ANALYSIS AND DESIGN OF MULTIPLE ORDER CENTRIFUGAL PENDULUM
VIBRATION ABSORBERS
By
Brendan James Vidmar

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mechanical Engineering
2012

ABSTRACT
ANALYSIS AND DESIGN OF MULTIPLE ORDER CENTRIFUGAL
PENDULUM VIBRATION ABSORBERS
By
Brendan James Vidmar
Impending government fuel economy standards are causing automobile manufacturers to
consider innovative ways to increase the average fuel economy of their fleets. In many cases,
as discussed in this study, automakers are required to push their engine operating conditions
to levels not previously considered. One constraint which limits the engines to operate
efficiently and salable in these conditions is torsional vibrations. Centrifugal Pendulum
Vibrations Absorbers (CPVAs) are devices used to reduce the levels of these undesirable
torsional vibrations without decreasing performance.
In this study, we consider nonlinear interactions in systems of order-tuned torsional vibration absorbers with sets of absorbers tuned to different orders. In all current applications,
absorber systems are designed to reduce torsional vibrations at a single order; however, when
two or more excitation orders are present and absorbers are introduced to address different
orders, nonlinear interactions become possible under certain resonance conditions. Under
these conditions, a common example of this phenomenon occurs for orders n and 2n where
crosstalk between the absorbers, acting through the rotor inertia, can result in instabilities that are detrimental to system response. In order to design absorber systems that avoid
these interactions and to explore possible improved performance with sets of absorbers tuned
to different orders, we develop predictive models that allow one to examine the effects of
absorber mass distribution and tuning. These models are based on perturbation methods
applied to the system equations of motion ultimately yielding system response features as

a function of parameters of interest, notably absorber and rotor response amplitudes and
stability. The model-based analytical results are compared with numerical simulations of the
complete nonlinear equations of motion and are shown to be in good agreement. In addition,
experimental absorbers at multiple orders were designed and tested on a controlled spin rig.
The experimental data is found to be in good agreement with the analytical predictions,
thus verifying the numerical and analytical studies. These results are useful for the selection
of absorber parameters to achieve desired performance. For example, they allow for approximate closed form expressions for the ratio of absorber masses at the two orders that yield
optimal performance. It is also found that utilizing multiple order absorber systems can be
beneficial for system stability, even when only forcing at one order.
In a related study, we develop relatively simple predictive formulations describing the
absorber and rotor dynamics. We do this by assuming specific forms about the absorber
response in order to simplify the steady state analysis of absorber systems. Utilizing these
assumptions, along with physically relevant scalings, the harmonics in the system can be
balanced and thus yield these predictive closed form expressions. Although unable to capture
all of the subtle system instabilities, these expressions are found to accurately capture both
the steady-state absorber response as well as the harmonically rich rotor response for a wide
range of absorber configurations, as confirmed by both numerical and experimental data.
This analysis provides accurate and simple descriptions of the system dynamics. These
descriptions are useful for selection of important system parameters when designing such
systems and are generalized to account for single or multiple order absorber systems.

ACKNOWLEDGMENTS

The five and a half years I have spent as a graduate student at MSU has been both a
gratifying, and at times, trying experience. I would not have been to able to get through
it without the support of my advisors, Steve Shaw and Brian Feeny. Professor Shaw is not
only a master of dynamical systems, but someone whose advice I would heed in all aspects
of life. Professor Feeny is one of the kindest men you will ever meet and his expertise was
invaluable. I would also like to thank Professors Xiaobo Tan and George Zhu for serving on
my committee.
Thanks goes Nick Miller, Scott Strachen, Pavel Polunin, Thomas Theisen and Ryan
Monroe whose fruitful discussions helped form many of the ideas in this study. A special
thanks must also be handed out to Abhisek Jain, who spent many a long night with me
conducting experiments in the lab. Professor Alan Haddow was instrumental in setting up
the experimental device, and continued to offer advice throughout the course of this study.
The undergraduate assistants who Professor Shaw continually was able to pull out of his
talent pool helped me in numerous ways, and if these students are representative of the
aspiring engineers in this country, I am excited for the future. These students were Jelena
Paripovic, Brian Rockwell, J. T. Whitman, Bara Aldasouqi, and Cody Little.
Bruce Geist of Chrysler LLC collaborated on much of this work, is a member of my
committee, and our weekly meetings allowed for an insight in the practical applications of
torsional vibration absorbers. I must also thank Chrysler LLC for funding throughout my
time here, and Kiranmaye Aluru for helping machine the experimental absorbers.
I would also like to thank Jeff Chottiner and John Brevick at Ford Motor Company
for allowing me some first-hand experience in automotive design and testing of torsional
iv

vibration absorbers as well as monetary support.
A final thanks goes out to the National Science Foundation (Grant Number CMMI1100260) who’s support funded much of the work.

v

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1 Introduction . . . . . . . . .
1.0.1 Motivation . . . . . . . . . . .
1.0.2 General Operation of CPVAs
1.0.3 Background . . . . . . . . . .
1.0.4 Dissertation Organization . .

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Chapter 3 Experimental Validation . . . . . . . . . . . . . . . . .
3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 System Parameter Values . . . . . . . . . . . . . . . . . . . . .
3.2.1 Rotor Inertia . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Absorber Tunings . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Absorber Dampings . . . . . . . . . . . . . . . . . . . . .
3.2.4 Rotor Damping . . . . . . . . . . . . . . . . . . . . . . .
3.3 Steady-State Experimental Results . . . . . . . . . . . . . . . .
3.3.1 Synchronous Response . . . . . . . . . . . . . . . . . . .
3.3.2 Non-Synchronous Response: Order n1 Absorbers Only .
˜
3.3.3 Non-Synchronous Response: Order n1 and n2 Absorbers
˜
˜
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4 A Harmonic Balance Approach to the Dynamics of
4.1 General Harmonic Balance . . . . . . . . . . . . . . . . . . . . .
4.2 Harmonic Balance Using a Power Series in Torque Amplitude .
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CPVAs
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81

Chapter 2 Modeling and Prediction of System
2.1 Equations of Motion . . . . . . . . . . . . . .
2.1.1 Absorber Paths . . . . . . . . . . . . .
2.2 Motivating Simulation Results . . . . . . . . .
2.3 Perturbation Analysis . . . . . . . . . . . . .
2.4 Synchronous Responses . . . . . . . . . . . . .
2.5 Non-Synchronous Responses . . . . . . . . . .
2.5.1 Minimizing Torsional Vibration . . . .
2.6 Non-Tautochronic Paths . . . . . . . . . . . .

vi

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4.3.1
4.3.2
4.3.3
4.4

4.5

Absorbers and Forcing at Two Orders . . . . . .
Multiple Absorbers with Single Order Forcing . .
Single Order Absorber with Single Order Forcing
4.3.3.1 Non-Tautochronic Paths . . . . . . . . .
Experimental Testing . . . . . . . . . . . . . . . . . . . .
4.4.1 Tautochronic Absorbers at Two Orders . . . . . .
4.4.2 Single Order Tautochronic Path Absorber . . . .
4.4.3 Single Order Circular Path Absorber . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5 Conclusions and Recommendations for
5.1 Summary of Results . . . . . . . . . . . . . . . .
5.2 Importance of Results . . . . . . . . . . . . . . .
5.3 Future Work . . . . . . . . . . . . . . . . . . . . .

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Future Work
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Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.1 Experimental Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.2 Harmonic Balance Higher Order Approximations . . . . . . . . . . . . . . . 139
Bibliography . . . . . . . . . . . . . . .

vii

. . . . . . . . . . . . . . . . . . . 145

LIST OF TABLES

Table 2.1

Definition of non-dimensional variables in Eqns. (2.4-2.5). . . . . . .

18

Table 3.1

Values of Experimental Parameters. . . . . . . . . . . . . . . . . . .

59

Table 4.1

Physical Multiple Order Epicycloidal Path Experimental Parameters.

98

Table 4.2

Circular Path Absorber Physical System Parameters. . . . . . . . . 111

viii

LIST OF FIGURES

Figure 1.1

(a) Bifilar Absorbers on a Helicopter Rotor. (http://www.b-domke.de
/AviationImages/Rotorhead/11358.html). (b) MSU Experimental
Bifilar Absorber Showing the Two-Point Suspension. For interpretation of the references to color in this and all other figures, the reader
is referred to the electronic version of this dissertation. . . . . . . .

2

Figure 1.2

Circular Path CPVA on Automotive Crankshaft. . . . . . . . . . . .

3

Figure 1.3

Schematic of a rigid rotor fitted with absorbers tuned to n and 2˜ . .
˜
n

6

Figure 2.1

Schematic of a rigid rotor fitted with absorbers tuned to n and 2˜ . .
˜
n

14

Figure 2.2

(a) Torques generated from individual cylinders, calculated by converting a characteristic gas pressure force acting on a piston to crank
torques. (b) Summed torque, which is dominated by a DC term and
second order harmonic. Higher order harmonics are also evident. . .

15

Absorbers paths which produce a linear tuning of n = 1.5, for three
˜
different nonlinear path parameters, λ = 0, λte , 1. . . . . . . . . . .

20

Simulated responses of a system composed of four tautochronic absorbers, two each at orders 1.5 (sn ) and order 3 (s2n ), subjected
to a torque composed of a linear combination of orders 1.5 and 3.
Note the different scales used for depicting the responses. (a) Torque
harmonic amplitude space, Γ3 vs. Γ1.5 , with boundaries of the
desired synchronous response indicated schematically; solid line depicts a subharmonic instability boundary, dotted (dashed) line depicts a symmetry-breaking instability boundary to nonsynchronous
responses of the order 1.5 (3) absorbers. (b) Mutually synchronous
response at point W. . . . . . . . . . . . . . . . . . . . . . . . . . .

23

(c) Response at point X; the order 1.5 absorbers have become nonsynchronous and are in a transition towards their amplitude limits.
(d) Response at point Y; similar to W, only with reversed relative
amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Figure 2.3

Figure 2.4

Figure 2.5

ix

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 3.1

(e) Response at point Z; the order 1.5 absorbers have become subharmonic and are in transition toward their amplitude limits, as shown
in (f). (f) Response at point Z, depicted over a long time interval;
the order n absorber instability eventually results in large amplitude
motion; the cusp amplitudes are the maximum possible amplitudes
for these absorbers. . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Predicted and simulated steady-state absorber amplitudes as a function of the order n torque amplitude; the order 2n torque amplitude
is zero. The cusp values are amplitude limits set by hardware constraints. System parameters: m3 /m1.5 = 0.1, = 0.05. . . . . . . .

33

Order n and 2n harmonic components of the rotor angular acceleration as a function of the order n applied fluctuating torque amplitude,
for the same conditions as Fig. 2.7. The solid line is the reference
response for the absorbers locked. . . . . . . . . . . . . . . . . . . .

34

Absorber amplitudes as a function of order n fluctuating torque amplitude; the order 2n torque amplitude is zero. System parameters:
m3 /m1.5 = 0.62, = 0.05. The absorbers reach their respective
cusps at approximately the same level of excitation. . . . . . . . . .

35

Order n = 1.5 absorber synchronous response stability boundary as
a function of the driving torques, for φ = 0. Torques outside the
range of this curve result in nonsynchronous responses of the order n
absorbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Order n applied torque versus the mass ratio of the different order
absorbers, when assuming a synchronous response. . . . . . . . . . .

41

Order n applied torque versus the mass ratio of the different order
absorbers. As evident, using a small amount of order 2n absorbers
can stabilize the order n absorbers. . . . . . . . . . . . . . . . . . .

42

Normalized rotor center to absorber COM length as a function of absorber arc length for different numbers of truncations of the absorber
path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Percent error between the perturbation equations and numerical simulations for a single order cycloidal path absorber system vs. absorber
tuning order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Schematic of the experimental setup. . . . . . . . . . . . . . . . . .

48

x

Figure 3.2

Photograph of the experimental setup. (a) The entire setup, showing
the motor, rotor, and corresponding motor control and data acquisition components. (b) Multiple order experimental absorbers. (c) Self
centering rollers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Geometrical schematic of the measured absorber angle via an optical
encoder with a rotating arm of length, L (λte provides a tautochronic
epicycloid and λ1 yields a cycloid. . . . . . . . . . . . . . . . . . . .

50

Calculated non-dimensional absorber arc length, s, versus measured
encoder angle, φ, for epicyloidal and circular path formulations. . . .

51

Amplitude of order n rotor angular acceleration versus the amplitude
of applied order n torque, with absorbers locked. . . . . . . . . . . .

54

Order n amplitude of rotor angular acceleration divided by the amplitude of fluctuating torque as a function of the applied torque order,
n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Free vibration response of both order absorbers. (a) Order n1 ab˜
sorber. (b) Order n2 absorber. . . . . . . . . . . . . . . . . . . . . .
˜

57

Figure 3.8

Rotor mean speed, Ω vs. the mean torque applied, T0 . . . . . . . . .

58

Figure 3.9

Experimental absorber amplitudes, s, vs. the amplitude of applied
order n torque, Γn , with perturbation and numerical simulation predictions. (n = 2.29, = 0.18486) . . . . . . . . . . . . . . . . . . . .

60

Experimental rotor angular acceleration, yy vs. Γn for multiple order epicycloidal absorbers compared against the perturbation and
numerical predictions (n = 2.29, = 0.18486). (a) First three rotor
response orders, n, 2n, 3n. (b) Order n. . . . . . . . . . . . . . . .

66

Figure 3.11

(c) Order 2n. (d) Order 3n. . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 3.12

Experimental stability results for a pair of order n1 absorbers. (a)
˜
Theoretical stability boundary with a sample of stable and unstable
experimental data points. (b) Experimental time response of the
non-synchronous absorber response corresponding to the data point
labeled (1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.10

xi

Figure 3.13

Figure 3.14

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

(c) Experimental time response of the sub-harmonic absorber response corresponding to the data point labeled (4) in Fig. 3.12. (d)
Short time sample of the sub-harmonic instability, showing the out
of phase absorber motion. . . . . . . . . . . . . . . . . . . . . . . . .

69

Experimental absorber amplitudes, s, vs. the amplitude of applied
order n torque, Γn , with perturbation and numerical simulation predictions. (n = 2.29, = 0.18486) . . . . . . . . . . . . . . . . . . . .

70

Tautochronic epicycloid absorber amplitude, |s|, versus amplitude of
fluctuating torque, Γn for different numbers of truncation for the
power series HB method. . . . . . . . . . . . . . . . . . . . . . . . .

82

Absorber amplitudes, |s1,2 |, vs. amplitude of applied order torques,
Γn,2n , from the different harmonic balance (HB) approximations,
with numerical simulations, for ϕ = 0 (tautochronic epicycloid). . . .

83

Rotor angular acceleration, yy , harmonic amplitude approximations
using the amplitude expansions, vs. the amplitude of applied order
torques, Γn,2n . (a) Full torque range showing approximations out to
O(Γ3 ). (b) The zoomed in region near the origin. . . . . . . . . . .
n

84

Harmonic amplitudes of the rotor angular acceleration, yy , predicted
using the general HB method vs. the amplitude of applied order
torques, Γn,2n . The absorber amplitudes A1 and A2 are assumed to
satisfy Eq. (4.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Absorber amplitude, |s|, vs. amplitude of applied order n torque,
Γn , from different order harmonic balance (HB) approximations, with
numerical simulations, for ϕ = 0 (tautochronic epicycloid). . . . . .

88

Rotor angular acceleration amplitude approximations using the amplitude expansions, yy , vs. the amplitude of applied order n torque,
Γn . (a) Full torque range showing approximations out to O(Γ3 ). (b)
n
The zoomed in region near the origin. . . . . . . . . . . . . . . . . .

89

Harmonic amplitudes of rotor angular acceleration |yy | vs. Γn , including approximations from the general HB method and simulations,
for ϕ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

xii

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Absorber amplitude, |s|/|s|max , vs. Γn , from the linear theory, the
two nonlinear harmonic balance (HB) approximations, and numerical
simulations, for ϕ = 0 and α = 0. . . . . . . . . . . . . . . . . . . .

91

Approximations of the harmonic amplitudes of yy obtained using the
amplitude expansions, vs. Γn , for ϕ = 0 and α = 0; comparison with
simulation results are also included. . . . . . . . . . . . . . . . . . .

92

Approximations of the harmonic amplitudes of yy obtained using the
general HB method, vs. Γn , for ϕ = 0 and α = 0; comparison with
simulation results are also included. . . . . . . . . . . . . . . . . . .

92

Absorber amplitude, |s|, vs. Γn , from the two HB approximations,
along with numerical simulations, for a perfectly tuned circular path
absorber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Approximations of the harmonic amplitudes of the rotor angular acceleration obtained using the series expansion HB method, vs. Γn ,
along with results from numerical simulations, for a perfectly tuned
circular path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Figure 4.13

Approximations of the harmonic amplitudes of the rotor angular acceleration obtained using the general HB method, vs. Γn , along with
results from numerical simulations, for a perfectly tuned circular path. 95

Figure 4.14

Absorber amplitude, |s|, vs. Γn , obtained using different order HB
approximations, along with results from numerical simulations, for a
detuned circular path. . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Approximations of the harmonic amplitudes of the rotor angular acceleration obtained using the series expansion HB method, vs. Γn ,
along with results from numerical simulations, for a detuned circular
path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Approximations of the harmonic amplitudes of the rotor angular acceleration obtained using the general HB method, vs. Γn , along with
results from numerical simulations, for a detuned circular path. . . .

97

Figure 4.15

Figure 4.16

Figure 4.17

Experimental absorber amplitudes, s, versus the amplitude of order
n torque, Γn compared against the harmonic balance approximations
for multiple order absorbers. (a) Series expansion HB method. (b)
General HB method. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

xiii

Figure 4.18

Experimental rotor angular acceleration, yy vs. Γn for multiple order
epicycloidal absorbers compared against the power series HB approximation. (a) All three rotor response orders, n, 2n, 3n. (b) Order
n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 4.19

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.18 . . . . . . 102

Figure 4.20

Experimental rotor angular acceleration, yy vs. Γn for multiple order
epicycloidal absorbers using the general HB approximation. (a) All
three rotor response orders. (b) Order n only. . . . . . . . . . . . . . 103

Figure 4.21

(c) Order 2n only. (d) Order 3n only. Coorsponding with Fig. 4.20 . 104

Figure 4.22

Experimental absorber amplitude, s, versus the amplitude of order n
torque, Γn compared against the harmonic balance approximations
for a single order tautochronic absorber. . . . . . . . . . . . . . . . . 105

Figure 4.23

Experimental rotor angular acceleration, yy vs. Γn for a single order
epicycloidal absorber compared with the power series HB approximation. (a) All three rotor response orders, n, 2n, 3n. (b) Order n. . 107

Figure 4.24

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.23 . . . . . . 108

Figure 4.25

Experimental rotor angular acceleration, yy vs. Γn for a single order
epicycloidal absorber compared with the general HB approximation.
(a) All three rotor response orders, n, 2n, 3n. (b) Order n. . . . . 109

Figure 4.26

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.25 . . . . . . 110

Figure 4.27

Picture of the experimental circular path pendulum “T-shaped” absorber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 4.28

Absorber arc length harmonic amplitude, |s| vs. Γn for a circular
path absorber driven exactly at tuning; experimental results compared with both harmonic balance approximations and simulations.

113

Figure 4.29

Harmonics of the rotor angular acceleration, yy vs. Γn , for a circular path absorber driven exactly at tuning; experimental results
compared against the power series harmonic balance approximation.
(a) All three rotor response orders n, 2n, 3n. (b) Order n. . . . . . 115

Figure 4.30

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.29 . . . . . . 116
xiv

Figure 4.31

Harmonics of the rotor angular acceleration, yy vs. Γn , for a circular path absorber driven exactly at tuning; experimental results
compared against the general harmonic balance approximation. (a)
All three rotor response orders n, 2n, 3n. (b) Order n. . . . . . . . 117

Figure 4.32

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.31 . . . . . . 118

Figure 4.33

Amplitude of the absorber arc length, s vs. Γn for a 5% overturned
circular path absorber with = 0.0519; experimental results compared with both harmonic balance approximations and simulations.

119

Figure 4.34

Harmonics of the rotor angular acceleration, yy vs. Γn , for a 5%
overturned circular path absorber, with = 0.0519; experimental results compared against the general harmonic balance approximation.
(a) All three rotor response orders n, 2n, 3n. (b) Order n. . . . . . 120

Figure 4.35

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.34 . . . . . . 121

Figure 4.36

Amplitude of the absorber arc length, s vs. Γn for a 5% overturned
circular path absorber with = 0.0189; experimental results compared with both harmonic balance approximations and simulations.

122

Figure 4.37

Harmonics of the rotor angular acceleration, yy vs. Γn , for a 5%
overturned circular path absorber, with = 0.0189; experimental results compared against the general harmonic balance approximation.
(a) All three rotor response orders n, 2n, 3n. (b) Order n. . . . . . 123

Figure 4.38

(c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.37 . . . . . . 124

xv

Chapter 1
Introduction
Vibration reduction in rotating machinery has long been an area of study in both industrial
and academic settings [11, 20, 21, 35]. An inherent difficulty in solving issues relating to torsional vibrations is, in many applications, the rotating inertia and associated damping are
desired to be as small as possible for maximum performance. Both of these properties, although necessary for good performance, can lead to troublesome torsional vibrations. Several
well-researched methods have been employed to attenuate these torsional vibrations, which
include adding inertia in the form of flywheels, utilizing energy dissipation through torsional
dampers, employing frequency tuned dual mass flywheels, and the use of order tuned Centrifugal Pendulum Vibration Absorbers (CPVAs). Due to the stringent efficiency demands
now present in some rotating machinery, such as internal combustion engines, adding inertia
and dissipating energy are not feasible. Also, as dual mass flywheels are tuned to a certain
frequency, that is an engine order excitation at a specific rotational velocity, their performance is limited to very specific conditions. CPVAs dissipate very little energy, are tuned to
an order, and, as will be explained in more detail further, can be very beneficial in reducing
the amplitude of torsional vibrations.
CPVAS are passive devices that have been shown to significantly reduce torsional vibrations in rotating machinery that arise from engine order excitation [12, 21]. Such machinery
includes internal combustion engines, helicopter rotors, turbines, and rotary aircraft engines. For several decades, CPVAs have been used in light aircraft engines and helicopter
1

rotors [21, 25] for vibration suppression. Recently, CPVAs have been shown to be very
beneficial for use in automotive engines [6, 33, 42]. As examples of CPVAs, Fig. 1.1 shows
a set of four bifilar (two-point suspension) CPVAs attached to a helicopter rotor, and an
experimental bifilar absorber used at MSU.

(a)

(b)
Figure 1.1: (a) Bifilar Absorbers on a Helicopter Rotor. (http://www.b-domke.de /AviationImages/Rotorhead/11358.html). (b) MSU Experimental Bifilar Absorber Showing the
Two-Point Suspension. For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation.

Previous use of CPVAs in aerospace engines has been to reduce torsional vibrations of a
rotor operating at a nearly constant angular velocity, in situations for which the most extreme
speed and load conditions are known. These conditions allow the inertia of the absorbers
to have sufficient inertia and be tuned so as to avoid the undesirable nonlinear behavior.
These non-linear dynamics include instabilities such as an amplifying “jump” bifurcation for
circular path absorbers, as well as non-synchronous motions for other paths that arise due
to non-linear absorber coupling through the rotor, each of which are detrimental to absorber
performance [3, 9, 39]. Recent research has been conducted into understanding the response
2

and assessing the effectiveness of CPVAs for use in high efficiency automotive engines, where
the mean speed of the rotor varies greatly throughout the range of operation [33,41], and for
which the inertia of the entire crank shaft is desired to be as small as possible for maximum
responsiveness. It should be noted that the rotary inertia of an engine powertrain has always
been desired to be small, but new demands have coupled this minimization of powertrain
inertia with very large engine torsional fluctuations. A crankshaft used in an experimental
Ford V8 automotive engine fitted with CPVAs is shown in Fig. 1.2 [32].

Figure 1.2: Circular Path CPVA on Automotive Crankshaft.

1.0.1

Motivation

With automotive manufacturers all battling to design more fuel efficient engines, vibration
suppression surprisingly plays an important role in maximizing this efficiency. The Corporate Average Fuel Economy (CAFE) standards state that passenger cars must meet a 30.5
mpg minimum and light trucks a 24.1 mpg minimum by this year. Also, by 2025, automotive manufacturers will be required to meet a combined fleet average of 54.5 mpg for
cars and light trucks, as imposed by the passing of the Energy and Independence Security
Act (EISA). With these impending government restrictions on fuel economy, car companies
3

are implementing many new engine technologies, two of which, as subsequently discussed,
require some sort of vibration attenuation to be as efficient and salable as possible. The first
engine technology is the so-called “displacement on demand” large displacement engines,
first used by General Motors in 1981 Cadillacs, modified by Mercedes-Benz in 1999, and also
employed by Chrysler and Honda since 2005 [1]. These internal combustion engines vary
the number of cylinders firing as engine speeds and loads change. The variable displacement
engines have been found to achieve a fuel economy saving of 10 − 20%, mainly through the
reduction of pumping losses, which occur when the engine is in its reduced cylinder mode.
Using current technology however, these multi-displacement system engines cannot operate
in the more efficient reduced cylinder mode for speeds below 1200 rpm, due to noise and
vibration issues. Resolving these low speed vibratory problems would result in even more
fuel economy improvement. The second type of new, more efficient engine technology is very
small displacement, usually turbo-charged, high torque engines, for example the Ford “Fox
I-3” engine . Running these engines at low speeds and high torques, that is, in the so-called
“lugging” mode, is very efficient, but results in very high levels of torsional vibrations. These
vibrations, which are felt by the operator and passengers, currently make these engines unsuitable for the marketplace. CPVAs have recently come to the forefront in the automotive
industry to reduce these torsional vibrations.
In most applications there exists a dominant source of engine order vibration, and absorbers are tuned to that order. In fact, all previous studies known to the author have
investigated the use of CPVAs tuned to attenuate torsional vibrations at a single order.
However, it is known that when absorbers are pushed to large amplitudes, where nonlinear
effects come into play, instabilities can occur, and the absorbers generate higher order torsional vibrations on the rotor. In this work we extend upon these studies and analyze the use
4

of multiple-order CPVAs in order to reduce vibrations at multiple engine orders. The goal
is to develop a fundamental understanding of the interplay between absorbers at different
orders, and use this knowledge to develop design strategies for systems of absorbers with
groups of absorbers tuned to different orders.

