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dissertation entitled

HARMONIC RESPONSE AND PASSIVE VIBRATION
ISOLATION OF RIGID BODIES

presented by
R. Matthew Brach

has been accepted towards fulfillment
of the requirements for

Ph.D. (kgmfin Mechanical Engineering

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MSU IaAn Affirmativ- ActioNEmll Oppoitunlty Intuition
Wanna-m

HARMONIC RESPONSE AND PASSIVE VIBRATION ISOLATION OF RIGID BODIES

By

R. Matthew Brach

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1995

ABSTRACT

HARMONIC RESPONSE AND PASSIVE VIBRATION ISOLATION OF RIGID BODIES

By

R. Matthew Brach

Passive vibration isolation is important in the automotive industry
where the undesirable effects of powerplant vibrational excitation need to
be isolated from the occupants and the automobile interior. Recently,
hydraulic engine mounts have proven to be effective in this capacity and
have replaced traditional elastomeric mounts in many automobiles. While
the characteristics of the mount itself are well-established, no
systematic investigations have been conducted that consider the nonlinear
response of the total system consisting of an engine on mounts.

Therefore, to better understand this type of vibration system, the
harmonic response and vibration isolation performance of a planar, three
degree of freedom rigid body on resilient linear supports is investigated.
The nonlinear differential equations of motion for the system are derived
using Lagrange's equations. Approximate solutions of these equations for
four different forcing cases are formulated using the method of multiple
scales. The frequency response of each degree of freedom for these four
cases is obtained from these approximate solutions. Specifically, the
response for this system where 1:1 and 2:1 internal resonances exist
between the system linear natural frequencies is found. The analysis
identifies regions where multiple steady-state solutions exist, and other

regions where no constant amplitude steady-state solutions exist. The

efficient transfer of energy from one mode to another is also investigated
and related to a novel type of vibration isolator.

This thesis also includes a review of the four common metrics used
to assess the performance of isolation. systems. These metrics are:
transmissibility, effectiveness, transmitted power, and the forces
transmitted to the support structure. The assumptions made in the
formulation of these metrics are presented, and the measure of the
transmitted forces is modified and applied to the planar problem presented

in the thesis.

Copyright by
R. MATTHEW BRACH

1995

DEDICATION

To my wife, Paula

ACKNOWLEDGEMENTS

This research was possible with the help of several individuals,
most notably my thesis advisor, Dr. Alan Haddow. Without his help,
encouragement, and patience, this work would not have been completed.
Thanks also goes to the members of my dissertation committee: Drs. Clark
Radcliffe, Philip FitzSimons and Charles MacCluer. Thanks also to the
Amoco Foundation for their financial support of my research efforts
through a fellowship from 1990 to 1993 and to my present employer, the
Ford Motor Company, which supported my efforts while I completed my
research.

I would also like to acknowledge Scott Emery for the many conversa-
tions, engineering and.otherwise, Paul Ferraiuolo for his understanding of
my commitment to this task, and my father for our conversations about my
research.

Special thanks to my wife Paula. Her patience through the six years
of this pursuit made it all possible. Without her encouragement I would
have never returned to school. Thank you.

And last but not at all in the least, thank you to my three children
Elizabeth, Olivia, and.Daniel for their patience with their daddy while he

"did his homework".

vi

TABLE OF CONTENTS

LIST OF TABLES ..................................................... ix
LIST OF FIGURES ..................................................... x
NOMENCLATURE ..................................................... xiii
INTRODUCTION ........................................................ 1
CHAPTER 1 LITERATURE REVIEW ........................................ 4
CHAPTER 2 SYSTEM BACKGROUND AND DEFINITION ........................ 10
2.1 REVIEW OF AUTOMOTIVE ENGINE ISOLATION STRATEGIES ....... 11
2.1.1 Center of Percussion Mounting .................... 11
2.1.2 Natural Frequency Placement ...................... 11
2.1.3 Torque Axis Mounting ............................. 13
2.1.4 Elastic Axes Mounting ............................ 14

2.2 SYSTEM DEFINITION: PLANAR RIGID BODY ON RESILIENT
SUPPORTS ............................................... 16
2.2.1 Problem Geometry ................................ 16
2.2.2 Equations of Motion .............................. 17
2.2.3 Scope of Analysis ................................ 24
CHAPTER 3 SOLUTION OF THE EQUATIONS OF MOTION ..................... 30
3.1 PERTURBATION ANALYSIS .................................. 30
3.2 CASE 1 (Vertical Forcing) .............................. 35
3.3 CASE 2 (Torsional Forcing) ............................. 39
3.4 CASE 3 (Torsional and Vertical Forcing) ................ 42
3.5 REMARKS ................................................ 46
CHAPTER 4 SYSTEM PARAMETERS AND TAYLOR SERIES VERIFICATION ........ 47
4.1 SYSTEM PARAMETER VALUES ................................ 47
4.2 TAYLOR SERIES VERIFICATION ............................. 49
4.2.1 Case 1 (Vertical Forcing) ........................ 51
4.2.2 Case 2 (Torsional Forcing) ....................... 52
4.2.3 Case 3 (Torsional and Vertical Forcing) .......... 56
4.3 REMARKS ................................................ 57
CHAPTER 5 SYSTEM FREQUENCY RESPONSE .............................. 58
5.1 ANALYSIS METHODS ....................................... 59
5.2 RESPONSE OF CASE 1 (Vertical Forcing) .................. 61

vii

5.3 RESPONSE OF CASE 2 (Torsional Forcing) ................. 67
5.4 RESPONSE OF CASE 3 (Torsional and Vertical Forcing)....70
5.5 RESPONSE OF CASE 4 (Torsional and Vertical Forcing,
Reduced amplitude Forcing) ............................. 73
5.6 MEASURES OF ISOLATOR PERFORMANCE ....................... 76
5.6.1 Transmissibility ................................. 77
5.6.2 Effectiveness .................................... 78
5.6.3 Transmitted Power ................................ 80
5.7 VIBRATION ISOLATION MEASURES FOR THE RIGID BODY
SYSTEM ................................................. 81
5.7.1 Transmissibility ................................. 82
5.7.2 Effectiveness .................................... 83
5.7.3 Power Transmission ............................... 83
5.7.4 Forces Transmitted to the Foundation ............. 84
5.8 ISOLATION PERFORMANCE OF THE PLANAR THREE DEGREE OF
FREEDOM SYSTEM ......................................... 84
5.9 REMARKS ................................................ 86
CHAPTER 6 SUMMARY AND CONCLUSIONS ................................ 88
6.1 SUMMARY ................................................ 88
, 6.2 CONCLUSIONS ............................................ 91
6.3 RECOMMENDATIONS ........................................ 92
APPENDIX A CENTER OF PERCUSSION ................................... 96
APPENDIX B DERIVATION OF THE POTENTIAL ENERGY OF THE RIGID BODY
SYSTEM ................................................ 101
APPENDIX C MATHEMATICA PROGRAM USED FOR THE DERIVATION OF THE
EQUATIONS OF MOTION ................................... 105
APPENDIX D NONDIMENSIONALIZATION OF THE EQUATIONS OF MOTION ...... 109
LIST OF REFERENCES ................................................ 114

viii

LI ST OF TABLES

Table 2.1

A subset of possible natural frequency/forcing

frequency combinations ............................................ 26
Table 2.2

Summary of forcing conditions ..................................... 27
Table 2.3

Summary of the cases investigated ................................. 28
Table 5.1

Comparison of linear and nonlinear amplitudes for case 3 .......... 73

ix

LIST OF FIGURES

Figure 2.1
Geometry of the rigid body system ................................. 17

Figure 2.2
Flow chart of the Taylor series evaluation process ................ 22

Figure 2.3
Physical dimensions of the rigid body ............................. 23

Figure 4.1
Time response envelopes from the cubic equations
Case 1 (Vertical forcing) ......................................... 52

Figure 4.2
Time response from the reduced equations
Case].(Vertica1fcrcing) ......................................... 52

Figure 4.3
Time response envelopes from the actual equations
Case 2 (Torsional forcing) ........................................ 54

Figure 4.4
Time response envelopes from the quadratic equations
Case 2 (Torsional forcing) ........................................ 54

Figure 4.5
Time response envelopes from the cubic equations
Case 2 (Torsional forcing) ........................................ 55

Figure 4.6
Time response from the reduced equations
Case 2 (Torsional forcing) ........................................ 55

Figure 4.7
Time response envelopes from the cubic equations
Case 3 (Torsional and Vertical Forcing) ........................... 56

Figure 4.8
Time response from the reduced equations
Case 3 (Torsional and vertical forcing) ........................... 57

Figure 5.1

Frequency response for X1, case 1, horizontal response ............ 62
Figure 5.2

Frequency response for'Xé, case 1, rotational response ............ 62
Figure 5.3

Frequency response for X3, case 1, vertical response .............. 63
Figure 5.4

Frequency response for'Xq, case 1, horizontal response ............ 65
Figure 5.5

Frequency response for'Xé, case 1, rotational response ............ 65
Figure 5.6

Frequency response for X3, case 1, vertical response .............. 66
Figure 5.7

Frequency response for X1, case 2, horizontal response ............ 68
Figure 5.8

Frequency response for'Xé, case 2, rotational response ............ 69
Figure 5.9

Frequency response for'Xg, case 2, vertical response .............. 69
Figure 5.10

Frequency response for X1, case 3, horizontal response ............ 71
Figure 5.11

Frequency response for X2, case 3, rotational response ............ 71
Figure 5.12

Frequency response for’Xg, case 3, vertical response .............. 72
Figure 5.13

Frequency response for X1, case 4, horizontal response ............ 74
Figure 5.14

Frequency response for X2, case 4, rotational response ............ 75
Figure 5.15

Frequency response for X , case 4, vertical response .............. 75
Figure 5.16

Single degree of freedom isolation system ......................... 78
Figure 5.17

Response plot of the forces transmitted to the base ............... 86
Figure A.l

Compound pendulum system ......................................... 98

xi

Figure A.2

Compound pendulum acted on by an impulse .......................... 99
Figure B.l

Geometry of the rigid body system ................................ 102
Figure B.2

Geometry associated with springs k1 and k2 ....................... 103
Figure 8.3

Geometry associated with springs [<3 and k4 ....................... 104

xii

NOMENCLATURE

Half width of the rigid body (m)

Half height of the rigid body (m)
Coefficient of viscous damping

Time derivative with respect to time Ti
Velocity effectiveness (dimensionless)

Force effectiveness (dimensionless)

Force transmitted to the foundation

Nondimensionalized form of Fi

Forcing amplitude (N)

Transmitted force (N)

Applied impulse (kg-m/s)

Mass moment of inertia (mg-kg)

Moments and products of inertia (ma-kg)
Radius of gyration about an axis through G
Spring constant (N/m)

Length (m)

Lagrangian (N-m)

Spring length (m)

Mass (kg)

Generalized forces

xiii

Radius (m)

Remainder of a Taylor series expansion
Kinetic energy (N-m)

Time scale

Velocity transmissibility

Potential energy (N-m)

Coordinate variables x, a, and y respectively
Angular displacement (radians)

Nondimensionalized form of fit

Coefficient produced by a Taylor series expansion
Phase parameter

Small parameter used in multiple scales analysis
Dimensionless damping coefficient

Ratio .3.
A

Detuning parameter
Angular velocity (rad/s)
Natural frequency (rad/s)

Forcing frequency (rad/s)

xiv

 

 

Equ

INTRODUCTION

This thesis documents an analytical investigation into the harmonic
response and vibration isolation of a rigid body mounted on resilient
supports. The rigid body system, with two translational and one rotational
degrees of freedom, is a planar analog of an automotive engine on mounts.

In the automotive industry, emphasis is being placed on producing
automobiles with a ride that is smooth and quiet. This has increased the
importance of the isolation iof engine vibrations from the rest of the
automobile. The analytical methods currently used in the automotive
industry to determine the position and orientation of engine mounts, i.e.
elastic axes theory, torque axis theory, and natural frequency placement,
have shortcomings which prevent them from establishing an accurate
prediction of the harmonic response of the system. Consequently, the
placement and orientation of the engine mounts predicted by these methods
will be less than optimum, and experimental tuning of the system will be
required prior to production. The most salient of these shortcomings is
the fact that linear system theory is used in the formulation of these
methods. The work in this thesis demonstrates that the response of this
system will often be nonlinear.

The equations of motion for this system are derived using Lagrange's

equations. It is desirable that these equations contain no simplifications

such that the general nonlinear response of the system can be
investigated. However, due to the complexity of the resulting equations,
the potential energy of the system is written using a Taylor series. This
simplifies the nonlinear terms in the equations of motion to polynomial
form enhancing the application of a perturbation solution.

The method of multiple scales is used to effect a solution of these
equations of motion. This technique ultimately leads to six nonlinear
algebraic equations, the solutions of which are the steady state response
amplitudes of the system. The response from these equations is pursued for
parameter values appropriate for an in-line four cylinder engine. The
response of this system is found for four different forcing conditions.
The stability of the solutions is established through examination of the
of the eigenvalues of the Jacobian matrix of these algebraic equations.
This system response is then used to analyze the isolation performance of
the system.

The contents of this thesis are divided into six chapters. The first
chapter presents a review of the literature pertinent to the major topics
addressed in the thesis: vibration isolation and harmonic response of
rigid bodies.

Since this investigation was motivated partially from interest in
the vibrational response of an automotive engine on mounts, the second
chapter begins with a review of existing automotive engine mounting
strategies. After this review, the problem geometry and system parameters
for a three degree of freedom rigid body on resilient supports are
presented. This system represents the planar equivalent of an engine
supported on mounts. The equations of motion of the system are derived

using the Lagrange formulation. Details of the assumptions and

 

we!

 

0f

3
approximations made in arriving at the governing equations of motion are
given. The resulting equations of motion for this system are nonlinear,
second order, ordinary differential equations. The approximate solution to
these equations is pursued using a perturbation technique. The chapter
ends with a presentation of the general scope of the analysis and details
the three cases to be investigated.

The third chapter presents the perturbation analysis of the
equations of motion using the method of multiple scales. This method is
applied to each of the three cases resulting in six nonlinear ordinary
differential equations for each case. These equations describe the slowly
varying amplitudes and phases of the system.

The fourth chapter presents the numerical values of the system
parameters used in the balance of the thesis to obtain the system
response. The second part of the chapter utilizes these parameter values
to demonstrate verification of the use of a Taylor series approximation.of
the potential energy used in the derivation of the equations of motion.

The fifth chapter presents the system response plots for each of the
three cases. These response plots are obtained by repeatedly solving
nonlinear algebraic equations for the steady-state amplitudes as a
function of the forcing frequency. The final section of the chapter
considers the vibration isolation aspect of the system. It begins with the
study of the various approaches for the quantification of vibration
isolation of rigid bodies presented in the literature. A modified version
of one of the measures is used to assess the isolation performance of the
planar rigid body problem considered herein.

The sixth chapter presents a summary of this thesis, the conclusions

of the investigation and recommendations for future work.

CHAPTER 1 LITERATURE REVIEW

The majority of the information. about the topics of ‘harmonic
response and passive vibration isolation of rigid ‘bodies has been
generated by the automotive industry. Therefore, the majority of the
papers reviewed deal with the simulation, harmonic response and vibration
isolation of multi-cylinder engines.

One of the earliest detailed.works addressing vibration.isolation of
rigid bodies is Vibration and Shock Isolation, (Crede, 1951). In this
book, vibration isolation is viewed from two perspectives. The first is
that of reducing the forces transmitted to the supporting structure and,
the second is the isolation of rigid body displacements. Crede (1951)
addresses these two problems utilizing the same approach, that of tuning
the system natural frequencies to be substantially lower than the forcing
frequency of the input. Crede (1951) considers one, two and three degree
of freedom linear, planar rigid body systems with various degrees of
symmetry. Since his approach is to place the natural frequencies of the
system far from the exciting frequency, much attention is placed on
obtaining expressions for the natural frequencies of the systems
considered.

Another book that addresses the topic of vibration isolation of
rigid bodies is Vibration Engineering (Wilson, 1959). This book examines
primarily the vibrational response of reciprocating engines and rotating
machinery. This work considers the vibrational characteristics of multi-
cylinder engines in detail. The equations of motion for a six degree of

freedom rigid body system are derived using the Newtonian formulation. The

l;

5

assumption is made that a displacement along any one of the three
coordinate axes produces no force in the springs along the other two
coordinate axes. This assumption reduces the number of terms that appear
in the equations of motion. The displacements of the systems are assumed
to be small. Therefore, the restoring forces and couples are written as
linear expressions of the displacement variables. This results in
equations of motion with only linear coupling terms.

As with the book by Crede (1951) , considerable attention is paid to
obtaining expressions for the natural frequencies of the various systems
considered. The vibration isolation aspect of the text focuses on the
prediction and management of the forces transmitted to the support
structure (which throughout the majority of the text is considered rigid).
Various other topics are also considered such as vibration of multi-
cylinder engines, elastic characteristics of the foundation, systems with
various degrees of symmetry, and information about rubber isolation
mounts.

An extensive review article about vibration isolation (Snowdon,
1979), looks at simple mounting systems, primarily single degree of
freedom systems. This article presents information and theory which
address vibration isolation systems for which a single degree of freedom
system is too simple to adequately describe system response. In addition,
the article presents information regarding the static and dynamic
properties of rubber and rubber-like materials, and some experimental
aspects of isolation measurement. The article contains a lengthy
bibliography of vibration isolation materials.

Since the time these three references were written, numerous papers

have been written that specifically address the harmonic response of

6

multi-cylinder engines. Various approaches have been taken to simulate
this rigid body'system. These include Radcliffe, et. al., (1983); Butsuen,
et. al., (1986); Matsuda, (1987); Furubayashi, (1991); Shiomi and
Mizuguchi, (1991); Xuefeng, (1991); and Hata, (1992). These papers focus
primarily on the modeling and simulation of the engine as a rigid body
system. In these papers, linear system theory is used to investigate the
response of an engine on a mount system.

Correspondinglyy a number of articles have investigated the harmonic
response of the engine mount. These include John and Straneva, (1966);
Schmitt and leingang, (1976); Sakamoto, et. al., (1981); Gennesseoux,
(1993). These articles look.specifically’at the function, characteristics,
and performance of the engine mount itself.

Of particular relevance to the topic of this thesis are those papers
which deal with the performance of vibration isolation systems utilizing
nonlinear characteristics in the suspension system. Eight papers were
found which explore this particular topic (Tobias, 1959; Henry and Tobias,
1959; Grootenhuis and.Ewins, 1965; Efstathiades and Williams, 1967; Shoup,
1971 and 1972; Shoup and Simmonds, 1977; Metwalli, 1986). The papers by
Shoup (1971 and 1972) and Shoup and Simmonds (1977) investigate single
degree of freedom systems possessing nonlinear force-displacement
characteristics. Metwalli (1986) considers a two degree of freedom system
representing a vehicle suspension model. In addition to nonlinear
stiffness in.the springs, nonlinear damping is also included in the system
model.

The article by' Grootenhuis and. Ewins (1965) investigates the
unforced response of a rigid body on resilient supports. The Lagrange's

equations are used to obtain the equations of motion for the six degree of

 

the
Con,
the
00115

that

7

freedom rigid body system. The equations of motion are computed with the
assumption that the rigid. body displacements are small. Hence the
expression for the potential energy is simplified and only linear terms
appear in the equations of motion. These equations are then used to find
the natural frequencies of the system. Specifically, the change in these
natural frequencies as a function of the location of the center of gravity
is demonstrated.

