GRAVITY’S EFFECT ON CENTRIFUGAL PENDULUM VIBRATION ABSORBERS
By
Thomas Michael Theisen

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Mechanical Enginering
2011

ABSTRACT
GRAVITY’S EFFECT ON CENTRIFUGAL PENDULUM VIBRATION
ABSORBERS
By
Thomas Michael Theisen
This work describes an analytical investigation of the effects of gravity on the dynamics
and performance of centrifugal pendulum vibration absorbers (CPVAs). CPVAs are passive
devices, consisting of movable masses suspended on a rotor, which act to reduce torsional
vibrations in internal combustion engines and other rotating systems. When the system
axis of rotation is not vertical, gravity acts directly on the absorbers, and can affect system
behavior at low rotor speeds. The investigation is based on a mathematical model that is
studied using perturbation methods and numerical simulations. The effects of gravity are
captured by a parameter γ, and are examined for different absorber designs and operating
conditions, paying particular attention to cases relevant to internal combustion engines. It is
found that the steady state system response can be divided into two cases, depending on the
order n of the applied torque that leads to torsional vibrations. In the general case, n = 1, 2,
the absorber response to gravity and the applied torque are essentially uncoupled, and the
system response can be systematically explored by the introduction of an equivalent system
order tuning parameter. In the special cases, n = 1, 2, nonlinear interactions from gravity
can alter the order n system response in a nontrivial manner. Specifically, we consider the
n = 2 case, in which a superharmonic response of the absorbers can occur, and the features
of this response depend on the number of absorbers N and other system parameters. These
predictive results will be of use in the design of CPVAs for automotive engines currently
being developed for low speed idle conditions.

To my family current and future.

iii

ACKNOWLEDGMENTS

I would like to take this opportunity to thank all those who have helped me along my journey.
This achievement would not have been possible without the support and encouragement of
friends, family and coworkers. I would like to thank Dr. Brian Feeny and Dr. Steven Shaw.
Dr. Feeny, without your knowledge of perturbation techniques I would not have been able
to learn and grasp the skills needed to complete this thesis. Dr. Shaw, it was your constant
encouragement and ability to look at problems from a different perspective that carried me
through difficulties. I’d also like to thank Alan Haddow, it is because of him that I initially
became involved and interested in CPVA’s. Thank you to Bruce Geist of Chrysler, your
insight on the application of CPVA was instrumental in the development of this thesis. I am
thankful to have worked with Ryan Monroe and Brendan Vidmar, without their expertise
and experience I would have been lost. Thanks to all the others who have been involve in my
research in too many ways to list, Venkat Ramakrishnan, Scott Strachan, Rickey Caldwell,
Nick Miller, Michael Farmer, Brian Rockwell and Jelena Paripovic. A special thanks to
Carl Coppola, without his help I may have never made it through Dr. Feeny’s nonlinear
vibration class. Thank you to the National Science Foundation and Chrysler Group LLC
for their support of this research. Lastly I’d like to thank my family, without their support
and guidance this would never have been achieved.

iv

TABLE OF CONTENTS

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

vii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction
1.1 History and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Equations of Motion
2.1 Model Development . . . . . . . . . . . . . . . .
2.1.1 Kinetic Energy . . . . . . . . . . . . . .
2.1.2 Potential Energy . . . . . . . . . . . . .
2.1.3 Lagrange’s Equations . . . . . . . . . . .
2.1.4 General Path Absorbers . . . . . . . . .
2.1.5 Transformation of Independent Variable
2.1.6 Nondimensionalization . . . . . . . . . .
2.1.7 Scaling . . . . . . . . . . . . . . . . . . .
3 Analysis and Results
3.1 Method of Multiple Scales . . . . . . . . . . .
3.1.1 Slow Flow Equations . . . . . . . . . .
3.1.1.1 Case 1 : n = 1, 2, 3 . . . . .
3.1.1.2 Case 2 : n = 2 . . . . . . . .
3.1.1.3 Case 3 : n = 3 . . . . . . . .
3.2 Time Trace of Response . . . . . . . . . . . .
3.3 Steady-State Analysis . . . . . . . . . . . . . .
3.3.1 Case 1: n = 1, 2, 3 . . . . . . . . . . . .
3.3.2 Case 2: n = 2 . . . . . . . . . . . . . .
3.3.2.1 2 Absorbers . . . . . . . . . .
3.3.2.2 4 Absorbers . . . . . . . . . .
3.3.2.3 3 Absorbers . . . . . . . . . .
3.3.2.4 Higher Numbers of Absorbers
3.3.3 Case 3: n = 3 . . . . . . . . . . . . . .
3.4 Ignoring Gravity . . . . . . . . . . . . . . . .
3.5 The Effects of τ . . . . . . . . . . . . . . . . .
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4 Conclusions and Recommendation for Future Work
4.1 Summary of Results and Conclusions . . . . . . . . . . . . . . . . . . . . . .
4.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . .

80
80
83

Appendices

85

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A: Additional Figures

. . . . . . . . . . . . . . . . . . . . . . . .

Appendix B: Elimination of Gravity Term

. . . . . . . . . . . . . . . . . . .

Appendix C: Higher Order Harmonics Induced by Absorber Motions
Appendix D: Pendulum Type Absorber

86
91

. . . .

94

. . . . . . . . . . . . . . . . . . . . .

97

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

105

LIST OF TABLES

2.1

Description of symbols used in Figure 2.1. . . . . . . . . . . . . . . . . . . .

12

2.2

Description of circular path. . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3

Description of epicycloid path. . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4

Description of cycloid path. . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.1

Difference values given as % between the level of alternating torque at which
the saddle-node bifurcation occurs, with respect to non-dimensional absorber
amplitude, for various values of γ, for n=1.5, with σ = 0%. . . . . . . . . . .

75

Difference values given as % between the level of alternating torque at which
the saddle-node bifurcation occurs, with respect to non-dimensional fluctuating torque amplitude, for various values of γ, for n=1.5, with σ = 0%. . . . .

75

Difference values given as % between the level of alternating torque at which
the saddle-node bifurcation occurs, with respect to non-dimensional absorber
amplitude, for various values of γ, for n=2, with σ = 0%. . . . . . . . . . . .

76

Difference values given as % between the level of alternating torque at which
the saddle-node bifurcation occurs, with respect to non-dimensional fluctuating torque amplitude, for various values of γ, for n=2, with σ = 0%. . . . . .

76

3.2

3.3

3.4

vii

LIST OF FIGURES

1.1

Picture of bifilar type absorber. For interpretation of the references to color
in this and all other figures, the reader is referred to the electronic version of
this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

Diagram of absorber model. . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Diagram of path consideration. . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.1

Time traces for two absorbers at order two without gravity. . . . . . . . . . .

35

3.2

Time traces for two absorber at order 2 with gravity. . . . . . . . . . . . . .

36

3.3

Time traces for four absorber at order two with gravity. . . . . . . . . . . . .

37

3.4

Description of different cases and the responses that occur. . . . . . . . . . .

38

3.5

Change in γ with respect to RPM for various distance from the center of the
rotor (Roj ) to the absorber point mass. . . . . . . . . . . . . . . . . . . . . .

40

Analytical amplitudes of the harmonics of the non-dimensional absorber amplitude versus non-dimensional fluctuating torque, σ = 0. . . . . . . . . . . .

42

Analytical amplitudes of the harmonics of the non-dimensional absorber amplitude versus non-dimensional fluctuating torque for varying γ, σ = 0. . . .

43

Analytical amplitudes of the harmonics of the non-dimensional absorber amplitude versus non-dimensional fluctuating torque comparison of numerical
simulations to analytical model (AM), σ = 0. . . . . . . . . . . . . . . . . . .

45

Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 0% detuning . . . . . . . . . . . . . . . . . . . .

47

3.10 Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 0% detuning, in the desirable operating region. .

48

3.6

3.7

3.8

3.9

viii

3.11 Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 2% detuning. . . . . . . . . . . . . . . . . . . .
3.12 Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 2% detuning, in the desirable operating region. .
3.13 γcrit versus forcing order, = .022276. . . . . . . . . . . . . . . . . . . . . .
3.14 Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
at order 1.5 , analytical model vs numerical simulations, γ = .066. . . . . . .

49
50
51
52

3.15 Order n harmonic of the non-dimensional absorber amplitude versus nondimensional fluctuating torque at order n = 2, for two absorbers, with σ = 0,
for various values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.16 Order n harmonic of the non-dimensional absorber amplitude versus nondimensional fluctuating torque near zero, at order n = 2, for two absorbers,
with σ = 0, for various values of γ. . . . . . . . . . . . . . . . . . . . . . . .

57

3.17 Order n harmonic of the non-dimensional absorber amplitude versus nondimensional fluctuating torque near zero, at order n = 2, for two absorbers,
comparison of simulations to analytical model. . . . . . . . . . . . . . . . . . 58
3.18 Order n harmonic non-dimensional rotor acceleration versus non-dimensional
fluctuating torque at order n = 2, for two absorbers, with σ = 0, for various
values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.19 Order n harmonic of the non-dimensional rotor acceleration versus non-dimensional
fluctuating torque near zero, at order n = 2, with σ = 0, for two absorbers
for various values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.20 Order n harmonic of the non-dimensional rotor acceleration versus non-dimensional
fluctuating torque near zero, at order n = 2, for two absorbers for various values of γ and 2% detuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.21 Order n harmonic of the non-dimensional rotor acceleration versus non-dimensional
fluctuating torque near zero, at order n = 2, for two absorbers, comparison of
simulations to analytical model. . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.22 Order n harmonic of the non-dimensional absorber absorber amplitude versus
non-dimensional fluctuating torque near zero, at order n = 2, with σ = 0 for
four absorbers, with γ = .05. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.23 Order n harmonic of the non-dimensional rotor acceleration versus non-dimensional
fluctuating torque near zero, at order n = 2, with σ = 0 for four absorbers,
for various values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.24 Order n harmonic of the non-dimensional rotor acceleration versus non-dimensional
fluctuating torque near zero, at order n = 2, with σ = 0, for three absorbers,
with γ = .066. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.25 Order n harmonic of the non-dimensional rotor acceleration versus non-dimensional
fluctuating torque near zero, at order n = 2, with σ = 0, for three absorbers,
for various values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ix

3.26 Evaluation of the magnitude of the special term χ in the n = 3 slow flow
equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.27 Comparison of order n harmonic of the non-dimensional absorber amplitude
versus non-dimensional fluctuating torque between n = 3 analytical model
and the general case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.28 Diagram of n = 1, 2 pendulum type circular path absorbers order n rotor
acceleration as a function of Γθ for various τ , τ has no effect, the locked
response is for reference only. . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.29 Diagram of two n = 2 pendulum type circular path absorbers order n rotor
acceleration as a function of Γθ for various τ , response are distinct for each
τ , the locked response is for reference only. . . . . . . . . . . . . . . . . . . .

79

A.1 Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

A.2 Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

A.3 Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .045. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A.4 Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .06. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

D.5 Diagram of pendulum type circular path absorbers. . . . . . . . . . . . . . .

99

x

Chapter 1
Introduction
Centrifugal pendulum vibration absorbers (CPVAs) are passive devices used to reduce engine
order torsional vibrations, that is, twisting vibrations at frequencies proportional to the mean
engine speed, in rotating systems. In these systems the constant of proportionality is referred
to as the engine order. A CPVA is essentially a mass that moves along a path relative to
the rotor, driven by centrifugal effects and the rotor torsional vibrations. The key feature of
CPVAs is that their natural frequency is also proportional to the mean rotor speed, and the
constant of proportionality for the absorber is referred to as the absorber order. When the
absorber is designed with its order close to the engine order, it will react in a manner that
counteracts the applied torque, thereby smoothing vibrations. The aim of this work is to
gain an understanding of the effects of gravity on centrifugal pendulum vibration absorbers
(CPVAs) and their effectiveness in reducing torsional vibrations in systems in which the
axis of rotation is horizontal. This problem is motivated by the proposed use of CPVAs
in automotive engines, for which gravitational effects become comparable with centrifugal
effects at and near engine idle speeds.
1

Automotive fuel economy standards in the United States continue to increase. One of
the ways that progress can be made towards reaching these goals is by lowering engine idle
speeds. The lower the engine idle speed, the less fuel it will consume during idling, thus
resulting in improvement of the vehicle’s fuel economy. However, as in most cases, this
fuel savings does not come for free. Lower idle speeds amplify engine torsional vibrations
that arise from the cylinder firing pulses. These torsional vibrations are undesirable from a
consumer stand point, and can even cause failure if severe enough. It has long been known
that CPVAs can reduce torsional vibrations using less inertia than flywheels. In all previous
studies of CPVAs gravity has been considered to be small relative to centrifugal effects,
and has thus been ignored. This thesis will consider the effects of gravity on the dynamic
response of CPVAs, showing that it can have a substantial impact when engine speeds are
low.

It is shown that it is important to include these gravitational effects when the parameter
γ = g/(R0 Ω2 ), which is the ratio of gravitational to centripital acceleration, is 0.02 or greater
for n = 1.5, and 0.015 or greater for n = 2, with reasonable inertia ratios in the range 5%20%. The corresponding range of engine speeds depends, of course, on the distance from
rotor center to absorber center of mass, R0 . For typical automotive engines, R0 is in the
range of 0.05-0.1 meters, which implies that one should consider gravitational effects for rotor
speeds below about 70-99 radians/second, that is, about 660-950 rpm for n = 1.5 and below
about 80-115 radians/second, that is, about 775-1100 rpm for n = 2. These are precisely
in the range of engine idle speeds. In this work it is shown that gravity, which provides
order one excitation directly to the absorbers, can be considered separately from the order
n oscillating torque acting on the rotor except when n = 1, 2. The n = 2 case is considered
2

in detail, since it has practical application to four cylinder engines, and it is shown that
the response depends on the number of absorbers employed. In many cases the response of
individual absorbers can have different amplitudes, even though they are subjected to the
same loading conditions from gravity and the rotor dynamics. The n = 1 case is left for
future study.