1.0.2

General Operation of CPVAs

CPVAs are masses suspended from a rotor in such a way that they are free to move along a
desired path, similar to that of a pendulum, and a general schematic of a rotor/CPVA system
is shown in Fig. 1.3. CPVAs have several inherent properties which make them ideal for use
in rotating machinery. First, they have a natural frequency that is proportional to the mean
rotational speed of the rotor, thus corresponding to a given order of rotation. This follows
since the effective stiffness on the absorbers is due to centrifugal effects, and this property
makes them effective over a continuous range of rotor speeds. Referring to the dimensions in
Figure 1.3, one can find that when the rotor spins at a constant rate Ω, the natural frequency
of the absorber is Ω

R0i /ρ0 − 1, where ρ0 is the radius of curvature of the absorber path

at the path’s vertex. The natural frequency of the absorber will be subsequently referred to
as nΩ where n is referred to as the absorber tuning order.
˜
˜
When tuned to the correct value of n, the absorbers oscillate in a manner that counteracts
˜
the corresponding order component of the fluctuating applied torque acting on the rotor.
The engine excitation orders, n, 2n, ... are inherent to a given machine. For example, in a
four stroke internal combustion engine, each cylinder fires once every two revolutions of the
crankshaft. This implies for an N cylinder engine, the dominant engine order is at N/2 = n,
but orders 2n, 3n and higher are also generally present, although often at a much smaller
amplitude. In fact, if there is significant cylinder-to-cylinder variation, one-half orders, such
5

Order n
˜
S1

mn

Order 2˜
n

R01 R1 (S1 )
S2
m2n

R02
R2 (S2 )
JR

T0 + Tn sin(nθ) + T2n sin(2nθ + φ) Rotor
Figure 1.3: Schematic of a rigid rotor fitted with absorbers tuned to n and 2˜ .
˜
n
as n/2, 3n/2, etc. can also be present. With this knowledge, we can model the forcing from a
four stroke internal combustion as a Fourier series in θ with frequency (order) n, for example,
with a leading order term of T sin(nθ), where the forcing order, n is equal to N/2, this is the
so called order n excitation. For some multiple displacement engines, as previously described,
the fluctuating torques applied when the engine is in “full” and “reduced’ cylinder mode can
both cause undesirable vibration effects. This results in vibration amplitudes at two different
orders which need to be attenuated. Also, in the high compression, small displacement
engines being developed, higher order harmonics can be of substantial magnitude. For
example, the oscillating torque for an engine which produces a torque in which the first two
harmonics are large can be modeled as T1 sin(nθ) + T2 sin(2nθ + φ). In general, the torque
can be expressed as a Fourier series in terms of the base order n. Previous investigations
have analyzed the performance of CPVAs tuned to a single order, when subjected to a single
order torque. This work extends upon those previous investigations by both predicting the
response of a system containing a rotor equipped with multiple order CPVAs subjected to
multiple order torques, as well as by developing methods to design such systems for optimal
6

performance.
As mentioned above, it is desired keep the total rotational inertia of the rotor/CPVA
system at a minimum to maintain good responsivity of the engine, that is, how quickly it
reacts to shifts in torque, for example, during vehicle launch. This requires using as small
an absorber mass m, and absorber distance from the center of rotation R0i , as possible.
The absorber works by using a combination of its mass, the distance from the center of
rotation, and the absorber amplitude S, to counteract the applied torque, and thus a minimization of the first two quantities results in a large absorber amplitude. This implies that
an understanding of the large amplitude, nonlinear dynamic behavior of the entire system is
essential for an effective design. It turns out that the path that the absorber center of mass
follows Ri (Si ), is very important to large amplitude behavior of the system [14, 39]. Also,
if more than one absorber is employed, nonlinear inter-absorber interactions can occur due
to kinematic coupling through the rotor. It will be shown that it is essential to consider the
effects of this inter-absorber coupling when designing multiple order absorber systems for
large amplitude behavior. In fact, if one has sufficient absorber inertia to keep amplitudes
small, and the system response can be described by a linear dynamic system model, then
one can use superposition to design the system, since there will be no interaction between
absorbers at different orders.

1.0.3

Background

We now turn to a summary of past contributions to our knowledge about CPVAs, and less
specifically, contributions to non-linear vibration absorbers. As will be shown, original studies investigated the steady-state linear response of CPVAs, slowly moved into the nonlinear
steady-state response for circular paths, went on to look at non-circular path steady-state
7

nonlinear analysis, and finally to transient non-linear analysis; all for single order CPVAs.
Frequency tuned vibration absorbers have long been understood and used as a means to
reduce vibrations, and are commonly found in undergraduate vibrations textbooks, e.g., [45].
Around the early 1930’s, the basic operational principles of order tuned vibration absorbers
were first brought to light in separate patents by Salomon and Sarazin [37, 38], which were
motivated by the need to alleviate crankshaft torsional vibrations in aircraft engines. A
thorough description of the problems associated with aircraft engine torsional vibrations, as
well as the development of the pendulum type vibration absorber, is presented in [26]. Shortly
after the Salomon and Sarazins’ patents, Chilton of Curtiss-Wright employed a bifilar CPVA
and E. S. Taylor designed a “puck” type pendulum absorber, both to alleviate propellor
snapping issues in 9 cylinder aircraft engines [26].
Shortly after Curtiss-Wright hired E. S. Taylor to become their principal vibration consultant, Pratt & Whitney decided to hire another M.I.T professor as its vibration consultant,
J. P. Den Hartog. Den Hartog was the first to investigate how damping and detuning (the
difference between forcing and absorber tuning orders) of a CPVA effects its linear steadystate response, and he extended on this to show that large amplitude oscillations of a circular
path CPVA cause a detuning that arises from nonlinear effects [12, 13]. Den Hartog’s Ph.D.
student, D. Newland, first investigated the dangerous amplifying “jump” bifurcation that
can occur for circular path absorbers (resulting from a saddle-node bifurcation), and recommended overtuning the absorbers (that is, making n > n) to alleviate this bifurcation.
˜
Overtuning the absorbers does increase the torque level that causes this instability, although
the absorbers are less effective at reducing torsional vibrations when overtuned. Further investigations have validated Newland’s ideas with controlled experiments, and extended upon
the analysis through the inclusion of Coulomb friction in the system model [34, 47].
8

This dangerous amplifying instability seen in circular path absorbers motivated research
into alternative path designs. It was known that this jump bifurcation is only possible for
softening nonlinearities (frequency decreasing as a function of amplitude), motivating Madden to propose a cycloidal path [25], which, in fact, induces a slight hardening nonlinearity.
The cycloidal path is the solution of the tautochrone problem in a gravitational field, described as follows: a particle released from any point on a cycloid will reach the vertex (the
bottom of the curve) at the exact same time in a gravitational field. In fact, the equation of
motion for a particle on this curve is a simple linear harmonic oscillator, that is, the response
has the same frequencies for all amplitudes. Huygens solved the corresponding tautochrone
problem in a centrifugal field, and found epicycloids to be the solution, but it wasn’t until
Denman in 1992 [14] that epicycloids were proposed for use in CPVAs. A special epicycloid,
known as the “tautochronic epicycloid,” renders the absorber frequency independent of amplitude (for constant rotor speed), thus making the absorber motion that of a linear harmonic
oscillator [14]. Denman was able to formulate the equations of motion for bifilar absorbers,
including the rollers, for a quite general path and was able to represent these paths as a
two parameter family of curves defined by their radius of curvature ρ2 = ρ2 − λ2 S 2 , where
0
ρ0 is the curvature at the vertex, which fixes the absorber’s linear tuning, and a parameter
λ that controls the large amplitude behavior of the curve. Monroe and Shaw [29] generalized Denman’s results to include the inertial effects of the rollers used in bifilar absorbers,
and showed that a tautochrone exists for this system. Lee and Shaw [24] investigated the
counteraction of applied torques through epicycloidal absorbers and found that, along with
applying a counteracting torque at the desired order, these absorbers produced higher order
harmonics back onto the rotor. As previously mentioned, epicycloidal paths yield a simple
harmonic oscillator when the rotor speed is constant, yet there exists a nonlinear coupling
9

between the absorbers and the rotor which produces these higher harmonics. Lee and Shaw
interestingly found that epicycloidal absorbers tuned to one half of the forcing torque would
undergo a subharmonic resonance which resulted in a pure harmonic torque applied to the
rotor. Chao and Shaw [7] incorporated design guidelines for systems with absorbers that
had small imperfections as well as systems of identical, subharmonic absorbers.
The cycloidal and tautochronic epicycloidal path absorbers avoid the dangerous jump
bifurcation encountered by circular path absorbers, but can still give rise to non-synchonous
response bifurcations for systems with multiple absorbers, including the aforementioned
subharmonic bifurcation. In general, these symmetry breaking bifurcations are detrimental
to system performance, as the absorbers no longer work together effectively acting as a
single large inertia. Chao et al. [8, 9] showed that a critical torque level exists which will
cause tautochronic absorbers to become non-synchronous. Shaw and Geist [39] subsequently
generalized those results to include linear and nonlinear detuning of the tautochrone, showing
where non-synchrounous and jump instabilities occur in the path parameter space, thus
allowing one to select absorber paths that avoid these instabilities. The authors showed
that both instabilities can be alleviated through overtuning the absorbers. Alsuwaiyan and
Shaw [5] provided conditions on this non-sychrounous bifurcation for non-tautochronic paths,
and also showed that overtuning the absorbers is beneficial to performance, although for
circular path absorbers, the jump bifurcation can still occur. All the aforementioned studies
consider only the steady-state response for CPVAs. Monroe and Shaw [27, 28] recently
categorized the transient response of CPVAs for a general range of paths. It was found
that an approximate value for the percent overshoot of a single absorber could be calculated
analytically, in terms of the absorber tuning, damping, and system nonlinearity. The only
studies known to the author that consider systems composed of absorbers tuned to multiple
10

orders are those where some absorbers are tuned to orders n ± 1 in order to reduce shaking
order n shake forces [6,10]. The only study that considers multiple order torques is that of Lee
and Shaw [23], who considered how a pair of identical order n absorbers can simultaneously
address torques of order n and 2n through nonlinear effects.
It is the goal of this PhD dissertation to extend upon these previous studies through a
comprehensive study of systems with multiple order CPVAs subjected to multi-order torques.
As explained in more detail in the next sections, the system nonlinearites, specifically the
nonlinear absorber/rotor coupling, play an important role in the dynamics of CPVAs at
multiple orders, and must be considered in order to achieve optimal performance of such
systems.

1.0.4

Dissertation Organization

The dissertation is organized as follows:
• Steady-State Dynamics of Multiple Order CPVA Systems. Chapter 2 analyzes the steady-state dynamics of multiple order CPVAs through the use of numerical
simulations, perturbation methods, and numerical continuation. System stability as a
function of the applied torques and other relevant parameters is investigated. Optimal
inertia ratios between the two order absorbers are obtained.
• Experimentation. Chapter 3 introduces the experimental setup utilized and provides
experimental validation of the previous analytical treatments. Synchronous and nonsynchronous multiple-order absorber responses are investigated.
• Harmonic Balance Methods Applied to CPVA Systems. Chapter 4 presents
an analysis on the dynamics of CPVAs through the use of harmonic balance methods.
11

The harmonic balance approximations are compared to numerical simulations as well
as experiments for a wide array of absorber systems.
• Conclusions. The final chapter reviews the work completed in this dissertations and
gives some recommendations for some future areas of study.

12

Chapter 2
Modeling and Prediction of System
Response
The equations describing the motion of a rigid rotor equipped with multiple order pendulum
absorbers are derived in this chapter. These complicated, nonlinear equations are then nondimensionalized and the non-dimensional parameters are scaled based on practical limits
for the ranges of these parameters. Perturbation methods are then applied to the scaled
equations of motion, and are shown to accurately capture the both the stable steady-state
dynamics as well as the instabilities present in the system. All analytical results are compared
to simulations of the fully coupled nonlinear equations of motion.

2.1

Equations of Motion

For the present investigation, we model the system at hand as a rigid rotor spinning about a
fixed axis with masses attached that move along pre-described paths relative to the rotor, as
shown in Fig. 2.1. The figure exhibits two different paths that the masses can follow, defined
by the distance from the center of the rotor to the center of the masses, referred to as Ri (Si ),
where Si is the masses’ arc length position relative to the rotating frame. The characteristics
of these curves, along with their distance from the center of the rotor determine the linear
and nonlinear tuning of the masses. The general system has N1 absorbers tuned to an order,
13

Order n
˜
S1

mn

Order 2˜
n

R01 R1 (S1 )
R02

S2
m2n

R2 (S2 )
JR

T0 + Tn sin(nθ) + T2n sin(2nθ + φ) Rotor
Figure 2.1: Schematic of a rigid rotor fitted with absorbers tuned to n and 2˜ .
˜
n
n1 , as well as N2 absorbers tuned to order n2 . For convenience it is assumed that N1 = N2
˜
˜
and that the order n1 absorbers are indexed from 1 : N1 and the order n2 absorbers indexed
˜
˜
from N1 + 1 : N1 + N2 , where we define M = N1 + N2 . The masses of the absorbers at
their respective order are assumed to be equal, i.e.,

mi = mn
mi = m2n

1 ≤ i ≤ N1
N1 + 1 ≤ i ≤ M

In order to motivate the modeling of the engine torque, a sample schematic of gas pressure
torques for a four-stroke, in-line, four cylinder engine is shown in Fig. 2.2.
This torque can be modeled as

T (θ) = T0 + Tn sin(nθ) + T2n sin(2nθ + φ)

in which T0 is the DC torque, Tn,2n are the fluctuating components of the torque, and φ is
the phase between the harmonics. With the above information, the kinetic energy for the

14

Torque Per Cylinder

T (θ)

π

0

2π

3π

T0

4π

Crank Angle
(a)

θ
(b)

Figure 2.2: (a) Torques generated from individual cylinders, calculated by converting a
characteristic gas pressure force acting on a piston to crank torques. (b) Summed torque,
which is dominated by a DC term and second order harmonic. Higher order harmonics are
also evident.
rotor/CPVA system can be found to be

1
˙
Tt = (JR θ2 +
2

N1

˙
˙˙
˙
(mi (Xi (Si )θ2 + Si + 2Gi (Si )θSi ))
i=1

M

+

(2.1)

˙
˙˙
˙
(mi (Xi (Si )θ2 + Si + 2Gi (Si )θSi )))
i=N1 +1

where
2
Xi (Si ) = Ri (Si ),
and
Gi (Si ) =

1 dX
Xi (Si ) − ( i (Si ))2 .
4 dSi

In order to obtain the differential equations describing the dynamics of the system we
use Lagrange’s method. The generalized forces are found to be

15

˙
˙
δW = −ca,i Si δS − (c0 θ − T0 − T (θ))δθ,
and performing the steps in Lagrange’s method yields N1 + N2 + 1 equations of motion,
found to be:

¨ 1 dXi (S )θ2 ] = −c S
˙
¨
˙
mi [Si + Gi (Si )θ −
ai i
2 dSi i

1 ≤ i ≤ M,

(2.2)

for the ith absorber and
N1

dGi
dX
¨
˙ ˙
¨
˙2
(Si )Si ))
(mi ( i (Si )Si θ + Xi (Si )θ + Gi (Si )Si +
dSi
dSi
i=1
M
dX
dGi
¨
˙ ˙
¨
˙
+
(mi ( i (Si )Si θ + Xi (Si )θ + Gi (Si )Si +
(S )S 2 ))
dSi
dSi i i
i=N1 +1

¨
JR θ +

(2.3)

˙
= −c0 θ + T0 + Tn sin(nθ) + T2n sin(2nθ + φ)
describing the dynamics of the rotor.
In order to get the equations of motion in a form suitable for non-linear analysis methods,
we non-dimensionalize the variables and change the independent variable from time to the
rotor angle, θ following the work of Chao et. al. [8]. To change the independent variable a
new dimensionless dependent variable y, is defined as

y=

˙
θ
Ω

This variable is used to then define relationships between derivatives with respect to time

16

and θ as follows:
¨ dy dθ Ω = y yΩ2
θ=
dθ dt
(˙) =

d( ) dθ
= ( ) yΩ
dθ dt

and
d2 ( )
(¨) =
dθ2

dθ 2 d( ) d2 θ
+
= ( ) y 2 Ω2 + ( ) y yΩ2
dt
dθ dt2

In summary, we have the following relationships, where ( ) =

¨
θ = Ω2 yy

(˙) = Ωy( )

d( )
,
dθ

(¨) = Ω2 yy ( ) + Ω2 y 2 ( )

This process transforms the equations of motion, Eqs. (2.2) and (2.3) into a set of periodically forced, non-autonomous equations, and thus the nonlinearity, Tn sin(nθ)+higher
harmonics, into a forcing term. The equations of motion are then rearranged into a nondimensional form as follows,

1 dxi
(s )y + µai si = 0
ysi + (si + gi (si ))y −
2 dsi i

1≤i≤M

N1
dx
1
(
(
[ i si y 2 + xi (si )yy + gi (si )(si yy + si y 2 )
N1 (1 + α)
ds
i=1 i
M
dg
dx
α
+ i s 2 y 2 ]) +
(
[ i si y 2 + xi (si )yy
dsi
N2 (1 + α)
ds
i=N1 +1 i
dg
+ gi (si )(si yy + si y 2 ) + i s 2 y 2 ]))
dsi
+ yy = −µ0 y + Γ0 + Γn sin(nθ) + Γ2n sin(2nθ + φ)

17

(2.4)

(2.5)

in which ( ) =

d( )
2
2
. The parameters xi (si ) = Ri (si )/Ri0 and gi (si ) =
dθ

dx (s )
xi (si ) − 1 ( i i )2
4 dsi

are path-related functions, described in detail in the next section, and the other variables
and parameters are defined and described in Table 2.1. While many of these parameters
have been considered in previous CPVA studies, the absorber inertia ratio α is of special
interest here, since it dictates the distribution of absorber inertia between the two orders for
a fixed amount of absorber inertia, that is, for a fixed value of . These equations form the
basis of the analysis that follows.
Table 2.1: Definition of non-dimensional variables in Eqns. (2.4-2.5).
Nondimensional
parameter

Definition

y

˙
θ
Ω

Description

2
N1 mn R0 (1+α)
JR

si

Si
R0

µ0

ca,i
mi Ω
co
JR Ω

Γ0

T0
JR Ω2

Γn,2n

Tn,2n
JR Ω2

α

m2n
mn

µa,i

18

Angular velocity of the rotor normalized by the rotor
mean speed, Ω
Inertia ratio of absorbers to
rotor
Absorber arc length normalized by its distance from
the rotor center
Normalized absorber viscous damping
Normalized rotor viscous
damping
Mean engine torque normalized by twice the rotor’s kinetic energy
Fluctuating component of
the engine torque normalized by twice the rotor’s kinetic energy
Ratio of masses at each order

2.1.1

Absorber Paths

In order to smooth out torsional vibrations with minimal absorber inertia under the most
severe fluctuating loads, the absorbers will move at large amplitudes, requiring careful selection of the absorber paths, which are specified by the functions xi in the equations of
motion. Denman [14] conveniently represented these paths as a two parameter family of
curves defined by their radius of curvature ρ2 (S) = ρ2 − λ2 S 2 , where ρ0 is the curvature at
0
the vertex, which fixes the absorber’s linear tuning order n, and λ controls the large ampli˜
tude behavior of the curve, thereby dictating the absorbers’ nonlinear dynamics. Typically,
the absorber tuning is taken to be slightly above the dominant engine order, for example,
for an N -cylinder four-stroke engine, the dominant engine order is n = N/2, and absorber
tuning is taken to be n = N (1 + σ)/2, where the detuning parameter σ is a few percent.
˜
The tradeoff is that smaller values of σ make the absorber work better, but it also makes
the absorber amplitudes large, inducing nonlinear effects, including the generation of higher
harmonics that can result in crosstalk among absorbers.
Selection of the nonlinear path parameter is more complicated. Basically, the nonlinear
part of the path dictates whether the absorber frequency (order) increases or decreases as
a function of absorber amplitude, and the degree to which this nonlinear detuning occurs
[3,39]. There exists a special value, λte , corresponding to the tautochronic epicycloid, which
renders this detuning zero, that is, selection of this path makes the absorber dynamics
essentially linear out to large amplitudes [14,39]. Figure 2.3 shows three curves for the same
linear tuning value, n = 1.5, with nonlinear path parameters of λ = 0 (circle), λ = λte , and
˜
λ = 1 (cycloid).
Conservative absorber designs select relatively large values of the detuning σ, keeping the

19

Ro
=

te

=0

=1

Figure 2.3: Absorbers paths which produce a linear tuning of n = 1.5, for three different
˜
nonlinear path parameters, λ = 0, λte , 1.
absorber response away from its resonance, and therefore linear [3, 36, 39]. More aggressive
designs, with improved performance for smaller absorber inertia, keep σ small and account
for nonlinear behavior, typically by taking λ ≈ λte . In this study, following the work of Shaw
and Geist [39], we define the path as a perturbation of the tautochronic epicycloid, which
allows for the analysis of different path types. This path can be easily described through the
radial distance from the rotor center to the absorber location along it’s arc length as,

xi (si ) = 1 − ns2 + hi (si ).
˜ i

20

The paths of each absorber at the different orders are assumed to be identical,

1 ≤ i ≤ N1

xi (si ) = xn (si )

N1 + 1 ≤ i ≤ M,

xi (si ) = x2n (si )

and the perturbation of the path away from the tautochronic epicycloid is described by
hi (si ). The form of the deviation from tautochronic paths is defined to be
N
hi (si ) =

2j+2
ϕj si

(2.6)

j=1
This is due to the symmetric nature of the curves and the fact that the second order corrections can be captured in the quadratic term already used tautochronic path formulation.
The exact form for ϕ1 has been found to be as follows,
1
n
ni
˜i
ϕi = ( )(˜ 2 + 1)2 (˜ 2 − λ2 (1 + n2 )),
i
12 i

where the complete path derivation and details can be found in Shaw and Geist [39].
The higher order coefficients of the amplitude expansions can easily be found performing a
series expansion in s of xi (si ).
As mentioned above, the design of absorber systems must account for various types of
instabilities that may occur. These instabilities arise since CPVA systems have multiple elements with identical natural frequencies, or, for the present study, natural frequencies in 2 : 1
ratios, low damping, and they are driven near resonance. One type of instability encountered
is a jump, due to absorber detuning (specifically, softening) at large amplitudes, resulting in
the absorber transitioning into a vibration amplifier (since its phase shifts such that it adds

21

to the fluctuating torque). This is a common problem with circular path absorbers, and it
is avoided in practice by selecting large detuning [3, 36]. Another, more subtle, instability
occurs for systems comprised of multiple identical absorbers. When aggressively tuned, these
absorbers experience cross-talk through the rotor inertia, resulting in an instability of the
desired synchronous response, and this occurs even for tautochronic absorbers, that is, when
the absorbers are designed to be as linear as possible. This instability can also be avoided
by detuning, which is accomplished by proper selection of the absorber path, and analysis
indicates how one can tune for optimal performance without encountering this instability
[39]. The issue of imperfections among absorbers is also detrimental to performance, and
tuning strategies to account for these effects, at least for small absorber amplitudes, are also
known [4].

2.2

Motivating Simulation Results

With the full equations of motion developed for CVPA systems with absorbers tuned to
orders n and 2n, one can begin to qualitatively map out the stability of the absorber response
using numerical simulations. Shown in Fig. 2.4 is the normalized order n and 2n torque
space, Γn and Γ2n , exhibiting some of the instabilities that can occur for zero relative phase
between the torque harmonics, φ = 0. This system has 4 absorbers, 2 each at orders n and
2n, that is, the absorbers are tuned to exactly match the torque orders. The mass ratio
between the different order absorbers is taken to be m2n /mn = 0.1.
Synchronous responses correspond to both sets of absorbers acting in relative unison, as
desired; these occur, for example, at points W and Y in Fig. ??. As the torque amplitudes
are increased, instabilities can occur, and these are quite rich if there are multiple absorbers

22

0.01

0.008

Nonsynchronous
Z

Subharmonic

2n

0.006

Y
0.004

Synchronous

NonSynchronous

0.002

0
0

W
0.005

0.01

0.015

0.02

X
0.025

n

(a)

0.2

sn
s

0.15

n

s2n

0.1

s2n

s

0.05
0
−0.05
−0.1
−0.15
−0.2
6250

6255

6260

6265

(b)
Figure 2.4: Simulated responses of a system composed of four tautochronic absorbers, two
each at orders 1.5 (sn ) and order 3 (s2n ), subjected to a torque composed of a linear combination of orders 1.5 and 3. Note the different scales used for depicting the responses. (a)
Torque harmonic amplitude space, Γ3 vs. Γ1.5 , with boundaries of the desired synchronous
response indicated schematically; solid line depicts a subharmonic instability boundary, dotted (dashed) line depicts a symmetry-breaking instability boundary to nonsynchronous responses of the order 1.5 (3) absorbers. (b) Mutually synchronous response at point W.
23

0.3

s

n

s

0.2

n

s

2n

0.1

s2n

s

0

//

−0.1
−0.2
−0.3
−0.4
6250

6255

6260

6265

(c)
0.08

sn

0.06

sn

0.04

s2n
s2n

s

0.02
0
−0.02
−0.04
−0.06
−0.08
6250

6255

6260

6265

(d)
Figure 2.5: (c) Response at point X; the order 1.5 absorbers have become nonsynchronous
and are in a transition towards their amplitude limits. (d) Response at point Y; similar to
W, only with reversed relative amplitudes.