In the paper by Efstathiades and Williams (1967), the coupled
response of two of the six degrees of freedom of a rigid body system with
supports at four corners is investigated. The corner supports possess
symmetric cubic force-displacement:characteristics. Internal.resonances of
the system are investigated using the method of harmonic balance. In
contrast, the resiliently supported rigid body system investigated in this
thesis demonstrates that supports with nonlinear force-displacement
characteristics are not required for a rigid body system to produce
nonlinear response. Nonlinear response is demonstrated analytically with
supports having linear symmetric force-displacement characteristics.

Two articles were found which looked at the effects of nonlinear
damping on the performance of isolation systems (Ruzika and Derby, 1971;
ElMadany and ElTamini, 1990). Nonlinear damping is not considered in this
thesis.

The response of a three dimensional rigid body system requires that
the equations of motion of the rigid body on resilient supports be written
considering six degrees of freedom. The two approaches to this problem are
the Newtonian formulation and Lagrange formulation. The former is
considered by Himelblau and Rubin (1961). In this paper, it is assumed

that the supports have independent linear force-displacement

8
characteristics along each of the three mutually perpendicular coordinate
axes. Only small translations and rotations are considered, which reduces
the complexity of the derivation by neglecting certain components of the
restoring force. The six resulting idifferential equations of" motion
contain only linear terms.

The Lagrange approach to the derivation of the equations of motion
is presented by Andrews (1960). No assumptions are made in this paper
regarding symmetry of the body or the location or orientation of the
resilient supports. The supports are assumed to have independent linear
force-displacement and linear viscous damping characteristics along each
of the three mutually perpendicular coordinate axes. The assumption of
small rotations of the rigid body is made, eliminating the need for Euler
angles to represent the rotational motion of the rigid body. Therefore,
the three angular velocities can then be‘written.as the time derivative of
its corresponding angular displacement. This enables the equations to be
written in a simpler form. The resulting differential equations of motion
are nonlinear.

The use of Lagrange's equations for the derivation of the equations
of motion is the method used later in this thesis to obtain the general
equations of motion for the resiliently supported rigid body system
considered herein. This system is a planar three degree of freedom
equivalent of the system considered in the above derivations by Andrews
(1960) and.Himelblau and.Rubin (1961). For this particular case, the small
angular displacement assumption. is :not needed. to ‘write the angular
velocity as a time derivative of an angular displacement. Hence, the
resulting equations of motion for this planar system are completely

general.

9
With the pertinent literature related to the topics of this thesis
identified, the analysis of the harmonic response of a rigid body on
resilient supports can be pursued. The following chapter initiates this
process by looking first at the methods used in the automotive industry to
isolate engines. The geometry of the system to be investigated in this
thesis is then.presented, and the derivation of the equations of motion of

the system is presented.

CHAPTER 2 SYSTEM BACKGROUND AND DEFINITION

Since the inception of the automotive industry, vibration of the
engine has been an identified problem. Various isolation schemes have been
devised to deal with this problem. Four methods have been identified in
the literature that are used in the automotive industry to address this
problem. They are: center of percussion mounting, natural frequency
placement, torque axis mounting, and elastic axes mounting. A review of
the literature indicates that the latter three techniques are more widely
used and investigated” All four techniques are described.here to establish
a comprehensive background about engine mounting strategies that consider
the motion of the engine as a rigid body only. The short review of each
method contains the pertinent references that present the development and
theory 'behind each. method. In. each case, discussion regarding the
assumptions made in the development of the method is presented and the
limitations of each method are also discussed. Vibration isolation methods
which consider the flexural motion of the automotive engine and drivetrain
are outside the scope of this thesis (Bolton-Knight, 1971).

Subsequent to this review, a simple problem of a planar rigid body
on resilient supports is introduced. This problem is the planar equivalent
to the grounded engine on mounts problem (the support structure is
considered rigid) restricted to three degrees of freedom. The general
equations of motion are derived in this chapter, with the perturbation
analysis and the system response following in later chapters. The bearing

of the results of the investigation of the response of this planar system

10

11
on the four existing engine isolation schemes is delineated in the sixth

chapter.

2.1 REVIEW OF AUTOMOTIVE ENGINE ISOLATION STRATEGIES

2.1.1 Center of Percussion Mounting

This technique utilizes the fundamental mechanical phenomenon known
as the center of percussion. One property of the center of percussion is
that for a compound pendulum acted on by an impulse applied through the
center of percussion and perpendicular to the line defined by the center
of mass and the fixed point, no reactive impulse results at the fixed
point. (A detailed presentation of the definition of the center of
percussion and the properties associated with this point is given in
Appendix A.) The application to engine mounting is direct. If the front
and. rear engine mounts are arranged such that their locations are
reciprocal centers of percussion (see Appendix A), then an impulse to one
mount from a road disturbance results in little or no reaction at the
other mount (Wilson, 1959; Timpner, 1965; Bolton-Knight, 1971). This
improves the overall isolation performance of the mounting system with
respect to impulsive inputs. Although the application of this isolation
scheme to actual mounting problems is typically not as simple as this
example would indicate, placement of the engine mounts consistent with

this theory will enhance their overall isolation effectiveness.

2.1.2 Natural Frequency Placement
Algorithms based on optimization routines have been developed which

aim to move the rigid body linear natural frequencies of the engine away

12
from the frequencies of the input sources, thereby changing the mode
shapes and reducing the transmitted forces. Numerous papers have been
written about this topic, including Johnson and Subhedar, (1979); Bernard
and Starkey, (1983); Geck and Patton, (1984); Spiekerman, Radcliffe, and
Goodman, (1985); Staat, (1986); and Saitoh and Igarashi, (1989).

In these studies the engine is modelled as a rigid body with six
degrees of freedom mounted on resilient supports attached to ground. The
design parameters include the mount stiffnesses, mount locations and
orientations. These methods depend largely on the ability to predict or
determine the natural frequencies of’ the engine. This task. can ‘be
difficult, and the results of this isolation method are only effective
near the identified frequency. As such, this technique is used primarily
to address vibrational problems which are due to the engine idle frequency
or one of its associated orders being close to a natural frequency. Of
course the trivial solution.of the optimization process, that of a natural
frequency being equal to zero, must be avoided. This is accomplished by
incorporating a function which penalizes large parameter changes. A
modification of this approach has been implemented which determines the
mount stiffnesses by directly minimizing the forces transmitted through
the mounts to the vehicle structure at idle (Oh, Lim and Lee, 1991;
Swanson, Wu, and Ashrafiuon, 1993).

More recently, optimization methods have been used.tx) analyze a
rigid body mounted on resilient mounts attached to a flexible support

(Ashrafiuon, 1993; Lee, Yim, and Kim, 1995).

13
2.1.3 Torque Axis Mounting

The torque axis, also referred to as the torque roll axis, is
defined as the resulting fixed axis of rotation of an unconstrained three
dimensional rigid body (i.e. either free or elastically supported on very
soft springs) when a torque is applied along an axis not coincident with
any of the principal axes (Timpner, 1965; Fullerton, 1984). For the case
of an automotive engine, the axis about which torque is applied is the
crankshaft axis. This axis is rarely coincident with a principal axis of
the engine. It is hypothesized (Timpner, 1966) that the disturbances
transferred to the vehicle can be reduced by positioning the engine mounts
such that the engine oscillates predominantly about this torque axis. The
two references above each give a method for determining the location of
the torque axis of an engine.

In order to better understand the concept of torque axis and
appreciate some of the shortcomings of this approach as used in the
automotive industry, consider the equations of motion governing the
rotational behavior of a rigid body in three dimensions acted on by the
moment vector H. These equations are given by (Greenwood, 1988):

”x = Inns. + Ixymy Wm) + lam. +4.0»)
* (In 'Iw)“*/"’z * In (“’3 “’33)

My =Ixy(“'5i”‘9“‘z) *Iyyd’y *Ivg("3‘zz*‘k“’v) (2.1)
4' (Ixx "122) “it": I Ixz (“’2 Tax)

”2 = Ixz (“ii ”99%) T 1,205» +q‘mz) 1' 12202
+ (Ivy-Imago» + 1"“): -wi)

For the case of an automotive engine, it is common to assume the
engine can be assumed to be a rigid body on resilient supports which are
attached to ground. It can be further assumed that the values of the

moments, both the applied moments and the moments generated by the

 

 

0-!)
L_J

a)
kg

9
LI};

Ara:

l4
reaction forces at the mounts, and inertia products are known or can be
calculated. Then, for a given set of initial conditions, a can
theoretically be solved for as a function of time. In general, the
magnitude and direction of the axis of rotation of the rigid body will
also be a function of time. Thus, the torque axis as defined previously,
does not exist.

It is further noted that equations (2.1) are derived under the
assumption that the inertia properties of the engine are independent of
time. This is not the case for an engine, because the motion of the
pistons, cranks, connecting rods, and the crankshaft make the inertia
properties periodic functions of time, independent of the coordinate
system. This further influences the time dependent nature of a in an
actual engine. However, the time dependence of the inertia properties is

typically neglected in practice (Bachrach, 1995).

2.1.4 Elastic Axes

Elastic axes for an elastically supported rigid body system are
those axes for which application of a force or torque, along or about the
axis produces only a corresponding translation or rotation, respectively.
In the coordinate system defined by the elastic axes, the system response
consists of decoupled translational and rotational modes.

The elastic axes of a rigid body on flexible supports are determined
using the flexibility matrix A. Analytical modal decoupling of the
flexibility matrix does not yield the elastic axes system because the
eigenvectors do not typically span a physical space. Hence, the
transformation to the elastic axes must be a physical coordinate

trans formation.

 

 

In

001
um

150

15

Since a physical coordinate transformation is required, a 2-
dimensional system with three degrees of freedom can be decoupled since
the six off-diagonal terms of the symmetric 3X3 flexibility matrix can be
eliminated by two independent translations and one rotation. Full
decoupling of the symmetric 6X6 flexibility matrix for a 3-dimensional
rigid body system with six degrees of freedom cannot be accomplished since
30‘ off-diagonal terms exist and only three independent coordinate
translations and three independent coordinate rotations can be defined
(Kim, 1991). Due to this limitation, effort has been spent in the area of
"partial" decoupling of the modes of vibration (Derby, 1973; Ford, 1985).

These four methods of automotive engine vibration isolation
represent those documented in the literature. A common assumption in the
implementation of these mounting strategies is that the response of the
system is linear. This assumption is made to simplify the analysis of the
system.

Recently, hydraulic engine mounts have proven to be effective
vibration isolators and have replaced traditional elastomeric mounts in
many automobiles. The experimental and analytical response Characteristics
of this class of engine mounts have been investigated and are inherently
nonlinear (Kim, Singh, and Ravindra, 1992; Brach and.Haddow, 1993; Kim.and
Singh, 1993). These nonlinear characteristics have been introduced
intentionally to improve the isolation performance of the engine on mounts
system. However, despite the apparent effectiveness of these mounts, no
investigations have been found which establish the response of the
complete system of an engine supported by such mounts. A clear
understanding,of the system response will improve the consideration.of the

isolation aspects of the system. Therefore, to better comprehend this

16

class of vibration system, the problem of determining the response and
isolation performance of a rigid body on resilient supports is considered.

The most general approach to this problem is that of a three
dimensional rigid body supported by arbitrarily located resilient
supports. This problem has six degrees of freedom: three translational and
three rotational. The problem considered presently is a two dimensional
rigid body on resilient linear supports. Linear supports are used since it
will be shown through this analysis that the geometry of the problem
introduces nonlinear response. The same procedure used in the solution of
this problem can be followed for supports with nonlinear characteristics.
This planar problem has three degrees of freedom: two translational and
one rotational. The problem was made planar to increase the tractability.
The following section introduces the specific geometry of the problem

investigated.

2.2 SYSTEM DEFINITION: PLANAR RIGID BODY ON RESILIENT SUPPORTS

2.2.1 Problem Geometry

The ensuing analysis is based on the geometry of a planar rigid body
mounted on resilient linear supports as shown in Figure 2.1. Each of the
mounts is characterized by two orthogonal stiffness values, one for the
axial stiffness in the x-direction and one for the lateral stiffness in
the y-direction. The dimensions shown in the drawing for the supports are
exaggerated to emphasize the geometry used in the analysis. The dimensions
utilized for the rigid body in the numerical analysis which follows were
chosen to closely approximate an actual 4-cylinder automotive engine. The

free lengths of the springs and the spring rates were chosen to be

17
representative of production elastomeric mounts. For this problem, the
stiffness values were chosen such that k1 - k4 and k2 = [(3 and the mounting
geometry is symmetric. The numerical parameter values chosen for all
problems will be given prior to the presentation of the response of the
system. It should be noted that the procedure used to analyze this system
with linear spring characteristics can also be used to analyze rigid body

systems with nonlinear spring characteristics.

 

 

 

 

 

Figure 2.1 - Geometry of the rigid body system

Figure 2.1 depicts the rigid ‘body in the static equilibrium
position. The static equilibrium position is used in this instance as a

more convenient point about which to write the equations of motion.

2.2.2 Equations of Motion
The equations of motion for this problem are derived using

Lagrange’s equations:

(1 3L _aL=. 2.2
Rim; 7?. Q! ‘ ’

where L - T - V, T is the kinetic energy, V is the potential energy, qj.are
the generalized coordinates, and Qj.are the generalized forces applied to

the system.

18
Using the notation given in Figure 2.1, the kinetic energy for the

rigid body, T, can be written directly:
T = :1..me + gmyz + gIGa’?‘ (2-3)

The potential energy for the problem, V, is complicated. For this
problem, the general equations of motion are desired, and therefore no
simplifications of the potential energy are made. The potential energy of

this system is the sum of the potential energies of the four springs. The

potential energy of a given spring, gkéz, where 6 is the deflection in the

spring and k is the linear spring constant, requires for this problem that
the deflection 6 be written in terms of the rigid body displacements x, y,
and a. For this problem, 5 for each spring is determined by writing the
expression for the length of the spring in the deflected state and
subtracting from this quantity the undeflected length of the spring,
(i.e. , the free length). The complexity of the expression for the
potential energy for this geometry is simpler than for the general case of
arbitrarily located springs. This is due to the fact that the attachment
points of the springs are collinear with the center of mass of the rigid
body and with each other. Appendix B presents more detail of the
derivation of the potential energy for this system.

The next step is to take the partial derivatives of this expression
with respect to x, y, and a. However,the resulting form of the equations
of motion will be complicated, and not conducive to the solution technique
of the method of multiple scales used later. It is more convenient to have
the equations appear in polynomial form. Therefore a Taylor series

expansion of the potential energy is performed prior to the implementation

 

He:
P6:
and

19
of the partial derivatives required by the Lagrange formulation of the

1

equations of motion . The multi-variable version of the Taylor series is:

f(x,y,a) = f(Xo,Yo,ao)

“*[[(11f‘xo)-::.;—ir * (TWO) 3; * (“110) 3;] f(x'7'“)]l<xo.yo.¢o)

+ E1?[[(x-x0) '33" 4' (Y'Yo) a * (0'30) 33] 2f(x:Y:a)]l(xo.Yo.¢o>

+ 31T[[(x-xo) .3; + (y-yc) 3; + (a-ao) gal3f(X.y,a)]l(xo,yo,ao) (21‘)

—1

+ 2%I[(X’Xo)g§ * (Y‘Yo) g; * (a‘ao) 5;]nf(x'y'a)]l<xo.Yo-ao>
n . J

+Ru(x,y.a)

where

RN(x,y,a) =%[(x-xo)§i +(y-y0)% +(a-ao) ga]"f(x,y,a) “5.11.0 (2.5)

and (E,n,C) is a point somewhere on the straight line segment between
(x0,yo,a0) and (x,y,a).

The derivation of the Taylor series expansion in two variables
(Hildebrand, 1976) was expanded to three variables to give the equations
above. This treatment can be further generalized for expansions of
variables of four or more variables. The Taylor series for the potential
energy for this problem is taken about the static equilibrium position,
(xo,yo,ao) = (0,0,0). Hence, this is the point where the series is
evaluated to establish the series coefficients. This position is chosen
since the reference coordinate system is fixed in space at the location of
the static equilibrium position as shown in Figure 2.1. Once the Taylor
series expansion of the potential energy is determined, Lagrange's

equations can be used to determine the equations of motion for the system.

 

1The order of the operation of the Taylor series and the partial
differentiation is not important, i.e. , these operations are commutative.
Hence, the application of the Taylor series approximation could be
performed on the appropriate terms after Lagrange's equations are obtained
and the same equations of motion would result.

 

 

CI

5 1;.

Ta';

20

With the application of the Taylor series, questions of the
convergence and accuracy of the series arise. The ratio test can be used
to Show that the Taylor series of a function f(x,y,a) about (xb,yb,ao), is
absolutely convergent for all real numbers if the function has derivatives
of all orders on some interval containing the point (x0,y0,a0). However,
for application of the Taylor series to this problem, the series must be
truncated to be useful, and the accuracy of this truncated series comes
into question. The remainder of the series Ru shown above, gives an
indication.of the order of magnitude of the error of the truncated series.

The question of what constitutes an acceptable error then arises.
Additionally, the exact.means of computation.also must be established. For
the remainder R", the choice of (6,0,0 needs to be made, and only then can
the numerical calculation of R% be performed. Using any method, a norm
could be established and the magnitude evaluated numerically, but the
interpretation of the magnitude of the norm still presents difficulty. One
alternative to this approach is to graphically compare the approximate
potential energy to the actual potential energy to ascertain the accuracy
of the truncated series over some appropriate range of system.coordinates.
Ultimately, it is only the applicability of the truncated series of the
potential energy to represent the actual potential energy which is
established.

The validity of the equations of motion derived using a Taylor
series approximation of the potential energy was established through a
comparison of the system response obtained by direct numerical integration
of the equations of motion using,a particular set of parameter values. The
system response obtained from the equations of motion derived using the

Taylor series approximation of the potential energy was compared to the

21
response given by the equations of motion derived with no approximation
for the potential energy. This process is depicted in the flow chart shown
in Figure 2.2.

Regardless of how closely the truncated series for the potential
energy might approximate the actual potential energy, the response of the
system using the equations obtained using the truncated series must
closely match the response of the system using the equations obtained
using the actual potential energy. If the response of the approximated
system does not match the response of the actual system satisfactorily,
terms must be added to the truncated series until the system responses
match. If the comparison shows that similar steady state responses are
obtained under identical forcing conditions, the truncated series was
considered acceptable.

The equations which are given by the above analysis do not include
damping. For this formulation, damping coefficients are introduced after
the application of the Lagrange equations. Damping is required in this
problem to bound the system response at resonance. Viscous damping
included in the Lagrange formulation through the use of Rayleigh's
dissipation function (Meirovitch, 1975) would introduce off-diagonal terms
in the system damping matrix. To avoid these coupling terms, a single
viscous damping term was added to each of the three equations. This
approach reduces the algebraic complexity of the analysis of the equations
of motion. Neglecting coupling damping terms in this analysis is
reasonable as only light damping is considered and prior experience has
shown that damping in these types of problems typically influences the
amplitude of response, not the type of response and only then, close to

resonant frequencies.

22

 

Lagrange Function, L = T - V

/

Taylor Expansion Exact v

 

 

 

 

 

Approximate V

 

 

 

 

 

 

 

Apply Lagrange's

 

 

 

Equations
Approximate nonlinear Exact nonlinear
equations of motion equations of motion

 

 

i i

Numerically integrate Numerically integrate
for a few test cases for a few test cases

 

 

 

 

 

 

 

 

 

Compare Solutions

i

 

 

Yes Indication is that the

 

Do the solutions agree?

i...

