EXPERIMENTAL MEASUREMENT OF THE RESPONSE OF CENTRIFUGAL
PENDULUM VIBRATION ABSORBERS
By
Abhisek Jain

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
Mechanical Engineering - Master of Science
2013

ABSTRACT
EXPERIMENTAL MEASUREMENT OF THE RESPONSE OF
CENTRIFUGAL PENDULUM VIBRATION ABSORBERS
By
Abhisek Jain
Centrifugal pendulum vibration absorbers (CPVAs) are devices designed to reduce torsional oscillations in order-excited rotational systems. The focus of this research is to develop
and test algorithms that allow one to determine the absorber motion (amplitude and phase)
using the readouts from an accelerometer attached to the absorber and an encoder sensing
the rotor motion. This requires a detailed analysis of the pendulum kinematics, which depends on the path followed by the absorber mass and the rotation of the absorber relative
to the rotor. The absorber kinematics are governed by a di↵erential equation that relates
the absolute absorber acceleration (measured signal from an accelerometer) and the rotor
speed and acceleration (measured signal from an encoder), to the motion of the absorber
relative to the rotor. This di↵erential equation is solved approximately using a harmonicbalance method, based on assumptions regarding the signiﬁcant harmonics in the rotor and
absorber dynamics. The resulting approximations are tested using numerical simulations
of the equations of motion, as well as in experiments in which the approximated absorber
motions are compared to direct encoder measurements of the response. Both simulations
and experiments show that absorber motions are successfully estimated from measurements
of the absolute acceleration of the absorber and rotor motions, although some deviation is
apparent when absorber motions have large amplitudes. This approach will allow test engineers to validate models that predict and assess absorber performance in rotational systems
(e.g., automobile engines), which will improve conﬁdence in the absorber design process.

To my Mom, Dad and Brother.

iii

ACKNOWLEDGMENTS

In my two years of graduate school at Michigan State University, I believe that I have doubled
my understanding of dynamics and this would not have been possible without the guidance
and support of my advisors, Dr. Steve Shaw and Dr. Brian Feeny. Professor Shaw is an
expert in dynamical systems and an amazing advisor. I thoroughly loved working on my
project under his guidance and the learning process have been extremely gratifying. Professor
Feeny has also been a great mentor for me throughout my research and his expertise has
been equally invaluable. I would also like to thank Professor Ranjan Mukherjee for serving
on my committee.
Special thanks goes to Brendan Vidmar who was a friend and a teacher for me for almost
a year in my graduate studies. He has helped me in understanding the CPVA system and
in setting up experiments for my project. The amount of knowledge I have acquired about
dynamics comes not only from my research but also from many discussions with Brendan.
The undergraduate students who worked in CPVA group also helped me in implementing
experimental setup and in conducting experiments. I thank J. T. Whitman, Bara Aldasouqi
and Cody Little for their help. I would also like to thank graduate students Scott Strachan,
Pavel Polunin, Venkat Ramakrishnan, Xing Xing and Mustafa Ali Acar and undergraduate
student Ming Mu for their help and support and also for numerous discussions in the subject
of dynamics.
I thank Bruce Geist of Chrysler LLC for his expert comments and discussion during our
weekly meetings. He has also helped and guided me in numerous aspects of my project. I also
thank Chrysler LLC for machining the experimental absorbers on which all my experiments
have been conducted. I thank Je↵ Chottiner and Vince Solferino from Ford Motor Company
iv

and Steve Walters and Mark Schrewe from Honda R & D Americas for providing practical
experience in CPVA absorbers. I must thank Analog Devices for providing me free samples
of their accelerometers used for the experiments in my research. I also thank the ECE shop
of MSU for designing the PCB board that was used to mount the accelerometers.
My ﬁnal thanks goes out to the National Science Foundation (Grant Number CMMI1100260), which provided funding for this work as well as to Honda R & D Americas for
providing ﬁnancial support for my graduate studies.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Basic Operation of a CPVA . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 2 Methods and Results . . . . . . . . . . . . . . . . . . . .
2.1 Kinematic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Absorber Arrangement . . . . . . . . . . . . . . . . . . . .
2.1.1.1 Hinged Circular Path Absorbers . . . . . . . . .
2.1.1.2 Biﬁlar Absorbers . . . . . . . . . . . . . . . . . .
2.1.2 Kinematic Di↵erential Equation for the Absorber Response
2.1.3 Approximate Kinematic Solutions . . . . . . . . . . . . . .
2.2 Experimental Setup and Methods . . . . . . . . . . . . . . . . . .
2.2.1 Experimental Rig . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Accelerometer Arrangement . . . . . . . . . . . . . . . . .
2.2.3 System Parameter Values . . . . . . . . . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . .

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Chapter 3 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . .

40

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

BIBLIOGRAPHY

55

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

vi

LIST OF TABLES

Table 1.1

Acclerometer vs encoder . . . . . . . . . . . . . . . . . . . . . .

4

Table 2.1

Parameter Values for Experiments . . . . . . . . . . . . . . . .

32

Table 2.2

Parameter Values for Dynamic Simulations [1] . . . . . . . .

33

vii

LIST OF FIGURES

Figure 1.1

Cycloidal path CPVAs on a helicopter rotor, picture taken by Steve
Shaw. For interpretation of the references to color in this and all
other ﬁgures, the reader is referred to the electronic version of this
thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Circular path CPVA on the crankshaft of a prototype internal combustion engine [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Comparison of a CPVA with a simple pendulum: left: simple pendulum. Right: centrifugal pendulum. . . . . . . . . . . . . . . . . . . .

5

Figure 2.1

Hinged circular path absorbers.

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

9

Figure 2.2

Biﬁlar absorber arrangement. . . . . . . . . . . . . . . . . . . . . . .

10

Figure 2.3

Model of a circular path absorber, hinged at point A. Axis of rotation of rotor is point O, R is the distance between rotor axis O
and absorber hinge point A, L is the distance between point A and
the absorber path vertex V (equal to the constant pendulum length),
ˆ
point C shows the location of the COM of the absorber, eR is the
unit vector normal to the absorber COM path at vertex V and mov˙ ˆ
ing with the rotor at ✓, e✓ is the unit vector tangent to the absorber
COM path at vertex V , is the angular displacement of the absorber
with respect to the rotor, ~c is the position vector of absorber COM
r
ˆ
measured from point O, eL is the unit vector normal to the absorber
COM path at a location which is at an angle from V and it moves
˙
ˆ
with the absorber at angular speed (✓ + ˙ ), and e is the unit vector
tangent to the absorber COM path at same location as eL and moves
ˆ
˙
at an angular speed of (✓ + ˙ ). . . . . . . . . . . . . . . . . . . . . .

12

Figure 2.4

Location of accelerometer from COM of a hinged absorber. . . . . .

13

Figure 2.5

x and y axes ﬁxed to accelerometer frame along which the absolute
acceleration is measured. . . . . . . . . . . . . . . . . . . . . . . . .