1.1

History and Background

The history and background of CPVAs, also sometime referred to as Adaptive Tuned Vibration Absorbers (ATVA), can be found in detail in the literature, for example, the works
of Alsuwaiyan [1], Schmitz [16], Nester [13], Vidmar [18], Wedin [19], Ker Wilson [22] and
Monroe [11]. For the purpose of this thesis, only selected parts of this background are
described.
CPVAs were first conceived in the early 1900’s. During WWII advancements in the power
to weight ratio of piston aircraft engines led to increased torsional vibrations, which were
severe enough to lead to failures in the engine unless addressed in some manner [22]. The
application of CPVAs as a way of reducing these torsional vibration is discussed by Den
Hartog [7], Ker Wilson [22] and Newland [15]. A key feature of CPVAs is that they are
tuned to a given engine order, as opposed to given frequency, and thus they remained tuned
at all engine speeds. Two of the main features discussed in the early development of CPVAs
were the means of suspending the absorbers, and methods of achieving the desired tuning.
The bifilar (two point) suspension design, as shown in Figure 1.1, allows the absorber to take
the place of existing engine counterweights, and achieve higher tuning orders in a compact
3

Figure 1.1: Picture of bifilar type absorber. For interpretation of the references to color in
this and all other figures, the reader is referred to the electronic version of this thesis.
package, as desired in piston engines.
Early absorber designs were based on a circular path for the absorber center of mass
(COM). The nonlinear effects of the circular path were first explored in a paper by Den
Hartog [8], who noted that the effective absorber tuning decreases as the absorber amplitude increases, an effect known as nonlinear softening. Newland [15] examined these effects
in detail, focusing on the performance of CPVAs, and showed that a jump instability, and
associated hysteresis in the response, can occur at moderate absorber amplitudes. Newland
suggested that slightly overtuning circular path absorbers is beneficial to their performance,
a practice still in use for circular path absorbers. Madden patented a bifilar absorber design
for application on helicopter rotors, using a new concept for the path of the absorber COM,
specifically, a cycloidal path. The cycloidal path absorber allows the pendulum’s tuning to
stay closer to a constant, compared to that of circular path, when the absorber amplitude
4

increases. When a cycloidal path absorber is experiencing high amplitudes of vibration, it’s
tuning does not decrease like that of a circular path absorber, it actually slightly hardens,
which avoids the nonlinear jump [10]. Denman expanded on these ideas for nonlinear absorber tuning by introducing the tautochronic epicycloidal path [5]. This is the path that
leads to constant order at all amplitudes, a result which was known to Huygens, and applied
to absorbers by Denman. These path issues were further explored in the works of Shaw and
co-workers, including Lee and Shaw [9], Alsuwaiyan [1], Chao and Shaw [3] and Geist [6], in
which a general class of paths that includes circles, epicycloids, and cycloids, were explored.
The general path formulation allows for the systematic study of a variety of paths with both
linear and nonlinear tuning.

The stability of the synchronous response of systems of multiple identical absorbers was
examined first by Lee, Chao, and Shaw [4], then Alsuwaiyan [1] and then Shaw and Geist [17].
These works focus on characterizing two distinct kinds of stabilities of a synchronous motion:
those that correspond to the dangerous nonlinear jump that can occur (described above),
and those that break the symmetry of the synchronous response. It is found that a nearly
cycloidal path absorber with a slight linear over tuning provides the best results with regards
to stability [1, 17]. Additionally it was found that the epicycloidal and cycloid path absorbers
do not undergo nonlinear jump behaviors, as do circular path absorbers. These analytical
predictions have been verified by simulation and experimental studies [1, 17, 13, 16]. The
effects of small imperfections, that is, small differences in tuning and damping, between
absorbers was also considered by Alsuwaiyan and Shaw [2], and it was found that this can
lead to localized responses in which subsets of absorbers are essentially inactive, leading to a
number of undesirable effects. A solution to avoid these problems is proposed by increasing
5

the linear detuning, such that it is large compared to the imperfections that may exist.
Nester provides the most systematic experimental study of circular absorber systems [13,
14]. The overall system response was compared against analytical predictions and simulations. The instability that corresponds to the dangerous nonlinear jump was explored
experimentally with a single absorber system. It was found that this instability occurs and
must be avoided, since the absorber becomes a vibration amplifier in this mode. A common
discrepancy that appeared between the analytical model, the numerical results, and the experimental results is that this jump occurred at a lower torque level than predicted. Systems
with four circular path absorber were also tested experimentally and non-unison responses
were observed. However, at low torque levels the non-unison behavior still exists, which does
not align with the analytical predictions, and is still not completely understood. In terms of
practical engine applications, tests were done with 7.5% overtuned (order 2.15) circular path
absorbers in an eight cylinder engine that was run in four cylinder mode, corresponding to
cylinder deactivation [13]. Accelerometer data was collected from the vehicle and showed a
significant decrease in vibration levels when the absorbers were employed.
Epicycloidal path absorbers were examined experimentally, and compared against analytical predictions, by Schmitz [16]. The effect of linear detuning was examined in detail for
these absorbers. Non-unison responses of a two absorber system were also seen during experiments, even at low levels of forcing when the absorbers were tuned close to the excitation
order. This is consistent with analytical predictions.
In a recent MS thesis, Wedin [19] examined a different arrangement for absorbers. In this
work, CPVAs were coupled with a dual mass flywheel, a system developed by LuK Group
and successfully marketed to Mercedes-Benz and BMW. Wedin’s thesis reviews many types
6

of absorbers, including the simple pendulum type, the bifilar type, and the roller type, and
epicycloid paths are also considered. Wedin’s work focuses on modeling and simulations in
an attempt to consider vibrations of additional downstream powertrain components, and
their effects on absorber performance.

Bifilar absorbers use rollers as part of their design. Denman developed a model that
considers the effects of roller inertia on the equations of motion (EOM), and how this can
be accounted for in linear absorber tuning [5]. These results were extended by Monroe,
who solved the nonlinear tuning problem for a bifilar absorber with rollers, and derived a
tautochronic absorber path that accounts for roller inertia [11].

The transient behavior of absorbers was recently examined by Monroe [11], who developed a method for predicting the overshoot for absorbers that respond when the rotor
is suddenly subjected to an applied oscillating torque. This is highly relevant to engines
that employ cylinder deactivation, where absorber overshoot must be taken into account in
designs, so that the absorbers do not exceed their rattle space limits. It was found that
overshoot can be characterized by two parameters: a damping parameter and a system
parameter that combines the excitation amplitude, the order detuning, and the system nonlinearity. Experiments were run to mimic cylinder deactivation in order to examine transient
responses and compare the analytical model and experiment; this was done using circular
path absorbers (the same as those used by Nester) . The results demonstrate the utility of
the simple analytical results in predicting the transient overshoot [11].
7

1.2

Thesis Outline

The remainder of this thesis is organized as follows: In Chapter 2 the mathematical model
for a rotor/absorber system is developed, the EOM motion are derived using a Lagrangian
approach, and the EOM are non-dimensionalized. This extends previous work by the inclusion of gravity, using an appropriate term in the potential energy. As part of the discussion,
the formulation of the absorber path is discussed and the EOM are derived for a general
path. However, for purposes of this research, only circular paths will be considered, since
they are sufficient to demonstrate the dynamical features of interest. In this Chapter it
is also shown that under certain symmetry-related conditions, the effects of gravity cancel
out, a result that is useful throughout the entire thesis. Once the EOM are obtained in the
desired form, they are scaled so that they can be analyzed using the Method of Multiple
Scales.
In Chapter 3, the Method of Multiple Scales is described and applied to the scaled EOM.
It is shown that, in contrast to previous work on absorbers, there are special cases that lead
to resonant interactions between the applied torque, the gravity, and the absorbers. These
cases, which depend on the tuning order and the number of absorbers, are identified and
the system steady-state response for the most interesting and relevant cases are analyzed in
detail. It is shown that the analytical predictions coincide well with numerical simulations
of the full EOM. The results are then analyzed to determine their potential implications for
system response, and to identify the conditions under which they are relevant in automotive applications. Chapter 4 provides a discussion of the results and a description of some
outstanding related problems.

8

Chapter 2
The Equations of Motion
In this chapter the equations of motion (EOM) for CPVAs will be derived and scaled in a
manner convenient for perturbation analysis. The equations of motion will include the effects
due to gravity, which is relevant in systems for which the axis of rotation is not perfectly
vertical. Previous works, specifically, those of Alusuwain [1] and Schmitz [16], outline the
process that is followed to derive the EOM for absorbers that utilize a general path. The
general path equations are made specific for a circular path, thus allowing for comparisons
with the results of Nester [13] and Vidmar [18].
The first step is to develop the kinetic and potential energies for the system, from which
the EOM can be derived using Lagrange’s equations. Once the EOM have been found they
are non-dimensionalized to identify important non-dimensional parameters, and to prepare
the EOM for perturbation analysis. Specifically, the equations are scaled to identify small
effects, so that they can be evaluated through the method of multiple scales (MMS). The
MMS provides equations that approximate the slowly varying (relative to the excitation)
dynamics of the amplitudes and phases of the CPVAs. These equations are analyzed for
9

steady-state responses that account for gravity. A stability analysis of these equations is left
for future work.

2.1

Model Development

The EOM for CPVAs are derived using Lagrange’s equations,

∂
∂t

∂L
∂ q˙
k

−

∂L
= Qk
∂qk

(2.1)

where qk are the generalized coordinates, Qk are the generalized forces, and L is the Lagrangian, which is equal to the kinetic energy minus the potential energy:

L = T − V.

(2.2)

Figure 2.1 shows a diagram of the model that is used to develop the EOM, and Table
2.1 provides a description of symbols used in the diagram. The model shows only one
absorber, but is easily generalized to account for N absorbers. In practical applications two
or more absorbers are desired for balancing and packaging. It is assumed that if more than
one absorber is being used, they are equally spaced around the rotor; this allows for some
beneficial simplifications in the EOM. The generalized coordinates can be seen to be θ for
the rotor angle and Sj for the absorbers. It is assumed that all absorbers are of bifilar
construction for this analysis, so that the absorbers do not rotate relative to the rotor. This
allows the absorber to be modeled by a point mass located at the absorber’s center of mass
(COM), and the absorber moment of inertia about its COM is simply included with that
of the rotor. Pendulum absorbers with single point suspensions rotate relative to the rotor,
10

and this changes the analysis [13]. However, a comparison can be made between these two
types of models, as described in Appendix 4.2.

Figure 2.1: Diagram of absorber model.

11

Symbol

Table 2.1: Description of symbols used in Figure 2.1.
Description

O

Center of rotor

C

Position of point mass when the path length is zero

M

Instantaneous position of point mass

Ro
R(S)

Distance from point O to point C
Instantaneous distance from point O to point mass

S

Path length

θ

Rotor angle

Tθ
g

Torque on rotor

ˆ
ı

Unit vector that is fixed in space and perpendicular to 
ˆ


ˆ

Unit vector that is fixed in space and perpendicular to ˆ
ı

eR
ˆ
eθ
ˆ

Unit vector that is attached to the rotor and perpendicular to eθ
ˆ
Unit vector that is attached to the rotor and perpendicular to eR
ˆ

eS
ˆ

Unit vector that is attached to the point mass and perpendicular to eφ
ˆ
Unit vector that is attached to the point mass and perpendicular to eS
ˆ

eφ
ˆ

Gravity

12

2.1.1

Kinetic Energy

The kinetic energy of the system can be separated into two parts, the kinetic energy of the
rotor and the kinetic energy of the absorbers,
N
T = Trot +

Tj

(2.3)

j=1
where Tj is used to identify the kinetic energy for the j th absorber. The kinetic energy of
the rotor is simple and given by,
1
˙
Trot = Jrot θ2
2

(2.4)

where Jrot is the inertia of the rotor plus that of the absorbers about their respective COM,
˙
and θ is the rate of rotor angular rotation. Each pendulum has a kinetic energy of,

1
Tj = mpj vj · vj
2

(2.5)

where mj is the mass of the j th absorber and vj is the velocity of the absorber. Referring
to Figure 2.1, the absorber velocity can be decomposed into two parts: the velocity of point
C relative to the origin O plus the velocity of the COM relative to point C.

vj(M/C) = S˙j eS
ˆ

(2.6)

˙e
vj(C/O) = R(Sj )θˆθ
This leads to the velocity of the j th absorber being given by,

˙e
vj = vj(M/C) + vj(C/O) = R(Sj )θˆθ + S˙j eS .
ˆ
13

(2.7)

In order to formulate the kinetic energy of the absorbers in terms of the state variables, some
relationships for unit vectors and the path function R(Sj ) are needed. First, for the unit
vectors note that,

eθ · eθ = 1,
ˆ ˆ
eS · eS = 1,
ˆ ˆ
eθ · eS = cos φ.
ˆ ˆ
From Figure 2.2 it can be seen that,

Figure 2.2: Diagram of path consideration.
14

(2.8)

cos (φ) =

dx
.
dS

(2.9)

Using the Pythagorean theorem it is seen that,

2
dSj = dx2 + dR2 ,

(2.10)

which leads to,
dx
=
dSj

dR(Sj ) 2

1−

dSj

.