Add terms to the Taylor

 

 

series performs satisfactorily

 

 

 

 

 

series and evaluate again.

 

 

 

Figure 2.2 - Flow chart of the Taylor series evaluation process

The generalized forces, Qj, associated with each of the system
coordinates remain to be found. These forces are found using the principle
of virtual work (Meirovitch, 1975) . For this system, the generalized force
associated with each coordinate is the force acting in that direction.

The equations of motion for the system can now be assembled. The
equations can be put into a simpler format using the following coordinate

reassignment: x 4 X1, :1 -’ X2, and y -’ X3. The equations for the system are:

23

W21 * C1X1 * (k2 *k3)X1 * 51X2X3 * 52X1X3 * 53X: * 54X1X22

3 2 (2.6a)
+ 55X1 + 56X1X3 =F1cosa1t
IGiZ * C2X2 +A2(k1+k4)X2 + #7X1X3 + 58X: (2 6b)
+ #9X1X22 + p1ox12x2 + 511X2X§ =F2cosS22t
.. 2
mX3 + c3X3 + (k1+k4)X3 1* I312X1 * .313X1X2 * B14X§X3 (2.60)
where
IG=.;.m(A2+BZ) (2.7)

The system parameters A and B are shown in Figure 2.3.

+— 2._.;

 

 

 

 

 

 

Figure 2.3 - Physical dimensions of the rigid body

The terms c1, oz, and c3 in equations (2.6) represent the viscous
damping associated with each of the degrees of freedom. The coefficients
for the linear terms are given here in terms of system parameters. The
coefficients for the nonlinear terms, represented in equations (2.6) by
fij's, are not given in terms of system parameters because this
representation of these values is exceedingly lengthy. However, specific
numerical values are assigned to the parameters of the problem in Chapter

4. In addition, Appendix G contains the Mathematica code used to derive

 

 

U0

Vii

24
these equations of motion. It is noted that due to the symmetry of the
problem, no linear coupling terms appear in the equations.
The equations of motion for the system can also be stated in a
nondimensional form, given by equations (2.8). A detailed presentation of

the procedure used to arrive at these equations is given in Appendix D.

2'2, + 2,31x1 + «ix, + é1x2x3 + ézx1x3 + 63x? + &4x1xzz

. (2.8a)
+515x? +¢3z6x1x§ =f1cosn1t
1(1 ”2);:2 +2fi2x2 + k’xz +&7x1x3 *9‘81‘3 +&9x1x2
3 2 2 (2.8b)
4’ &10X12X2 4' &11X2X32 = f2COSta
x+2‘X+2x+‘x2+‘x +‘x2
3 #3 3 “3 3 0‘12 1 “13 1X2 “1!. 2X3 (2.36)

+ &15x12x3 + 5‘16"; = f3cosn3t

The parameter v is the ratio % where the dimensions A and B are shown in

Figure 2.3.

For the purposes of this investigation, the nondimensionalized form
of the equations are not used. No parameter studies are performed whereby
use of the nondimensionalized equations would.be advantageous. Hence, the
physical parameters of the system are utilized throughout the analysis.

This also makes direct comparison to the physical system simpler.

2.2.3 Scope of Analysis

The equations of motion for this system, given by equations (2.6),
fall into the category of weakly nonlinear systems. A system is weakly
nonlinear if, to the lowest order approximation, the system is always

viewed as a simple harmonic oscillator; the nonlinearity enters as a

25
correction term (Kahn, 1990). For this system, this point can be seen by

letting the fl} = 0 in equations (2.6). The resulting system is a set of

three uncoupled, simple harmonic oscillators. However, as will be shown
later, even if the amplitudes of the response are small, nonlinear
response can still be realized through appropriate tuning of the natural
frequencies.

Prior to presenting the perturbation analysis of the equations, it
is useful to establish the scope of the investigation within the context
of the perturbation analysis. For analysis of a nonlinear system, the
relationships between the linear natural frequencies, referred ix) as
internal resonance conditions, and the forcing frequencies are typically
the defining attributes between problems. For this geometry, the possible
relationships between the three linear natural frequencies (01, oz, and
03), the forcing frequencies (91, 92, and S23), and the presence or absence
of forcing (F3, F}, and F3) is considered. Table 2.1 presents a subset of
the many combinations. The number of possible entries in this table would
increase significantly if the forcing (F1,I3, and.F3) were also included
in generating the cases.

All of the cases listed. in 'Table 2.1 lead to resonance-like
conditions in which all modes are excited. There are many more possible
combinations. Several different combinations, based on a physical system,

are investigated.

 

 

 

 

 

 

 

 

 

 

fl Internal Resonance RelationShips External Resonance Relationships

ll o1=02, @1203 91:01, 91:"2’ Q3=o3
01=02,01=03 91*“1'923‘5’933‘3
w1=02, u1~o3 S21~o1, 92*”2’ £23=o3
o1=oz,o1~03 n1~o1,92~02,n3+03
o1zoz,o1zo3 91*”1’92102’ 93-103
o1zaz,o1=o3 91+01,92z02,93+o3
01:02, 01:03 .91zo1, {224-02, “3‘2;

 

 

 

Table 2.1 - A subset of possible natural frequency/forcing
frequency combinations

In this physical system, the characteristics and parameters for the
rigid body were chosen to represent an in-line four cylinder automotive
(four-cycle) engine at hot idle. For this engine, all the first order
forces2 and moments are theoretically balanced while certain second order
forces and moments are unbalanced (Den Hartog, 1984). Therefore, in the X1
direction (horizontal) for first order,.F1-=0. However, in actual engines,
it has been found that the first order moments in the X2 direction
(torsional), actually do not completely balance (Whitekus, 1994) and so
there is a component FZ oscillating at the idle speed :22. For the X3
direction (vertical), unbalance appears at the second order, 93 = 292 =

twice idle frequency. Table 2.2 depicts these results.

 

2First order refers to phenomena occurring at engine rotational speed
and is also referred to as primary. Second order refers to phenomena
occurring at twice the engine rotational speed and is also referred to as
secondary.

27

 

lst Order

2nd Order

 

Magnitude of this
force at this order
is equal to zero.

Unbalanced Horizontal
Force

Magnitude of this
force at this order
is equal to zero.

 

Magnitude of this
force at this order
is equal to zero.

Unbalanced Vertical
Force

Magnitude of this
force at this order
is not equal to zero.

 

Magnitude of this

Unbalanced Moment force at this order

 

 

is not equal to zero.

 

Magnitude of this
force at this order
is not equal to zero.

 

 

Table 2.2 - Summary of forcing conditions

Typically, the six rigid body linear natural frequencies of an
engine are between 5 Hz and 25 Hz. It is not unreasonable to expect that
there could be nearly simple integer relationships between the system
linear natural frequencies. Several papers that list the six linear
natural frequencies for actual engines (Johnson and Subhedar, 1979; Geck

indicate that such

1984; 1985)

and Patton, Spiekerman, et. a1.
relationships do occur in actual systems. Guided by these works and the
information in Table 2.2, the relationships between the linear natural
frequencies chosen for this study are the following:

01 z oz,

201 = 03 and (22 z (02, :23 z 03

with

F1 =ef1 =0, F2 =ef2 #0, and F3 =ef3 =0 (first order) (15)

and

F1=5f1=0: F2 =6 2 =0, and F3 =ef3 #0 (second order) (16)

28

The second order component of F2 is neglected. This reduces the
number of forcing terms, whiCh correspondingly simplifies the multiple
scales analysis. Inclusion of this forcing terms would create multiple-
term, multiple-frequency excitation. Hence, excluding this term improves
the understanding of the system response.

Therefore, the response of the system is analyzed in three cases of
increasing complexity. This process of investigating the simpler of the
three cases, cases 1 and 2, validates the methods required to analyze the
problem and creates confidence in the results generated in the more
complex case 3.

The first case, case 1, investigates the rigid body system with
forcing in only the X3-direction.(Fé = 0, 53 ¢ 0). The second case, case
2, investigates the system with forcing in only the XZ-direction.(P} a 0,
F3 - 0). The third case, case 3, investigates combined vertical and
torsional forcing (FE ¢ 0, F3 ¢ 0). Table 2.3 gives a summary of these

three cases.

 

 

 

 

 

 

 

 

= J
Natural Forcing
Frequencies Frequencies Force Magnitudes
Casel 01=02, 201=o3 Q3=o3 F1-O, 172-0, Flio
Case2 01:02, 201:03 92:02 F1-0, [72:30, F3-0
Case3 01=2,201=L92=0,93=o FL-O,F2¢O,Fg¢0‘

 

 

 

Table 2.3 - Summary of the cases investigated

Having derived the nonlinear differential equations of motion for
the system, the solution for the three cases designated can be sought. A
perturbation technique is used to find an approximate solution. Chapter 3

presents the perturbation analysis for each of the three cases. Chapters

29
4 and 5 present time and frequency response plots resulting from the

approximate solution for each of the three cases.

CHAPTER 3 SOLUTION OF THE EQUATIONS OF MOTION

This chapter presents the perturbation analysis of the equations of
motion presented in Chapter 2. The first part of the chapter presents the
analysis germane to all three cases. The analysis is then completed
through investigation of each individual case. For each case, a set of six
nonlinear differential equations is obtained through which the frequency
response of the system can be investigated. This aspect of the
investigation is presented in Chapter 5. Prior to this, Chapter 4 presents
verification of the approximation used for the potential energy of the
system. Chapter 4 also presents the system parameter values used in the

balance of the analysis.

3.1 PERTURBATION ANALYSIS

The modal interactions in this three degree of freedom planar rigid
body system are examined through perturbation analysis. The method of
multiple scales is used to effect a solution (Nayfeh and Mook, 1979). The
nondimensionalized equations, as presented in equations (2.8), are not
analyzed directly. The approach used is to perform the multiple scales
analysis on equations (2.6) and (2.7) and then to assign numerical values
to the constants such as the fli's, A, and B, etc. The frequency response
of the system can then be obtained. Two reasons motivated this approach.

The first reason is that as presented previously, a Taylor series
expansion is used to simplify the expression for the potential energy. A
general evaluation of the series approximation (treating all system

constants as parameters) would yield unwieldy equations of uwmion and

30

31
reduce the effectiveness of the series to simplify the equations of motion
and facilitate multiple scales analysis.

The second reason is that due to the complexity of the equations of
motion, aui analytical solution, although desirable, is precluded even
though multiple scales analysis is used to simplify the equations
describing the response of the system. Hence the solutions for all three
cases are generated numerically using the results from the multiple scales
analysis. Numerical values for the parameters are therefore required.
Accordingly, the system 'parameters in equations (2.6) are assigned
numerical values prior to the analysis. However, each equation is
simplified by dividing through by the coefficient of its respective
acceleration term. The equations then take a form similar to that of
equations (2.8).

The method.of multiple scales is used to obtain an approximation for
X1,.X2, and X3. The basic assumption in this method is that the time t is
viewed as being constructed from a succession of independent time scales.

This is shown by the equation:

Tn=ent ~for 0<e<1 and n=0,1,2,,,, (3.1)

The small parameter s is introduced to distinguish responses with
frequencies of close to 01, oz, and 03 occurring at the fast time scale

Tb:=t, and response that occurs at the slower time scale T1 = at. The

response at the slower time scale represents the modulations in the
response at the faster time scale produced by the nonlinearities, the
damping, and the excitation. Hence, these three terms must appear at the
same order of s. This is accomplished by ordering terms in equations (2.6)

and (2.7) and letting:

32

Ci—z -f '-13 a “if '-2

.E - ep, or i- , an .fG or 1-

131 . 5; .

”Hz-=60” for J=1,..,6,12,..,16 and TE=€aj for J=7,..,11(3.2)
Fl: ‘ Fk ‘

_ =€fk for k=2 and _ =osf|< for k=3

16 111

Thus, equations (2.6) then take the form:

X1 + 26H1X1 + 4X1 '* 5‘11X2X3 "' ea2X1X3 I €a3x§ (3.3a)
+ EG4X1X22 * €a5X13 4' €a6X1X32 = 0

22 4' 2€M2X2 'f' (AgXZ 4’ EQ7X1X3 + €08X23 4‘ £G9X1X§ (3.31))
4' 6&10X12X2 4' 6&11X2X32 = Efzcosgzt

" 2 2 2
X3 4' 2€p3X3 4' (13X3 4’ €012X1 4' 6&13X1X2 4‘ €Cf14X2X3 (3.3C)
+ ea15X12X3 + ea16X: = ef3cosn3t
Following the practice of the method of multiple scales and neglecting
terms of 0(52) , the solutions to equations (3.3) are expressed in the form:
Xj(t,'€) =on(T0,T1) 1‘ EXj1(T0,T1) where To = t and T1: EC (3-4)

The derivatives with respect to time must also account for the different

time scales. Accordingly, they become:

 

 

dam dam) 2
7%— =(Do'f'ED1) (Xi), dc; = (DO +ZeDoD1) (Xi) (3.5)
where
a x- a? X-
Dk(Xj) = 7:73?) and D:(Xj) = (2") (3.6)
k

Substituting equations (3.4)-(3.6) into equation (3.3) and equating

coefficients of like orders of 5 gives:

33

Z 2
BOX") +m‘x1o = 0
2 2
D0X20+6§X20 =0 (37)

2 2
Do x3o * “5X30 = 0

05x11 +m§X11 = -ZDoD1X1o ’2I11Dox1o “a1xzox3o ‘azxwxso -asx§’o
-.,x,,x§, -asxfo -.,x,,x;o

05x21 +m§xz1 = -200D1X20 -2#2D0X20 -a7x1ox3o 'aaxgo —a9x10x220 (3 , 8)
-a1ox120x20 -a11x20x§0 +f2cosnzTo

D§X31 * 0§X31 = '2DoD1X3o '2H3Dox3o '0‘121‘120 ‘a1sx1oxzo 'a1axzzox3o

2 3 ‘
" a15X1oX3o '016X30 4‘ f3COSQ3T0

The solution to equations (3.7) can be written in the form:

xio =A; (T1)e”“’i‘°’ we “-9)

for j = 1,2 and 3 and co denotes the complex conjugate of the preceding
term or terms. The Aj are complex functions of T1 and are determined later
in the analysis.

substituting equations (3.9) into equations (3.8) leads to:
Dozx“ +w§x11 = -2A1/iqei°'1° ~2p1A1iqeiqT°
-a1(A2A3e(Q+“5)T° +AZA3eu5-Qno)
-a2 (A1.43,,ei (6‘ "5’10 +A1A3ei“5-qn°)
-a3(A§e3i“P'0 ,3Agzzei0270) (3.10)
"at. (A1Azzei(q+2¢.p)To +2A1A21—128iqT0 +Z1Azzei(-q+2¢.2)10)

-a5 (afesiWO +3Af21ei“"‘°)

' T — ' T - ' +2 T
‘G6(A1A3281(q +205) 0 +2A1A3A3e‘0' 0 +A1A3261(-0| Q5) 0) +CC

34

Dgxz1+a§xz1 = -2A2’1aéei“2'° -2p2A21a2ei"9'°
_a7(A1A3ei(q+n5)To +Z1Aseiu5—qn0)
-a8(A23e3iQT° +3A§ZzeiQTo)
-a9(A1A22ei(q+2fieno +2A1A222e‘m0 +21A§ei"‘“*2“9’T°) (3.11)
-a10 (A1‘12A‘13ei‘2m‘"‘2’To +2A1A1A2eiqto +A1ZAZei (20' Weno)
_a‘11(AZAszei(aB+Z¢-5)To +2A2A3Z3eiqto +ZZA328R-o24r205n0)

f2 NT
20
+78 +C

C

2 z /, i T , i T 2 2i T -
DOX31+Q3X31 = “2143.16.36 “5 O’ZM3A31658 as 0-a12(A1e ”I 0+A1A1)

. T - ° - T
-a13 (A1A2e‘w' +02) 0 +A1A2€u0l 02) 0)

' 2 T - ' T - ' - T
«11,, (Ag/13.9" “2‘“5’ °+2A2A2A3e”‘5 ° +A§A3e"2""‘ “5’ 0) (3.12)
° 2 + T - ' T - 2 - T

3‘ T - ' T f ' T
-a16(A;e ”‘5 0+3A32A3em!’ 0) +7§em3 0«icc

where the Aj denote the complex conjugate of the Aj. Since the expansions

3, secular producing terms are

which are sought are uniformly valid in time
identified and eliminated from the right hand side of equations (3.10),
(3.11), and (3.12). In this way the expressions for the AJ- are determined.

The balance of the perturbation analysis addresses each of the three
cases separately since the secular terms for each case are distinct as
they depend on the relationships between the linear natural frequencies

and the forcing frequencies. The reduced equations for each of the three

cases will be derived prior to presenting the numerical results.

 

3For the notation used here, the approximation for Xi(t;e) of equation
3.4 should be such that small terms remain small for all time. That is, if
xio is the principal term in the approximation and Kit is the correction to
it, we want |exi1|<|xi0| for all time. When this is accomplished, the
resulting approximation is said to be uniformly valid in time (Kahn,

1990) .

35

3.2 CASE 1 - (Vertical Forcing)

Consistent with the method of multiple scales, it is appropriate to

introduce three detuning equations:
where 01 is an external detuning parameter and a2 and 03 are internal
detuning parameters. Using these relationships in equations (3.10)-

(3.12), the secular terms are eliminated by letting:
{-2.41’10. -2p1A11q)e“‘"’° -a1(AZA3ei“5-Q)T°)
-a2(Z1A3e”“‘-""'"”°) —a3(3A§ZZei“‘T°) (3.14)
-a4(2A1A2A-zeiqT° +A1A22eHZQ-qno)
-a5(3AfZ1e““T°) -a6(2A,A3Z3e““T°) =0

l. . i T — 1‘ " )T
{-2A2142-2u2A21a2N "2 ° '07(A1A39 “5 q 0)
— i T - i 1 - 2 NZ - )1
-a8(3A22Aze (.2 0) -a9(2A1A2A28 0| 0 +A1Aze “B (I! 0) (3.15)
_ - - ' - 1
- 010(2A1A1A281QT0 4’ A12A28‘I (2“ 02) 0)

" (111(2A2A3Z38H‘210) = 0

' ' T
(-2A3lia3 -2p3A3iag)e1gT° -a12A12e2‘q 0
i( + )T - i T
’d13(A1A28 0‘ “2 0) -014 (2A2A2A3e as 0) (3-16)

_ - _ . f .
‘ a15(2A1A1A3e‘gT°) -a16(3A32A3eNST°) + 73.810310 = 0
With further use of equations (3.13), equations (3.14)-(3.16) can be
simplified to:

/ , , -‘ i(02-a3)T1
-2A11.w, '2fl1A1lq - a1A2A3e

"" -' T " -' T
-a2A1A3e 1031 ’3Q3A22Aze 1021 (317)
-a4(2A1A2A2 +Z1A226

"' 305141221 " 2G6A1A3Z3 3 0

-2i02T1)

36

. , - ‘ .. 1' _
-2A2/ia2 -2p2A21a2 -c27A1A3e‘(a2 a3) 1 - 3a8A22A2
- a9 (2A1A222ew2'1 +A1A22e-WZT1) (3.18)

— — Z. t —
— am (2111.41.42 +A1ZA2e "’2 1) - 2a11A2A3A3 = 0

03T1 i(03-02)T1

/. . 2 i
~2A3165“2#3A3l&3 -a1ZA1e ‘G13A1Aze ‘ (3 19)
_ _ _. f - '
" 2014A2A2A3 - 2a15A1A1A3 - 3016A32A3 + 73: 6'01T1 =’ 0

To facilitate the solution of equations (3.17)-(3.l9), it is

advantageous to put them into autonomous form. This can be done by
letting:
Bk i¢kT1
AI: = 79

and solving for the ¢k such that all of the exponential terms have the same
exponent. Utilizing the values of ¢k which result from this process, the

Ak can be rewritten as follows:

 

.01‘03
A1=_2.e
B .2231, (3.20a.b.C)
A = 26 2
2 7
B3 1011"]
A3378

These relationships can be substituted into equations (3.17)-(3.l9)
with Bk - Bkr + inj for k -= l, 2, and 3. Here the subscript r designates
the real component of the term and the subscript j designates the
imaginary component of the term. Taking the resulting equations,
separating the real and imaginary terms gives the following six equations

governing the behavior of the uj's in time T1:

37

U/ = U2 (01-03)-p1u1- a1 (U3U6‘U4U5)’ a2 (U1U6-U2U5)
‘ 7 Zr»: 36’:

3
-_8%’ (ugul. +u2) -.;:_’.'. (nan: +3u2u42 +2u1u3u4) (3- 213)
3&5
-.8_.(u12u2+u;)- za_(u2u52 +u2u62)
/ u a a
U2 = '71. (O1 ‘03) -[.l1l.12 +2—1- (1.13115 +U4U6) +2: (U1U5 +UZU6)
3a
+3.2(u3 +u3u 2)+.8_ (3u1u32+u1uf +2u2u3u4) (3-21b)
3a
+-8_a_:'(u$ M11112 2)+Za.6_(u1u52 +u1ug)
03/= 7(o1+202-o3) ~pzu3- Z.Z.(u1ug- u2u5)-.:_ :(uguuruz’)
a9
(uzug +3u2uf +2u1u3u4)
:75 (3.21c)
-317: (ufu‘+3u§u,,+2u1uzu3)

a‘11

-272 (U411: +u4uf)

/ U3 “7 3‘13 3 2
”I. = ’7- (01*202“°3) “#2111. *2; (“1‘15 “12%) *3; (”3 *usuz.)

