODE and AUTO. The main aim was to verify the qualitative predictions
from the analytical work. Instead of quantitative accuracy, we focused our attention of the
qualitative behavior associated with the analytical work. To achieve this goal, we conducted
these numerical studies by considering the effect of each parameter on the overall response
predicted from equation (4.21). As the results show, there are remarkable agreements be-
tween the analytical and numerical results. However, these numerical studies were restricted
to small length ratios 5 << 1. As we mentioned in the previous sections, the length ratio
.5 serves as the primary parameter for the current problem. The length ratio 5 represents
not only the magnitude of the inertia force, but also contributes to the strength of the
nonlinearity. When the length ratio 6 increases, the amplitude of the response increases.
At some value of f, the original equation develops an overly strong nonlinearity, which

the method of multiple scales is not able to handle. Although, according to the analytical

173

approximation, the mass ratio S is the only source of nonlinearity, the numerical results
show that, for large 5, there exists a nonlinearity in addition to the one arising from the
inertia of the end mass m., attached to the slider end. In order to demonstrate this, let
us consider the frequency response curve plotted in Figure 6.29. Note that there is a very
small frequency shift associated with the primary resonance in Figure 6.29. This frequency
shift implies that even without any mass attached to the slider end, there exists a small
nonlinearity of the softening type associated with the primary resonance. Because this
detuning of the frequency associated with the primary resonance is very small compared
with case 7, the nonlinearity associated with case 10 is very weak. Another interesting
observation arises from the type of nonlinearity associated with the principal parametric
resonance. To show this, let us consider Figures 6.30 and 6.31 which contain the branches
of the nontrivial responses associated with the principal parametric resonance for cases 4

and 7, respectively. From these ﬁgures, we ﬁnd that these branches bend slightly to the

a, : overall response

 

 

 

 

 

0.6 I I I I I
0.5 — —
0.4 - E = 0.10 ._
S = 0.00
#2 = 0.02
as 0.3 "' \ ”3 = 000 —1
‘ [I4 = 0.00
0.2 — _
0.1 " _
0 1 1 1 1 1
0 0.5 1 1.5 2 2.5 3

Frequency ratio 0

Figure 6.29: Frequency response curve for case 10

right. This corresponds to a hardening nonlinearity. By considering the slopes associated

174

a, 2 overall response

 

2.5 I I I T I I I
2 " -I
f : 0.01
1.5 - S = 0.00 ..
a p; = 0.02
" p3 = 0.00
1 *- [£4 = 0.00 -
0.5 >- a
0 1 1 1 1 L 1

 

 

 

 

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency ratio 9

Figure 6.30: Nontrivial responses associated with the principal parametric resonance (case

 

 

 

 

4)
a, = overall response

1.8 I I I I I I I
1.6 *- -
1.4 *- -
1.2 i- 6 = 0.05 d

S = 0.00
a 1 _ p; = 0.02 "
8 O 8 I— ”3 = 0.00 _
. [£4 = 0.00

0.6 — _
0.4 r -
0.2 r —

0 l l J l l l l

1.6 1.8 2 2.2 2.4 2.6 2.8 3
Frequency ratio 9

Figure 6.31: Nontrivial responses associated with the principal parametric resonance (case
7)

I /_1 1

175

with these branches, we conclude that the effects of the nonlinearity is very weak. Hence,
we know that equation (4.21) possesses a hardening type nonlinearity when there is no
end mass m4 attached to the slider end. In the distributed parameter approach, the mass
ratio S is not the only source of nonlinearity. The geometric nonlinearity does share this
role with the mass ratio S. Despite these disagreements, the analytical work does provide a
reasonable prediction regarding the qualitative behavior of equation (4.21), especially when

S is nonzero.

 

S t 112 S 5 #2
0 0.35 0.04 0 0.25 0.02
AUTO 0.5 0.25 0.04 0.5 0.20 0.02
1.0 0.15 0.04 1.0 0.10 0.02
0 no limit 0.04 0 no limit 0.02
MMS 0.5 0.05 0.04 0.5 0.025 0.02
1.0 0.036 0.04 1.0 0.018 0.02

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 6.3: Limits of MMS and AUTO

Both the analytical method and the numerical simulations have difﬁculties in obtaining
the overall response curves over a wide range of frequencies (0 = 0.005 to 3.00). Table 6.3
contains information concerning the limitations for both the analytical method MMS and
the numerical tool AUTO. Since the effect of friction parameters #3 and 114 are not signif-
icant, we let #3 = 114 = 0 in generating Table 6.3. Since the length ratio é is the primary
parameter, we select it as the key parameter to monitor the limits of these two approaches.
Regarding the limits of AUTO, the data provided in the table indicate the maximum
length ratio 5 for which AUTO will generate the overall frequency response curves. For
the analytical solutions MMS, the data are the maximum values of 6 which will not pro-
duce a detuning associated with the primary resonance, larger than unity so that the peak

amplitude of the response can be observed for a positive operating frequency 9. It should

176

be noted that signiﬁcantly larger values of E can be handled by AUTO if one restricted

attention to a smaller frequency range.