(2.11)

˙e
Now R(Sj )θˆθ can be restated in terms of R(Sj ) and its derivatives, as follows,

R(Sj )ˆθ · eS =
e ˆ

R2 (Sj ) − R2 (Sj )

dR(Sj ) 2
.
dSj

(2.12)

Note that in some previous formulations the following relationship is used,

R2 (Sj )

dR(Sj ) 2
dSj

1
=
4

dR2 (Sj )

2
.

dSj

(2.13)

˜
The term in equation (2.12) is conveniently expressed by the following path function, G(Sj ),
is convenient for the EOM, defined using equations (2.12), and (2.13), and given by,

˜
G(Sj ) =

1
R2 (Sj ) −
4

dR2 (Sj )
dSj

2
.

(2.14)

In terms of the expressions given above, the system’s kinetic energy is given by,

1
1
2
˙
˙
˙ ˜
T = Jrot θ2 + mpj [R2 (Sj )θ2 + S˙j + 2θS˙j G(Sj )].
2
2
15

(2.15)

2.1.2

Potential Energy

When gravity can be ignored, for example, for rotors running at high speeds (Ω >>

g/R),

or for rotors with a vertical axis of rotation, the absorber system can be accurately modeled
without a potential energy component. When gravity must be accounted for, for example,
in systems with a horizontal axis of rotation running at low speeds, the model must include
a gravitational potential energy. The rotor speeds over which gravity can be ignored, as well
as those for which gravity must be included, are characterized in detail subsequently. For
this work it is assumed that the rotor spins about a fixed, perfectly horizontal axis, which
implies that the gravitational potential energy comes solely from the vertical position of the
absorbers, and is given by,
N
V pj

V =

(2.16)

j=0
where
Vpj = mpj ghj

(2.17)

where g is gravity and hj is the vertical distance from the fixed rotor center O to the COM.
For this purpose, it is convenient to express the distance R(Sj ) in terms of an x component
Xpj (Sj ) and a y component Ypj (Sj ), such that the absorber position vector can be expressed
as (see Figure 2.1),
−R(Sj )ˆφ = Ypj (Sj )ˆR + Xpj (Sj )ˆθ .
e
e
e

(2.18)

These components are described in more detail subsequently for different paths. In order to
find hj , one must project R(Sj ) onto the fixed reference frame (ˆ and ), as shown in Figure
ı
ˆ
16

2.1. The required relationships between the rotating vectors (ˆR , eθ ) and (ˆ, ) are given by,
e ˆ
ı ˆ

eR = sin (θj )ˆ + cos (θj )ˆ,
ˆ
ı


(2.19)

eθ = cos (θj )ˆ − sin (θj )ˆ.
ˆ
ı


(2.20)

2π
θj = θ + ψj where ψj = (j − 1) ,
N

(2.21)

where

is the rotor angle, θj , for absorber j is defined in terms of the segmental angle ψj , to give
the angular position of the j th absorber relative to θ = 0. Using these in equation (2.18)
gives,
−R(Sj )ˆφ = [Ypj (Sj ) sin (θj ) + Xpj (Sj ) cos (θj )]ˆ
e
ı

(2.22)

+ [Ypj (Sj ) cos (θj ) − Xpj (Sj ) sin (θj )]ˆ.

The potential energy for the j th absorber is thus given by,

Vpj = mpj g[Ypj (Sj ) cos (θj ) − Xpj (Sj ) sin (θj )].

2.1.3

(2.23)

Lagrange’s Equations

Now that the kinetic and potential energies have been determined, Lagrange’s method can
be used to find the EOM. Note that dissipation and applied torques will subsequently be
included by generalized forces. The Lagrangian is given by,
N
N
1
1
2 (S )θ2 +S˙ 2 +2θS˙ G(S )]]−
˙2 +
˙
˙ ˜
[ mpj [R
[mg[Ypj cos (θj )−Xpj sin (θj )]],
L = Jrot θ
j
j
j
j
2
2
j=1
j=1
(2.24)
17

where the generalized coordinates are θ and Sj . The generalized forces Qk are, for the rotor,

˙
Qθ = −co θ + T0 + Tθ sin (nθ + τ ),

(2.25)

which includes rotor viscous resistance and applied torques, and for the j th absorber damping
is accounted for using a viscous model,

QS = −caj S˙j .
j

(2.26)

In these generalized forces, n is the engine order, co is the rotor resistance coefficient, T0 is
the mean torque, Tθ is the amplitude of the alternating torque at order n with a phase of
τ (allowing this torque to be phased relative to gravity), and caj is the absorber damping
coefficient. The engine order is based on the number of cylinders in an engine, for example,
for a four stroke engines with Nc cylinders, the engine order is half of the number of cylinders,
n = Nc .
2

Using Lagrange’s method the rotor EOM is found to be,
N
¨
Jrot θ +
i=1

2
˜
2 (S )θ + dR (Sj ) S˙ θ + G(S )S + dG(Sj ) S˙ 2
¨
˙ ˜
¨
mpj [R
j
j
j j
j
dS
dS

(2.27)

˙
− g(Xp (Sj ) cos(θj ) + Yp (Sj ) sin(θj )] = −co θ + T0 + Tθ sin (nθ + τ ),
and the absorber EOM are given by,
dR2 (Sj ) 2
dXp (Sj )
dYp (Sj )
¨
˙
¨ + G(S )θ − 1
˜
mpj [Sj
θ + g(−
sin(θj ) +
cos(θj ))] = −caj S˙j .
j
2 dS
dS
dS
(2.28)
18

These form the basis of the analysis that follows.

2.1.4

General Path Absorbers

A quite general formulation is used to describe the path for the absorber COM, as described
in detail by Denman [5] and used in subsequent studies [17, 1, 16, 11, 19]. The general path
contains a two parameter family of epicycloids that range from circles to cycloids, both of
which are special cases of epicycloids. It is beneficial to describe these paths by the local
radius of curvature at every point on its path,

2
ρ2 = ρ2 − λ2 Sj ,
oj
j
j

(2.29)

where ρj is the instantaneous radius of curvature, ρoj is the radius of curvature of the path
at the vertex, and λj ∈ [0, 1] describes the amplitude dependent characteristics of the path.
When λj = 0, the path is a circle, and when λj = 1 the path is a cycloid. Another case of
importance is the tautochronic path, described below, for which,
nj
˜
,
(˜ 2 + 1)
nj

λj =
t

(2.30)

where nj is the tuning order of the absorber.
Nonlinearities of this system manifest themselves in two different ways: nonlinear coupling through the rotor and a nonlinear restoring force acting on the absorbers. Circular
paths provide a softening type of nonlinear restoring force, while cycloids are hardening.
The tautochronic path is of importance because it results in a purely linear restoring force
acting on the absorber. The detailed expressions for the (x, y) components for these paths
19

are given in Tables 2.2, 2.3, and 2.4. In terms of these two path parameters the linear tuning
order is found from small amplitude vibration analysis, and is given by,

nj =
˜

(Roj − ρoj )
,
ρoj

as described in [5, 1].
Table 2.2: Description of circular path.
Circular Path
λj
Sj
Xp (Sj )

Yp (Sj )

0
ρoj φj
λ2 Sj
j
2 (sin (φj ) cos (λj φj ) − ρoj cos (φj ))
1−λj
ρoj
λ2 Sj
j
Roj +
2 (cos(φj ) cos (λj φj ) + ρoj sin(φj ) − 1)
1−λj
ρoj

Table 2.3: Description of epicycloid path.
Epicycloid Path
λj

0 < λj < 1

Sj

ρoj
sin (λj φj )
λj

Xp (Sj )

Yp (Sj )

λ2 Sj
j
2 (sin (φj ) cos (λj φj ) − ρoj cos (φj ))
1−λj
ρoj
λ2 Sj
j
Roj +
(cos(φj ) cos (λj φj ) + ρo sin(φj ) − 1)
1−λ2
j
j
ρoj

20

(2.31)

Table 2.4: Description of cycloid path.
Cycloid Path
λj

1

Sj

ρoj sin (φj )

Xp (Sj )

1 ρ (φ + 1 sin (2φ ))
j
2 oj j 2

Yp (Sj )

1
Roj − 4 ρoj (1 − cos (2φj ))

In Tables 2.2, 2.3, and 2.4 the angle φ is used to describe the effective angle of rotation
of the absorber as measured from the radial line passing through the rotor center and the
absorber vertex [5]. In each case, φ can be determined as a function of Sj , and then Xp (Sj )
and Yp (Sj ) can be expressed as functions of Sj . Once Xp (Sj ) and Yp (Sj ) have been defined,
it can be verified that
R2 (Sj ) = Xp (Sj )2 + Yp (Sj )2 .

2.1.5

(2.32)

Transformation of Independent Variable

The EOM have, as usual, time as the independent variable. However, due to the applied
torque dependence on the rotor angle θ, it is convenient to use θ as the independent variable.
This requires a change of coordinates, as described in detail, for example, in [1]. For the
rotor speed we define a non-dimensional variable ν as,

ν=

˙
θ
,
Ω

(2.33)

where Ω is the the mean speed of the rotor. Thus, fluctuations in rotor speed about its mean
are described by deviations in ν from unity. The chain rule is used to find the relationships
21

between derivatives with respect to time and derivatives with respect to θ, which are given
by,
2
¨ d θ = d ( dθ ) = d (Ων) = Ω d (ν) = Ω dν dθ = Ω2 νν ,
θ=
dt dt
dt
dt
dθ dt
dt2

(2.34)

for the rotor angular acceleration, and generally by,

d
d dθ
d
=
= νΩ = νΩ( ) ,
dt
dθ dt
dθ

(2.35)

d2
d2 dθ 2 d d2 θ
=
( ) + (
) = ν 2 Ω2 ( ) + Ω2 νν ( ) .
2
2 dt
2
dθ dt
dt
dθ

(2.36)

˙=

¨=

Equations (2.27) and (2.28) can now be transformed to θ dependent variable form, and are
given by,

Jrot Ω2 νν +

N

2
2 (S )Ω2 νν + dR (Sj ) ν 2 Ω2 S + G(S )[ν 2 Ω2 S + Ω2 νν S ]
˜
mpj [R
j
j
j
j
j
dS

j=1
˜
dG(Sj ) 2 2 2
ν Ω Sj − g(Xp (Sj ) cos(θj ) + Yp (Sj ) sin(θj )]
+
dS
= −co νΩ + T0 + Tθ sin (nθ + τ ),

(2.37)
2
2 Ω2 S + Ω2 νν S + G(S )Ω2 νν − 1 dR (Sj ) Ω2 ν+
˜
mpj [ν
j
j
j
2 dS
dXp (Sj )
dYp (Sj )
g(−
sin(θj ) +
cos(θj ))] = −caj νΩSj .
dS
dS

2.1.6

(2.38)

Nondimensionalization

The next step is to non-dimensionalize these equations to reduce the number of parameters.
We begin with the dynamic states of the system, and then turn to parameters. The rotor
speed has been nondimensionalized by the introduction of ν, and the nondimensional absorber displacement is defined by sj = Sj /Roj . Then, dividing equation (2.37) by Jrot Ω2
22

and equation (2.38) by mpj Ω2 Roj yields the following non-dimensional EOM,

νν +

bo
N

N
[x(sj )νν +
j=1

d˜(sj ) 2 2
g
dx(sj ) 2
ν sj + g (sj )[ν 2 sj + νν sj ] +
˜
ν sj
ds
ds

(2.39)

− γ(xp (sj ) cos(θj ) + yp (sj ) sin(θj ))] = −µo ν + Γ0 + Γθ sin (nθ + τ ),
1 dx(sj )
ν 2 sj + νν sj + g (sj )νν −
˜
ν
2 ds
dyp (sj )
dxp (sj )
sin(θj ) +
cos(θj )) = −µaj νsj ,
+ γ(−
ds
ds

(2.40)

in which several nondimensional parameters and functions have been introduced, as follows:

Io
bo =
,
Jrot
co
Jrot Ω

N

2
Io = mo Ro ,
j
= µo ,

mo =

m pj ,
j=1

T0
= Γ0 ,
Jrot Ω2

Sj
= sj ,
Roj

R2 (Sj )
= x(sj ),
2
Ro
j

Tθ
=Γθ ,
Jrot Ω2

Yp (Sj )
= yp (sj ),
2
Ro
j

˜
G(Sj )
= g (sj ),
˜
Roj

g
Roj Ω2

ca
= µa ,
m pj Ω

Xp (Sj )
= xp (sj ),
2
Ro
j

= γj .
(2.41)

The functions xp (sj ) and yp (sj ) are specified by the selection of the path. Using the fact
Roj
that Sj = sj Roj , as well as the relation ρo = 1 + n2 from equation (2.31), it can be found
˜j
j
that for epicycloidal paths,

xp (sj ) =

1
(1 − λ2 )
j

[sin (

− λ2 sj cos (
j

arcsin (sj λj (1 + nj 2 ))
˜
λj

arcsin (sj λj (1 + nj 2 ))
˜
λj

23

)

)],

1
− (s2 λj )2
j
(1 + nj 2 )2
˜

(2.42)

and

yp (sj ) = − 1 −

1
(1 − λ2 )
j

[cos (

arcsin (sj λj (1 + nj 2 ))
˜
λj

)

1
− (s2 λj )2
j
2 )2
(1 + nj
˜

arcsin (sj λj (1 + nj 2 ))
˜
− λ2 sj sin (
) − 1].
j
λj

(2.43)

Taking the derivatives of equations (2.42) and (2.43) we obtain, after simplifications,
arcsin(λj (1 + nj 2 )sj )
˜

xp (sj ) = cos

λj

(2.44)

and

arcsin(λj (1 + n2 )sj )
˜j
.
yp (sj ) = sin 
λj


(2.45)

Additionally, a Taylor Series expansions of xp (sj )2 + yp (sj )2 = r2 (sj ) in sj is useful for
some calculations, and is given by,

x(sj ) = 1 − n2 s2 + κ1 s4 + . . . ,
˜ j
j j

(2.46)

1
n
nj
˜j
κ1 = (˜ 2 + 1)2 (˜ 2 − λ2 (1 + n2 )).
j
j
12 j

(2.47)

where

It is noted that there are special forms of the path functions for circular and cycloidal paths,
as mentioned above. However, the general epicycloid expresssions with λ = .01 and λ = .99
model the circular and cycloidal paths, respectively, quite accurately, and are used here.