24

0.1

s

n

sn
s2n

0.05

s2n

s

0

−0.05

−0.1
6250

6255

6260

6265

(e)
0.06

Order 2 Cusp
Order 4 Cusp

Z

0.04

s

0.02
0
−0.02
−0.04
−0.06
2.6

2.7

2.8

θ

2.9

3

3.1
4

x 10

(f)
Figure 2.6: (e) Response at point Z; the order 1.5 absorbers have become subharmonic
and are in transition toward their amplitude limits, as shown in (f). (f) Response at point
Z, depicted over a long time interval; the order n absorber instability eventually results in
large amplitude motion; the cusp amplitudes are the maximum possible amplitudes for these
absorbers.
25

at each order. Three types of instabilities are indicated. The solid line represents the order
4 absorbers undergoing a subharmonic transition [22, 30] in which they go out of phase and
respond at order 2, an example of which occurs at point Z in Fig. 2.4. The dashed-dotted
line represents the order 2 absorbers becoming nonsynchronous with an amplitude and phase
shift, but remaining at order 2; such a response occurs at point X. The dashed line represents
a condition for which the order 4 absorbers become nonsynchronous (not encountered here).
Note that this instability is seen to occur beyond the subharmonic transition of the order
two absorbers for the present system, but this may not always be the case. Each of these
instabilities is detrimental to system performance, since the absorber amplitudes grow and
reach physical limits, imposed by either hardware or the mathematics of the tautochronic
absorber paths [39]. These preliminary results provide a general overview of the behavior
of multiple order CPVA systems, and the analytic results developed below will provide a
method for carrying out parameter studies of this behavior, to which we now turn.

2.3

Perturbation Analysis

In order to make the equations of motion suitable for analytical investigation, specifically by
perturbation methods, we follow a scaling similar to that of Chao [8]. Taking advantage of a
small dimensionless inertia ratio allows one to scale the responses, expand the equations of
motion, and ultimately uncouple the rotor dynamics from those of the absorbers to leading
order in . In the subsequent analysis, it is assumed that the following non-dimensional
quantities are small, specifically O( ): the torque amplitudes at orders n and 2n, the absorber
and rotor damping coefficients, and the absorber detunings (˜ i − kn)/(kn) for k = 1, 2.
n
These conditions are met in applications, and perturbation analyses following this scaling

26

have proven useful for matching experimental results [27, 40, 47] and for absorber system
design [3, 27, 39].
This scaling results in a system for which the rotor runs at nearly constant speed with
small fluctuations, that is, y = 1 + w, where y is the rotor speed normalized by its mean
value. Solving the rotor equation of motion, Eqn. (2.5), for yy and performing expansions
in , the rotor acceleration can be expressed as yy = w , where

˜
˜
w = Γn sin(nθ) + Γ2n sin(2nθ + φ)

N1
1

(2˜ 2 si si + n2 si gi0 − si2 gi0 )
ni
˜i
+
N1 (1 + α)
i=1

M
α(2˜ 2 si si + n2 si gi0 − si2 gi0 ) ,
n2
+
˜2
N1 +1
where gi0 (si ) =

(2.7)

1 − n2 (1 + n2 )s2 . Inserting w into the absorber equations, Eqn. (2.4),
˜i
˜i i

utilizing the above the scalings and expanding in

yields M equations for the absorbers in

which the rotor dynamics has been eliminated to leading order. These equations have the
following form, convenient for application of standard perturbation methods:

si + n2 si = fi (s1 , .., sM , s1 , .., sM , µi , ni , λ, T (θ)) + O( 2 ),
˜i
˜

27

(2.8)

where
N1
1
3 − µ s + [s + g(s )][ 1 (
(−2n2 si si − n2 g(si )si
fi = 3ϕi si ˜ai i
i N
i
i
i
1+α
1 i=1
+

+

dg(si ) 2
s i)+
dsi

M
i=N1 +1

α
(−2n2 si si − n2 g(si )si
i
i
1+α

(2.9)

dg(si ) 2
˜
˜
s i )) − Γn sin(nθ) − Γ2n sin(2nθ + φ)].
dsi

Note that the absorber amplitudes are not scaled to be small in this analysis, since the
absorber path is taken to be close to the tautochrone, which yields a linear oscillator equation for the absorber motion out to large amplitudes. Thus, the equations of motion allow
for large amplitude absorber motions, but their motion is sufficient to counteract the applied
torque, resulting in attenuation of rotor vibration. The order

terms in the absorber equa-

tions capture, to leading order, the effects of absorber damping, and the coupling between
the absorbers’ through the rotor dynamics, which is driven by the applied torque and the
dynamics of the other absorbers. Once solutions of Eq. (2.8) are known, the rotor response,
which is the quantity of principle interest for vibration reduction, can be reconstructed using
Eq. (2.7). Perturbation analysis can be performed on Eq. (2.8), in order to obtain approximate solutions of the absorbers’ amplitudes and phases as a function of θ. In this study, we
employ the method of multiple scales (MMS) [31] to construct these approximate solutions.
As is common for capturing dynamics near resonances, we introduce detuning parameters
which allow for small deviations of the absorbers’ orders relative to their respective forcing
orders, as well for variations in the 2 : 1 relationship between the absorbers. For forcing at

28

orders n1 = n and n2 = 2n the detuning parameters are defined as,

n = n1 + σ1
˜
2n = n2 + σ2
˜
In order to make the equations for amplitudes and phases autonomous, we use the standard MMS coordinate change, φi = βi − σi . Using these assumptions, the results of MMS
produce 2M equations representing the slow time dynamics of the amplitudes and phases,
ai and φi , of the absorbers. These complicated expressions capture the nonlinear absorber
interactions, and are given in a condensed form here in order to demonstrate their general
form,
µ
Γn
cos(φi )F1 (ai )
ai = [− i ai +
2
n1
Γ
1 ≤ i ≤ N1
− 2n ai sin(2φi ) + Pi ]
4
Γn
3
sin(φi )F2 (ai )
φi = [−σ1 − ϕi a3 −
i
4
n 1 ai
n 5 a2 n
Γ
1
− 2n ai cos(2φi ) −
( 1 i − 1 ) + Ri ]
4
N1 (1 + α) 4
2

(2.10)

(2.11)

1 ≤ i ≤ N1
µ
ai = [− i ai
2
Γ2n
+
cos(φi )F1 (ai ) + Vi ]
n2

(2.12)
N1 + 1 ≤ i ≤ M

Γ
3
φi = [−σ2 − ϕi a3 − 2n sin(φi )F2 (ai )
i
4
n2 ai
5 a2
n
n
α
−
( 2 i − 2 ) + Xi ]
N1 + 1 ≤ i ≤ M
N1 (1 + α) 4
2

(2.13)

where the functions (Pi , Ri , Vi , Xi ) contain the coupling terms, which depend on all absorber

29

amplitudes and phases, and are found to be:

1
Pi (ai , aj , φji ) =
N1 (1 + α)

N1

1
[− n3 ai a2 sin(2(φj − φi ))
4 1 j
j=1

+ n1 aj G1 + n1 (n2 + n4 )a3 H1 ]
1 j
1
+

α
2N2 (1 + α)

M

[ai aj n2 cos(2φi − φj )F1 (aj )]
2
j=N1 +1

1
Ri (ai , aj , φji ) =
N1 (1 + α)

N1

1
[ n3 a2 cos(2(φj − φi ))
4 1 j
j=1
a3
aj
2 + n4 ) j H ]
+ n1 G2 + n1 (n1
1 a 2
ai
i
M
α
[aj n2 sin(2φi − φj )F 1(aj )]
−
2
2N2 (1 + α)
j=N1 +1

α
Vi (ai , aj , φji ) =
N2 (1 + α)

M

1
[− n3 ai a2 sin(2(φj − φi ))
4 2 j
j=N1 +1

+ n2 aj G1 + n2 (n2 + n4 )a3 H1 ]
2
2 j
1
−
2N2 (1 + α)

N1

[a2 n3 cos(2φj − φi )(F1 (ai ) + F2 (ai ))]
j 1
j=1
M

α
N2 (1 + α)

1
[ n3 a2 cos(2(φj − φi ))
4 2 j
j=N1 +1
a3
aj
2 + n4 ) j H ]
+ n2 G2 + n2 (n2
2 a 2
ai
i
N1 a2
1
j
−
[ n3 sin(2φj − φi )(F1 (ai ) + F2 (ai ))]
2N2 (1 + α)
a 1
j=1 i

Xi (ai , aj , φji ) =

Where the Hi and Gi terms are integrals defined below.

30

2π
1
G1 (ai , aj , φji ) =
cos(x) sin(x − φj + φi )[1 − ni (1
2π 0
+ n2 )a2 cos2 (x)](1/2) [1 − ni (1 + n2 )a2 cos2 (x − φj + φi )](1/2) dx
i j
i i
2π
1
G2 (ai , aj , φji ) =
cos(x) cos(x − φj + φi )[1 − ni (1
2π 0
+ n2 )a2 cos2 (x)](1/2) [1 − ni (1 + n2 )a2 cos2 (x − φj + φi )](1/2) dx
i j
i i
2π
1
H1 (ai , aj , φji ) =
cos(x) sin2 (x) sin(x − φj + φi )
2π 0


2 )a2 cos2 (x − φ + φ ) (1/2)
1 − ni (1 + ni i
j
i 

dx
2 )a2 cos2 (x)
1 − ni (1 + ni j
2π
1
cos(x) sin2 (x) cos(x − φj + φi )
H2 (ai , aj , φji ) =
2π 0


2 )a2 cos2 (x − φ + φ ) (1/2)
1 − ni (1 + ni i
j
i 

dx
2 )a2 cos2 (x)
1 − ni (1 + ni j
The functions F1 and F2 , which are related to the individual absorber nonlinear behavior,
are defined as [39]:

F1 (ai ) =

2π
1
sin2 (x)[1 − n2 (1 + n2 )a2 cos2 (x)](1/2) dx
i
i i
2π 0

F2 (ai ) =

2π
1
cos2 (x)[1 − n2 (1 + n2 )a2 cos2 (x)](1/2) dx.
i
i i
2π 0

Nonlinear interactions arise from the fact that the absorbers are driven near their individual resonances at orders n and 2n (as required for good performance) and, in addition,
the fact that nonlinearities in the absorber motions result in combinations of harmonics that
promote crosstalk, as shown below. These nonlinear interactions become important at large

31

absorber amplitudes, and thus the nearly tautochronic scaling is required to capture these
effects; that is, this coupling does not appear at leading order if one considers amplitude
expansions for the absorbers, as is done for circular paths [3]. We now turn to results derived
from this analysis, and compare them with direct simulations.

2.4

Synchronous Responses

In order to exhibit the basic effects that the different order absorbers have on each other,
we first assume the absorbers at their respective order behave in a mutually synchronous
manner. This is dynamically equivalent to a system with only two absorbers, one at each
order; however, the use of a single absorber at each order is seldom realizable in practice due
to space and balance issues. Assuming a synchronous response, the perturbation equations,
Eqs. (2.10) to (2.13) above are simplified through the removal of the summations across
multiple absorbers as well the complicated non synchronous expressions.
As a motivating example, Fig. 2.7 shows the amplitudes of both absorbers, as obtained
from the perturbation equations along with simulation results, versus the amplitude of the
order n applied torque, with the order 2n torque taken to be zero; this corresponds to
sweeping along the horizontal axis in Fig. 2.4(a). This is of interest to situations when the
order 2n absorbers are used to address the second harmonic in the rotor response induced
by the nonlinear dynamics of the order n absorbers. We initially assume a stable response
in the perturbation equations for Fig. 2.7, but, as will be shown, nonsynchronous responses
can occur.
It is of interest to note that although the system is being excited only at order n, the
order 2n absorbers actually reach their cusp amplitude at a torque level much below that

32

Non−Dimensional Absorber Amplitude (s)

0.4
0.35
0.3
0.25
0.2

s1 Sims
s1 Theory
s3 Sims
s3 Theory
s1 cusp
s3 cusp

0.15
0.1
0.05
0
0

0.005
0.01
0.015
0.02
0.025
Amplitude of Non−Dimensional Torque ( n)

Figure 2.7: Predicted and simulated steady-state absorber amplitudes as a function of the
order n torque amplitude; the order 2n torque amplitude is zero. The cusp values are
amplitude limits set by hardware constraints. System parameters: m3 /m1.5 = 0.1, = 0.05.
which causes the order n absorbers to reach their amplitude limit. This is a result of the
non-linear interaction between the absorbers, specifically the fact that the order n absorbers
impart a harmonically rich torque on the rotor, composed primarily of orders n (used to
address the order n torque), 2n, and 3n. It is clear from these results that the order 2n
absorbers do not have sufficient inertia for these loading conditions. If one were to analyze
this system using linear response theory, which ignores the nonlinear interactions, the results
would show the order 2n absorbers to have a much smaller amplitude, since the order 2n
absorbers are being driven well away from their resonance. Thus, the growth of the order 2n
absorbers as the order n torque is increased is clear evidence of the importance of nonlinear
effects. One goal of the present work is to provide a tool for determining the appropriate

33

inertia balance, described by the parameter α, for such sets of absorbers.
As the main goal of the absorbers is to reduce torsional fluctuations of the rotor, one can
investigate how well they perform by looking at the angular acceleration amplitude of the
rotor. This is shown in Fig. 2.8, for the same conditions as Fig. 2.7.

Rotor Angular Acceleration Amplitude

0.03
0.025

Order n
Order 2n
Absorbers Locked

0.02
0.015
0.01
0.005
0
0

0.005
0.01
0.015
0.02
Amplitude of Non−Dimensional Torque (

0.025
)
n

Figure 2.8: Order n and 2n harmonic components of the rotor angular acceleration as a
function of the order n applied fluctuating torque amplitude, for the same conditions as Fig.
2.7. The solid line is the reference response for the absorbers locked.

In order to benchmark the effectiveness of the absorbers, one considers the rotor response
with the absorbers locked in place, so as to account for their added flywheel inertia, but not
their dynamic effects; this torsional vibration level is represented by the solid line in Fig.
2.7. As seen from Fig. 2.8, the absorbers significantly reduce the order n component of the
rotor torsional vibrations, as desired. It is also interesting to note that, even though the
rotor is being driven only at order n, order 2n torsional vibrations are present, as described
34

above. In fact, these would be even more prominent if not for the presence of the order 2n
absorbers. For all results to follow, the rotor vibration results are very similar to those of

Non-Dimensional Absorber Amplitude (s)

Fig. 2.8, and are thus not shown.

0.4

s1
s1
s3
s3
s1
s3

0.35
0.3
0.25
0.2

Sims
Theory
Sims
Theory
cusp
cusp

0.15
0.1
0.05
0

0

0.005 0.01 0.015 0.02 0.025 0.03
Amplitude of Non-Dimensional Torque (Γn )

Figure 2.9: Absorber amplitudes as a function of order n fluctuating torque amplitude; the
order 2n torque amplitude is zero. System parameters: m3 /m1.5 = 0.62, = 0.05. The
absorbers reach their respective cusps at approximately the same level of excitation.

The results of Fig. 2.7 show that this design does not have appropriate inertia balance
between the order n = 1.5 and 2n = 3 absorbers, since the order 1.5 absorber is not near its
limits at a torque level where the order 3 absorber reaches its limits. An ideal design would
have the absorbers reach their limits at the same torque level. In fact, the perturbation
equations offer an analytical prediction for the mass ratio that provides this condition, determined as follows using the steady-state perturbation equations. First, damping is taken
to be zero, so that the absorber response phases are all zero or π, and then the absorber
amplitudes are set to be at their respective cusp values, or some percentage thereof (which

35

are functions of only their tuning orders n [14, 39]). This results in a pair of equations that
˜
involve the following parameters: the absorber mass ratio, the peak torque level (correspond˜
˜
ing to the value when both absorbers reach their cusps, Γn,m and Γ2n,m ), the two forcing
orders and the values of the absorber detunings. These equations provide a very useful tool
for selecting important absorber design parameters, and determining the maximum torque
level the resulting system can handle. These equations are given by:

˜
Γn = −

a1,m n1

−2n1 + a2 n5
1,m 1

4F2 (a1,m )(1 + α)

˜
+2a2,m F1 (a2,m )n2 α + 4σ1 (1 + α) − Γ2n (1 + α) ,
1
and

α=

˜
a2 (−2(F1 (a1,m ) + F2 (a1,m )))n3 − 4(a2,m n1 σ2 + F2 (a2,m )Γ2n )
1,m
1
˜
a3 n6 + a2,m n1 (−2n1 + 4σ2 ) + 4F2 (a2,m )Γ2n
2,m 1

,

where, for the tautochronic epicycloids, the maximum absorber amplitude is defined as:

ai,m =

ni
˜

1
.
n2 + 1
˜i

Fig. 2.9 repeats the conditions shown in Fig. 2.7 with the same total absorber inertia, but
with a different ratio of absorber inertias. Here the absorber inertia ratio is calculated so
that both sets of absorbers reach their maximum values at the same torque level for the
given absorber tuning, specifically, α = 0.62, a surprisingly large value given the fact that
there is no applied torque at order 2n. Note that the corresponding maximum torque level
is predicted to be Γn,max = 0.025, closely matching the value found by simulations. These
results demonstrate the utility of the perturbation predictions. Note that the optimal mass

36

ratio can also be obtained when both orders of excitation are present, but one must assume
some relationship between the different forcing order amplitudes, and found by solving the
above equations simultaneously.
The above results show the effects of the multiple absorbers on each other, as well as
the dependency of the system response on the ratio of the absorber inertias. The matter of
how these parameters affect torsional vibration levels will be considered after we describe
non-synchronous responses.

2.5

Non-Synchronous Responses

We now turn to some instabilities that can occur when multiple absorbers are used at each
order. It should be noted than in order to carry out an analytical study on the stability of
the synchronous response, one must compute the Jacobian of the set of the non-synchronous
perturbation equations, evaluate this Jacobian on the synchronous solution, and then calculate the eigenvalues of this matrix. This, in fact, is not a trivial task for single order absorber
systems [8, 39], and may perhaps be impossible for sets of multiple order absorbers due to
the fact that an analytical synchronous response solution is not evident to the author. With
this in mind, the stability calculations done in this study are carried out using numerical
continuation software, and prove to be computationally inexpensive and very accurate.
Using the full, nonsynchronous perturbation equations for a system of four absorbers
(two at each order), a numerical continuation study using AUTO [15, 16] is carried out for
the conditions used in Fig. 2.7. The order n torque is swept until the bifurcation to nonsynchronous response of the order n absorbers is found, in which the absorber response is similar
to that in Fig. 2.5 (c). This instability is then investigated as both the order n and order

37

2n torque amplitudes are varied. As clear from Fig. 2.10, the nonsynchronous response of
the order n absorbers, caused by the order n torque, is actually due to the same bifurcation
as the subharmonic response of the order n absorbers when driven primarily by an order
2n torque. Thus, the solid and dashed-dotted bifurcation lines depicted in Fig. 2.4(a) are
actually a single curve, bounding synchronous and nonsynchronous responses of the order
n absorbers, although the character of these nonsynchronous responses appears different in
the two limiting cases, Γn = 0 and Γ2n = 0. Note that in all these simulations the order 2n
absorbers remain mutually synchronous. The maximum (cusp) amplitude of the absorbers
is shown in Fig. 2.10 as well, and it is clear that this instability occurs near the maximum
absorber amplitudes. Simulations of the full non-linear equations confirm the accuracy of
these bifurcation curves, as shown in Fig. 2.10. Also shown in Fig. 2.10 are contours of the
root mean square (RMS) values of the rotor response, generated from analytical approximations by inserting the absorber amplitudes obtained from the perturbation equations into
the scaled rotor response assumption, Eqn. 2.7. One can see that at torque levels above this
instability line, the rotor response increases dramatically in amplitude, as the absorbers go
unstable and grow towards their amplitude limits.
The curve in Fig. 2.10 provides very useful guidelines for designing multiple order absorber systems, as one must design the system to be in the stable regime, and desires to
minimize the rotor vibration amplitudes. Typically, one would have a range of the two
torque amplitudes along with their relative phase, and one would select absorber parameters
so that the synchronous response was stable for all operating conditions. Examples of this
situation are not presented here, but are straightforward to obtain using the tools developed
herein.

38

−3

8

−3

x 10

x 10

9

7

8

(Order 2n)

6

7

5

6

4

5

3
2
1

4

Rotor RMS
Stability Curve
Cusp
Stable (Sims)
Unstable (Sims)
0.005

3
2
1
0.01
0.015
(Order n)

0.02

Figure 2.10: Order n = 1.5 absorber synchronous response stability boundary as a function of the driving torques, for φ = 0. Torques outside the range of this curve result in
nonsynchronous responses of the order n absorbers.

2.5.1

Minimizing Torsional Vibration

In order to attenuate torsional vibrations at multiple orders, CPVA’s tuned to the multiple
orders must be used. Interestingly, it was shown above that if only the order n torsional
vibration is desired to be decreased, a system using multiple orders of CPVA’s can be more
effective than using only absorbers tuned to order n (we used 2n absorbers in addition to n,
but further adding 3n might offer even more benefit). In this part of the study we keep the
total inertia of all absorbers constant as α (defined in Table 2.1) is varied, which makes the
performance comparisons meaningful, since one expects to get better performance if more
total absorber inertia is added. Of particular interest here is the level of torsional vibration,
since its reduction is the ultimate goal of the absorber systems. This is computed from the
perturbation results for the absorber amplitudes, specifically using the scaled rotor angular
acceleration, w , after the absorber amplitudes have been determined using Eqns. (2.10)-

39

(2.13). The measure we use for the rotor torsional vibration is percent correction, defined
as the ratio of the order n component of rotor angular acceleration amplitude when the
absorbers are locked to that when they are free.
As α is varied the limiting mechanisms of the absorbers depend on the arrangement
of absorbers. If one has only a single absorber at order n, the limiting factor is the cusp
amplitudes of the absorbers and the results of Section 2.4 apply; in this section these limits
are depicted in the Figures that follow as dashed-dot curves for the limit of the order n
absorbers and as solid curves for the limit of the order 2n absorbers. If there are multiple
absorbers at order n, the limiting factor will be the instability of the synchronous response,
and the critical value Γ∗ is depicted as a dotted curve in the Figures, as calculated by
n
numerical continuation of the perturbation equations using AUTO.
To illustrate the main points of this section, Figure 2.11 shows a contour plot of the
torsional vibration amplitude as a function of the order n torque amplitude Γn and the
mass ratio α, where the absorbers are assumed to remain synchronous, which is analogous
to having a single absorber at each order, and there is no torque at order 2n (Γ2n = 0). In
this case the order n absorbers reach their cusps along the dashed-dot line, and the order
2n absorbers reaching cusps along the solid line. Note that these two conditions meet at
α ≈ 0.62, which is the optimal value found in the analysis of Section 2.4. For reference,
Γ∗ is shown as the dotted curve in Fig. 2.11, although it is not relevant to this case. This
n
indicates that the limiting condition is the loss of stability of the synchronous response when
multiple absorbers are used at order n, which will be considered below. As is evidenced in
Fig. 2.11, when desiring to attenuate vibrations at order n only and assuming a synchronous
absorber response, using all the available inertia towards order n absorbers yields the largest
correction. The addition of order 2n absorbers decreases the correction, although only by
40

a small amount, and will be shown to provide a benefit in the stabilization of the order n
absorbers when the stable response assumption is lifted.

(Order n)

0.03

0.025

Order n % Correction
*
n

90

Cusp (order 2n)
Cusp (order n)

85
80
75

0.02

70
65

0.015

0.2

0.4

0.6

0.8

1

Figure 2.11: Order n applied torque versus the mass ratio of the different order absorbers,
when assuming a synchronous response.

Fig. 2.12 shows the contours of torsional vibration correction for the same conditions
as Fig. 2.11, but with two absorbers at order n. Note that the limiting condition here is
Γ∗ , and that the loss of stability causes a sharp increase in the torsional vibrations at the
n
limit, which is well below the cusp limits. In the stable regime, the contours of the percent
correction are exactly the same as those in Fig. 2.11, although different contour lines are
labeled. This clearly indicates the importance of the stability of the synchronous response.
An interesting, and surprising, observation is the fact that the order 2n absorbers provide a
23.5% increase in the order n operating torque range, that is, the order 2n absorbers actually
stabilize the synchronous response of the order n absorbers. In this case the optimal value
of α for maximizing the torque range is approximately 0.4. This is well below the “equal
cusp” value of 0.62, but still a significant distribution of absorber inertia at order 2n, given
41

the fact that the torque is purely order n. Considering the contours, one obtains a better
correction for small values of α, but α allows for a significant increase in the torque operating
range. Practically, this trend allows the tuning on the order n absorbers to be much more
aggressive (closer to the torque order), and thus results in better performance in reducing
the order n torsional vibrations.

(Order n)

0.03

Order n % Correction
*
n

50

0.025

0
−50

0.02
−100

0.015

−150
0.2

0.4

0.6

0.8

1

Figure 2.12: Order n applied torque versus the mass ratio of the different order absorbers.
As evident, using a small amount of order 2n absorbers can stabilize the order n absorbers.