14

Figure 1.2

Figure 1.3

viii

Figure 2.6

Model for a general path biﬁlar absorber attached to a rotor. Axis of
rotation passing through point O (perpendicular to the plane of paˆ
ˆ
per) with COM of absorber at point C, eR and e✓ are unit vectors as
deﬁned in Figure 2.3, ✓ is the instantaneous angular displacement of
rotor from a reference position, s is the instantaneous distance of the
absorber from the vertex V and measured along the path traversed
by the COM, V is the vertex of the path, ~c is the position vector
r
of absorber COM measured from point O, X(s) is the instantaneous
distance of the absorber from the rotor axis O and measured along
unit vector e✓ , Y (s) is the instantaneous distance of the absorber
ˆ
from the rotor axis O and measured along unit vector eR , and the
ˆ
rectangular box in the ﬁgure represents the absorber moving along
the speciﬁed path. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Figure 2.7

Location of accelerometer from COM of a biﬁlar absorber. . . . . . .

17

Figure 2.8

Experimental rig. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Figure 2.9

Closeup of unmounted absorber mass, with accelerometer and encoder. 28

Figure 2.10

Procedure to measure the absorber COM distance from the rotor center. 29

Figure 2.11

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

31

Normalized absorber displacement amplitude estimated from linear
(4) and cubic (2) analysis compared with exact values (#) at 400
RPM, over a range of absorber response amplitudes. . . . . . . . . .

35

Normalized absorber displacement amplitude estimated from linear
(4) and cubic (2) analyses compared with exact values (#) at 300
RPM, over a range of absorber response amplitudes. . . . . . . . . .

35

Absorber displacement amplitude S estimated from linear (4) and
cubic (2) analyses compared with encoder values (#) at 300 RPM. .

38

Absorber displacement amplitude S estimated from linear (4) and
cubic (2) analyses compared with encoder values (#) at 400 RPM. .

38

Absorber displacement amplitude S estimated from linear (4) and
cubic (2) analyses compared with encoder values (#) at 500 RPM. .

39

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

ix

Figure 2.17

Comparison of cubic estimation from simulations and experiments. .

39

Figure 3.1

Comparison of simulated absorber dynamics with the numerical solution of di↵erential equation (2.8). . . . . . . . . . . . . . . . . . .

42

x

Chapter 1
Introduction
This study considers the development of a method for experimentally measuring the response
of centrifugally driven pendulum vibration absorbers (CPVAs). These absorbers consist of
masses suspended from a rotor (hinged or suspended through rollers) such that they are free
to oscillate relative to the rotor. When correctly tuned they are capable of reducing machineorder torsional (twisting) vibrations in rotating machinery. These absorbers are currently
being developed for automotive engine applications, where reduction of crankshaft torsional
vibrations will allow for low-speed, high-torque operation, resulting in signiﬁcant improvements in fuel economy. The development of these absorbers requires extensive testing, and
while it is quite simple to measure rotor torsional vibrations, currently no convenient method
exists for determining the motion of the absorbers in a rotating machine. The present work
is aimed at developing the tools needed to measure absorber dynamics using an accelerometer ﬁxed to the absorber mass, where the signal can be transmitted via a slip ring or by
telemetry [3].
In this chapter we will state the motivation of this research and will later include some
background on centrifugal pendulum vibration absorbers. The second chapter contains the
theory developed for this research and the validation of the theory with simulations and
experiments. The third and ﬁnal chapter contains conclusions and inferences from this
research, and some directions for future work in this area.
1

1.1

Motivation

CPVAs are devices that reduce undesired torsional vibration in a rotating systems. Torsional
vibrations occur in rotors when the torque source is oscillatory in nature. When the torque
input to the system is periodic, with frequency proportional to mean speed (known as engineorder excitation), CPVAs can be implemented to absorb the torsional vibrations in the rotor
at the problematic order. A typical example of order excitation occurs in an automobile
engine, where the excitation arises from cylinder pressure, resulting in an order equal to half
the number of cylinders.
A basic CPVA device consists of a rotor and an absorber mass hinged to the rotor.
The rotor is free to rotate about its axis and the absorber is restricted to move along a
certain path with respect to the rotor. CPVA systems have been researched for several
decades [4, 5, 6, 7, 8, 9] and have been implemented in various rotational systems, including
helicopter rotors and automobile engines [6, 8, 9, 2, 1]; see Figures 1.1 and 1.2. Once
properly designed and implemented, there is no need to measure the response of a CPVA
system. However, such measurements are very important during the development of these
systems. The rotor dynamics of the CPVA system can be easily measured using encoders.
The absorber dynamics can be easily measured using encoders in a laboratory environment if
there is ample space, but it is not feasible to implement encoders for absorbers in experiments
that are tightly spaced, or in prototype engines and drive trains which may have harsh
environments. These issues provide motivation to ﬁnd alternative ways to measure absorbers
in a rotating machine.
The goal of this research is to develop a strategy by which one can conveniently measure
the dynamics of these absorbers in running machines such as internal combustion engines.
2

Figure 1.1: Cycloidal path CPVAs on a helicopter rotor, picture taken by Steve Shaw. For
interpretation of the references to color in this and all other ﬁgures, the reader is referred to
the electronic version of this thesis.

Figure 1.2: Circular path CPVA on the crankshaft of a prototype internal combustion engine [2].

3

In this research, we develop an analysis method by which one can estimate the dynamics
of absorbers using accelerometers ﬁxed to the absorber mass. The beneﬁts and limitations
of using accelerometers over encoders are listed in Table 1.1. The idea is to measure the
absolute acceleration of the absorber, and also independently measure the rotor dynamics
using an encoder (which is already implemented in many rotating systems) and use these
measurements to estimate the motion of the absorbers relative to the rotor. The success of
this approach will help with the testing of proposed absorber designs, and will also aid in
basic research related to these absorbers.
Table 1.1: Acclerometer vs encoder
Accelerometer

Encoder

Small size, and mass. Can directly be Comparatively large,
attached to absorber

typically at-

tached to rotor and requires extra space
and arrangement on absorber

Measurement has to be further pro- Provides a direct and independent meacessed to determine the absorber dy- surement of absorber dynamics relative
namics relative to rotor

1.2

to rotor

Basic Operation of a CPVA

The physical arrangement of a centrifugal pendulum is similar to that of a simple pendulum.
However, the restoring force arises from a centrifugal ﬁeld instead of gravity, as depicted in
Figure 1.3.
The left panel of Figure 1.3 shows a simple pendulum with mass P and length L hinged
4

Simple'pendulum'

Centrifugal'pendulum'

Figure 1.3: Comparison of a CPVA with a simple pendulum: left: simple pendulum. Right:
centrifugal pendulum.

with a stationary base at point A. The restoring force for any deviation from the equilibrium
position is provided by the gravitational ﬁeld g, which points downward, and the small
q
g
amplitude natural frequency of oscillation is !n =
L . The right panel of Figure 1.3
is that of a centrifugal pendulum with a base rotating about an axis passing through O,

perpendicular to the plane of paper, at a constant speed ⌦. The pendulum is hinged at a
point A to the rotating base, referred to here as the “rotor”. The radially outward arrows
show the centrifugal ﬁeld of magnitude ⌦2 r, r being the radial distance of the point of interest
from the rotation axis O. When the rotor is spinning at a constant speed, the equilibrium
position of the absorber with respect to the rotor is radially outward. The absorber starts
working as soon as there is a disturbance in the mean speed of the rotating system, which
arises, for example, from a ﬂuctuation in the applied torque. For small oscillations in angle
5

about equilibrium, the linearized natural frequency of the absorber with the rotor spinning
at a constant speed ⌦ is given by
!n =

r

R
⌦
L

where R is the distance between the points O and A (see Figure 1.3), and L is the length
q
R is called the tuning order of the absorber and is
of the ideal pendulum. The factor
L

denoted by n. The term order is used to replace frequency in CPVA systems, and it is equal
˜
to the number of complete oscillation cycles occurring in each rotor revolution.

