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dissertation entitled

Performance, Stability, and Localization
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Abdallah Saleh Alsuwaiyan

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PERFORMANCE, STABILITY, AND
LOCALIZATION OF SYSTEMS OF VIBRATION

ABSORBERS

By

Abdallah Saleh Alsuwaz'yan

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1999

ABSTRACT

PERFORMANCE, STABILITY, AND LOCALIZATION OF
SYSTEMS OF VIBRATION ABSORBERS

By

Abdallah Saleh Alsuwaiyan

This work addresses the dynamics of systems of identical and nearly-identical
tuned vibration absorbers. Of particular interest are the details of the dynamic
response and the manner in which these relate to the performance of the absorber
system. The results are analytical in nature, and based on both linear and nonlinear
dynamic models of absorber systems. In all cases the results obtained are verified
using extensive simulations.

The main focus of the thesis is on the reduction of torsional vibrations via the use
of centrifugal pendulum vibration absorbers (CPVAs). However, many of the results
extend directly to the case of translational absorbers, and this is pointed out and
exploited in some special cases. CPVAs are small masses that move along designer-
specific paths relative to the rotor whose vibration is to be suppressed. Until recently,
designs of CPVA systems were based on the response of a single absorber mass or the
dynamically equivalent case of multiple absorber masses moving in unison. Recent
studies of multiple absorbers that ride on a specific absorber path, namely, the so-
called tautochronic path, showed that the unison response does not always prevail,
due to dynamic instabilities that arise from nonlinear effects. In the present work,
the more general case where systems of multiple absorbers ride on general paths is

considered. The existence and stability of unison motions and the general effects

of path type and path mistuning are investigated by utilizing a physically relevant
scaling of the system parameters that allows for the application of the asymptotic
method of averaging. The existence and stability of the unison response and some
types of non-unison responses are considered in detail. It is found that the stability
of the unison motion depends on both the path type and the level of mistuning. For
the commonly used circular paths, and paths close to them, two types of instabilities
were found. One is the well-known classical jump, which maintains the unison nature
of the response, and the other is a bifurcation that gives rise to one or more types
of non-unison responses. Steady-state responses other than unison were found to
exist for over-tuned circular paths and paths close to them. In cycloidal paths, no
instabilities occur for realistic ranges of mistuning, and therefore these shortcomings
are completely avoided.

Another theme of this thesis is the investigation of localization phenomena in
systems of vibration absorbers. Localization corresponds to a response in which the
system’s vibration energy is concentrated in a single absorber (or a small subset of
absorbers), resulting in much higher amplitudes of vibration than expected for that
(those) absorber(s). This behavior is investigated for both free and forced vibrations.
In the linearized models, it occurs for the free vibration modes as well as the har-
monically forced, steady-state responses for both translational and torsional vibration
absorber systems. Steady-state localization, as a particular form of non-unison mo—
tion, is also found to occur for the nonlinear forced vibration of systems of CPVAs
riding on over-tuned circular paths and paths close to them.

Based on the findings of this thesis, one can conclude that slightly positively
mistuned cycloidal paths are the best choice for practical implementation of sys-
tems of nearly-identical CPVAs, and, similarly, that a slight hardening nonlinearity
is desirable for systems of nearly-identical translational absorbers. This avoids the

instabilities while not sacrificing system performance.

To my great father Saleh M. Alsuwaiyan

iv

ACKNOWLEDGMENTS

All prayers and glories are due to the all mighty ALLAH whose help is the key to
any accomplishment.

My appreciations and thanks are due to my advisor, Prof. S. W. Shaw who was
very helpful and cooperative. He enriched me with valuable information and guidance
throughout my work. He is like an ocean full of knowledge. I learned a lot from him.

My thanks are also due to my Ph.D committee members, Prof. J. Schuur,
Prof. A. Haddow, and Prof. B. Feeny. In addition to being a member in my Ph.D
committee, every one of them taught me a graduate course which I found to be very
useful and helpful.

Here, I find it a good opportunity to extend my thanks to a great Professor
from whom I also learned a lot. He is Prof. H. Khalil of the Electrical Engineering
department.

My sincere thanks are also due to my sponsoring agency, the general organization
for technical education and vocational training (GOTEVT) of Saudi Arabia, for giving
me this opportunity to learn and achieve my Ph.D degree. I also deeply appreciate
partial support from the US. National Science Foundation.

I will never forget to thank my parents and my wife’s parents whose continuous
prayers and directions were like the light that showed me the way.

Last but not least, I keep my deep thanks for my wife for her support throughout
my studies here in the US. I ask ALLAH to reward her the best in this life and in
the hereafter.

TABLE OF CONTENTS

vi

LIST OF FIGURES viii

LIST OF TABLES xi

1 Introduction 1

1.1 History and Background ......................... 3

1.1.1 History ............................... 3

1.1.2 Background ............................ 6

1.2 Objective and Dissertation Organization ................ 8

Performance and Dynamic Stability of General-Path CPVAs 11

2.1 Mathematical Model ........................... 12

2.1.1 Equations of Motion ....................... 12

2.1.2 General Path Representation ................... 18

2.1.3 Scaling ............................... 19

2.1.4 The Averaged Equations ..................... 21

2.1.5 Existence and Stability of the Unison Response ........ 23

2.2 Numerical Examples and Discussion ................... 30

2.2.1 Effect of Path Type ........................ 31

2.2.2 Circular Paths ........................... 35

2.2.3 Cycloidal Paths .......................... 41

2.2.4 Note on the Absorber System Performance ........... 45

2.3 Summary and Conclusions ........................ 47
Non-synchronous Steady-State Responses of Tuned Pendulum Vi-

bration Absorbers 51

3.1 Mathematical Formulation ........................ 52

3.1.1 Existence of Certain Steady State Solutions .‘ ......... 54

3.1.2 Stability of the Steady State Solutions ............. 56

3.2 Numerical Example and Discussion ................... 58

3.2.1 Circular Path ........................... 58
3.2.2 Cycloidal Paths .......................... 66
3.3 Conclusions ................................ 66

4 Localization of Free Vibration Modes in Systems of Nearly-Identical

Vibration Absorbers 67
4.1 Formulation ................................ 68
4.1.1 Translational Vibration Absorbers ................ 68

4.1.2 Centrifugal Pendulum Vibration Absorbers ........... 70

4.2 Analysis and Discussion ......................... 72
4.2.1 General Features of the System ................. 72

4.2.2 Perturbation Method Formulation ................ 73

4.2.3 Examples ............................. 74

4.3 Conclusions ................................ 75

5 Forced Localized Response of Vibration Absorbers 79
5.1 Linear Forced Response ......................... 79
5.1.1 Mathematical Formulation .................... 79

5.1.2 Numerical Examples and Discussion ............... 82

5.2 Nonlinear Forced Response ........................ 89
5.3 Conclusions ................................ 93

6 Summary, Conclusion, and fixture Work 95
6.1 Summary ................................. 95
6.2 Conclusion ................................. 99
6.3 Future Work ................................ 99
BIBLIOGRAPHY 103

vii

1.1
1.2
1.3
2.1
2.2
2.3
2.4
2.5

2.6
2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

2.15

LIST OF FIGURES

Translational vibration absorber. .................... 8
Torsional vibration absorber ........................ 10
Frequency response of the mass M of Figure 1.1 ............ 10
Schematic diagram of multiple CPVAs arrangement ........... 13
CPVAs ................................... 14
Mistuning levels below which non—unison motions do not exist. n=2. . 29

Effect of absorber path type on the stability of the unison response. . 32
Path type effect on the absorbers performance. 6 = 0.05, n = 2.
(a) Perfectly tuned, (b) a = 0.4 ...................... 34
Circular path. Upper: 0% mistuning; lower: 4% mistuning ...... 36
Critical torque level versus percentage mistuning for circular path CP-
VAs (a) Bifurcation from unison, (b) Jump ............... 37
Effect of mistuning, circular path CPVAs - analytical. Stability is not
indicated here ................................ 38
Effect of mistuning on rotor acceleration ( circular path CPVAs ) . . 40
N=4 circular path CPVAs with mistuning level of 0.5%, (a) before
bifurcation to non—unison (Pa = 0.0076), (b) after bifurcation to non-
unison (F9 = 0.0082) - from numerical simulation ............ 41
Cycloidal paths. Upper 0% mistuning, middle 5% mistuning, lower
10% mistuning .............................. 42
Rotor angular acceleration, vv’(0) for mistuned cycloidal paths; (a) 0%
Mistuning (b) 2.5% Mistuning (c) 5% Mistuning (d) 10% Mistuning . 43

Numerically simulated absorbers responses for 0' = —0.1 (N =4 cy-
cloidal path absorbers) .......................... 45
Numerically simulated absorbers responses for a = —0.2 (N =4 cy-
cloidal path absorbers) .......................... 46
Rotor angular acceleration, vv’(0), for [‘9 = 0.02 for mistuned circular

paths — from numerical simulations ................... 48

viii

2.16 Rotor angular acceleration, vv’(0), for mistuned cycloidal paths, Pa =

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

4.1

4.2

4.3

4.4

4.5

5.1

5.2

5.3

5.4

5.5
5.6

0.02 - from numerical simulations, ( N =4 ) ...............
Steady-state solution conditions for a perfectly tuned circular path,
(a) M=1, (b) M=2, Pa = 0.007. .....................
Steady-state solution conditions for a mistuned circular path with a =
0.1, (a) M=1, (b) M=2, Pa = 0.006 ....................
Response curves for three absorbers for the case M=1, N =4. Circular
paths with a = 0.1 .............................
Response curves for one absorber for the case M=1, N =4. Circular
paths with a = 0.1 .............................
Response curves for two absorbers for the case M=2, N =4. Circular
paths with a = 0.1 .............................
Response curves for three absorbers for the case M=1, N=4. Circular
paths with a = 0.5 .............................
Response curves for one absorber for the case M=1, N =4. Circular
paths with a = 0.5 .............................
Response curves for two absorbers for the case M=2, N =4. Circular
paths with a = 0.5 .............................
Typical localized response. a = 0.1 circular path. N =4, M=1, Pa =
0.004. ...................................
Typical localized response. a = 0.5 circular path. N =4, M=1, F9 =
0.004. ...................................

Modes of vibration for example 1(a). (*) Exact,(o) second order per-
turbation .................................
Modes of vibration for example 1(b). (2:) Exact, (0) second order
perturbation. ...............................
Some modes of vibration for example 2 . (*) Exact, (0) second order
perturbation ................................
Absorbers amplitudes verses fluctuating torque level for example 5.1.

Absorbers amplitudes verses fluctuating torque level for example 5.2.

Absorbers amplitudes verses fluctuating torque level for example 5.3.

Effect of damping level on localization .................
Effect of mistuning differences on localization .............
Non-dimensional rotor acceleration verses fluctuating torque level for

example 5.1. (From numerical simulations) ...............

ix

49

59

60

61

61

62

62

63

63

78
83
85
87
88

90

5.7

5.8

Localized response curves. N = 6 circular path absorbers with c =
1/75, and a = 2.0. (a) The case M=1. (b) The case M=2. ......
Localized response curves. N = 6 circular path absorbers with c =

1/75, and a = 4.0. (a) The case M=1. (b) The case M=2. ......

91

92

2.1

2.2

4.1
4.2
5.1
5.2
5.3

LIST OF TABLES

Effect of the number of absorber on bifurcation to non-unison torque
level. Circular paths. (From theory). .................. 35
Theoretical ranges of absorbers amplitudes and torque levels at which
non unison motions exist for N = 4 absorbers with cycloidal paths . . 44

Data for example 1 ............................. 74
Data for example 2 ............................. 74
Data for example 5.1 ........................... 82
Data for example 5.2 ........................... 84
Data for example 5.3 ........................... 86

xi

CHAPTER 1

Introduction

Tuned vibration absorbers are one of the most successful and widely used methods for
reducing excessive vibration levels in mechanical systems. The linear theory of simple
vibration absorber systems is very well established. However, there remains much
to learn about the dynamics of systems composed of multiple absorbers, especially
when nonlinear behavior is taken into account. In this thesis we focus on the linear
and nonlinear dynamic response of systems of nearly identical absorbers of both
translational and torsional types.

Many methods can be employed to reduce vibration levels, including the addition
of inertia, such as lumped masses or flywheels [I], or the implementation of tuned
vibration dampers [2, 3, 4]. These methods, however, each have some shortcomings.
Increased inertia has several undesirable side effects, while tuned dampers dissipate
energy and are typically valid only for a range near a single resonance frequency (or a
small set of them). In cases where the excitation frequency is known and essentially
fixed, lightly damped, tuned absorber systems are very effective. Such system are the
focus of this investigation.

The thesis considers the general problem of a primary inertia that is subjected
to a fluctuating load and to which are attached several identical, or nearly identical,

tuned vibration absorbers that are used to reduce the vibration amplitude of the

primary mass. Such arrangements of absorbers are quite common and are used to
balance inertias and forces in the system. It is typically expected that these absorbers
move in a synchronous, or unison, manner, thereby rendering the system dynamically
equivalent to that with a single absorber mass. The overall objective of this thesis
is to explore the eflects of weak nonlinearities and small imperfections among the
absorbers. Of particular interest are dynamic instabilities and bifurcations, non-
unison responses, and localized responses in which one absorber moves with a much
larger amplitude than the others. Since the mathematical structure of the dynamic
equations are very similar, both translational and torsional vibration absorbers are
considered. However, the main emphasis is on torsional absorbers, since it is more
common to use light damping in these systems.

Much of the work in this thesis deals specifically with centrifugal pendulum vi-
bration absorbers (CPVAs), a particular type of torsional absorber. CPVAs are a
very effective method for reducing torsional vibrations in rotating machinery. They
consist of small masses mounted on the rotor whose vibration is being addressed.
During operation, these masses move along specific paths relative to the rotor, and
these paths are designed to dynamically counteract the applied torque that causes
torsional vibration. CPVAs have been successfully used in IC engines and helicopter
rotors. These absorbers take advantage of the centrifugal field of rotation in order to
self-tune to a given order of excitation over a range of rotation rates. The selection
of the path of the absorber masses allows one to set a desired linear tuning, as well
as to design for a beneficial large-amplitude, nonlinear behavior. In this system other
nonlinearities arise due to large-amplitude coupling effects. These are accounted for
in the analysis, but are shown to be of higher order.

The remainder of this chapter consists of a brief history and some background on
vibration absorbers, followed by an overview of the content of the main body of the

thesis.

1.1 History and Background

1.1.1 History

A good review of the developments of tuned vibration absorbers for vibration and
noise suppression is given by Sun et al. [5], from which the following summary was
distilled. Since its invention almost a century ago, many designs for the tuned vibra-
tion absorber have been developed and successfully used. The simplest and the most
favorable is a spring-mass oscillator. Some other designs included an ER-fluid rotary
dynamic absorber, a dynamic absorber of ring type with a distributed support spring,
a beam-type absorber, and a magnetic dynamic absorber that uses eddy currents to
provide damping. Tuned absorbers with multi degrees of freedom, which allow for
vibration reduction at several excitation frequencies, have also been studied.

As mentioned before, CPVAs have been efficiently used to reduce torsional vibra—
tions in rotating and reciprocating machinery. The original ideas for CPVAs go back
to the early 1900’s, although it was not until WWII that they came into wide use. A
thorough history on the theory and implementation of CPVAs can be found in the
works by C. T. Lee [6], V. Garg [7], and C. P. Chan [8]. What follows is a brief history
summary that will provide a better understanding of the objectives of this work.

CPVAs were used in IC (internal combustion) engines as early as 1929 [9]. They
have been effectively employed to reduce torsional vibrations in light aircraft en-
gines [3], diesel cam-shafts, and automotive racing engines and there are continuous
efforts to investigate and improve their performance. It should be noted however, that
until around 1980, all designs used circular paths for the absorber masses. In recent
years, other paths that offer improved performance have been introduced. Cycloidal
path absorbers are used in helicopter rotors [10], and epicycloidal path absorbers have
been proposed for use in automotive engines [11, 12, 13]. There are many possible

physical arrangements for CPVA systems. The treatise by Wilson [3] offers a thor-

ough background and overview of CPVA systems, as well as detailed analyses of their
application to flexible rotating shafts using linear vibration theory.

One of the earliest considerations of nonlinear effects is found in the paper of Den
Hartog [14], where the shortcomings of circular paths are described. The paper also
outlines a remedy for this problem, by intentionally mistuning the path so that it
comes into more favorable tuning as the amplitude grows. Newland [15] expanded on
this idea by providing a detailed analysis of the failure of circular paths and offering
a guideline for the level of mistuning. After that, much of the work focused on linear
vibration applications, although the patent of Madden [10], in which cycloidal paths
are put forward, was an important step forward into the nonlinear regime. Subse-
quently, the work of Denman and co-workers [11, 12] pushed the subject even further
by exploring more paths as well as implementation in an experimental automotive
four-cylinder engine.