This completes the formulation of the terms needed for the EOM. Next, the EOM are
scaled to identify small parameters, so that perturbation analysis can be performed.
24

2.1.7

Scaling

In order to proceed with perturbation analysis, the EOM (2.39) and (2.40) need to be scaled
using a small bookkeeping parameter

that reflects the relatively small effects of forcing,

damping, gravity, and nonlinearity, when compared to centrifugal and inertial effects. The
scalings of these small parameters and variables by

are chosen so that these effects are

captured by a relatively simple perturbation analysis. Several terms in equations (2.39)
and (2.40) need to be scaled by

in order for the desired overall scaling to be achieved.

Specifically,
bo = B , ν = 1 + W w , ν = W w , sj = P pj , sj = P pj , sj = P pj , κ1 = κ0 −B
˜
Γθ = F Γθ , γj = G γj , µaj = L µaj , nj = n(1 + Q σj ) , Γ0 − µo ν = O( D ),
˜
˜
˜
(2.48)
which are described here. The inertia ratio bo is small parameter the rotor inertia J is
considerably larger than the total absorber inertia Io . The applied oscillating torque Γθ is
small since it is scaled by the rotational kinetic energy of the system. Due to this torque,
˙
the rotor velocity θ is not a constant but has small variations compared to its mean value,
accounted for by W w. The absorber oscillation amplitudes sj are small compared to the
radial distance Roj . The path nonlinearity is scaled so that it will be retained (this scaling
is not standard, since the power of

is negative, but provides useful results). The gravity

parameter γ is small compared to the centripetal acceleration Roj Ω2 . The absorber damping
parameter µa is small, since it is proportional to the damping ratio, and these absorbers are
lightly damped. It has been shown in previous work that a small level of detuning between
the excitation and absorber orders is beneficial, and this is captured by Q σ. The detuning
˜
will be stated by σ or as a percent detuning, 100 ∗ n−n . For a nearly constant rotor speed
n
25

to be maintained, the difference between the mean applied torque Γ0 and bearing resistance
torque µo ν must be small and, in fact, it is assumed that the scaling exponent D is such
that this difference does not appear in the resulting model (this assumption is required for
the analysis to carry through, and fortunately, it allows for useful results to be obtained).
These scalings and series expansions are used in the absorber path functions, resulting
in the following simplified expressions,
dx(sj )

x(sj ) = 1 − n2 2P p2 + κ0 3P p4
j
j
j
j
(n2 + n4 ) 2P p2
j
j
j
g (sj ) = 1 −
˜
2

dsj

d˜(sj )
g
dsj

= −2n2 P pj + 4κ0 2P p3
j
j
j

= −(n2 + n4 ) P pj
j
j
(2.49)

(1 + n2 )2 3P p3
(1 + n2 )2
dxp (sj )
j
j
j
2P p2
Pp −
xp (sj ) =
=1−
j
j
6
dsj
2
(1 + n2 )
dyp (sj )
j 2P 2
yp (sj ) = −1 +
pj
= (1 + n2 ) P pj
j
2
dsj

It is also convenient to assume that each absorber in the system is identical in terms
of mass, damping, and path, even though the absorbers may not have identical responses.
Under this assumption,

Roj = Ro

ρoj = ρo

λj = λ

γj = γ

σj = σ

κ0 = κ0
j

nj = n

µ aj = µ a .
˜
˜
(2.50)

Substitution of equations (2.48), (2.49), and (2.50) into equations (2.39) and (2.40), an
expanded form for the EOM is obtained, as follows,
n2 pj P + pj P + 2n2 σpj P +Q + µa pj L+P + w W − γ sin (θj ) G − 2κo p3 3P
˜
˜
j
+ (1 + n2 )˜ pj cos (θj ) P +G + HOT = 0
γ
26

(2.51)

and

1
˜
(−Γθ sin(nθ + τ )) F + (w +
N

N
j=1

1
pj +
N

N

γ sin (θj )) 2P + HOT = 0,

(2.52)

j=1

where HOT refers to terms of higher order in . It is crucial that the parametric excitation
term (1 + n2 )˜ pj cos (θ) P +G be retained, since it describes an important effect of gravity
γ
on the system. To capture this effect, along with the forcing, damping, detuning, and path
nonlinearities, the following values are select for the scaling: P = 1 , G = 1 , W = 1,
2
2
1
1
3
Q = 2 ,L = 1 , B = 2 , D = 2 and F = 1.
2

The final result for the expanded EOM, that retains the important effects, is given by,

1
˜
γ
pj + n2 pj − γ sin (θj )) 2 + (2n2 σpj + µa pj + w − 2κo p3 + (1 + n2 )˜ pj cos (θj )) = 0
˜
j
(2.53)
for the absorbers, and

1
˜
(w − Γθ sin (nθ + τ ) +
N

N
pj +
j=1

1
N

N
γ sin (θj )) = 0,

(2.54)

j=1

for the rotor, where the higher order terms have been dropped.

It is convenient at this point to note that with this scaling one can decouple the rotor
EOM from those of the absorbers. This is done by solving for w in equation (2.54) and
1
substituting the result into equation (2.53). The result is divided by 2 and expanded in ,
27

yielding,
1
˜
[pj + n2 pj − γ sin (θj )[+ 2 [2n2 σpj + µa pj − 2κo p3
˜
j
1
˜
+ (1 + n2 )˜ pj cos (θj ) + Γθ sin (nθ + τ ) −
γ
N

N
j=1

1
pj +
N

N

(2.55)
γ sin (θj ) = 0.

j=1

The leading order terms can be used to solve for pj , as

pj = γ sin (θj ) − n2 pj .
˜
˜

(2.56)

1
This is used to replace pj in the order 2 term in equation 2.55, and the result is simplified
using the method described in Appendix 4.2, which shows that N sin(θj ) = 0. Addition1
1
ally, it is convenient to define 2 = ˆ. This results in the final equation that used for the
detailed analysis presented in the next chapter,
(pj + n2 pj − γ sin (θj )) + ˆ(2n2 σpj − 2κo p3 + (1 + n2 )˜ pj cos (θj )
˜
γ
j
+

1
N

N

˜
n2 pj + µa pj + Γθ sin (nθ + τ )).
˜

(2.57)

j=1

Note that this equations contains leading order effects of damping, order detuning, as well as a
cubic nonlinearity. Additionally, it should be noticed that the equation has small parametric
forcing due to gravity, hard (nonresonant) direct forcing due to gravity, and soft forcing due
to the (resonant) oscillating torque. In the next chapter we determine approximations for
the response of the system using perturbation methods applied to equation (2.57).

28

Chapter 3
Analysis and Results
In this chapter the method of multiple scales will be used to develop the slow flow equations.
For different resonance cases based on the order of forcing. The most interesting case is
when the forcing order equals two. In this case it will be shown that the solution to the
slow flow equations varies based on the number of absorbers that are being used. It will
be shown that from the slow flow equations the absorber amplitude can be solved for as
a function of the detuning (σ), damping (µa ), path nonlinearity (κ0 ), gravity (γ), forcing
order (n) and absorber spacing (ψj ). It should be noted that κ0 is dependent on the type
of path defined, which has is defined by λ and the tuning of the absorbers (˜ ). Additionally
n
the rotor acceleration can be estimated from the slow flow equations in conjunction with the
scaled equations. This allows the effect of gravity on performance to be analyzed easily.

3.1

Method of Multiple Scales

The method of multiple scales is used to determine the slow flow equations. Nayfeh and
Mook [12] describe this method in their book in more depth. Only the math for our specific
29

application will be shown. In the method of multiple scales it is assumed that the response
involves different time scales. In this problem the EOM have changed from a time dependence
to a rotor angle dependence. This change has no effect on the procedure. The response can
be defined by

pj = X0 (θ0 , θ1 , θ2 , ...) + X1 (θ0 , θ1 , θ2 , ...) + 2 X2 (...) + ...
j
j
j

(3.1)

pj = D0 X0 + D0 X1 + D1 X0 + O( 2 )
j
j
j

(3.2)

2
2
p = D0 X0 + D0 X1 + 2 D0 D1 X0 + O( 2 )

(3.3)

which leads to

and

where j is the absorber index, D0 is the partial derivative with respect to the rotor angle
scale θ0 , and D1 is the partial derivative with respect to the rotor angle scale θ1 , and so forth
for higher-order expansions. Equations (3.1), (3.2), (3.3) and (2.22) can then be plugged into
equation (2.57). This will give a rather large equation but only the first two orders of
considered important for this analysis. Separating coefficients of like powers of

leads to

0 : D2 X + n2 X = γ sin(θ + ψ )
0j ˜
j
0 0j
2 N
1 : D2 X + n2 X = −2D D X − 2n2 σX − n
X0
1j
0 1 0j
0j
0 1j
k
N
k=1
3
˜
˜
− (1 + n2 )˜ cos(θj )X0 + 2κ0 X0 − µa D0 X0 − Γθ sin(nθ + τ )
γ
j
j
j
which are partial differential equations.

30

are

(3.4)

(3.5)

The solution for X0 in equation (3.4) is harmonic and contains both a homogeneous
j
solution as well as a particular solution. For this analysis a cosine will be used for the
homogeneous response and will be written in exponential form with amplitude aj and phase
αj which are functions of θ, as they are constants of integration of a partial differential
equation. This leads to the following solution for equation (3.4):

¯
¯
Xi0 = Cj einθ0 + Cj e−inθ0 + Λj eiθ0 + Λj e−iθ0

(3.6)

1
1
iα
−iαj
¯
C j = aj e j & C j = aj e
2
2
i˜
γ
−i˜
γ
iψ
−iψj
¯
Λj =
e j & Λj =
e
2)
2)
2(1 − n
2(1 − n

(3.7)

where

and Cj is a function of θ1 . Now the solution found in equation (3.6) can be plugged into
equation (3.5) such that

1 : D2 X + n2 X = −2C ineinθ0 − 2n2 σ(C einθ0 + Λ eiθ0 )
1j
j
j
0 1j
j
n2
−
N

N

iψ
¯
¯
(Ck einθ0 + Λk eiθ0 ) − ∆e j (Cj eiθ0 (1+n) + Cj eiθ0 (1−n) + Λj ei2θ0 + Λj )
k=1

3
2¯
¯
¯2
+ 2κ0 (Cj ei3nθ0 + 3Cj Cj einθ0 + 6Cj Cj Λj eiθ0 + 3Cj Λj eiθ0 (1−2n)
2
¯
+ 3Cj Λj eiθ0 (1+2n) + 3Cj Λ2 eiθ0 (2−n) + 3Cj Λ2 eiθ0 (2+n) + Λ3 ei3θ0
j
j
j
˜
iΓ
¯
¯
+ 6Cj Λj Λj einθ0 + 3Λ2 Λj eiθ0 ) − µa (Cj ineinθ0 + Λj ieiθ0 ) + θ eiτ einθ0 + C.C.
˜
j
2
(3.8)
where here ( ) =

d( )
, and
dθ
∆=

(1 + n2 )˜
γ
2

31

(3.9)

3.1.1

Slow Flow Equations

In the method of multiple scales the slow flow equations are determined by eliminating the
secular terms in equation (3.8). The order of the absorbers relative to the order of excitation
has an impact on what terms will be secular. There are four different cases that can be
considered. The special cases are n = 1, n = 2 and n = 3. Other orders of the absorbers fall
˜
˜
˜
into the general case. The case when n = 1 will not be explored in this work.

3.1.1.1

Case 1 : n = 1, 2, 3

This is the general case; it directly applies to engines with 3, 5, 7, 8 or more cylinders. When
all of the secular terms are collected and equated to zero the following is found:
N
˜
iΓ
n2
2¯
¯
(2nCj i + 2n2 σCj +
Ck − κ0 (6Cj Cj + 12Cj Λj Λj ) + n˜a Cj i − θ eiτ )einθ0 = 0
µ
N
2
k=1
(3.10)
¯
¯
Plugging in Cj , Cj , Λj , Λj and ∆ and separating into real and imaginary parts, the slow
flow equations are obtain as

RE :

− aj αj + naj σ +

n
2N

N

3
[ak cos(αk − αj )] − κ0 a3
j
4n
k=1

˜
˜2
Γ
3 κ0 aj γ
−
− θ sin(αj − τ ) = 0
2 n(1 − n2 )2 2n
IM :

n
aj +
2N

(3.11)

N

˜
Γ
µa
˜
[ak sin(αk − αj )] +
aj − θ cos(αj − τ ) = 0.
2
2n
k=1

When γ = 0 the slow flow equations give the same results as those found by Alsuwaiyan [1].
32

3.1.1.2

Case 2 : n = 2

This is the case of a four cylinder engine. When all the secular terms are collected and
equated to zero the following is found:
2 N
iψ
2 σC + n
2¯
¯
(2nCj i + 2n
Ck − ∆Λj e j − κ0 (6Cj Cj + 12Cj Λj Λj )
j
N
k=1
˜
iΓ
+ n˜a Cj i − θ eiτ )einθ0 = 0
µ
2

(3.12)

¯
¯
Plugging in Cj , Cj , Λj , Λj , from equations (3.7) and ∆ from equation (3.9) and separating
into real and imaginary parts produces the slow flow equations as

RE :

IM :

n
− aj αj + naj σ +
2N

N

3
[ak cos(αk − αj )] − κ0 a3
j
4n
k=1
˜
˜2
Γ
3 κ0 aj γ
(1 + n2 )˜ 2
γ
−
sin(2ψj − αj ) = 0
− θ sin(αj − τ ) −
2 n(1 − n2 )2 2n
4n(1 − n2 )
N
˜
Γ
n
µa
˜
aj +
[ak sin(αk − αj )] +
aj − θ cos(αj − τ )
2N
2
2n
k=1
(1 + n2 )˜ 2
γ
+
cos(2ψj − αj ) = 0.
4n(1 − n2 )

(3.13)

Notice that in this case the slow flow equations have an additional term compared to the
general case. This extra term contains ψj , meaning the absorber response is dependent on
the relative position of the absorber. This means the equations can change for systems if
the number of absorbers changes. This will be discussed in more length later.