Chapter 7

Summary and Conclusions

High operating speeds, superior reliability and accurate performance are major charac-
teristics of modern industrial machinery and commercial equipment. The traditional rigid
body analysis which presumes low operating speeds becomes insufﬁcient for describing the
performance of such high speed machinery. A thorough understanding of the dynamic
behavior of machine elements undergoing high speed operations is necessary in these situa-
tions. It is the purpose of this dissertation to present an analytical approach for describing
the elastic mechanism problem and provide a theoretical explanation for some of the dy-
namic phenomena observed in both numerical and experimental works. One of the simplest
and most popular mechanisms, the slider-crank mechanism, is selected as the prototype
system to demonstrate this analytical work.

In this dissertation, we study the ﬂexural vibration associated with the elastic connecting
rod of an otherwise rigid slider crank mechanism by using the Method of Multiple Scales
and the numerical techniques. Similar perturbation methods, such as the regular expansion
method[8, 9] and the Method of Averaging[30, 31], have been applied to study this problem.
However, up to the present time, all these investigations have provided only linear results.
In particular, analyses of the stability associated with the steady state response and detailed

studies of the nonlinear response have not been carried out.

177

178

This problem is equivalent to the ﬂexural vibration of a simply supported beam subjected
to (1) inertia force and the motion of its foundation, arising from the motion of the crank
shaft, (2) axial loads which come from the presence of the friction force on the slider mass and
the inertia force associated with the slider mass, (3) concentrated frictional moments which
come from the presence of bearing friction on the joints between links. The single mode
ﬂexural response of the connecting rod can be approximated by a single ordinary differential
equation with multi-frequency harmonic excitations and time-varying coefﬁcients which are
also of multi-frequency. These frequencies are composed of the crank speed plus harmonics
of it.

Several different types resonances can arise in this problem. The ﬁrst type is caused by
the time-varying coefficients, and is referred to as the parametric resonances. The second
type occurs when one of the harmonic excitation terms has the same frequency as one of
the natural ﬂexural frequencies, and correspond to as the primary and/or superharmonic
resonances. A third type, the subharmonic resonances can occur when a natural frequency
is a multiple of one of the excitation frequencies.

Two different models are used to study this problem. In the ﬁrst model, the dynamic
response of the elastic connecting rod is described by a set of partial differential equations.
The dynamic behavior is studied by obtaining approximate solutions of these partial differ-
ential equations using the Method of Multiple Scales and Galerkin truncations. In second
model, the dynamic behavior of the connecting rod is described by a single nonlinear dif-
ferential equation obtained by direct modal truncation of the partial differential equations.
This latter procedure follows the works by Viscomi et al.[49] and Badlani[9], although we
have included the effects of bearing and slider friction. We analyze the response predicted by

this model by obtaining approximate solutions of the governing ordinary differential equa-

179

tion using the Method of Multiple Scales. In both of these two models, several resonances,
such as the primary resonance, the principal parametric resonance and superharmonic res-
onances, were studied in detail and compared. In addition, extensive numerical studies
were carried out, including direct simulation as well as the generation of frequency response
curves using path following software.

The major features of this work are outlined here. The length ratio 5 turns out to be
the parameter which has the most direct inﬂuence on the response. The magnitude of the
response and the strength of the nonlinearity both depend on the this parameter. The
reordering of the length ratio is the essential step needed to extend the analysis to include
nonlinear effects. Since the external force is expressed as an asymptotic expansion in 6,
the validity of the application of the asymptotic method in studying this problem relies
on the order of length ratio 5. The mass ratio S is proved to be the strongest source of
nonlinearity. When there is no end mass attached to the slider, there exists only a very weak
geometrical nonlinearity associated with the geometrical effects of ﬂexure. This point is
clearly shown by the numerical studies and the analysis of the distributed parameter system.
The presence of the end mass makes the system behave like a system possessing a softening
type of nonlinearity. As a consequence, the frequency response curve will bend to the left.
Regarding this parameter, our analysis is in conﬂict with previous investigations[30, 31]. Our
results, which are conﬁrmed by the numerical studies, indicates that the peak amplitude of
the response is independent of the mass ratio. Moreover, the amplitude of the superharmonic
resonances can be slightly reduced by choosing a proper end mass. The eﬂects of three
damping mechanisms are included in the model, these are represented by the parameters
11;, [.13 and 114. The internal material damping 11;, as expected, has a favorable effect on

the overall response. This parameter affects the response by reducing the amplitude of the

180

response and shrinking the width of the instability regions. The bearing friction, [13, has
an adverse effect on the overall response. Although, the effect of bearing friction is not
as signiﬁcant as that of 11;, the presence of 113 will increase the amplitude of the response
and increase the parameter regions where instabilities occur. This occurs since the bearing
friction at the connection acts like a constant bias moment plus a ﬂuctuating moment which
adds a constant bias to the response in addition to providing an additional source of periodic
input. The slider friction parameter 114 shows a variety of effects. The slider friction force
is composed of two actions: rigid body motion and elastic deformation. Friction based on
the rigid body motion has an adverse effect on the response. Friction based on the elastic
deformation has a favorable effect on the response. Although the effect of 114 may not as
significant as 11;, it has a not favorable effect on the overall response. The effects of shear
deformation and rotary inertia, included in the distributed parameter system analysis, affect
the response by shifting the entire frequency response curves leftward in frequency.