In a study that considered a wider range of possibilities for the absorber paths,
Lee and Shaw [16] investigated the performance of a single CPVA mass for the case
of perfectly tuned absorber paths with a quite general nonlinear character. They
confirmed the well known shortcomings of circular paths and demonstrated the im-
provements offered by cycloidal and epicycloidal paths. (The epicycloidal path is
special, since it yields essentially linear absorber dynamics over a large amplitude
range.) Those results were generalized to also include the effects of intentional linear
mistuning by Shaw et al. [17]. It was shown that circular path absorbers with some
positive mistuning work quite well and behave very linearly over a wide amplitude
range (these are widely used in practice), but that perfectly tuned cycloids offer even
better performance.

In a treatment of a purely nonlinear absorber system, Lee et al. [18] considered a
pair of absorbers riding on what they called sub-harmonic epicycloidal paths, which

are tuned to an order equal to one~half that of the applied torque excitation. In

this case the desired response is a sub-harmonic motion in which the absorbers move
exactly out of phase with respect to one another. This arrangement offers ideal per—
formance in terms of vibration reduction, but requires more space for implementation.
Chao and Shaw [19] investigated the effects of intentional mistuning on these sub-
harmonic absorber systems and showed that their dynamic stability can be made
quite robust by a slight over-tuning at the linear or nonlinear order.

In the area of identical, multi-absorber systems, Chao et al. [20] studied the sta-
bility of the unison motion for systems of multiple CPVAs riding on perfectly tuned
epicycloidal paths, and considered the post-bifurcation dynamics of these systems [21].
They showed that the unison response could become unstable, resulting in a type of
nonlinear localized response in which one absorber moves with a much larger am—
plitude than the others. It was shown that this response actually slightly improved
the vibration reduction characteristics of the system, but that it significantly reduced
the torque operating range of the system. Chao and Shaw [22] also considered the
stability and performance of multiple pairs of sub—harmonic absorbers, including the
effects of imperfections and mistuning. They showed that multiple pairs of absorbers
can be made to behave like a single pair, again by imposing a very slight over-tuning
at the linear order.

Another topic that arises in the study of systems of absorbers is the phenomenon
of mode localization. This follows since the absorber system possesses several nearly-
identical subsystems that are weakly coupled through the primary inertia, which is
typically much larger than the total absorber inertia. Such a system is ripe for localiza-
tion, since it is known that if the degrees of freedom of a nominally periodic structure
are weakly coupled, and there exist some small imperfections in the structure, then
the free vibration modes typically localize in terms of spatial energy distribution. This
results in confined regions of the structure where vibration amplitudes are much larger

than predicted using the perfectly tuned model. This type of localization was first

considered in the field of solid state physics by Anderson [23], who showed that in a
randomly disordered linear chain of particles, the quantum-mechanical wave function
of the chain can exhibit spatially confined modes of motion. One of the earliest stud-
ies of the phenomenon of localization in the field of structural dynamics was made
by Hodges [24]. Subsequently, Pierre and Dowel] [25] investigated the localization
phenomenon for a chain of coupled oscillators, and Pierre et al. [26] theoretically and
experimentally investigated localization of the free modes of vibration of disordered
multi-span beams constrained at irregular intervals. Wei and Pierre [27, 28] studied
both free and forced vibration localization in nearly periodic mistuned assemblies
with cyclic symmetry. Also, it has recently been found that localization can occur
in nonlinear systems, even when the subsystems are perfectly tuned. In this case,
the mistuning is caused by the amplitude dependence of the subsystems’ frequencies.
Samples of work on localization phenomenon in both linear and nonlinear systems
can be found in [29, 30, 31].

The present work fills in some important gaps in the results known for multi-
absorber systems, and offers another application of mode localization. In particular,
the case of multiple identical absorbers with general paths, including a range of lin-
ear mistunings and nonlinearities, is considered. In addition, mode localization is
investigated for systems of absorbers in which small imperfections exist among the

absorbers.

1.1.2 Background

For background purposes, the well-known results from linear vibration theory for both
translational and torsional single-mass vibration absorbers are given in this section.

Figures 1.1, and 1.2 show the single-mass translational and torsional vibration
absorbers, respectively. In the translational vibration absorber, the absorber para-

meters, which are the spring stiffness, kc, and the absorber mass, ma are chosen such

6

that the absorber’s natural frequency, {ii—g, is equal to that of the excitation, war.
This, as shown on the frequency response curve of Figure 1.3, theoretically results
in complete elimination of the steady-state vibration of the primary mass, M at the
driving frequency. Damping alters this ideal picture, but small damping only slightly
alters the results. (Optimal damping parameters can be selected to reduce the vibra—
tion amplitude across the frequency range; this is the common tuned damper [32].) In
torsional vibration absorbers, the parameters R and 7' shown in Figure 1.2 are chosen
such that the square root of their ratio, i.e., fl, is equal to the order of the applied

torque n, where the applied torque is approximated to be harmonic of order n. i.e.,
T = To sin(nflt)

where Q is the mean rotation rate of the primary inertia. For example, in four-
stroke IC (internal combustion) engines, n is equal to half the number of cylinders.
When this is done, the path is said to be tuned to order it, since the linearized
natural frequency of the pendulum in the constant rotation case is equal to n times
the rotation rate. The frequency response curve here is similar to that shown in
figure 1.3 with the difference that here the pendulum vibration absorber will eliminate
the rotor’s torsional vibration for any rotor speed, (I. (Here the system is tuned
to a given order, rather than a given frequency.) Also, the lower resonance peak
corresponds to a rigid body mode and is therefore at zero frequency.

It should noted here that these results are obtained using the linear theory for
a single absorber mass, or the dynamically equivalent case of multiple, identical ab-

sorber masses moving in unison.

/////////
K

 

M Fa sin(wd,t)

 

 

 

 

 

 

 

Figure 1.1. Translational vibration absorber.

1.2 Objective and Dissertation Organization

The present research has been aimed toward providing a better understanding of
the important problem of determining the conditions under which systems of nearly
identical vibration absorbers behave like their single~mass counterparts, and the con-
sequences of situations in which this assumption does not hold. This is done by

analyzing and studying the following topics, which form the chapters of the thesis:

0 Forced, unison response of general path CPVAs.
The unison response and its stability for multiple absorbers is investigated.
General CPVA paths with linear mistuning are considered, and guidelines for

designs are provided. This study is presented in Chapter 2.

e Non-unison steady state responses in CPVA systems.
The existence and stability of a certain class of non-unison steady state re-

sponses in CPVA systems is investigated. This study is presented in Chapter 3.
o Localization in vibration absorbers.

o Localization in the linear free vibration modes of absorber systems. Sys—
tems of nearly identical translational and torsional vibration absorbers are
investigated for the possibility of the existence of free vibration mode local-
ization. This problem is different from what is available in the literature in
the sense that the coupling between the absorbers is not direct, but rather
arises in an indirect manner through the primary inertia. This study is

presented in Chapter 4.

0 Linear and non-linear forced, localized response of general path CPVA
systems. Here, the existence and stability of forced, localized responses in
systems of CPVAs is investigated. The results of this study are presented

in Chapter 5.

NORMAUZEDIMG‘JITUDEGPRIWDMSSM

m

1* <15
RN)
Y 16

l

O

 

Jo

Figure 1.2. Torsional vibration absorber.

 

.]
l

3
I

i

a

C
'

 

 

.5
'n

 

i;

 

d

0.8

 

 

 

 

l l l l l l
0 0.2 0.4 0.6 0.8 1 1 .2 1 .4 1 .6 1 .8
RATIO OF FORCNG FREQ. TO ABSORBER NAT. FREQ. (0.] a.)

Figure 1.3. Frequency response of the mass M of Figure 1.1

10

CHAPTER 2

Performance and Dynamic

Stability of General-Path CPVAs

In this chapter, the performance and dynamic stability of systems comprised of mul-
tiple, identical centrifugal pendulum vibration absorbers that have general paths are
considered. The study is carried out by considering a scaling of the system parame-
ters, based on physically realistic ranges of dimensionless parameters, which allows
for application of the method of averaging. It is found that the performance of theses
systems is limited by two distinct types of instabilities. In one type, the system of
absorbers lose their synchronous character, while the other is a classical nonlinear
jump behavior that aflects all absorbers identically, and leads to highly undesirable
system behavior. These results are used to evaluate the common types of absorber
paths, namely circles, and cycloids including intentional levels of mistuning. The
results are also used to make some recommendations about the selection of paths
to achieve design goals in terms of absorber performance and operating range. The

analytical predictions are confirmed by numerical simulations.

11

2.1 Mathematical Model

2.1.1 Equations of Motion

A system of N CPVAs mounted on a rotor of inertia J, as shown schematically in
Figure 2.1, is considered. These absorbers, each with a mass of mg, ride on paths
specified by the arc length variables Sg’s. These arc length variables are symmetric
about their vertices. Their origins are taken to be at their vertices. The distance
from any point on the 2"" absorber path to the center of rotation of the rotor, 0, is
specified by the variable Rg. At the vertex of the i“ absorber path, this distance is
Rgo, i.e., 12,-(S.- = 0) = Rgo. For each path, this distance is an even function of the
the arc length variable 5;, i.e., 12,-(5;) = R;(-S.-). This is due to the symmetry of
each path. The rotor is subjected to an external torque that has a mean, To, and a

fluctuating, T(0), components.

The kinetic energy of the system consists of the kinetic energy of the rotor and

that of the absorbers. The rotor’s kinetic energy is

l .
Tr = — 02,
2.]

where 0 is the rotation rate of the rotor. The kinetic energy of the 2"" absorber is

where 27,- is the velocity of the 2"" absorber. This velocity is given by

6; = 12.0 E9 + S; 53,,

12

 

 

Figure 2.1. Schematic diagram of multiple CPVAs arrangement.

where é'g and 65.. are unit vectors in the 0 and 5'.- direction respectively. Looking at
Figure 2.2, it can be seen that the dot product 59.53.. is equal to 12,-3%. It is also clear

from Figure 2.2 that the following relationship holds:

((1502 ((112,)? + (R:d¢:)’, or
[W
9.63.. — d5, .

The total kinetic energy will then be

 

N o o o 0 ~
T = -;—{J(i2 + Zm; (X,-02 + S? + 203,09},

t=l
where
X.(S,) = R?(S.-),

~ (2.1)
G.(S.-) = \/Xi(5i)-i(%f(5£))2-

 

13

 

Figure 2.2. CPVAs

To reach to the dynamic equations that govern the system motion, Lagrange’s
method is applied with the reasonable assumption that potential energy is small
and could be ignored. This is due to the fact that potential energy arises only from
gravitational effect, which is small compared to centrifugal eflects for any reasonable

rotation speed of the rotor. Lagrange’s method is applied as follows:

d 3T 3T
It.(3—ql) —5a—Q(, f— I,...,N+1,

where q1 = 0, and q, = 5,, j = 2,3, ..., N +1, and i = 1,2, ...,N are the generalized

coordinates. Qi’s are the generalized forces. The generalized forces are:

14

 

. N . dR, 2
Q1 = ‘009 + aniSiRe' 1 - (a?) + To + T(0)
i=1 1

N
= —c09 + Z CuisiGi + To + TM),

i=1

Qj = —Caij, j = 2,3,...,N+ I,

where 6.3;, and co are the damping coefficients for the 2"” absorber and the rotor

bearing respectively. The equations of motion will then be
m,[§.- + e.(S.-)é — §§J§;(s,)02] = —c..-S.-, (2-2)

15+ 2?; malt—{giKSaSeé + X.(S.-)é' + é.(s.-)§.-
5%(50331 = N 6.34303. — coé + T. + T(0).

i=1

(2.3)

The 1"" absorber is indirectly coupled to other absorbers through the dynamics of the
rotor, as is clear from equation (2.2), which describes its dynamics. Equation (2.3)
represents the torque balance on the rotor.

These equations represent an autonomous dynamical system, because the varying
component of the applied torque, T (0), is expressed as a function of the rotor angle 9.
For purposes of subsequent analysis, it is convenient to convert the equations in such
a manner that the rotor angle serves as the independent variable, replacing time. To
this end, a new non dimensional variable v is defined as the ratio of the rotor angular

velocity to the nominal rotor angular velocity , i.e.,

(2.4)

:2
III

:3] so.

This dynamic variable will be used to represent rotor speed and acceleration. Using

the chain rule, one can obtain the following relationships between derivatives with

15

respect to time and derivatives with respect to 0:

" _ 3,7 = 02v% = szv’,

(I) = a = “vii? = nvc)’,

('5) = 9%! = mug—’1]; + mag},
= flzvv’(.)’ + 02v2(.)”.

(2.5)

 

After equations (22,23) are nondimensionalized, and the independent variable is

changed from t to 0, the equations of motion become
+ is:- + a.<s.-)1v' - 77s.») = was; (2.6)
251:1 bilfidffSi-vz + z,(s.-)vv’ + §i(st)s:-vv’

+§i(3i)3?02 + £(s;)s?v2] + vv’ (2.7)

= 2:11 bifla;§e(si)32v - #01) + F0 + l‘(0),

where
3; = Si/Rio, §:(s.-) = Gi(Si)/Rio, bi = Ii/Ja
I; = ng30, pa.- = cag/mgfl, p0 = co/Jfl,
1‘0 = To/Jflz, I‘(0) = T(0)/Jflz,

and

$3.03") = Xi/Rtgo’

“3") = fids‘) ’ Hid—fast»?

Note that the system now is non-autonomous, but its order has been reduced, since

 

(2.8)

only first derivatives in v appear. Assuming that all absorbers have equal masses

and equal damping, and all paths have the same value of R,- at each vertex, i.e.,

16

mi = m, Cat = Ca, Rio = R0 Vi E [1, N], equation (2.7) becomes

iii Ezdgfisfivz + $i(8i)vv' + §;(s;)s£vv’
+§i(8i)Si-'v2 + §}(s.~)s§v2] + vv’ (2.9)
= if 2&1 fla§i(3i)Si-v - #ov + [‘0 + I‘(0),
where

E
J,

No

a
mail ’

b0 2 Io = moRz, ya = and mo = Nm.

The fluctuating torque generally contains several harmonics. In most situations
only one or two harmonics have significant amplitude, and therefore we approximate
the fluctuating torque by its dominant harmonic, taken to be of order n, as follows:
I‘(0) = [‘9 sin(n0). For example, in four-stroke IC engines, n is equal to half the

number of cylinders.

Note on the Damping:

The damping on the main rotor does not have a role on the system other than setting
its nominal speed by balancing the mean component of the applied torque. However,
the absorber damping has a great effect on the system performance. Due to the
fact that this damping is complicated and depends on the way the absorbers are
implemented, its modeling is quite difficult. In this work, similar to the linear case,
the absorber damping is modeled as an equivalent viscous damping that does not
depend on the mass of the absorber. With this, the quantity [ii/1 = 36:5 is a fixed
physical quantity and does not depend on the number of absorbers as long as their

total mass is fixed. This indicates that the non-dimensional damping coefficient, pa,

is proportional to the number of absorbers.

17

2.1.2 General Path Representation

As described in Denman [11], it is convenient to represent the general path for the

1"" absorber by the local radius of curvature at any point on the path, given by
p? = 9?. - ”5?,

where Pia is the path’s radius of curvature at the vertex, which dictates the small
amplitude nature of the path. A dictates the large amplitude character of the path,
and can take any value from zero to one. Some special cases of interest are: A = 0
describes a circular path, A = A.3 = fl??? describes an epicycloidal path with its
base circle of radius (Rgo -— pgo) centered at the rotor center (the so-called tautochronic
path of order ii,- [11]), and A = 1 describes a cycloidal path. Note that the value of
A dictates the nature of the amplitude-dependent frequency of the absorber when it
oscillates freely along its path. In fact, when the rotor speed is constant, the epicy—
cloid case separates softening nonlinearities, like the circular path, from hardening
nonlinearities, like the cycloidal path.

The order of the path is given by the square root of the ratio between the distance
from the rotor center to the center of the path vertex circle and the path vertex radius

of curvature, that is

(2.10)

 

This fixes the linearized natural frequency of the 1"“ absorber, when the rotor spins
at a constant rate (I, to be 13.12. This frequency is used for tuning the absorber at
small amplitudes, but it aflects the large amplitude dynamics as well.

In the equations of motion an expression for 35(3) for the general absorber path

is needed (see equations (2.6,2.9)). The following expression can be reached by ex-

18

panding in 3;,

15435) = 1 — 7353.? + 7.3: + 009?), (2-11)

where

7.- : ($)(fi?+1)”(fi?- A2(1+ as).

Note that this expanded form is used in the analysis, but the full form of x;(s.-) is

used in the numerical simulations.