33

3.1.1.3

Case 3 : n = 3

This is the case for a six cylinder engine. When all the secular terms are collected and
equated to zero, the following is found:
n2
(2nCj i + 2n2 σCj +
N

N

2¯
¯
Ck − κ0 (6Cj Cj + 12Cj Λj Λj + Λ3 )
j
(3.14)

k=1
˜
iΓ
+ n˜a Cj i − θ eiτ )einθ0 = 0.
µ
2

¯
¯
Plugging in Cj , Cj , Λj , Λj from equations (3.7) and ∆ from equation (3.9) and separating
into real and imaginary parts leads to the slow flow equations,

RE :

IM :

N

3
[ak cos(αk − αj )] − κ0 a3
j
4n
k=1
˜
˜2
Γ
κ0 γ 3
˜
3 κ0 aj γ
−
− θ sin(αj − τ ) −
sin(3ψj − αj ) = 0
2 n(1 − n2 )2 2n
4n(1 − n2 )3
N
˜
Γ
n
µa
˜
aj − θ cos(αj − τ )
aj +
[ak sin(αk − αj )] +
2N
2
2n
k=1
κ0 γ 3
˜
+
cos(3ψj − αj ) = 0
4n(1 − n2 )3
n
− aj αj + naj σ +
2N

(3.15)

Notice that in this case the slow flow equations are also dependent on ψj , similar to Case 2.
This case will be examine in more detail later to show that it falls in to Case 1.

3.2

Time Trace of Response

Before the steady-state analysis is done for the different cases in the next few sections, it
is beneficial to look at the responses seen in numerical simulations of the full equations of
motion. This will show that, unlike in previous work, additional work will needed in order
34

to fully understand the results and compare them to analysis. Consider a system that has
= .022276, µa = .014921, n = 2, σ = 0 and N = 2. The absorber response will be found
for the case without gravity (γ = 0) and with gravity (γ = .05).

The time trace without

Figure 3.1: Time traces for two absorbers at order two without gravity.
gravity, (Figure 3.1), shows that absorbers have the same amplitude and are synchronous.
The time trace that includes gravity, (Figure 3.2), shows that the two absorbers have the
same response form but are phased differently. Using a fast Fourier transform (FFT) in
MATLAB the amplitude and phases of the responses can be found. When this is done the
35

Figure 3.2: Time traces for two absorber at order 2 with gravity.

amplitudes of the responses at order one and order two are the same. The phase difference
of the order two part of the response is zero. The phase difference of the order one part is
π. When equation (2.57) is examined it can be seen that the order one part of the forcing
(˜ sin (θj )) is dependent on the position of the absorber. This introduces a phase difference
γ
between the order one responses of the two absorbers.
The number of absorbers and the order forcing has an impact on the different kinds of
response that can be seen. Similar to the case of two absorbers, when the forcing order does
36

not equal two the response will have an order one and order n term. The order one response
will have a phase difference that will be the same as the difference in the positions of the
absorbers around the rotor, ψj . However, the case when the forcing order is two does not
follow this. Figure 3.3 shows this case with four absorbers. A FFT is used to break down
each response into its components. In this case the order one is the same as the other cases.
However, in this case the phase differences between the order n responses is not equal to
zero. This is due to the superharmonic response of the absorbers to gravitational forcing.
This will be addressed more in future sections.

Figure 3.3: Time traces for four absorber at order two with gravity.
37

3.3

Steady-State Analysis

It is ideal to look at the steady-state of the system. In the absence of gravity unison motions
have typically been sought by setting the absorber amplitudes and phases equal for all
absorbers. This assumes that each absorber will have the same amplitude and phase, which
has already been shown to not always be the case when gravity is present. This has an effect
on the analysis but can easily be accounted for in the analysis of the slow flow equations.
The chart in Figure 3.4 shows all of the special cases that were found, where S means that
each absorber response is the same, V means that a quantity in the response varies, and D
means that they are different with the number meaning the number of different amplitudes
or phases. Each of the special cases of slow flow equations, that contain information about
order n response, will now be analyzed.

Figure 3.4: Description of different cases and the responses that occur.

38

3.3.1

Case 1: n = 1, 2, 3

The steady-state condition corresponds to aj = 0 and αj = 0. Applying this to equation
(3.11) and setting aj = a and αj = α for a unison response at order n and gives the following:

RE :

IM :

na[σ +

3κ0 γ 2
˜
1
3
−
] − κ0 a3
2 2n(1 − n2 )2
4n

˜
Γ
− θ sin(α − τ ) = 0
2n
˜
Γ
µa
˜
a − θ cos(α − τ ) = 0
2
2n

(3.16)

These equations can be combined into one equation:

(

˜
Γ
κ0 γ 2
˜
1 3
µa 2
˜
3
a) + (n(σ + −
)a − κ0 a3 )2 = ( θ )2
2
2 2 n2 (1 − n2 )2
4n
2n

(3.17)

It has been shown by Newland [15] that a certain amount of overtuning of the absorber is
ideal in order to avoid the dangerous jump that can occur. This jump is harmful because the
absorbers become amplifiers, and instead of reducing the vibration they will increase it. The
desirable operating range of the absorber system precedes this jump. Equation (3.17) has a
term that will be looked at more in depth and can be considered as an equivalent detuning:

σe = σ +

κ0 γ 2
˜
1 3
−
2 (1 − n2 )2
2 2n

(3.18)

The gravity term is part of this equivalent detuning. It is desirable to have an equivalent
1
over tuning that is positive and greater than 2 . It is known that σ will be positive for an
over-tuned absorber. Thus the gravity factor is actually bringing the absorbers toward exact
tuning and undertuning. As an engine slows down the effects of gravity will increase because
39

γ will increase. This can be seen in Figure 3.5.

Figure 3.5: Change in γ with respect to RPM for various distance from the center of the
rotor (Roj ) to the absorber point mass.

Since γ increases as the engine slows down the equivalent detuning will decrease. This
can be beneficial but also dangerous. Absorbers will preform best when tuned exactly to
the forcing order, i.e. σ = 0. However, when absorbers are perfectly tuned they are more
susceptible to the nonlinear jump. As the engine speed decreases, Γθ will increase according
to equation (2.41). There will be more alternating torque that needs to be corrected as
40

the engine slows down, but the absorbers’ equivalent detuning tends towards the equivalent
detuning of a perfectly tuned absorber allowing the system to absorb more of this alternating
torque. Since the absorbers are tending towards the equivalent detuning of a perfectly tuned
absorber, the predicted nonlinear jump will occur earlier than if gravity is left out of the
analysis.
˜ γ ˜
Equation (3.17) can be solved for Γθ (˜ , µa , σ, κ0 , n) to give
˜
µa
˜
1 3 κ0 γ 2
3
˜
Γθ = 2n ( a)2 + (n2 (σ + −
)a − κ0 a3 )2
2
2 2 n(1 − n2 )2
4n

(3.19)

˜
Now Γθ versus |s| can be plotted. In the plots that follow the magnitude of the response is
broken into harmonic components instead pf the total response which is a combination of
the harmonics that includes their phasing. a is only the amplitude of the order n response.
Equations (3.6) and (3.7) show that there is an order 1 as well as an order n response. The
order one component is dependent only on the order of the absorbers and gravity. Thus it
is a constant over all Γθ . This is shown in Figure 3.6.

Now the effect of varying γ will

be examined. Since the order one part is linearly based on γ it is uninteresting. However,
γ has an effect on the order n part. For this only the order n curve will be examined, as
shown in Figure 3.7. It can be seen that as γ increases the amplitude starts to increase
more dramatically and the jump occurs early. It has been shown that a similar effect can be
achieved by decreasing the tuning of the absorbers [1, 13]. This makes sense considering the
equivalent detuning of equation (3.18) that was talked about earlier.

It should be noted

that the effect of gravity is very small for γ less than about 0.025, this will be shown to be
true later in Section 3.4. This shows that the assumption used in previous models, that the
41

Figure 3.6: Analytical amplitudes of the harmonics of the non-dimensional absorber amplitude versus non-dimensional fluctuating torque, σ = 0.

effect of gravity was negligible, is true at least under certain parameter conditions. It can
be seen in Figures A.1, A.2, A.3 and A.4, located in Appendix 4.2, that even as the value of
(based on mass ratio) changes this assumption still holds true.
An important part of absorber design and analysis is the possibility that the absorber
will hit the cusp. This has been addressed by Denman [5] but needs to be considered in the
case of gravity. The cusp will tell us what the maximum value of the absorber path can be.
When gravity is included, the absorber response can be thought of as having two parts: an
42

Figure 3.7: Analytical amplitudes of the harmonics of the non-dimensional absorber amplitude versus non-dimensional fluctuating torque for varying γ, σ = 0.

order one part and an order-n part. The phasing between these two parts, as well as what
order-n actually is, plays a role in what the maximum absorber response can be. However,
an upper bound on the absolute maximum absorber displacement could be obtained by a
simple addition of the order one and order n magnitudes.
Now that the analytical model has been shown, it is ideal to compare the analytical
model to numerical simulations to verify the findings. This was done using the software
Mathematica. For the numerical simulations the full non-dimensional equations of motion
43

before scaling were used, Equations (2.35) and (2.36). This means that the full path equations
stated in Table 2.2, 2.3 and 2.4 were used, not their truncated Taylor series expansions. Once
the simulations were completed the responses needed to be analyzed. As was discussed in
section 3.2, the response of each absorber contained components at different orders. These
were primarily be at order one, due to gravity, and order n, due to the engine forcing. The
amplitudes of these components were found using an FFT.
Figure 3.8 shows amplitudes of the harmonics of absorber responses of the analytical
model versus numerical simulations for the case of a three cylinder engine. Thus n = 1.5.
The other parameters used were µa = 0.014921,

= 0.022276 and γ = 0.05. Since this

analysis is for circular path absorbers λ = 0. Using equation (2.47) and (2.48) it can be
found that κ0 = .2956. κ0 is dependent on the tuning of the absorbers (˜ ), the type of path
n
(λ) and the inertia ratio ( ). κ0 will not be solved for each case that is examined but the
information has been provided so that it can be determined in for each case tested. The
responses of both absorbers are the same in amplitude. As was talked about earlier, the
relative phases between the two can be different depending on the order. Looking at the
order one component of the response, there are two lines based on the numerical simulations.
One of these lines aligns with order one analytical model, while the other does not. The
results that do not align well (the upper set of stars in the figure) correspond to when the
absorbers have jumped. For the order n component of the response, the simulation matches
the analytical model for the lower branch (LB). The results for the upper branch (UB) are
not as accurate but this was expected. As was found by Nester [13] the jump occurred earlier
in numerical simulations than predicted by the analytical model.
The overall goal of CPVAs is to reduce the rotor oscillation that is seen in the crankshaft
44

Figure 3.8: Analytical amplitudes of the harmonics of the non-dimensional absorber amplitude versus non-dimensional fluctuating torque comparison of numerical simulations to
analytical model (AM), σ = 0.

and in other components. The non-dimensional rotor acceleration, νν , will give insight into
how CPVAs perform compared to an engine without absorbers. The non-dimensional rotor
acceleration can also be looked at by rewriting equation (2.54) as

1
˜
νν = (Γθ sin (nθ + τ ) +
N
45

N

(−˜ sin (θj ) + n2 pj ))
γ
j=1

(3.20)

where the assumed solution for pj is

pj = aj cos (nθ + αj ) +

γ
˜
sin(θ + ψj )
(−1 + n2 )

(3.21)

As long as the assumption that more than one absorber is used, then the gravity term will
cancel at the leading order out according to Appendix 4.2 and hence the rotor response
has no order-1 contribution. Additionally, trigonometric identities are used to simplify the
equation leading to

1
˜
νν = [(Γθ cos (τ ) −
N
+

1
N

N

˜
(n2 aj sin (αj ))) sin (nθ) + (Γθ sin (τ )
j=1

N

(3.22)

(n2 aj cos (αj ))) cos (nθ)]
j=1

Now it is important to note that Γθ is shown as a function of pj and other parameters in
equation (3.19). For this analysis τ = 0 will be used. However some analysis has been done
˜
on what affect τ has in Section 3.5. It is convenient to plot |νν | vs Γθ , as in Figures 3.9
and 3.10.

46

Figure 3.9: Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 0% detuning

It can be seen in Figure 3.9 and Figure 3.10 that when gravity is included in the model
with perfectly tuned absorbers the non-dimensional rotor acceleration increases as γ increases.
47

Figure 3.10: Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 0% detuning, in the desirable operating region.