The results presented in this dissertation represents a successful extension of previous
eﬂorts in that the effects of nonlinearities have been analyzed in a systematic manner. The
major drawback of this approach is that tedious manipulations are involved in analyzing
the higher order resonances. This difﬁculty may be overcame with more advanced symbolic
manipulation programs. Also, with the advent of high speed computers, the ﬁnite element
approach becomes more and more favorable for speciﬁc design purposes. However, like
most numerically based approaches, the ﬁnite element approach can only provide pointwise
information in parameter space regarding the response. A recent ﬁnite element study[12]
used more than one hundred simulations with a commercially available package to obtain a
response curve. We must admit that the ﬁnite element approach will provide more reliable

and accurate solutions. However, this analytical approach provides important information

181

useful for the prediction of the inﬂuence of various parameters, and a theoretical explanation
for the nonlinear phenomena associated with these systems.

Future works in this area can be separated into several distinct directions. First, this
analysis can be directly extended to study the elastic behavior associated with the ﬂexible
coupler of an otherwise rigid four-bar mechanism. Second, for a more complete understand-
ing of the elastic behavior of the slider crank mechanism, the ﬂexibility of the crank shaft
needs to be included. Moreover, the effects of oﬂ-set, clearances in the bearings, piston
cylinder clearances, external force on the slider mass, and out-of-pane deformation should
also be included. An application of Hsu’s method[24] to study the linearized model, with
multi-degree-of-freedom, should also be considered. A detailed study to the slider crank
mechanism with low frequency ratio 0 and large length ratio 5 is recommended.

As for the general improvement in the analysis techniques for elastic mechanism prob-
lems, a more thorough understanding of the inertia force is required. In all previous analyti-
cal works, the dynamics of the ﬂexible link is formulated relative to a coordinate system that
follows the rigid body motion of the beam. This approach presumes an inﬁnitesimal strain
and is sometimes referred to as the shadow beam approach. In future work, it is intended
to include large overall motion and strains by employing geometrically exact formulations.
Recent results from Simo and Vu-Quoc[38, 39] may prove useful for this extension.

In addition, future work should include careful systematic experiments to verify the
modeling assumptions and to examine the nature and consequences of the behaviors pre-

dicted in this study.

APPENDICES

Appendix A

Construction of the Linear Operator

In this appendix, we consider the ﬂexural vibration of a simply supported beam sub-
jected to the following conditions: (1) a distributed force is applied along the span of this
beam, (2) concentrated bending moments are applied at both ends, and (3) motion of
the supporting foundation. This is modeled by a linear partial differential equation sub-
jected to nonhomogeneous boundary conditions. Because of the linear nature of the partial
differential operator, it can be converted into a partial differential equation with homo-
geneous boundary conditions. In this appendix, we solve this problem by using concepts
from vibration theory [16, 46] rather than rigorous mathematical arguments. Therefore,
the transverse deﬂection will be described by a linear partial differential equation subjected
to a set of homogeneous boundary conditions, which can then be truncated by Galerkin’s
method with the usual linear mode shape functions. To achieve this goal, we ﬁrst convert
the nonhomogeneous kinematical (or rigid) boundary conditions[16] into homogeneous ones
by considering the motions of the foundation. Next, we consider the natural boundary con-
ditions and obtain the equivalent partial differential equation with homogeneous boundary
conditions. Based on this equivalent equation, and its associated boundary conditions, we
then apply Galerkin’s procedure to transform the continuous system into a discrete system.
After this procedure has been presented completely, we then provide an example problem

to demonstrate this procedure.

182

183

The transverse deﬂection of this simply supported beam is described by

017F4(Dozv) + 01%”: = Q1(5€, t), (A-I)

with the following boundary conditions:

v(0,t) = Q;(t), (A.2)
v(1,t) = Q3(t), (A.3)
vu(0,t) = Q40). (A-4)
vm(1,t) = 050), (A-5)

where Do represents derivative with respect to the time variable, Q1(a:, t) represents a dis-
tributed load applied along the span of the beam, Q;(t) and Q3(t) stand for the motion of
the left and right points, respectively, and Q4(t) and Q5(t) represent the bending moments
applied to the left and right ends, respectively. We note that the ﬁrst two boundary con-
ditions are referred to as the kinematical boundary conditions, while, equations (AA) and
(A5) are referred to as the natural boundary conditions. Our goal is to ﬁnd an equivalent

ordinary differential equation which takes the form,

0:17I4(Do2wn) + wzwn = qn(t) (A.6)

describing the amplitude of the transverse deﬂection of this simply supported beam. First,

we compensate for the motion of the foundation by letting

”(3,1): 217(3', t) + 02(1) (1 - at) + 030) . (A-7)

184

Substituting this expression into equation (A.1) and its associated boundary conditions (A.2)

to (A.5), we obtain

017F4D021I) 'l' alwxxxx = Q($a t) : Ql(zat) - 017F4D02IQ2(1— it) '1' Q3] (A8)

and the following boundary conditions:

117(0,t) = 0, (A.9)
2D(1,t) = 0, (A.10)
w..(0.t) = Q40). (Aoll)
zsz(1,t) = 625(1). (A.12)

This procedure transforms the kinematical boundary conditions to zero, and replaces the
effect of the beam motion by a modiﬁcation of the distributed force acting on the beam. We
note that equation (A.8) and its associated boundary conditions (A9) to (A.12) correspond
to a simply supported beam subjected to concentrated bending moments applied at each
end. Next, we apply the principle of virtual work to handle the nonhomogeneous bending

moments (A.11) and (A.12). First, we compute the virtual work done by the inertial force,
1