2.1.3 Scaling

Since the ratio of the total absorbers’ inertia, 10, to the rotor inertia, J, is small, we

define a small parameter c as

This parameter will form the basis used for the scaling. Note that in the unperturbed
case, 6 = 0, the rotor dynamically decouples from the absorbers.

Based on the purely linear theory of CPVAs, the order of the path should be
tuned to the frequency of the disturbance torque. However, certain nonlinear effects
are nicely handled by incorporating a small level of intentional linear mistuning on

the path. To account for the mistuning, the path order in this analysis is taken to be
fig = n(1+ 690;),

where 0; represents a measure of the mistuning of the 1"" absorber path. Such mis-
tuning is always intentionally built into existing circular path absorbers in order to
counteract some undesirable nonlinear effects [15].

The preferred absorber configuration has small damping, since it remains at the
set tuning at all rotational speeds. Also, the fluctuating torque amplitude is small

compared to the kinetic energy of the rotor, rendering the non-dimensional torque

19

amplitude small. Therefore, the parameters an, #0, F0, and F9 can also be taken to

be small and are scaled by c as follows:
pa = e'fla, yo = (lilo, F0 = (To, Pa = c'f‘g.

In addition, the absorber oscillations are assumed to scale with the fluctuating torque

level in some manner, and so we take
3,- = (”25. (2-12)

We now turn to the matter of balancing the desired terms in the equations of motion
so that the applied torque, the damping, and the nonlinearities come into play at the
same order. We begin with a couple of preliminary expansions.

Note that when 6 = 0, i.e., be = 0, equation (2.9) states that the rotor spins at
a constant angular speed, (I. For 0 < 6 << 1, the rotor will have small fluctuations
about the constant angular speed, 9. Therefore, it is convenient to expand the rotor
speed as follows:

v(0) = 1 + cwvw(0) + HOT, (2.13)

where HOT means higher order terms.

The path function can be expanded in terms of c as well. Evaluating g, using
equation (2.8), along with equation (2.11), and expanding it and 7; in powers of 6",
the following expressions can be reached
n2 + n‘)s?

g;(3.-) = ——§—-—'- + HOT (2.14)

7.- : 70+0(€°)

20

where

70 =(11—2)(n2 +1)2(n2 — W1 + n2». (2.15)

Note that 70 < 0 (softening) corresponds to A > A6 while 70 > 0 (hardening) corre-
sponds to A < Ac.

The final preliminary step is to note that in order for the constant torque terms
to balance, I‘" = no must hold. This states that the constant applied torque is
offset by the bearing damping torque arising from the mean rotation rate. Note that
if a constant load torque is introduced into the equations of motion, it is simply
counterbalanced here by F0, such that the mean spin rate is maintained.

Substituting the above scaled parameters, the constant torque balance, the expres—
sions for §;(s.-), 7;, and 12(0) into equation (2.9), expanding, matching terms according
to r = (p-I-V), and keeping the (r-I—u) order terms, one finds that the non-dimensional

rotor acceleration is given by

N ~ r+y N
vv'(0) = (KI—[1‘72 n22,- + F9 sin(n0)} + 2 2712211;- + HOT. (2.16)
‘ 1

J: i=1

6

 

Using this result in equation (2.6), a suitable choice of the scaling orders is found

to be

‘0
N
H
p—t
\D
‘1
ll
N10.)
'3
||
p—A
v
8
||
NIOJ

This leads to the desired form of the absorber equations, to which we now turn.

2.1.4 The Averaged Equations

With the above scaling results, the rotor dynamics can be eliminated (using the 5"

terms in equation (2.16)), resulting in the following uncoupled equations that describe

21

the absorbers dynamics,

II 2 _ 3 2 , ~ I
z,- + n z.- — ([2702,- — 2n 0,2. — p32,-

(2.17)
—-,{7 2:35;, n22,- - 1“. sin(n0)] + HOT.
It should be noted here that at the order considered, the only nonlinear effect that
appears is the one due to the path, i.e 702?. Missing at this order are all of the non-
linear terms that arise from the kinematic coupling of the rotation and the absorber
motion. The nonlinear path term is zero for epicycloidal paths because 70 = 0 (see
equation (2.15) along with the definition of A). In this case, the model reduces to the
linearized model, with which it is impossible to capture any nonlinear effects. There-
fore, in order to analyze the case of epicycloidal paths, one has to employ a scaling
wherein nonlinearities other than the path nonlinearity are retained. Chao et al. [20]
have done this, and analyzed the case of perfectly tuned epicycloidal paths in some
detail. In the present study, epicycloidal paths are not considered.

Note that the absorbers’ dynamics are now uncoupled from the rotor dynamics
to leading order, and equations (2.17) represent a set of weakly non-linear, weakly
coupled oscillators. These have the very special feature that they all have the same
unperturbed natural frequency, and are all resonantly excited by the fluctuating ap-
plied torque. Furthermore, the absorbers are all coupled to one another in an identical
fashion, and when the mistuning is zero (0',- = 0 Vi), this forms a system with a special
symmetry (see [8]).

The averaging method will be used to determine approximate steady-state solu-
tions of these equations. To obtain the standard periodic form, the usual transfor-
mation to polar coordinates is used,

2,- = a;sin(n0 + (fit), (2 18)

z:- = na;cos(n0+¢g).

22

The standard periodic form for the equations is then found to be,

af = if. cos(n0 + (1);) + HOT,

(2.19)
<15:- = -;€;f.- sin(n9 + (fir) + HOT,
where
f.- = 2700? sin3(n0 + ¢g) — 2nzaga; sin(n0 + (13,-)
—fiana,- cos(n0 + (1),) — fi if; nzaj sin(n0 + 43,) (2°20)

—f‘9 sin(n0).

The functions f,- are periodic in the independent variable 0, with a period of (25).
Averaging these equations over one period, one reaches the following averaged equa-
tions -
«’1:- = c[—’i,—°&.- + g:- sin(4_3.-)
+5"? xiv-:13“ 51' sin(4_3,- — 91' )l
+H OT,
€1.ch- = £[-§ff&? + n(a.~ + fin,- i (2.21)
+9:- cos(gf>,~)]
+& 21213; 711' (305(‘f’i — (fail

+HOT,

 

J

where an overbar indicates the averaged value of the corresponding variable. These

equations are the basis of the analysis of the system dynamics.

2.1.5 Existence and Stability of the Unison Response

In this section we consider the case in which all absorbers are identical and move in
a perfectly synchronous manner. This is the desired response of the absorbers, so its
existence and stability are of interest.

One objective of this study is to determine the effects of intentional linear mis-

tuning, and so we fix an identical level of mistuning for all absorber paths, as follows,

23

05:0 Vie [1,N].

When a unison response occurs, all the absorbers have the same vibration amplitude
and phase, i.e.,

a, = 01': 7'2,
651' = (Fj =¢z-

When these are substituted into equations (2.21), they become pairwise identical and

reduce to the following,

‘

r’z = £[—é‘7‘z + :1 "3111022) + HOT’

 

r2462- - 4-21.4: r, r3+ n(a +- at", ] (222)
+‘;_:' COS(¢2)] + HOT.
The steady-state conditions are given by,
111 = .9.
2 r’ s’n(¢"’ (2.23)
fn—rz — n(a + %)rz = g: COS(¢z)-

Eliminating the phase in the standard manner, and solving for the torque amplitude

in terms of the absorber amplitude (for ease of plotting, etc.), one obtains,

 

 

Fg— — 2n\/[—r,,]2 + [1:733 — n(or + %.)r,]2 (2.24)

These results relate in a simple manner the absorber response, in terms of amplitude
and phase, to the system path parameters, the damping level, and the fluctuating
torque amplitude.

In order to analyze the stability of this unison response, the Jacobian of the

system given by equations (2.21) must be evaluated at the steady state conditions.

24

This yields a matrix of the form

"A1 A2 . . . A2“
A2 A1 A2 . . A2

A: . . . . . . (2.25)
A2 . . . A1 A2
_A2 . . . . Arm,“

 

 

where A1 and A2 are 2 by 2 matrices with the following entries:

 

 

A111 = —%’1
A1 — 13—(N 1) + Ecosfifi)
12 — 2N 7’2 2n 2 a
_ 37o i:‘0 "(N — 1)
A121 T — 2n 7‘, — 2727‘: cos(¢z) — 2Nr, ’
I‘a .
A122 — —% sm(¢z),

n n
= : _— z, = —, A2 = 0.
A211 0, A212 2N? A221 2er 22

If all the eigenvalues of the Jacobian matrix have negative real parts, the unison
response is exponentially stable. For a system with a J acobian of a form similar to that
of the system considered, it can be shown that each eigenvalue of the 2 by 2 matrix
[A1 — A2] is an eigenvalue of the Jacobian matrix A repeated (N — 1) times, and the
remaining two eigenvalues are the eigenvalues of the matrix [A1 + (N — 1)A2] [33].
For the stability evaluation, we will use the fact that the eigenvalues of a 2 by 2
matrix have negative real parts if and only if its determinant is positive and its trace
is negative.

The matrix [Al — A2] is given by

—&21 3&7er -— nor.
A1 - A2 = (2.26)
s—eu —e

25

Its trace is equal to —p., and is always negative. Its determinant is given by

”2

2
Det[A1 — A2] = fffffoir: — 3yoar: + (71202 + -’$151). (2.27)

The matrix [A1 + (N — 1)A2] is given by

-“2—" %r3 - n(a + fir,
A1 + (N -1)A2 = 1 l (2.28)
"(a + 5):: - if?” if
Its trace is also equal to —pa and its determinant is given by
Det[A1 + (N — 1)A2] = f—gjér; — $700 + 2a)rf + [n20 + 71202 (2 29)

+%(n2 +119]-

Since the traces are both negative, no Hopf bifurcations to quasi-periodic motions
are possible. The conditions at which stability changes occur are captured by setting
Det[A1 — A2] = 0 and Det[Al + (N — 1)A2] = 0. It is quite simple to solve these
conditions for the corresponding critical values of the absorber amplitude, yielding
values of r, at which bifurcations occur. The attendant critical torque levels can then
be found from equation (2.24).

It should be noted that solutions coming from equation (2.27) represent critical
amplitudes at which the unison motion becomes unstable, but continues to exist,
rendering some type of non-synchronous, steady state response. On the other hand,
solutions coming from equation (2.29) represent the condition at which the unison
response is annihilated in a saddle-node bifurcation, representing a sudden jump in
the absorbers’ motion to another response branch, which may or may not be of
unison type. One can see that Det[A1 + (N — 1)A2] = 0 corresponds to instabilities
that preserve the unison nature of the response by considering the stability of the

equivalent single-absorber mass system represented by equation (2.22), and noting

26

that it gives the same instability condition.
When equations (2.27, 2.29) are equated to zero and solved for r,, the following
results are obtained.

For the bifurcation to a non-unison response, i.e., Det[A1 — A2] = 0:

4n
r2 3' =
b f «537.

 

-2 .g. a
30 — (may ._ 3%.“202 + %)) ] . (2.30)

 

For the jump condition, i.e., Det[A1 + (N — 1)A2] = 0

2n 0 + l
sz % g —( 2), (2'31)
’70
where the approximation is noted because we have neglected the damping term in
the torque equation, since it is very small compared to other terms. These can be
used in equation (2.24) to find the critical torque levels.

The following general observations can be made by considering these two critical

conditions:

From the first condition:

0 For '70 > 0, bifurcation to non-unison can exist only when the level of mistuning
is greater than some positive value set by the damping level and n. To be able
to see this observation, the damping term was not neglected in this equation as

was done in the second equation. ( More about this below).

0 For 7., < 0, bifurcation to non—unison can exist only when the level of mistuning
is negative. Fortunately, negative mistunings are not of practical importance as

will be shown later. So, this is an advantage for the paths with 70 < 0.

27

From the second condition:
0 For 70 > 0, ajump exists only when a > —%.

e For 70 < 0, a jump exists only when 0’ < —%.

It should be recalled here that '70 > 0 includes paths where 0 S A < A,, (see
equation (2.15)), that is, paths ranging from circular, up to, but not including, epicy-
cloidal paths. Similarly, 70 < 0 includes paths where A. < A S 1, that is, paths
ranging from, but not including, epicycloidal up to cycloidal.

To see how the absorber damping, as modeled in this work, can alter the value of
the critical mistuning level above which bifurcation to non—unison can exist, as was
mentioned in the first observation above, the argument of the inner square root in
equation (2.30) is equated to zero and solved for that critical mistuning level, ac...

When this is done, the following equation is obtained:

J5-
ac, = ——p,,.

2n

Note that the number of the absorbers, N, is an implicit parameter in this equation.
This is because the scaled damping coefficient, [1,, is proportional to N ( see note on
the damping above) which means that increasing the number of absorbers increases
the damping level. Figure 2.3 represents a plot of this equation for different number
of absorbers for the case when the applied torque order, n, is 2. For a better absorbers
performance, the damping level should be kept small, as will be shown later. In addi-
tion, as it is clear from Figure 2.3, for practical [2,, values, the the critical mistuning
levels below which bifurcations to non-unison can not exist are small. This implies
that in the presence of reasonable mistuning levels, it is practically not possible to

avoid bifurcations to non-unison.

28

 

0.3 I I I I I I I I I

 

025? ---—-- N-B .

 

 

 

0.2 -

Mistuning (0')

0.1-

 

 

 

 

0.05 '-
0 l l_ 1 1 l 1 l l
0 0.005 0.01 0.015 0.02 0. ~ 0.“! 0.035 0.04 0.045 015
[La
N

Figure 2.3. Mistuning levels below which non-unison motions do not exist. n=2.

Application to Some Common Absorber Paths
Circular paths:

For circular paths, A = 0, which when used in equation (2.15) gives the following

expression for the non-linear path coefficient, 70:
1
70 = 1—2—n2(1 + n2)2. (2.32)

When this is substituted in equations (2.27,2.29), the following expressions are ob-
tained:

~2
Det[A1 — A2]“, = 25—6an + n2)"r: — inzfl + n2)2ar: + (n202 + [:1—0)’ (2.33)

29

3

Det[AI+(N—I)A2]a‘r = 2-5—6

"2(1+n2)4rfi—-n2(1+n2)2(1+2dlri'l'—l"2(1+2a)2+"¢]’
(2.34)

corresponding to the bifurcation to non-unison and the jump conditions, respectively.

Cycloidal paths:

Here A = 1 and 70 is given by,
1 2 2
7.: —12(1+n). (2.35)

When this is substituted in equations (2.27,2.29), the following expressions are ob-
tained:

‘2

1
Det[A1— A2]Cyc == (-—1 + n 2") r2+ 1(1 + n 22) or: + (71202 +# )—4—, (2.36)

256n 2

Det[A1+(N—1)A2].,. = ——(1+n2)‘r;+— 8"‘(1+n2) (1+2a)r§ +- :[n2(1+2a)’+i1.l,

(2.37)

256n 2

corresponding to the bifurcation to non—unison and the jump conditions, respectively.

2.2 Numerical Examples and Discussion

Here, we fix the values of all parameters except the path parameters and the fluctu-
ating torque level. CPVA systems with an inertia ratio (6) of2’— 0are considered. The
order of the applied fluctuating torque (n) is taken to be 2. The non-dimensional
absorber damping coefficient is taken to be 5% = 0.02, and the dimensionless rotor

damping coefficient is taken to be 11., = 0.005.

30

Note on Numerical Simulations

Throughout this work, in all numerical simulations, no expansions in terms of 6 were
used to simplify the equations of motion (equations (2.6,2.9)). Also, the following
exact representations of the circular, epicycloidal, and cycloidal paths are used [16]:

Circular Paths:
2a,? {1 — cos [(13.2 + l)s.-]}

3i(3i)=1— (fl? + 1)2

 

Epicycloidal Paths:

Cycloidal Paths:

 

_ -2 3 2 sin“[(fi?+1)3il 2
31(80-1—("1 + 2103.- +( 203'? + 1) )

Sin-2 [(1712 + 1)s.-] sin [2 sin"1 [(122,2 + 1)Sil]
4m: + 1):

 

2.2.1 Effect of Path Type

Path Type Effect on the Stability of the Unison Response:

Using the above numerical values in equations (2.30,2.31), along with equation (2.24),
the critical torque levels were obtained as functions of the path coefficient (A) for
various levels of mistuning. The results are presented in Figure 2.4. It is worth
mentioning here that the plot shows only positively mistuned paths, ranging from
circular up to, but not including, epicycloidal paths (0 S /\ < Ac). This is because, as
seen earlier, for other paths neither bifurcations to non-unison nor jumps are present
for positive mistuning levels. Also, as shown below, negative mistuning levels are not
of practical importance and should always be avoided.

It is clear from the figure that the dependence of the critical torque levels on the

31

 

   
   

 

 

 

 

 

 

 

 

 

 

0-1 fl I I I I I l I I
— BIFURCATDON TO NON-UNISON
0.09 - - - - JUMP ‘
0.08 - -‘
0.07 " l .-
l
l
I... 5.0 9’. MISTUNING ( a n 1.0) ,
" 1
g 0.05 , .
*‘ 0.04 -
0.03 ‘
0.02 q
0.01 1
o J L 1 1 l l l L l
0 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9 1

PATH COEFFICIENT ( A.)