An interesting thing happens when the absorbers are over-tuned and γ is varied. As γ is
˜
increased there is an improvement of the non-dimensional rotor acceleration in the desirable
operating range. However, the jump occurs at a lower Γθ and this needs to be accounted for
in design. This effect can be seen in Figure 3.11 and more easily in the magnified figure of
the desirable operating range, Figure 3.12.
48

Figure 3.11: Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 2% detuning.

49

Figure 3.12: Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
for various values of γ, with 2% detuning, in the desirable operating region.

1
As was mentioned before the best performance is at σ = 0, or σ = 2 . Therefore from
equation (3.18) there is a critical value of γ when the γ term balances with the intentional
detuning of the absorbers. From equation (3.18) and (2.48) it is found that

1
γcrit = 2

2σn2 (1 − n2 )2
.
3κ0
50

(3.23)

In Figure 3.13 it can be seen that γcrit is usually quite large even for small detuning. However,
referring back to Figure 3.5 it is seen that γ can grow rapidly over a small decrease in RPM.
Thus a small change in engine speed can change a well designed absorber from equivalently
over-tuned to equivalent under-tuned as equation (3.18) states. The cases where γcrit is
small would involve low forcing orders. An example of a practical application that would fall
into to this area would be a slightly detuned absorber for a three cylinder engine. However,
γcrit is also dependent on , as

increases so will γcrit , the inverse also being true.

Figure 3.13: γcrit versus forcing order, = .022276.
51

The analytical model can be validated using numerical simulations. Figure 3.9 and
3.11 show that a jump occurs. The portion were the jump has occurred has been omitted
during the numerical comparison. Additionally, referring back to equation (3.22), the gravity
portion that would be at order one has canceled out. The parameters are the same as was
used in Figure 3.8 but in this case we have tested with 0% and 2% detuning.

Figure 3.14: Non-dimensional rotor acceleration versus non-dimensional fluctuating torque
at order 1.5 , analytical model vs numerical simulations, γ = .066.

Figure 3.14 shows that the simulations follow the analytical model. There is an improve52

ment in the rotor acceleration cause by the restoring force of the absorbers that is verified by
the simulations. As was expected by the analytical model, over tuning the absorber increases
the fluctuating torque that the system can handle before the jump occurs. This, however,
does not come without drawbacks. The low end performance is sacrificed as was confirmed
by the simulations. The effect gravity has on absorbers in the general case is the same as
detuning. Thus when designing absorbers, the equivalent detuning, Equation (3.23), needs
to be considered and can be broken into the following cases.
• If σe = 1 → Perfectly Tuned
2
– Optimal correction in rotor acceleration
1
• If σe > 2 → over-tuned
– Less correction in rotor acceleration in some regions.
– Increased operating fluctuating torque range.
1
• If σe < 2 → under-tuned
– Less correction in rotor acceleration.
– Reduced operating fluctuating torque range.

3.3.2

Case 2: n = 2

For this case equation (3.13) will be used to evaluate the steady-state conditions. In this
˜
case it is very difficult to combine the amplitude and phase equations and solve for Γθ .
Instead mathematical software will be needed to solve the equations numerically and obtain
the desired information. For the results shown, Mathematica was used. The equations were
53

simplified for their assumed steady-state and then a root solver was used to find amplitudes,
phases and forcing. This case needs to be broken down even further because equation (3.13)
is dependent on ψj which depends on the number absorbers that are being used. The cases
of 2, 3 and 4 absorbers will be examined here.

3.3.2.1

2 Absorbers

When there are two absorber there are two sets of equations that define the steady state
response:

RE :

IM :

n
n
3
a1 + a2 cos(α2 − α1 ) − κ0 a3
1
4
4
4n
˜
Γ
˜
(1 + n2 )˜ 2
γ
3 κ0 a1 γ 2
− θ sin(α1 − τ ) −
sin(−α1 ) = 0
−
2 n(1 − n2 )2 2n
4n(1 − n2 )

na1 σ +

˜
Γ
n
µa
˜
a2 sin(α2 − α1 ) +
a1 − θ cos(α1 − τ )
4
2
2n
2 )˜ 2
(1 + n γ
+
cos(−α1 ) = 0
4n(1 − n2 )

(3.24)

and

RE :

IM :

n
n
3
na2 σ + a2 + a1 cos(α1 − α2 ) − κ0 a3
2
4
4
4n
˜
Γ
˜
3 κ0 a2 γ 2
(1 + n2 )˜ 2
γ
−
− θ sin(α2 − τ ) −
sin(2π − α2 ) = 0
2 )2 2n
2 n(1 − n
4n(1 − n2 )
˜
Γ
n
µa
˜
a1 sin(α1 − α2 ) +
a2 − θ cos(α2 − τ )
4
2
2n
2 )˜ 2
(1 + n γ
+
cos(2π − α2 ) = 0
4n(1 − n2 )
54

(3.25)

Now assuming that a1 = a2 and α1 = α2 the two sets of equations can be reduced to one:

RE :

IM :

γ
3
3 κ0 a˜ 2
n
a − κ0 a3 −
2
4n
2 n(1 − n2 )2
˜
Γθ
(1 + n2 )˜ 2
γ
sin(α1 − τ ) −
sin(−α) = 0
−
2n
4n(1 − n2 )
˜
Γ
(1 + n2 )˜ 2
γ
µa
˜
a − θ cos(α1 − τ ) +
cos(−α) = 0
2
2n
4n(1 − n2 )

naσ +

(3.26)

As before the order 1 will be a straight line that increases linearly as γ increases. Since the
order 1 response is linear it has been left out of the figures below.

The response in Figure

3.15 is similar to that of the general case seen in Figure 3.7. The difference between the two
can be seen clearly in Figure 3.16. There is a superharmonic resonance of the absorbers with
respect to gravity. This is evident because when Γθ = 0 a response of the absorber at order
two occurs. The magnitude of this response is less than that of the magnitude of the order
one response.

55

Figure 3.15: Order n harmonic of the non-dimensional absorber amplitude versus nondimensional fluctuating torque at order n = 2, for two absorbers, with σ = 0, for various
values of γ.

56

Figure 3.16: Order n harmonic of the non-dimensional absorber amplitude versus nondimensional fluctuating torque near zero, at order n = 2, for two absorbers, with σ = 0, for
various values of γ.

57

Figure 3.17: Order n harmonic of the non-dimensional absorber amplitude versus nondimensional fluctuating torque near zero, at order n = 2, for two absorbers, comparison
of simulations to analytical model.

Figure 3.17 shows numerical simulations of the full equations compared to the analytical
model. In this figure the upper branch is not shown, although the steady-state solutions that
correspond to the upper branch do exist. They are omitted for clarity. As was predicted
by the analytical model, when the non-dimensional fluctuating torque is zero the absorbers
show a response. In general the simulation matches the analytical model. The simulations
58

differ from the analytical results more as the non-dimensional gravity term increases. This
is because the analysis employs the assumption the γ parameter is small. Thus as γ gets
larger the analysis would be expected to break down.

Figure 3.18: Order n harmonic non-dimensional rotor acceleration versus non-dimensional
fluctuating torque at order n = 2, for two absorbers, with σ = 0, for various values of γ.

Similar to previous analysis the non-dimensional rotor acceleration versus the non-dimensional
fluctuating torque can be solved for. The same equations, (3.21) and (3.22), will be used.
When γ = 0, then a = 0 even when Γθ = 0. This means that the non-dimensional rotor
59

acceleration will not be equal to zero, even when Γθ = 0. This is a CPVA system that performs worse than a system without absorbers for values less than a certain Γθ . A potential
case when this could happen though is with start-stop technology. In such a case the engine
would be shut off thus eliminating the alternating torque due to the cylinders firing, causing
Γθ to tend towards zero. The engine would slow down leading to a increase in γ. This could
potentially cause the system to go through the range, were the system with CPVAs performs
worse than one without, before the engine comes to a stop.

60

Figure 3.19: Order n harmonic of the non-dimensional rotor acceleration versus nondimensional fluctuating torque near zero, at order n = 2, with σ = 0, for two absorbers
for various values of γ.

Another interesting occurrence takes place if the absorbers are over-tuned instead of
perfectly tuned. As was in the general case an improvement is seen in the non-dimensional
rotor acceleration when the absorbers are initially over-tuned and γ is increased. It should
be noted that the γcrit value when n = 2 is different than in the general case. This would
be solved from Equations (3.26) This is due to the superharmonic resonance that the n = 2
61

absorbers will experience. This makes γcrit very difficult to find mathematically, but it can
be found by varying γ in the equations and plotting.

Figure 3.20: Order n harmonic of the non-dimensional rotor acceleration versus nondimensional fluctuating torque near zero, at order n = 2, for two absorbers for various
values of γ and 2% detuning.
62

Figure 3.21: Order n harmonic of the non-dimensional rotor acceleration versus nondimensional fluctuating torque near zero, at order n = 2, for two absorbers, comparison
of simulations to analytical model.

Figure 3.21 demonstrates that the simulations follow the analytical model. The analytical model predicted that there is a region of low fluctuating torque where the rotor
acceleration is greater with absorbers in action than if they are locked down. Of more interest though is the effect of over tuning the absorbers. The results found in Figure 3.14,
which is for the general case, show that over tuning the absorbers does not improve the non63

dimensional rotor acceleration for all values of non-dimensional fluctuating torque before the
jump occurs. However, in Figure 3.21, when the absorbers are for a second order engine,
over tuning provides an improvement of the non-dimensional rotor acceleration across all the
non-dimensional fluctuating torque before the jump. Additionally the simulation supports
the analytical prediction that there is a range of Γθ in which the non-dimensional rotor
acceleration approaches zero. If the idle condition of a four cylinder engine could be set with
a two absorber system designed to correspond to this point, then the performance could be
improved dramatically.
Since the equations were not solvable analytically it does make designing for this point
more difficult, but it can still be done. An engine would need to be tested to determine
Γθ vs. Ω. Once this is done a relationship between the two is known. Now an absorber
design needs to be assumed. From this a graph similar to Figure 3.5 can be developed. A
plot similar to Figure 3.21 can also be made. When this graph is made it should be done
for a variety of different values of detuning at a constant value of γ that corresponds to the
desired engine idle speed. Now the value of Γθ where νν is minimized can be determined
from the graph. This can then be compared to the Γθ vs. Ω chart. For some engine speeds
Γθ may be much larger than the Γθ corresponding to the minimum rotor acceleration. In
this case a higher Ω would be chosen or a larger inertia ratio and the process repeated until
the optimal conditions were found.

3.3.2.2

4 Absorbers

In the case of four absorbers there are four sets of equations developed from equation (3.12).
For equally spaced absorbers, ψ1 = 0, ψ2 = π , ψ3 = π, ψ4 = 3π . Normally when looking
2
2
at the steady-state it is assumed that aj = a and αj = α. If this is tried it can be found that
64

the four sets of equations do not simplify to one set. Instead, two sets of equations can be
simplified to one set of equation, while the other two sets are equal to another set of equations.
This means that the steady-state of four absorbers will behave differently than other cases.
In the case of four absorbers absorber one will align with absorber three, while absorber two
will align with absorber four. When it is assumed that a3 = a1 , a4 = a2 , α3 = α1 and
α4 = α2 at steady-state, the four sets of steady-state equations can be simplified to two sets
of equations as,

RE :

IM :

n
3
n
na1 σ + a1 + a2 cos(α2 − α1 ) − κ0 a3
1
4
4
4n
˜
Γ
˜
3 κ0 a1 γ 2
(1 + n2 )˜ 2
γ
−
sin(α1 ) = 0
− θ sin(α1 − τ ) +
2 n(1 − n2 )2 2n
4n(1 − n2 )
n
µa
˜
a2 sin(α2 − α1 ) +
a −
4
2 1
(1 + n2 )˜ 2
γ
+
cos(α1 ) = 0
4n(1 − n2 )

˜
Γθ
cos(α1 − τ )
2n

(3.27)

and
RE :

IM :

n
3
n
na2 σ + a2 + a1 cos(α1 − α2 ) − κ0 a3
2
4
4
4n
˜
Γ
˜
3 κ0 a2 γ 2
(1 + n2 )˜ 2
γ
−
sin(α2 ) = 0
− θ sin(α2 − τ ) −
2 )2 2n
2 n(1 − n
4n(1 − n2 )
˜
Γ
n
µa
˜
a1 sin(α1 − α2 ) +
a2 − θ cos(α2 − τ )
4
2
2n
2 )˜ 2
(1 + n γ
cos(α2 ) = 0
−
4n(1 − n2 )

(3.28)

These equations can be solved numerically as has been done previously to get the nondimensional absorber amplitude versus non-dimensional fluctuating torque, and non-dimensional
rotor acceleration versus non-dimensional fluctuating torque. However, as is the case with
three absorbers, when these equations were solved, unreasonable results were sometimes
found. It is believed that this is because of the numerical solver that was used. The unrea65

sonable results were never found in numerical simulation and have been omitted. As was
the case with two absorbers, when there is no non-dimensional fluctuating torque there is a
response at order two by the absorbers. It can be seen in Figure 3.22 that the two sets of
absorbers can have different amplitudes. This is confirmed by numerical simulations.

Figure 3.22: Order n harmonic of the non-dimensional absorber absorber amplitude versus
non-dimensional fluctuating torque near zero, at order n = 2, with σ = 0 for four absorbers,
with γ = .05.

As was the case with two absorbers, the same equations (3.28) can be plugged into (3.22)
66

and allow for the non-dimensional rotor acceleration to be examined.