6U1 = —a11r4/ (D0211?)6wds (A.13)
o

and the virtual work done by the elastic force and the external loads

1 1
w2 = —a(/ a1w§,dx)+ / Q(:1:,t)6112d:1:
0 0

1 1
= _a,w,,,6w,,+a1wmm—al / wmméwdz+ / magma. (11.14)
0

0

185

From the principle of virtual work, we must have

6(U1 + U;) = 0 (A.15)
which yields
1 1 1
/ alvr4002u‘26u‘2da: = — / wuuéwda: + / Q(:1:,t)61Dd:c
o o 0
— 011701361131; + (11101-33361?) . (A.16)

We now assume 17) = w,,(sin mrz) and employ the boundary conditions for equation (A.16).

After carrying out all the integrations, we obtain
mr
alw4D02wn + 33w. = qn(t) — 017[ Q5(t) cos n7r — Q4(t)] (A.17)
where

1
Qn(t) = 21) Q(:1:,t)sinmra:ds

1 1
= 2/ Q1(z,t) sin mrxdz — 2011761002] [Q;(1— 2:) + Qs] sin n7rxda:(A.18)
o 0

represents the component of the external force projected onto the n-th mode.
To demonstrate the procedure described above, let us consider the following equation,

from section 3.1:

a11r4(D02v3)+ 01031:“: = Q1013, 1) (A19)

186

where

2
—ala2lex:r1:1:x " 201/122D0v11'rx1: + (6031,21):

Q1(:r:,t)

— 01(v1xvfm)x + (60, via): + afaavr‘Dozvm
— alrr4(2DoD;v1 + 2D0D1 v; + Dlzvl) + 011r4£ffl2v1 cos 9T0
+ 2011461 0(D0u;) cos 0T0 - 0171'4619211; sin 9T0

— 01x46?92(—1—-:—z) sin3 9T0 + oznr‘Eiinx cos2 9T0 sin 9T0 (A.20)

with the following boundary conditions:

v3(0, t) = Q;(t) = 0 , (A.21)
v3(1,t) = Q3(t) = %(n7r)2f2,2,£1 sin 0T0 , (A.22)
vgm(0, t) = Q4(t) = —(3§—‘5 + ala2v1mx)x - 2,1132D0'1‘11n — 21132510 cos 9T0 ,(A.23)
”31:1:(lat) = Q50) = -(2§£ + 0110201qu + 21132 00131:: - 21‘32519 C0891b .(A.24)

After applying the procedure, we obtain

0021737. + wiﬁsn = -2jwn(DzAn + #22 n4Anle-TPanTo)
- In...(mr)‘w.A?./i.ezp(jw..ro) - a?)(mr)"w.A..A¥exp(jw.To)
+ 2A,,(ala; + ala3)(n7r)2n4exp(jwnTo)
- 1% — (% + sour)?In‘(n«)2A3./i.ezp(jw.ro)
+ 3.1.4... A.exp<jw.To) + (§)u..(mr)4m¥exp(3mo)

+ jf2€f€$P(3jQTo) + flffAneszWTo) + ifs/13516171909710 — ijnTo)

187

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ [‘40 Ai(n1r)4A1(% - wn)e:cp(jﬂTo — 2jwnTo) + N.S.T. + c.c. (A.25)
where
f _ (22 2271202 NM!” «212694
1 ‘ _8_ + 12 ’ 8(1)}, _ my 6w, _ my
3526 «271296 + 11293 cos mr + 294 cos MI
8003. - 92V 6(‘03 - 92V (‘03. - 92) (‘03. - 92)
29%1 27.2mm mag (A 26)
"2003. - WY" (“’3 - 92)2 "404.2. - (Q + w71)?) ’ '
f _ ()2 n306 398 mrfl8
2 ' ‘87.? ' 27.14.23, _ my": 8n1r(w,2, — 92)3 60.23, _ my
94 mrfl‘ + 02 cos MI 396 cos mr
161.1133, — 92) 24(1)}, — o2) 4n1r n1r(w,2, — o2)
Q4A1 2QSA1 04A}
’ 2n7r(w,2, — 4122 ) n31r(w§ - my n37r(w3, — 522)
04131 cos mr 96A?
21.3«(33 _ 4m) + 1.57.003, _ mxwg - 492) ’ (A27)
f _ 3211592 + 32110“ «311394 «3511304
3 ‘ 2(a)}, _ 92) 16(1)}, — :22) 6(1)}, — 122) 2(w3, — o2)
31rnfl3wn «31139302,, «3512393121,, 9n1r (221.123,
3003. - 92) 3003. - n2) (“’3 - 92) 16(0),? - m)
3 3 2 2 3 2 2 2
1r n R can 3(n1r) 59 can mrQ cos n1r (A.28)

 

‘20s}; _ I22) " 205,2, — a2) ’ 4 ’

and N .S .T. represents those terms which have no effects in considering the secular terms of
the particular solution of 113... When the frequency ratio 9 is near to 9311, the superharmonic

resonance takes place. Under this condition, equation (A.25) reduces to

D0263" + 023133,, = —2jw,,(D;An + p;,n4An)exp(jw,,To)

188

. . 2 .
+ (01103 + ala;)(n7r)2n4ezp(]w,,To) «1» 357241132 AneerwnTo)

- (72521241135 - (é + S)(n.)211311,..p(j..,2~0)

+ fllffAnexp(jwnT0) + jf21£?€$p(3jQTo)

- J'ﬂ4o(mr)4A.2. fineszwnTo) - j(%i‘2£)(n1r)2w,,§12A,,exp(jwnTo)

+ %’E(nr)wn§?e$p(3jQTo) + N.S.T. + c.c.