Figure 2.4. Effect of absorber path type on the stability of the unison response.

nonlinear path parameter is not very significant, until one approaches the epicycloid.
From the stability point of view, no benefits are gained by changing from the
easily-manufactured circular paths to other paths; however, the performance must
also be considered before drawing general conclusions, and this is done next. Note
that the level of mistuning does have a significant effect on the stability levels, but,
again, performance must be taken into account. Also, note that near the epicycloidal
path, that is, for A z Ae = 0.89 the critical torque levels become large. Here the
results are not reliable because, as mentioned earlier, the theory does not work for

epicycloidal paths, due to the scaling. In both cases — increased mistuning and

32

A :3 Ac — the response becomes more like that of a linear system, and thus more
stable. In the case of increased mistuning, this is due to the fact that we are moving
away from a resonance condition. For A z Ac, the nonlinear part of the path is

balanced between softening and hardening; this is the tautochronic condition [11].

Path Type Efiect on the Absorbers Performance:

The amplitude of the non-dimensional rotor angular acceleration, vv’(0), is used
as a measure of the performance of the absorbers. Assuming unison motion, this
amplitude is calculated using equation (2.16) for different torque levels and different
path types. Two cases are considered here, namely, perfectly tuned paths (0 = 0),
and positively mistuned paths with a = 0.4. The results for these two cases are
shown in Figures 2.5(a) and (b) respectively. Examining theses figures, it can be
seen that the benefits gained by employing a wide range of paths that are not
circular are small and does not balance the advantage of the ease in manufacturing
circular paths have. However, as one approaches epicycloidal paths, theses benefits
become appreciable. The main advantage that will be gained as one approaches
the epicycloidal paths is the high operating range as it is clear from the figure.
Although, as mentioned earlier, our model reduces to the linear model for the case
of epicycloidal paths, the results when the absorbers are in unison is in a very
good agreement with numerical simulations. This means that the linear model is
an acceptable one for the case of epicycloidal paths as long as the absorbers are in
unison. However, it has been shown by Chao et al. [20] that at a certain torque
level, the absorbers unison response becomes unstable. This, of course, can not be
captured by the linear model simply because it is a non—linear effect. This instability
is observed through numerical simulation and its effect can be seen in Figure 2.5(a)
where the unison motion of the absorbers with epicycloidal paths becomes unstable

at approximately [‘9 = 0.026. The theory presented in this work, states that neither

33

(a) PERFECTLY TUNED PATHS b-O ). NERTIA RATD (e )-0.05

 

 

 

 

 

 

 

 

 

 

 

 

   

o.“ .I v I . v T
, " ' —- 1.0 (01mm
0.04 ’ . ........ m2
: ’ - ‘ M.‘
g 000 ----- 1.0.0
a Hail (Epicvddd)
i 0.02» u—u 21.1(Cydoid)
O O Saul-tion (Circular)
0.01 , u n Summon (Eplcydold)
0 0 Simuldion (Cydold)
0
0 0.01 002 0.03 0.04
Fe
(mmsrmeo PATHS (o-0.4 ). Nam mmm-aos
0.“ I b : I i I I T ' a f
1 ' '
a.“ r- l
: ' i
Z I | '
sh 0.03 - ‘ I l
:‘J ' I
3 .
_$>_ 002 ~ : !
0.01 - 4 "'"
o l l l 1 l l

 

 

 

 

0 0.01 0.02 0.03 0.04 015 0.!” 0.07

Figure 2.5. Path type effect on the absorbers performance. 6 = 0.05, n = 2. (a) Per-
fectly tuned, (b) a = 0.4.

jumps nor bifurcations to non-unison exist for paths raging from but not including
epicycloidal to cycloidal, as mentioned earlier. This is clear in Figure 2.5 as the
absorbers with cycloidal path continue to move in unison and does not undergo any
jump.

Our goal is to investigate the operating range of absorber systems, as they are
limited by the critical torque levels, and to evaluate the effectiveness of the absorbers
by computing the angular acceleration of the rotor, which is desired to be small. We
focus on circular and cycloidal absorber paths and distill some general conclusions

regarding choices of path parameters.

34

Table 2.1. Effect of the number of absorber on bifurcation to non—unison torque level.

Circular paths. (From theory).

 

 

N Mo = 0.2) Fg(a = 0.4) Mo = 0.8)
2 1.048 1.792 3.415
6 1.062 1.798 3.418
10 1.089 1.809 3.424
14 1.131 1.826 3.428

 

 

 

 

 

 

2.2.2 Circular Paths

We begin by demonstrating the accuracy of the analytical results, and then turn to a
more systematic investigation. Figure 2.6 depicts the unison absorber response versus
torque level, showing both theoretical results from equations (233,234) and numerical
simulation results for N = 4 absorbers with 0 and 4% mistuning levels. Note that
the method is very accurate on the lower branches, that is, those of greatest interest.
Also, the error grows as amplitude increase; this is due to both the scaling and the
application of averaging. Figure 2.7(a) shows the critical torque levels above which
the unison motion becomes unstable and Figure 2.7(b) shows the critical torque levels
above which the jump occurs, both for different mistuning levels, again for N = 4.
These results demonstrate the validity of the analytical approach employed.

It is clear from equation (2.33) that the only parameter that could be changed to
delay the existing bifurcation to non-unison, without introducing some further inten-
tional mistunings to the absorbers’ paths, is [26, and this can be done by increasing
the number of absorbers, N. However, the dependence on this parameter is very
small. To see this, three different mistuning levels of the paths in the present numer—
ical example are considered. They are 1%(0 = 0.2), 2% (a = 0.4), and 4% (a = 0.8).
With all other numerical values fixed, increasing the number of absorbers from 2 to

14 in each case increases the critical torque levels at which bifurcation to non-unison

35

0.25

 

.0
70

0.15

p
d

 

NON DIM. AMPLM (I: ABSORRBER MOTDN

 

 

 

 

 

 

 

a.” 1- —.
-—' neonnsmaua)
-—---- DEORY(UNSTABLE)
o 0 summon
o l l I l I I 4 l 1
0 0.005 0.01 0.015 002 0.025 0.00 0.035 0.04 0.045 0.05
109006 LEVELU‘.)

Figure 2.6. Circular path. Upper: 0% mistuning; lower: 4% mistuning

takes place by only 8%, 2%, and 0.4%, respectively (see Table 2.1). The practical
method for delaying bifurcation to non-unison, as mentioned earlier and as can be
seen from Figure 2.7(a), is to increase the level of mistuning in the paths. In fact,
this will significantly delay both of the bifurcation points. The response curves for
N = 4 with different mistuning levels, depicted in Figure 2.8, also show that the jump
point shifts rightward as the level of mistuning is increased. Figure 2.8 also shows
that as the level of mistuning is increased, the absorbers’ amplitudes become smaller
for a given torque level, which implies that the absorbers are cancelling less of the
applied torque. This indicates that there is a tradeoff between high operating range
and better absorber performance. It should also be noted that as the mistuning level

approaches —2.5% (0* = -0.5), the torque level at which the jump occurs approaches

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.08 I l I I I W
-095 - m 0 SIMULATION O 3
1..” o 0 THEORY
g 0.04 '- . -
Eam- . .
§002- . .
o
.— . .1

0.01 F g

o l l l l l
0 1 2 a 4 5 6
LEVELOFMISTUNING ( as) (a)

O 07 I I I T I
“0.05 ~ 0 4 smuumoN *
130 05 _ o 0 THEORY 9 .
g 0.04 - 0 -
m 0.03 - Q -
g... . . .
2" 0.011 2" q

o J 1 4L 1 l
0 1 2 3 4 5 s
LEVEL OF MISTUNING ( 96) ( b)

Figure 2.7. Critical torque level versus percentage mistuning for circular path CPVAs
(a) Bifurcation from unison, (b) Jump

zero, indicating that the absorbers will jump no matter how small the applied torque
is. After the jump, as will be shown later, the absorbers’ motions actually add to the
applied torque and increase the vibration levels as compared with the rotor without
absorbers. Therefore, one should always avoid negative mistuning levels.

The stability of the various steady-state curves in Figure 2.8 are nearly as ex-
pected, with one exception. The lower branches are stable up to a torque level just
prior to the jump, where the bifurcation to non-unison occurs. The middle branch is,

of course, everywhere unstable, and the upper branch is everywhere stable.

37

 

0.2 f I I I I

0.18 - '1

O O
L5 L.
& G
I
1

.0
.0
N

p
8

.c
8

AMPUTUDE OF ABSORBER MOTION]
o

p
2

0.02

 

 

 

 

0 . . 0.00 0.04 . 0.08
TORQUE LEVEL ( F.)

Figure 2.8. Effect of mistuning, circular path CPVAs - analytical. Stability is not
indicated here.

Figure 2.9 shows a plot of the the amplitude of the non-dimensional rotor accel-
eration, vv’(0), versus the applied torque level. The previous comment about the
tradeoff between performance and range is clear from this figure as well, since the
performance degrades as the range is increased. However, in all cases shown, the ab-
sorbers reduce the vibration levels when compared to the system with the absorbers
locked at their respective vertices (where they play the role of a simple flywheel).

An interesting observation in Figure 2.9 is the presence of a peak acceleration
in every theoretical curve just before the jump point whenever a bifurcation to

non-unison exists. It has been observed that these peak acceleration points represent

38

the points where bifurcations to non-unison response take place. This can be shown
mathematically, as follows. From equation (2.16) and the first of equations (2.18),

we have

vv’(0) = c'[n2rz sin(n0 + 45;) + F0 sin(n0)] + 005””)

Also, since [2,, is small, it is seen from equations (2.23) that d), is close to zero or 1r.

Before the jump, (13,, z 7r, as will be shown later. The above equation thus becomes
‘U‘UI(0) z 6"[f’9 — nzrz] sin(n0)
The acceleration amplitude is given by
|vv'(0)| a: e'[Fg — nzrz].

Differentiating this with respect to F9, and making use of the second of equa-

tions (2.23) with <0, = 7r, the following expression is obtained:

dlvv’(0)| ~ ri _ n2
df‘o 2722(0‘ + %) — §7ar3

 

].

~

Solving this for the r, value at which |vv’(0)| is a maximum, i.e., where this expression

2n 0
r, z — —,
3 70

which is exactly the same as the expression for "zbo'f given by equation (2.30) when

is zero, one finds

the term with the damping coefficient [1,, is ignored. This feature of the response is
not well understood.
The general response for circular paths is observed to be a unison response with

a smooth increase in absorber amplitude and angular acceleration, up to a point

39

 

0.07 I T j 7' / l O '

 

— THEORY(STA8LE) o 0 o
----- - THEORY ( UNSTABLE) o

°°°’ o o SIMULATION o O I;
- - - ABSORBERS LOCKED 1 , ’

 

 

 

p
8

.o
2

p
8

      

9
8

-

AMPUTUDE OF NON-DIM. ROTOR ACCELERATION

0.01

 

 

1 1 ‘

0 0.01 0.02 0.03 0.04 0.05 0.06
TORQUE LEVEL ( F“)

Figure 2.9. Effect of mistuning on rotor acceleration ( circular path CPVAs )

at which the acceleration peaks and the system bifurcates to a non-unison motion.
Typically, the response beyond this torque level is captured by the undesirable upper
branch of the unison response. This is due to the bifurcation being sub-critical, or
there being a very small basin of attraction for the post-bifurcation response. In the
present case with N = 4, the only level of mistuning where it was possible to observe
the non-unison response was at 0.5%, as demonstrated in Figure 2.10. Figure 2.10(a)
shows the absorbers’ unison response at Fa = 0.0076 (before bifurcation to non-
unison), and Figure 2.10(b) shows the absorbers’ non-unison response at [‘9 = 0.0080

(after bifurcation to non-unison), wherein one absorber moves at a lower amplitude

40

 

 

 

 

 

    

 

 

 

00‘ I I I I j I I I I
‘3‘
E 0.05 - -
g 0 W
gs... -
.041 l j— l l l l l l 1
0.2 0.4 0.5 0.5 1 1.2 1.4 1.5 1.8 2
ROTOR ANGLE (0 II: ) (a )
o-‘ T I I I I I T i I
7 absorbers
~— 1 absorber 3
S 0.05
g 0
£2
LU
g .805
.0.‘ l l l l 1 l L l I
0.2 0.4 0.5 0.5 1 1.2 1.4 1.5 1.5 2
ROTOR ANGLE (o h: ) ( b )

Figure 2.10. N=4 circular path CPVAs with mistuning level of 0.5%, (a) before
bifurcation to non-unison (I‘g = 0.0076), (b) after bifurcation to non-unison (Pa =
0.0082) - from numerical simulation.

and lags the other three, which move in relative unison close to the unstable uni-
son response. This is similar to the post-critical response observed for epicycloidal

paths [20].

2.2.3 Cycloidal Paths

The cycloidal path offers a slightly hardening nonlinearity ( 70 < 0, as given by equa-
tion (2.35)), and this avoids many of the problems and shortcomings associated with

circular paths. For cycloidal absorber paths, neither the bifurcations to non-unison

41

nor jumps are present when 0' _>_ 0, as is clear from equations (2.36237). Figure 2.11
shows theoretical and numerical simulation results for N = 4 absorbers with 0%, 5%,

and 10% mistuning levels, respectively. Similar to circular paths, increasing the mis-

 

 

 

 

 

 

 

 

 

02 I I I I I I I
o o SIMULATION
0.1a ~ THEORY 0
o
o
0.16 " -I
o
a 0.12 - o .
(I)
01 - o .
‘6 ' o
g 0.08 '-
8 ° 1
n. 0.06 - O 0 -
3 o
0
° 1
0.04 ~ 0
O
0
0.02 - . . .
. ‘ I l 1 j l l l
0 0.01 0.02 0.00 0.04 0.05 0.06 0.07 0.00

TOROUE LEVEL Ir.)

Figure 2.11. Cycloidal paths. Upper 0% mistuning, middle 5% mistuning, lower 10%
mistuning

tuning level in cycloidal paths will decrease the amplitude of the absorbers’ motion for
the same torque level. Figure 2.12 shows theoretical and numerical simulation results
for the amplitude of the non-dimensional rotor angular acceleration versus torque
level for N = 4 absorbers with 0%, 2.5%, 5%, and 10% mistuned cycloidal paths,

respectively. These results indicate that the mistuning should be kept as small as

42

possible in order for the absorbers to effectively cancel the fluctuating torque. In this

case, in contrast with circular paths, the range is not limited by a jump bifurcation.

0.1

 

30.08»

gm.

 

 

Simulation
Locked

 

 

 

 

 

 

 

 

 

 

 

3 0.04) 1
30.02)
0 0.02 0.04 0.08 0.08 0.1
TOROUE LEVEL (1‘. )
0.2 -
Thory
0.15' Slmulatlon
Locked
0.1)
0.05: 1
o A L A
0 0.05 0.1 0.15 0.2

0.1

008’
I

0.08

0.04 '

0.02

0.2

0.15'

0.1 '

0.05

 

 

 

 

 

 

 

0.02 0.04 0.08 0.“

 

 

 

 

 

 

 

0.1
Thory
Slnuatlon
— Locked
0.05 0.1 0.15 0.2

Figure 2.12. Rotor angular acceleration, vv'(0) for mistuned cycloidal paths; (a) 0%
Mistuning (b) 2.5% Mistuning (c) 5% Mistuning (d) 10% Mistuning

It should be noted here that the agreement between theory and simulation is not

as good as it was for circular paths. As mentioned earlier, for a < —-;-, the theory

predicts jumps. But, this could not be found in the numerical simulations. The

reason for this is that cycloidal paths are much closer to epicycloidal paths where, as

Table 2.2. Theoretical ranges of absorbers amplitudes and torque levels at which non
unison motions exist for N = 4 absorbers with cycloidal paths

 

 

 

 

 

 

 

Mistuning (a) rzl r12 1‘ ()1 F92
-0.10 0.30 0.50 0.012 0.022
-0.20 0.42 0.71 0.014 0.032
-0.30 0.51 0.88 0.014 0.039
-0.40 0.58 1.01 0.012 0.045

 

 

mentioned earlier, nonlinearities other than the path nonlinearity are also important.
In any case, mistunings where a < —% are not of practical importance (as described in
the following section), and the theory works very well for practical levels of mistuning.
An interesting range of mistuning that deserves further mention is —% < a < 0.
For every mistuning level in this range, the theory predicts an amplitude of motion
range where the unison response is unstable. This range can be found using equa-
tion (2.36). For cycloids, it was possible to numerically find some stable steady state
non-unison responses in these ranges. Table 2.2 shows the theoretical ranges of ab-
sorber amplitudes and torque levels at which the unison response is unstable for a
system with N = 4 absorbers, and for different mistuning levels. Figures 2.13 and
2.14 show the numerically simulated steady-state absorbers’ responses for a’ = —0.1,
and a = —0.2, respectively for sets of torque levels that run through the unstable
ranges. These results indicate that a rather complicated set of bifurcations takes place
in these ranges, resulting in a variety of possible non-unison steady-state responses.
Other negative mistuning levels are not important because, similar to circular

paths, the absorbers will not be working properly as is shown in the following section.