This order tuning feature of a centrifugal pendulum is special because it allows the CPVA
to operate in reducing engine-order torsional vibrations (deﬁned below) at all rotor speeds.
There are a variety of rotational systems where the mean speed of the rotor is variable, and
the rotor is subjected to a ﬂuctuating torque that is synchronous with the rotor, resulting
in engine order excitation, which has frequency n⌦, where n is referred to as the excitation
order. If we design the centrifugal pendulum such that its tuning order n is close to n,
˜
the pendulum will e ciently absorb vibrations occurring at that order, resulting in lowering
the torsional vibration of the rotor at that order [4, 5, 6, 7]. This property makes them
highly suitable for a variety of rotating systems, such as internal combustion engines, where
the torsional excitation arises from cylinder gas pressure acting through the pistons and
connecting rod. For a four stroke engine with N cylinders, the excitation order is n = N .
2
In summary, the CPVA system behaves like the classical tuned absorber, with excitation
and response focused on a given order (which is ﬁxed for a given machine), rather than a
frequency. The dynamics of these absorbers has been studied extensively [5, 6, 7, 8].
Here we focus on a kinematic measurement problem, for which the dynamics are not
essential. The basic problem can be viewed from the right panel of Figure 1.3. Of interest is
6

knowing the pendulum response

relative to the rotor so that the experiments can be used

to correlate models and validate analysis, and also to allow the absorber response range to
be monitored. Since it is often di cult to measure directly, we develop a method that allows
one to compute the relative absorber response

˙
from measurement of the rotor response ✓

(taken from an encoder) and the absolute acceleration of the pendulum mass P (taken from
an accelerometer).
For small amplitude absorber motions this is a linear problem. However, these absorbers
are designed to undergo large amplitude oscillations, in which case the kinematics become
nonlinear. In fact, the absorbers can be designed so that they follow quite general paths
(not just circles) which further complicates the kinematics.
We now turn to the details of this problem and its solutions.

7

Chapter 2
Methods and Results
In this chapter we develop a theory that will enable us to map the measurements from an
accelerometer into absorber response. This mapping will require addition information about
the rotor dynamics that can be easily measured using encoders. The analytical approach is
developed using basic kinematic relationships, which are then used to derive some convenient
approximate expressions for the absorber response. Once the theory is developed, we describe
the experimental setup and methodology required to perform experiments. In the third
section we show results obtained from simulations and experiments and compare these with
the analytical approximations.

2.1

Kinematic Analysis

To develop the model of interest, we assume that the rotor is rigid and spins with ﬂuctuations
about a constant mean speed ⌦. It is also assumed that the ﬂuctuations in the rotor and
the absorber response are periodic. We express the dynamics of the absorber (amplitude
and phase) in terms of the absolute acceleration of absorber, the rotor dynamics, and other
design parameters. We discuss two di↵erent absorber types in section 2.1.1, and describe
the models developed for these types in section 2.1.1.1 and section 2.1.1.2, respectively.
8

2.1.1

Absorber Arrangement

Based on their means of attachment to the rotor, the absorbers can be categorized in two
ways: i) single-point hinged compound pendulums (mainly for circular path absorbers) as
shown in Figure 2.1 and, ii) two-point suspensions using rollers, also called the biﬁlar arrangement, as shown in Figure 2.2. For each type it is assumed that the accelerometer is
ﬁxed to the absorber mass, so that the accelerations measured are two components of its
absolute acceleration oriented with the absorber.

Figure 2.1: Hinged circular path absorbers.

A compound pendulum pivoted about a ﬁxed point (point A in Figure 2.1) on the absorber is called a single-point hinged absorber. Due to the construction of these absorbers,
the path traversed by the center of mass (COM) of these absorbers relative to the rotor is
a circle and the absorber body undergoes a rotational motion relative to the rotor. Because
of this rotation, an accelerometer (see Figure 2.5) ﬁxed to the absorber will measure its
9

absolute acceleration along a coordinate system ﬁxed to the absorber. These absorber ﬁxed
coordinates (discussed in section 2.1.1.1) will be selected as per convenience.

Figure 2.2: Biﬁlar absorber arrangement.
A biﬁlar absorber is a compound pendulum restricted at two distinct points (either on
a moving base, such as the rotor, or on a base ﬁxed to ground) in such a way that the
COM of the pendulum moves along a speciﬁc path with respect to the base (see Figure 2.2).
Typically, these absorbers are supported using a pair of rollers, as shown in Figure 2.2. Here
the absorber and the base each have identical but inverted cutout curves between which
cylindrical rollers are placed (see Figure 2.2). The rollers are assumed to roll without slip.
The constraint between the absorber and the base makes the COM of the absorber move
along a desired path with respect to the base, the path itself prescribed by the design of
the cutout curves. This arrangement of the absorber makes the motion of the absorber with
10

respect to its base purely translational, i.e., it does not rotate relative to the base (in contrast
with the hinged absorber). This fact implies that the directions along which the axes of the
accelerometer (see Figure 2.5) will be aligned are ﬁxed relative to both the rotor and the
absorber. This biﬁlar arrangement provides a convenient way to construct a general path
along which the absorbers are allowed to move relative to the rotor. Biﬁlar absorbers are
most often used in practice. In fact, the absorbers shown in Figure 1.1 and 1.2 are biﬁlar.
The goal of this e↵ort is to develop a method for both biﬁlar and hinged arrangements.
An analytical approach is formulated and approximate solutions are developed for both types
of absorbers. The method developed for biﬁlar epicycloid path absorbers is experimentally
veriﬁed.