 

 

 

 

where
f _ -91n“ + 111r2n6 + 5n4 cos mr 5n2A1 A'f
1’ ‘ 2304 1728 72 144 63
and
f _ 583113 171m5 + 61113 cos n7r 37nA1 nA1 cos mr A?
2‘ " 368642 27648 23041 115201 901 360n1r '

This case is fully investigated in section 3.5.

Next, we consider the case in which the frequency ratio fl is near to 3wn

(11.29)

(11.30)

(11.31)

. Under this

condition, a subharmonic resonance takes place and equation (A.25) reduces to

D02b3n 'I’ ”36371 = “zjwn(DlAn + #2; n4An)exp(jwnT0)
+ An(ala; + ala3)(n1r)2n4ezp(jw,,To)
15 1 2 4 2 2 - .
— {—8- — (5 + S)(n1r) ]n (mr) AnAneprwnTo)

2 . - ‘
+ 331241132 An€$P(JWnTO) + flefAneszwTo)

+ jfzg.1 AﬁfleszQTo — 2jwnTo) — jp4o(n7r)4wnAi14nemp(jwnTo)

. 81 .
- J(E)(n7r)2wnﬂ4oéfAneprwnT0)

9 - .
+ (§)(mr)3wn114061AieprQTo - 2wnTo) + N.S.T. + c.c.

(11.32)

189

where

1899124 8712116 189n4cosn1r 405n2A1 27A?

256 64 8 16 5

 

 

f12 =

and

f3: = 189 n5 —9—7ran7 + -2—7n31r3 + £23.97? — 9 5

"6?" 16 16 8

This case is studied in detail in section 3.6.

ZWTI COS nn’ .

(11.33)

(11.34)

Appendix B

Superharmonic and Principal Parametric

Resonances for 5 = 661

Recalling that in analyzing the particular solution of 13;”, we need to distinguish between
the following cases: (1) Q is near to 21.)”, (2) Q is near to 11;, and (3) Q is away from 262,,
and “3211. The ﬁrst case corresponds to the principal parametric resonance, while the second
one corresponds a superharmonic resonance. When these two resonances take place, it is
necessary to reorder the damping parameters 11; and 114 so that they will appear, together
with the detuning parameter 01, in the ﬁnal resonant condition. To achieve this goal, we
rescale the damping parameters 11; and 114 according to equations (3.49) and (3.10), and
then redo all the relevant computations. In this appendix, we provide a detailed solution
procedure for obtaining the ﬁnal resonant condition.

Expanding equations (2.51), (2.52) and (2.53) with these newly reordered damping
parameters, and equating the corresponding coefﬁcients of the like-power terms in c, we
obtain the following equations:

Order c3

(1133 + 111,1);3); = (1303 )x = —a11r4§192 cos 9T0 , (B.1)

0177400201 + 01121,", = 017F46102(1 - ((2)8111 9T0 , (B.2)

190

191

Order 6“

4 2 _ 4
Q17" Do v; + 0102:”: - (€03le):c - 20117? (0013101)

+ anr‘ffﬂz cos 0T0 sin 0T0 -— 20111;, Dovlmn . (B.3)

Applying the same procedure to the associated boundary conditions, we obtain the following

boundary conditions:

Order 6
v1 = 111” = 0 , at x = 0, (B4)
121 = ‘01:”; 2 0 , at X: 1, (3.5)

Order 62
v; = 021-;- : 0 , at X = 0, (B.6)

v?

(1121, + _2_:r) = v; = 0,021.1,- = —2ﬂ329 , at X = 1, (13.7)

Order c3
1131: + 1211,1221, = 0112451925 cos 9T0 + 2a17r4/140519 sin 9T0 . at x = 1, (3.8)

Note that we have dropped all the terms involved in products with so, in obtaining the
2

equations, since we have already shown that so, = 11;, + 351 = 0 in section 3.1. We ﬁrst

project equation (B.2) onto the n-th mode by setting v1 = 131,,(sinn1rx) and integrating

from :1: = 0 to a: = 1, resulting in equation (3.29). The general solution of 131,, is then given

 

 

..._.—. ___. __ ._ A . _ _ - _ — — i A . i .___

192

in equation (3.30). Now, we solve the expression for e03 and obtain

(“31: ‘l’ vlzv2x)

€03

0112461920 — :1: + 5') cos 9T0 + 2011141140510 sin 0T0. (B.9)

Hence, equation (B.3) reduces to

00217271 ‘I' “9:927: = —2j#21wnn4Anezp(jwnT0)
- 2jwn(DlAn)e:rp(jw,,To)
1 — cos mr . .
+ (——)679261P(2JQT0 - J7r/2)