2.6

Non-Tautochronic Paths

All of the previous analysis was conducted for a tautochronic path, which renders the absorber motion linear out to large absorber amplitudes. Previous researchers have conducted
extensive analysis on the dynamics of non-tautochronic path absorbers [3, 34, 36, 47], specifically circular and cycloidal path absorbers, all for single order absorber systems. In order to
conduct analysis on these non-tautochronic paths, one must scale the absorber amplitudes to
42

be small, which is not required for perturbation analysis of the tautochronic path absorbers.
Since the torque generated by the absorbers onto the rotor becomes harmonically rich at
larger absorber amplitudes, this amplitude scaling results in perturbation equations which
fail to capture the non-linear multiple order absorber crosstalk to leading order of the small
parameter. As mentioned earlier, and detailed in [39], a perturbation at leading order away
from the tautochrone can be implemented in order to analytically investigate absorber paths
which deviate from the tautochrone without having to assume small absorber amplitudes.
The question that then remains is if this O( ) correction to the path can accurately capture
the system’s dynamics throughout the range of curves possible using the two-parameter family of paths as explained above. In order to answer this question, the two extremes of the
family of curves are investigated, specifically circular and cycloidal paths. Referring back to
Fig. 2.3, one can see that, strictly geometrically speaking, the cycloidal path does not differ
greatly from the epicycloidal path, and in fact can almost be exactly replicated by keeping
terms to O(s8 ) in the perturbation away from the tautochrone, hi (si ), as shown in Fig. 2.13.
It is also noteworthy that, as fully derived in [14], the nonlinear path parameter which yields
a tautochronic epicycloid, λte , can be expressed in terms of the absorber tuning as,

λte =

n2
˜
.
n2 + 1
˜

Recalling that λ = 1 yields a cycloidal path, as n → ∞ the tautochronic epicycloid
˜
approaches a cycloid. One then expects the absorber tuning order to effect the accuracy of
the results, causing a general formulation of the dynamics non-tautochronic path absorbers
to be difficult.
We begin by the analysis on a single order absorber/rotor system in order to asses both

43

1
Full Cycloid
4

N=1, O(s )

0.95

N=2, O(s6)
N=3, O(s8)

x(s)

0.9

0.85

0.8

0.75
0

0.2

0.4

0.6

s / smax

0.8

1

Full Cycloid

0.84

N=1, O(s4)
N=2, O(s6)

0.83

N=3, O(s8)

x(s)

0.82
0.81
0.8
0.79
0.78
0.8

0.85

0.9

0.95

s / smax

1

1.05

Figure 2.13: Normalized rotor center to absorber COM length as a function of absorber arc
length for different numbers of truncations of the absorber path.
the ability of the perturbation equations to accurately estimate the dynamics of the system,
as well the effect of the absorber tuning on this ability. Shown in Fig. 2.14 are the percent

44

difference in absorber amplitudes obtained from the perturbation equations to those found
by numerically integrating the full equations of motion versus the absorber tuning order for
cycloidal paths. It is know from previous researchers, and can be confirmed by numerical
simulations, that the cycloidal path causes the the absorber frequency to increase with amplitude, a so-called hardening nonlinearity. We therefore perform the comparisons between
the perturbation equations and the numerical simulations at a torque level which produces
an absorber amplitude equal to 90% of it’s cusp amplitude, as found from simulations. It
is clear from Fig. 2.14 that using the tautochronic path scaling to capture the dynamics
of cycloidal path absorbers is yields large errors as compared to numerical simulations at
large absorber amplitudes, although these errors are decreased as the absorber tuning order
is increased. It is therefore necessary to perform a “strongly” nonlinear analysis in to accurately predict the dynamics of cycloidal path absorbers for a general tuning order. This
analysis would require the use of action/angle coordinates and could be quite involved. For
this reason, further investigation of multiple order, cycloidal path absorber systems is not
undertaken in this study.
A similar analysis on the ability to analytical investigate circular path absorbers is not
given here, as previous research has shown the detrimental instabilities that can occur for
such paths [34,36,47]. Circular path absorbers are only beneficial in their ease of implantation
and should only be considered as a last resort.

45

Percent Error of Perturbation Results as
Compared to Numerical Simulations

17
16
15
14
13
12
11
10
9
8

1.5

2

2.5

3

3.5

4

Absorber Tuning Order

4.5

5

Figure 2.14: Percent error between the perturbation equations and numerical simulations
for a single order cycloidal path absorber system vs. absorber tuning order.

46

Chapter 3
Experimental Validation
The analytical predictions presented in the previous chapter are investigated through the
use of a customized experimental setup here. The experimental setup is described and a
set of parameter identification tests are conducted. Synchronous, steady state experiments
are performed on multiple order absorbers, as well non-synchronous stability tests. The
experimental data is found to agree with the theoretical results.

3.1

Experimental Setup

The experimental rig consists of a shaft (the rotor) with a mounting section for up to four
absorbers, driven by a feedback-controlled electric motor. The setup is equipped with four
absorbers here, two tuned to order n1 and two tuned to order n2 , where n2 = 2˜ 1 . For the
˜
˜
˜
n
subsequent discussion, the order n1 absorbers are referred to as absorbers 1 and 2 and the
˜
order n2 absorbers as 3 and 4. A schematic diagram of the experimental setup is shown in
˜
Fig. 3.1, outlining the sensors and controls, explained subsequently. A photograph of the
experimental setup, equipped with the multiple-order absorbers is shown in Fig. 3.2, giving
the reader a view of the actual components described below.
The instantaneous angle of the absorbers φ (relative to the rotor) is measured via an
optical encoder. In order to compare the measured absorber angle to the analytical formulation, we must convert this angle into an arc length, S (in meters). For an epicycloidal

47

Control
Box

Servo
Motor

Torque
Measurement

Encoders
Rotor

PC

Absorbers
Figure 3.1: Schematic of the experimental setup.
path, this measured angle can be described in terms of the absorber arc length, as shown in
Fig. 3.3, as
φ = tan−1

x(S)
y(S)

,

(3.1)

where the x(S) and y(S) distances for an epicycloidal path can be shown to be

x(S) =

y(S) = L +

ρ0
1 − λ2

ρ0
1 − λ2

(sin(ξ) cos(λξ) −

λ2 S
cos(ξ)),
ρ0

(cos(ξ) cos(λξ) +

λ2 S
sin(ξ) − 1),
ρ0

where ξ = (1/λ) sin−1 (Sλ/ρ0 ), which is the local tangent angle (in radians) to the pendulum
COM curve with respect to the x axis , and L is the length of the encoder arm. In order
to get a relationship between the measured encoder angle and the absorber arc length,
48

(a)

(b)

(c)
Figure 3.2: Photograph of the experimental setup. (a) The entire setup, showing the motor,
rotor, and corresponding motor control and data acquisition components. (b) Multiple order
experimental absorbers. (c) Self centering rollers.
Eq. (3.1) must be solved for S in terms of φ. This can only be done numerically, when given
the values of the encoder arm length, L, non-dimensional large-amplitude path coefficient,
λ, and the path vertex radius of curvature, ρ0 (in m). These values are indeed known,
and interestingly, when one plots the absorber arc length versus the encoder angle for the
49

Ro
=

te

=0

=1
L

y(S)

x(S)
Figure 3.3: Geometrical schematic of the measured absorber angle via an optical encoder
with a rotating arm of length, L (λte provides a tautochronic epicycloid and λ1 yields a
cycloid.
epicyloidal path formulation and compares this to the obtained absorber arc length for a
circular path (S = Lφ), the differences are negligible, as shown in Fig. 3.4. In fact the error
in calculating the arc length from a circular path assumption when the absorber is at its
amplitude limit is only 0.112%. Thus we can use the circular path assumption in order to
translate the measured encoder angle into an absorber arc length.
˙
The instantaneous rotor speed θ is also measured by an optical encoder, from which
the mean and harmonic components, as well as the angular acceleration, can be distilled.
The torque T (θ) applied to the rotor is supplied by an input voltage to the armature, and
is quantified by measuring the current sent to the motor; a current to voltage conversion

50

0.16
0.14
0.12

s

0.1

s, Epicycloidal Path
s, Circular Assumption

0.08
0.06
0.04
0.02
0
0

5

10

15

20

(deg)

25

30

35

Figure 3.4: Calculated non-dimensional absorber arc length, s, versus measured encoder
angle, φ, for epicyloidal and circular path formulations.
is set in the control box which allows the corresponding torque voltage to be displayed in
LabVIEW. Using the inertial properties of the motor given by the manufacturer, the torque
(in Newton-meters) can then be obtained from the voltage measurement. All three of these
signals (absorber angle, rotor speed, and torque) are fed into a PC running data acquisition
and control software (coded in LabVIEW), which allows for real-time viewing and postprocessing of data in the time or order (frequency divided by the mean speed) domains. The
custom written LabVIEW code also allows for PID feedback control of the mean rotor speed
to maintain a nearly constant mean speed, upon which the fluctuating torque is applied. An
error analysis has been conducted on the experimental sensors in order to obtain quantitative
measures of the accuracy of the data and is presented in Appendix A.1. An important
feature of this setup is that the fluctuating components of the applied torque are based on

51

the rotational angle of the rotor, which allows for accurate engine order excitation, for which
multiple engine order harmonics can be generated, and also allows for external data from, say
a dynamometer, to be used to generate the torque pulses. The ability to change the applied
order of the torque allows one to easily investigate the influence of mistuning between the
absorber and applied torque order, without having to change the absorber hardware. Further
details of this rig are described in [40].

3.2

System Parameter Values

In order to quantify certain physical parameters associated with the experimental setup that
cannot be directly measured, specifically the rotor inertia, absorber tunings, and absorber
and rotor damping, a set of parameterization experiments are first conducted. The unknown
parameters listed above can be extracted by running the experiment under certain conditions
and utilizing the available measured quantities, which will now be explained in detail.

3.2.1

Rotor Inertia

To obtain the rotor inertia, J, the attached absorbers are locked into place at their respective
vertices, which reduces the equation of motion for the rotor to,

¨
˙
J θ + c0 θ = T0 + T sin(nΩt),

(3.2)

˙
where θ is the rotational angle of the rotor, () = d()/dt, c0 is the rotor damping, and T0 ,
T are the amplitudes of applied mean and fluctuating components of torque, respectively.
Assuming the rotor damping force balances the mean torque, the rotor angular velocity,
˙
θ can be measured for different levels of applied fluctuating torque, T , when applied at a
52

constant order, n. The order n rotor angular acceleration fluctuations can then be computed
from the measured velocity, and are plotted versus different values of the applied order n
torque, as shown in Fig. 3.5. A line is then fit to this data and the slope of this line is
1/Jmeas , from which we obtain the rotor inertia. It should be noted that when measuring
the inertia, the absorbers are kept on and locked instead of being removed. This is due to
the fact that the bifilar absorbers do not rotate with respect to the rotor and, therefore the
inertia of the absorber about their center of mass can be added to the rotor inertia. The
rotor inertia used for analysis is then easily calculated from the measured inertia as
M
J = Jmeas −

2
(mi Ri,0 ).

i
The rotor inertia is different for each of the subsequent experimental investigations, as we
add and remove absorbers for the different tests and the overall inertia ratio is desired to be
kept the same, done by adding inertia to the rotor. Therefore this test must be repeated each
time and, for this reason the obtained rotor inertia values for each experiment are presented
in the corresponding experimental investigation sections below.

3.2.2

Absorber Tunings

The absorbers were designed to be tuned to a certain order, but one must experimentally
verify the actual absorber tuning order. It can be shown [19] that amplitude of rotor angular
acceleration, when compared against the forcing order n, for a given torque level, will have
a minimum at the tuning order of the pendulum. Shown in Fig. 3.6 is the amplitude of the
order n harmonic of the rotor angular acceleration divided by the measured amplitude of
the applied order torque versus the order of the applied fluctuating torque for an order n1
˜

53

Amplitude of Rotor Angular Acceleration

250

200

Experimental Data
Fit

150

100

50

0
0

2

4

6

Torque (Nm)

8

10

12

Figure 3.5: Amplitude of order n rotor angular acceleration versus the amplitude of applied
order n torque, with absorbers locked.
absorber. The angular acceleration amplitude is divided by the measured applied fluctuating
torque because, although the desired amplitude of torque is kept constant along the entire
order range (as specified by the user), the motor may produce small fluctuations about the
specified torque. From Fig. 3.6, it is seen that the absorber’s tuning order is n1 = 2.3. Similar
˜
experiments were carried out on the other three absorbers in order to quantify each absorber’s
tuning order, where the results are presented in Table 3.1. When performing these order
sweeps, care must be given to the level of applied fluctuating torque used. All previously
studied absorber paths (circles, epicycloids, cycloids) have some sort of large amplitude,
non-linear behavior with circles producing a softening non-linearity and epicycloids and
cycloids a hardening one. If the amplitude of fluctuating torque is too large, then these
inherent non-linearities effect the measured tuning order, moving it lower or higher than the
actual value, depending on the path. If the amplitude of fluctuating torque is too small,

54

the absorbers will not overcome their frictional damping force, and thus not oscillate. It is
therefore somewhat of an art form to correctly apply the right amount of fluctuating torque

Rotor Angular Accel / Torque [rad/s2/Nm]

for generating responses that are good for estimating the tuning order.
12
11
10
9
8
7
6
2.2

2.25

2.3

Order of Fluctuating Torque, n

2.35

Figure 3.6: Order n amplitude of rotor angular acceleration divided by the amplitude of
fluctuating torque as a function of the applied torque order, n.

3.2.3

Absorber Dampings

An equivalent viscous damping is used to model the energy dissipation mechanisms in the
experimental absorbers. In order to quantify this damping, a logarithmic decrement scheme
[45] is employed. In order to achieve a free vibration response experimentally, an amplitude of
oscillating torque which produces an absorber amplitude of ≈ 12◦ for the order n1 absorbers
˜
and ≈ 7◦ for the order n2 absorbers is suddenly shut off, while the rotor maintains a
˜
nearly constant speed. These absorber amplitudes were chosen to allow for a sufficient
amount of absorber decay time while at the same time keeping the absorbers in their linear

55

response regime. The experimentally obtained absorber amplitudes versus there respective
times of decay are shown for both order absorbers in Fig. 3.7. As commonly found in all
undergraduate vibrations texts, the ratio between N peaks in a single degree of freedom,
free vibration decay is given by

δ=

1
ln
N

xi
xi+N

= ζωn τd ,

where ζ is the non dimensional damping ratio, ωn = ni Ω, and τd = 2π/ωn
˜

1 − ζ 2 . One

can then solve for the non-dimensional damping ratio from

ζ=

δ
δ 2 + 4π 2

.

In order to overcome the quantization error inherent in the optical encoders, as well as
to calculate the average viscous damping coefficient over the entire decay, nn xi near the
beginning of the decay is used with an xi+N near the end of the decay in order to get an
average damping value.
The obtained non-dimensional damping values for each absorber are presented in Table 3.1, for a mean rotor speed of 300 RPM, as the non-dimensional damping is defined as
µa,i = ca /(mΩ), where ζa,i = µa,i /(2˜ i ).
n

3.2.4

Rotor Damping

The analytical treatment of the system at hand involves an assumption that the rotor damping force balances with the mean torque, T0 = c0 Ω, which eliminates the effect of the rotor
damping in the first order of . The rotor damping effect is included in the numerical simula-

56

Absorber 1 Free Vibration
10

5

0

−5

−10
4.8

5

5.2

5.4

5.6

5.8

Time (s)

6

6.2

6.4

(a)
Absorber 3 Free Vibration
6
4
2
0
−2
−4
−6
5.8

6

6.2

6.4

Time (s)

6.6

6.8

7

(b)
Figure 3.7: Free vibration response of both order absorbers. (a) Order n1 absorber. (b)
˜
Order n2 absorber.
˜
tions, though, which requires the quantification of the experimental rotor damping. In order
to accomplish this, we run the rotor at multiple constant speeds with the absorbers locked

57

and measure the applied mean torque required to obtain these speeds. When the rotor spins
at a constant speed, Ω, Eq. (3.2) reduces to

c0 Ω = T0 .
The measured mean speed is then plotted against the measured mean torque for a range of
speeds, as shown in Fig. 3.8. If the rotor damping force is not a function of mean speed, then
the data should be linear in the mean torque with a slope of 1/c0 . Looking at Fig. 3.8, the
data is not precisely linear, but a linear assumption suffices here, as the rotor damping has a
small effect on the system dynamics. The dimensional value of this damping is in Table 3.1.
40
35

Experimental Data
Linear Fit

Speed (rad/s)

30
25
20
15
10
5
0
0.5

0.6

0.7

0.8

Mean Torque

0.9

1

Figure 3.8: Rotor mean speed, Ω vs. the mean torque applied, T0 .
All of the experimental parameters obtained by the methods described above are presented in Table 3.1.

58

Table 3.1: Values of Experimental Parameters.
Parameter
Definition
m1,2
absorber 1 and 2 mass
m3,4
absorber 3 and 4 mass
α
inertia ratio between multiple order absorbers
R1,0
rotor center to absorber 1 and 2 COM
R3,0
rotor center to absorber 3 and 4 COM
n1
˜
absorber 1 tuning order
n2
˜
absorber 2 tuning order
n3
˜
absorber 3 tuning order
n4
˜
absorber 4 tuning order
µa,1
absorber 1 viscous damping
µa,2
absorber 2 viscous damping
µa,3
absorber 3 viscous damping
µa,4
absorber 4 viscous damping
c0
rotor damping

3.3

Value
0.4312 kg
0.0482 kg
0.106 kg
0.12316 m
0.123 m
2.3
2.3
4.6
4.6
0.1
0.1054
0.191
0.186
0.0144 Nms

Steady-State Experimental Results

Utilizing the experimental setup described above, steady-state absorber and rotor responses
as functions of the amplitude of applied fluctuating torque were tested for sets of multiple order absorbers. Synchronous and non-synchronous absorber responses are compared
against the perturbation results derived in the previous chapter (Eqs. (2.10) to (2.13)). The
experimental rotor angular acceleration fluctuations are compared with predictions obtained
by extracting the first three rotor harmonics from the scaled and expanded form of the rotor
angular acceleration, as given in Eq. (2.7). Numerical simulation results are also presented
for each experimental investigation.

59

3.3.1

Synchronous Response

To experimentally test the synchronous response of multiple order absorbers, one absorber at
each order is unlocked and free to oscillate. The amplitude of the fluctuating torque, applied
at n = 2.29, is varied and the response of the absorbers at each order is measured through
optical encoders. Shown in Fig. 3.9 is the experimentally obtained non-dimensional absorber amplitudes versus the amplitude of applied order n torque, along with the predictions
obtained through the perturbation equations (labeled as theory) and numerical simulations.

0.18
0.16

s1, Experiment

s1, Cusp

s3, Experiment

0.14

s1, Theory

0.12

s3, Theory
s1, Simulations

s

0.1

s3, Simulations

0.08
0.06

s3, Cusp

0.04
0.02
0
0

0.05

0.1

0.15

0.2

Γ (Order n)

Figure 3.9: Experimental absorber amplitudes, s, vs. the amplitude of applied order n
torque, Γn , with perturbation and numerical simulation predictions. (n = 2.29, = 0.18486)

As visible from Fig. 3.9, the perturbation equations accurately predict the amplitudes of
both order absorbers for the entire torque range. The internal 2 : 1 resonance, which drives
the order n2 absorber is accurately captured in the experimental data, as the forcing is
˜
directly resonating the order n1 absorber. There exists a small error between the predictions
˜
and the experimental results at low torque levels, most likely due to the absorber damping
60

having features that are not included the model, explained subsequently. First, the level of
absorber damping generally is a function of its amplitude [47], whereas a single effective value
is used for all amplitudes in the present study, in order to match the analytical models. In
fact, it is known that some level of nonlinear damping is present in the system, and this has
been investigated in detail for the Coulomb friction aspects of the circular path absorbers [47].
In general, the motion of bifilar absorbers involves frictional rubbing between the absorber
and flange, as well as rolling resistance of the rollers. The rubbing is particularly problematic
when the rotor axis is vertical and the rotor is run at low speeds, since gravity will promote
rubbing. (In fact, gravity is an important effect when running the system with a horizontal
rotation axis at the speeds feasible with this equipment [44]). In order to overcome some
of the difficulties associated with these effects, self-centering rollers are implemented for
the tautochronic bifilar absorbers as shown in Fig. 3.2(c) [2]. In order to center the bifilar
absorber with respect to the flange with these rollers, the absorbers must oscillate, and this
requires a certain threshold of oscillating force acting on the absorber, thus giving a physical
explanation to the damping as a function of absorber amplitude. In fact, this low amplitude
effect is similar to the dry friction measured for circular path absorbers [47]. Also, as detailed
in Section 3.2.3, the absorber damping coefficient used for experimental data is measured by
performing a logarithmic decrement on the absorber response when the oscillating part of the
applied torque is suddenly switched to zero from an amplitude that causes the absorber to
oscillate at an amplitude of about 12◦ . The decrement is performed over the entire absorber
decay, and the method assumes a single degree of freedom system ( → 0), whereas the
actual dynamics involve coupled oscillations of the rotor and absorber.
Looking at the corresponding experimental rotor harmonics, as shown in Figs. 3.10
and 3.11, the predictions again accurately match the data. The experimental order n rotor
61

harmonic deviates a small amount from the predictive results at low levels of applied fluctuating torque. This is due to the fact that the experimental absorber amplitude at this level
is smaller than the predictions, thus not absorbing as much of the torsional oscillations as
the model predicts. The order 2n and 3n experimental data match the predictions across
the entire range of torque, and this deviation of accuracy in the order n harmonic due to the
absorber amplitude error at small torque levels is not observed. Most likely, this is due to
the amplitudes of the higher harmonics being relatively small at these low levels of torque.
Overall, the synchronous multiple order predictions track the experimental data throughout the range of applied fluctuating torque, confirming the accuracy of the analytical and
numerical predictions.

3.3.2

Non-Synchronous Response: Order n1 Absorbers Only
˜

For the following experimental results, both order n1 = 2.3 absorbers are unlocked, and the
˜
order n2 absorbers are removed, yielding an inertia ratio of = 0.226. Combinations of two
˜
orders of applied fluctuating torque at orders n = 2.3 and 2n = 4.6 are applied. The applied
torque is increased until the non-synchronous bifurcations captured analytically in Section 2.5
are experimental observed. Four points covering the Γn,2n surface are investigated, as shown
in Fig. 3.12.
As visible from Fig. 3.12(a), the analytical stability curve, generated from a numerical
continuation of the perturbation equations, accurately predicts the torque amplitudes which
cause the absorber instabilities. The points labeled (1) in Fig. 3.12(a) exhibit the synchronous
and non-synchronous absorber responses generated from a torque at order n only. The
square data point corresponds to the experimental torque which causes the non-synchronous
response. It can be seen from Fig. 3.12(b) that their is phase difference between the two
62

absorbers, and in some cases, this instability causes an absorber to increase in amplitude and
reach it’s corresponding amplitude limit. The period doubling, sub-harmonic bifurcation is
labeled as the square point (4) in Fig. 3.12(a), and arrises when the applied torque is at
order 2n only. This bifurcation causes a drastic increase in the absorbers amplitudes, and
they quickly reach their amplitude limits, as visible from Fig. 3.12(c). This sub-harmonic
bifurcation results in the absorbers oscillating completely out of phase with each other, as
captured in Fig. 3.12(d), which exhibits a short time recording of the sub-harmonic response.

3.3.3

Non-Synchronous Response: Order n1 and n2 Absorbers
˜
˜

For the next set of experimental results, all four of the experimental absorbers are unlocked
(two at each order). As in the previous section, the absorber instabilities in the Γn,2n torque
space are experimentally compared to the analytical results obtained in Section 2.5. For an
accurate comparison to the previous section, the ratio of the absorber inertia to that of the
rotor was desired to be the same value. In order to accomplish this, due to the added inertia
of the order n2 absorbers, inertial weights were added to the rotor. The corresponding
˜
measured inertia ratio was found to be, = 0.218. This is a 3% decrease in the inertia ratio
used in the previous section and will suffice for our investigation.
Figure 3.14 exhibits the theoretical stability boundary as a function of the two applied
fluctuating torque orders, with experimental results for stable and unstable absorbers responses. The experimental data agrees with the analytical prediction, with the order n1
˜
absorber bifurcations being the same as detailed in the previous section. Interestingly, the
addition of the order n2 absorbers does not increase the level of fluctuating torque which
˜
causes the absorber bifurcations. In fact, the experimental results motivated further study
into this phenomenon.
63

It was found that both the inertia ratio, , as well as the absorber damping effect the
stabilization of the order n1 absorbers through the use of higher order absorbers. Specifically,
˜
as the inertia ratio between the absorbers and rotor changes, the optimal inertia ratio of
different order absorbers changes as well. The optimal different order absorber inertia ratio
of ≈ 0.38 given in Section 2.5 is valid for the specific inertia ratio of absorbers to rotor used
in that case ( = 0.05). The absorber damping plays an important role in the stabilization as
well, and if the absorber damping is too large, the addition of order n2 absorbers will decrease
˜
the torque level which causes the non-synchronous response of the order n1 absorbers. For the
˜
experimental parameters used, it was found that a non-dimensional damping ratio coefficient,
ζ, of 0.018 was the damping threshold which caused the order n2 absorbers to no longer
˜
increase in the stabilization of the order n1 absorbers. The absorbers used in this study have
˜
a damping ratio of ζ ≈ 0.2, for which no apparent benefit in terms of system stabilization
is gained. This does not render the results in Chapter 2 practically unattainable however.
The damping ratio in the experimental absorbers used in this investigation should be much
larger than that used in applications, specifically automotive, for two reasons. First, the
spin rig used in this study spins in the vertical plane, causing the effects of dry friction to
be larger than in an horizontally spun rotor. Second, the damping ratio is a function of the
rotor mean speed, Ω, for which the tests conducted in this study are at a much lower speed
than in most applications.