2.1.1.1

Hinged Circular Path Absorbers

Figure 2.3 shows a schematic model of a hinged type absorber. The caption in Figure 2.3
deﬁnes the parameters noted in the ﬁgure. Angles ✓ and
absorber-rotor system,

are the degrees of freedom of the

being the absorber coordinate measured relative to a line rotating

with the rotor, which we are interested in estimating, and ✓ being the rotor angle. For
convenience, the x and y axes of the accelerometer (shown in Figure 2.5) are assumed to be
ˆ
ˆ
aligned with unit vectors e and eL , respectively, from Figure 2.3. The position vector ~ a
r
of any general point on the pendulum mass (shown in Figure 2.3) can be written as

~ a = ~ c + x e + y eL
ˆ
ˆ
r
r

(2.1)

where x and y are the o↵sets from the COM to the point of interest; see Figure 2.4.
Di↵erentiating ~ a in equation (2.1) twice with respect to time, we the obtain acceleration
r
11

q
`
eR

`
eq

V

L

`
eL
C

A
Æ
rc

R

f

`
ef

O

Figure 2.3: Model of a circular path absorber, hinged at point A. Axis of rotation of rotor is
point O, R is the distance between rotor axis O and absorber hinge point A, L is the distance
between point A and the absorber path vertex V (equal to the constant pendulum length),
ˆ
point C shows the location of the COM of the absorber, eR is the unit vector normal to the
˙ ˆ
absorber COM path at vertex V and moving with the rotor at ✓, e✓ is the unit vector tangent
to the absorber COM path at vertex V , is the angular displacement of the absorber with
ˆ
respect to the rotor, ~c is the position vector of absorber COM measured from point O, eL
r
is the unit vector normal to the absorber COM path at a location which is at an angle
˙
ˆ
from V and it moves with the absorber at angular speed (✓ + ˙ ), and e is the unit vector
tangent to the absorber COM path at same location as eL and moves at an angular speed
ˆ
˙
of (✓ + ˙ ).

12

~ of this point on the absorber as
a
⇣

¨e
~ = R ✓ˆ✓
a

˙ ˆ
✓2 e

R

⌘

x

✓⇣

⌘
⇣
⌘2 ◆
¨
˙
ˆ
ˆ
✓ + ¨ eL + ✓ + ˙ e
+ L+ y

⇣

¨
(✓ + ¨)ˆ
e

˙
ˆ
( ✓ + ˙ ) 2 eL

⌘

(2.2)

a

Accelerometer)
loca+on)

Figure 2.4: Location of accelerometer from COM of a hinged absorber.

The acceleration expression in equation (2.2) needs to be expressed in terms of the unit
ˆ
ˆ
vectors corresponding to the absorber ﬁxed axes, namely, eL and e . Using the following
ˆ
ˆ
transformation from unit vector pair eR and e✓

ˆ
ˆ
eR = eL cos

ˆ
e sin

ˆ
ˆ
ˆ
e✓ = eL sin + e cos
13

we obtain
~ = a e + aL eL
ˆ
ˆ
a

(2.3)

where
⇣
⌘
⇣
⌘
¨
˙
¨
a = R ✓ cos + ✓2 sin
+ L+ y ✓+ ¨
⇣

¨
aL = R ✓ sin

˙
✓2 cos

⌘

L+ y

⇣

˙
✓+ ˙

⌘2

x

⇣

˙
✓+ ˙

⇣

⌘2

⌘
¨+ ¨
x ✓

(2.4)
(2.5)

The expression of the acceleration in equation (2.3) will be used for estimating the relative
absorber angle

in terms of other measured quantities, as discussed in section 2.1.3. Now

we derive a similar expression for the biﬁlar arrangement of absorbers.

Figure 2.5: x and y axes ﬁxed to accelerometer frame along which the absolute acceleration
is measured.

2.1.1.2

Biﬁlar Absorbers

Figure 2.6 shows a schematic model for a biﬁlar absorber, where the rectangular box in
Figure 2.6 represents the absorber with its COM at point A. The rotor rotates about the
14

q
e`
R
V e` s
q

Y Hs Æ
L r
c

O
XH

C

s)

Figure 2.6: Model for a general path biﬁlar absorber attached to a rotor. Axis of rotation
passing through point O (perpendicular to the plane of paper) with COM of absorber at
ˆ
ˆ
point C, eR and e✓ are unit vectors as deﬁned in Figure 2.3, ✓ is the instantaneous angular
displacement of rotor from a reference position, s is the instantaneous distance of the absorber
from the vertex V and measured along the path traversed by the COM, V is the vertex of
the path, ~c is the position vector of absorber COM measured from point O, X(s) is the
r
instantaneous distance of the absorber from the rotor axis O and measured along unit vector
e✓ , Y (s) is the instantaneous distance of the absorber from the rotor axis O and measured
ˆ
along unit vector eR , and the rectangular box in the ﬁgure represents the absorber moving
ˆ
along the speciﬁed path.

15

axis passing through point O. The parameters for the model are deﬁned in the caption
of Figure 2.6. Coordinates ✓ and s represent the two degrees of freedom of this absorberrotor system. The functions X(s) and Y (s) for a useful class of paths that includes circles,
ˆ
ˆ
cycloids, and epicycloids can be found in [6]. Note that eR and e✓ are, for convenience,
taken to be the directions along which the axes (x and y) of the accelerometer are aligned
(see Figure 2.5).
The position vector of any general point on such an absorber can be written in terms of
ˆ
ˆ
eR and e✓ as
~ a = ~ c + x e✓ + y eR
ˆ
ˆ
r
r

(2.6)

where x and y are o↵sets from the COM to the point of interest; see Figure 2.7.
Di↵erentiating equation (2.6) with respect to time twice, we obtain the acceleration of
the point as follows
~ = a✓ e ✓ + aR e R
ˆ
ˆ
a

(2.7)

where
⇣
¨
a✓ = X

⇣
¨
aR = Y

⌘
˙
¨
˙ ˙
(X + x ) ✓2 + 2 Y ✓ + Y + y ✓
˙ ˙
2X ✓

¨
(X + x ) ✓

˙
Y + y ✓2

⌘

(2.8)
(2.9)

where, by the chain rule,
dX
˙
X=
s
˙
ds
d2 X 2 dX
¨
X=
s +
˙
s
¨
ds
ds2
dY
˙
Y =
s
˙
ds
d2 Y 2 dY
¨
Y =
s +
˙
s
¨
ds
ds2
16

(2.10)

˙ ¨
Equation (2.7) provides an expression which relates measured variables (aR , a✓ , ✓, ✓) to
desired absorber response, expressed as s, through the speciﬁed path functions (X(s), Y (s)).

a

Accelerometer)
loca+on)

C

Figure 2.7: Location of accelerometer from COM of a biﬁlar absorber.

2.1.2

Kinematic Di↵erential Equation for the Absorber Response

Equations (2.3) - (2.5), and equations (2.7) - (2.10), deﬁne di↵erential equations in terms of
the unknown variables

˙ ¨
and s, respectively, with coe cients involving a , aL , ✓, ✓, x , y ,

˙ ¨
L and R for hinged absorbers, and a✓ , aR , ✓, ✓, x , y and the path parameters for biﬁlar
absorbers. The known coe cients such as ai measured from accelerometers, ✓ measured from
rotor encoders and the time independent parameters can directly be substituted into the
di↵erential equations (2.3) and (2.7). These second-order di↵erential equations are strongly
nonlinear and have several time-periodic terms that arise from the measured acceleration and
rotor responses. Finding the desired time-periodic solution of these equations is the central
17

problem of the analysis. One could use numerical techniques (e.g., shooting methods) for
their solution. However, it is convenient to have approximate analytical expressions for the
time-periodic response. Thus we use a harmonic balance method to solve (2.3) and (2.7),
and for this we require some modiﬁcations of these equations. This will be developed in
section 2.1.3. We will further discuss solutions of the di↵erential equations (2.3) and (2.7)
in the last chapter of this thesis.