2n1r
1 . .
" 1(1 + 25)(n7r)2Anélﬂzexp(anTo + JQTO)

1 - . .
— 3(1 + 2S)(n1r)2An£1928$P(-JQT0 + anTo)
_ %(1+ 25)(n7r)2A1£1028$p(2jQT0 - J'W/2)

— p4o(nx)2£1Sl/Inexp(jQTo — jwnTo — j7r/2)

(W)2
2

 

p4o§IQAlezp(2jQTo) . (3.10)

Note that the difference between this equation and equation (3.34) in section 3.1. When 0

is near to 2112", we describe 9 as Q = 2(6),, + 601), then equation (B.10) reduces to

0021327. + (423.17% = ‘2jwn(DlAn)e$p(jwnT0) - 2jwnn4l121An€$PanTol
—(1 + 2S)(nn)21inglw§exp(jwnro + 2 jam)
-2p40(n1r)2£1wn/Inexp(jwnTo + 2j01T1 -— j7r/2)

+N.S.T.+ c.c. (B.11)

193

which is equation (3.51) in section 3.2. When 9 is near to w—gl, we express 0 as Q =

20"“ + 601) in equation (3.51). In that case, we obtain

120212271 + 033527; = —2j(DlAn)wnea:p(jw,,To) — 2jwnn4p21Anexo(jwnTo)
-cosn1r . .
+ (n7r)2114o£151Alezp(2jﬂTo) +1—61292€$p(2]QT0 — J7r/2)

_ 4(1 + 25)(nn)2A15192ezp(2jﬂT: — j1r/2) + N.S.T. + c.c. (B.12)

which is equation (3.117) in section 3.4.

Appendix C

Solution Procedure for the Case 5 = 625;

In this appendix, we present a complete solution procedure for the case in which the
length ratio 5 is scaled by 5 = 625;. Moreover, in order to have the damping parameters
11;, p3 and 114 included in the ﬁnal resonant equation, we scale these parameters according
to equations (3.8), (3.9) and (3.10). Expanding equations (2.51), (2.52) and (2.53) with
these ordering relations, and equating the coefficients of the like-power terms, we obtain

the following equations:

Order 6
“1.21: = (801)2' = 0 a (C1)
Order 62
vi".
(”2x + 7):: = 60; = 0 a (C2)
Order 63
(”3.2: + lev2z)z = (603 )1: = O 1 (C3)
4 2 ”fa:
a17r D0 ”1 ‘I' allexxx = [(1122: "l' 3")”1141: 1 ((3.4)
Order 6“
[u Eir— Egg - 4 D 2 92 QT
4:: + 8 + 2 + 01:03:: 11: + al(”1:r”1.7:1::1:)1: — 0117f ( 0 U2 — £2 COS 0) I (C-5)

194

195

(111T4D02‘02 + 0111233351- : 017F4£QQ2(1 — 2:) sin 9T0 — 201774DoD1'01 ,

Order (5

al1r4Do2v3 + 011 v3.1:zxz = -017|’4(D12vi + 2DoDzvi + 20013202)
+ O¥O3W4(D02lex) + (€04v1x)x — 01(vlrvixxlx

2
— (11 (122)131-3333 _ 201H22D0v131‘1’3 ,

subjected to the following boundary conditions:

Order (1
ulzvlzvlmzo, atx=0,
u12=v1=v1m=0, atx=1,
Order 62
112:1)22‘0231-7—0, atx=0,
viz
(uk + —2—)x 2 v2 = 0mg“ 2 -2n329, at x = 1 ,
Order 63

u3=v3=0,

3
'0
”31:1: = -(% + a102lexx)x + 2H32 00011:, at X = 0 a
(“31: + lev2z) = ‘03 = 0 7
vi}
”31:1: : —("— "l' 0102lezr)x + ”132007113, at X = 1 a

6

(0.6)

((3.7)

(C.8)

(0.9)

(0.10)

((3.11)

((3.12)

(0.13)

196

Order 6‘
”fr: v31: 4
(”41: + Y + —2" + '01::va) + 01(v1xvlz1‘x) = "2&1"r [140(D0u2)

— Sa11r4(Dozu2) + Saur‘fgﬂz cos 9T0 — 2011r4p4o£2 sin 9T0, at x = 1(C.14)

Because we have already shown that co, = (u23+ 321) = 0 in section 3.1, all terms involving
products with this quantity have been dropped in obtaining these equations. Moreover,
all terms involving products with 603 = (1:3,; + levgz) are dr0pped also, because 603 =
(11.33; + levgx) = 0 for the current case. To show this, we solve equations (C3) and obtain
its general solution which is

(“3.1: + vlzv2z) : Co ((3.15)

where co is a function independent of the spatial variable a: and it must be equal to zero,
because of the boundary condition (C .13). We now proceed to solve the ﬁrst order equation

for ﬂexural vibration which is described by equation (CA). Equation (C.4) becomes
002131" + with" = 0 (C.16)

after being discretized by Galerkin’s procedure with v1 = 131n(sin mrz). Hence, the general

solution of 131,, takes the form
131,, = An(T1,T2) ezp(jwnTo) + c.c. . (ON)

2
We then solve for 112 from the ex ression uh + 311 = 0 to obtain
p 2

_ 1 2.2 sin2n1rz
u2——4(mr) v1n(x+ 2M ) (0.18)