3.4

Conclusions

The experimental setup utilized to investigate the dynamics of multiple order absorber systems has been detailed. The indirect measurement of the parameters associated with this

64

setup that cannot be directly calculated is detailed. Experimental results are shown to agree
with the theoretical predictions given in the previous chapter. Specifically, synchronous
multiple order experimental absorber responses are shown to closely match the theoretical
and numerical predictions. Experimental non-synchronous responses for sets of order n1
˜
absorbers quantitatively match the stability boundaries calculated through the use of a numerical continuation of the perturbation equations. The experimental bifurcations of the
order n1 absorbers are then investigated with the addition of order n2 absorbers. The ex˜
˜
perimental and theoretical results are again found to be in agreement, proving the utility of
the analytical predictions for use in absorber design.

65

0.16
0.14
0.12

yy′

0.1
0.08

Order n, Experiment
Order 2n, Experiment
Order 3n, Experiment
Order n, Theory
Order 2n, Theory
Order 3n, Theory
Order n, Simulations
Order 2n, Simulations
Order 3n, Simulations
Absorbers Locked

0.06
0.04
0.02
0
0

0.05

0.1

0.15

0.2

0.15

0.2

Γ (Order n)
(a)

0.16
0.14
0.12

Order n, Experiment
Order n, Theory
Order n, Simulations
Absorbers Locked

yy

′

0.1
0.08
0.06
0.04
0.02
0
0

0.05

0.1
Γ (Order n)
(b)

Figure 3.10: Experimental rotor angular acceleration, yy vs. Γn for multiple order epicycloidal absorbers compared against the perturbation and numerical predictions (n = 2.29,
= 0.18486). (a) First three rotor response orders, n, 2n, 3n. (b) Order n.
66

yy

′

0.1

Order 2n, Experiment
Order 2n, Theory
Order 2n, Simulations
Absorbers Locked

0.05

0
0

0.05

0.1
Γ (Order n)

0.15

0.2

0.15

0.2

(c)

0.14
0.12
0.1

Order 3n, Experiment
Order 3n, Theory
Order 3n, Simulations
Absorbers Locked

yy

′

0.08
0.06
0.04
0.02
0
0

0.05

0.1
Γ (Order n)
(d)

Figure 3.11: (c) Order 2n. (d) Order 3n.

67

0.25
Stability Boundary (Theory)
Unstable Response (Experiment)
Stable Response (Experiment)

0.2

Γ (Order 2n)

(4)
0.15

(3)

0.1
(2)

0.05

(1)

0
0

0.05

0.1

0.15

0.2

Γ (Order n)
(a)
Absorber 1
Absorber 2
Cusp

0.15
0.1

s

0.05
0
−0.05
−0.1
−0.15
9.5

10

Time (s)

10.5

(b)
Figure 3.12: Experimental stability results for a pair of order n1 absorbers. (a) Theoretical
˜
stability boundary with a sample of stable and unstable experimental data points. (b)
Experimental time response of the non-synchronous absorber response corresponding to the
data point labeled (1).

68

0.15
0.1

Absorber 1
Absorber 2
Cusp

s

0.05
0
−0.05
−0.1
−0.15
4

5

6

7

Time (s)

8

9

10

(c)

0.15
0.1

Absorber 1
Absorber 2
Cusp

s

0.05
0
−0.05
−0.1
−0.15
9

9.2

9.4

9.6

9.8

10

10.2

(d)
Figure 3.13: (c) Experimental time response of the sub-harmonic absorber response corresponding to the data point labeled (4) in Fig. 3.12. (d) Short time sample of the sub-harmonic
instability, showing the out of phase absorber motion.

69

Γ (Order 2n)

0.2

0.15

0.1

0.05
Stability Boundary (Theory)
Unstable Response (Experiment)
Stable Response (Experiment)

0
0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Γ (Order n)
Figure 3.14: Experimental absorber amplitudes, s, vs. the amplitude of applied order n
torque, Γn , with perturbation and numerical simulation predictions. (n = 2.29, = 0.18486)

70

Chapter 4
A Harmonic Balance Approach to the
Dynamics of CPVAs
In this chapter harmonic Balance (HB) methods are applied to the scaled equations of
motion Eqs. (2.7) and (2.8) as given in Section 2.3. The purpose of this analysis is that
it provides explicit formulas that predict the steady-state system response, including its
harmonic content, although without stability information. Two assumptions about the form
of the absorber amplitudes are used in the approximations, and relatively simple formulas
are derived which accurately predict absorber amplitudes and corresponding rotor response.
Predictions are validated using numerical simulations and experimental data.

4.1

General Harmonic Balance

The rotor/absorber system is dynamically complicated, as inherent nonlinearities in the absorber motion and the absorber/rotor coupling do not allow for a closed form solution for
the system dynamics. Current methods of analyzing CPVAs involve numerical simulations
and perturbation analysis. The perturbation methods do allow for accurate studies of the
system dynamics as a function of system parameters, and can capture the detrimental nonlinear instabilities inherent in the system [3, 8, 39]. For example, when the absorber path
is a circle, the absorbers behave similar to a simple pendulum and become detuned at large
71

amplitudes. This detuning can cause a jump bifurcation in the pendulum’s amplitude, which
is accompanied by a phase shift in the absorber response causing amplification of the rotor
vibrations [3, 36]. Denman [14] grouped a set of absorber paths into a two parameter family
of curves, one parameter controlling the linear tuning of the absorber and the other controlling the large amplitude nonlinear behavior. The two extremes of this family of curves,
in terms of the nonlinear parameter, are circles (softening path) at one end and cycloids
(hardening path) at the other. Between these two extremes is a family of epicycloids, one
of which is the so called tautochronic epicyloid, which renders the absorber motion, when
decoupled from the rotor, linear over all feasible amplitudes (that is, out to the cusp on
the curve). Implementing non-circular paths can alleviate the jump bifurcation, but nonsynchronous absorber responses can still occur, due to the multiple harmonics imparted onto
the rotor from the absorber/rotor nonlinear coupling [3, 8, 39, 46]. One important aspect of
these nonlinear interactions is that the second harmonic (2n) can cause, through parametric
coupling terms, subharmonic instabilities in the system.
The perturbation methods referenced above provide an accurate representation of the
system dynamics, but involve quite complicated calculations and resulting expressions [9,39,
46]. In this chapter, we derive relatively simple formulas for the absorber amplitude and the
harmonic amplitudes of the corresponding rotor angular acceleration as a function of system
parameters. In order to obtain these formulas, the harmonic balance method is applied to
a scaled version of the equations of motion. The harmonic balance method is an efficient
way to estimate the steady-state dynamics of nonlinear systems, as commonly applied in
nonlinear electric circuits [18]. We first utilize scaling techniques in order to decouple the
absorber dynamics from those of the rotor, and two different assumptions about the form of
the amplitude of the absorbers are used to perform the harmonic balance.
72

We begin by applying the harmonic balance methods to a quite general system that
involves a rotor equipped with absorbers tuned to two orders and subjected to a torque
composed of two orders. In the first harmonic balance method, the so-called general method,
the unknown steady state amplitudes of the degrees of freedom are assumed to be unknowns
and take no particular form. It will be shown that due to the complicated non-linearities
inherent in the system, usable closed form solutions for the absorber amplitude and rotor
angular acceleration as a function of the applied torque cannot be obtained by this method.
To overcome this, the applied torque is expressed as a function of the absorber amplitudes
and other parameters, which allows for accurate absorber design studies to be carried out.
In the second approach, we assume the absorber amplitude to be in the form of a power
series in the (non-dimensional) torque amplitude. This formulation allows for the absorber
amplitude to be obtained explicitly as a function of the system parameters, by collecting
harmonics of the absorber response, as well as powers of the applied torque. The utility of
this method is in the fact that relatively simple formulas for absorber and rotor response
can be obtained in terms of the other relevant parameters in the system. It will be shown
that due to the assumption about the absorber amplitude, certain solution branches for the
the absorber amplitude will not be captured, and there is some degradation in the accuracy
of this theory at large absorber amplitudes.
We begin with the harmonic balance analysis of the general system consisting of absorbers
tuned to two orders as well as forcing at two orders. This formulation will lead to the most
general form of the approximations, which can then be greatly simplified in some special cases
of interest. As previously stated, the application of harmonic balance to this system, although
providing accurate predictions to the steady-state dynamical response of the absorbers and
rotor, will fail to capture instabilities of the system response [3, 8, 23, 39]. For this reason,
73

for this investigation we assume a response of the absorbers in which each group is mutually
synchronous, implemented by grouping the inertia of all the absorbers at each order into a
single effective absorber, thus simplifying Eqs. (2.7) and (2.8) into the form

w = Γn sin(nθ) + Γ2n sin(2nθ + φ)
+

1
2
(2˜ 2 s1 s1 + n2 s1 g10 − s1 g10 )
n1
˜1
1+α

(4.1)

2
˜2
+ α(2˜ 2 s2 s2 + n2 s2 g20 − s2 g20 ) ,
n2
sj + n2 sj = fj (sj , sj , .., s , µ, nj , nj , λ, T (θ)) + O( 2 ) j = 1, 2,
˜j
˜

(4.2)

where,
1
fj = hj /2 − µa,j sj + [sj + g0j (sj )][
˜
(−2n2 s1 s1 − n2 g01 (s1 )s1
1
1
1+α
+

dg01 (s1 ) 2
dg (s ) 2
s1 + α(−2n2 s2 s2 − n2 g02 (s2 )s2 + 02 2 s2 ))
2
2
ds1
ds2

˜
˜
− Γn sin(nθ) − Γ2n sin(2nθ + φ)],
¨
in which w = yy = θ2 is the non-dimensional rotor angular acceleration, which we use
Ω
as the measure to quantify rotor torsional vibration. Also, due to the complicated nature
of the path functions, specifically gi (si ), which is explained in detail in Chapter 2 1 , one
must expand Eqs. (4.1) and (4.2) in the absorber arc length si in order to obtain explicit
expressions; here we keep terms up to O(s5 ).
i
It has been previously found, from extensive simulation and experimental studies, that
the motion of the absorbers is very closely resembled by a single harmonic, for all paths.
1 This

term is specifically the radial distance from the rotor center to the absorber center multiplied by the dot product of the unit vectors in the rotational direction of the vertex of the absorber
path and the tangent of this curve as the absorber moves along the path.

74

Using this as a guideline for the harmonic approximation of the response, we assume the
absorbers each respond at single harmonic, that is, they have the form,

s1 = A1 sin(nθ) + B1 cos(nθ),

(4.3)

s2 = A2 sin(2nθ) + B2 cos(2nθ).

(4.4)

and

Equations (4.3) and (4.4) which are inserted into Eq. (4.2) and harmonics are balanced.
The balancing of harmonics results in four equations in the four unknowns, A1,2 , B1,2 .
Commonly, the absorber amplitudes would be solved for in terms of the forcing, but in this
case the nonlinearities do not allow for a closed form solution for the unknown harmonic
amplitudes. However, by neglecting damping the phases are simplified such that one can
eliminate the B1,2 terms. The applied torques can then be solved for in terms of A1,2 and
other relevant system parameters as follows,
−(16A1 (A2 n3 (−8 + A2 (˜ 2 + n4 ))α
2 n2 ˜ 2
X1
˜
−2Γ2n n(1 + α) + 2n2 (2 + 2α − 2 + A2 n4 )
1 ˜1
+
X1
−2(1 + α)(2˜ 2 − 3A2 ϕ1 )))
n1
1
+
,
X1

(4.5)

−(8(−8A2 (˜ 2 (1 + α) + 4n2 (−1 + α(−1 + )))
n2
X2
−8A2 n˜ 2 + 3A2 A2 n˜ 2 n2 (1 + n2 )
˜2
1 n1
1 2 n1 ˜ 2
X2
4A3 (4n2 n4 α + 3(1 + α)ϕ2 ) ))
˜2
2
,
X2

(4.6)

˜
Γn =

˜
Γ2n =

75

where,
Xi =(−64 + 5A4 n4 (1 + n2 )2 + 24A2 (˜ 2 + n4 ))(1 + α)
˜i
i ˜i
i ni ˜ i
i = 1, 2.
The above equations, with some manipulations that are subsequently explained in further detail, allow for accurate prediction of the absorber response as a function of torque
amplitudes. This is essentially accomplished by choosing amplitudes for the absorber amplitudes and then solving for the corresponding torque amplitudes. In order to gauge the
effectiveness of the absorbers in terms of reducing torsional vibrations of the rotor, we consider the rotor angular acceleration, defined as w in Eq. (4.1). While each absorber responds
at single harmonic, the rotor response is composed of multiple harmonics. Observations, justified by the following analysis, shows that the rotor response can be well approximated by
three harmonics. These multiple harmonics are imparted onto the rotor by nonlinear interactions with the absorbers, and they can be quantified by inserting the absorber amplitude
approximations, Eqs. (4.3) and (4.4), into the rotor angular acceleration equation, Eq. (4.1),
and collecting harmonics. This results in terms for the rotor angular acceleration at three
different harmonic orders, with harmonic amplitudes given by,

wn =

−A1 n2 (−64 + A4 n4 (1 + n2 )2 + 8A2 (˜ 2 + n4 ))
˜1
1 ˜1
1 n1 ˜ 1
64(1 + α)

w2n =

˜
+ Γn ,

(4.7)

˜
16Γ2n (1 + α) − n(−16A2 n2 + A2 n
1 ˜1
16(1 + α)

(4.8)

× (−64 + A4 n4 (1 + n2 )2 + 8A2 (˜ 2 + n4 ))α),
˜2
2 n2 ˜ 2
2 ˜2
and
3A3 n2 n2 (1 + n2 )(16 + 3A2 (˜ 2 + n4 ))
˜1
1 ˜1
1 n1 ˜ 1 .
w3n =
128(1 + α)

76

(4.9)

Substitution of the torques from Eqs. (4.5) and (4.6) into these equations allows one to
express the rotor angular acceleration harmonics in terms of the absorber amplitudes. While
the expressions are straightforward, the actual calculation is not so. Typically one specifies
the system and input parameters, including the torque amplitudes, and solves for A1,2 from
Eqs. (4.5) and (4.6), using numerical methods. However, a well known approach to avoiding
numerics is to vary the absorber amplitude and solve for the corresponding torque, since
it is known explicitly. However, this is not possible when two torques are present. A way
to perform this operation in this case is to assume a linear relationship between the two
˜
˜
torque harmonic amplitudes and a constant relative phase between them, i.e Γ2n = β Γn ,
which is a good assumption for some automotive applications [17]. With this assumption,
one is still left with the problem of obtaining A2 from Eq. (4.6), which is a cubic equation.
One approach, which will be shown to be limited in accuracy, but does allow one to obtain
explicit expressions, is to keep only the terms linear in A2 , which yields
˜
A2 n˜ 2 + Γn (1 + α)β
1 n1
.
A2 = −
n2 (1 + α) + 4n2 (−1 + α(−1 + ))
˜2

(4.10)

˜
One must now only specify the range of amplitudes for A1 , and then Γn can be obtained, from which all other results follow. This process will be presented subsequently
using examples.

77

4.2

Harmonic Balance Using a Power Series in Torque
Amplitude

We now turn to a harmonic balance approach in which the absorber amplitudes are assumed
to have a special form, specifically as a power series in the applied torque amplitude. This
approach works since the torque amplitudes are non-dimensionalized by the twice the rotor
kinetic energy (JΩ2 ) and are therefore small in practice. This series expansion approach
yields convenient closed form solutions for the harmonic amplitudes for the absorber and the
rotor angular acceleration as a function of the applied torques and system parameters.
The absorber amplitudes are expressed as,
k
k
˜n
˜n
s1 = (
Ci Γi ) sin(nθ) + (
Di Γi ) cos(nθ),
i=1
i=1
and
k

k
˜ i + G Γi )) sin(2nθ) + (
˜
˜n
˜
s2 = (
(Ei Γn
(Fi Γi + Hi Γi )) cos(2nθ).
i 2n
2n
i=1
i=1
˜
We have included the Γn terms in the s2 equation in order to capture the effects of the internal resonances caused by the multiple harmonics imparted onto the rotor by the absorbers.
However, we do not include terms of the form

Γ2n in the s1 expansions, which would be

needed to capture the subharmonic vibration absorber [9, 22]. Inserting the above expres˜
sions into Eq. (4.2), collecting orders of Γn,2n , and balancing harmonics, linear equations
for the coefficients are obtained. The terms that are linear in the applied torque, C1 , D1 ,

78

E1 , F1 , G1 and H1 , are found to be,

C1 =

˜
4n(1 + α)(˜a,1 + µa,1 α + 2G1 Γ2n n2 α) 2
µ
˜

(4.11)

P1

1
(−(2(1 + α) (−2˜ 2 (1 + α) + 2n2 (1 + α − )
n1
D1 =
−P1

(4.12)

˜
˜
+ 4Γ2n H1 n3 α + Γ2n n(1 + α) )))
E1 = F1 = 0
G1 =

2˜a,2 n 2
µ
4α
4˜2 n2 2 + (˜ 2 + n2 (−4 + 1+α ))2
µa,2
n2

4α
(˜ 2 + n2 (−4 + 1+α ))
n2
H1 = −
4α
4˜2 n2 2 + (˜ 2 + n2 (−4 + 1+α ))2
µa,2
n2

(4.13)

(4.14)

where,
2
˜
P1 = 4˜ 4 (1 + α)2 − 16Γ2 (G2 + H1 )n6 α2 2 − n2 (1 + α)
n1
2n 1
˜
× (8˜ 2 (1 + α − ) + (Γ2 − 4˜2 )(1 + α) 2 ) + 4n4
n1
µa,1
2n
˜
˜
× ((−1 + )2 + α2 (1 − 2Γ2 H1 2 ) − 2α(−1 + + Γ2 H1 2 ))
2n
2n
Note that the above equations do not capture the perturbation away from the tautochrone,
ϕi , since they capture the linear response of the absorber. Solving for the higher order
expansions in Γn,2n , truncating at k = 5, one finds C2,4 , D2,4 , E3,5 , F3,5 , G2,4 , H2,4 = 0,
and the expressions for C3,5 , D3,5 , E2,4 , F2,4 , G3,5 , and H3,5 are complicated functions
of , n1,2 , n, µa,1,2 , and ϕ1,2 . These expressions are given in Appendix B.2, and although
˜
they are complicated in their general form, they simplify in many practical cases. Also, the
parameters in the expressions are always known explicitly, which allows one to simply insert
values into the expressions for an accurate representation of the absorber amplitude. Once

79

the expressions for the constants are known, the absorber amplitudes can easily be found
through

k
|s|1 =

k
˜n
Ci Γi )2 + (

(
i=1

˜n
Di Γi )2 ,

˜n
˜
Fi Γi + Hi Γi )2 .
2n

k
˜n
˜
Ei Γi + Gi Γi )2 + (
2n

(

(4.16)

i=1

k
|s|2 =

(4.15)

i=1

i=1

With the amplitude of the absorbers known, operations similar to those above can be
performed in order to get an accurate representation of the rotor response, outlined as
follows: The absorbers’ responses are expanded using Eq. (4.1) and powers of the applied
torque as well as harmonics of each order are collected. The amplitudes of the first two
harmonics of the rotor angular acceleration, wlin , when one keep only linear terms in the
torque amplitudes, are given by,

˜
wlin,n = Γn

˜
wlin,2n = Γ2n

2
(C1 n4 + (1 + D1 n2 + α)2 )
(1 + α)2

(16G12 n4 α2 + (1 + α + 4H1n2 α)2 )
.
(1 + α)2

(4.17)

(4.18)

The higher harmonics that are imparted onto the rotor response by system nonlinearities
obviously cannot be captured by the linear response assumption. In order to capture these
higher harmonics, and obtain a more accurate representation on the order n rotor response,
one must include higher orders of Γn,2n in the approximation. The inclusion of these higher
order terms yields additional terms in all orders of the rotor response; these are presented
in their most general form in Appendix B.2 for reference.

80

4.3

Examples

The accuracy of the equations formulated in the previous section will now be investigated
through some cases studies of special interest. We first look at the most general system
utilizing the general form of the approximations developed above, and extend the study to
include some special cases with simplifying assumptions, as well as the effects of perturbations
away from the tautochronic path.

4.3.1

Absorbers and Forcing at Two Orders

Assuming the system at hand consists of two absorbers, with one absorber having a tuning
twice of that of the other, as well as applied torques at each of these orders, one can use
Eq. (4.5) in its presented form, along with the solution for A2 as a function of A1 as
presented above, to capture the absorber amplitude as of function of torque. The absorber
amplitude approximations derived from assuming a power series in the applied torque can
also be utilized to look at the linear response assumption, done by simply truncating the
series expansions at k = 1 in Eqs. (4.15) and (4.16). The nonlinear responses can also be
used through the inclusion of the higher order terms in the expansions. For the subsequent
examples, the following non-dimensional parameters are used: n1 = 1.5 (corresponding to
˜
a three-cylinder four-stroke engine), n2 = 3, α = 0.5, µa,1 = 0.1, µa,2 = 0.2,
˜

= 0.05,

˜
˜
and the relationship between the multiple torques is assumed to be Γn = 0.25Γ2n with zero
phase between the two torques, which act directly at the resonant frequency of the absorbers
(n = n1 = 1.5, 2n = n2 = 3). There exists a maximum amplitude for each absorber, |s|max ,
˜
˜
referred to as the “cusp amplitude” (since it corresponds to a cusp on many paths), which is
the maximum amplitude the absorbers can achieve, due to the fact that non-circular paths

81

becomes undefined beyond this point. This amplitude is found by solving for |s|max from
g(|s|max ) = 0. In practice, stoppers are implemented at an amplitude that are typically
close to, but less than this cusp amplitude. Due to the expansions in the absorber arc length
that are employed, this amplitude limit is not captured in the approximations, but is evident
in the numerical simulations.
We take a detour here to explain the comparisons between the two HB methods. As
previously mentioned, the absorber amplitude is expanded to O(s5 ). The accuracy of the
i
power series in the applied torque HB method does not improve with orders of torque greater
than O(Γ3 ), as shown in Fig. 4.1, due to the breakdown of the amplitude expansions of g(s)
and other path functions near the cusp of the path.
1.5

Cusp
s / s, max

1

|s|, Lin
|s| (O (Γ3) )

0.5

|s| (O (Γ5) )
Simulations
0
0

0.01

0.02

0.03
0.04
Γ (Order n)

0.05

0.06

Figure 4.1: Tautochronic epicycloid absorber amplitude, |s|, versus amplitude of fluctuating
torque, Γn for different numbers of truncation for the power series HB method.