9(3u1u2 +u1u‘ 2+2u2u3u4)
(3 . 21d)

£19

+ a1
‘8—
0‘11

-2_

O

(3 u1zu3 +u§:3 +2u1u2u4)

3

2

'5

a
4 (11321.16 +UEU6)

115/ = "601’113'45 _;_1_2_u1u2 ”:2 (“1’44 “12%) '
a5 (.5 (3.21e)

(1 3a
'2‘: 0112116 «ax-fun ‘T: (ufu, mg)

38

/ “12 2 2 “13
u = ‘U5°1‘i‘3u6*2— (u -u )+_4.__ (u1u3—u2u4)

“11. 2 Z “15 2 2
+T§(u3u5+u4u5) +2? (u1u5+uzu5) (3.21f)

3016 3 2 f3
+1.; (m5 +u5u6) ~27;
where the following variable reassignments were used:

8.” -’ U1, 811.4 uz, 82'. 4 U3, B2]. 4 U4, 33'. -b us, B31.» L16 (3.22)

The relationship of the ori coordinates to the physical system can be
ascertained from equations (3.9), (3.20), and (3.22). Equation (3.9)
establishes that the A]. are the time varying amplitudes (with time T1) of
the variables Xj. The A]. are complex quantities, having real and imaginary
components, which account for the magnitude and the phase of the response
variables X1" Equations (3.20) give the relationships between the Aj and
the Bj such that the secular equations (3.17), (3.18), and (3.19) are
autonomous. The factor of 1‘: is introduced in the equations defining Aj in
terms of the Bj to account for the amplitude differences which result from
the use of complex notation. The Bj are complex quantities and have real
and imaginary components. These components are reassigned to the
variables uj according to equation (3.22) . The resultant magnitudes of the
u. are the envelopes of the response in time To. The relationship between

J

the u. and the Xj in equation form is:

J
.. - / 2 2 93 _ -1112 (3.23a)
X1 .. x10 - u1 +1.12 cos[7.t +y1] where y1 — tan [Tl—1]

/ 2 2 93
X2 z x20 = L13 +u‘ cos[7t +12] where yz

tan'1 ['11:] (3.23b)
U3

39

u5

J u
X3 = x30 = u52+u62cos[n3t «find where y3 _-. tan-1L1] (3.23c)

3.3 CASE 2 - (Torsional Forcing)

As in case 1, it is appropriate to introduce three detuning
equations:

92 -= "2 + 501, 02 = a1 - £02, «:3 =201 - 503 (3.23)
where 01 now quantifies the nearness of the forcing frequency 92 to 02.
Using these relationships in equations (3.10)-(3.12), the secular terms

are eliminated by letting:

/- - i 1 '- i< - )T
(‘2A11‘0I “AH/3116108 0‘ 0 -a1(A2A3e “5 “2 0)
- i( - )T — i 1'
-a2(A1A3e «5 0‘ 0) -a3 (314221426 (.2 0)

_ . T _ . 2 T (3.24)
—az,,(2A1A?_Aze'"’I ° +A1A22e” are!) 0)
-a5(3AfZ1e“'"T°) -a6(2A,A3Z3e“‘"'°) =0
/ . . iQTO "' i(‘n5‘¢-I|)To
(-2A21a2-2p2A21wz)e -a7(A1A3e )
— i T " i T "’ i(2 - )T
“08(3A22A28 (.2 0) 'a9(2A1A2A28 0! 0 +A1A§e (.2 ”I 0) (3 25)

"' . - ' - T
' a10(2A1A1A281QT0 4' A12Aze‘(2q a!) 0)

f .
7gefllz'fo = 0

- ' T
‘ G11 (21421431436102 0) 4'

(-2A3’ia5-2p3A3ia3)eiu5T° -a12A12e2'qT°
° + T -' ° T
’ (113(A1A281U‘n Q) 0) ' (114 (2A2A2A3e‘05 0) (3'26)
-a15(2A1Z1A3eN5T°) '016(3A32;38H510) =0
With further use of equations (3.23), equations (3.24)-(3.26) can be

simplified to:

40

l. . — i(a - )T
'2A1101 '2111A1ll0' -a1A2A3e Z 03 1

'dzZ1A3e-‘O3T1 '3a3A22226-102T1 (3.27)
’at. (2A1A222 *Z1Azze-zwzn)
' 3615141221 -2a6A1A3Z3 9- 0
-2 ’1 —2 A ' - AA i‘°2‘°3”‘ -3 2.?
A202 #221“: “7139 aaAzz
’09(2A1A2228‘02T1 *Z1A228-102T1) (3-28)
_ — 2' T - f '
-a10(2A1A1AZ +A12A2e "’2 1)-2a11A2A3A3 + 7.2.ew1T1 = 0
-2 ’1 -2p A 1 -a Azem-3T1 -a AA e"°3"’2’T1
“5 3 3 (‘5 121 1312 (3.29)

" 2G14A222A3 "' 2a15A1Z1A3 ' 301613523 = 0
Equations (3.27)-(3.29) can be put into autonomous form by letting

Bk “01
AI: = file

and solving for the ¢k such that all of the exponential terms have the same

exponent. Utilizing the values of ok which result from this process, the

Ak can be rewritten as follows:

3 ia- I
16(102M

A1 = 7.
82 ia1T1

A2 = Te (3.30a,b,c)
B ' - T

A3 = 7.3-eICZU1 202+03) 1

These relationships can be substituted into equations (3.27)-(3.29)
with Bk -= Bkr + inj for k = l, 2, and 3. Here the subscript r designates
the real component of the Bk term and the subscript j designates the
imaginary component of the same term. Taking the resulting equations,
separating the real and imaginary terms gives the following six equations

governing the amplitude modulations in time T1:

41

C
I

“1 2 “2
1 ' u2(“1”“2) ’H1u1'm (”3‘16 ‘“4“5) ‘72; (“1‘16 ““2“5)

3
-3; (ugul, +uz) -332: (uzug +3u2uf +2u1u3u4) (3. 318)

3a a
-.8;I§(u12u2 +ug) 25% (uzu: +u2u:)

a1 :12
-"'U O‘O " U+ uu+uu 4' UU+UU
1(12)”12m(3546)m(1526)

C
N \
I

a
(u3 +u3uf) +3.2; (31.11 u: +u1uf +2u2u3u4) (3-31b)

.
$1.22”

3
+ (u

Q
VI

3
3 2 a6 2 2
1+u1u2) ~12a (u1u5 +u1u6)

El

/ “7 3“8 2 3
L13 = u4a1--;::3--z_(1.11116-uzu5)23..“—2 (u3 u‘ +114)

-_8_ (u2u32 +3u2uf +2u1u3u4) (3.310)

“10 2
-372 (u1 L14+3u2 uk+2u1u2u3)- $1.1. (u4u52 +u4u2)

/ “7 3“8 3 2
”z, = "“301 -p::4+Z—a§ (111115 “4206) 4' mfig 1”1311).)

(3 u1 1.132 +3 u1u4 2+2u2u3u4)

m (3.31d)
+31% (3u12u3 +u22u3 +2u1 112a,.)
C‘11 Z 2 f2
762““ ”3"“ ' '27.;
a a
115/ = u6(201-202+a3) -p3u5 -71_a:u1u2 -2155 (U1 11,. +u2u3) 3 31 )
( . e
a a 3a
-217: (u3zu64-ufu6) 22:1; (u12u6 +u22u6) 2.8.5 (uszué 411:)
l
u = -u5 (201 -202 +03) -p3u6+ (L112- u§)+ Zl(u1u3 -u2u,.)
6 2‘— ‘5 (3.311?)

““15 3“ “16 3 2
+z1_a:(u§u5+u4u5) +2.“; (u12 u5+u2 2:15) +1: (u5 +u5u6)

42
where the following variable reassignments were used:

B1r " ”1’ 31°

’4 uz, B2" 4 113, 82].» U4, 83'. -+ us, 831.4 U6 (3.32)

The relationship between these uj and the Xj for this case can be written

in equation form:

2 2 - u 3.
X1 z x10 =‘/u1 «r'u2 cos[92t +y1] where 71 = tan 17:] ( 338)

“3

X3 z x30 =1/u: +u§cos[292t +73] where y3 = tan“ [1:6] (3.33c)

uS

3.4 CASE 3 - (Torsional and Vertical Forcing)

In this case, forcing in both the X2 and X3 coordinate directions is
present. Hence, both of the forcing terms in equations (3.11) and (3.12)
are nonzero. Rather than introduce an additional detuning relationship,
the fact that $23 =- 292 is used to keep the number of detuning relationships
to three as in cases 1 and 2. Physically, this relationship represents
the fact that the forcing frequency associated with X3, 93, is of second
order and the forcing frequency associated with X2, 92, is of first order,
and hence these two frequencies are always in the ratio of 2:1. Therefore
the detuning equations for this case are:

92 = 02 + £01, 02 -= a1 - £02, 03 = 201 - 603, (3.31m)
with the additional dependent detuning relationship

93 - 202 + 2501. (3.34b)

43

Using these relationships in equations (3.10)-(3.12), the secular terms

are eliminated by letting:

/. . i T - i( - )T
(-2A11w, -2p1A11w.)e 0' 0 -a1(A2A36 “5 Q 0)
_ .- Neg-0|)To _ 2- iapTo

a2<A1Ase_ . T )_“3(3.A§A2e T ) (3.35)
-a4(2A1A2A2eW' 0 +A1A2281( 02-0” 0)

-a5 (3A.?Z1eiqTo) -a6(2A1A3Z3e‘qT°) =0

l' ' i 1 - i‘ ‘ >7 2- i T
(’ZAle‘ZFZAzl‘QM “2 0 ‘“7(A1A3e “5 0' 0) -“3(3A2Aze «.2 o)
.— ' T _ . _ T
-a9(2A1A2A2e'0‘ 0 +A1Azze'(2“2 1“) 0)

- ’ — ' _ (3.36)

- “10(2A1A1A261QT0 +A12Azel(2"’l 02)To)

_ - f .
" G11 (2A2A3A381QT0) + 7136102") .2 0
l. . 1 T 21 T i( + )T

{-214le 'ZM3A3165)3.“5 0 ‘012A126 0‘ 0.. Q13 (A1Aze 0‘ “2 O)
- i T - i T (3.37
-a1(, (21424422436 05 0) :a15(ZA1A1A3e “5 O) )

._ - f .
-a16(3A32A3e‘°’—"T°) + 73e‘°3T° =0

With further use of equations (3.34), equations (3.35)-(3.37) can be

simplified to:

/ . . - i(a «13111 - —ia3T1
’2A1101 ~2p1A11w. - “12422438 2 ' (121412438

‘ 3G3A2222€
- 3a5A1221 " 206A1A3Z3 = 0

40271 ““1. (214114222 +Z1Age-Zi02T1) (3.38)

‘ZAz/iQ ’szAziQ -a721A3e‘(02-a3"1 ’308A22X2
" ' T
‘09(2A1A2A2’e‘02 1

" G10 (2A1Z1A2 4' 4412228

_- _' T
1* A1Azze ‘02 1)

2102‘“) (3'39)

_ f - T
- 2011A2A3A3 4‘ 72.6101 1 = 0

44

/ . . 2 i T i( ~a )T
“2143 J. (:5 '2H3A3lfs ‘ a12A1e 03-1 ' 0131412428 03 Z 1
‘ 2014242142233 ‘: 2015A1A1A3 (3 .40)

- f '2 - T
-3a16A32A3 +73-en(a1202+03)1=0

Equations (3.38)-(3.40) can be put into autonomous form by letting

Bk W71
Ak=7e k

and solving for the ¢k such that all of the exponential terms have the same

exponent. Utilizing the values of ¢k which result from this process, the

Ak can be rewritten as follows:

B ' - 1
1e'(0102>1

141:7
_ 32 mm (3 40a b c)
A2 “'76 ° 1 7
B '2 -2 + T
A3=—;e‘(a1 02 ”3’1

These relationships can be substituted into equations (3.38)-(3.40)
with Bk - Bkr + inj for k -= l, 2, and 3. Here the subscript r designates
the real component of the Bk term and the subscript j designates the
imaginary component of the same term. Taking the resulting equations and
separating the real and imaginary terms gives the following six equations

governing the amplitude modulations in time T1:

:1 a
u1/ = “2 (01 ’02) -p1u1 -2iw, (£131.16 -u4u5) -7112; (u1u6 -u2u5)
3
-32 (u3zu4 41143) ';% (1121.132 +3UZUZ +2111 1.13114) (3'423)

3a (1
-3350112112 +ug) -Z.% (112L152 +u2u5)

4S

/ a1 02
U = 'u1 (“1 '02) “#1U2 * (“3'45 “14%) * (“1‘15 “12%)
2 301%:

30:
+3? (U3 +u3ul. 2)+.8_ (3u1u§ +u1u 2+2u2u3u4)

3“ “s 2
+3.3 (11? +u1u§) +2_:.;'. (u1u5 +u1uz)

/ _ “7 3“8 2 3
U3 - 11401 112113-22E (u1u6-u2u5) -3172 (u3u4 +u4)
“9 2 3 2 2
-.872.(u2u3+ u2u4+ u1u3u4)
a
'3%(U12U4+3U22U4+2U1U2U3)

_ “11
To;

2 2
(u,.uS +u4u6)
/ _ “7 “8 3 2
“a ‘ -U3o1-y2uum(u1us H12%) *3; (“3 ”13%)
+ a9 (31.! u2+u u2+2u u u)
'8— 1 3 1 z. 2 3 lo
+.3._(3u 2u3+u22u3+2u1u2u4)

“11 2 2 f2
-23 (m3uS +u3u6) - m

/ a
U5 = U6(201’202 +03) ”#3115 1'213111 U2

2
(uu+uu)-.Za_ W(uu+uu)
T14 23 2'17 6 6

a 3a
'T-l: (”1 “6 fl122116) -—3—a;6 (1152 U6 +u2)

U6, -'-' “115(201-2024'03) -p3u6+ G12 (U1 ”1.12)
Zas

“13 / “11. 2 2
+ (u1u3-u2u4) + (u u5+u us)
2:5" Z (‘5 3 lo

“15 2 2 3“16 3 2
+2E (u1u5 +u2u5) +_8_“5 (us +u5u6) -7?

where the following variable reassignments were used:

311- " “1' B11 " ”2' Bar " “3' sz " "4'

Bsr " “5v 33'

”6

(3.42b)

(3.42c)

(3.42d)

(3.42e)

(3.42f)

(3.43)

The relationship between these uj and the Xj for this case can be written

in equation form:

46

/ 2 2 - U 3.44
X1: x10 = u1+u2cos[92t+y1] where 71 tan1 FE] ( a)

.3“
1
ll
1"?
3

u
~ x20 = «a; +uf cos[ta +12] where 72 '1 3%] (3.44b)

~ _ 2 2 _ -1 u6 (3.44c
X3 ~ x30 - u5 +1.16 cos[2£22t+y3] where 73 - tan is. )

3.5 REMARKS

The equations governing the response of the system in the slow time
scale, T1, for each of the three cases have been derived. These equations
can now be used to investigate the system response for each of the three
cases. This is done in Chapter 5. Prior to presenting this information,
Chapter 4 presents the parameter values for the physical system to be
investigated and gives information that validates the use of the Taylor

series in the derivation of the equations of motion of the system.

CHAPTER 4 SYSTEM PARAMETERS AND TAYLOR SERIES VERIFICATION

4.1 SYSTEM PARAMETER VALUES

Prior to presenting the numerical results for each of the three
cases, discussion of the parameter values to be used in the numerical
analysis is needed. It is desirable that the parameter values for the
numerical analysis be realistic. As such, the physical problem
investigated here is an in-line four cylinder automotive engine. Certain
aspects of the geometry were set to facilitate the tractability of the
problem. The location of the center of mass at the geometric center of
the rigid body and the symmetry of the support locations were set in this
fashion. The physical dimensions of the rigid body are representative of
the dimensions of an engine looking along the crankshaft axis. The
geometry depicting these parameters are Figures 2.1 and 2.2. The values

chosen for the physical problem are:

Mass m: 219.400 kg
Inertia 16: 14.155 kg-m2
Height 28: 0.842 m
Width 2A: 0.254 m
k1: 1,732,312.8 N/m
k2: 433,078.2 N/m
k3: 433,078.2 N/m
k‘: l,732,312.8 N/m
Spring free length: 0.102 m

The values for IG, [(1, [(2, k3, and k4 were chosen such that the tuning
between the rigid body linear natural frequencies is: 201 - 202 - 03.
These relationships between the linear natural frequencies need not be

strict equalities. Indeed, the preceding analysis of Chapter 2 allows for

47

48
such inexact tuning through the parameters 02 and 03. However, we chose
to set 02‘=‘§ - 0, realizing that only small changes in the main features
of the response plots would occur for small changes in these
relationships.

These parameter values are also used to obtain numerical
coefficients of the terms of the Taylor series approximation of the
potential energy about the static equilibrium point. These coefficients
are then divided appropriately by either m or IG producing the «1's in

equations (3.3). The resulting numerical values are:

a1 = 41915.45 (19 = -58,055.75
:12 - 154,817.68 am - -474,311.20
(:3 = -1248.52 (111 - 180,387.67
“4 ' £33353? ' "2.333%?
a51= , . ‘H3 - - .