2.1.3

Approximate Kinematic Solutions

If an accelerometer is ﬁxed on the absorber and the rotating system is in motion, the accelerometer will provide time signals of the absorber acceleration components in directions
ˆ
ˆ
ˆ
ˆ
eL and e (Figure 2.3) for hinged absorbers, and eR and e✓ (Figure 2.6) for biﬁlar ab˙ ¨
sorbers. An encoder attached to the rotor will provide (✓, ✓) by standard di↵erentiation
methods in MATLABTM or LabViewTM . In the present analysis, we intend to measure the
absorber and rotor dynamics at steady state, i.e., when the ﬂuctuation over the mean rotor
speed is periodic, and the absorber response is periodic. The various assumptions taken in
transforming the expressions for the acceleration are as follows (detailed justiﬁcations follow
below):

ˆ
ˆ
(a) The oscillatory contribution to the acceleration is more useful in the e and e✓ compoˆ
ˆ
nents, for the hinged and biﬁlar absorbers, respectively, than the eL and eR components.

(b) The nonlinear terms can be approximated by expansions to cubic degree in the absorber
response coordinates

and s.
18

(c) For a single harmonic excitation order, the rotor response is represented by three significant harmonics, and the absorber response consists of a single signiﬁcant harmonic.

The justiﬁcations for these assumptions are detailed below.
Regarding assumption (a), the expressions for the acceleration given in equation (2.3)
and equation (2.7) consist of two components that are orthogonal to each other, but as per
assumption (a) only one of these components is needed to determine the absorber dynamics.
ˆ
ˆ
In fact, the radial components eL and eR are dominated by centripetal e↵ects (represented
˙
˙
in the (R + L)✓2 and Y (s)✓2 terms) and are thus less useful for measurement of the absorber
motion, since one would have to consider a small oscillation riding on a signal with a large
ˆ
ˆ
mean value. Thus we consider the e and e✓ components of acceleration a and a✓ , for
hinged and biﬁlar absorbers, respectively, to estimate the absorber dynamics.
To simplify the expression of the acceleration as per assumption (b), we Taylor expand
the path functions. For a hinged absorber we expand sin

and cos

in terms of , as follows:

3

sin

3!

=

+ ···

2!

+ ···

2

cos

=1

(2.11)

For biﬁlar absorbers we expand X and Y in terms of s, making use of the fact that the
curve followed by the absorbers is symmetric about X = 0, so that the expansions take the
following form
X = x + X 1 s + X 3 s3 + · · ·
Y = y + Y0 + Y 2
19

s2

+ ···

(2.12)

where the values of the coe cients Xi and Yi depend on the geometry of the absorber path,
and we have again assumed that the accelerometer is installed at position ( x , y ) from the
absorber COM. Coe cients X and Y for various paths can be found in [6]. We substitute
ˆ
ˆ
these expansions of equations (2.11) and (2.12) into the acceleration coe cients of e and e✓
given in equation (2.3) and equation (2.7), respectively, and using assumption (b) we obtain
expressions for a and a✓ as

¨
a ⇠ R+L+ y ✓+ L+ y ¨
=

a✓ ⇠ X 1 s
¨
=

⇣

˙
x ✓+ ˙
✓
˙2
R✓

⌘2

+
3◆

3!

¨
R✓

2

(2.13)

2!

˙
¨
(X1 s + x ) ✓2 + Y0 + y ✓+
⇣
˙ ˙
¨
(6X3 ss + 4Y2 s✓)s + 3X3 s + Y2 ✓
˙
¨

˙
where we note that ✓ = ⌦ +
speed, plus oscillations

⌘
˙
X 3 s✓ 2 s2

(2.14)

(t) will have a DC (mean) component of ⌦, the mean rotor

(t), and all other time-varying terms will be oscillatory with zero

mean. Equations (2.13) and (2.14) are simpliﬁed versions of exact di↵erential equations (2.3)
and (2.7). The main results of the analysis are obtained by doing a balance of the order n
harmonics in these equations. For small amplitudes, the linear truncation may be su cient,
and we also make use of it for comparison purposes, and to determine the limitations of
the linearized analysis. The cubic truncations in equations (2.13) and (2.14) are useful for
capturing nonlinear kinematic e↵ects of the paths at moderate amplitudes.

20

Regarding assumption (c), the torque acting on the rotor in the cases of interest is
composed of mean and periodic components, with frequencies proportional to the mean
speed ⌦. If n is the fundamental order of this excitation, the torque can be expressed as

T ⇠ T0 +
=

N
X⇣

⇤
Tk eikn⌦t + Tk e ikn⌦t

k=1

⌘

where k’s allow for multiple harmonics, and the complex conjugate of T is indicated by T ⇤ .
Note that these torques arise from loads that depend on the rotor angle ✓, and are then
approximated by a Fourier series in which ✓ ⇡ ⌦t is assumed. In general, the torque is
dominated by a single order and thus using assumption (c) we claim that the rotor response,
˙
represented by ✓, is dominated by its mean component plus three harmonics. In complex
form the rotor speed can be expressed as

˙=
✓ ⇠⌦+

3
X⇣

Qk eikn⌦t + Q⇤ e ikn⌦t
k

k=1

⌘

(2.15)

where the complex conjugate of Q is indicated by Q⇤ . Note that the harmonic amplitudes
Qk will be determined from measurements of the rotor response.

Similarly, it is observed in simulations and experiments that the absorber response is
dominated by a single harmonic at order n [7, 10, 11]. The absorber response is therefore
approximated by the following expressions, for hinged and biﬁlar absorbers, respectively:

⇠ P ein⌦t + P ⇤ e in⌦t
=

(2.16)

s ⇠ Sein⌦t + S ⇤ e in⌦t
=

(2.17)

21

It is the goal of this study to determine complex coe cients P and S from more easily
measured quantities, namely, the rotor harmonics, typically obtained from an encoder, and
the accelerometer output.
Since the absorber response is dominated by a single harmonic (order n), for each type
of absorber only two unknowns are of interest, namely, the real and imaginary parts of P
or S, or, equivalently, the amplitude and phase of the order n component of the absorber
response. Thus, only two equations are needed to obtain expressions for these unknowns.
This allows us to consider only the ﬁrst harmonic of the series expressions for the acceleration
components, which can be expressed as,

a ⇠ 1 ein⌦t + ⇤ e in⌦t
=
1

(2.18)

a✓ ⇠ ⇥1 ein⌦t + ⇥⇤ e in⌦t
=
1

(2.19)

Harmonic assumptions (2.16 - 2.19) are inserted into equations (2.13) and (2.14), which
are then expanded in terms of their harmonic components. The equations to be solved for
the unknowns P and S are obtained by retaining only the ﬁrst harmonic of the resulting
expansion, resulting in a pair of equations for the real and imaginary parts of P and S. It
is important to note that due to nonlinear kinematic e↵ects, harmonics will couple to one
another, resulting in equations that involve order n harmonics from all sources, as well as
higher order harmonic coe cients from the rotor response. These equations are detailed in
Appendix A, speciﬁcally equations (A.12 - A.13) for hinged absorbers and equations (A.44
- A.45) for biﬁlar absorbers. Each set of equations is a pair of coupled cubic equations in
the unknown coe cients, and therefore they cannot be solved in closed form. If one retains
22

only linear terms, the equations become two linear equations in two unknowns which can be
inverted in closed form. For solving the equations for the unknowns of interest, P and S,
the following information is used: The harmonic coe cient

1

or ⇥1 is determined from a

Fourier analysis of the signal from the accelerometer; quantities Qk and ⌦ are determined
from a Fourier analysis of the measured rotor response; and the excitation order n and system
geometric variables are known from the physical conﬁguration of the system.