Figure 4.2 exhibits the approximate absorber amplitudes as a function of the applied
torques using both HB methods, with numerical simulations of the coupled, nonlinear equa-

82

0.4

|s|

0.3

0.2

s1, Lin
s2, Lin
s1 Series in Γ
s2 Series in Γ
s1 General HB
s2 General HB
s1 Simulations
s2 Simulations

s1, Cusp

s2, Cusp
0.1

0
0

0.005

0.01

0.015

0.02

Γ (Order n, 4 X Order 2n)

0.025

Figure 4.2: Absorber amplitudes, |s1,2 |, vs. amplitude of applied order torques, Γn,2n , from
the different harmonic balance (HB) approximations, with numerical simulations, for ϕ = 0
(tautochronic epicycloid).
tions of motion for baseline comparison. As is visible from Fig. 4.2, both of the harmonic
balance approximations provide an accurate representation of the absorber amplitude for
the entire range of torque. There exists a small error between the approximations and the
numerical simulations which occurs when the order n2 absorber approaches its amplitude
˜
limit. The higher order approximations capture some of the non-linear effects of the system
at larger amplitudes, which is visible from the hardening of the absorber amplitude near the
cusp. Interestingly, the linear approximation is accurate for the order n1 absorber for the
˜
torque range used here, which is due to the fact that we are using the tautochronic path, and
that this absorber does not get very close to its cusp value. Below we show that the accuracy of the linear response degrades as the perturbation away from the tautochrone becomes
non-zero. The accuracy of the linear approximation of the order n2 absorbers degrades sig˜
nificantly even at a very small torque level. This is due to the parametric excitation of the
order n2 absorber by the order n1 absorber, resulting from the nonlinear coupling between
˜
˜
83

the absorbers and rotor. The amplitude of the parametric excitation here is the square of
the order n1 absorber amplitude, for which the linear approximation will obviously break
˜
down, but is captured by the higher order approximations.

yy

′

−3

x 10
15
Order n, Linear
Order 2n, Linear
Order n, Series in Γ
Order n, Simulations
10
Order 2n, Series in Γ
Order 2n, Simulations
Order 3n, Series in Γ
Order 3n, Simulations
5

0
0

0.005

Absorbers
Locked
Order
2n
Cusp

0.01
0.015
0.02
Γ (Order n, 4 X Order 2n)

0.025

(a)
−4

20

x 10

yy

′

15

Absorbers
Locked

10

5

0
0

0.002

0.004 0.006 0.008
0.01
Γ, (Order n, 4 X Order 2n)
(b)

0.012

Figure 4.3: Rotor angular acceleration, yy , harmonic amplitude approximations using the
amplitude expansions, vs. the amplitude of applied order torques, Γn,2n . (a) Full torque
range showing approximations out to O(Γ3 ). (b) The zoomed in region near the origin.
n
84

The approximations of the harmonic amplitudes of the non-dimensional rotor angular
acceleration, yy , versus the applied torques are shown in Fig. 4.3 for the power series HB
method. The linear harmonic balance approximation for the order n torque is in good
agreement with the numerical simulations for a limited torque range, which breaks down
when the order n1 absorber is at ≈ 50% of it’s amplitude limit. Including the higher
˜
order terms in the approximation, the order n, as well as the order 2n and 3n harmonic
approximations, are also in good agreement with the numerical simulations, again deviating
in accuracy at the torque level which causes the absorber amplitude approximations to lose
some accuracy.
Figure 4.4 exhibits the harmonic amplitudes of the rotor angular acceleration as a function
of the applied torque amplitudes using the general HB method, with A2 expressed as a
function of A1 , as given in Eq. (4.10). This assumption makes this method not as accurate in
predicting the rotor harmonic amplitudes. In fact, it can easily be shown that for a linearized
absorber/rotor model with zero damping, the absorber counteracts all of the applied torque
when perfectly tuned, resulting in only a DC response from the rotor [12]. Due to nonlinear
effects, however, the rotor angular acceleration is not identically zero, and the harmonic
amplitudes grow like nonlinear powers of the torque amplitudes, that is, like Γm with m ≥ 2.
n

4.3.2

Multiple Absorbers with Single Order Forcing

It is of interest to consider the case of a rotor subjected to forcing at a single order and
fitted with multiple order absorbers. This has automotive applications, for example, in
multi-displacement engines [33]. In order to investigate the accuracy of the approximations
˜
for this situation, one can set Γ2n = 0 in the above expressions (Eqs. (4.5), (4.6), (4.8),
(4.11) and (4.12)), resulting in some simplification of the formulas. Equation (4.5) and the
85

15e10−3
n, General HB
n, Simulations
2n, General HB
2n, Simulations
3n, General HB
3n, Simulations

yy

10

Order
Order
Order
Order
Order
Order

5

0
0

0.005

0.01

0.015
Γn , 4Γ2n

0.02

0.025

Figure 4.4: Harmonic amplitudes of the rotor angular acceleration, yy , predicted using
the general HB method vs. the amplitude of applied order torques, Γn,2n . The absorber
amplitudes A1 and A2 are assumed to satisfy Eq. (4.10).
˜
expression for A2 are as presented above without the Γ2n terms, and the linear terms in the
expansions in Γn reduce to,

C1 =

µa,1 n 2
˜
µ2 n2 2 + n2 + n2 −1 + 1+α
˜a,1
˜1

D1 = −

2

,

n2 + n2 −1 + 1+α
˜1
µ2 n2 2 + n2 + n2 −1 + 1+α
˜a,1
˜1

2

,

˜
and the higher order terms can easily be found by setting Γ2n = 0 in the expressions provided
in the appendix. In this case the order n2 absorber is driven only through the nonlinear
˜

86

crosstalk with the order n1 absorber, and its amplitude is described through the E2 and F2
˜
terms given in the appendix. An example of the absorber amplitude approximation, as a
function of Γn , and results from simulations, are shown in Fig. 4.5. It interesting to note
that the order n2 absorber grows in amplitude and reaches its cusp limits even though the
˜
system is being forced only at n = n1 . This is due to the aforementioned nonlinear crosstalk
˜
between the multiple order absorbers, which is accurately captured by both harmonic balance
methods. As in Fig. 4.2, the approximations decrease in accuracy near the cusp amplitudes,
but are valid over a large torque range. The rotor responses for both HB methods (expansions
in Γn and general) are presented in Figs. 4.6 and 4.7, and exhibit the same trends as Figs. 4.3
and 4.4. The importance of the ratio of the absorber masses as a design parameter is made
evident in these figures, as both absorbers reach their respective amplitude limits for the
same level of torque in Fig. 4.5. This is a desirable feature of the response, which does not
hold for the example presented in Fig. 4.2.

4.3.3

Single Order Absorber with Single Order Forcing

We now turn to the simplest case, that is, a single order absorber with the rotor subjected
to a single order torque. The results are obtained from the above approximations by setting
α = 0 (the mass ratio between the different order absorbers), which results in the following
expressions for the linear terms in the expansions in Γn ,
µa n1 2
−µ2 n2 2 − (−n2 + n2 + n2 )2 ,
a 1
1 ˜1
1

(4.19)

(−n2 + n2 + n2 )
1 ˜1
1
D1 =
.
−µ2 n2 2 − (−n2 + n2 + n2 )2
˜1
a 1
1
1

(4.20)

C1 =

and

87

0.4
0.35
0.3

s1, Lin
s1 Series in Γ
s2 Series in Γ
s1 General HB

0.25
|s|

s1 Cusp

s2 General HB

0.2

s1 Simulations
s2 Simulations

0.15

s2 Cusp

0.1
0.05
0
0

0.005

0.01

0.015
0.02
Γ (Order n)

0.025

0.03

Figure 4.5: Absorber amplitude, |s|, vs. amplitude of applied order n torque, Γn , from
different order harmonic balance (HB) approximations, with numerical simulations, for ϕ = 0
(tautochronic epicycloid).
˜
Therefore, the first harmonic approximation for the amplitude of s, out to first order in Γn ,
that is, |s|lin , is thus given by truncating at k = 1 in which

2
2 ˜
(C1 + D1 )Γ2 = (
n

2
n4 + n4 ( − 1)2 + n2 (2˜ 2 ( − 1) + µ2 2 )
˜1
n1
a
1

˜n
)Γ2 .

The higher order terms in Γn for the absorber amplitude, C3 and D3 , can also be included
and simplified by making the above assumptions to the expressions given in the appendix.
The relationship between Γn and the absorber amplitude A1 obtained from the general
harmonic balance method also simplify as follows,

˜
Γn =

32A1 (−2˜ 2 + n2 (2 − 2 + A2 n4 ) + 3A2 ϕ)
n1
1 ˜1
1
,
4 n4 (1 + n2 )2 + 24A2 (˜ 2 + n4 ))
(−64 + 5A1 ˜ 1
˜1
n1 ˜ 1
1

88

(4.21)

0.03

Order n Linear
Order n, Series in Γ
Order n, Simulations
Order 2n, Series in Γ
Order 2n, Simulations
Order 3n, Series in Γ
Order 3n, Simulations

0.025

yy′

0.02
0.015

Order
2n Cusp
Absorbers
Locked

0.01
0.005
0
0

0.005

0.01

0.015
0.02
Γ (Order n)
(a)

0.025

0.03

−3

3

x 10

2.5
Absorbers
Locked

1.5

yy

′

2

1
0.5
0
−0.5
0

0.005

0.01

0.015

(b)
Figure 4.6: Rotor angular acceleration amplitude approximations using the amplitude expansions, yy , vs. the amplitude of applied order n torque, Γn . (a) Full torque range showing
approximations out to O(Γ3 ). (b) The zoomed in region near the origin.
n
and the first three rotor harmonics simplify to,
1
˜
(64Γn − A1 n2 (−64 + A4 n4 (1 + n2 )2
˜1
1 ˜1
64
+ 8A2 (˜ 2 + n4 ))),
1 n1 ˜ 1
89

wn =

0.025
0.02

yy′

0.015

Order n, General HB
Order n, Simulations
Order 2n, General HB
Order 2n, Simulations
Order 3n, General HB
Order 3n, Simulations

Order 2n
Cusp
Absorbers
Locked

0.01
0.005
0
0

0.005

0.01

0.015
0.02
Γ (Order n)

0.025

0.03

Figure 4.7: Harmonic amplitudes of rotor angular acceleration |yy | vs. Γn , including approximations from the general HB method and simulations, for ϕ = 0.
w2n = A2 n˜ 2 ,
1 n1

(4.22)

3 3 2 2
A n n1 (1 + n2 )(16 + 3A2 (˜ 2 + n4 )).
˜
˜1
w3n =
1 n1 ˜ 1
128 1

(4.23)

and,

The amplitude of the absorber as a function of Γn , using the harmonic balance approximations and compared to simulations of the full nonlinear equations of motion is shown in
Fig. 4.8. The linear absorber amplitude approximation is accurate here to ∼ 50% of the
torque which causes the absorber to reach its amplitude limit, and, in this case, with a single
absorber only, the general harmonic balance method is more accurate than the power series
in the applied torque method near the amplitude limits. This is due to the breakdown of
the expansions in the applied non-dimensional torque near the absorber cusps.
The two HB approximations of the harmonic amplitudes of yy versus Γn , and simulation
results, are shown in Figs. 4.9 and 4.10. The linear harmonic balance approximation is again

90

|s| / |s|, max

1.5

1

s, Lin
s, Series in Γ
s, General HB
s, Simulations

0.5

0
0

0.01

0.02

0.03
0.04
Γ (Order n)

0.05

0.06

Figure 4.8: Absorber amplitude, |s|/|s|max , vs. Γn , from the linear theory, the two nonlinear
harmonic balance (HB) approximations, and numerical simulations, for ϕ = 0 and α = 0.
shown to be in good agreement with the numerical simulations until the torque level approaches around 50% of that which causes the absorbers to reach their cusp limits, following
the deviation of the absorber approximations from the simulations. In order to increase the
accuracy of the order n approximation, as well as to estimate the amplitudes of the higher
harmonics on the rotor, we again must include the higher order terms in the expansions. As
shown in Fig. 4.9, the higher order approximations capture some of the nonlinear effects of
the order n torque as well as the order 2n and 3n torques, and provide reliable results out
to torque amplitudes at which the absorbers reach ≈ 70% of their cusp amplitude.

4.3.3.1

Non-Tautochronic Paths

The above results show good agreement between the theory and the numerical simulations
for the tautochronic path, up to certain torque levels near to where the absorbers reach
their cusps. We now investigate the accuracy of the method for non-tautochronic paths,

91

0.05
0.04

yy′

0.03
0.02

Order n, Linear
Order n, Series in Γ
Order n, Simulations
Order 2n, Series in Γ
Order 2n, Simulations
Order 3n, Series in Γ
Order 3n, Simulations
Absorbers Locked

0.01
0
0

0.2

0.4
0.6
Γ / Γ, max (Order n)

0.8

1

Figure 4.9: Approximations of the harmonic amplitudes of yy obtained using the amplitude
expansions, vs. Γn , for ϕ = 0 and α = 0; comparison with simulation results are also
included.
0.05
0.04

yy′

0.03

Order n, General HB
Order n, Simulations
Order 2n, General HB
Order 2n, Simulations
Order 3n, General HB
Order 3n, Simulations
Absorbers Locked

0.02
0.01
0
0

0.2

0.4
0.6
Γ / Γ, Max (Order n)

0.8

1

Figure 4.10: Approximations of the harmonic amplitudes of yy obtained using the general
HB method, vs. Γn , for ϕ = 0 and α = 0; comparison with simulation results are also
included.
specifically a circle. Utilizing the perturbation away from the tautochrone in the form of
h = ϕs4 , we investigate the absorber and rotor responses for a circular path absorber. Shown
92

in Fig. 4.11 is the estimated absorber amplitude versusΓn obtained using both harmonic balance approximations. As found by previous researchers, the circular path absorbers exhibit
an instability at a certain torque level which causes a jump in their amplitude [36]. This
occurs since, when the rotor spins at a constant speed, circular path absorbers behave similar
to a softening Duffing oscillator with bistability in the force response. This can be seen in
the numerical simulations in Fig. 4.11. However, the results from the power series harmonic
balance method fails to capture the large amplitude solution branch; this is expected, since
this approach does not allow for multiple solution branches. In contrast, the general harmonic balance approach does capture the bistability, which is due to the fact that the form
of the amplitude is not constrained to be of a certain form, thus allowing multiple solutions
to be obtained. Figure 4.12 shows the rotor angular acceleration harmonics as a function of
Γn for the power series HB method along with simulation results. One can see that the numerical simulations capture the amplification of the torsional vibrations due to the absorber
jump instability, but, as noted above, this HB approximation fails to capture the jump phenomenon. Figure 4.13 shows the rotor response as estimated from the general HB method
along with simulations. This approximation is much more accurate than the power series
approach; it captures the jump instability, and is also very accurate in estimating the higher
order harmonics. In fact, one can easily calculate the torque level which causes the absorber
to go unstable, denoted here by Γ∗ . This is done by setting the derivative of Eq. (4.21)
n
with respect the absorber amplitude, A1 , equal to zero, solving for the corresponding value
of A1 , and inserting it back into Eq. (4.21). Performing these operations for the conditions
in Fig. 4.11, it is found that Γ∗ = 0.0083242, which matches the numerical simulation data
n
quite well.
It should be noted that in practice absorbers are often overtuned, that is, designed
93

0.25

|s|

0.2

0.15
slin
s, Series in
s, General HB
s, Simulations

0.1

0.05

0
0

0.005

0.01

0.015

0.02

0.025

0.03

Γn
Figure 4.11: Absorber amplitude, |s|, vs. Γn , from the two HB approximations, along with
numerical simulations, for a perfectly tuned circular path absorber.
with n > n, in order to alleviate the absorber jump instability [3, 36]. This is especially
˜
important for circular path absorbers, where detuning extends the stable torque range, but
at the expense of absorber effectiveness [3, 36]. Since increasing the detuning makes the
absorber response more linear over a large torque range, the power series HB method likewise
improves in accuracy, and becomes quite accurate when the absorber is overtuned as little as
2%. We present an example case for a circular path absorber overtuned by 5%, (˜ 1 = 1.575).
n
The estimated absorber amplitude versus Γn for both HB methods, and comparisons with
simulations, are shown in Fig. 4.14. With the detuned absorber, the response is stable out to
a very large level of applied torque, specifically, about 6.3 times the torque level which caused
the perfectly tuned absorber to jump. Both harmonic balance estimations are very accurate

94

0.06
0.05

yy′

0.04
0.03

Order n, Linear
Order n, Series in Γ
Order n, Simulations
Order 2n, Series in Γ
Order 2n, Simulations
Order 3n, Series in Γ
Order 3n, Simulations
Absorbers Locked

0.02
0.01
0
0

0.005

0.01

0.015 0.02
Γ (Order n)

0.025

0.03

0.035

Figure 4.12: Approximations of the harmonic amplitudes of the rotor angular acceleration
obtained using the series expansion HB method, vs. Γn , along with results from numerical
simulations, for a perfectly tuned circular path.
0.06
0.05

yy′

0.04
0.03

Order n, General HB
Order n, Simulations
Order 2n, General HB
Order 2n, Simulations
Order 3n, General HB
Order 3n, Simulations
Absorbers Locked

0.02
0.01
0
0

0.005

0.01

0.015 0.02
Γ (Order n)

0.025

0.03

0.035

Figure 4.13: Approximations of the harmonic amplitudes of the rotor angular acceleration
obtained using the general HB method, vs. Γn , along with results from numerical simulations, for a perfectly tuned circular path.
out to large torque amplitudes here, and it can be shown the same accuracy follows for all
detuned absorber paths. The rotor response for the power series HB approach is shown in
95

Fig. 4.15, and for the general HB method in Fig. 4.16. As expected from the accuracy of the
absorber response approximations, both methods are very accurate in predicting the rotor
harmonic approximations over a large torque range.

0.3

yy′

0.25
0.2
0.15

s, Lin
s, Series in Γ
s, General HB
s, Simulations

0.1
0.05
0
0

0.01

0.02
0.03
Γ (Order n)

0.04

0.05

Figure 4.14: Absorber amplitude, |s|, vs. Γn , obtained using different order HB approximations, along with results from numerical simulations, for a detuned circular path.

4.4

Experimental Testing

Utilizing the experimental setup described in Chapter 2, we investigate the accuracy of
the above analytical and numerical approximations. Single order tautochronic and circular
path absorber systems are tested, as well as multiple order tautochonic absorbers. The experimental data is found to confirm the accuracy of the harmonic balance approximations
developed above for all cases considered. The experimental observations are presented in
the same format as the above analytical and numerical investigations, beginning with multiple order absorbers and concluding with single order, non-tautochronic absorbers. When
showing experimentally measured quantities we use their normalized versions, matching the
96

0.04

yy′

0.03

0.02

Order n, Linear
Order n, General HB
Order n, Simulations
Order 2n, General HB
Order 2n, Simulations
Order 3n, General HB
Order 3n, Simulations
Absorbers Locked

0.01

0
0

0.01

0.02
0.03
Γ (Order n)

0.04

0.05

Figure 4.15: Approximations of the harmonic amplitudes of the rotor angular acceleration
obtained using the series expansion HB method, vs. Γn , along with results from numerical
simulations, for a detuned circular path.
Order n, General HB
Order n, Simulations
Order 2n, General HB
Order 2n, Simulations
Order 3n, General HB
Order 3n, Simulations
Absorbers Locked

0.05
0.04

yy′

0.03
0.02
0.01
0
0

0.01

0.02
0.03
Γ (Order n)

0.04

0.05

Figure 4.16: Approximations of the harmonic amplitudes of the rotor angular acceleration
obtained using the general HB method, vs. Γn , along with results from numerical simulations, for a detuned circular path.
non dimensional quantities used in the analysis.

97

4.4.1

Tautochronic Absorbers at Two Orders

Epicycloidal path absorbers tuned to orders n1 = 2.3 and n2 = 4.6, as described in Chapter 3
˜
˜
are tested to investigate the accuracy of the multiple order harmonic balance methods. The
physical parameters used in the subsequent experimental results are summarized in Table 4.1.
Table 4.1: Physical Multiple Order Epicycloidal Path Experimental Parameters.
Physical Parameter
Definition
m1
Order n1 Absorber mass
˜
m2
Order n2 Absorber mass
˜
α
Different order absorber inertia ratio
R1,0
Rotor center to absorber 1 COM (at vertex) distance
R2,0
Rotor center to absorber 2 COM (at vertex) distance
µa,1
Absorber 1 viscous damping
µa,2
Absorber 2 viscous damping
Absorber to rotor inertia ratio
Ω
Rotor mean speed

Value
0.4313 kg
0.0482 kg
0.0106
0.12316 m
0.12 m
0.1
0.191
0.18486
300 RPMs

Figure 4.17 shows experimental absorber amplitudes versus the amplitude of the fluctuating part of the applied torque. The experimental torque is applied at a single order only,
n = 2.29, which excites the order n1 absorber resonantly, and the internal 2 : 1 resonance
˜
drives the order n2 absorber. As visible from Fig. 4.17, both of the harmonic balance meth˜
ods accurately predict the experimental absorber amplitudes. As expected, the general HB
method captures the large amplitude hardening of the order n1 absorber more accurately
˜
than the power series HB method, although the power series method is more accurate at
lower torque levels, for both absorbers, due to the inclusion of damping in that approach. At
low torque levels, both the numerical simulations and the HB approximations differ slightly
from the experimental results, which could be due to a number of factors, perhaps most
importantly, assumptions made about the absorber damping. As described in Chapter 3,

98

a constant effective absorber damping value is used for each absorber, whereas in fact the
absorber damping is estimated using coupled absorber/rotor free vibration data, and it is
known that the damping level depends on amplitude.
Considering the harmonics of rotor angular acceleration, as shown in Figs. 4.18 and 4.19
for the power series HB method and Figs. 4.20 and 4.21 the general HB method, the experimental results track both predictions quite well, with negligible differences between the two
methods in terms of approximating the experimental results. Interestingly, recalling Fig. 4.7,
the accuracy of the general HB method in approximating the rotor harmonics for a similar
system was significantly less than that for the power series HB method. The explanation for
the increase in accuracy for the general HB method here can be attributed to the fact that
in Fig. 4.7, the order n1 absorber was being forced precisely at its tuning order, whereas
˜
here there is a small (0.05%) detuning. Although this detuning is very small, the general
HB method improves drastically in accuracy, as noted in Section 4.3.3.1.
There exists some deviation in the order n rotor harmonic approximations at small torque
levels, specifically, the experimental data is larger in amplitude than the predictions. This is
due to the fact that the absorber amplitude at this level is smaller than the approximations,
thus not absorbing as much of the torsional oscillations as the model predicts. Also, near
the torque level which causes the order n2 absorber to reach it’s cusp, the order 2n and
˜
3n experimental rotor harmonics are smaller in amplitude than predicted. This can be
attributed to two effects. First, the experimental rotor harmonics are computed by taking
a digital Fourier Transform of the measured time response of the rotor speed. As not all
experimental data samples are taken for the same time period, some leakage of the higher
rotor harmonic amplitudes could occur in the Fourier Transform computation, causing one to
calculate a lower amplitude of the desired harmonic than measured. Second, when decoupling
99

0.2

s1 Experiment
s2 Experiment
s1, Lin

0.15

s2, Lin

s

s1 Series in Γ
0.1

s2 Series in Γ
s1 Simulations

0.05

0
0

s3 Simulations

0.05

0.1
Γ (Order n)

0.15

0.2

0.15

0.2

(a)

0.2

s

0.15

0.1

s1, Experiments
s2, Experiments
s1, General HB
s2, General HB
s1, Simulations
s2, Simulations

0.05

0
0

s1, Cusp

s2, Cusp

0.05

0.1
Γ (Order n)
(b)

Figure 4.17: Experimental absorber amplitudes, s, versus the amplitude of order n torque,
Γn compared against the harmonic balance approximations for multiple order absorbers. (a)
Series expansion HB method. (b) General HB method.

100

yy′

0.1

0.05

0
0

0.1

Order n, Experiments
Order 2n, Experiments
Order 3n, Experiments
Order n, Linear
Order n, Series in Γ
Order 2n, Series in Γ
Order 3n, Series in Γ
Order n, Simulations
Order 2n, Simulations
Order 3n, Simulations
Absorbers Locked

0.05

0.1
Γ (Order n)
(a)

0.15

0.2

Order n, Experiments
Order n, Series in Γ
Order n, Simulations
Order n, Linear
Absorbers Locked

′

yy

0.05

0
0

0.05

0.1
Γ (Order n)
(b)

0.15

Figure 4.18: Experimental rotor angular acceleration, yy vs. Γn for multiple order epicycloidal absorbers compared against the power series HB approximation. (a) All three rotor
response orders, n, 2n, 3n. (b) Order n.
the absorber dynamics from those of the rotor, one must assume that the amplitude of the
applied mean torque balances the mean rotor damping torque, that is, T0 = c0 Ω, thus
101

0.1
0.08

Order 2n, Experiments
Order 2n, Series in Γ
Order 2n, Simulations
Absorbers Locked

yy′

0.06
0.04
0.02
0
0

0.1
0.08

0.05

0.1
Γ (Order n)
(c)

0.15

0.2

0.15

0.2

Order 2n, Experiments
Order 3n, Series in Γ
Order 3n, Simulations
Absorbers Locked

yy′

0.06
0.04
0.02
0
0

0.05

0.1
Γ (Order n)
(d)

Figure 4.19: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.18
eliminating the effects of the rotor damping from the model. Numerical simulations have
shown that if one increases the rotor damping and compensates for this by increasing the
mean torque level, the corresponding rotor angular acceleration amplitudes will be lower.
102

0.15

yy′

0.1

0.05

0
0

0.15

Order n, Experiment
Order 2n, Experiment
Order 3n, Experiment
Order n, General HB
Order 2n, General HB
Order 3n, General HB
Order n, Simulations
Order 2n, Simulations
Order 3n, Simulations
Absorbers Locked

0.05

0.1
Γ (Order n)
(a)

0.15

0.2

0.15

0.2

Order n, Experiment
Order n, General HB
Order n, Simulations
Absorbers Locked

yy′

0.1

0.05

0
0

0.05

0.1
Γ (Order n)
(b)

Figure 4.20: Experimental rotor angular acceleration, yy vs. Γn for multiple order epicycloidal absorbers using the general HB approximation. (a) All three rotor response orders.
(b) Order n only.
This is a higher order effect that we do not investigate here, but warrants further study.
Overall, both of the HB methods are accurate in predicting the absorber and rotor
103

yy′

0.1

Order 2n, Experiment
Order 2n, General HB
Order 2n, Simulations
Absorbers Locked

0.05

0
0

0.12
0.1

yy′

0.08

0.05

0.1
Γ (Order n)
(c)

0.15

0.2

0.15

0.2

Order 3n, Experiment
Order 3n, General HB
Order 3n, Simulations
Absorbers Locked

0.06
0.04
0.02
0
0

0.05

0.1
Γ (Order n)
(d)

Figure 4.21: (c) Order 2n only. (d) Order 3n only. Coorsponding with Fig. 4.20
response of a system equipped with internally resonate multiple order absorbers, confirming
the utility of the HB methods for designing such systems.

104

1

s/s

max

0.8

s,
s,
s,
s,
s,

Experiments
Simulations
Series in Γ
Linear
General HB

0.6
0.4
0.2
0
0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Γn
Figure 4.22: Experimental absorber amplitude, s, versus the amplitude of order n torque,
Γn compared against the harmonic balance approximations for a single order tautochronic
absorber.