44

I"0 - 0.011 I“ a 0.012
0.15 v . 0.15 1

 

 

0.1 . 1

0.05

 

-0.05 .

 

—0.1 » «

 

 

 

 

 

 

0 0.5 1 1.5
ROTOR ANGULAR POSITION (all)

I‘0 I 0.023 I‘" - 0.024

 

 

0.15

0.1 ’

0.05 *

-0.05

 

—0.1 '

 

 

 

 

 

 

Figure 2.13. N umerically simulated absorbers responses for a = —0.1 (N =4 cycloidal
path absorbers)

2.2.4 Note on the Absorber System Performance

In order to determine how the performance of the absorbers is affected by the level
of mistuning, one can consider equation (2.23), along with the fact that [1,, is small.
It can be clearly seen that the absorbers’ steady state phase angle 03, is either close
to 0 or close to 7r. For paths with 70 > 0, for example circular paths, if a > —% and
the absorbers’ amplitude of vibration is small enough, which is the case here, then (b,

is close to 7r. Then, from equation (2.16), along with equation (2.18), one concludes

45

F0 - 0.0125 F0 - 0.014

 

 

0.2

0.1 '

 

 

 

 

 

 

0 0.5 1 1.5
ROTOR ANGULAR POSITION (0h)

ro-omz

 

 

0.2 -

 

 

 

 

 

 

 

 

0.5 1 1.5

Figure 2.14. Numerically simulated absorbers responses for a = —0.2 (N=4 cycloidal
path absorbers)

that the absorbers are producing a torque which is opposite to the applied torque. If
the the absorbers’ amplitude of vibration is increased, say, by increasing the applied
torque level, then it will reach a value where cos(¢z) will jump from near (-1) to
near (+1). This corresponds to the jump onto the upper part of the response curve,
at which point the absorbers add to the fluctuating applied torque to the rotor. For
a S —%, cos(¢,) is always near (+1) and the absorbers add torque to the rotor. This is

demonstrated by the simulation results for N = 4 absorbers, as shown in Figure 2.15.

46

This figure shows the rotor non-dimensional angular acceleration, vv’(0), versus rotor
angle for steady state responses on the lower and the upper portions of the response
curve for a 5% mistuning, and a torque level of 0.02. It also shows vv’(0) for the same
torque level but with a mistuning level of —5% (where the absorbers are in phase
with the applied torque), and for the absorbers locked at their vertices. Note that
the absorbers actually increase the vibration level when on the upper branch of the
response curve.

For paths with 70 < 0, for example cycloidal paths, if a > —%, then ¢z is close
to 7r and the absorbers are reducing the torsional vibrations as long as they move in
unison. Since for —% < a < 0 there are ranges of applied torques where non-unison
motions exist, one can not conclude that the absorbers are working properly in these
torque ranges. For a Z 0, the absorbers are reducing torsional vibrations at all torque
levels. This is because for a Z 0, neither bifurcations to non-unison nor jumps are
present. When a g —-;-, equation (2.23) indicates that 4), is near 0, which means that
the absorbers are always adding torsional vibrations to the rotor. Figure 2.16 shows
numerical simulation results for the non-dimensional rotor angular acceleration for
N = 4 cycloidal path absorbers with 0%, 5%, and —5% mistuning levels, subjected
to the same torque level of 0.02. The eflect of negative mistuning mentioned above

is clearly demonstrated here.

2.3 Summary and Conclusions

The following points summarize the findings of this study:

47

 

0.1 I T

 

G———O 5.0%MISTUNING(LOWERPORTDNOFRES.CURVE)
- - - ABSORBERS LOCKED

—— 4.0% MISTLNING

8—0 5.0% MISTUNING ( UPPER PORTDN OF RES. CURVE)

 

 

 

 

 

 

—000 1 L
0 05 1 1.5

ROTOR WAR PCBITICWHII)

 

Figure 2.15. Rotor angular acceleration, vv’(0), for F9 = 0.02 for mistuned circular
paths - from numerical simulations

For paths ranging from circular up to, but not including, epicycloidal paths, 0 g A <
Ac:

0 There are positive mistuning levels below which no bifurcations to non-unison

are present. These levels are usually very small and are parameter dependent.
0 Jumps are always present for paths with a > —%.

0 Paths other than circular do not have significant benefits over the easily-

manufactured circular path.

0 For paths with a > —%, the absorbers reduce torsional vibrations for absorber

48

 

 

 

 

 

 

 

 

 

 

o.“ I I I T I I I I I
o—e 0.094 IIIISTLNING
0.00 _ +-—+ 5.0% MISTUNING ‘
- - - ABSORBERS LOOKED
—- .5011 MISTUNNO
§ 0.02 I " ~
g 0.01‘
.5 " ‘
g -0.01 ~
—0.02 -
.000 -
4.04 I l l 1 l l l l l
0 0.1 0.2 0.0 0.4 0.5 0.0 0.7 0.0 0.0 1

ROTOR AMULAR POSITIONflh)

Figure 2.16. Rotor angular acceleration, III/(0), for mistuned cycloidal paths, F9 =
0.02 - from numerical simulations, ( N=4 )

responses that are on the lower portions of the response curves, but they increase

torsional vibration for responses on the upper portions of the response curves.

For paths ranging from, but not including, epicycloidal paths up to cycloidal paths,

Ac<ASI:

0 Neither bifurcations to non-unison nor jumps are present for perfectly tuned

and positively mistuned paths.

0 For negatively mistuned paths, there are torque ranges where non-unison mo-

tions exist.

49

o The method presented predicts jumps for paths with a < -%, but these are not

found in the numerical simulations.

For all paths considered:

0 As the mistuning levels are positive and increased, the bifurcation to non-unison
and the jump points, if they exist, are delayed, and they approach each other,

resulting in increased operating ranges.

0 When the operating ranges are increased by increasing the mistuning levels, the

effectiveness of the absorber system is reduced.

0 With 0‘ S —%, the absorbers actually increase, rather than reduce, the levels of

torsional vibration.

In light of these observations, one can conclude that for any type of absorber path,
when the operating torques are kept very small, it is best not to have any mistuning,
i.e., perfectly tuned paths are the best choice. However, if one wants to increase the
operating range, positive mistuning levels should be selected, keeping in mind that
the absorbers’ performance will be reduced. Negative mistuning levels should always
be avoided.

When the performances of absorber systems with circular and cycloidal paths are
compared, it is concluded that absorbers with cycloidal paths are preferred because, in
addition to their much larger working ranges, they neither undergo jump bifurcations
nor bifurcations to non-unison steady state responses. This implies that only one
steady-state response exists at each torque level, and this response is equivalent to
that predicted by using a model with a single absorber mass, making design analysis
much easier. Also, they are dynamically robust and therefore suitable for practical

implementations.

50

CHAPTER 3

Non-synchronous Steady-State
Responses of Tuned Pendulum

Vibration Absorbers

In this chapter, an approach is taken that allows one to investigate the dynamics
of multi-absorber systems that have quite general paths for the absorber masses,
including those used in practice. Of particular interest here are the existence and
stability of certain classes of non-unison responses. It is shown via the method of
averaging that these steady state responses may exist and be stable for certain types
of paths, including the commonly-used mistuned circular path. Furthermore, it is
shown that these responses can co-exist with a stable unison response, even for very
small torque levels. The analytical results are compared with numerical simulations
and good agreement is found.

The chapter is organized as follows. The mathematical model, which is identical
to that considered in chapter 2, is first briefly described. Some assumptions are given
that allow one to derive a relatively simple set of equations that capture the absorber
dynamics. These are presented and the method of averaging is applied to investigate

the existence and stability of various types of steady state responses. A numerical

51

example is studied in some detail and compared against the analytical predictions.

The chapter then closes with some conclusions.

3.1 Mathematical Formulation

From chapter 2, the equations that describe the dynamics of N torsional vibration

absorbers, each of mass m, mounted on a rotor of inertia J are:

22’ + nzz, = ([2702? — 2n20,-z,- — [2,22
—,{,— i=1 n22,- — F9 sin(n0)] + HOT (3.1)
i = [1, N],

where HOT, as indicated before, refers to higher order terms. Again (.)’ represents
differentiation with respect to the rotor angular orientation, 0. n denotes the average
angular speed of the rotor, upon which the torsional vibrations are superimposed.
The scaling and the definitions of system variables and parameters are the same as
those given in chapter 2. The following description briefly summarizes them. The
base scaling parameter used is c, which is the ratio of the absorbers’ inertia to that of
the rotor, c = N mRfi/ J , which is small in practice. The scaled absorber displacement
variables 2,- are given by z,- = cisg, where 3,- represents the non-dimensional arc length

variable for the position of the i"2 absorber, i.e., s,- = -L where S,- and R0 are as

3,,
indicated in Figure 2.1. [2,, is the scaled absorber damping coefficient, pa = ([1,, = 3.90,
where c, is the physical damping constant for each absorber (assumed to be identical).
The applied torque is assumed to be harmonic of order n, i.e., [‘9 sin(n0), where F9 is
the amplitude of the fluctuating component of the applied torque, nondimensionalized

by the kinetic energy of the rotor and scaled by c, as follows, Pa = £3“; = Cgfa. The

parameter 0,- accounts for the effects of mistuning between the applied torque and

52

the tuned frequency of the absorber, and is defined by
71,-: n(1+ 50,-),

where n,- represents the dimensionless tuning frequency of the i222 absorber path (the
actual frequency is given by 73,9), and is geometrically given by the square root of
the ratio between the distance from the rotor center to the center of the path vertex
and the radius of curvature of the absorber path at the vertex. Such mistuning is
intentionally built into circular path absorbers, in order to counteract some unde-
sirable nonlinear effects at moderate vibration amplitudes. The parameter 7,, is a
parameter that describes the nonlinear nature of the absorber path. Its appearance

in the equations comes from moderate amplitude kinematic effects, and it is given by,
=1—1-(22)(n +1)2(n2A—2(1 + n2)). (3.2)

where A is a convenient parameter that describes various path types by taking on
values from zero to one. Some special cases of interest are: A = 0 describes a circular
path, A '2 “PIT describes an epicycloidal path with its base circle centered at the
rotor center, and A = 1 describes a cycloidal path. It should be noted here that
this formulation is not suitable for studying epicycloidal paths where 70 = 0. This is
because for epicycloidal paths, this model will reduce to the linearized one (see [20]
for the case of epicycloidal absorbers).

The nonedimensional rotor angular acceleration is given by

vv'(0)- -— 551% :2 n z,- + F9 sin(n0) + F9 sin(n0) )} +— N2: 2n2 2.1-z;- + HOT. (3.3)

Nj=1Nj=l

This is the measure used to assess the effectiveness of the absorber system, since

torsional acceleration is a measure of deviation from constant rotation speed. In

53

particular, once the absorbers’ motion 2,- are know, this allows for a quick estimate
of the the level of torsional vibration.

The method of averaging is used to determine approximate steady-state solutions
of equations (3.1). To reach to the averaged equations, the following standard trans-
formation to polar coordinates is used,

2'. = a;sin(n0+¢i) (3.4)

2: = na,-cos(n0+¢,~),

and averaging is applied, yielding

a: = eI-Ea. + ‘5: sins.) + a 2:52....- 6. sins.- — 4.11 + HOT

519-51- = 51-2335? + "(as + 2+0)?“ + 22.2.2 C050131)] + % 212192151 005(4-51 '— 21)] + HOT:
(3.5)

where an over-bar indicates the averaged value of the corresponding variable.

3.1.1 Existence of Certain Steady State Solutions

To capture a certain class of non-unison steady state responses, the absorbers are
divided into two groups. One group consists of M absorbers moving in relative unison
with an amplitude of 61 and a phase angle of $1, and the other group consists of the
remaining (N - M) absorbers, also moving in relative unison with an amplitude of
motion of ii; and a phase angle of (in. Note that the unison response is a special case
in which M = 0. Of course, other, more general, types of responses may occur, but
these are left for future study. To find the steady state amplitudes and phase angles
(65,25) j = 1,2, these conditions are imposed on equations (3.5) and solved. This

task is simplified by employing Cartesian coordinates for the slowly varying system.

54

To do that, the following invertible Van der Pol transformation is used:

(2)0122) ...,

I: cos(n0) sin(n0) J

—n sin(n0) n cos(n0)

where

Performing this transformation, averaging the resulting equations over one period,
and splitting the absorbers into the two groups described above with the assumption
that the absorber paths in the first group have an identical mistuning of 6; and those
in the second group have an identical mistuning of 52, the following equations (in
Cartesian coordinates ) that describe the steady state response of the absorbers are

reached:

0 = -%°-a, + n(&, + 51%).}, — %(&,6, + 0?) + {MN — My), + E: + HOT
0 = —i‘,—°0, — n(&, + fig-)0, + $3,303,112, + a?) — 5%,(N — M)i22 + HOT
0 = —%a, + n(&, + 1’2—2'NA‘J)0, — 3373(030, + 0;) + {NM}, + 2:: + HOT
0 = —F‘,—°5, — n(&2 + $127,102, + $0,172 + 03) — 2?va + HOT

 

J

(3.7)
After solving these equations, the results are then transformed back to amplitude and
phase coordinates for physical interpretation of the results. The polar and rectangular

coordinates are related in the usual way, as follows:
“2' = i1? + 5.2, $1=tan"(—2), j= 1.2. (3.8)

which yield the amplitudes and phases for the two groups of absorbers. To ensure
the capture of all possible real solutions, a graphical / numerical method is used. This

method proceeds as follows. Eliminating one pair of the (ii, 13), here taken as (112, {32),

55

in the above four equations gives a lengthy pair of equations in terms of the remaining
variables, here (1'21, 131). The zero contours for each of these equations are plotted in
the (111,131) plane, and the points where the zero contours intersect represent the
steady-state solutions for (111,131). The corresponding solutions for (112,52) can be
found by using the equations that were used to eliminate them from the original four
equations. Note that solutions that satisfy (£21,131) = (112,52) are unison solutions,

and these will be captured by the analysis for any value of M.

3.1.2 Stability of the Steady State Solutions

The stability of the steady state solutions is obtained by numerically computing the
eigenvalues of the Jacobian matrix of the system (3.5), evaluated at 6,- = 51;, 4;,- = $1
Vi E [1,M], and d,- = 52,ng = (I); Vi E [M +1,N]. Due to the symmetry of this

problem, the Jacobian matrix J has the following block structure:

 

 

A B
J = (3.9)
C D
where the matrices A, B, C, and D are as follows:
”A1 A2 . . . A2”
A2 A1 A2 . . A2 Bl . B 1
A = ,B =
A2 . . . A1 A2 81 . Bl 2MX2(~_M)
- A2 0 o o 0 A1‘ 2MX2M

 

 

56

 

 

'01 D2 . . . D2'
Cl . Cl D2 D1 02 . . D2
0: . . . D =
Cl . Cl 2(N_M)X2M D2 . . . 01 02
_D2 . . . . Dl . 2(N_M)X2(N-M)

The matrices A1, A2, B1, C1, D1, and D2 are 2 by 2 with the following entries:

A111 = — 94,=A112 Jcos(¢1)+ 2—N(M — l)&1 + 5%,-(N — M)&2cos(d>1- $2),
A121 = :%,7’&1 — affiKM ‘1)51 + (N — Mla2lCOS(¢1 4’2) + JCOSWIII,
A122: —-—[’2:§ sin(d>1) + ffiéfiN - M) sin($1 — (722)],

A211: 0 A212— - - 7001, A221" — 52—70 A222 = 0,

3111 = —',’v-sin(<i51 — 032) 3112 = "'fiaz 003(451‘ (2’2),

3121: JrMcos(¢1—¢2)13122- ’ fi‘SIHWI — (7’2)
C'111': ffiSinUIh — (131)10112— ——N02 COS(¢2- 4’1)

0121 = aNn—hcosf422 - 17’010122 = figflflfi — 4’1)

Dlu = —& ,D112 = g'fcosfiffig) ‘I’ n N;M l 52 +2; —NMal C08(¢2_ ¢l)
01,, = 32113312 —;‘§{5’,‘V[(N —- M -1)&2 + M0,] cos(¢g2 — <23.) + 2,3: cos(<752)}.
D122-— ——[2::' sin(<f>2) “I" ”31120261 Sin($'l '— $1 )I1

D211 = 0, D212 = —%521 0221: 2N1D222- 0-

57

3.2 Numerical Example and Discussion

Here, the system used in the last chapter will again be considered here. The numerical

i.e., c = 1 the order of

values are as follows: N = 4 CPVAs with an inertia ratio of i ;5,

20’
the applied fluctuating torque is 2, i.e., n = 2, the non-dimensional absorber damping
coefficient is 0.02, i.e., % = 0.02, the dimensionless rotor damping coefficient, a
quantity needed for numerical simulations, is 0.005, i.e., p0 = 0.005. Two path types
are studied, namely, circular and cycloidal paths. For numerical simulations, the full

equations of motion and the exact representations of the paths given in chapter 2 are

used.