66 = -1,897,270.97 (11,. - 11,638.05
(17 - -76,188.63 6,5 - -l,897,270.97
68 .. 444.52 616 .. 189,727.10

The values of the viscous damping coefficients c1, c2, and c3 were
chosen such that the dimensionless viscous damping factors of the
linearized system equal 0.01. This leads to the following dimensionless

modal damping coefficients specified as pi in equations (3.2):

(H a 0.63
pa - 0.63
”3 - 1.26

The magnitude of the forcing, F} and F“, were chosen to be of the
same order of magnitude of force found in production engines. Limited
information was found regarding these values and the following values
reflect the information available (Fullerton, 1984):

F} - 150.0 N-m
P3 - 1000.0 N

49
The frequencies associated with the forcing terms F} and.F3, 92 and
93 respectively, were chosen to correspond to the first order and second
order associated with a hot idle condition of an in-line four cylinder
engine which is approximately 600 rpm. Thus the values of 92 and 93 are:

92 a 20"

The parameter 6 introduced in equations (3.3) must be specified to
enable the numerical evaluation of the results of the multiple scales
analysis of the rigid body system. For this problem, 5 is not a naturally
occurring parameter of the system. It was introduced as part of the
multiple scales analysis as a means to distinguish between motions at the
different time scales. The method of multiple scales as used here

1

introduces corrections that are of 0(a) for the time interval _.>t3>0
E

(Kahn, 1990). Hence for the value of e chosen here, 6 = 0.01, this time
interval is 1005. The time intervals associated with the transient
response for the three cases investigated here are on the order of 103 and

are well within this interval.

4.2 TAYLOR SERIES VERIFICATION

Prior to presenting the response curves of the three cases,
verification is presented for the use of the Taylor series approximation
of the potential energy used in the Lagrange formulation of the equations
of motion for this system. Time response plots, obtained from numerical
integration (4th-order Runge-Kutta) for a given set of parameter values,
are shown for each of the three cases. The comparisons demonstrate good

agreement between the response predicted by the numerical integration of

50

the reduced equations obtained from multiple scales analysis (hereafter
referred to as "reduced equations") and the envelopes obtained from the
numerical integration of the equations of motion obtained from the
approximation of the potential energy using a Taylor series retaining up
to quartic terms. These equations are hereafter referred.tx> as "cubic
equations" as it is cubic terms that appear in the equations of motion
after the required partial derivatives of the quartic terms of the
potential energy approximation are performed.

In addition, for case 2, further comparison is made with the
envelopes obtained from the numerical integration of the equations of
motion obtained from the approximation of the potential energy using a
Taylor series retaining up to only cubic terms. These equations are
hereafter referred to as "quadratic equations" as it is the quadratic
terms that appear in the equations of motion after the required partial
derivatives of the cubic terms of the potential energy approximation are
performed. Comparison is also made with the envelopes obtained from the
actual equations of motion with no approximation in for the potential
energy, hereafter referred to as "actual equations".

It is shown clearly in this case that the time response obtained
from the quadratic equations is different than the time response obtained
from the actual equations. Hence, although not an exhaustive test, it is
reasonable to assume that it is valid to neglect terms in the Taylor
series higher than quartic. Hence all of the reduced equations are derived

from multiple scales analysis of the cubic equations of motion.

51
4.2.1 Case 1 (Vertical Forcing)

Figure 4.1 shows the time response plots of the numerical
integration of the cubic equations. This figure depicts the upper and
lower envelopes of the time response for each of the three system
coordinates. Figure 4.2 shows the time response of the numerical
integration of the reduced equations, equations (2.28). Very close
agreement is seen between these two response plots, particularly in the
steady state amplitudes. However, small differences are seen between the
transient aspects of the responses. These differences are due to the
different initial conditions used to generate these two sets of responses.
The initial conditions used to generate the responses in Figure 4.1 are
specified in terms of the initial displacement and velocity of the system
coordinates. The initial conditions used to generate Figure 4.2 are
specified in terms of the real and imaginary parts of the complex response
defining each of the system coordinates. Since the multiple scales
analysis will be used to predict the steady state amplitudes of the
systems, no effort was taken to show full agreement of the transient
aspects of the time response.

Comparison of Figures 1 and 2 also show that the reduced equations
fail to predict the bias in the steady-state response predicted in the
cubic equations as shown in Figure 4.1. This is expected due to the
approximation of the multiple scales analysis. The terms that govern this
behavior are 0(52), and are not accounted for in the analysis performed

here.

RESPONSE

O
O
O
U1

-0.01

-0.015-

-0.02

52

 

 

 

 

 

 

 

 

 

 

 

4

 

 

4 6 8
t

10

12

FIGURE 4.1 - Time response enve10pes from the cubic equations

0.015~

0.005

RESPONSE
O

0.005-

-0.01

-0.015-

-0.02

0 0

Case 1 (Vertical forcing)

 

fir

 

 

 

 

 

 

-L 4

 

 

.02

0.04 0.06 0.08
T1

FIGURE 4.2 - Time response from the reduced equations

Case 1 (Vertical forcing)

4.2.2 Case 2 (Torsional Forcing)

This case is used to present the correlation of the response

predicted by the reduced equations and the actual equations.

Figure 4.3

shows the envelope of the time response obtained from the actual equations

53
of motion. This response is contrasted with the time responses shown in
Figures 4.4, 4.5 and 4.6.

Figure 4.4 is the time response for the system obtained from the
quadratic equations. Comparison of this response with the response from
the actual equations given in Figure 4.3 shows that retention of the cubic
terms in the potential energy is not sufficient to predict the correct
transient or steady state response of the system. However, the time
response shown in Figure 4.5, obtained from the cubic equations, shows
good agreement with the time response from the actual equations of motion
shown in Figure 4.3. Here the comparison of the transient and the steady
state aspects of the time response is valid as the initial conditions are
identical between the two simulations. Good agreement is seen between
these two simulations.

Figure 4.6 gives the time response of the reduced equations. This
time response shows good agreement with the response of the actual
equations of motion given in Figure 4.1 and with the time response of the
cubic equations of motion shown in Figure 4.5.

The response for this particular set of parameter values clearly
shows the importance of retaining the quartic terms in the Taylor series
approximation of the system potential energy. This particular comparison
illustrates the fact that it is not only differences in the transient or
steady-state responses of the system which can be realized, but the
qualitative nature of the response can be different. The time response
obtained from the actual equations of motion, shown in Figure 4.3, shows
that the steady state response of the system is time varying for this set
of conditions. In contrast, the time response obtained from the quadratic

equations, shown in Figure 4.4, incorrectly predicts that the system has

54
a fixed value steady state. The cubic equations do qualitatively and

quantitatively predict the correct time response as shown in Figure 4.5.

 

 

 

 

 

 

 

£31
(I)
2
0
On
U)
Ed
(I

-0.05 - ~

-o.1 9

-0.15 1 . . .
0 2 4 6 8 10 12
t

FIGURE 4.3 - Time response envelopes from the actual equations
Case 2 (Torsional forcing)

 

 

 

RESPONSE

 

 

 

 

 

6 8 10 12
t

FIGURE 4.4 — Time response envelopes from the quadratic equations
Case 2 (Torsional forcing)

RESPONSE

RESPONSE

55

 

0.15 . - ' ”v *

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2 4 6 8 10 12

FIGURE 4.5 — Time response envelopes from the cubic equations
Case 2 (Torsional forcing)

 

 

 

 

 

 

 

 

-0_15 . . . . .
0 0.02 0.04 0.06 0.08 0.1 0.12

T
1
FIGURE 4.6 - Time response from the reduced equations
Case 2 (Torsional forcing)

56
4.2.3 Case 3 (Torsional and Vertical Forcing)

The results of the comparison of the time response of the cubic
equations to the time response of the reduced equations are identical to
the comparison performed in case 1. Figure 4.7 shows the envelopes of the
time response obtained from the numerical integration of the cubic
equations. Comparison of these with the time response obtained from the
reduced equations, shown in Figure 4.8, shows good agreement for the
steady-state response. Some differences are seen in the transient
responses of the two sets of equations. This is again due to the different

n tial conditions used to generate the two response plots as explained

above in case 1.

 

A
1 D
i
N\

 

 

 

 

-0.15 . . A 1 .
0 2 4 6 8 10 12

(3

FIGURE 4.7 - Time response envelopes from the cubic equations
Case 3 (Torsional and vertical forcing)

 

S7

 

 

 

 

 

 

 

 

 

 

0.15 e 1
X2
0.1~ .
0.05» X .
1 X
E73 . . \ / 3
5 o» ~w--—w-u-IIIIE§.N.DMGIDGI»‘3»«—- ,/
a .
[I]
m
—0.05L .
—o.1 .
_O.15 . 1 A . 1
0 0.02 0.04 0.06 0.08 0.1 0.12

T
1
FIGURE 4.8 — Time response from the reduced equations
Case 3 (Torsional and Vertical forcing)

4.3 REMARKS

The results of this chapter are important in that they establish
that use of the Taylor series for the potential energy expression in the
derivation of the equations of motion leads to system response consistent
with the actual equations of motion. Further, the reduced equations
obtained from multiple scales yields system response consistent with both
the actual equations of motion and the Taylor series equations retaining
cubic terms. The reduced equations can.now be used further to investigate
the frequency response of the rigid body system for each of the three
cases. This will be done by utilizing the reduced equations derived for
each of the three cases and looking for the steady-state response of the

system. This information is presented in the following chapter.

CHAPTER 5 SYSTEM FREQUENCY RESPONSE

Having completed the verification of the reduced equations, the next
step in the investigation into the harmonic response of the rigid body
system is to obtain the frequency response of the system. The frequency
response can be obtained from the reduced equations produced by the
multiple scales analysis. Steady state motions of the system are obtained
by solving the reduced equations when the time derivatives of the system
are set equal to zero. This can be done for each of the three cases and
the frequency response plots obtained by plotting these solutions, and the
associated stability information, for an appropriate range of the detuning
the parameter, o1, i.e., the nearness of the forcing frequency to the
associated natural frequencies. The stability of the solution is
determined through examination of the eigenvalues of the Jacobian matrix
for the system of reduced equations.

The detuning relationships for each case are given in Chapter 3.
However, for all cases analyzed herein, c:1 is the only detuning parameter
varied in producing the response plots. In each case, this parameter
represents the tuning of the forcing frequency relative to the natural
frequency'aq for j - 2,3 corresponding to the coordinate being forced. The
detuning parameters 02 and 03 are both set to zero for all cases yielding
perfect tuning of the internal resonances, a1 - 02, 201 - 03..A fourth
case, which is a variation of case 3 having reduced values for the
magnitude of the forcing, will also be investigated.

Prior to proceeding‘with the response plots, the process and.methods

used in solving the algebraic equations for the steady state amplitudes

58

59
and obtaining the frequency response are discussed. The response plots for

each of the four cases follows.

5.1 ANALYSIS METHODS

As stated, the frequency response of the system for each of the four
cases is found by solving the reduced equations with the derivatives set
to zero for an appropriate range of 01. Thus the problem of establishing
the frequency response is reduced to that of repeatedly solving six
nonlinear algebraic equations. For example, the algebraic equations
governing the steady state amplitudes for case 1 are given by equations
(5.1a)-(5.lf), i.e., equations (3.21) with the time derivatives set to

zero .

u2 “1 “2
7— (01'03)'“1u1'm(u3u6 ”“4“5) '75 (“1'46 'Uzus)

3a a

-37.; (U§UA*UZ) -_8.% (u2u32 +3u2uf+2u1u3u4) (5-13)
3“5 2 3 “6 2 2

"83 (”1”2*u2) ‘27:. (“ans “12%) = 0

U1 (:1 dz
‘7(°1"“3) ’F1U2 *2; (“3‘15 “14%) ‘27; (“1‘15 “12%)

3a 0!

+333 (u§+u3uf) +79% (3u1u§+u1uf+2u2u3u4) (5-1b)
3&5 3

4.

a
‘83 (“1 “-11"? ’72: ("“152 “11513) = 0

u 0: 3a
711(01-6202 -o3) -p2u3 -z% (1.11116 -u2u5) -338 (113211,. +11?)

2 2 “10 2 2 1
(u u +3u 11 +211 11 u,,)- (u u4+3u u4+2u1u2u3) (5- C)
2 3 2 7. 1 3 '37? 1 2

$1.138

2 2
(uku5 +u4u6) = 0

60

u 01 3a
'73: (O1 *202 '03) “#zuzfizzg- ("1'15 “12%) +33:- (“33 *usuf)
+.;.§2. (3 u1u32+u1uf +2u2u3u4) «931—0 (3u12 u3 +u2u3 +2u1u2u4) (5.1-d)

“11 Z

-2? (1.13115 +u3ug) = 0

uo- u-a12uu-a13(uu+uu)a (uu2 6+u2u)
61’135775-12m1623zzg-W3 (.6

3 (5.1e)
015 2 2 G16 2 3 _
25E (u1u6+u2u6) ~17; (u5u6+u6) - 0
a a a
““501 '“3u6+z1_a: (”12 ”17?) ’23 (U1U3 ’Uzul.) ”’21,:- (“zfus *ufus)
(5.1f)

I
“15 2 2 3016 3 2 f3
4.2—“S (u1u5 +u2u5) +173 (us +u5u6)-.2_a;

:=0

Obtaining a response plot from the six algebraic equations consists
of two phases. In the first phase, solutions to the equations are found
for several values of 01. Typical values of a, are: o1== O (i.e., Q3 - 03)
and 01 = i300 (i.e., 93 = 40" i 3 rad/s). These solutions are found using
a FORTRAN program based on the IMSL routine DNEQNJ (IMSL, 1980). This
routine produces the solution of a set of nonlinear algebraic equations
using the Levenberg-Marquardt optimization algorithm.

After the various solutions are found at the selected values of 01,
they are used as seeds in a program.which traces the solution paths, i.e.
the frequency response, by iterating on.the values of'01. The solution for
the previous value of a1 is used as the seed in the solution.algorithm for
the next step. This program is also based on the IMSL routine DNEQNJ, but
has the added feature that it loops on a parameter. In all cases examined
herein this parameter is 01. The stability of each of the solutions is

established by evaluating the eigenvalues of the Jacobian matrix of the

61
nonlinear algebraic equations. This procedure was found to be an efficient
and convenient method for finding the solutions to these algebraic
equations, establishing the associated stability, and thereby determining
the system response.

The six solution components found for a given value of 01 form the
three pairs of the real and imaginary components of the three coordinate
variables X3, X2, and X3, i.e., corresponding to the coordinates x, a, and
y, which are the horizontal, angular and vertical displacements,
respectively. The collective magnitudes and the corresponding stability
information for an appropriate range of Oj‘values form the response plots
for the system. It .is these response plots for each of the four cases

which are presented next.

5.2 CASE 1 (VERTICAL FORCING)

The algebraic equations to be solved.for the steady state amplitudes
for case 1 are given by equations (5.1a)-(5.lf). Following the procedure
outlined in the previous section, frequency response plots for the
magnitudes of the coordinates X3, X2, and.X3 are given. For this case, two
response plots for each of X}, Xé, and X3 are presented. The first set of
response plots, Figures 5.1, 5.2, and 5.3, include several of the unstable
solutions with comparatively large magnitudes. These large values of
response represent solutions which are not physically realizable and are
only a mathematical curiosity. (The dimensions associated with X1 and X3
are given in meters in the response plots and in radians for X2.) Response
plots with a larger ordinate range are not shown for any of the subsequent

cases and are shown here for completeness.

62

 

0.08~

0.06~

0.04-

0.02-

 

 

 

 

 

0
-1500

 

~1000 1000 1500

Figure 5.1 -Frequency response for X1, case 1, horizontal response
stable solution, ---- unstable solution

 

 

-_.—
’_--
_—____-____-_______---_--_.___——_.—_-_—_—o-—

 

0

A

 

 

-1500

-1000

500

1000

1500

Figure 5.2 — Frequency response for Xé, case 1, rotational response
stable solution, ---- unstable solution

 

Before discussing some of the important features of these figures,
the actual interpretation of the response curves is addressed. Recall

equations (3.23):

63

 

 

 

 

 

 

o . 12 ‘
\\\‘\“

0 . 1 ______________ \\\ ’2‘

‘‘‘‘‘‘‘‘‘‘ \\\\ ””/’

- “““ \\ ,«’

0.08 ............... ;

n \ _______ a”
R 0 o 0 6 " -1
0.04 - .
0 . 02 [ d

-1500 -1000 -500 0 500 1000 1500
O1

Figure 5.3 - Frequency response for X3, case 1, vertical response

stable solution,

X z x = ‘/u2 +u2cos[ 93 t +y] where
1 10 1 2 '1? 1

X = X = u2 +uzcos[ 93 t + ] where
2 20 3 I, 7- 72

X3 ~ x30 - 11115 +u6 cos[fl3t +73] where

and equation (3.13a):
Q3:= 03-+ 601

From equations (5.2) and (5.3),

oscillating at the forcing frequency {23, X1

---- unstable solution

. ”-1... :2. (22>
1 ‘11

y - tan"1 3: (5'3)
2 113

y = tan'1 :9. (5'4)
3 ”5

it can be seen that instead of

and X2, which are the

magnitudes of the response in the horizontal and rotational directions,

respectively oscillate at one half this value. Figures 5.4 and 5.5, depict

64

the response curves for X1 and X2, respectively. Therefore, these plots are
unlike the usual "linear" response curves where the frequency of the
response is always the same as the forcing frequency. Finally, it is noted
that for all cases investigated, a 1 Hz change in the forcing frequency
corresponds to a o1'va1ue of 628.32. We are now in a position to discuss
the features of the response curves shown in Figures 5.4-5.15.

The second set of response plots, shown in Figures 5.4, 5.5, and
5.6, have a reduced range of response amplitude to show more explicitly
the transitions between the linear and nonlinear solutions. It is seen
that away from the external resonance, i.e. a1 + 0, the motion of the
system follows that of the linear system. For this case, the linearized
system is a set of three uncoupled linear oscillators. Therefore, as the
forcing acts only on coordinate X3, the linear solution for the X1 and X2
coordinates is, to this level of approximation, the trivial solution for
all 01. Slowly increasing the frequency of the forcing starting from far
below this resonance leads to unstable linear response for the X, and X2
coordinates which occurs at o1==-470. At this value of the detuning, the
X1 and X2 coordinates abruptly shift from the trivial solution up to a
nonzero steady state response. The linear solution for the X3 coordinate
also becomes unstable at this value of a1 and the steady state response
jumps up to a different stable solution. These non-trivial steady state
solutions continue until a1:= 971 where all of the coordinates experience
downward jumps back to their respective linear solutions.

When starting far above the resonances and decreasing the frequency
of the forcing, transition from the linear solutions to the nonlinear
solutions occurs at a, = 450. These solutions continue until a large

downward jump back to the linear solutions is experienced at a, = -860.