In the following section, we demonstrate results for a biﬁlar absorber system using three
levels of approximation. The ﬁrst is the linear approximation, which can be solved in closed
form; the second is the cubic approximation, which requires the solution of the polynomial
equations given in Appendix A and B; the third approximation comes from direct simulation
of the system dynamic equations, which are labeled as “exact”. The closed form solution for
linear approximations will be discussed here.

If we linearize the expressions for a and a✓ in equations (2.13) and (2.14) and substitute
the harmonic expansions given in equations (2.15 - 2.17), we obtain expressions for

1

and

⇥1 , respectively, that are linear in terms of the absorber amplitudes P and S. Expressing
1

and P in terms of real and imaginary parts, with the subscripts “r” and “i”, respectively,

and inverting the expressions, yields

Pr =
Pi =

( r
( r

c1 ) f 4
f1 f4
c1 ) f 3
f2 f3

( i c2 ) f 2
f2 f3
( i c2 ) f 1
f4 f1

(2.20)

where f1 , f2 , f3 , f4 , c1 and c2 are expressions of known variables and measured quantities
(see Appendix (A.1.1) for expression for the fi ’s and c1,2 ).
23

Doing the same for ⇥1 and S, we ﬁnd

Sr =
Si =

(⇥r
(⇥r

C1 ) F 4
F1 F4
C1 ) F 3
F2 F3

(⇥i C2 ) F2
F2 F3
(⇥i C2 ) F1
F4 F1

(2.21)

where F1 , F2 , F3 , F4 , C1 and C2 are expressions of known variables and measured quantities
(see Appendix (A.2.1) for expressions for the of Fj ’s and C1,2 ).
Equations (2.20) and (2.21) provide closed-form approximate solutions for the real and
imaginary parts of the absorber response, from which the amplitude and phase can be determined. These expressions have linear relationships between the acceleration component
of interest and the absorber complex amplitude. However, in order to consider a range
of response amplitudes for the dynamic system, one varies the amplitude of the ﬂuctuating
torque applied to the rotor, denoted here by , and the absorber and accelerometer responses
are nonlinear in terms of . Therefore, when one plots results for the absorber amplitude
obtained from the linear approximation, the curve will be linear if plotted versus the acceleration component, but nonlinear if plotted versus

. In our study we vary

, since this

corresponds to what is done when investigating the system dynamics.
While the linear approximation yields convenient (if lengthy) closed form expressions,
more accurate approximations can be obtained by considering nonlinear relationships. A ﬁrst
step in this direction is to use the cubic approximations given in equations (2.13) and (2.14).
Applying the harmonic approximations to these equations leads to polynomial (coupled
cubic) equations for Pr and Pi for hinged absorbers, and for Sr and Si for biﬁlar absorbers.
These equations, which are given in Appendix A.2.2, cannot be solved in closed form, and
therefore numerical solutions are required. It is found that these cubic approximations
24

improve the accuracy of the results when compared to the linear approximations, but the
gain is generally not worth the increased complexity in the equations.
If one is going to use numerical methods, it is suggested that the kinematic di↵erential
equations for a (given in equation (2.4)) or a✓ (given in equation (2.8)) be solved by more
direct methods. These expressions do not involve any approximations, but they require
numerical solution of a di↵erential equation with unknown initial conditions. Speciﬁcally,
one would feed into these equations data from the accelerometer and rotor encoder (in the
form of harmonic amplitudes and phases), and seek the periodic solution for

for hinged

absorbers, or for S for biﬁlar absorbers (recalling in the latter case that X and Y are functions
of S). This would require a method for numerically solving a boundary value problem, from
which one would ﬁnd a very good approximation for the absorber response. This issue is
explored further in the ﬁnal chapter of the thesis.

25

2.2

Experimental Setup and Methods

In this section we ﬁrst describe the experimental setup and equipments required for the
experiments, and then describe the methods used to obtain the estimated absorber response
from experiments.

2.2.1

Experimental Rig

Figure 2.8: Experimental rig.
The experiments were performed using the existing rotor rig shown in Figure 2.8. This
rig is described in detail in [9, 12]. It consists of a vertical shaft that is driven by a servo
motor. The servo motor is controlled through a control system that takes its input from a
26

LabViewTM program.
The servo motor can be instructed through the program to run at a desired mean rotor
speed ⌦ using a DC torque. An oscillating torque can be superimposed on the DC torque,
resulting in torsional oscillations. The program allows for engine order excitation that mimics
many applications, including automotive engines. The rig is designed to hold up to four
torsional absorbers, but in the present study a single absorber is employed. The rotor
˙
dynamics (✓) are measured through data captured by an encoder. The absorber used in
this study is shown in Figure 2.9. The absorber is mounted onto the rig using a pair of
rollers. The absorber and encoder can be locked to the rotor (for reference measurements)
or free to oscillate, in which case the angular displacement of the absorber relative to the
rotor is measured using an encoder, as shown in Figure 2.9. In the biﬁlar case, it has been
shown that the output of the encoder is nearly exactly proportional to the S displacement
of the absorber, even out to large amplitudes [13]. The output from this encoder provides a
measurement of the absorber motion that is independent of the accelerometer measurement,
allowing for a means of validating the approach developed in this work.

2.2.2

Accelerometer Arrangement

Accelerometers measure absolute acceleration along certain predeﬁned axes, x and y as
shown in Figure 2.5, relative to the accelerometer. We measure the absolute acceleration of
the absorbers using accelerometers by mounting them directly on the absorbers in such an
orientation that the x and y axes of the accelerometer (see Figure 2.5) are aligned with the
desired axes on the absorber, namely, eL and e for hinged absorbers (Figure 2.1) and e✓ and
ˆ
ˆ
ˆ
eR for biﬁlar absorbers (Figure 2.2). Figure 2.9 shows an unmounted absorber mass, with
ˆ
27

accelerometer and encoder mounted. The accelerometer is mounted on the absorber using
super glue. To avoid surface damage of the absorber, the absorber is covered with masking
tape and the accelerometer is mounted on the masking tape as shown in Figure 2.9.

Figure 2.9: Closeup of unmounted absorber mass, with accelerometer and encoder.

2.2.3

System Parameter Values

To demonstrate the present method, we need to compare the expressions for acceleration
from Equations (2.18) and (2.19) with the corresponding measured harmonics obtained from
the accelerometer during experiment, from which the absorber response, given by P or S,
can be determined. The quantities needed for the calculations are the distance c of the
absorber COM from the rotor axis, the coe cients Xi ’s and Yi ’s as given in Equation (2.12),
and the accelerometer o↵set values x and y .
The distance c is measured using a vernier caliper. In our experimental setup, the rotor
is cylindrical and thus, for convenience and accuracy, the distance from the further edge of
28

the rotor to the absorber COM (c + r) is noted and then the radius of rotor r is deducted
from this measured value (see Figure 2.10). The accelerometer o↵sets x and y are also
measured using a vernier caliper.