197

where 131,. is given in equation (C.17). With the solution of 11.2, we next solve the expression

for 60, and obtain

vi‘ v3
:1: :r:
604 = ("41: + — + — + vlzv3x)

8 2
= a11r4(£292)(1 - a: + 5) cos 9T0 — alleleu — 2al7r4p40£29 sin 0T0
+éal7r4(Do2i2fn)[— sin mm:2 +(n1r)2(1+ 25 -— 32)]

+a11r4(n7r)4p40(D0231n)131n . (C.19)
We can then convert equation (C.6) into
D0262" + will?" = -2jwn(Dli‘21n)e:cp(jwnTo) + £6292 sin ”TO ((3.20)

which is equation (3.66). In analyzing the particular solution of this equation, we need to
distinguish the following cases: (1) Q is near to can and (2) Q is away from can. The case in
which 0 is near to can will be investigated in section 3.3. At the present time, we assume
that Q is away from can and continue to consider the higher order equations. With all the

information obtained up to the present, equation (C.7) becomes

017F4Doz’03 + 0103”“,- = -201W4(D0D201) + O¥0137r4(D02012,-x) - 01(levimhc
- aiaz‘vluum - 201M; Dolez’xz - 2Oil 1r 41140529711221: Sin QTo
- aﬁvﬂvlmx): + a11r2[§2ﬂ2(1 — :1: + 5')le cos 9T0];
+ c211r2[é(D()""t“)¥,,)[—-(sinmm)2 +(n1r)2(1— 2S — x2)]v1$]3

+ a11r4p40(mr)2(Dof21n)ﬁlnv1m (C21)

198

subject to boundary conditions (C.12) and (C.13). This represents the ﬂexural vibration
of a simply-supported beam subjected to a distributed load and to concentrated moments
applied at each end. This is a simpliﬁed case of the more general case handled in the Ap-
pendix A. Therefore, by applying the results obtained in the Appendix A, equation (C.21)

can be converted into

1
D0253" + a): = —2(Dng’f)1n) - a103n2(002f21n) + 11167313?”

1
+ alagx2n4fnn — 2p22n4(Dof)ln) — (E + S)1r2n2w§{2131n cos 9T0

3 , 1 mr 2 S mr 2 . .
— 114192)?” E - (—12—)— - -L4—-2-) + (71,7?)25201711140Q SIR QT0

— ju4o(mr)’A,2.ﬁ..ezp(jwnTo) (0.22)

by Galerkin’s procedure with 2);; = 133,, sin mrz. This equation is the equation describ-
ing the resonance condition for the principal parametric resonance. We note that the
there are two friction type forces in this equation, both arise from slider friction. The
[ (n7r)2£2i>1np4oﬂ sin 0T0 ] term arises because the action of rigid body motion; while the

l ‘jﬂ4o(nW)2A3, Anexp(jwnTo) ] represents the contribution of elastic deformation.

Appendix D

Solution Procedure for the Case 6 = 6353

In this appendix, we present a detailed solution procedure for equations (2.51), (2.52)

and (2.53) expanded with 6 = 6363, 112 = 62112,, 113 = 52113, and 114 = 1140. We expand

equations (2.51), (2.52) and (2.53) with these new ordering relations and solve them se-

quentially. After application of the procedure in Appendix A to the partial differential

equation describing the third order ﬂexural vibration, we obtain the ordinary differential

equation which is used in section 3.3.

Expanding equation (2.51), (2.52), (2.53) and their associated boundary conditions with

these ordering relations, and equating the coefﬁcients of the like-powered terms in c, we

obtain the following equations:

Order 6
“11:1: : (601 )1: = 0
Order 62
vi"..-
(u21: 'l" ‘3")1: =(602)1: : 0 9
Order 63

(“31: + vlzv23)x = (803 )1: = 0 a

2
'0
0114002221 + a1 ”11:1:1'1: = [(1123 + fmxh ,

199

(13.1)

(13.2)

(D.3)

(13.4)

200

Order 64
”1‘1: ”31: 4 2
“4:: + —8_ + 3— + ”11:va + 01(lelem)x = 0177 (D0 ”2) , (1)-5)
01114002112 + 0102““ = -2017r4DoD1v1 , (D.6)
Order 65

4 2 4 2
01% Do 123 + 010333;; = —al7r (D1 ’01 + 2000201 + 2D002’02)
2 4 2 2 2
'l' 0103“ D0 ”11:1: + (3041,1117: " al(lev1xx)x - alaZvlzxxxxx

— 2011/122D0v1mu + (111463920 — 1:) sin (ITO , (D.7)

subjected to the following boundary conditions:

Order 6
u1=v1=v1m=0, atx=0, (D.8)
u13=v1=v1u=0, atx=1, (13.9)

Order 62
u2=v2=v2u=0, atx=0, (D.10)

v12..-

(‘ng + —2—-) = 02 = 0 ,‘Ugrx = —2p320 , at X = 0 , (D.11)

Order £3

“3 = '03 = 01
vi.