65

 

0.025 . . I

0.015»
>< 0.01 ~

0.005-

 

 

 

 

 

 

-1500 ~1000 -500 0 500 1000 1500

Figure 5.4 -Frequency response for X1, case 1, horizontal response
stable solution, ---- unstable solution

 

 

0,015,

0.005»

 

 

 

 

 

0

 

-1500 -1000 -500 0 500 1000 1500

Figure 5.5 — Frequency response for X2, case 1, rotational response
stable solution, ---- unstable solution

 

Response similar to the response described here has been

demonstrated analytically for two degree of freedom systems (Nayfeh and

66

 

0.014L

0.012-

0.008»
0.006L
0.004r

0.002r

 

 

 

A

0 A 1 A 1
—1500 -1000 -500 0 500 1000 1500
O1

 

Figure 5.6 — Frequency response for X3, case 1, vertical response
stable solution, ---- unstable solution

 

Mook, 1979; Haddow, et. al., 1984; and Perkins, 1992). Additionally, in
Haddow, et. al. (1984), this response was also verified experimentally.
The system investigated here exhibits some of the same features shown in
these references. In particular, these systems display the saturation
phenomenon. This is the situation whereby increased forcing magnitude is
not accompanied with an increase in the magnitude of the response variable
in the direction of the forcing. Thus the mode is "saturated". This
phenomena occurs over the region of nonlinear solutions and need not be
associated with large displacements. In contrast, the case where only an
external resonance is present (Nayfeh and Balachandran, 1989), the
additional parametric resonance (actually and autoparametric resonance,
see Nayfeh and Mook, (1979)) makes the response plot asymmetric about the
o1 - 0 axis. This agrees with the results shown in Perkins (1992) where

asymmetry is also shown.

67

This system response shows some interesting implications from the
vibration isolation point of view. The response of'X3 (vertical), shown in
Figure 5.6, shows that over the range -400:s<fi s 550, the nonlinear system
shows drastically reduced response amplitudes than for the corresponding
linear case. This however comes with two trade-offs. The first is that the
response amplitudes for X} (horizontal) and X: (rotational), are nonzero.
This is in contrast with the linear solution where the response of both of
these would be zero for the entire range of 01. The second is that over the
ranges -860 5 a, s -400 and 550 5 a1 5 971 the nonlinear response is
greater than the linear response. However, the peak nonlinear response is
still much less than the peak linear response. In particular, where the
linear response reaches a maximum, the nonlinear response reaches a local
minimum which is approximately seven times less than the linear solution.

Hence, it is seen that the energy has been distributed among the
various coordinates rather than being concentrated in the coordinate
response associated with the direction of the forcing. This type of
response could be utilized effectively in circumstances where response of
a system near a particular frequency, and in a coordinate direction
different from the one being excited, is more tolerable than response in
the direction of the forcing. For example, this might occur if a
supporting structure for a system has different stiffness values in

different coordinate directions.

5.3 CASE 2 (TORSIONAL FORCING)
The six algebraic equations governing the steady state amplitudes
for case 2 are given by equations (3.31a)-(3.31f) with the terms

containing the time derivatives, i.e., the left hand side of these

68
equations, set equal to zero. Following the analysis procedure outlined in
Section 5.1, the frequency response plots for the coordinates X1,.X2, and
X3 are generated.
For this case, the linearized system is again described by three

uncoupled linear oscillators. Since the forcing appears in the equation

for the X2 coordinate (see equations (3.3) with 1:3 =0), the linear

response for the XI and X3 coordinates is the trivial solution. Figures 5.7
and 5.9 show that the nonlinear response of these two coordinates does
occur. In Figure 5.8, the linear solution for the X2 coordinate is plotted
in addition to the nonlinear solution. As can be seen in this figure, the
amplitude of these two solutions match very closely except in the region

4005015 100.

 

0.005 1 , 7
.0045 r i 1
0.004 . Q ~
.0035 . i! -
0.003 - .’ «

>3— .0025 L " .
0.002 ~ 2 ‘
.0015 r t
0.001 - § 4
.0005 - ;

 

 

 

-400 -200 0 200 400
01
Figure 5.7 — Frequency response for X1, case 2, horizontal response
stable solution, ---- unstable solution

 

In contrast to the response plots for case 1, the response for this

case does not contain any sudden jumps from the linear to the nonlinear

69

 

 

400

200

-200 0
°1

 

linear solution

 

 

 

 

 

 

0 +
-400
Figure 5.3.. Frequency response for Xé, case 2, rotational response
——- stable solution, --- unstable solution, -
0.004 - - f .
.0035 ~ : 1
‘1"
0.003 ~ / .
.0025 » f i -
g? 0.002r 1
.0015- ‘
0.001» ‘
.0005» 2 1
-200 0 200 400
O1
---- unstable solution

stable solution,

 

0
-400
Figure 5.9 - Frequency response for X3, case 2, vertical response

solutions. All three of the response plots have smooth transition for this

range of 01.

70
As seen in Figures 5.7, 5.8, and 5.9, for -36 5 a1 5 31, no stable
steady state solutions exists. As such, the response is never expected to
reach a steady state (a condition where the modal amplitudes are
constant). Figure 4.6 shows the envelope of the time response for this
system obtained from numerical integration of the reduced equations for

o1=0 CH = 0. This time response illustrates the lack of a steady state

solution with constant modal amplitudes.

5.4 CASE 3 (TORSIONAL AND VERTICAL FORCING)

The six algebraic equations governing the steady state amplitudes
for case 3 are given by equations (3.41) with the terms containing the
time derivatives set equal to zero. Once again, we follow the procedure
outlined in Section 5.1 to obtain the frequency response plots.

In this case, forcing in the equations for the X2 and X3 coordinates
is present (see equations (3.3)). The forcing frequencies are such that
for the condition investigated, external resonances are present for both
of these coordinates. This is the case as the forcing frequency on the X3
coordinate, Q3, arises from a second order type of unbalance whereas the
forcing frequency on the X? coordinate is of first order, i.e., o3 - 292.
The internal resonances (01 - 02 and 201 - 03 are also assumed. The response
plots for this system are given in Figures 5.10, 5.11, and 5.12.

Figure 5.10 shows the frequency response of the X, coordinate. Here
it is seen that over the majority of the range -500 5 a1 5 500, the
response of this coordinate is multi-valued. In the range -110 5 01:5 50,
the response is single valued. In the regions -380 s a} s -250 and 220 s
cm s 450 the response is triple valued. For -450 s a, s -380 and 450 5 a1

5 510, the response is double valued. The responses of coordinates X} and

71

 

0.025

0.015 -

 

0.005 ~ \ J . .

 

 

 

0 4.
-600 -400 -200 0 200 400 600
°1
Figure 5.10 - Frequency response for X1, case 3, horizontal response
stable solution, ---- unstable solution

 

 

0.06

0.04

0.02

 

 

 

 

o . 1 . A
-600 -400 -200 0 200 400 600
O'

1

Figure 5.11 — Frequency response for Xé, case 3, rotational response
-——— stable solution, ---- unstable solution

X3, shown in Figures 5.11 and 5.12 respectively, also exhibit these same
features.
For this case, verification of the response magnitudes for each of

the three system coordinates was made. The verification is made through

72

0.014 1, e . a

 

0.012 ’ :I,\ 1

 

 

 

 

-600 -400 -200 0 200 400 600
O
1

Figure 5.12 — Frequency response for X3, case 3, vertical response
stable solution, ---- unstable solution

 

comparison with the response amplitudes predicted through numerical
integration of the cubic equations to those predicted by the steady state
‘values obtained from the reduced.equations. The comparisons were made only
for selected values of 01: a1 - i300 and a1 - 0. These amplitudes are
indicated in Figures 5.10, 5.11, and 5.12 by the o. This comparison shows
that for the response amplitudes predicted ath - 0, very good correlation
is shown. Correlation at a} - £300 is also quite good. Additionally, the
correlation. shows good. agreement 'between. qualitative nature of the
solutions.

For this case, again the linearized system for this system
configuration is three uncoupled linear oscillators. Since two of the
coordinates contain a forcing term, the resulting motion will contain the
response from only these coordinates, but for all values of the input

forcing frequencies the response would be single valued. Table 5.1 lists

73
a comparison of the maximum amplitudes of the linear and nonlinear

responses of the system for this case.

 

 

 

 

Linear Response Nonlinear Response |
|X1LnEx 0.000 0.022 "
|X2|m 0.135 0.125 ll
nghgg 0.014 0.010

 

 

 

 

Table 5.1 - Comparison of linear and nonlinear amplitudes for case 3

As in case 1, it is seen.here that a large reduction in amplitude is
realized in the X3 coordinate response for the range -200 s a, s 200. It
is in this range that the nonlinear system produces small response
compared to the linear response. This difference in amplitudes can be

utilized to enhance the isolation response of the system.

5.5 CASE 4 (REDUCED AMPLITUDE TORSIONAL AND VERTICAL FORCING)

Currently, limited information is available pertaining to the
magnitudes of the forcing present in an in-line four cylinder engine. The
available information (Fullerton, 1984; Geck and Patton, 1984) presents
data only for selected force components. In an effort to account for any
uncertainty in the amplitudes, an additional load condition with smaller
force magnitudes was selected and the response of the system obtained.
Information from the above references was used to determine the reduction
in force amplitude. The magnitudes of the forcing selected are:

P} = 30.000 N-m (previously 150.000 N-m)

F3 - 500.000 N (previously 1000.000 N)

74

Figures 5.13, 5.14, and 5.15 show the frequency response plots for
the coordinates X1, X2, and X3. As shown in these response plots, the
solution is still nonlinear. The comparison in the response plots with
those of case 3 reveals that the onset of the nonlinear response in case
4 occurs at a smaller value of 01 than for case 3. This nonlinear response
is also sustained for a smaller range of o1.'This can be seen most clearly
by comparing the responses of the X3 coordinate for the two cases. In
considering a sweep of the frequency parameter 01 from negative to positive
(left to right in the figures), it can be seen that the onset of the
nonlinear solution.for case 3 occurs at o1==-250, whereas for case 4 this
value is o1== -160. For increasing values of 01, the nonlinear solution is

sustained in case 3 until 01 = 510, whereas for case 3 this value is

01.2230.

0.012 - - . f - e

 

0.008-

 

..0.006 -
><
0.004 L . ; I .

0.002-

-q
’— “
I ~
” ~‘
’ ‘5
—_,"' ~
-—

 

 

\
0 y A . i
-400 -300 -200 -100 0 100 200 300 400

 

Figure 5.13 - Frequency response for X3, case 4, horizontal response
stable solution, ---- unstable solution

 

75

 

0.025»

010.015‘

0.0051

 

 

 

 

O . - . 1 .
-400 -300 -200 -100 0 100 200 300 400

“1
Figure 5.14 - Frequency response for X2, case 4, rotational response
stable solution, ---- unstable solution

 

0.008 . . +447 . . . -

 

0.007 » ,-
0.006- / \
0.005-
n0.004~
0.003-

0.002»

 

0.001—

 

 

A

0 a L l i I -
-400 -300 -200 -100 0 100 200 300 400
O
1

 

Figure 5.15 — Frequency response for X3, case 4, vertical response
stable solution, ---- unstable solution

 

Finally, it can be seen that there is a qualitative change in the
response due to the reduction in forcing amplitude. Although the response
for case 3 is similar to that of case 4 in that there are corresponding

regions where the solutions for the coordinates are single valued, double

76

valued, and triple valued, case 4 response does not exhibit the single
valued response in the region containing c:1 - 0. Close comparison shows
that some interesting changes in the solutions occur as the forcing
amplitudes are varied. One of the solutions which is continuous through
the region given approximately by -150 s <31 5 50 for case 4 as shown in
Figure 5.12, is discontinuous in case 3.

Aside from the differences in response due to the reduction in
forcing amplitude just listed, excitation of X1 again occurs. Therefore,
even at lower forcing amplitudes, the reduction in the resonance peak

associated with the X3 coordinate can be accomplished.

5.6 MEASURES OF ISOLATOR PERFORMANCE

Having presented a collection of response plots for the
displacements x, a, and y of the rigid body, it is interesting to
investigate the associated forces transmitted to the foundation. This
section opens with a brief discussion of isolation performance.

Various metrics have been applied to the evaluation of vibration
isolation systems. Three of the measures prevalent in the literature are
transmissibility, isolation effectiveness, and power transmission.
Additionally, in many of the citations mentioned in this thesis, the
actual force transmitted to the base (or foundation) is the measure of the
performance of an isolation system. It is the purpose here to consider

each of these measures and understand their use.

77
5.6.1 Transmissibility
The concept of transmissibility for damped, linear single degree of
freedom systems is defined in many undergraduate vibrations textbooks. See
for example Rao, (1986) or Thomson, (1972). For a single degree of freedom

system under a harmonic force excitation applied to the mass,

F
transmissibility is defined as T}==._I (see Figure 5.16(a)). This quantity

 

 

is also referred to as force transmissibility (Sykes, 1956; Ungar and
Dietrich, 1966). Here F3 is the force applied to the mass and FT is the
force transmitted through the spring (and damper if needed) to the
foundation (see Figure 5.16(a)). It should be noted that this definition
of force transmissibility is made under the assumption that the structure

to which the spring and damper are attached is rigid. Transmissibility for

a single mass undergoing base excitation is defined as

 

.;| (see Figure

5.16(b)). Here Y is the amplitude of the input displacement and X is the
amplitude of the displacement of the system mass.

The concept of transmissibility is defined more explicitly by Sykes,
(1956) and Ungar and Dietrich, (1966) , utilizing methods from modal theory
(Ewins, 1986). They further qualify the concept of transmissibility with

velocity transmissibility; TV. The ‘velocity transmissibility’ for' an

VR
V6,

isolator positioned between a source and a receiver is defined as IV =

 

 

where V3 is the velocity amplitude at the receiver and V6 is the velocity

amplitude on the source side of the isolator (see Figure 5.16(b)). It can

78
be Shown that 13 = I}, at any one frequency for I» and.13 as defined above

for the single degree of freedom systems shown in Figure 5.16.

If;

 

 

 

 

 

 

 

 

i 1.121;.
1 m m 7757
I

1

Fr TICK,
7VC//7i//: VJ’ 4/'iZ3<rhm

 

(a) (b)

Figure 5.16 - Single degree of freedom isolation system

5.6.2 Effectiveness

These same authors also define another measure of isolator
performance called effectiveness (Sykes, 1956; Ungar and Dietrich, 1966).
This quantity is also referred to as isolator effectiveness, isolation
effectiveness, insertion ratio and insertion loss (Rubin and Biehl, 1967).
Effectiveness is the nondimensional measure of the reduction of the
vibration of a system. It is the ratio of the structural vibration of the
system with no isolator (the source and receiver are rigidly connected) to
the structural vibration of the isolated system. If this ratio is less
than unity, the isolator reduces the amplitude of the receiver velocity
(acceleration or displacement). If this ratio is greater than unity, the
isolator increases the amplitude of the receiver velocity (acceleration or
displacement).

Recently, the notion of effectiveness has been expanded to systems

that include multiple parallel isolators connecting one mass to a

79
foundation (Swanson, et. al.; 1994). For this system, both.21 velocity
effectiveness and a force effectiveness are defined. For velocity
effectiveness, this is done by defining a matrix, Ev, that represents the

level of velocity effectiveness in the isolated mounting system:

v"=15vi (5.2)

s V s

where the velocity effectiveness matrix, Ev, is given by:

Ev = [I+(ze +zs)"zezi“zs] (5.3)
and where \i‘ is the vector of structure velocities for the unisolated

system at the mounting points to the structure, v; is the vector of the

structure velocities for the isolated system at the same mounting points,

and Ze, Z and Zi are the impedance matrices of the engine (rigid body),

8,
structure and isolators respectively. For the force effectiveness, this is

done by defining a matrix, Ef, given by:

fr. =EFfi (5.4)

where the force effectiveness matrix, Ef, is given by:
Ef = [I +2.12e *Zs)’1ZeZ{1] (5.5)

where fh is the vector of the forces transmitted to the structure at the
attachment points, fri is the vector of the forces applied by the engine to

the isolators in the isolated system, and Ze, Z and Zi are as defined

8’
previousLy. The matrix equivalent of the scalar magnitude, the matrix

norm, is used to extend the scalar notion of effectiveness previously

developed to systems with multiple parallel isolators. Instead of

80
maximizing a scalar quantity as is done for a single degree of freedom
system, the spectral norm of the velocity or force effectiveness matrix is
used. Here the maximum and minimum singular values of the velocity
effectiveness matrix or the force effectiveness matrix.provide a frequency

dependent measure of isolator performance.

5.6.3 Transmitted Power

Another metric for vibration isolation systems is that of power
transmission, or power flow (Pinnington, 1987; Qu and.Qian, 1991). In the
article by Pinnington (1987), the expression for the time averaged power
transmission due to a harmonic input is written for multiple input,

multiple isolator system. This expression is given by:
w? = %1m([F’]T[A] [F]) (5.6)

where [F] is the vector of applied forces, [A] is the accelerance matrix,
and a is the frequency of the applied force. The superscript T denotes the
transpose and the superscript * denotes the complex conjugate. Further
qualification of this expression is presented for broad band application
where spectral densities are used. Correlation between the analytical
method and an experimental system is presented in the cited work.

All of the metrics presented up to this point make the assumption
that the system under investigation is linear. No provision is made in any
of the metrics for nonlinear system response.

Moreover, except for the expanded isolation effectiveness approach
utilized by Swanson, et. al. (1994), the effectiveness and
transmissibility metrics assume that the chosen system response variable

(force, velocity, displacement, etc.) is rectilinear in nature.

81
Application of these metrics to general rigid body vibration isolation
problems would be inappropriate, because the system response is typically
not rectilinear.

Finally, it is important to note that for the transmissibility and
effectiveness metrics as applied, it is assumed that the disturbance and
the response are collinear. Thus, a single isolator does not isolate in
more than one direction. As such, with the use of these metrics, the
vectorial aspects of the response are neglected and the actual
effectiveness of the isolation system cannot be captured. Moreover, small
changes in the harmonic force acting on a rigideody on resilient supports
can lead to pronounced changes in the system response. This situation can
affect the direction of the reaction force at the point of connection
between the isolator and the foundation. Therefore, the general response
of the system must be investigated in order to completely capture the
system response and isolation performance.

The next section presents the results of an investigation of the
vibration isolation analysis of the planar three degree of freedom rigid
body. The failure of three of the metrics as applied previously and
described in this section to characterize the isolation performance of the
system is described. An alternative vibration isolation.metric applied to

the problem is established and the reasons are stated.

5.7 VIBRATION ISOLATION MEASURES FOR THE RIGID BODY SYSTEM

The application of the three metrics as presented in Section 5.6 do

not fully characterize the isolation performance of the planar three

82
degree of freedom rigid body system. Each of these metrics will be

considered here relative to the planar three degree of freedom problem.

5.7.1 Transmissibility

The theory of transmissibility was derived with two assumptions. The
first is that the system response acts collinear with the forcing term and
the second is that the response of the system is linear. The second of
these assumptions implies that the system response consists of a single
frequency. For the three degree of freedom planar rigid body system
considered here, neither of these assumptions is valid. As shown in the
response plots earlier in this chapter, the response of the system can
occur in degrees of freedom which are not directly forced and which are
not collinear with the input forcing. The response of the system is also
comprised of components at more than one frequency.

One of the conveniences of' force transmissibility is that it
utilizes the magnitude of the applied force as a normalizing quantity in
establishing the metric. For the single degree of freedom case, this is
convenient as this ratio has a magnitude of unity for the static case.
This serves as a reference point against which the system performance can
be measured. In a like manner, independent force transmissibilities for
each of the degrees of freedom could be established for the case under
investigation. However, this ratio becomes meaningless for the degrees of
freedom which are not forced but have nonzero response, as a division by
zero will result. This is true regardless of the frequency or amplitude of
the forcing.

Velocity transmissibility is not applicable to this problem due to

the assumption that the system investigated here is grounded. Therefore,

83
the velocity at the attachment of the spring to the support has zero
velocity. This results in a division by' zero, regardless of the forcing
frequency. Therefore, both force transmissibility and velocity
transmissibility fail to completely characterize the isolation performance

of this planar system.