3.2.1 Circular Path

Here, perfectly tuned (a = 0), slightly mistuned (a = 0.1), and moderately mistuned
(a = 0.5) paths are analyzed (negative mistuning is not of practical interest (see
chapter 2). Figures 3.1(a) and 3.1(b) show typical graphical solutions for the steady
state responses at a certain torque level for the perfectly tuned path for steady-state
responses of types M = 1 and M = 2, respectively. Figures 3.2(a) and 3.2(b) show
similar results for the case a = 0.1, which has more solutions. Every intersection
point of the curves represents a steady state response. Figures 3.3-3.8 show absorber
response amplitudes versus torque amplitude, in each case showing the hysteretic
jump behavior of the unison response. The points Plu, P2u, and P3u, in Figures 3.1
and 3.2 represent points that correspond to unison responses. Plu is a point on the
lower unison response curve, P2u is a point on the middle unison response curve, and
P3u is a point on the upper unison response curve (see chapter 2 for more details
on the unison response branches). The only steady state solutions that could be
found for the perfectly tuned paths are the unison responses, see Figures 3.1(a) and

3.1(b). When positive mistuning is introduced, a solution that is usually employed

58

 

 

 

 

 

 

 

 

 

 

r I I I I I T
05- 4
o.4~ P3" .
02 ~ .
N 0 " P1u ‘
-04 - -
_06 l- mu u

_03 l l l l l l l

-03 -0.6 -0.4 -0.2 0 02 0.4 0.6 0.8
711 (a)

I I I— I I I I
05 ~ 1
0.4 - ~
02 - .

N

z: 0 _ _
-0.2 r ~
-o.4- -
-0.6- -

“0.8 l l L l I l 1

~08 -0.6 -0.4 -02 0 0.2 0.4 0.6 0.8

111 (b)
Figure 3.1. Steady-state solution conditions for a perfectly tuned circular path,
(a) M=1, (b) M=2, Pa = 0.007.

59

0.6
0.4
02

-o.4
-o.s
-os

-08

0.5
0.4

02

:3 0
-02

-o.4

-0.6

-03

 

I

 

P10

P3u

«'

 

 

 

 

I

1

 

l

 

 

 

 

-0.8

-0.6

-0.4

-oz

111

Figure 3.2. Steady-state solution conditions for a mistuned circular path with a = 0.1,

(a) M=1, (b) M=2, r. = 0.006.

60

(b)

H.w1.c.at.m(amm)

OJ. I I I I f I

 

 

 

 

 

 

a“ ’ o o summon
UNISON o
(m ~ -

UNISON ' '~-..,

 

 

 

 

our «
a.“ r- ." -1
on ’.I.I.
UNISON . . ...-"" 1
A I l l I l I
'0 on am noon 0.003 0.01 no12 0.014

memeur.)

Figure 3.3. Response curves for three absorbers for the case M=1, N =4. Circular
paths with a = 0.1.

HM-I.uo.1mPA1H(1Am)

 

 

M” — 'meonnsuaus) 1

0.10 . o O SIMULATION //
0.14 . 4

'~.-.
--
_.
‘-
-.‘.

 

 

 

._I.
0.12- """"" ‘
.

's
0.1- ‘s. -I
.4

0.“ *-

 

 

 

 

0.“ " q
0.” l' ..
a l 1 J 1 L 1
0 0.” 0.” 0.” out 0.01 0.012 0.014
roams [EVEL (1;)

Figure 3.4. Response curves for one absorber for the case M=1, N =4. Circular paths
with a = 0.1.

61

HHOIO.LWPATH(2W)
0.0. U U V V I I

a“... """ WY(UNSTAIE) WW

.
-.‘.‘
'§._.
~'-.
"‘,-
.
~
.-.

 

 

 

 

 

o

u.

‘
I

p
d
N
T

‘ .

9
d
I
I
l

O O
8 8
Y I
I!
Ii
Ii
'i
i
j
I
i
i
.'
!
I
i
j
J I

MUTUOE OF ”$88888 MOTION
II
I

 

 

 

\ -,-.—'— '.,’
’.'\. _____ -’.’
" \ ’ ’
0.00 " .1 ’ '\ ’ a q
0’ ’-
.\I l.
a.” l' . ' ‘l
UNISON ~.s _ ,-’
a l l I J l l
0 0.“)2 0.” 0.000 0.” 0.01 0.012 0.004
TWLEVELU‘.)

Figure 3.5. Response curves for two absorbers for the case M=2, N =4. Circular paths
with a = 0.1.

Nd. $1.0-05m(8m)
0.25 y I u u r

THEORY ( STABLE )
------ meonkusrm
o o smuu'nou

 

 

 

 

 

 

0.2 -

OJS '

MOE

0.1-

0% '-

 

 

 

 

Figure 3.6. Response curves for three absorbers for the case M=1, N =4. Circular
paths with a = 0.5.

62

H,“fl.¢-O.5WPATH(G£MR)

 

 

 

 

 

 

 

 

0.” I V r W I
— THHNIY(8TABLE)
------ maonkusrmm
O 0 mm
0.2!- .
mason
................ m
~-~.-.~"~.m
0.15" _._. -."~.~" "
f ‘. ‘.‘ “
w I: ..... ~ ' -.-‘.:..:::'I:::-I—-o;~.\
..... ’ (70"-' .\
_’ ~ ~ ~ ~ o
0.1» _,~’ ‘~--.~ .' ..
I '\.‘ II
'\ .’
‘ \."_I
\ ‘.
.\' .
I- \ - I
0.05 '\. I
x ,'
's. ."
um ‘ \_ ,-’
\. ,1
5 -’
0 n n l "1 1
0 0.” 001 0. 0.” 0."« 0.0:!
Tonouemeur.)

Figure 3.7. Response curves for one absorber for the case M=1, N =4. Circular paths
with a = 0.5.

H,“o-0.5,GIMPATH(2W)

 

 

 

 

 

 

 

 

oz: . , . j t
—— THW(8TABLE)
------ meonvaS'rmE)
O 0 mm
0.2. //M
............. W
go.‘5. ‘I"‘.~ ‘
c. ‘. -~'~.‘
M 'J‘x-n.‘ _- ‘5‘
~""~°-Z'.""""“"'_'.-.'.-.~.-.-e-u'A-.
''''''' -~‘._.._._.—.—. \-
3 , --------- -. ~.
"\I \
OJ l- -’ "\. I 4
8 I x -
I. \_ .’
\ /’
I
I
.’
0.“? -’ ‘
\_ I.’
\'\- WW .I.
‘ \ _l"
.1
\ ‘. "_l
a l 1 ‘ -m' '.1 n 1
0 0 005 0.01 0.015 0. 0 06 0.03
TWELEELU‘.)

Figure 3.8. Response curves for two absorbers for the case M =2, N =4. Circular paths
with a = 0.5.

63

0.0.1, H. thug-om

 

 

O O summon

 

 

 

a.)

WRPOBTIOfHSI

 

 

 

 

.0.
.0.
41.00
41ml- -
‘0‘ g 4. l 1 A L g l l
0 oz 0.4 o o as 1 1.2 1.4 1 o 1 a 2
mm (on)

Figure 3.9. Typical localized response. a = 0.1 circular path. N =4, M=1, Pa = 0.004.

11-05. N-4. M-1.I'.-0.W

 

 

 

 

 

 

 

 

 

_02 l L l l l J l l I
O 0.2 0.4 0.0 0.8 1 1.2 1.4 1.8 1.8 2
mm MUMR P0811131 (HI)

Figure 3.10. Typical localized response. a = 0.5 circular path. N=4, M=1, [‘9 =
0.004.

64

to overcome some undesirable non-linear effects, and to increase the working range
of the absorbers, solutions other than unison exist, as shown in Figures 3.2(a) and
3.2(b).

Figures 3.3-3.8 each show the unison response branches, along with some of the
non-unison solutions. These figures indicate the accuracy of the asymptotic, approx-
imate solution procedure, both in terms of existence and stability. Figures 3.3 and
3.4 show the response curves for the case a = 0.1, M = 1, with Figure 3.3 indicating
the response of the group of three absorbers and Figure 3.4 showing the response of
the single absorber. An example of such a localized response for a particular torque
level is depicted in Figure 3.9. For the same path parameters, Figure 3.5 shows the
response of the group consisting of two absorbers moving in unison, i.e., M = 2 for
the case a = 0.1. Note here that all such non-unison solutions are unstable.

Similarly, Figures 3.6, 3.7, 3.8 are the analogous plots for a case with larger mis-
tuning, a = 0.5. These figures clearly show the existence of non-unison solutions
and the way they bifurcate from the unison solution. Figure 3.10 shows a simulation
result at a particular level of torque excitation.

It is important to note that these non-unison responses co—exist with the stable
unison response, but result in one or more absorbers undergoing a much larger am-
plitude of oscillation than predicted for the unison response. Also, in practice one
introduces some level of mistuning in order to extend the operating range of the
absorbers for a given amount of absorber mass; this can be observed by comparing
the horizontal scales in Figures 3.3-3.5 with those of Figures 3.6-3.8, and noting that
the lower unison response branch is the one of interest. However, the larger level of
mistuning actually increases the possibility of encountering a non-unison response at

a given torque level.

65

3.2.2 Cycloidal Paths

Steady state solutions other than unison were not found for perfectly and positively
mistuned cycloidal paths. They exist only for negatively mistuned cycloidal paths.
Since negatively mistuned paths are not of practical importance (see chapter 2 for

more details), no results are presented here.

3.3 Conclusions

Based on the above results, the following conclusions can be drawn:

0 For perfectly tuned circular CPVA paths, the only steady state solutions that
were found were the ones that correspond to unison motions. When mistuning is
introduced, other steady state solutions that depend on the level of mistuning
appear. Those solutions must be taken into account when designing CPVA

systems.

0 For perfectly tuned or positively mistuned cycloidal CPVA paths, no solutions
other than those that correspond to unison motions exist. This is one of many

advantages that cycloidal paths have over circular paths [10, 17].

o The steady state responses where a subset of absorbers is not in unison with
the remaining absorbers can often correspond to a non-linear localized response.
The strength of this localized response and the range over which it is stable
depend on the level of mistuning. This type of steady state solutions will be

reconsidered in chapter 5.

66

CHAPTER 4

Localization of Free Vibration
Modes in Systems of
Nearly-Identical Vibration
Absorbers

In this chapter, the linear free vibration of systems in which groups of nearly identical
vibration absorbers are employed is considered. It is demonstrated that the phenom-
enon of mode localization occurs in these types of systems. In these systems the
absorbers are not directly coupled to one another via flexibility elements, but rather
the coupling is through the inertia of the primary mass. This coupling is of the order
of the ratio of the absorber inertia to the primary mass inertia, and is typically small
in applications. An eigenvalue/eigenvector perturbation technique is used to find ap-
proximations of the modes of free vibration, and it is shown to be accurate when the
ratio of coupling to mistuning is small. Both translational and torsional absorber
systems are considered, and the results obtained raise some interesting questions re-
garding the steady-state response of the overall system, and the performance limits of

the absorber system, when it is subjected to periodic excitation at a frequency close

67

to that of the absorbers’. This chapter is organized as follows. Section 4.1 describes
the two types of absorber systems and formulates the equations of motion for each
in such a manner that the perturbation technique can be readily applied. Section 4.2
describes the analysis and presents sample results, and the chapter closes with some

conclusions in Section 4.3.

4.1 Formulation

4.1.1 Translational Vibration Absorbers

Consider a structure of mass M on which are mounted N vibration absorbers of
masses m.- and spring stiffnesses kg, (1' = l.., N), as shown in Figure 4.1. We consider
the case in which the natural frequency of the primary mass-spring system is much
smaller than that of the absorbers, and use the limiting case in which the primary
mass has no stiffness to ground, so that the overall system has a rigid body mode.
In the analysis, this mode will be uncoupled via a simple change of coordinates. A
similar system (only with stiffness to ground) has been considered by Weaver [40],
who considered the response to random excitation of a system with a large number
of substructures having a distribution of natural frequencies.

The equations of motion for free vibration of this system are

M1 + ime + a) = o, (4.1)

1:]
m(z,- + y) + 1652; = 0. (4.2)

It is assumed that the absorbers have equal masses ( i.e., m.- = m ), and the stiffnesses
will be used to introduce mistuning among the absorbers. The dynamics of the big

mass can be uncoupled from the absorbers’ by substituting 3'] from equation (4.2) into

68

*1 k: k”

m=n~y

 

I__:Q

 

 

 

Figure 4.1. Translational vibration absorbers

equation (4.1), and rearranging. The following equations which describe the responses

of the absorbers are obtained
2. + (1 + ——)w;+’z.- + i-"f-‘i 2 dz.- = 0, (4.3)

where mo = N m is the total mass of all absorbers, and w? = k,- / m are their individual,
uncoupled frequencies.

In typical applications the absorber mass is significantly smaller than the primary
mass. Therefore, the ratio of the absorber masses to the structure mass, denoted as c,
i.e., c = $3 is introduce as a small parameter. Since the absorbers are coupled to each
other only through the primary mass, 6 represents the degree of coupling between the

individual absorbers. Equation (4.3) can now be expressed in matrix form as follows

I§+A§ =6, (4.4)

 

 

 

Figure 4.2. Torsional vibration absorbers

where

A = A0 + 6A
(4.5)

A0=Diag[w?].
6A is an N X N matrix with each element equals to fiwfiom, wnom is the nominal
frequency of the absorbers. Since the diagonal matrix A0 has readily obtained eigen-
values and eigenvectors, equation (4.4) is in a form suitable for the application of an
eigenvalue perturbation method. Before doing so, it is first shown that the dynamics

of a system of torsional vibration absorbers can also be expressed in this form.

4.1.2 Centrifugal Pendulum Vibration Absorbers

A system of N torsional vibration absorbers which are mounted on a rotor of inertia

J, as the one that was introduced in chapter 2 is considered. The free vibration

70

equations of motion for this system, for arbitrary absorber amplitudes, are as follows:

12‘s

2 d3,(3‘)é2 = o (4.6)

31 + 91(805 -

" N d ' ' ~ .. d 1' .
J9 + Z mfiilifbifiao + $486)” + 94308: + 2313083 = 0, (4-7)

1:]
where the functions m;(s,-) and g.-(a:,-) are as defined in chapter 2. For reasons that
will be clear later, the 1"" path order, fig, which as indicated before represents the
variability of the tuning of each absorber path from the nominal tuning of order n,
is defined slightly in a different manner than its definition in chapter 2. It is defined
here as follows:

ii.- = n(1+ 0;) 0.- << 1.

Again assuming that the absorbers have equal masses and the value of R.- at the
vertex of the path is the same for all the absorbers ( i.e., m,- = m, and Rio = R, ),
using the definitions of the functions g;(s,-) and m;(s.-), and linearizing equations (4.6)
and (4.7) about 3.- = 0, and 0 = Q, where Q is the nominal speed of the rotor, the

following linear equations for the system dynamics are obtained

.15 + £7: mRZ(5 + 3;) = 0 (4.8)

5:!
g.- + ('9' + (22,133.- = o. (4.9)

Comparing equations (4.8) and (4.9) with equations (4.1) and (4.2), they are clearly

seen to be equivalent. This implies that equation (4.4) also applies to torsional vibra-

tion absorbers systems. In this case, the small parameter 6 represents the ratio of the
NmR2

total moment of inertia of the absorbers to the rotor moment of inertia ( c = ——J—-“ ,

and w.- is replaced here by (Mg.

71

4.2 Analysis and Discussion

4.2.1 General Features of the System

The system of equations (4.1, 4.2) has some interesting properties when the absorbers
are identical. When k,- = 1: Vi, the absorbers have identical natural frequencies. In
this case the overall N + l degrees-of-freedom system has two modes in which the
absorber masses move in a synchronous manner, and these correspond to the modes of
an equivalent two-degrees-of-freedom system (one is the rigid body mode, the other is
oscillatory). The remaining N -— 1 modes have identical frequencies and mode shapes
that correspond to the absorber masses moving in such a manner that they exert zero
net force on the primary mass, rendering it stationary. The selection of the modes
in this degenerate case is highly non-unique. Therefore, the level of imperfections
cannot be used as a small parameter in a perturbation scheme, since one does not
have a specified set of modes that can be used as the starting point in the perturbation
scheme. This unperturbed system has absolute sensitivity to the mistuning in the
sense that different arrangements of mistuning lead to completely different sets of
modes.