Figure 3.23: Order n harmonic of the non-dimensional rotor acceleration versus nondimensional fluctuating torque near zero, at order n = 2, with σ = 0 for four absorbers,
for various values of γ.

From Figure 3.22 and Figure 3.23 it can be seen that the numerical simulations align with
the analytical model. The multiple amplitude responses are confirmed by the simulations.
It can be seen in Figure 3.23 that as γ gets large the simulations start to vary from the
analytical model, but this is likely because the analysis was an assumption that γ is small.
67

3.3.2.3

3 Absorbers

As was done previously, the three absorber set up is used to develop the sets of equations
from equation (3.13). In this case, as was the case with four absorbers, it cannot be assumed
that all the amplitudes and phases are the same for the steady-state response. This leads to
three sets of steady-state equations that are coupled but can be solved numerically:

RE :

IM :

n
n
n
3
na1 σ + a1 + a2 cos(α2 − α1 ) + a3 cos(α3 − α1 ) − κ0 a3
1
6
6
6
4n
2
˜
Γ
˜
3 κ0 a1 γ
(1 + n2 )˜ 2
γ
−
− θ sin(α1 − τ ) +
sin(α1 ) = 0
2 )2 2n
2 n(1 − n
4n(1 − n2 )
˜
Γ
n
n
µa
˜
a2 sin(α2 − α1 ) + a3 sin(α3 − α1 ) +
a1 − θ cos(α1 − τ )
6
6
2
2n
2 )˜ 2
(1 + n γ
cos(α1 ) = 0
+
4n(1 − n2 )

(3.29)

n
n
3
n
a2 + a1 cos(α1 − α2 ) + a3 cos(α3 − α2 ) − κ0 a3
2
6
6
6
4n
˜
Γ
˜
4π
3 κ0 a2 γ 2
(1 + n2 )˜ 2
γ
− θ sin(α2 − τ ) +
sin(α2 −
−
)=0
2 n(1 − n2 )2 2n
3
4n(1 − n2 )
˜
Γ
n
µa
˜
n
a1 sin(α1 − α2 ) + a3 sin(α3 − α2 ) +
a2 − θ cos(α2 − τ )
6
6
2
2n
2 )˜ 2
4π
(1 + n γ
+
cos(α2 −
)=0
3
4n(1 − n2 )

(3.30)

and

RE :

IM :

na2 σ +

68

and

RE :

IM :

n
n
n
3
a3 + a1 cos(α1 − α3 ) + a2 cos(α2 − α3 ) − κ0 a3
3
6
6
6
4n
˜
Γ
˜
3 κ0 a3 γ 2
(1 + n2 )˜ 2
γ
8π
−
)=0
− θ sin(α2 − τ ) −
sin(α3 −
2 n(1 − n2 )2 2n
3
4n(1 − n2 )

na3 σ +

n
n
µa
˜
a1 sin(α1 − α3 ) + a2 sin(α2 − α3 ) +
a −
6
6
2 3
(1 + n2 )˜ 2
γ
8π
)=0
+
cos(α3 −
2)
3
4n(1 − n

˜
Γθ
cos(α3 − τ )
2n

(3.31)

Now the equations can be solved numerically and allow for plots of the non-dimensional
absorber amplitude versus the non-dimensional alternation torque. It can be seen that
even when there is only forcing due to gravity that there is a response at order two by the
absorbers. Also at this point all of the absorbers have the same amplitude. As the nondimensional alternating torque increases, three different response amplitudes can be seen by
the absorbers.

69

Figure 3.24: Order n harmonic of the non-dimensional rotor acceleration versus nondimensional fluctuating torque near zero, at order n = 2, with σ = 0, for three absorbers,
with γ = .066.

70

Figure 3.25: Order n harmonic of the non-dimensional rotor acceleration versus nondimensional fluctuating torque near zero, at order n = 2, with σ = 0, for three absorbers,
for various values of γ.

From Figure 3.24 and Figure 3.25 it can be seen that the numerical simulations align
with the analytical model but not as well as in the other special cases.
71

3.3.2.4

Higher Numbers of Absorbers

Some general statements about cases that contain high numbers of absorbers can be made.
If the number of absorbers is even then there exist a steady-state for which the number of
distinct amplitudes at order two is half the number of absorbers. This is because of the
arrangement of absorbers. When there is an even number of absorbers each absorber will
be π radians from another absorber. The term in equation (3.13) that imposes multiple
amplitudes at order two can be reconciled if ψ = π. However, if there are an odd number of
absorbers this π phasing between two absorbers does not happen and thus the equation can
not be reduced. This leads to each absorber having its own independent amplitude at order
two.

3.3.3

Case 3: n = 3

An interesting occurrence is when the absorbers are tuned to order three. Similar to when
κ0 γ 3
˜
=χ
the absorbers are tuned to order two, an extra term with a magnitude of
4n(1−n2 )3
appears in the slow flow equations. However, this extra term that is in equation (3.15) is
typically very small. Breaking down the term it is seen that κ0 is dependent on , n, and
˜
λ. Assuming that n ≈ 3, and that if to maximize κ0 then λ = 0, the extra variable can be
˜
solved to be a function of only two variables, γ and . Now the magnitude of this term will
be examined by varying γ and

in Figure 3.26.

This term only becomes relatively large when γ is large and

is small. However, even in

such instances the value of the magnitude is of the order 10−3 . This is orders of magnitude
smaller than the other terms and thus the term can be dropped, which means that the cases
when n = 3 falls into the general case. This can be shown graphically as well. Since it is
72

Figure 3.26: Evaluation of the magnitude of the special term χ in the n = 3 slow flow
equations.

known where the extra term is largest that is where the equations will be tested. In Figure
3.27 only select points of the n = 3 analytical model are being compared with the line. This
allows for the any differences in the lines to be seen.
It can be seen in Figure 3.27 that the numerical solution of the n = 3 analytical model
coincides with that of the general case. It should be noted that as was the case when n = 2,
there is a potential for different amplitudes based on the number of absorbers in the system.
However, since it has been shown that this term is small the potential for different amplitudes
73

Figure 3.27: Comparison of order n harmonic of the non-dimensional absorber amplitude
versus non-dimensional fluctuating torque between n = 3 analytical model and the general
case.
is very small and is negligible. Additionally when this term is dropped the non-dimensional
rotor acceleration also becomes the same as in the general case.

3.4

Ignoring Gravity

In order to determine at what values of γ the effects of gravity can be ignored, the saddle
point at which the nonlinear jump occurs is examined. The difference between the value
74

of the non-dimensional fluctuating torque at which the saddle point occurs without gravity
and with gravity was found. This was examined for both the non-dimensional absorber
amplitude and the non-dimensional fluctuating torque amplitude. Additionally this will be
examined for the case of n = 1.5 and n = 2, with µa = .014921.
Table 3.1: Difference values given as % between the level of alternating torque at which
the saddle-node bifurcation occurs, with respect to non-dimensional absorber amplitude, for
various values of γ, for n=1.5, with σ = 0%.
n

1.5

Inertia Ratio(b)

5%

10%

15%

γ=.01

0.49%

0.29% 0.16% 0.15%

γ=.02

1.98%

1.04% 0.66%

0.5%

γ=.03

4.42%

2.26%

1.12%

γ=.04

7.82%

4.01% 2.69% 2.01%

γ=.05

12.12% 6.27% 4.17% 3.14%

1.5%

20%

Table 3.2: Difference values given as % between the level of alternating torque at which the
saddle-node bifurcation occurs, with respect to non-dimensional fluctuating torque amplitude, for various values of γ, for n=1.5, with σ = 0%.
n

1.5

Inertia Ratio(b)

5%

10%

15%

γ=.01

0.48%

0.25% 0.18% 0.12%

γ=.02

1.90%

1.0%

γ=.03

4.28%

2.24% 1.52% 1.12%

γ=.04

7.53%

3.98% 2.68% 2.01%

γ=.05

11.68% 6.18% 4.18% 3.14%

0.68%

20%
0.5%

For the case of a three cylinder engine Tables 3.1 and 3.3 show that the impact of gravity
decreases as the inertia ratio increases. It is desired that the difference be less than 2.5%.
75

Thus for all the inertia values examined γ must be less than .02. In typical applications Ro
is between .05 m and .1 m. Thus when an engine is running above 950 RPM and 660 RPM
respectively, gravity can be ignored.
Table 3.3: Difference values given as % between the level of alternating torque at which
the saddle-node bifurcation occurs, with respect to non-dimensional absorber amplitude, for
various values of γ, for n=2, with σ = 0%.
n

2

Inertia Ratio(b)

5%

10%

15%

20%

γ=.01

1.04%

0.4%

0.24% 0.15%

γ=.015

2.33%

0.94%

0.53% 0.35%

γ=.02

4.15%

1.59%

0.93% 0.63%

γ=.03

9.33%

3.61%

2.11% 1.42%

γ=.04

16.56%

6.42%

3.73% 2.54%

γ=.05

25.86% 10.07% 5.83% 3.96%

Table 3.4: Difference values given as % between the level of alternating torque at which the
saddle-node bifurcation occurs, with respect to non-dimensional fluctuating torque amplitude, for various values of γ, for n=2, with σ = 0%.
n

2

Inertia Ratio(b)

5%

10%

15%

20%

γ=.01

1.04%

0.43% 0.24% 0.15%

γ=.015

2.33%

0.94% 0.53% 0.35%

γ=.02

4.15%

1.62% 0.93% 0.63%

γ=.03

9.33%

3.64% 2.11% 1.42%

γ=.04

16.56% 6.46% 3.73% 2.54%

γ=.05

25.87% 10.1% 5.83% 3.96%

The change in the saddle point was examined for the case of a four cylinder engine in
Tables 3.2 and 3.4. It is desired that the difference be less than 2.5%. Thus for all the inertia
76

values examined γ must be less than .015. In typical applications Ro is between .05 m and
.1 m. Thus when an engine is running above 1100 RPM and 775 RPM respectively, gravity
can be ignored.

3.5

The Effects of τ

The effect of phasing the forcing due to the engine relative to the origin (τ ), which is fixed by
gravity, was briefly examined using perturbation results. Different values of τ were explored.
Figure 3.28 shows that for any number of absorbers n = 1, 2, τ does not have an impact
on the steady state response of the non-dimensional rotor acceleration. Figure 3.29 shows
that when n = 2, τ will have a significant impact on the steady state response of the nondimensional rotor acceleration. Therefore, when considering the effects of gravity, it is crucial
to know the relative phases of gravity and the order n applied torque. These results will be
seen for any number of absorbers but it is shown only for the simplest case of 2 absorbers.

77

Figure 3.28: Diagram of n = 1, 2 pendulum type circular path absorbers order n rotor
acceleration as a function of Γθ for various τ , τ has no effect, the locked response is for
reference only.

78

Figure 3.29: Diagram of two n = 2 pendulum type circular path absorbers order n rotor
acceleration as a function of Γθ for various τ , response are distinct for each τ , the locked
response is for reference only.

79

Chapter 4
Conclusions and Recommendation for
Future Work

4.1

Summary of Results and Conclusions

The purpose of this investigation was to systematically investigate the effects gravity has
on the dynamics and effectiveness of CPVAs. This was done using a mathematical model,
which was studied by perturbation analysis and numerical simulations. Gravity provides
both direct and parametric forcing of CPVAs at engine order 1 (one cycle per revolution),
which can interact with engine order n vibrations under certain circumstances, leading to
rich dynamic behavior.
The thesis provides a brief background of CPVAs, including previous relevant work and
various applications. The analysis of gravity begins by developing model equations of motion for a rotor spinning about a horizontal axis, subjected to torsional excitation of order
n, fitted with N identical general path CPVAs, and the entire system is acted on by gravity.
80

The model equations were rescaled and transformed into a form suitable for perturbation
analysis. This procedure included small parameter assumptions that are consistent with values encountered in applications. The absorber equations of motion were uncoupled from the
rotor equation to leading order in the small parameter, which allows one to find approximate
response solutions for the absorbers, from which the rotor dynamics can then be recovered.
The method of multiple scales was used to obtain the slow flow equations governing the amplitudes and phases of the absorbers. A numerical root solver was used to solve the for zeros
of the slow flow equations, yielding steady-state conditions that were used for parameter
studies. Direct numerical simulations of the full EOM were used to verify the theoretical
results. Stability of these responses was not considered, except by observations from these
simulations. When the fluctuating torque is zero the absorbers experience equal magnitudes
of an order two super harmonic response from gravity, however the absorbers phases are
not equal. For the cases when N > 2 and n = 2 the root solver frequently had difficulties
with convergence; the initial conditions used in the root solver were extremely important to
successfully finding a solution.