2r

c+r

Absorber%

Rotor%
Figure 2.10: Procedure to measure the absorber COM distance from the rotor center.

The coe cients Xi and Yi for the biﬁlar absorber have been derived from the exact
expressions for X(s) and Y (s), as given in [6]. These coe cients, up to the cubic order are
as follows,
X1 ⇠ 1
=
X3 ⇠
=

1
6⇢2
0

Y0 ⇠ c
=
Y2 ⇠
=

1
2⇢0

(2.22)
(2.23)
(2.24)
(2.25)

where ⇢0 is the radius of curvature of the absorber at its vertex. It is determined using the
29

value of absorber tuning n (measured as described below) and distance c as follows [5, 6],
˜

⇢0 =

c
1 + n2
˜

(2.26)

The parameters R and L for hinged absorbers can also be estimated distance c and the
absorber tuning n using [5, 6, 7],
˜

L=

c
1 + n2
˜

R=c

L

(2.27)
(2.28)

To determine these values, we need to estimate the absorber tuning order n. The ab˜
sorbers are designed to be tuned to a certain order, but one can experimentally verify the
actual absorber tuning order. It can be shown [11] that the steady-state amplitude of the
rotor angular acceleration, will have a minimum at n as the excitation order while n is varied.
˜
Shown in Figure 2.11 is the amplitude of the order n harmonic of the rotor angular acceleration normalized by the measured amplitude of the order n applied order torque versus the
excitation order n. The tuning order estimated from this experiment is n ⇡ 2.31.
˜
It should be noted that the absorber tuning will be a↵ected by the placement of the
accelerometers with respect to the absorber COM, and the accelerometer mass relative to
the absorber mass. If y is the o↵set of the accelerometer mass and ma is the mass of
accelerometer, the COM of the absorber will move from its original position due to the
30

2
Rotor ang. accel/ Torque (rad/s /Nm)

Estimation of tuning from experiments
0.8
0.6
0.4
0.2
2.15

2.2

2.25
2.3
2.35
Order of fluctuating torque, n

2.4

Figure 2.11: Order n amplitude of the rotor angular acceleration divided by the amplitude
of ﬂuctuating torque as a function of the applied torque order, n.

attachment of the accelerometer by

, given by

=

ma y
mp + ma

(2.29)

where mp is the mass of the absorber. With this deviation in the location of the absorber
COM, the absorber tuning order becomes n0 , given by
˜
n
˜
n0 = q
˜
1+ ⇢
0

(2.30)

where ⇢0 is the original radius of curvature of the absorber path at the vertex. In our system
n and n0 di↵er by
˜
˜

n ⇠ 0.336 %, given by
˜=

n=
˜

|˜ 0
n

n
˜

31

n|
˜

⇤ 100

(2.31)

The resulting parameter values measured and estimated from experiments are summarized in Table 2.1

Table 2.1: Parameter Values for Experiments
Rotor center to absorber COM (c)

0.123 m

Absorber tuning (˜ )
n

2.31

Radius of curvature at vertex (⇢0 )

0.0196

Accelerometer o↵set ( x )

0.000 m

Accelerometer o↵set ( y )

0.025 m

Absorber mass (mp )

0.432 kg

Accelerometer mass (ma )

0.0023 kg

The corresponding values for Xi and Yi are obtained using the parameter values listed
in Table 2.1 and Equations (2.22-2.25).

In order to simulate the dynamics equations of motion that correspond to the experimental conditions, we require additional dynamic parameters including the rotor inertia J, the
absorber mass mp , the rotor damping c0 , and the absorber damping ratio µa . The dynamic
parameters required for simulation are measured and estimated as given in [1] and are listed
in Table 2.2.
32

Table 2.2: Parameter Values for Dynamic Simulations [1]
Parameter

Value

Absorber mass (mp )

0.4312 kg

Rotor inertia (J)

0.01075 kg-m2

Absorber viscous damping (µa )

0.105

Rotor damping (c0 )

0 .0144 Nms

33

2.3

Results

This section describes results obtained from the proposed method, for both simulations and
experiments.

2.3.1

Simulations

The above sections outline a method for mapping data from an absorber-mounted accelerometer and a rotor encoder into the amplitude and phase of the absorber motion relative to
the rotor. Before considering experimental data, the method is tested using simulation data.
Since this is a purely kinematic exercise, it is possible to examine the method using some prescribed motions for the rotor and absorber and compute the resulting absorber acceleration
components. However, in order to mimic the experimental procedure, the motions are computed using numerical integration of the known equations of motion for the rotor-absorber
system, which are well known and can be found in [6, 7, 8, 9]. The typical outputs from the
˙
simulation are the rotor response (✓) and the absorber motion ( or s). The components
of absolute acceleration, which would be measured by an accelerometer in an experiment,
˙
need to be constructed from simulation data of ✓ and

or s, and their derivatives, using

equations (2.4) and (2.8). In order to test the results based on the linear and cubic approximations, the signals from the acceleration and the rotor response are Fourier decomposed,
keeping a single harmonic for the acceleration component of interest (a or a✓ ) and three
harmonics for the rotor response. This ﬁlters out noise and provides the harmonic coe cients that are used to determine the complex amplitude of the ﬁrst harmonic of the relative
absorber motion.
The estimated data from the linear and cubic approximations are then plotted along
34

Estimation of absorber amplitude at 400 RPM (simulation)
1

S/Smax

0.8
0.6
0.4
Exact
Cubic approximation
Linear approximation

0.2
0
0

0.05

0.1

0.15

Γ

0.2

0.25

0.3

0.35

Figure 2.12: Normalized absorber displacement amplitude estimated from linear (4) and
cubic (2) analysis compared with exact values (#) at 400 RPM, over a range of absorber
response amplitudes.
Estimation of absorber amplitude at 300 RPM (simulation)
1

S/smax

0.8
0.6
0.4
Exact
Cubic approximation
Linear approximation

0.2
0
0

0.05

0.1

0.15

Γ

0.2

0.25

0.3

0.35

Figure 2.13: Normalized absorber displacement amplitude estimated from linear (4) and
cubic (2) analyses compared with exact values (#) at 300 RPM, over a range of absorber
response amplitudes.

with the reference results obtained directly from simulations. The resulting plots are shown
in Figure 2.12.