0n the other hand, the case of zero coupling and nonzero mistuning is unique,
since the modes are represented by the ideally localized responses in which only one
absorber is active and the others are stationary. These are naturally suited for use as
the seed modes in a perturbation scheme [25].

Singular perturbation schemes have recently been used to capture localized modes
for all relative ranges of mistuning and coupling [39, 41]. However, in the present case
this is not feasible, due to the completely degenerate nature of the system in the case

of zero mistuning.

72

4.2.2 Perturbation Method Formulation

The normal modes of vibration are represented by the eigenvectors of the matrix A in
equation (4.4). These can be determined here using a standard perturbation method
of the eigenvalue problem [42]. The following expressions are used for the eigenvalues

(Ag) and the eigenvectors (6g), up to second order in approximation
A; = A01 + 5A; + 52M,

~ ~ ~ 2..
v1: Us + 6121+ 6 v1,

where Aog,1'3g,g are the eigenvalues and eigenvectors of A0 respectively. The terms in

the perturbation expansion are solved for by matching terms and are given by

N N
~ ~ 2~ ~ .
6'01: E :Vikvoka 6 v1 = E :Uikvoln 1 = 11'°°1N
k=l k=l

 

”7'. ~. "7‘. ~.__ .~T. ~.
M, = ———y°'l§fi‘]v°', w. = y°'[5A]5‘ff,- 5A‘ym5v', _—_ 1, .,N
yoa'voi yoivoi

where, for j 31$ i, the coefficients are

1 3733514115,,-
ij '-

37.5% ’\01' _ ’\°.i

 

 

 

1 (Shays,- — 3707,.[6A]66g
771” : ~ ~ 1
J yfgvoj Aoj - Aoi
and for j = i,
0 613? 66g
Vii = , ii = - .. - -
’7 20500;

Here the symbol 370,- represents the left eigenvectors of A0. Note that these expressions
are, as expected, singular if the frequencies are repeated. That is, this perturbation

scheme accounts for the effects of coupling, but the results become invalid if the

73

Table 4.1. Data for example 1.

 

 

Absorber Mistuning (a) Mistuning (b)
l 2% 1%
2 -2.5% 4.2%
3 3.5% 1.7%
4 -4.5% -2.2%
5 -1.0% -0.5%
6 0.5% 0.2%

 

 

 

 

 

Table 4.2. Data for example 2.

 

Absorber 1 2 3 4 5 6 7 8 9 10
%Mistuning 1.2 0.0 1.6 -0.4 0.4 0.8 -1.2 -1.8 -0.8 -1.6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

mistuning among the absorbers is small [25].

These perturbation results can be used to find the frequencies and modes of vi-
bration. This is easy in the present case, since the eigenvectors of the matrix A0
can be taken as the canonical unit vectors (for the case of distinct eigenvalues). Two

numerical examples are given here.

4.2.3 Examples

The purpose of these examples is to demonstrate the localization phenomenon and

to show the accuracy of the perturbation method.

SIX ABSORBERS WITH MODERATE COUPLING

In this example, we consider a system of six absorbers with the data given in Table 4.1,
and with a coupling coefficient of %. Two cases of mistuning are considered, as

indicated in Table 4.1.

74

Figures 4.3 and 4.4 show the modes of free vibration obtained by the second order
perturbation method and by the exact solution of the full eigenvalue problem. As
expected, the perturbation results start to deviate from the exact solution in the case

of small mistuning.

TEN ABSORBERS WITH SMALL COUPLING

Here we consider a system of ten absorbers with the data given in Table 4.2. The
coupling coefficient is 71-5, a value taken from an existing light aircraft engine. Fig-
ure 4.5 shows the free vibration modes for this system of absorbers. The modes are
seen to be highly localized, and, as expected, the perturbation method works very

well in this case.

4.3 Conclusions

Consideration of results obtained from the examples clearly shows that localization
indeed occurs for this class of systems, and that the perturbation method is a reliable
tool for obtaining the modes of vibration for a range of small coupling relative to
mistuning. These results also indicate that some interesting and unexpected behav-
ior may be found in the forced response of system of tuned vibration absorbers, in
particular since these systems are excited at a frequency that is very close to the

frequencies of the absorbers. This topic is the subject of the next chapter.

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 ' 1 O . 1‘
0.5 ' 0 5
C
o r . . . . 0 [ . . .
-° 5 -o.5 » '9
-1 L -1 ‘ ‘
1 2 3 4 5 6 1 2 3 4 5 6
1 - 1 r
0.5 l 0 5 t
O 0
O 0 O
-0.5 ’ 1 —0 5 P
—‘I ‘ Q _1 . J
1 2 3 4 5 6 1 2 3 4 5 6
1 i 1 u .
0.5 0 5
Q 0 O
* O
o 00 e
-0.5 I -0.5 '
..1 4 - _1 -
1 2 3 4 5 6 1 3 4 5 6

Figure 4.3. Modes of vibration for example 1(a). (*) Exact,(o) second order pertur-
bation

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 . - 1 . .
.5. a . .51 1
o . e 9 ' 5 ,1
o l- . 0 6 6 c)
-0.5 > 1 -0.5 ' 3
_1 . . . . _1 A 4 A
1 2 3 4 5 6 1 2 3 4 5 6
1 - . 1 , i
0.5 1 0.5? o
* an
-0.5 1 -0.5 - 1
..1 . A j A _1 A A j
1 3 4 5 6 1 2 3 4 5 6
1 . 4 . 1 T r
5 £5
0 5 1 0.5 ’
O o
1* O 11 Q 1* <>
0 * 1 0* 9 'I
O 0 1*
-0 5 O “ —o 5 1
_‘I 4 A _1 l
1 2 3 4 5 6 1 2 3 4 6

Figure 4.4. Modes of vibration for example 1(b). (at) Exact, (0) second order pertur-
bation.

77

 

 

 

 

 

 

 

 

 

 

 

 

1 - O
0.51 1

2 ' O

o O O O O O O O
-0.5(

_1 .

2 4 6 8 1o

1 ~ - O
0.5:

W O 0 O I 0 g 0 9
-o.5- 1
_1 . .

2 4 6 s 10

1 _. .

0.51 1

09 O O O O O ' O

O
-0.5-
-1 1 O
2 4 6 8 10

Figure 4.5. Some modes of vibration for example 2 . (4:) Exact, (0) second order

perturbation

78

 

 

 

 

 

 

 

 

 

1’ u
0.5 '
0 L 0 O . . 0
O
—O.5 -
_1 -
2 4 6 10
1 -
0.5 r
Y o
0 . Q . ‘
—0.5 4
_1 L . L
2 4 6 10
1 A
0.5» l
00 O O 0 O O f
-0.5 - 1

 

 

 

10

 

CHAPTER 5

Forced Localized Response of

Vibration Absorbers

In the previous chapter, localization of the absorbers’ modes of free vibration was
considered. It was found that when the ratio of coupling to imperfection is small, free
vibration modes do localize. This result motivates the investigation of the localization
phenomenon in the more important case, namely, when the system is under opera-
tion and subjected to periodic excitation. The study done in this chapter focusses
on torsional vibration absorbers. The translational vibration absorbers case can be

analyzed in a similar manner.

5.1 Linear Forced Response

5.1.1 Mathematical Formulation

Similar to the system studied in chapter 2, a system of N torsional vibration absorbers
is considered. After scaling the system variables and parameters in the same way, and

linearizing the equations that describe the absorbers dynamics, the following linear

79

equations are Obtained:

N
21’ + "221' = Cl-ana'izi - fiazgf - 71,-: ngzg — F9 sin(n0)] + HOT, (5.1)

i=1

where all variables and parameters are defined in chapter 2. Rearranging and writing

these equations in matrix form, one obtains
2 ~
1N2" + nzDiagN (1 + 2w.) 2 + 6%0nes(N)§ + €11,le = ergo)” 3111619) (5.2)

where IN denotes the N X N identity matrix, DiagN(:c) denotes an N X N diagonal
matrix with z in each diagonal entry, Ones(N) denotes an N X N matrix with every
element equal to one, and (1)N denotes an N X 1 vector with every element equal to
one. To formulate the problem so that the steady state responses, 2'”, can be found,
we assume that E = inch”, where j = \/—_l, and 5,, 6 C”. When this is substituted

in equations 5.1, the following equations are obtained:

(A '1' B) 288 = f1 (5'3)
where
. .17.; 1 ~ 1 f‘
A = Dzag (20,- +];) , B = NOnes(N), f = 23(1)}(1 9.

To find the steady state solutions, this NX N coupled linear system of equations
must be solved. When this system is examined, it can be seen that it is not
difficult to uncouple it. This can be done by employing the following procedure.
For m E [1, N], m 74 i, subtract the m“ equation from the 1"“ equation for every
value of m to get an expression for each z,,,,, in terms of 2,,g. Substitute these expres-

sions back in the equation for the 1"“ absorber to reach to the following uncoupled set

80

of equations:

 

 

(Reg +j Img)z,,g = f, (5.4)
where ' 2
1 1 N 4010' + ’f}
R8; = — + 20'; +— N z 1 ( )2 1
N j=ld¢i 4012+(éf)

 

.. N ._ 1' 1 ~
Imi=&1+'1%‘z OJ (:2 1f=—F9
" 1:14-21 40} + (if)

The scaled steady state vibration amplitude of the 1"" absorber will then be

2 - = f 5.5
l ml \/(Reg)2+(lmi)2 ( )

 

 

When damping is small compared to mistuning, this amplitude becomes

 

[2331' = f (5.6)
(N + 20'+ +N 21:11.19“ 0,)

Examining this equation, it can be clearly seen that the amplitudes of vibration for
all the absorbers will be the same as long as the levels of mistuning are the same.
However, due to the fact that perfectly identical paths are not possible to manufac-
ture and there are always some imperfections, one should not omit the possibility of
the existence of some localized response, due the presence of the ratio (5;) in equa-
tion (5.5). This equation says that when the mistuning levels of all absorbers are close
to zero and a sub-group of absorbers have mistuning levels that are lower than the
remaining absorbers, then those with the low mistuning levels will localize, i.e., their
amplitudes of vibration will be significantly higher than those of the remaining ab-
sorbers. The strength of this localized response depends on how close the levels of
mistuning of the sub-group that is localizing are to zero and how far they are from

those of the remaining absorbers.

81

Table 5.1. Data for example 5.1

 

 

 

 

 

 

 

Absorber a. a(%) b. a(%) c. a(%) d. a(%)
1 0.01 (0.16) 0.010 (0.16) 0.010 (0.16) 0.010 (0.16)
2 0.07 (1.16) 0.008 (0.13) 0.008 (0.13) 0.008 (0.13)
3 0.05 (0.83) 0.050 (0.83) 0.009 (0.15) 0.009 (0.15)
4 0.06 (0.99) 0.060 (0.99) 0.060 (0.99) 0.011 (0.18)

 

 

In the presence of small damping ( recall that damping should be small for good
absorber performance, see chapter 2 for more details ), this result will not be signif-
icantly affected. However, when damping is large compared to the mistuning levels,
localization will not occur.

It should be noted here that localization is expected to occur only when the levels
of mistuning of all absorbers are close to zero and there are relatively significant
variations among them. This suggests that a solution to avoid localized responses is to

introduce some positive, nearly identical intentional mistuning among the absorbers.

5.1.2 Numerical Examples and Discussion
Example 5.1: Small damping compared to imperfections

In this example, a system of four absorbers with the following numerical data is
considered: inertia ratio 6 = 0.1662, n=2 ( this data is taken from the 2.5 liter, in-line,
four stroke, four cylinder engine considered by Denman [11]), and i1, = 0.008N. Four
cases are considered with the mistuning levels shown in Table 5.1. The first case
corresponds to a situation where one absorber has a smaller mistuning level than the
remaining three. The second case corresponds to a situation where two absorbers
have smaller mistuning levels than the other two. The third case corresponds to a
situation where one absorber has a larger mistuning level than the other three. The

fourth case corresponds to a situation where all the mistunings are small and close

82

 

0.1 w . - 0.1

 

 

 

 

 

        
    

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

O
— Th 0
0 08 ’ D 0 Simulation 0 03
- -- Unlson
2 0.06 ' 0.06 '
<
3 0.04 ’ 0.04 ’
i 0.02 0.02
o ' a A A
0 0.01 0.02 0.03 0.04
TORQUE LEVEL (1". ) ( a)

0.1 - - . 0.1
0.08 ' 0.08 .
0.06 ~ 0.06
0.04 ' 0.04 .
0.02 - 0.02

0 -
0 0.01 0.02 0.03 0.04

 

(c)

Figure 5.1. Absorbers amplitudes verses fluctuating torque level for example 5.1.

to each other. The amplitudes of the steady state responses of the absorbers versus
the applied torque level for these four cases are shown in Figure 5.1. It should be
mentioned here that the simulation results are obtained by numerically solving the
full non-linear equations represented in chapter 2 for the case of epicycloidal path,
which is the closest to being linear over a wide range of amplitudes. It is clear
from Figure 5.1 that localization indeed occur for this system. The severity of the
localization depends very much on the mistuning differences between the absorbers.

Absorbers with smaller mistunings do localize as long as there are other absorbers

83

Table 5.2. Data for example 5.2

 

Absorber a. a(%) b. a(%)

1 0.002 (0.03) 0.001 (0.02)
2 0.014 (0.23) 0.007 (0.12)
3 0.010 (0.17) 0.005 (0.08)
4 0.012 (0.20) 0.006 (0.10)

 

 

 

 

 

 

with larger levels of mistuning. It is also clear from the figure that the strength of
the localized response depends on the number of absorbers that are localizing. The
most severe case is when only one absorber has a smaller level of mistuning than the

remaining absorbers, i.e., case (1) Figure 5.1(a).

Example 5.2: Large damping compared to imperfections

This example is limited to the most severe case where only one absorber may localize,
i.e., only one absorber has a smaller level of mistuning than the remaining absorbers.
Here, the damping level is kept the same as that in example 5.1. The two cases
shown in Table 5.2 are considered. The first is the case where the mistunings among
the absorbers are one fifth those in example 5.1(a). The second is the case where
the mistuning levels are one tenth those in example 5.1(a). The amplitudes of the
absorbers’ steady state responses versus the applied torque level for these two cases
are shown in Figure 5.2. It is clear that localization gets weaker as the levels of

mistuning get smaller.

84

 

 

 

 

 

 

 

 

 

 

0.08 a O 0.“
— Theory —" "my
D O Slmulatlod D 0 Simulation
0.07' "'" W00 0.07,. ._. m
0
O
a.“ ’- od 0.“ .-
O
O
0.“ r O 1 0.“ - O
O
0.04 " O om .
g 0.03 - 1 g 0.03 ..
i 012 '- 3 092 ..
0.01 '- 0.01 '-
0 l A I o l I L
0 0.01 0.02 0.03 0.04 0 0.01 QM 0.03 0.04

 

 

 

TOROUE LEVELU‘.) (a)

 

 

 

romuueveur.) (b)

Figure 5.2. Absorbers amplitudes verses fluctuating torque level for example 5.2.

Example 5.3: Small damping compared to imperfections- with intentional

mistuning

In this example, three nominal intentional mistuning levels of 0.1, 0.2, and 0.3 are
applied to all four absorber paths of example 5.1. The imperfections among the
absorbers’ paths are similar to those of example 5.1(a), and shown in Table 5.3. The
amplitudes of the absorbers’ steady state responses versus the applied torque level for
these three cases are shown in Figure 5.3. It is clear from this figure that the system
becomes more robust when a positive intentional mistuning that is large compared
to the imperfections in the absorbers’ paths is introduced. This robustness depends

on the magnitude of the introduced intentional mistuning.

85

Table 5.3. Data for example 5.3

 

 

Absorber a. a b. a c. 0‘
1 0.11 0.21 0.31
2 0.17 1.27 0.37
3 0.15 0.25 0.35
4 0.16 0.26 0.36

 

 

 

 

 

 

Effect of damping level on localization

To clearly see the effect of the damping level on localization, example 5.1(a) and
example 5.2 above are reconsidered with variable damping levels. The ratios of the
maximum to the minimum absorber amplitudes, gaff, versus the scaled damping
level, [1,, are plotted for these examples and shown in Figure 5.4. This figure clearly
demonstrates the effect the level of damping has on the localized response. In all
cases, increasing the damping level decreases the strength of the localized response.
However, example 5.1(a) is the most severe case because its absolute amplitudes of
mistunings are larger than the other two ( it is 5 times larger than those of exam-
ples 5.2(a) and 10 times larger than those of examples 5.2(b) ).