It was determined that it is important to include the effects of gravity when the ratio of
gravity to centrifugal acceleration, γ = g/(R0 Ω2 ), is sufficiently large (assuming reasonable
absorber/rotor inertia ratios in the range 5%-20%). For example, it is found that gravity
must be accounted for when γ > 0.02 for n = 1.5, and for γ > 0.015 for n = 2. The
corresponding range of engine speeds depends, of course, on the distance from rotor center
to absorber center of mass, R0 . For typical automotive engines, R0 is in the range of 0.05-0.1
meters, which implies that one should consider gravitational effects for rotor speeds below
about 70-99 radians/second, that is, about 660-950 rpm for n = 1.5 and below about 80-115
81

radians/second, that is, about 775-1100 rpm for n = 2. These are precisely in the range of
engine idle speeds.
In addition to these general observations, it was found that nonlinear dynamic interactions can occur under certain conditions. These arise from interplay between gravity (of
order 1) and fluctuating torques (of order n) when they are internally resonant. For most
values of n, specifically, for n = 1, 2, 3, the system response is essentially linear, that is, the
responses of order 1 and n can be determined separately and combined to obtain the overall
response. In these cases it was shown that the effects of gravity can be accounted for by
using an equivalent detuning parameter that is smaller than the detuning between order n
and the absorber order n. This means that gravity has the effect of lowering the torque
˜
threshold at which the dangerous jump instability occurs for circular path absorbers. For
n = 3 it was shown that, even though there are subharmonic interactions, these effects are
very small and can generally be ignored. The special cases of n = 1, 2 proved to be the most
interesting. In this work we focused on the n = 2 case, since it has particular relevance to
four cylinder engines. The equally rich n = 1 case is left for future study.
For n = 2 it was determined that the number of absorbers N has an effect on the types
of responses that can occur. The simplest response occurs when there are two absorbers,
N = 2, in which case the steady-state responses of both absorbers have the same amplitude
and phase at order n. The order one response of the absorbers, from gravity, has the same
amplitude, but with a phase difference of π, due to the angular spacing of the absorbers.
It is also shown that the n = 2 absorbers have a superharmonic response to the order one
gravitational forcing. The superharmonic response leads to an interesting finding regarding
the rotor acceleration, which is a measure of rotor torsional vibration. It was found that if
82

the absorbers are over-tuned (as is the case in practice) and the gravity parameter exceeds a
certain level, for zero relative phase between the applied torque and gravity there is parameter region in which the nondimensional rotor acceleration shows a minimum value as the
fluctuating torque is varied, and this minimum is lower than the value experienced without
gravity. Thus, it may be possible to design absorbers so that gravity actually reduces second
order torsional vibration levels under engine idle conditions. Cases for N = 3, 4 absorbers
were explored throughly, while general comments were made for N > 4. It was also found
that when N > 2, depending on the value of the fluctuating torque, different absorber amplitudes and phases are possible. When N > 2, the amplitude of the order one response was
the same for all absorbers, but the phases are not equal. More interesting is the observation that in this case synchronous order two responses were not found to occur. In general,
even numbers of absorbers will display N unique responses among sets of absorbers, that
2
is, there are N groups of mutually synchronous absorbers. For N odd it was observed that
2
the absorbers display N unique responses.

4.2

Recommendations for Future Work

While a number of interesting results were obtained in this study, several important topics
remain to be explored. A list of possible topics for future work along these lines includes the
following:

• Compare the analytical model and numerical simulations against physical experiments;
these tests are being planned for the MSU test facility, with a horizontal axis of rotation.

• Symmetry techniques may be very useful to sort out the cases where the absorber re83

sponses are synchronous and non-synchronous, and the details of the non-synchronous
responses.
• Investigate the effects of gravity on absorbers with different absorber paths, including
cycloids and tautochronic epicycloids; the basic resonance cases will be the same, but
the quantitative effects may differ.
• The present study focused on the case where the applied order n torque had a zero
phase relative to gravity, that is τ = 0; the effects of this phase need to be studied in
more detail.
• Complete a stability analysis of the slow flow equations that include gravity, in order to
carry out a parameter study to determine specific parameter conditions for the jump,
non synchronous, and subharmonic instabilities.
• Explore the effects of absorber damping on the results; this would be motivated by
specific case studies of practical interest.
• Further explore the torque condition that minimizes the rotor acceleration, and see if
it has a practical application to engine idle.
• Numerical solution of the steady-state responses for order 2 absorbers was extremely
sensitive when there were more than two absorbers; the source of this sensitivity, and
its resolution, would be beneficial to future investigations.
• This work assumes that the absorbers are identical and equally spaced around the
rotor, but in some cases absorbers of different orders may be employed, and the effects
of gravity on these systems may prove interesting as well.

84

Appendices

85

Appendix A

Additional Figures

86

The following figures examine the effect that arise from varying the value of

from .015

to .06 has on the absorber response. Figures A.1 through Figure A.4 are from perfectly
tuned absorbers σ = 0.

Figure A.1: Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .015.

87

Figure A.2: Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .03.

88

Figure A.3: Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .045.

89

Figure A.4: Non-dimensional absorber amplitude versus non-dimensional fluctuating torque
at order n for = .06.

90

Appendix B

Elimination of Gravity Term

91

For the analysis there are always two or more identical absorbers equally spaced around
the rotor, a typical arrangement for balancing. In this case the gravitation effects that
appear in the scaled rotor equation can be simplified, essentially by symmetry properties of
the system. Consider the term that shows up twice in equation (2.55),
N

N

[sin(θj )] =
j=1
j=1

1
2π
sin(θ + (j) ) = eiθ
N
2i

N

N
j2π
j2π
1 −iθ
i N
−i N
− e
,
e
e
2i
j=1
j=1

(B.1)

where ψj in θj has been shifted without affecting the results. By changing the exponential
form back to its trigonometric form the following is found,
N

N
N
j2π
j2π
j2π
i N
e
cos(
i sin(
=
)+
).
N
N
j=1
j=1
j=1

(B.2)

It can be shown that
(N +1)X
X
)
cos( N2 ) sin(
2
cos(jX) =
sin( X )
j=0
2

(B.3)

(N +1)X
X
sin( N2 ) sin(
)
2
sin(jX) =
.
sin( X )
j=0
2

(B.4)

N

and
N

X = 2π can be inserted into equations (B.3) [20] and (B.4) [21] which will simplify equation
N
(B.2) and consequently equation (B.1). Note that the summations in equations (B.3) and
(B.4) contains the j = 0 term. When this is accounted for it is found that
N
[(sin(θj ))] = 0,
j=1

92

(B.5)

which simplifies the equations of motion.

93

Appendix C

Higher Order Harmonics Induced by
Absorber Motions

94

When the absorbers move along their paths in an order n response, they produce a torque
on the rotor that contains harmonics of order n, as well as higher harmonics. These can be
modeled as terms of the form,
N
zm sin(m(θ + j
j=1

2π
))
N

(C.6)

where m is a multiple of n. Here we show that for some harmonics these terms add, rather
than cancel, as was the case for gravity, for which n = 1. In this case the number of absorbers
also becomes an important part of the analysis. In application if there is no forcing from
the engine (Γθ = 0) harmonics will appear from the coupling between the absorbers, which
are driven by gravity, and the rotor. Typically order 2 is of comparable size to order 1,
while other higher orders are significantly smaller. It is important to realize is that since the
harmonics stem from gravity they remain to have a phase dependence in them.

Case 1: N=m
It is known that
(N +1)X
X
sin( N2 ) sin(
)
2
sin(jX) =
sin( X )
j=0
2
N

(C.7)

where in this case X = m2π and, as before, accounting for the j = 0 term, it is found that,
N
N
zm sin(m(θ + j
j=1
95

2π
)) = 0
N

(C.8)

Case 2: N=m
Here the summantion is given by,
N
j=1

mj2π
zm sin(mθ + (
)=
N

N
zm sin(N θ + j2π) = N zm sin(N θ)

(C.9)

j=1

Notice that in this case the term does not cancel out, they actually add up. In most cases
zm is a very small value and even if multiplied by the number of absorbers the value is still
small.

Implications
It has been seen that in applications of absorbers with gravity, the absorbers produce a
significant amount of 2nd order torque that appears in the rotor acceleration. Thus in the
case of 2 absorbers there is a noticeable amount of 2nd order torque of non-dimensional rotor
acceleration, even when there is no alternating torque from the engine. For the case of 3 and
4 absorbers zm is typically very small and thus even when summed can be ignored.

96

Appendix D

Pendulum Type Absorber

97

The future work that can be done on this topic contains experimental testing, in which
case a pendulum style absorber would likely be used. A pendulum style absorber would have
a radius of gyration that would alter the analysis and are circular path. As was done by
Nester [13] and Vidmar [18] the equations of motion can be normalized so that they can be
compared to the results derived in this work.

Equations of Motion
Largrange’s equations will be used to find the EOM as was done in this work. The kinetic
and potential Energy can be found from the model. Figure 4.2 shows the model that is used
to develop the equations of motion.

1
1
1
˙
˙ ˙
T = JR θ2 + Jpj (θ + φ)2 +
2
2
2

N

˙
˙ ˙
˙ ˙ ˙
mpj R2 θ2 + 2 (θ + φ)2 + 2Rθ (θ + φ) cos φ (D.10)

j=1

Where Jpj is the mass moment of inertia of the pendulum absorber and can be stated as
Jpj = mj ρ2 . Where ρ is the radius of gyration. The potential energy can be found to be.
N
g

[mpj (R cos θj + (cos φ cos θj + sin φ sin θj ))].
j=1

98

(D.11)

Figure D.5: Diagram of pendulum type circular path absorbers.

99

Using Lagrange’s Equations the EOM can be found. The rotor equation is given by
N

N
2 + 2 + ρ2 + 2R cos φ)]]θ + [
¨
¨
[JR +
[mj (R
[mj (ρ2 + 2 + R cos φ)]]φ
j=0
j=0
N
N
˙ + φ)φ sin φ] + g
˙ ˙
−
[mj R (2θ
[mj [R sin θj + (cos φ sin θj − sin φ cos θj )]]
j=0
j=0

(D.12)

˙
= −co θ + T0 + Tf sin nθ
and the absorber equations are to be found,
¨
˙
¨
[mj ( 2 + ρ2 + R cos φ)]θ + [mj ( 2 + ρ2 )]φ + mj R sin φ θ2

(D.13)

˙
+ mj g (sin φ cos θj − cos φ sin θj ) = −caj φ
When the absorber is analyzed it can be found that the natural frequency of the absorber,
when the rotor spins at a constant rate, is porportional to the mean angular velocity. This
allows for the natural frequency to be expressed in an order which is given by,

R
.
2 + ρ2

n=
˜

(D.14)

Non-dimensionalizing and Comparison

The same process is used as was for the general path that was done previously.

ν=

˙
θ
Ω

100

(D.15)

Additionally the following is true.
˙
φ = νΩ(φ)

(D.16)

¨
φ = ν 2 Ω2 (φ) + Ω2 νν (φ)

(D.17)

¨
θ = Ω2 νν

(D.18)

It is desired to change the angle φ into a non-dimensional arc length s. It is necessary in
order to compare the pendulum style circular path absorbers to the analysis that has already
been done that a new parameter β be introduced and solved for later.

s=

Lφ
β

(D.19)

When these equations are inserted into the equations of motion and expanding trig functions
to the third power the following is found,
N m
s2 β 2
pj 2
2 + ρ2 + 2R (1 − j )]νν
[
[1 +
(R +
JR
2 2
j=1
N m β
m Rβ 3 s2
j
pj
2 + ρ2 ) − pj
+
[(
(R +
)(νν sj + ν 2 sj )]
2J
JR
2 R
j=1
N m Rβ 3 s2
N m Rβ 2
s3 β 2
s3 β 2
pj
j
pj
2 (s − j )] −
2 s (2s − j )]
−
[(
)(νsj ) j
)ν j j
[(
2J
JR
6 2
3 3
R
j=1
j=1
N gm R gm
N gm s β gm s3 β 3
gmpj s2 β 2
pj j
pj
pj
j
pj j
+
[(
+
−
) sin(θj )] −
[(
−
) cos(θj )]
2
2
2
2
2 J Ω2
JR Ω
JR Ω
2 JR Ω
JR Ω
6 R
j=1
j=1
Tf
c
T0
+ 0 ν=
+
sin(nθ)
JR Ω
JR Ω2 JR Ω2
(D.20)
101

s3 β 2
β 2 s2 R + 2 + ρ 2
R 2
j
j
2+
2s + R
(s −
)ν
(−
+
sj ν ν + ν j
)νν
2 + ρ2 j
2
2 + ρ2 )
2
R
6
β(
2
gβ 2 s3
gβs2
g sj
g 2
j
j
+(
−
) cos(θj ) − (
−
) sin(θj )
2 + ρ2 )Ω2 6 ( 2 + ρ2 )Ω2
2 + ρ2 )Ω2 2( 2 + ρ2 )Ω2
(
β(
+

ca
νs = 0
mp Ω( 2 + ρ2 ) j
(D.21)

These equations can be compared to equations (2.39) and (2.40). When this is done it can
be seen that β can be solved for by expanding the g (sj ) and comparing the νν terms.
˜
2 (n2 + n4 )s2
β 2 s2 R + 2 + ρ 2
R 2
j
j
gi (s) = (1 −
˜
)=
)
(−
+
2
R
β( 2 + ρ2 )
2 2

(D.22)

It can easily be seen that
β = (1 + n2 )
˜

(D.23)

Similar comparisons can be made fore other terms as well. The following parameter definitions were derived by Vidmar [18],

bo =

mo β
(R + 2 + ρ2 )
JR

µa =

ca
mp Ω( 2 + ρ2 )

µo =

co
JR Ω

Γ0 =

T0
JR Ω2

(D.24)

T
Γθ =
JR Ω2
102

Carrying out a similar analysis for the gravitational terms yields the following relationships,
gβs2
g 2
j
−γ
sin(θj ) = −(
−
) sin(θj )
dsj
β( 2 + ρ2 )Ω2 2( 2 + ρ2 )Ω2
gβs2
(1 + n2 )2 s2
˜
g 2
j
−
)
−γ(1 −
) = −(
2 + ρ2 )Ω2 2( 2 + ρ2 )Ω2
2
β(
g 2
γ=
β( 2 + ρ2 )Ω2
dxpi (s)

(D.25)

These results provide the relationships required to compare results for compound pendulum
absorbers with single point suspensions with those derived for point mass absorbers.

103

BIBLIOGRAPHY

104

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106