4.4.2

Single Order Tautochronic Path Absorber

Utilizing the order n1 epicycloidal path absorber used above, with the order n2 absorber
˜
˜
locked, we investigate the validity of the HB approximations for a single order tautochonic
absorber. The parameters defined in Table 4.1 apply to this case, except for two: first, the
inertia ratio which changes to

= 0.1642, due to only one absorber being active and the

other locked; and well the order of the applied torque is taken to be exactly at the absorber’s
tuning order, that is, n = n = 2.3.
˜
The experimentally obtained absorber amplitude versus Γn is compared against both HB
methods and numerical simulations in Fig. 4.22.
Again, both HB methods accurately capture the experimental observations for a wide

105

range of applied torque. The general method better captures some of the absorber hardening
near its amplitude limits, but both methods degrade in accuracy at this limit. Just as in
the multiple order absorber case, the experimental absorber amplitudes are smaller than the
predictions at small torque levels, which is consistent with the previous results. One can
see the effect of this error in estimating the absorber amplitude at small torque levels in the
corresponding order n rotor harmonic, as shown in Figs. 4.23 and 4.24 for the power series
HB method and in Figs. 4.25 and 4.26 for the general HB method.
The fact that the experimentally measured absorber amplitude is smaller than the predictions at the small torque levels again causes a similar error in the order n rotor response.
Interestingly, this trend does not seem to carry over to the higher order harmonics, mainly
due to their respective amplitudes being relatively small at this torque level. The accuracy of
both methods in approximating the higher order rotor harmonics is similar for this system.
The accuracy difference between the two methods is visible in the estimation of the order
n harmonic, where the accuracy of the general method is much lower than the power series
HB method. This is surprising, since in the multiple order absorber case, a .05% detuning
causes the general method to be much more accurate in modeling the order n torque, where
here we are forcing directly at the absorbers tuning order. Again, both methods are effective
overall in predicting the absorber and rotor response. Also, when the absorber is exactly
tuned, the absorber damping has a more significant effect, and it is included in the power
series method but not in the general method.

4.4.3

Single Order Circular Path Absorber

We now turn to an experimental investigation of circular path absorbers, which exhibit
nonlinear behavior at much smaller amplitudes, due to the softening nature of the path. A
106

yy′

0.1

0.05

0
0

0.05

0.1
Γ (Order n)
(a)

0.15

Order n, Experiments
Order n, Simulations
Order n, Linear
Order n, Series in Γ
Absorbers Locked

yy′

0.1

Order n, Experiments
Order 2n, Experiments
Order 3n, Experiments
Order n, Simulations
Order 2n, Simulations
Order 3n, Simulations
Order n, Linear
Order n, Series in Γ
Order 2n, Series in Γ
Order 3n, Series in Γ
Absorber Locked

0.05

0
0

0.05

0.1
Γ (Order n)
(b)

0.15

Figure 4.23: Experimental rotor angular acceleration, yy vs. Γn for a single order epicycloidal absorber compared with the power series HB approximation. (a) All three rotor
response orders, n, 2n, 3n. (b) Order n.
perfectly tuned circular path absorber is first tested, followed by investigation a detuned
circular path absorber for different inertia ratios. In order to experimentally test circular
107

yy′

0.1

Order 2n, Experiments
Order 2n, Simulations
Order 2n, Series in Γ
Absorber Locked

0.05

0
0

0.1
Γ (Order n)
(c)

0.15

0.2

0.15

0.2

Order 3n, Experiments
Order 3n, Simulations
Order 3n, Series in Γ
Absorber Locked

yy′

0.1

0.05

0.05

0
0

0.05

0.1
Γ (Order n)
(d)

Figure 4.24: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.23
path absorbers, the T-shaped compound pendulum absorber used in [32, 47] and shown
in Fig. 4.36 is used. It must be noted here that, unlike bifilar absorber suspensions, this
configuration allows the pendulum to rotate with respect to the rotor. One must then include
108

yy′

0.1

0.05

0
0

Order n, Experiment
Order 2n, Experiment
Order 3n, Experiment
Order n, Simulations
Order 2n, Simulations
Order 3n, Simulations
Order n, General HB
Order 2n, General HB
Order 3n, General HB
Absorber Locked

0.05

0.1
Γ (Order n)

0.15

(a)

Order n, Experiment
Order n, Simulations
Order n, General HB
Absorber Locked

yy′

0.1

0.05

0
0

0.05

0.1
Γ (Order n)
(b)

0.15

Figure 4.25: Experimental rotor angular acceleration, yy vs. Γn for a single order epicycloidal absorber compared with the general HB approximation. (a) All three rotor response
orders, n, 2n, 3n. (b) Order n.
the rotational inertia of the pendulum with respect to its center of mass when defining the

109

yy′

0.1

Order 2n, Experiment
Order 2n, Simulations
Order 2n, General HB
Absorber Locked

0.05

0
0

0.1
Γ (Order n)
(c)

0.15

Order 3n, Experiment
Order 3n, Simulations
Order 3n, General HB
Absorber Locked

yy′

0.1

0.05

0.05

0
0

0.05

0.1
Γ (Order n)
(d)

0.15

0.2

Figure 4.26: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.25
tuning order of the absorber, which can be found to be,

n=
˜

RL
,
L2 + ρ2
110

where R is the distance from the rotor center to pivot point of the absorber, L is the distance
from the pivot point to the absorber center of mass, and ρ is the pendulums radius of gyration.
The analysis presented in the previous sections is adequate for a point mass, or a bifilar
design, where the rotational inertia of the absorber with respect to its center of mass can be
grouped into the rotor inertia, J. Previous analysis [33, 47] has shown that one can account
for this rotational inertia and formulate the non-dimensional absorber and rotor equations
of motion in the same form as Eqs. (2.2) and (2.3) by including a simple dimensional scaling
constant when describing the non-dimensional absorber arc length, defined as,

s=

Lφ
,
β

where φ is angle the absorber swings at with respect to its vertex position, and β is the
dimensional scaling constant. One then finds that by choosing β = L(1 + n2 ), the equa˜
tions of motion for the compound pendulum absorber can be put into the necessary form
for comparison with bifilar absorbers. For the subsequent plots, the experimental physical
parameters used for these absorbers are presented in Table 4.2.
Table 4.2: Circular Path Absorber Physical System Parameters.
Circular Path Parameter
Definition
m
Absorber Mass
R
Rotor Center to Absorber Pivot Point Distance
L
Absorber Pivot Point to Absorber COM Distance
n
˜
Absorber Tuning Order
ρ
Absorber Radius of Gyration
µa
Absorber Viscous Damping
Ω
Rotor Mean Speed

Value
0.225 kg
0.118 m
0.051 m
1.315
0.03 m2
0.017
300 RPMs

We begin with a circular path absorber driven exactly at its tuning order, with n = n =
˜
111

Rotor
Absorber
Figure 4.27: Picture of the experimental circular path pendulum “T-shaped” absorber.
1.315, with an inertia ratio of

= 0.1045. Figure 4.29 shows the experimentally obtained

absorber amplitude compared against both harmonic balance approximations and numerical
simulations versus the amplitude of the applied fluctuating torque, represented by Γn .
As with the comparisons with the numerical simulations presented in Section 4.3.3.1
above, the general method tracks the experimental data very accurately and also captures
the jump bifurcation. The power series method is accurate up to near the jump bifurcation,
but, as noted in Section 4.3.3.1, it fails to capture the bistability, and thus misses the large
amplitude response entirely. The corresponding rotor harmonics are presented inFigs. 4.29
to 4.32 and, as expected, the inability of the power series HB method to accurately predict the
absorber motion carries over to the rotor harmonics. For all subsequent circular path studies
we therefore present only the rotor harmonic approximations obtained from the general HB
method.
112

0.5
0.4

s

0.3
0.2
0.1
0
0

0.01

0.02
0.03
Γ (Order n)

s, Experiments
s, General HB
s, Linear
s, Series in Γ
s, Simulations
0.04
0.05

Figure 4.28: Absorber arc length harmonic amplitude, |s| vs. Γn for a circular path absorber driven exactly at tuning; experimental results compared with both harmonic balance
approximations and simulations.
As evident from Figs. 4.31 and 4.32, the general HB method accurately captures the
increase in torsional vibrations due to the jump to the upper response branch of the absorber.
On that branch the absorbers are in phase with the applied torque, thus amplifying the rotor
torsional vibration at order n. Their exists some error between the HB approximations and
the experimental data after the jump occurs, due to the relatively large absorber amplitudes,
and the fact that expansions in the absorber amplitude were employed in the analysis.
We now look at the response of a circular path absorber when the applied torque is
detuned from the absorber order. We effectively detune the absorber by forcing at an order
less than the absorber’s tuning order, specifically we take n = 1.2492, which results in a
detuning of 5%. Shown in Fig. 4.33 is the absorber amplitude versus applied torque for a
system with an inertia ratio of

= 0.0519, and again the general HB method accurately

predicts the absorber amplitude throughout the torque range. This method also accurately
captures the rotor harmonics, as presented in Figs. 4.34 and 4.35, although the experimental
113

data for the order 3n harmonic does not exhibit a jump in amplitude as predicted by theory,
that is, the amplitudes of this harmonic are the same on both the upper and response
branches. The reason for this anomaly is not presently understood.
We now decrease the inertia ratio to,

= 0.0189, by adding weights to the rotor to

increase its rotational inertia, in order to investigate the effects this parameter has on the
analytical predictions. Figure 4.36 shows the absorber amplitude versus the amplitude of
applied fluctuating torque for this case. This added inertia causes the absorber to jump at
a lower non-dimensional torque level, but otherwise there is no discernible difference in the
accuracy of the HB method. Once again, there is a slight error in the rotor harmonics, as
shown in Figs. 4.37 and 4.38, most notably in the order 2n and 3n harmonics. These errors
are not significant, however, as these harmonics are relatively small in this case, since in
this case the torque exerted on the rotor by the absorber has a smaller effect on the rotor
dynamics relative to that caused by the fluctuating torque.

4.5

Conclusions

Approximate, closed form expressions for pendulum absorbers and the corresponding rotor
dynamics have been developed. One form of the assumptions yields approximation predictions for the applied fluctuating torque as a function of absorber amplitudes. The other
form of the approximations, using the assumption that the absorber amplitude is a power
series in the applied non-dimensional torque, gives predictions for the absorber amplitude
as a function of the applied torque. Both methods also give predictions for the harmonically rich rotor dynamics. The methods are found to agree with numerical simulations and
experimental results for a wide range of absorber configurations.

114

Order n, Experiment

0.12

Order n, Linear
Order n, Series in Γ

0.1

Order n, Simulations
Order 2n, Experiment

0.08

Order 2n, Series in Γ

yy′

Order 2n, Simulations
Order 3n, Experiment

0.06

Order 3n, Series in Γ
Order 3n, Simulations

0.04

Absorber Locked

0.02

0
0

0.01

0.02

0.03

0.04

0.05

0.06

0.04

0.05

0.06

Γ (Order n)
(a)

0.12
0.1

Order n, Experiment
Order n, Linear
Order n, Series in Γ
Order n, Simulations
Absorber Locked

yy

′

0.08
0.06
0.04
0.02
0
0

0.01

0.02

0.03
Γ (Order n)
(b)

Figure 4.29: Harmonics of the rotor angular acceleration, yy vs. Γn , for a circular path
absorber driven exactly at tuning; experimental results compared against the power series
harmonic balance approximation. (a) All three rotor response orders n, 2n, 3n. (b) Order
n.
115

0.04
0.035
0.03

Order 2n, Experiment
Order 2n, Series in Γ
Order 2n, Simulations
Absorber Locked

yy′

0.025
0.02
0.015
0.01
0.005
0
0

0.01

0.02

0.03
0.04
Γ (Order n)

0.05

0.06

0.05

0.06

(c)

0.035
0.03
0.025

Order 3n, Experiment
Order 3n, Series in Γ
Order 3n, Simulations
Absorber Locked

yy′

0.02
0.015
0.01
0.005
0
0

0.01

0.02

0.03
0.04
Γ (Order n)
(d)

Figure 4.30: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.29

116

Order n, Experiment
Order n, General HB
Order n, Simulations
Order 2n, Experiment
Order 2n, General HB
Order 2n, Simulations
Order 3n, Experiment
Order 3n, General HB
Order 3n, Simulations
Absorber Locked

0.1
0.09
0.08
0.07

yy′

0.06
0.05
0.04
0.03
0.02
0.01
0
0

0.01

0.02

0.03

0.04

0.05

0.06

Γ (Order n)
(a)

0.12
0.1

yy

′

0.08
0.06
0.04
0.02
0
0

0.01

0.02

Order n, Experiment
Order n, General HB
Order n, Simulations
Absorber Locked
0.03
0.04
0.05
0.06
0.07
Γ (Order n)
(b)

Figure 4.31: Harmonics of the rotor angular acceleration, yy vs. Γn , for a circular path absorber driven exactly at tuning; experimental results compared against the general harmonic
balance approximation. (a) All three rotor response orders n, 2n, 3n. (b) Order n.
117

0.04
0.035
0.03

yy′

0.025
0.02
0.015
0.01
0.005
0
0

0.01

0.02

0.03
Γ (Order n)

Order 2n, Experiment
Order 2n, General HB
Order 2n, Simulations
Absorber Locked
0.04
0.05
0.06

(c)

0.03

Order 3n, Experiment
Order 3n, General HB
Order 3n, Simulations
Absorber Locked

0.025

yy′

0.02
0.015
0.01
0.005
0
0

0.01

0.02

0.03
0.04
Γ (Order n)

0.05

0.06

(d)
Figure 4.32: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.31

118

0.4
0.35
0.3
s

0.25
0.2

s, Experiments
s, General HB
s, Linear
s, Series in Γ

0.15
0.1
0.05
0
0

0.01

0.02
0.03
Γ (Order n)

0.04

Figure 4.33: Amplitude of the absorber arc length, s vs. Γn for a 5% overturned circular
path absorber with = 0.0519; experimental results compared with both harmonic balance
approximations and simulations.

119

Order n, Experiments
Order n, General HB
Order 2n, Experiments
Order 2n, General HB
Order 3n, Experiments
Order 3n, General HB
Absorber Locked

0.08
0.07
0.06

yy′

0.05
0.04
0.03
0.02
0.01
0
0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.025

0.03

0.04

0.045

Γ (Order n)
(a)
0.08

Order n, Experiments
Order n, General HB
Absorber Locked

0.07
0.06

yy

0.05
0.04
0.03
0.02
0.01
0
0

0.005

0.01

0.015

0.02

0.035

0.04

Γn
(b)
Figure 4.34: Harmonics of the rotor angular acceleration, yy vs. Γn , for a 5% overturned
circular path absorber, with = 0.0519; experimental results compared against the general
harmonic balance approximation. (a) All three rotor response orders n, 2n, 3n. (b) Order
n.
120

0.035
0.03

Order 2n, Experiments
Order 2n, General HB
Absorber Locked

0.025

yy′

0.02
0.015
0.01
0.005
0
−0.005
0

0.01

0.02
0.03
Γ (Order n)

0.04

(c)

0.025

Order 3n, Experiments
Order 3n, General HB
Absorbers Locked

0.02

yy′

0.015
0.01
0.005
0
−0.005
0

0.01

0.02
0.03
Γ (Order n)

0.04

(d)
Figure 4.35: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.34

121

0.4
0.35
0.3

s

0.25
0.2
0.15

s, Experiments
s, General HB
s, lin
s, Series in Γ

0.1
0.05
0
0

0.005

0.01

0.015
0.02
Γ (Order n)

0.025

0.03

Figure 4.36: Amplitude of the absorber arc length, s vs. Γn for a 5% overturned circular
path absorber with = 0.0189; experimental results compared with both harmonic balance
approximations and simulations.

122

0.04
0.035
0.03

yy′

0.025

Order n, Experiments
Order n, General HB
Order 2n, Experiments
Order 2n, General HB
Order 3n, Experiments
Order 3n, General HB
Absorbers Locked

0.02
0.015
0.01
0.005
0
0

0.005

0.01

0.015

0.02

0.025

0.03

0.015
0.02
Γ (Order n)

0.025

0.03

Γ (Order n)
(a)

0.04
0.035

Order n, Experiments
Order n, General HB
Absorber Locked

0.03

yy′

0.025
0.02
0.015
0.01
0.005
0
0

0.005

0.01

(b)
Figure 4.37: Harmonics of the rotor angular acceleration, yy vs. Γn , for a 5% overturned
circular path absorber, with = 0.0189; experimental results compared against the general
harmonic balance approximation. (a) All three rotor response orders n, 2n, 3n. (b) Order
n.
123

−3

15

x 10

Order 2n, Experiments
Order 2n, General HB
Absorbers Locked

yy′

10

5

0
0

0.005

0.01

0.015
0.02
Γ (Order n)

0.025

0.03

(c)
−3

6

x 10

Order 3n, Experiments
Order 3n, General HB
Absorbers Locked

5
4

yy′

3
2
1
0
−1
0

0.005

0.01

0.015
0.02
Γ (Order n)

0.025

0.03

(d)
Figure 4.38: (c) Order 2n. (d) Order 3n. Coorsponding with Fig. 4.37

124

Chapter 5
Conclusions and Recommendations
for Future Work

5.1

Summary of Results

In this dissertation, the dynamics of rotational inertias equipped with pendulum-type absorbers tuned to multiple orders were investigated using analysis, simulations, and experiments. To the author’s knowledge, this is the first study which maps out the response
characteristics of a multiple order absorber system subjected to multiple order torques. It
includes important information about the response amplitudes, phases, and stability, of the
absorber response, as well as the rich harmonic content of the rotor response. These results
are very useful for design strategies that yield optimal performance for rotors fitted with
pendulum absorbers.
The analytical findings are based on a dynamic model derived in Chapter 2. Here the
differential equations which govern the motion of a system composed of arbitrary numbers of
order n1 and order n2 absorbers attached to a rigid rotary inertia that is subjected to multi˜
˜
order torque harmonics were derived using energy methods. A formulation for the path of the
absorbers, which dictate is linear tuning and nonlinear behavior, was employed that allows for
small deviations from perfect linear tuning at small amplitudes, as well as from the desirable
tautochronic path at large amplitudes. Non-dimensionalization and scaling were employed
125

to simplify the system parameters. Key small parameters were identified, specifically the
non-dimensional mean and fluctuating torques, and the absorber and rotor non-dimensional
viscous damping, which facilitate analysis, primarily by allowing one to decouple the absorber
dynamics to leading order in these parameters. Once the absorber dynamics are determined
from these equations, one can substitute these into the rotor equation to obtain its response.
Approximate solutions of the absorber equations were solved using perturbation methods
(in Chapter 2) and harmonic balance techniques (in Chapter 4).
In Chapter 2 perturbation methods were used to predict and evaluate system response,
using an expanded versions of the absorber equations of motion. These perturbation methods
yielded equations for the slowly varying absorber amplitudes and phases, which can be used
to estimate the steady state response characteristics, including stability. It was shown that
these slowly varying absorber amplitude and phase equations accurately captured the effects
of the internal resonance resulting when n2 ≈ 2˜ 1 , which arises due to the non-linear
˜
n
kinematic coupling between the absorbers and the rotor. Using numerical continuation
methods on the perturbed equations, it was shown that a torque of sufficient amplitude
applied at order n, with a set of multiple order n1 absorbers tuned close to n, can cause
˜
these absorbers to bifurcate to a non-synchronous response. It was also shown that a order 2n
torque can cause a set of multiple order n1 absorbers, when 2n ≈ n1 , to undergo a period
˜
˜
doubling bifurcation, arising from parametric excitation effects, resulting in an unstable
response in which the absorbers become non-synchronous and their amplitudes grow until
they approach their limiting values. Interestingly, it was shown, by considering a general
torque space with both order n and 2n excitation harmonics, that these are actually the
same bifurcation, since both bifurcations result in non-synchronous behavior of the order
n1 absorbers. These instabilities are detrimental to system performance, most importantly
˜
126

since the absorbers reach their limits at lower torques than they would if the response
remained synchronous. Another interesting finding of the perturbation investigation is that
the addition of absorbers at order n2 generally increases the level of torque which causes the
˜
non-synchronous bifurcations. This occurs through the attenuation of the higher order (most
notably, order 2n) harmonics imparted onto the rotor by the nonlinear dynamics of the order
n1 absorbers, which move at order n in steady state. This stabilization allows the order n1
˜
˜
absorbers to be tuned closer to resonance, thus increasing their ability to attenuate vibration
at order n. The perturbation equations also were shown to allow for a closed form solution
of the optimal ratio of inertias between the two different order absorbers, corresponding to
the situation in which both sets of absorbers reach their respective amplitude limits at the
same level of torque, which maximizes the system’s allowable torque range.
In Chapter 3 an experimental spin rig is described, along with results that verify the
analytical predictions of Chapter 2. The rig consists of a computer controlled motor driving a rotor to which up to four absorbers can be fitted. For examining the multi-order
response and stability results of Chapter 2, two absorbers at order n1 = 2.3 and two at
˜
order n2 = 4.6, both with nearly-tautochronic paths, were fitted to the rotor. These ab˜
sorbers use a bifilar suspension with self-centering rollers, which minimize rubbing friction
between the absorbers and their supporting flanges. The motor was programmed to provide
a torque composed of two orders, that is, a two term Fourier series, of general form. Steadystate experiments were conducted for one absorber at each order free, in order to compare
against the synchronous response theoretical developments. Experimental runs were then
made with all four absorbers unlocked, which investigated the non-synchronous instabilities
uncovered analytically. The experimental data was shown to agree well with the analytical
and numerical predictions, and to the author’s knowledge, this is the first time that period
127

doubling sub-harmonic absorber bifurcations were captured experimentally. The response
amplitudes and experimental instabilities were found to match the numerical continuation
results, proving the utility of these analytical tools in absorber design.
Chapter 4 describes an alternative to numerical simulations and the perturbation analysis
for describing the system response, specifically, a harmonic balance (HB) approach. The
advantage of this approach over perturbation methods is that it yields simple expressions
for response amplitudes and phases, although it cannot provide stability information. Two
HB approaches were employed. The general method employed provides accurate predictions
of single order absorber and rotor dynamics for a wide range of absorber paths, but requires
assumptions which cause the method to decrease in accuracy when analyzing multiple order
absorber systems. The other method, which employs expansions in the torque amplitude,
yields relatively simple, closed form expressions for the absorber and rotor dynamics as a
function of the applied torques, but fails to accurately capture some of the large-amplitude
non-linear absorber dynamics. Numerical simulations and experimental tests using the spin
rig were employed to verify the accuracy of these closed-form solutions. The data was
shown to agree with the predictions provided by these solutions in most cases. Two types
of absorbers were consider to experimentally verify the HB results: the epicycloidal path
absorbers of one and two orders described above, and simply pivoted circular path absorbers
of order n1 = 1.315. The power series HB method provides more accurate comparisons to
˜
multiple order, nearly-tautochronic absorber data while the general method is more beneficial
when implementing a non-tautochronic path absorber. These closed-form expressions allow
designers to quickly and easily investigate the effects of system parameters on absorber
performance, which allows for a preliminary design tool for sizing absorbers. Of course,
prospective designs should be tested for dynamic stability using either simulations or the
128

perturbation methods of Chapter 2.

5.2

Importance of Results

The results here advance our fundamental understanding of the system’s nonlinear dynamics, and these provide guidance for improving their performance. The predictive nature of
the results will help shorten the design cycle for absorber systems, and the experimental
validation insures the fidelity of the analytical results.

5.3

Future Work

As described in this study, the addition of multiple order absorbers have multiple benefits.
They reduce the higher order harmonics generated by single order absorbers, they allow for
vibration attenuation of systems subjected to multiple order torques, and, in some cases, they
can stabilize the response of multi-order absorber systems. This study investigated absorber
tuned to two different orders, it would be interesting to examine nonlinear interactions and
the potential benefit of adding absorbers tuned to three or more orders (e.g., n, 2˜ , 3˜ ).
˜ n n
Absorbers of order n generate three harmonics of noticeable magnitude on the rotor, with
the first order (n) used to counteract the applied torque, while the others actually add to
the overall fluctuations of the rotor response. Second order absorbers have a particular
role on the system response, since they affect second harmonics and thus can modulate the
effects of parametric excitation. While third order absorbers will not have such a specific
role, they may still be beneficial due to the fact that third harmonics are prominent on the
rotor. An interesting design problem would be to extend the present analysis to consider
the nonlinear interactions in sets of absorbers tuned to three orders, namely, how does one
129

most effectively allocate the available inertia to each order absorber in order to achieve an
optimal design, for example, to minimize the root mean square of the rotor response. The two
order absorber problem studied in this dissertation borders the line in which perturbation
analysis becomes too complicated for the benefit gained, and adding three order absorbers
may send the complication over this line. A combined harmonic balance and simulation
study of three-order absorber systems would be the approach recommended that the author.
It would also be beneficial to conduct a study where sets of multiple order absorbers are
detuned, and map out the benefits and drawbacks such systems. In general, detuned absorbers are more stable but less effective in reducing torsional vibrations. A potentially beneficial setup would incorporate detuned order n2 absorbers, which are implemented purely
˜
for stabilization of the order n1 absorbers, but detuned so as to increase the torque range
˜
the n2 ’s can safely operate in.
˜
At the end of Chapter 2, a brief study is carried out on alternatives to the tautochronic
path, specifically cycloids. The analysis shows that the accuracy of the perturbation equations is not accurate for low order absorbers, in terms of their ability to capture the large
amplitude dynamics. This is due to the hardening nature of the cycloidal path, which causes
the absorber nonlinearity to no longer be weak. The accuracy of the perturbation methods
increases at the tuning order of the absorber increases, since a cycloid approaches the tautochronic epicycloid as n becomes larger. For an accurate analysis of multiple order cycloidal
˜
path absorbers, one must perform a strongly non-linear oscillator analysis, which would be a
rather complicated study involving averaging using the system described in terms of actionangle coordinates. Again, it would be recommended to perform a numerical study when
investigating these strongly non-linear absorbers.
Currently, several automotive manufacturers are conducting research into the implemen130

tation of pendulum absorbers, and the ideas in this study may be put into practice in the
near future.