As a direct result of this, one may think of having relatively higher damping level
in order to avoid localization. However, one should keep in mind that the damping
level directly influences the absorbers’ performance. The lower the damping level, the
more effective the absorbers (see chapter 2). A better solution to avoid localization
is to introduce a small intentional mistuning in the absorbers’ paths which makes the

system to be more robust, as was shown in Figure 5.3. However, again one has to

keep in mind that mistuning also affects the performance, as discussed in chapter 2.

86

QM

0.05

_o
2

AMPLITUDE OF ABSORBER MOTION
p o
B 8

0.01

Figure 5.3. Absorbers amplitudes verses fluctuating torque level for example 5.3.

Effect of differences in imperfections on localization

Figure 5.5 shows the effect the differences in imperfections among the absorbers have
on localization for N = 4 absorbers with the same numerical data given above. Two
cases of absolute mistunings are assigned to three absorbers, namely, 0.07 and 0.01.
In each case, the level of mistuning of the fourth absorber is varied from zero to the
corresponding value of its group. It is clear that as the difference in mistuning between
the fourth absorber and the other three becomes larger, localization becomes stronger.

It is also clear that localization strength depends on the absolute magnitudes of the

0
0 0.02

a -0.1
fl

 

 

Theory
00 Simulation
-- Unison

 

 

 

 

J

 

 

0.04

TOROUE LEVEL(I‘°) (a)

0.06

0.05

0.04

0.03

0.02

0.01

0

 

 

 

 

(In-0.2
9
<7
I
.9
I.
’.
0 0 02 0.04

(b )

87

0.05

0.04

0.03

0.02

0.01

a -o.3

 

 

 

 

0.02
(0)

0.04

 

   

 

—— EXAMPLE 5.1 (a) _
- — - EXAMPLE 5.2(a)
- - - — - EXAMPLE 5.2(b)

 

 

 

 

 

 

. '._ , .
\ \
' \
‘.\ ,
\
3' \ \ '1
‘ \
\. \
\. \
\ \
2— ‘\ \\\ -l
.\ “
\‘ ‘~‘~
1 , ",' --------- ,_._._.Z.‘;.‘=T::. ::: _ - ,_ - , _, — _ —— _ ,_ -
0 0.02 0.04 0 06 0.08 0.1 0.12

SCALED DAMPING COEF.

Figure 5.4. Effect of damping level on localization

mistuning. Localization is stronger for higher absolute magnitudes of mistunings.
Mainly, there are two reasons why localized responses should be avoided. The
first is that localization deceases the system’s performance as shown in Figure 5.6.
The second is the fact that the localizing absorber(s) will hit its amplitude limits at
a smaller level of applied torque than if there were no localization. This means that

localization decreases the system’s operating range.

88

 

U!
..
q
..
..
..
.4
-
q

04 : VARIELE;o,-o.o7 v i e [1 .31 ’

 

SIS
amps

 

 

 

 

 

 

L A J L 1 L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 1
(am—amt!) I am ( a )

 

O
..
..
..

 

0|

 

 

 

 

SIS
Amy’s-

 

 

 

O
..
. l-
..
-
.
E

Figure 5.5. Effect of mistuning differences on localization

5.2 Nonlinear Forced Response

In this section, non-linear localization phenomenon in systems of absorbers with iden-
tical paths is investigated. The mathematical tool used to perform this task is the
same one used in chapter 3 to investigate the existence and the stability of steady
state responses other than the unison response in systems of multiple identical CP-
VAs. Therefore, the mathematical model will not be presented here.

Based on the findings of chapter 3, path types where responses other than the

unison response can occur are the positively mistuned paths ranging from circular

89

 

 

 

 

 

 

 

 

0.014 . , r , .
au-——1» EXAMPLE 5.1 (a)
13—5 EXAMPLE 5.1 (b) .
0-012 ' o—-——o UNISON
g 0.01 -
§ 0.008 r-
<
§
0
a; 0.006 -
2
Q o
i 0.004 - ”
0.002 - ., i
/: z s
O ‘ "45’: 1 1 1 1 1
0 0.01 0.02 0.03 0.04 0.05 0.06

TOROUE LEVEL (r0)

Figure 5.6. Non-dimensional rotor acceleration verses fluctuating torque level for
example 5.1. (From numerical simulations)

to, but not including, epicycloidal paths, i.e., paths with 0 S /\ < 71337,. Here,

the search will be more specific, and we will only seek localized solutions. The six

L

75, a value taken from an existing light

absorber system with the inertia ratio of
aircraft engine, that was considered in chapter 4, will again be studied here as an
example. The absorbers of this system ride on circular paths. The order of the
applied torque, n, is 3. The procedure presented in chapter 4 is used and the values

of M that are of interest here are l and 2. This means that we are looking for two

types of localized responses. The first is where one absorber has a higher amplitude of

90

(a) CIRCULAR PATHSa-Ifls, N-G; Mute-2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

z 0 V f I V V
g cm . ONE ABSORBER .
O
O O
, 00000000000000 flaw
$0.0m °°°°° J o O SImulatIon
u.
g 0.04 ~ -
E 0. 02 _ FIVE ABSORBERS q
E
o A l A A L L
0 0.002 0.004 0.006 0.000 0.01 0.012 0.014
TOROUE 0‘.)
(b ) CIRCULAR PATHSa-Ifls, N-a; M-2;o-2.
0.0a » 1W0 ABSORBERS .
oooooooooooooooooo:~ Raw

3 0.06 - o o Simulation
1:.
g 0.04 - -
I; 0.02 . FOUR ABSORBERS ,
0.
2
4!

 

 

 

o A L A 1 A j
0 0.002 0.004 0.006 0.000 0.01 0.012 0.014
TOROUE 0‘.)

Figure 5.7. Localized response curves. N = 6 circular path absorbers with c = l/ 75,
and a = 2.0. (a) The case M=1. (b) The case M=2.

91

 

 

(a) CIRCULAR PATHSC-Ifls. N-6; Mute-4.

 

 

 
 
    

 

 

 

 

 

 

 

 

 

 
 
     

 

 

 

 

 

 

 

5 0.12 r ‘ ‘ ' -
F ONE ABSORBER 0
g 0.1 ’ f e o o "
30.03.000000000000000 1 'Is‘heory
<
‘5 0.06 ~ .
3 0.04~ .
$0.02 _ FIVE ABSORBERS .

o l A l L A l

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

TOROUE (r0)
(0) CIRCULAR PATHSo-lns. N-S; M-2;o.4.
5 0.12 p U V I I U V q
F TWO ABSORBERS 1
0.1 l- U a

g f 0 o O O
. o o O 0 Theory
go.00~°°°°°°°°°°° . Slmulollon
‘5 0.00 - .
50.04 P -
5: 0.02 _ FOUR ABSORBERS .

o A L l l L A

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

TOROUE 0‘.)

Figure 5.8. Localized response curves. N = 6 circular path absorbers with c = l/ 75,
and 0‘ = 4.0. (a) The case M=1. (b) The case M=2.

92

vibration than the remaining five which move in relative unison. The other is where
two absorbers move in relative unison and have a higher amplitude of motion than
the remaining four. Two levels of mistuning are considered, namely, 0' = 2 (2.67%),
and a = 4 (5.33%). Figure 5.7 show the results for the cases M=1 and M22 for the
paths with a’ = 2 and Figure 5.8 shows the corresponding cases for the paths with
a’ = 4. Examining these figures, it can be clearly seen that nonlinear forced localized
responses do exist and are stable over certain ranges of the applied torque. It is
also clear that the strength of these localized responses and the ranges over which
they occur are in direct relationship with the level of mistuning, i.e., they increase as
the level of mistuning is increased. It should be mentioned here that these localized
responses coexist with the stable unison response over a wide operating range ( see

chapter 3 for more details).

5.3 Conclusions

The following conclusions can be drawn from the findings of this chapter:

0 In the presence of small imperfections between the perfectly tuned absorber
paths, linear localization does exist in the forced response of CPVA systems.
The relative strengths of the localized responses depend on both the ratio of
the level of damping to the level of imperfections and the variations of imper-
fections among the paths. Systems with lower damping to imperfection ratio
are expected to have stronger localized responses. Also, systems with higher

differences in imperfections are expected to have stronger localized responses.

0 An effective solution to avoid linear localization in CPVA systems is to introduce
a small, but large when compared to imperfections, intentional mistuning in the

absorber paths. This renders CPVA systems robust against localization.

93

e For positively mistuned circular paths, non—linear localized responses exist. The
strength of these responses and the applied torque ranges over which they exist
and are stable depend on the level of mistuning. As the level of mistuning is
increased, the localized responses become stronger and occur over more ranges

of the applied torque.

e For positively mistuned paths ranging from circular to but not including epicy-
cloidal, i.e., paths with 0 S A < «fl, there is no way to avoid the possibility
of encountering nonlinear localization. However, with paths ranging from but
not including epicycloidal to cycloidal, i.e., paths with :33- < A S 1, localized

responses do not exist.

94

CHAPTER 6

Summary, Conclusion, and

Future Work

6.1 Summary

In this dissertation some important extensions of previous research efforts in the
field of mechanical vibration reduction have been investigated. These extensions
focussed on some important aspects of systems of multiple vibration absorbers.
While the main focus was on CPVA systems, the main results are applicable to
certain translational absorber systems as well. To achieve the objectives of this work,
three main tasks were performed. The first task was to investigate the existence
and stability of the desired response in which multiple vibration absorbers move in
a synchronous manner, and the effects that linear mistuning and nonlinearities in
the absorbers have on the absorbers’ performance. The second task was to explore
the existence and stabilities of other types (specifically, non-unison) of responses.
The third task was to investigate systems of multiple vibration absorbers for the
possibility of the existence of the phenomenon of localization. Linear free vibration
modes, linear forced vibration responses, and non-linear forced responses of vibration

absorbers were considered for localization. What follows summarizes the findings of

95

this dissertation.

Performance and Stability of Unison Response of Multiple
CPVAs Riding on General Paths

It was found that there are two types of instabilities one should take into consideration
when designing CPVA systems. The first is one where the absorbers loose stability
of their synchronous motion, and the second corresponds to jumps in the absorbers’
response curves. Both of these instabilities were found to exist in all path types
ranging from the commonly used circular paths up to, but not including, epicycloidal
paths, i.e., paths with 0 S A < fig. For proper absorber performance, both of
these instabilities should be avoided. It was determined that an effective method for
increasing the operating range, by delaying these instabilities, is to introduce some
intentional positive mistuning on the absorbers’ paths. However, the introduction of
this mistuning reduces the overall performance of CPVA systems in terms of vibration
reduction. This means that there are trade-offs between absorber performance and
wide operating ranges. The above instabilities were not found to exist for paths
ranging from, but not including, epicycloidal up to cycloidal paths, i.e., paths with
«37:: < A S 1. This suggests that cycloidal paths are more robust and most suitable
for practical implementations. Although epicycloidal paths were not considered in
this work, they are not the best choice. This is true because it has been shown by
Chan et al. [20] that the first instability type mentioned above, where the absorbers
loose their synchronous motion, does exist for this absorber path, and this has been

observed in numerical simulations.

96

Existence and Stability of Responses Other Than Unison in
Systems Of CPVAs Riding on General Paths

Here it was found that certain types of stable, non-unison responses exist for posi-
tively mistuned path types with 0 S A < «£1. These responses result from the
bifurcations to non-unison that take place at the points where unison responses first
become unstable. It should be mentioned here that these solutions can coexist with
the stable unison response over a wide range of operating conditions leading to an
undesirable situation in terms of absorber robustness. With paths ranging from, but
not including, epicycloidal up to cycloidal, i.e., paths with :5”? < A S 1, it was
found that such responses do not exist. This is another and important advantage

that cycloidal paths have over other paths.

Localization in Vibration Absorber Systems

Modes of Linear Free Vibration

It was found that as long as there exist some imperfections between the weakly coupled
vibration absorbers, then the free vibration modes will indeed localize. The strength
of this localization depends on the ratio of the coupling between the absorbers to the
relative imperfections among their paths. When this ratio is small, the localization is
strong and it becomes weaker as this ratio is increased. The parameter that measures
the strength of the coupling between the absorbers is the ratio of absorbers’ inertia
to the main inertia, which is always small. As a result, localization of free vibration

modes is almost always expected to exist in these systems of absorbers.

Forced Response Localization

Linear forced response localization

It was found that linear forced localized responses exist in systems of CPVAs that ride

97

on perfectly tuned paths with some small imperfections. The strengths of these 10-
calized responses depend on the variations of the imperfections among the absorbers’
tunings and on the ratio of these imperfections to the absorbers’ damping level. As
the variations in the imperfections and the ratio of these imperfections to the damp-
ing level gets larger, the localized response becomes stronger. When slight positive
intentional mistuning is introduced in the absorbers’ paths, the strength of the lin-
ear localized responses will be highly weakened. Thus, slight over-tuning is a good
solution for avoiding this type of localized responses in these systems of absorbers.

However, again this comes at the price of reduced absorber performance.

Nonlinear forced response localization

Nonlinear localized responses in systems of identical CPVAs were investigated for ex-
istence and stability. This is a special case of the work summarized above where
non-unison solutions were investigated. The same conditions of existence apply
here, i.e., they exist only for systems of absorbers that ride on positively mistuned
paths ranging from circular up to, but not including, epicycloidal, i.e., paths with
0 S A < J%. The effects of intentional mistuning on these localized responses
were also studied. It was found that as the level of mistuning is increased, the local-

ized responses become stronger and occur over larger ranges of applied torques.

Note on linear verses nonlinear forced localization

For all path types considered in this work, i.e., paths ranging from circular up to, but
not including, epicycloidal and from, but including, epicycloidal up to cycloidal, the
linear and the nonlinear localized responses are distinct in nature. This is because, as
mentioned above, nonlinear localization exists only for systems of identical absorber
paths when a bifurcation to non-unison exists, i.e., paths ranging from circular up
to, but not including, epicycloidal, with some positive level of mistuning. For these

systems, linear localization is not expected to occur because of the absence of relative

98

mistunings among the paths. Even in the presence of some small relative imperfec-
tions among the paths, linear localization is not expected to occur as well. This is
due to the presence of the intentional over-tuning.

The only path type where linear and nonlinear localization can be linked is the
perfectly tuned epicycloid. This is because it has been shown by Chao at al. [20]
that for this path type, a bifurcation to non-unison does exist, to a stable, nonlinear
localized response. The present work shows that in the presence of some small im-
perfections in zero mistuned paths, linear localization exists irrespective of the path
type. As mentioned in chapter 2, due to the scaling used, the mathematical model of
this work fails to capture the nonlinear effects for the case of epicycloidal paths. So,
in order to link the linear and the nonlinear localization for this path type, a different

scaling should be employed.

6.2 Conclusion

Based on the findings of this work, it can be concluded that the most suitable path
type for CPVA systems is a slightly over-tuned cycloidal path. This is because, in
contrast with other common paths, i.e., circular and epicycloidal, absorbers riding on
cycloids do not undergo any kind of instability. Furthermore, they are robust against
linear and nonlinear localization in their steady-state response. These facts make this

path type the best choice for practical implementation.

6.3 Future Work

The following are some directions suggested for future work:

0 The effect that absorber path mistunings have on the absorbers’ stabilities and

performances for the case of epicycloidal paths should be investigated using a

99

different scaling than what has been used herein. One can start from the work

done by Chao et al. [20] to achieve this goal.

0 Although the the method presented here allowed for drawing very useful and im-
portant conclusions for the paths ranging from, but not including, epicycloidal
up to cycloidal, the accuracy was not as good for these paths as it was for the
case of circular and nearly circular paths. This is due to strength of the nonlin-
earity and the scaling. A method that allows one to capture non-linear effects
other than those that come from the path will, off course, be more accurate. It
is believed that this could be done by employing a technique that allows one to
formulate the problem without scaling the absorbers’ amplitudes. An example
of such a technique is to formulate the problem as done in Chao et al. [20] and

analyze it using action angle coordinates.

e A more general study of CPVA systems should account for the multi-harmonic
nature of the applied torque, which is a much better model for IC engines and
other rotating machinery. This is because the torque acting on the crankshaft
is periodic, but not harmonic, in the crankshaft rotation angle, although it can

be well approximated by its first several harmonics.

0 It would be of interest to investigate the possible connection between linear and
nonlinear localization. It would require that one find a method that can describe

the entire range of localized behaviors and use it to obtain general results.

0 The effect of rotor flexibility on CPVA systems is another subject for research.
Although some preliminary work showed that this effect is of higher order, i.e., it

is not relatively important, it deserves further consideration.

0 A very important direction of future work in this field is the experimental veri-

fication of the findings of the present work and all previous related theoretical

100

works. This is of great importance because in all theoretical developments the
system dynamics is idealized in several aspects to obtain analytical estimates of
system behavior. A very important example of such idealization is the damping,

which is taken to be small and of viscous type. Work on this topic is underway.

101

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102

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