”31:1: : _(_ + 010201311): ‘l' 2[1321)01’11: , at X : 0 a (D12)

6

201

(“31: + 011.023) 3 v3 = O 9

3
v 1:
”31:1: = "(—é + ala2lexx)1: + 2/‘32D0v11: 3 at X = 1 3 (D13)
Order 64
”f1: v31: 4 2
(”4: + _8 + —2 + 01:03:) + 01(lelemx) = —Sa11r (Do “2)
— 2a17r4;140(D0u2) ,at X = 1.03.14)

Note that in obtaining these equations and boundary conditions, we have dropped all those
terms involving products with 120., = (112;; + 11%,, / 2) and 603 = (113,,- + levgr), since these two

quantities are zero. Equation (D.4) can be truncated with 121 = 131n(sin mm) to obtain
002131.. + with. = 0 (D15)
which admits the following solution:
131,, = An(T1 ,T2)e.7:p(jwnTo) + c.c. (D.16)

where c.c. stands for the complex conjugate term of the preceding terms. Substituting this

2
I g 0 v I 0
solution of 121,, into 112,, + $1 = 0 and solvmg for 112, we obtain

sin 2mm:

2” ) (13.17)

112 = -%(n1r)2f)fn :1:

where 13,, is given in equation (D.16). Now, we solve equation (D5) and obtain

1)4 v2 - ,
“41: + J83 + —;£ + levaa: = -01(vlzvlmx) + 017T4ll40(n7r)4(Dovln)vln

202

+ é-alw4(D0213fn)[(n1r)2(1+ 2S — 1:2) — (sin n1r:1:)2] (D.18)

which is the forth order relative elongation along the median line. In obtaining the solution

of 132“, we need to consider the following equation:
. 2
D0262n + «23132,. = —2jwn(D1An)ezp(]wnTo) - $11329 cos n7r + c.c. (D.19)

where —nlwp3,ﬂ cos mr comes from the nonhomogeneous boundary condition. To remove
the secular term from the particular solution of 132", we must have (DlAn) = O. This
implies that An must be independent of the time scale T1. Collecting this information,

equation (D.7) reduces to

alx4(D02v3) + 01111335,,“ = 0114(3920 — 1:) sin 9T0 — 0114(2DOD2v1)
‘l‘ 012a3ﬂ'4(002v1x1:) -' 01(levgxx)1: _ aga'zlerz‘xxx
_' 2all‘2gDovlz1rx1: - al(v¥xle1:1:)x + 01774H4o(nw)4(D0f)1n)ﬁlnvl1:1:

+ $011r4(D0213f,,)[ [(n1r)2(1 + 2.5' — 1:2) -— sin n112]v1x]$ (D.20)

subjected to boundary conditions (D.12) and (D.13). This equation and its associated
boundary conditions correspond to a simply supported beam subjected to distributed load
applied along its span and concentrated moments applied to each end. By using the proce-
dure described in Appendix A, equation (D.20) and its associated boundary conditions can

be transformed into the following equivalent system:

002133.. + 1.23133“: —2jwn(D1An)exp(jwnTo)— jy4o(n7r)4A,2,/lnea:p(jwnTo)

+ An(n1r)2n6(alag + ala3)ezp(jwnTo)

203
4
. . .Tl .
- 21w..n“1122 AneprwnTo) + 2] F1132 AneerwnTo)

_ [18? _ (E15. + S)(n1r)2](nx)2n4A3,.dnexp(jwnTo) + N.S.T. + c.c. (D.21)

which is the resonant equation considered in section 3.3. We note that the term involving
with the friction parameter #4 in the last equation represents the slider friction force due to
the foreshortening. We note that the [p4o(n7r)4A3,/ln ] term in the last equation represents

the contribution of the slider friction acting on the connecting rod.

Appendix E

Coefﬁcients f7 and f8

In this appendix, the values of f7, f8 and f9 in equation (4.36) are given. First, we

present the function f7, which is the coefﬁcient multiplying exp(jTo),

_ 5112 612504 2122 4124
f7 ‘ 8 +(1_112)2’1—112‘1..112'

«4124 «45124 «43294
16(1— (1 + 12)2)+ 4(1 — (1 + (1)2) + 4(1 — (1 + 122)2)'

 

 

 

 

 

 

+

For the superharmonic resonance occuring at 9 z (%), the coefﬁcient f7 becomes

f _ E 1r4 3125' fig «452
7‘ ‘ 72 1008 32 252 252'

 

 

For the subharmonic resonance taking place at 0 z 3, the coefficient f7 becomes

f _§§ 2714+243125 27145 271452.
7” 8 80 32 20 2o ’

 

 

(13.1)

(13.2)

(13.3)

Next, let us consider the coefficient function f3. The coefﬁcient multiplying exp(3 j OT 0)

is given by

f _ -592+ 19“ + 1594 + 91598 596
8 ’ 87r 4(1—492) 2(1—492) (1—02)3—27r(1—92)2

 

 

204

205

 

 

 

594 + 10‘ + «5'04 1396
161(1—92) 6(1— 122) 2(1—122) 16(1 _4122)(1 422)
13596 135296

 

 

4(1— 402)(1 — Q2) _ 4(1 _ 492)(1_ ()2) ° (13.4)

This coefficient function becomes active only when the superharmonic resonance 9 z (35)

occurs. Under this condition, it becomes

—5 171r 1r3 46113 «35 1352

f3, = 64—1 + 2160 ’ 5760 + 23040 ' 1440 " 1440 '

 

 

 

 

(E.5)

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