5.7.2 Effectiveness

The notion of effectiveness is derived using the assumptions that
the system response is linear and collinear with the input forcing.
Therefore, response in unforced degrees of freedom not collinear with the
input and response at frequencies other than the forcing frequency will
not be included in the system evaluation. Additionally, the notion of
effectiveness as defined by Swanson, et. al., (1994) which expands the
notion of effectiveness to multi-degree of freedom systems, makes the two
assumptions that the isolators act in parallel and that the system
response is linear. Therefore, although this system does allow for
response in more than one degree of freedom, it is inadequate in
characterizing the performance of the isolation system for the planar

three degree of freedom system investigated in this thesis.

5 . 7 . 3 Power Transmission

The use of power transmission was developed under the assumption
that the system response is linear. In the article by Pinnington (1987),
the assumption is made that the frequency of the response and the input
forcing to the system are identical. Therefore, although this system does
account for response in more than one degree of freedom, it does not

account for system response at more than one frequency.

84

5.7.4 Forces Transmitted to the Foundation

The use of forces transmitted to the foundation to characterize the
isolation performance of a rigid body system has been applied previously
(Spiekerman, et.al., 1985; Ashrafiuon, 1993; Swanson, et. al., 1993). In
each of these applications, the- assumption is made that the system
response is linear. Therefore, these models do not contain the complete
system response performance. The nonlinear phenomena of multi—frequency
response and response in unforced degrees of freedom will not be included
in the system response. Hence, the resulting forces are not accounted for

correctly.

5.8 ISOLATION PERFORMANCE OF THE PLANAR THREE DEGREE OF FREEDOM SYSTEM

Assessment of the isolation performance of the system in case 4 is
now developed. The metric chosen for application to this system is the
transmitted forces. Two modifications are required to adapt this metric to
account for the nonlinear response of the system. The first modification
is to develop the capability to account for the presence of more than one
frequency in the system response. In this case, the system response
contains two frequencies, but is periodic. Therefore, the root-mean-square
(rms) of the response is used to account for multiple frequencies.

The second modification to this metric is made to account for
response in all degrees of freedom. This is done by using the forces
developed in each of the springs in the system. These forces are due to
the response of each of the three degrees of freedom and therefore will

account for the total motion of the rigid body.

85
Using these two modifications, the force used in the metric can be
computed. First, the x and y components of the forces for each of the
springs (F1), and F1), for [(1, Fax and FZy for k2, 1:3" and F3), for [(3, and F,“
and F6y for 1‘1.) are computed. The total force from each pair of springs on
the foundation is due to the deflection of the pairs of springs k1 and k2,

and [<3 and [(4, given by F12 and F34 respectively. The forces are given by:

 

F12(t) =J(F1x +F2x)2 1" (F1y +F2Y)2 (5'7)

 

Fun) = Jag, +p,,)2 + (F3, .pMZ (5.8)

These forces are a function of time. Therefore, the rms value of each of
these forces over one period is computed and then they are summed. This is

given by:

E = (F12)ms*(F34)rms (5'9)
It is this force F that is used to assess the isolation performance of the
rigid body system.

Figure 5.17 shows a plot of the magnitude of the force given in
equation (5.9) as a function of the detuning parameter 01 for case 4. This
figure also contains the frequency response of the linear system. Here,
the linear system response was computed using the same method after
setting the coefficients of the nonlinear terms to zero.

From Figure 5.17, it can be seen that in the region between
o1=i125, the force transmitted to the foundation for the nonlinear
system, which can be either of two stable solutions, is reduced in

comparison with the forces transmitted by the equivalent linear system. In

particular, the large peak in the response for the linear system is

86

eliminated" This performance advantage is lost in the regions 228 > a, >
125 and -125 > 01 > -228, where the transmitted forces of the nonlinear
system are greater than the forces of the corresponding linear system. For
a, > 228 and o1‘< -228, the linear and nonlinear solutions are identical
and no advantage is realized from the nonlinear system. In these regions,
the transmitted forces are much lower than those associated with the
linear resonance condition and are typically of less concern.

This analysis illustrates the need for proper system modelling.
Modelling this system using linear theory leads to the incorrect system
response which predicts that a large peak in the transmitted forces will

result surrounding a} = 0.

20000 . . ”i . - -

 

15000

)

10000

FORCE (Newtons

U1
0
O
O

 

 

 

4

o 1 . 1 . A .
-400 -300 -200 -100 0 100 200 300 400
O

1

 

Figure 5.17 - Response plot of F, the force transmitted to the
foundation; stable solution, ---- linear solution

 

5.9 REMARKS

This chapter brings together the two major aspects of this

investigation, that of the frequency response of a rigid body system and

87

the isolation performance of the same system. This chapter shows that for
systems responding nonlinearly, caution must be used in assessing the
isolation performance of the system. In particular, the metrics commonly
used for this purpose do not adequately capture multiple frequency or
multiple direction response. A modified version of the metric which
monitors the forces transmitted to the base is proposed, and offers one
alternative with improved effectiveness over the more commonly used
measures. Using this metric, it is shown that the nonlinear response of
the system can be used to eliminate the peak associated with the
transmitted forces which occurs at resonance for linear systems.

The proposed metric will not likely satisfy the diverse requirements
of all rigid body isolation systems. Indeed, the performance of an
isolation system may require that system specific attributes be considered
in formulating the measure of performance, similar to the process used in
this investigation” This investigation delineates some of the difficulties
which can accompany an investigation into vibration isolation of rigid

bodies.

CHAPTER 6 SUMMARY AND CONCLUSIONS

This chapter first presents a summary of the major sections of this
thesis. The subsequent section presents the conclusions arrived.at in the

investigation. Finally, recommendations for future work are offered.

6.1 SUMMARY

The thesis begins with a literature search into the topics of the
harmonic response and vibration isolation of rigid bodies. This literature
search establishes a foundation on which to pursue the investigation. The
major topics included in the search are vibration isolation, harmonic
response of multi-cylinder engines, the harmonic response of engine
mounts, nonlinear vibration isolation systems, and the response of three
dimensional rigid body systems.

Prior to introducing the rigid body system which is the focus of
this thesis, a comprehensive review of the techniques used for isolation
of automotive engines is presented. The techniques included in the review
are center of percussion, natural frequency placement, torque axis, and
elastic axes. This review is included to encompass the complete scope of
the area of vibration isolation, particularly as it pertains to automotive
engines. The following observations are made about each of these
techniques.

The center of percussion technique is useful in the response of a
rigid body on resilient supports to an impulsive load. This method does

not address the harmonic response of the system.

88

89

Natural frequency placement, associated chiefly with optimization
procedures, attempt to move the system natural frequencies away from
forcing frequencies. This method is based on system response formulated
from linear system theory. In light of the nonlinear response demonstrated
by the system in this investigation, judgment must be used to ensure that
the assumption of linear system response is valid prior to applying this
technique.

The torque axis technique contends that a three dimensional rigid
body under an applied torque not coincident with a principal axis will
tend to oscillate about a spatial axis. Assessment of this system using
the Euler equations of rigid body rotational motion as a guideline,
indicates that this axis as applied in this technique does not exist.

Elastic axes theory asserts that an elastically supported rigid body
responding to an applied force or torque along or about an elastic axis,
will produce only a corresponding translation or rotation. It has been
shown in previous work (Kim, 1991) that a physical set of axes which
result in decoupled motion in all six degrees of freedom of a three
dimensional rigid body does not exist. Furthermore, this notion is based
on linear system theory and the response of the system demonstrated in
this thesis indicates that the response a rigid body system on resilient
supports can be nonlinear.

With this foundation established, the specific system investigated
in this thesis is introduced. The system is planar, three degree of
freedom rigid body supported by resilient elements which are attached to
a rigid base. The system parameters of the rigid body used in the analysis
were chosen to model the planar equivalent of an in-line four cylinder

automotive engine. The equations of motion of the system are derived using

90

Lagrange's equations. A Taylor series expansion of the potential energy is
used. The acceptance criteria which establishes the necessary number of
terms required in the series was found to be important. It was found that
acceptance of a Taylor series based only on a comparison between the
approximate potential energy and the actual potential energy could fail to
predict correct system dynamics. The acceptance criteria used in this
investigation is the correlation between the predicted system response of
the actual equations of motion and the equations of motion obtained using
the Taylor series approximation.

Three cases of increasing complexity are investigated using the
equations of motion previously derived for the rigid body system. These
cases are distinguished by the type of forcing present. The forcing
conditions utilized.are: vertical forcing, torsional forcing, and.combined
vertical and torsional forcing. The method of multiple scales is used to
produce the reduced equations which govern the envelopes of the response.
These equations are then used to establish the steady state frequency
response of the system. Repetitive solution of these equations for a
varying frequency parameter is then used to establish the frequency
response of the system for each of the three cases.

The frequency response of three cases investigated shows the
nonlinear behavior of the system. In particular, a region was found in
case 2 where a constant amplitude steady state does not exist. Cases 1 and
3 showed that a reduction in amplitude can be realized in the coordinate
being forced over the linear response of the system. This reduction in
amplitude is accomplished by excitation of the other modes.

Prior to the investigation of the isolation performance of the

system, an overview of established isolation metrics relative to the

91
response characteristics of the three degree of freedom rigid body system
investigated in this thesis is presented. The metrics assessed are
transmissibility, isolation effectiveness, transmitted power, and forces
transmitted to the foundation of the system. The assumptions made in the
development of these metrics are discussed and the inability of these
metrics to characterize the isolation performance of the three degree of
freedom rigid body system investigated in this thesis is discussed. The
metric of monitoring the forces transmitted to the base is modified to
produce a measure of isolator performance which can account for nonlinear
system behavior. These modifications account for multi-frequency and
multi-directional system response. The application of this metric to the
rigid body system shows that the resonance peak of the transmitted forces

associated with the linear response of the system can be eliminated.

6 .2 CONCLUSIONS

There are three conclusions resulting from this investigation. The
first conclusion is the recognition that the response for a three degree
of freedom rigid body can be nonlinear even if the assumption is made that
the force-deflection characteristics of the supports are linear. Moreover,
this nonlinear behavior does not require large response amplitudes.

The second conclusion of this investigation is that the traditional
approaches to isolation performance using the measures of effectiveness,
transmissibility, transmitted forces, and transmitted power, have
shortcomings when applied to nonlinear systems. Specifically, these
metrics do not account for the multi-frequency response of nonlinear
systems. In addition, effectiveness and transmissibility, do not account

for system response in coordinate directions different from the direction

92
of the input force. However, these two aspects of nonlinear system
response have been incorporated into the metric of assessing the forces
transmitted to the foundation. This alternative metric has been.applied to
the rigid body system.

The third conclusion is that the standard analytical methods used to
determine automotive engine mount placement and.orientation, i.e., natural
frequency placement, torque axis, and elastic axes, may be ineffective
when considering the general motion of the engine on it mounts. This
ineffectiveness is due to the fact that these methods do not account for

nonlinear system behavior.

6.3 RECOMMENDATIONS
Two groups of recommendations are proposed. The first two

recommendations expand the analysis contained herein without requiring

changes in the scope of the problem. Specifically:

i. The present analysis does not investigate the influence of the
mistuning of the natural frequencies of the system. An
analysis of this type will provide insight into the practical
application of this theory where exact tuning of the system

linear natural frequencies typically does not occur.

This analysis does not investigate the affect of the variation

[-h
H.

of the underlying geometry of the system. Studies of the
influence of the physical parameters on the response of the
system will lead to. broader understanding of the system
response. This in turn can lead to Optimization of the

isolation performance of the system.

93

The second group of recommendations would result in an increase in

the scope and complexity of the investigation. Four recommendations are

made. They are:

1.

Ho
Ho

Ho

The analysis should be expanded to analyze a three dimensional
rigid body system. This system would have six degrees of
freedom instead of the three degrees of freedom investigated
in this thesis. This analysis should allow for arbitrary
number, location, and orientation of the resilient supports.
This would be a significant step toward the application of
this analysis to an automotive engine.

The three degree of freedom system should be investigated
experimentally and the results of the analysis presented in
this thesis should.be correlated to the experimental results.
Successful correlation would be the first step in establishing
this analysis as a useful tool for the study of rigid body
isolation systems.

This investigation looks only at forces acting on the rigid
body. Expansion of the analysis to include base excitation
would.have practical significance. The combination of forcing
applied to the rigid body and from the base would facilitate
the analysis of an engine on mounts where the engine is
subjected to both road forces and forces due to engine
unbalance.

It is assumed in this investigation that the base to which the
resilient supports are attached is rigid. This may be a
reasonable assumption for some rigid body systems. However,

for other systems, automotive engines in particular, this

94
assumption is not valid. Therefore, this analysis should be

expanded to incorporate a non-rigid base.

APPENDICES

APPENDIX A

APPENDIX A

CENTER OF PERCUSSION“

Consider a rigid body in a horizontal plane, fixed at one point, and
moving under applied loads as shown in Figure A.l(a). This system is
referred to as a compound pendulum (Greenwood, 1988). The free body
diagram for this rigid body is shown in Figure A.l(b) which includes the
reaction force at O and applied loads. The external force system can be
replaced by the resultants ma and.Ia. The vector quantity ma can be broken
down into its components r02 and ra which act through the center of mass
in the normal and tangential directions, respectively, as shown in Figure
A.l(c). Another resultant force diagram can be obtained by moving the
force mra to point Q along the line 00 (actually beyond point C) such that
the resulting moment created about 0 equals 13a. 13a can then be removed
from the resultant force diagram as shown in Figure A.l(d).

This condition can be written as:

Isa + mrza - mrafl (A.l)

Replacing IG by kgm where kc is the radius of gyration of the rigid body

about G:

kfan:+-au2a a mraz (A.2)

which results in

 

1'The topic of the center of percussion is covered in various forms in
many books on dynamics and vibrations. The material presented here most
closely follows that presented in Meriam (1978) and Greenwood (1988).

95

96

k: +r2=r£ (A.3)
Solving this for 2:
2 z 2
“kg” =ko (AA)
_r' 7

where kb is the radius of gyration of the rigid body about 0. The point Q
defined by this procedure is called the center of percussion of a body of

fixed point 0. Note that 2M0 0.

,4

 

(b)

Figure A.l

The center of percussion has two interesting properties as shown by
the following analysis. The first property involves the response of the
rigid body to an impulsive load and the second shows a property about the
natural frequency of a compound pendulum.

Consider the same compound pendulum as shown in Figure A.l(a),
initially at rest, impacted by an impulse F 1 OG, as shown in Figure A.2.

Since by definition, the impulse is equal to the change in momentum:

97

 

 

 

 

Figure A. 2

(A.5)

F = P2 “P1 = 111(ch "’01)

= r6 where 9 is the angular velocity of the

slw.

Since vG1=0, let v52 =vG - .

line 06. By definition, the angular impulse of F about G, M, is
(A.6)

(A.7)

Also,
5: =11.2 -H1 =156

Using the two previous results along with the fact that H1 - 0 and

H2 =HG = 169, 6 can be solved for:
a = 131’ (A.8)
7?
5 =r6 it can be shown that __5 =k‘32 =rb.
m

Using this relationship and that

With b = 2 - r, the same relationship for the location of Q is obtained as

before:

98

2
12% (A.9>
r

fl =

 

This demonstrates that no impulsive reaction occurs at the point 0 for the
applied impulse F. Note that an impulsive reaction at 0 will occur if F

is applied in any direction other than perpendicular to OC.
Now consider the natural frequency for small motion of the compound

pendulum:

mgr (A. 10)

Consider the natural frequencies of the compound pendulum for motion about
points 0 and 0. Using the previous result, the following relationships are

obtained:

 

 

(4.) = mgr
° \ 70— (A.ll)

_ m b

(WHO '4—150

 

Using the fact that Io =k: +r2, Io =k62 + b2, and that k: = rb, it can be

shown that ((4,)0 = («5)0. This relationship illustrates the second property

of the center of percussion which is that point 0 is the center of
percussion for the body when Q is the center of oscillation and vice
versa. Points 0 and Q are then referred to as reciprocal centers of

percussion.

APPENDIX B

APPENDIX B

DERIVATION OF THE POTENTIAL ENERGY OF THE RIGID BODY SYSTEM

The general formulation of the equations of motion for this system,
shown in Figure B.1, requires that the potential energy for the system be
written in terms of the rigid body displacements (x,y,a). For this
formulation, no assumptions are made regarding small motions and the full,
general motion of the system is considered. To this end, each of the four

springs in the system is analyzed separately.

 

 

 

 

 

Figure B.1
Geometry of the rigid body system

To simplify the geometry of the analysis, Figures B.2 and B.3 do not
explicitly show the spring element. The lengths and displacements are
shown using only lines. It is the change in length of each of the springs
which is required. The center of mass of the rigid body, G, is shown, but
the outside dimensions of the rigid body are not shown. The distance
between the center of mass and the attachment point of the springs,

denoted A, is indicated in both figures as the radius of a circle.

99

100

H- A(1-coso()
L‘— (a

 

 

 

 

////////7

Figure B.2
Geometry associated with springs k1 and k2

Consider the geometry associated with springs R1 and k2 with free
lengths £1 and 22 respectively, as shown in Figure B.2. Using the
dimensions indicated, the expressions for the new lengths of spring It, and

spring [(2, denoted L.l and L2, respectively, can be written:

 

L1 "[[21fl‘351'JR-7Hy]2 + [A (l ~cosa) -x]2 (3'1)

 

L2 =1/[22+A(1-Cosa) -x]2 + [Asina 4.3,]2 (B.2)

Consider the geometry associated with springs [<3 and [(4 with free
lengths £3 and £4 respectively, as shown in Figure B.3. Using the
dimensions indicated, the expressions for the new lengths of spring [(3 and

spring k , denoted L3 and L , respectively, can be written:

 

L3 =J[23*A(1 -cosa) «+sz + [ASina-y]2 (B.3)

 

L4 = \/[24 ~Asina «I-y]2 + [A (1 'cosa) +x]2 (3.4)

101

A(1-cosa) -~

 

 

 

 

 

Figure B.3
Geometry associated with springs k3 and kb

The expression for the potential energy of the system, V can now be

written:
v = %k1(L1-£1)2 + %k2 (La-122)2 + %k3(L3-£3)2 + gk, (L, -2,)3 (8.5)

where all of the parameters have been defined previously.

APPENDIX C

APPENDIX C

MATHEMATICA PROGRAM USED FOR DERIVING THE EQUATIONS OF MOTION

The following is the source code used in the Mathematica (Wolfram,
S., 1991) program to derive the equations of motion for the systems
investigated in this thesis. The notation used below is consistent with

that shown in the figures of Chapter 2 and Appendix B.