131

Appendices

132

CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK133

A.1

Experimental Error Analysis

In this appendix we will attempt to quantify the experimental measurement error in the
CPVA testing device. The main measurements of interest include:
1. Absorber COM position, S
˙
2. The speed of the rotor θ
3. The applied torque which includes the mean T0 and fluctuating Tθ (at order of interest),
where the position of the absorber(s) and the rotor speed are each measured with an encoder.
The applied torque on the rotor is measured from a feedback current that the motor outputs.
Each of these measurements will be discussed in more detail in the following sections. This
document follows the NIST experimental uncertainty guidelines [43].
Uncertainty in Absorber position: A U.S. Digital optical encoder is used to measure
absorber position. These encoders contain 4 channels which are A, -A, B, and -B. The two
negative channels are literally the negative of A and B, respectively. To eliminate the noise
in the encoder signal the positive channel is subtracted from the negative channel and then
divided by 2. The encoder has two sets of 360 equally spaced lines. The lines in each set
are spaced 90◦ apart, the so-called quadrature spacing, in order to obtain the direction of
rotation. The 360 lines yield 720 measurements for a complete encoder rotation, meaning
that with two sets of lines we get 1 degree resolution from this encoder. Therefore, assuming
4
the pulses are equally spaced on the encoder and we don’t miss any counts, the uncertainty
in the absorber position is
us = ±

1◦
.
4

(1)

Speed of rotor: The encoder on the rotor has 1000 pulses per revolution, which output
a frequency corresponding to speed of rotation. This frequency signal is then sent to a
frequency to voltage (F2V) converter in order to supply to the analog to digital converter
with a usable signal. The F2V converter measures the instantaneous frequency between two
pulses in pulses/sec. This frequency is divided by 1000 to obtain the frequency of the rotor
in Hz. This device is rated to be able to measure up to 25 kHz between pulses which means
approximately 25 Hz for 1000 pulses. An important check here was to make sure the F2V
converter could detect the frequency of the rotor during a torsional disturbance. This means
the rotor is now rotating at a mean speed with a oscillating part superimposed. In terms of
pulses going into the F2V this means that these pulses are no longer evenly spaced along the
time axis. Now their time spacing is modulated (i.e. they get closer and further apart, etc.)
due to the fluctuation about the mean rotor speed. The bandwidth of the F2V converter
turned out to be sufficiently large to resolve the frequency of the rotor pulses. Using the
specification sheet from the device manufacturer, the accuracy calibration of the device is
given as a maximum ±0.1% of the frequency span to be measured. Since the frequency at
the tuning order for the absorbers used in this study is about 10 Hz for mean speeds at
400 rpm, an estimate for the uncertainty in the F2V conversion using a 10 Hz span which
corresponds to an output voltage of 3 V (of the 5 V range) is

u˙
= ±(3V)(.001) = ±3mV.
θ,F 2V

(2)

After the signal leaves the F2V converter it is digitized by the National Instruments DAQ
board (PCI-6281). According to the specification sheet, the device has an absolute accuracy
of 1.05 mV with the built-in low pass filter turned off. The quantization step size for the 18

134

bits at ±(10V) range is

2 ∗ (10V)
QN I =
= 7.6294 ∗ 10−5 V.
218 − 1

(3)

To obtain the quantization error from this we divide Q in half because at worst our actual
signal amplitude could have been exactly in the middle of the quantization step (i.e. Q/2),
in which case it would be rounded up and anything below is rounded down. Note that this
also assumes the maximum amplitude of our signal is approximately 10 volts which is the
maximum analog input voltage to the DAQ board. Although, this isn’t always the case, we
will approximate the quantization error as

QN I,error = QN I /2 = ±0.0381mV.

(4)

Since the quantization error is about 3% of the absolute accuracy given in the NI DAQ
specification sheet, we will assume that the quantization error was accounted for in this
specification. Therefore the uncertainty in digitizing the speed signal uDAQ is

u˙
= ±1.05mV.
θ,DAQ

(5)

Finally, the last bit of uncertainty is the overall noise floor of the signal’s Digital Fourier
Transform (DFT). This can be determined simply by taking an DFT of the speed signal in
LabView and then plotting the magnitude of the Fourier coefficients. One then estimates
the largest amplitude at which white noise is present (i.e. constant amplitude level at all

135

frequencies). This is determined to be

u˙
= ±0.02mV.
θ,F F T

(6)

The total uncertainty in the rotor speed is

u˙ = u˙
+u˙
+u˙
= ±3.07mV.
θ,T
θ,F 2V
θ,DAQ
θ,F F T

(7)

Torque Uncertainty: The torque measurement comes from a current feedback signal
that is multiplied by two calibration constants, one which converts the motor current to a
measurable voltage, and another, given by the motor manufacturer, which gives the motor
torque in Newton-meters from the inputted current. This signal is digitized by the motor
control box and the National Instruments DAQ board. The quantization for the motor
control box which is 8 bits at a ±(10V) range is

Qbox =

2 ∗ (10V)
= 78.43mV.
28 − 1

(8)

Just like before we can approximate the quantization error as

Qbox,error = Qbox /2 = ±39.22mV.

(9)

Digitizing the torque signal with the National Instruments DAQ board will have the same
uncertainty as calculated in equation (5). Similar to the rotor speed signal, the white noise

136

level for the torque signal is estimated in the same way. This uncertainty is found to be

uT ,DFT = ±2mV.
θ

(10)

The total uncertainty in the torque signal is then

uT ,T = u ˙
+ Qbox,error + uT ,DFT = ±42.27mV.
θ
θ
θ,DAQ

(11)

Combined Uncertainties for the Rotor Angular Acceleration and Rotor Inertia.
Using the uncertainties above, this section computes the combined uncertainty for the rotor
acceleration and the rotor inertia.
Rotor Angular Acceleration: The rotor angular acceleration at order n is computed as
follows
¨
˙
θn = nΩθn ,

(12)

˙
where n is order of excitation, Ω is the mean speed that the fluctuation is about, and θn is
the magnitude of the DFT of the rotor speed signal at order n. Assuming the fluctuations
˙
¨
in Ω and θn are uncorrelated, the combined uncertainty for θn is computed following the
NIST standards as follows:

u2 =
¨
θn

¨
∂ θn 2 2
un +
∂n

¨
∂ θn 2 2
uΩ +
∂Ω

¨
∂ θn 2 2
u˙ ,
˙
θn
∂ θn

(13)

which is essentially a first order Taylor series approximation of a function whose inputs are
measured quantities and whose output is the desired quantity one wishes to resolve from the
combined measurements. The function is squared in order to yield positive values for the

137

combined uncertainty. We assume here that there is no error in the order of the torque signal
un = 0 which is generated in LabView 1 . To estimate error in the ability of the PID to
maintain a constant speed (uΩ ), an experiment is run with the rotor spinning at a constant
rate with PID active. The mean and standard deviation of the resulting rotor speed signal
is computed using basic statistics and the standard deviation yields uΩ which is

uΩ = ±2.39mV.

(14)

The uncertainty in the rotor speed at order n (u ¨ ) had already been calculated in equation
θn
(7). Taking the derivatives in equation (13) and using equations (14) and (7), the uncertainty
in the angular acceleration at order n is

˙
u2 = (nθn )2 u2 + (nΩ)2 u2 ,
¨
˙
Ω
θn
θ,T

(15)

¨
which will provide error bars on a θn measurement of ±u ¨ .
θn
Rotor Inertia: To compute the rotor inertia we use Newton’s law to obtain

J=

Tθn
,
¨
θn

(16)

¨
where Tθ is the magnitude of the applied fluctuating torque at order n and θn is the
n
angular acceleration of the rotor at order n calculated according to equation (12). Assuming
¨
the fluctuations in Tθ and θn are uncorrelated, the uncertainty in the rotor inertia is
n
u2 =
J
1 Or

1 2 2
uT +
¨
θn
θn

Tθ 2 2
n
u¨ ,
¨2
θn
θn

(17)

the error in the generated order can be included in the deviation of the mean speed, Ω

138

where uT

θn

and u ¨ are the uncertainties calculated in equations (11) and (15), respectively.
θn

The error bars on the rotor inertia calculation will then be ±uJ .

B.2

Harmonic Balance Higher Order Approximations

˜
The form of the expressions for the higher order expansions in Γn,2n developed in Section 4.2
in their most general form are found to be as follows:

2
2
˜ ˜
E2 = (4n˜ 2 ((C1 − D1 )(−8µa,2 n(1 + α) + G1 Γ2 n2
n1
2n 2
× (1 + α + n2 (1 + α + 16H1 n2 α))) + 2C1 D1
˜2
˜
× (˜ 2 (1 + α)(−4 + 3Γ2 H1 (1 + n2 ) ) + 8n2
n2
˜2
2n
1
2˜
˜
˜
× (2 + α(2 + (−2 + G2 Γ2 n4 + 3Γ2 H1 n4 ) )))))( ),
1 2n ˜ 2
2n
2
Q
F2 =
(

2C1 D1
˜ ˜
−8˜a,2 n(1 + α) + G1 Γ2 n2 (1 + α + n2 (1 + α + 16H1 n2 α))
µ
˜2
2n 2

˜
+ ((˜ 2 (1 + α)(−4 + Γ2 H1 (1 + n2 ) ) + 8n2 (2 + α(2 + (−2
n2
˜2
2n
2˜
2
2
˜
˜
+ 3G2 Γ2 n4 + Γ2 H1 n4 ) )))((C1 − D1 )(−8˜a,2 n(1 + α)
µ
1 2n ˜ 2
2n
2
˜ ˜
+ G1 Γ2 n2 (1 + α + n2 (1 + α + 16H1 n2 α))) + 2C1 D1 (˜ 2
˜2
n2
2n 2
˜
× (1 + α)(−4 + 3Γ2 H1 (1 + n2 ) ) + 8n2 (2 + α(2
˜2
2n
n2
2 Γ2 n4 + 3Γ2 H 2 n4 ) ))))) 4n˜ 1 ,
˜
˜ ˜
+ (−2 + G1 2n 2
2n 1 ˜ 2
Q

139

where,
˜ ˜
˜2
Q = ((8˜a,2 n(1 + α) − G1 Γ2 n2 (1 + α + n2 (1 + α
µ
2n 2
˜ ˜
+ 16H1 n2 α)))(8˜a,2 n(1 + α) + G1 Γ2 n2 (1 + α
µ
2n 2
˜
n2
+ n2 (1 + α + 16H1 n2 α))) 2 + (˜ 2 (1 + α)(−4 + Γ2
˜2
2n
× H1 (1 + n2 ) ) + 8n2 (2 + α(2 + (−2 + 3G2
˜2
1
2˜
˜
˜
˜ ˜
˜2
n2
× Γ2 n4 + Γ2 H1 n4 ) )))(˜ 2 (1 + α)(−4 + 3Γ2 H1 (1 + n2 ) )
2n
2
2n
2n 2
2˜
˜
˜
+ 8n2 (2 + α(2 + (−2 + G2 Γ2 n4 + 3Γ2 H1 n4 ) )))).
2
2n
1 2n ˜ 2
2 n
˜
˜
C3 = −(2Γ2n (G1 + Γ2 G3 )n2 (16C1 E2 n3 α − C1 (˜ 2 (1 + α)
2n
1
+ n4 (1 + 4D1 n2 + α) + 12D1 (1 + α)ϕ1 ) − D1 (−16F2 n3 α
˜1
2
+ 3D1 n2 (1 + n2 )(1 + α) + 4D1 (n2 n4 + 3(1 + α)ϕ1 )))
˜1
˜1
˜1
2
2 n
˜
− (C1 n((C1 + D1 )˜ 2 (1 + n2 ) + 4F2 (−2n + Γ2n
˜1
1
˜
˜
˜
× (H1 + Γ2 H3 )˜ 2 (1 + n2 ))) + 4E2 (2D1 n2 + C1 Γ2n (G1 + Γ2 G3 )
n2
˜2
2n
2n
× n˜ 2 (1 + n2 ) + 2(1 + α)))(2˜ 2 (1 + α) − 2n2 (1 + α − )
n2
˜2
n1
1
˜
˜
˜
+ 4Γ2n (H1 + Γ2 H3 )n3 α + Γ2n n(1 + α) )) ,
2n
P3

140

2
4C1 D1 n2 n4
˜1
2˜
2˜
2˜
2˜
D3 = ((1 + α) (C1 n2 + 3D1 n2 + C1 n4 + 3D1 n4 +
1
1
1
1
1+α
3 n2 n4
4D1 ˜ 1 16C1 E2 n3 α 16D1 F2 n3 α
2
−
−
+ 12C1 D1 ϕ1
+
1+α
1+α
1+α
3
˜
˜
˜
+ 12D1 γ1 + (4(2Γ2n (G1 + Γ2 G3 )n2 α − µa,1 (1 + α))
2n
2 n
˜
˜
× (2Γ2n (G1 + Γ2 G3 )n2 (16C1 E2 n3 α − C1 (˜ 2 (1 + α)
1
2n
+ n4 (1 + 4D1 n2 + α) + 12D1 (1 + α)ϕ1 ) − D1 (−16F2 n3 α
˜1
2
+ 3D1 n2 (1 + n2 )(1 + α) + 4D1 (n2 n4 + 3(1 + α)ϕ1 )))
˜1
˜1
˜1
2
2 n
˜
˜
− (C1 n((C1 + D1 )˜ 2 (1 + n2 ) + 4F2 (−2n + Γ2n (H1 + Γ2 H3 )
˜1
1
2n
˜
˜
× n2 (1 + n2 ))) + 4E2 (2D1 n2 + C1 Γ2n (G1 + Γ2 G3 )n˜ 2 (1 + n2 )
˜2
˜2
n2
˜2
2n
˜
˜
+ 2(1 + α)))(2˜ 2 (1 + α) − 2n2 (1 + α − ) + 4Γ2n (H1 + Γ2 H3 )
n1
2n
1
˜
× n3 α + Γ2n n(1 + α) ))) ,
P3
where,

G3 = ( (G1 (H1 n2 (1 + n2 )(1 + α) + 2G2 (4n2 n4 α + 3(1 + α)ϕ2 )
˜2
˜2
˜2
1
2
+ H1 (8n2 n4 α + 6(1 + α)ϕ2 ))(˜ 2 (1 + α) + 4n2 (−1 + α(−1 + )))
˜2
n2
− µa,2 n(1 + α)(G2 (˜ 2 (1 + α) + n4 (1 + α + 16H1 n2 α)
˜
˜2
1 n2
2 n
˜2
+ 12H1 (1 + α)ϕ2 ) + H1 (3˜ 2 (1 + α) + n4 (3 + 3α + 16H1 n2 α)
2
+ 12H1 (1 + α)ϕ2 )) ))

1
4R3

141

3 2 ˜4
2 2 ˜4
2 + G H n4 + 8G1 n n2 α + 8G1 H1 n n2 α
(G H n
˜
H3 =
1 1 ˜2
8˜a,2 n 1 1 2
µ
1+α
1+α
1

2
˜2
˜2
+ 6G3 ϕ2 + 6G1 H1 ϕ2 − ((G1 (H1 n2 (1 + n2 )(1 + α) + 2G2
1
1
2
n2
˜2
× (4n2 n4 α + 3(1 + α)ϕ2 ) + H1 (8n2 n4 α + 6(1 + α)ϕ2 ))(˜ 2 (1 + α)
˜2
+ 4n2 (−1 + α(−1 + ))) − µa,2 n(1 + α)(G2 (˜ 2 (1 + α) + n4
˜
˜2
1 n2
2 n
× (1 + α + 16H1 n2 α) + 12H1 (1 + α)ϕ2 ) + H1 (3˜ 2 (1 + α)
2
4α
1
+ n4 (3 + 3α + 16H1 n2 α) + 12H1 (1 + α)ϕ2 )) )(˜ 2 + n2 (−4 +
˜2
n2
))) ,
1 + α R3
where,
4α
))2 )))
R3 = ((1 + α)2 (4˜2 n2 2 + (˜ 2 + n2 (−4 +
µa,2
n2
1+α
The first through third harmonics in the rotor response are defined as,

yyn =

1
3˜ ˜
˜
˜
Γ2 (n4 (−8C3Γ2 + C1 Γ2 n2 (1 + n2 )
˜1
n
n 1
2 n
64(1 + α)

2˜ ˜
3˜
+ C1 (−8 + D1 Γ2 n2 (1 + n2 )))2 + (D1 Γ2 n2 n2 (1 + n2 )
˜1
˜1
n 1
n ˜1
2˜ ˜
˜n
+ D1 n2 (−8 + C1 Γ2 n2 (1 + n2 )) − 8(1 + D3 Γ2 n2 + α))2 ),
˜1
n 1
1
˜
˜ ˜
(n2 (−4C1 D1 Γ2 n2 + n(E2 Γ2 (−8
yy2n =
n 1
n
2
4(1 + α)
2˜
˜
˜
˜
˜
˜2
˜2
+ 3G2 Γ2 n2 (1 + n2 ) + Γ2 H1 n2 (1 + n2 )) + Γ2n (−8Γ2 G3
2n
2
1 2n ˜ 2
2n
˜n ˜
˜
˜2
˜2
˜2
+ G3 Γ2 n2 (1 + n2 ) + G1 (−8 + 2F2 Γ2 Γ2n H1 n2 (1 + n2 )
1 2n ˜ 2
2
2˜
˜ ˜
˜
+ Γ2 H1 n2 (1 + n2 ))))α)2 + (−Γ2 Γ2 (2E2 G1 H1 + F2 (G2 + 3H1 ))
˜2
n 2n
1
2
2n
2 n
˜
× n2 n2 (1 + n2 )α + Γ3 n2 (8H3 − H1 (G2 + H1 )˜ 2 (1 + n2 ))α
˜2
˜2
˜2
2n
1
2
2˜
2˜
˜n
˜
+ 2Γ2 n(−C1 n2 + D1 n2 + 4F2 nα) + 2Γ2n (1 + α + 4H1 n2 α))2 ),
1
1
142

and
2
2 ˜
9(C1 + D1 )3 Γ6 n4 n4 (1 + n2 )2
˜1
n ˜1
yy3n =
.
64(1 + α)2

143

BIBLIOGRAPHY

144

BIBLIOGRAPHY
[1] Engine-cylinder deactivation saves fuel. http://www.jdpower.com/content/detail.
htm?jdpaArticleId=925, 2012.
[2] B. Aldasouqi and B. J. Vidmar. The design of self-centering rollers. Technical report,
Michigan State University, 2012.
[3] A. Alsuwaiyan and S. W. Shaw. Performance and dynamic stability of general-path
centrifugal pendulum vibration absorbers. Journal of Sound and Vibration, 252:791–
815, 2002.
[4] A. Alsuwaiyan and S. W. Shaw. Steady-state responses of systems of nearly-identical
torsional vibration absorbers. Journal of Vibration and Acoustics, 125:80–87, 2003.
[5] Abdallah Saleh Alsuwaiyan. Performance, Stability, and Localization of Systems of
Vibration Absorbers. PhD thesis, Michigan State University, 1999.
[6] V. J. Borowski, S. W. Shaw, H.H. Denman, D. L. Cronin, J.P. Hanisko, L.T. Brooks,
D.A. Mikulec, W.B. Crum, and M.P. Anderson. Reducing vibration of reciprocating
engines with crankshaft pendulum absorbers. 1991 SAE Technical Paper 911876, 1991.
[7] C. P. Chao. The Performance of Multiple Pendulum Vibration Absorbers Applied to
Rotating Systems. PhD thesis, Michigan State University, 1997.
[8] C. P. Chao and S. W. Shaw. Stability of the unison response for a rotating system
with multiple centrifugal pendulum vibration absorbers. Journal of Applied Mechanics,
204(5):769–794, 1997.
[9] C. P. Chao and S. W. Shaw. The dynamic response of multiple pairs of subharmonic
torsional vibration absorbers. Journal of Sound and Vibration, 231:411–431, 2000.
[10] D. L. Cronin. Shake reduction in an automobile engine by means of crankshaft-mounted
pendulums. Mechanism and Machine Theory, 27:517–533, 1992.
[11] J. P. Den Hartog. Vibration in industry. Journal of Applied Physics, 8:76–83, 1937.

145

[12] J. P. Den Hartog. Tuned pendulums as torsional vibration eliminators. Stephen Timoshenko 60th Anniversary Volume, pages 17–26, 1938.
[13] J. P. Den Hartog. Mechanical Vibrations. Dover Publications, Inc. New York, 1985.
[14] H. H. Denman. Tautochronic bifilar pendulum torsion absorbers for reciprocating engines. Journal of Sound and Vibration, 159:251–277, 1992.
[15] Eusebius Doedel. Auto [software]. http://cmvl.cs.concordia.ca/auto/, 1996.
[16] Bard Ermentrout. Xppaut [software]. http://www.math.pitt.edu/~bard/xpp/xpp.
html, 2002.
[17] B. K. Geist. Small displacement high torque engines. personal communication, 2012.
[18] R. J. Gilmore and M. B. Steer. Nonlinear circuit analysis using the method of harmonic
balance-A review of the art. Part I. Introductory concepts. International Journal of RF
and Microwave Computer-Aided Engineering, 1(1):22–37, 1991.
[19] A. G. Haddow and S. W. Shaw. Centrifugal pendulum vibration absorbers: An experimental and theoretical investigation. Nonlinear Dynamics, 34:293–307, 2003.
[20] C. M. Harris and C. E. Crede. Shock and Vibration Handbook. McGraw-Hill Book Co.,
1976.
[21] KerWilson. Practical Solutions of Torsional Vibration Problems. Chapman and Hall,
Ltd., London, 1968.
[22] C.-T. Lee, S. W. Shaw, and V. T. Coppola. A subharmonic vibration absorber for
rotating machinery. ASME Journal of Vibration and Acoustics, 119:590–595, 1997.
[23] C. T. Lee and S.W. Shaw. Nonlinear dynamic response of paired centrifugal pendulum
vibration absorbers. Journal of Sound and Vibration, 203:731–743, 1997.
[24] C.T. Lee and S.W. Shaw. On the couteraction of periodic torques in rotating systems
using centrifugally driven vibration absorbers. Journal of Sound and Vibration, 191:695–
719, 1996.
[25] J. Madden. Constant frequency bifilar vibration absorber. Technical report, United
States Patent No. 4218187, 1980.
146

[26] K. D. McCutchen. No short days. the struggle to develop the r-2800 “double wasp”
crankshaft. Journal of the American Aviation Historical Society, Summer:124–146,
2001.
[27] R. Monroe and S.W. Shaw. Transient dynamics of centrifugal pendulum vibration
absorbers. In The 2011 ASME International Design Engineering Technical Conferences,
2011.
[28] R. J. Monroe. Transient dynamics of nonlinear oscillators with applications to centrifugal pendulum vibration absorbers. PhD thesis, Michigan State University, 2011.
[29] R. J. Monroe, S. W. Shaw, A. G. Haddow, and B. K. Geist. Accounting for roller
dynamics in the design of bifilar torsional vibration absorbers. Journal of Vibration and
Acoustics, 133, 2011.
[30] A. Nayfeh. Nonlinear Interactions: Analytical, Computational, and Experimental Methods. John Wiley & Sons, 2000.
[31] A. H. Nayfeh and D. T. Mook. Nonlinear Oscillations. Wiley-Interscience, 1979.
[32] T. Nester, A. H. Haddow, S. W. Shaw, J Brevick, and V. J Borowski. Vibration
reduction in variable displacement engines using pendulum absorbers.
[33] T.M. Nester, A.G. Haddow, S.W. Shaw, J.E. Brevick, and V.J. Borowski. Vibration
reduction in variable displacement engines using pendulum absorbers. In Proceedings
of the SAE Noise and Vibration Conference and Exhibition, 2003.
[34] Tyler M. Nester. Experimental investigation of circular path centrifugal pendulum
vibration absorbers. Master’s thesis, Michigan State University, 2002.
[35] E. J. Nestorides. A Handbook on Torsional Vibration. Cambridge University Press,
1958.
[36] D. E Newland. Nonlinear aspects of the performance of centrifugal pendulum vibration
absorbers. ASME Journal of Engineering for Industry, 86:257–263, 1964.
[37] B. Salomon, 1933.
[38] R. R. Sarazin, 1932.

147

[39] S. W. Shaw and B. K. Geist. Tuning for performance and stability in systems of nearlytautochronic torsional vibration absorbers. Journal of Vibration and Acoustics, 132,
2010.
[40] S. W. Shaw, P. M. Schmitz, and A. G. Haddow. Dynamics of tautochronic pendulum
vibration absorbers: Theory and experiment. Journal of Computational and Nonlinear
Dynamics, 1:283–293, 2006.
[41] S.W. Shaw, M.B. Orlowski, A.G. Haddow, and B. Geist. Transient dynamics of centrifugal pendulum vibration absorbers. In Proceedings of the 12th International Symposium
on Transport Phenomena and Dynamics of Rotating Machinery, number 2008-20119,
Honolulu, HA, 2008.
[42] M. Swank and P. Lindemann. Dynamic absorbers for modern powertrains. SAE Technical Paper, (2011-01-1554), 2011.
[43] B. N. Taylor and C. E. Kuyatt. Guidelines for evaluating and expressing the uncertainty
of NIST measurement results. http://physics.nist.gov/Pubs/guidelines/TN1297/
tn1297s.pdf, 1994.
[44] T. Theisen. Gravity’s effect on centrifugal pendulum vibration absorbers. Master’s
thesis, Michigan State University, 2011.
[45] W. T. Thomson and M. D. Dahleh. Theory of Vibration. Prentice Hall, 1998.
[46] B. J. Vidmar, S. W. Shaw, B. F. Feeny, and B. K. Geist. Analysis and design of multiple order centrifugal pendulum vibration absorbers. In The 2012 ASME International
Design Engineering Technical Conferences, 2012.
[47] B.J. Vidmar, B.F. Feeny, S.W. Shaw, A.G. Haddow, B. Geist, and N.J. Verhanovitz.
The effects of Coulomb friction on the performance of centrifugal pendulum vibration
absorbers. Nonlinear Dynamics, 69(1-2):589–600, 2011.

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