Ig = 14.155;

m = 219.4;

kl a 1732312.8;
k2 - 433078.2;
k3 433078.2;
k4 1732312.8;
x0 - 0.0;

y0 - 0.0;
alphaO = 0.0;
11 - 0.102;

12 = 0.102;

13 = 0.102;

14 = 0.102;

A - 0.127;

T = (l/2)*m*xdot22 + (l/2)*m*ydot“2 + (l/2)*Ig*alphadot“2;

L1 = Sqrt[(ll + A*(Sin[alpha]) + y)“2 +
(x - A*(1 - Cos[a1pha]))A2];

L2 - Sqrt[(12 — (x - A*(1 - Cos[a1pha])))“2 +
(A*(Sin[alpha]) + Y)A2];

L3 - Sqrt[(l3 + A*(l - Cos[a1pha]) + x)“2 +
(Y ‘ A*(Sin[alpha]))“2];

L4 = Sqrt[(14 + y - A*(Sin[a1pha]))‘2 +
(x + A*(1 - Cos[a1pha]))‘2];

Val - (1/2)*k1*(L1-ll)“2;
Va2 - (1/2)*k2*(L2—12)‘2;
Vbl - (1/2)*k3*(L3-13)“2;
Vb2 - (1/2)*k4*(L4-14)‘2;

V1 - Val + Va2 + Vbl + Vb2;

102

(at

El

E2

E3

E4

E5

E6

E7

E8

E9

E10
E11
E12
E13
E14
E15
E16
E17
E18
E19
E20
E21
E22
E23
E24
E25
E26
E27
E28
E29
E30
E31

103

Expand the potential energy in a Taylor Series

about equilibrium point *)

D[Vl,{x,2}] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,{y,2}] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,{alpha,2}] /. {x->x0,y->y0,alpha->a1pha0};
D[Vl,x,y] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,alpha] /. {x->x0,y->y0,alpha->a1pha0};
D[Vl,y,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,{x,3}] /. {x—>x0,y->y0,alpha->alpha0};
D[Vl,{y,3}] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,{alpha,3}] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,x,y] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,x,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,y,y] /. {x->x0,y->y0,a1pha->alpha0};
D[Vl,alpha,y,y] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,alpha,alpha] /. [x->x0,y->y0,alpha->alpha0};
D[Vl,y,alpha,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,y,alpha] /. {x->x0,y->y0,alpha->a1pha0};
D[Vl,{x,4}] /. {x->x0,y->y0,a1pha->alpha0};
D[Vl,{y,4}] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,{alpha,4}] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,x,x,y] /. {x->x0,y->y0,alpha->a1pha0};
D[Vl,x,x,x,a1pha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,y,y,y] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,y,y,y,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,alpha,a1pha,alpha,x] /. {x->x0,y->y0,alpha->a1pha0};
D[Vl,alpha,alpha,a1pha,y] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,x,y,y] /. {x—>x0,y->y0,alpha->alpha0};
D[Vl,x,x,alpha,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,y,y,alpha,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,x,y,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,y,y,alpha] /. {x->x0,y->y0,alpha->alpha0};
D[Vl,x,y,alpha,alpha] /. {x->x0,y->y0,alpha->a1pha0};

V - ((1/2)*(El*x“2 + E2*y“2 + E3*alpha“2 + 2*E4*x*y + 2*E5*x*alpha +
2*E6*y*alpha) +

(1/6)*(E7*x“3 + E8*y‘3 + E9*alpha“3 + 3*E10*x‘2*y +
3*E11*x“2*alpha + 3*E12*x*y“2 + 3*E13*y“2*alpha +3*El4*x*alpha‘2 +
3*E15*y*alpha“2 + 6*E16*x*y*a1pha) +

(1/24)*(E17*XA4 + E18*yA4 + E19*alpha“4 + 4*E20*x“3*y +
4*E21*x“3*alpha + 4*E22*y“3*x + 4*E23*y“3*alpha + 4*E24*alpha“3*x +
4*E25*alpha“3*y + 6*E26*x‘2*y“2 + 6*E27*x“2*alpha“2 +
6*E28*y“2*alpha“2 + 12*E29*x“2*y*alpha +

12*E30*x*y22*alpha + 12*E31*x*y*a1pha“2));

PTx - D[D[T,xdot],t,NonConstants -> {xdot,ydot,alphadot,x,y,alpha}];
PTy - D[D[T,ydot],t,NonConstants -> {xdot,ydot,alphadot,x,y,alpha}];
PTalpha - D[D[T,alphadot],t,NonConstants ->

{xdot,ydot,alphadot,x,y,alpha}];

104

PVx - D[V,x};
PVy = D[V,y];
PValpha - D[V,alpha];

EQl = PTx + PVx;
EQ2 = PTy + PVy;
EQ3 = PTalpha + PValpha;

Print ["EQl = ". EQll;
Print ["502 = ". EQ2];
Print ["EQ3 - ", EQ3];

APPENDIX D

APPENDIX D

NONDIMENSIONALIZATION OF THE EQUATIONS OF MOTION5

The equations of motion of the system presented previously are the
starting point for the nondimensionalization. These equations are listed

again for convenience.

I11581 * C1X1 * (k2*k3)x1 * 51X2X3 * 52X1X3 * 53X:

(D.la)
4» max: + 552:? + 062(1):; =1:1 cosm:
1(A2+BZ)X + X +A2(k +k x + xx + X3+ XX2
3111 2622 11.)25713582 5912(D1b)
4‘ p10X1ZX2 + p11X2X32 =F2C0592t
u 2 2
1sz3 * C5X3 * (k1+k,,)X3 * I312X1 * p13X1X2 * BMXZX: (D.lc)

4‘ B15X12X3 4' I316X32 =’ F3COSQ3C

The first step in the nondimensionalization process is to designate
the characteristic length, mass, and time. The characteristic mass is the
mass of the rigid body m, and A is chosen as the characteristic length.
Both of these dimensions are sound choices since if either of these were
equal to zero, the problem would not make physical sense. The choice for

a characteristic time is not as obvious. Three possible candidates are:

m m m . O O O
, , or . (To find these, terms which include time,
E1 *El. 122 +753 BiA

 

5The method.used.here to nondimensionalize the equations of motion is
presented in Murdock (1991).

105

106
such as [(1, k2, R3, [(4 and any of the fli's should be determined. Mass and
length can be eliminated from these by multiplying or dividing by the
already designated characteristic mass and length.) The third candidate
is not a naturally appearing expression in the problem as are the first
two expressions, therefore it can be eliminated. Either of the first two

expressions will be a suitable characteristic time, therefore the

characteristic time for this set of equations is chosen to be .
E2 *3

Having made these choices, the coordinates and parameters of the problem

can be nondimensionalized.by multiplying and dividing by these quantities

raised to appropriate powers. The following are obtained:

x2 := X3 has units of radians which are already dimensionless

2 m 1
r: k k - a. =
“’1 (“VIZ—Isa: 1

1 _ 1‘1”“

/.= 2 . m .1.

2._ . m .1_k1+k4
“3"(1‘1’k4) 172; '5 ’51:;

(Mass does not appear explicitly in the nondimensionalized

equations.)

107

.— O O 'I' -
IG '-III(A2+B) _ _ -—(l+-—) -—(1+V) (where V=_)

 

 

 

t:=T- [(21:18
2:1 =Fi._§1.%.1(z_n;_k_3
c1 91-2111 =c1-E- £2115
kgm? * l m 1
021 1’2” ”2'5 my:

 

kg -2 .= .1.
C3 :1 ”3' €350.72;

 

 

The nondimensionalization of the coefficients of the nonlinear
terms, the 55's, can be accomplished in a manner similar to that of the
ci '5 shown above. This can be accomplished by determining the units of each
individual term required to keep the units of the equations consistent.

The nondimensionalized version of these values are shown as 515’s in the

nondimensionalized form of the equations. The nondimensionalized form of

the equations is :

i1 4.23.11,“ + fig)“ +&1X2X3 *&2X1X3 +&3X23 +é6x1x22 (D 28)

+ ésx? + &6x1x§ . f1cosn1t

1 2" '- / - ~ 3 .. 2
_1+v x+2 X+kx+axx +ax +axx

* £110X12X2 + &11X2X32 3 fzcosnzt

108

-- . 2 ~ 2 ~ ~ 2
x + 2 X + x + + +
3 #3 3 “3 3 “12"1 a13x1x2 cl11.X2X3 (D.2c)

+ é15xfx3 + 9163: = f3 c0393 t

LI ST OF REFERENCES

LI ST OF REFERENCES

Andrews, G. J., (1960). Vibration Isolation of a Rigid Body on Resilient

Supports. The Journal of the Acoustical Society of America, v 32, n
8: 995-1001.

Ashrafiuon, H., (1993). Design Optimization of Aircraft Engine-Mount
Systems. Journal of Vibrations and Acoustics, v 115: 463-467.

Bachrach, Ben I., (1995). Personal communication at the Ford Motor
Company, April 25, 1995.

Bernard, James E. and Starkey, John M., (1983). Engine Mount Optimization.
SAE Paper 830257, Warrendale , PA.

Bolton-Knight, B. L., (1971). Engine Mounts: Analytical Methods to Reduce
Noise and Vibration. Instn. Mech. Engrs., C98/7l.

Brach, R. Matthew and Haddow, A. G., (1993). On the Dynamic Response of
Hydraulic Engine Mounts. SAE Paper 931321, SAE, Warrendale, PA.

Butsuen, T., Ookuma, M., and Nagamatsu, A., (1986). Application of Direct
System Identification Method for Engine Rigid Body Mount System, SAE
Paper 860551, SAE, Warrendale, PA.

Crede, Charles E., (1951). Vibration and Shock Isolation. John Wiley 8
Sons, New York, New York.

Den Hartog, J. P., (1984). Mechanical Vibrations, Dover Publications,
Inc., New York.

Derby, Thomas F., (1973). Decoupling the Three Translational Modes from
the Three Rotational Modes of a Rigid Body Supported by Four Corner-
located Isolators. Shock and Vibration Bulletin, Bulletin 43, Part
4: 91-108.

Ewins, D. J., (1986). Modal Testing? Theory and Practice. Research Studies
Press, Ltd., Letchworth, Hertfordshire, England.

Efstathiades, G. J. and Williams, C. J. H., (1967). Vibration Isolation

Using Nonlinear Springs. J. Mech. Sci., Pergamon Press, Ltd., v 9:
27-44.

109

110

Ford, D. M., (1985). An Analysis and Application of a Decoupled Engine
Mount System for Idle Isolation. SAE Paper 850976, SAE, Warrendale,
PA.

Fullerton, Raymond R., editor, (1984). Front Wheel Powertrain Mounts:
Design Guide. Ford Motor Company.

Furubayashi, M. , Soma, N. , Nakada, T. , and Iwahara, M. , (1991). Estimation
of the Engine Exciting Force and the Rigid Body Vibration Mode of
the Powerplant, 1991 Small Engine Technology Conference Proceedings,
Published by Society of Automotive Engineers of Japan, Tokyo, Japan,

Geck, Paul E. and Patton, R. D., (1984). Front Wheel Engine Mount
Optimization. SAE Paper 840736, Warrendale, PA.

Gennesseoux, A. , (1993) . Research for New Vibration Isolation Techniques:
From Hydro-Mounts to Active Mounts. SAE Paper 931324, SAE,
Warrendale, PA.

Greenwood, D. T. , (1988). Principles of Dynamics. Prentice-Hall, Englewood
Cliffs, New Jersey.

Grootenhuis, P. and Ewins, D. J., (1965). Vibration of a Spring Supported
Body. Journal Mechanical Engineering Science, v 7, n 2: 185-192.

Haddow, A. G., Barr, A. D. S., and Mook, D. T., (1984). Theoretical and
Experimental Study of Modal Interaction in a Two-degree-of-freedom
Structure. J. of Sound and Vibration, v 97, n 3: 451-473.

Hata, Hideyuki, (1992). Body Vibration Analysis Using a System Model
Approach. JSAE, v 46, n 6.

Henry, R. F. and Tobias, S. A., (1959). Instability and Steady-state
Coupled Motions in Vibration Isolating Suspensions. Journal
Mechanical Engineering Science, v 1, n 1: 19-29.

Hildebrand, F. B., (1976). Advanced Calculus for Applications. Prentice-
Hall, Englewood Cliffs, New Jersey.

Himelblau, H. Jr. and Rubin, S. , (1961). Vibration of a Resiliently
Supported Rigid Body. Shock and Vibration Handbook, Harris and Crede
Editors, Volume 1, Chapter 3, McGraw-Hill, New York, New York.

International Mathematics and Statistical Library, Inc. , (1980) . Reference
Manual. Houston, Texas.

John, R. A. and Straneva, E. J., (1966). A New Generation of General
Aircraft Engine Mountings, SAE Paper 660222, SAE, Warrendale, PA.

Johnson, Stephen R. and Subhedar, Jay W. , (1979) . Computer Optimization of
Engine Mount Systems. SAE Paper 790974, SAE, Warrendale, PA.

lll

Kahn, P. B., (1990). Mathematical Methods for Scientists and Engineers.
Wiley-Interscience, John Wiley & Sons, New York, New York.

Kim, B. J., (1991). Three Dimensional Vibration Isolation Using Elastic
Axes. M. S. Thesis, Michigan State University, East Lansing, MI.

Kim, G. and Singh, R., (1993). Nonlinear Analysis of Automotive Hydraulic
Engine Mount. Journal of Dynamic Systems, Measurements, and Control ,
v 115: 482-487.

Lee, J. M., Yim, H. J., and Kim, J., (1995). Flexible Chassis Effects on
Dynamic Response of Engine Mount Systems. SAE Paper 951094, SAE,
Warrendale, PA.

Matsuda, A., Hayashi, Y., and Hasegawa, J., (1987). Vibration Analysis of
a Diesel Engine at Cranking and Idling Modes and Its Mounting
System. SAE Paper 870964, SAE, Warrendale, PA.

Meirovitch, L., (1975). Elements of Vibration Analysis. McGraw-Hill, New
York, New York.

Meriam, J. L., (1978). Engineering Mechanics, Volume 2, Dynamics. John
Wiley & Sons, New York, New York.

Metwalli, S. M., (1986). Optimum Nonlinear Suspension Systems. Journal of
Mechanisms, Transmissions, and Automation in Design, v 108, June
1986: 197-202.

Murdock, J. A., (1991). Perturbations: Theory and Methods. John Wiley &
Sons, New York, New York.

Nayfeh, A. H. and. Balachandran, B., (1989). Modal Interactions in
Dynamical and Structural Systems. Appl. Mech. Rev., v 42, n 11, Part
2: 175-201.

Nayfeh, A. H. and Mook, D. T., (1979). anlinear Oscillations. Wiley-
Interscience, John Wiley & Sons, New York, New York.

Oh, T., Lim, J. and Lee, S. C., (1991). Engineering Practice in Optimal
Design of Powertrain Mounting System for 2.02 FF Engine. Proceedings
for the Sixth International Pacific Conference on Automotive
Engineering, Published by Korea Society of Automotive Engineers,
Inc., 1638-3, Socho-dong, Seoul, South Korea: 607-616.

Perkins, N. C., (1992). Modal Interactions in the Non-linear Response of
Elastic Cables Under Parametric/External Excitation. International
Journal of Non-Linear Mechanics, v 27, n 2: 233-250.

Pinnington, R. J., (1987). Vibrational Power Transmission to a Seating of
a Vibration Isolated Motor. Journal of Sound and Vibration, v 118,
n 3: 515-530.

112

Qu, J. and Qian, B. , (1991) . On the Vibrational Power Flow from Engine to
Elastic Structure through Single and Double Resilient Mounting
Systems. SAE Paper 911057, SAE, Warrendale, PA.

Radcliffe, C. J., Picklemann, and Spiekerman, C. E., (1983). Simulation
of Engine Idle Shake Vibration, SAE Paper 8302259, SAE, Warrendale,
PA.

Rao, S. S., (1986). Mechanical Vibrations. Addison-Wesley, Reading, MA.

Rubin, S. and Biehl, F. A., (1967). Mechanical Impedance Approach to
Engine Vibration Transmission Into an Aircraft Fuselage, SAE Paper
670873, SAE, Warrendale, PA.

Saitoh, S. and Igarashi, M., (1989). Optimization Study of Engine Mounting
Systems, 1989 ASME 12th Biennial Conference on Mechanical Vibration
and Noise, Montreal, Quebec, Canada, September 17-21.

Sakamoto, H. , Yazaki, K., and Fukushima, M. , (1981). Reduction of
Automobile Booming Noise Using Engine Mountings That Have an
Auxiliary Vibrating System, SAE Paper 810399, SAE, Warrendale, PA.

Schmitt, Regis, V. and Leingang, Charles, J. , (1976). Design of Elastomer-
ic Vibration Isolation Mounting Systems for Internal Combustion
Engines, SAE Paper 760431, SAE, Warrendale, PA.

Shiomi, Kazuy'uki and Mizuguchi, Hiroshi, (1991). Isolating Vibration
through Use of Traveling Center of Engine Oscillation, 1991 Small
Engine Technology Conference Proceedings, Published by Society of
Automotive Engineers of Japan, Tokyo, Japan: 573-582.

Shoup, T. E., (1971). Shock and Vibration Isolation Using a Nonlinear
Elastic Suspension. AIAA Journal, v 9, n 8: 1643-1645.

Shoup, T. E., (1972). Experimental Investigation of a Nonlinear Elastic
Suspension, AIAA Journal, v 10, n 4: 559-560.

Shoup, T. E. and Simmonds, G. E., (1977). An Adjustable Spring Rate
Suspension System, AIAA Journal, v 15, n 6: 865-866.

Singh, R., Kim, G. and Ravindra, P. V., (1992). Linear Analysis of
Automotive Hydro-Mechanical Mount with Emphasis on Decoupler
Characteristics. Journal of Sound and Vibration, v 158, n 2: 219-
243.

Snowdon, John C. , (1979). Vibration Isolation: Use and Characterization.
Journal of the Acoustical Society of America, v 66, n 5: 1245-1274.

Spiekerman, C. E., Radcliffe, C. J., and Goodman, E. D., (1985). Optimal
Design and Simulation of Vibration Isolation Systems. Journal of
Mechanisms, Transmission and Automation in Design, v 107: 271-276.

113

Staat, L. A”, (1986). Optimal Design of Vibration Isolators Based on
Frequency Response. M. S. Thesis, Michigan State University, East
Lansing, MI.

Swanson, D. A., Wu, H. T., and Ashrafiuon, H., (1993). Optimization of
Aircraft Engine Suspension Systems. Journal of Aircraft, v 30, n 6:
979-984.

Swanson, D. A., Miller, L. R., and.Norris, M. A., (1994). Multidimensional
Mount Effectiveness for Vibration Isolation” Journal of Aircraft, v
31, n 1: 188-196.

Sykes, A“ 0., (1956). The Evaluation.of Mounts Isolating Nonrigid Machines
from Nonrigid Machines. Shock and Vibration Instrumentation, New
York: ASME.

Thomson, William T., (1972). Theory of Vibration with Applications.
Prentice-Hall, Englewood Cliffs, NJ.

Timpner, F. F., (1965). Design Considerations for Engine Mounting, (966B)
SAE Paper 650093, Warrendale, PA.

Tobias, S. A., (1959). Design of Small Isolator unite for the Suppression
of Low-frequency’Vibrationu Journal Mechanical Engineering Science,
v 1, n 3: 280-292.

Ungar, E. E. and Dietrich, C. W., (1966). High-frequency Vibration
Isolation, Journal of Sound and Vibration, v 4, n 2: 224-241.

Whitekus, J. D., (1994). Personal communication.at the Ford Motor Company,
August 3, 1994.

Wilson, William Ker, (1959). Vibration Engineering: A Practical Treatise
on the Balancing of Engines, Mechanical Vibration, and Vibration
Isolation. Charles Griffin and Company Ltd., London.

Wolfram, S., (1991). Mathematica: A System for Doing Mathematics by
Computer. Addison-Wesley, Redwood City, CA.

Xuefeng, Jiang, (1991). A Study of Inclined Engine Mounting System Design
Parameters, Proceedings for the Sixth International Pacific
Conference on Automotive Engineering, Published by Korea Society of
Automotive Engineers, Inc., 1638-3, Socho-dong, Seoul, South Korea:
339-345.

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