Figures 2.12 and 2.13 shows result obtained from simulations of a bi-

ﬁlar absorber moving along a tautochronic epicycloidal path [8] with the rotor running at
a mean speed of ⌦ =400 and 300 rpm respectively. The simulations were performed at a
35

low speed since we intend to compare it with experiments, which are also performed at low
speeds due to the hardware constraints of the experimental rig. The scaled amplitude of the
oscillating torque applied to the rotor varies along the horizontal axis and the corresponding
absorber response is plotted along the vertical axis. The absorber response in the plot is
the non-dimensional absorber amplitude S S which corresponds to the absorber amplitude
max
normalized by the maximum amplitude the absorber can achieve (Smax is set by the cusp
of the epicyloid [5]). The scaled oscillating torque amplitude is a non-dimensional quantity
, which corresponds to the ratio of the oscillating torque to twice the rotor kinetic energy,
i.e.,

=

|T |
.
J⌦2

The plot shows three di↵erent curves, representing the absorber amplitude

obtained in three di↵erent ways. The (#) curve represents the solutions from numerical simulations of EOM of the CPVA system. This data is analogous to the measured and ﬁltered
amplitude of the absorber obtained directly from the absorber encoders in the laboratory.
The (2) and (4) curves represent estimated values of the absorber amplitude from the cubic
(equation (2.14), appendix A.2.2) and linear (equation (2.21)) approximations, respectively.
The result in Figure 2.12 shows that the predictions from the analysis are accurate when
compared with the exact results from simulations for a moderate range of absorber amplitudes, say up to 60% of the cusp value. The approximate results begin to deviate from the
simulation data at large absorber amplitudes, higher than 60% of the cusp value. The errors
grow at a higher rate for estimations using the linear approximation as compared to estimations using the cubic approximation. This deviation is due to the Taylor series and ﬁnite
harmonic assumptions used to approximate the non-linear system dynamics. Of course, one
could improve the method by including more terms in the approximations. However, a more
reasonable approach is to numerically ﬁnd solutions of the di↵erential equations for a or

36

a✓ , which is discussed in the ﬁnal chapter of this thesis. We now experimentally examine
the proposed method.

2.3.2

Experimental Results

The absorber response that is estimated from the accelerometer and rotor encoder data
using the present method will be compared to the absorber response taken directly from
the absorber encoders. The response data is stored and post-processed using a LabViewTM
program. The program computes the Fourier components of the signals taken from the
accelerometer (for ⇥1 ), the rotor encoder (for ⌦ and the Qi s), and the absorber encoder
(for veriﬁcation). The known and measured quantities are used as follows: For the linear
approximation, they are substituted into equation (2.21) and solved in close form for Sr and
Si . For the cubic approximation, they are substituted into equations (A.44 - A.45), which
are solved numerically for Sr and Si . The absorber encoder is used to generate the reference
result, which is the amplitude of the order n harmonic of the directly measured absorber
response.
Results for a range of absorber amplitudes are obtained by sweeping the amplitude of
the oscillating part of the torque acting on the rotor. Runs were carried out at 300, 400 and
500 RPM. The results obtained from each sweep, using the linear and cubic approximations,
as well as the reference results from the absorber encoder, are shown in Figures 2.14, 2.15
and 2.16. The trends are the same as those observed in the simulations. Here the linear
approximation works well up to about 50% of the maximum absorber amplitude, and the
cubic approximation works well at all amplitudes we were able to measure, that is, up to
about 80% of the allowable absorber amplitude. This agreement validates the approach
37

derived in this thesis.
Experimental estimation of absorber amplitude at 300 RPM
1

S/Smax

0.8

Exact data from encoder
Estimation from cubic approximation
Estimation from linear approximation

0.6
0.4
0.2
0
0

0.05

0.1
Γ

0.15

0.2

Figure 2.14: Absorber displacement amplitude S estimated from linear (4) and cubic (2)
analyses compared with encoder values (#) at 300 RPM.

Experimental estimation of absorber amplitude at 400 RPM
1

S/Smax

0.8
0.6
0.4
0.2
0
0

0.05

Exact data from encoder
Estimation from cubic approximation
Estimation from linear approximation
0.1
0.15
0.2
Γ

Figure 2.15: Absorber displacement amplitude S estimated from linear (4) and cubic (2)
analyses compared with encoder values (#) at 400 RPM.

Figure 2.17 shows a comparison of the simulation and experimental results, both obtained
using the cubic approximations. This comparison shows that the cubic estimation of the
absorber response from simulations and experiments are in agreement up to about 80% of
the peak absorber amplitude, which is at the limit of the experimental setup.
38

Experimental estimation of absorber amplitude at 500 RPM
1

S/Smax

0.8
0.6
0.4
Exact data from encoder
Estimation from cubic approximation
Estimation from linear approximation
0.1
0.15
0.2
Γ

0.2
0
0

0.05

Figure 2.16: Absorber displacement amplitude S estimated from linear (4) and cubic (2)
analyses compared with encoder values (#) at 500 RPM.

Comparison of absorber estimation using cubic approximations
(Simulation vs Experiment)
1

S/S

max

0.8
0.6
0.4
0.2
0
0

Estimation from cubic approximation (Experiment)
Estimation from cubic approximation (Simulation)

0.05

0.1

0.15

Γ

0.2

0.25

0.3

0.35

Figure 2.17: Comparison of cubic estimation from simulations and experiments.

39

Chapter 3
Conclusions and Future Work
The results generated from simulations as well as from experiments support the proposed
method, of using an absorber-mounted accelerometer, as an e↵ective means of determining
the response of a centrifugal pendulum absorber in a running machine. The approximations
developed are very good at low and moderate absorber amplitudes, but degrade at large
amplitudes, due to the truncated nonlinear expansions and the approximations made in the
harmonic balance analysis.
The proposed approach o↵ers the beneﬁt of small packaging space, and the potential for
using telemetry [3] for data transfer, makes it appealing for applications. The main use of
this approach will be in the development and testing of passive absorber systems, but it
could also be used for active or semi-active pendulum absorber systems.
The present study provides a method for mapping the absorber acceleration into the
information about the steady state response of the absorber. Some topics for improving
and/or continuing this work include the following:

(i) In our present analysis, we had assumed that the torque in the system is periodic with
one harmonic. This method may be further modiﬁed and extended to include cases
where the torque has multiple harmonics.
(ii) The method can be generalized to the case of the transient response of absorbers in a
40

running system. In this case the issue of the initial conditions used in Equations (2.4)
and (2.8) is crucial.
(iii) This approach can also be extended to estimate the absorber damping using ring down
tests. For this, a transformation of the absorber acceleration into a log decrement
of the absorber response is required. Absorber damping is a key aspect in absorber
performance and is often di cult to predict in a running engine. If we can propose a
method of estimating the absorber damping coe cient using accelerometers, it will be
important in the absorber design process.
(iv) A detailed investigation of how uncertainties in measurements and the various approximations used a↵ect the precision af the results would be beneﬁcial for the understanding
the limitations of the approach.
(v) When absorbers are operated at large amplitudes, the linear approximation developed
herein breaks down. One can use the cubic approximation, also described in this thesis, but it requires numerical solution of polynomial equations. If one is going to use
numerical methods, it is probably worthwhile to deal more directly with the kinematic
di↵erential equations (2.4) and (2.8), and not resort to harmonic balance. Here we outline a proposed method for doing so for the case of biﬁlar absorbers; the approach for
hinged absorbers is the same. For this direct approach, one would take measured signals
˙
for a✓ (from the accelerometer) and ✓ (from the rotor encoder) and Fourier decompose
each into a ﬁnite set of harmonics, essentially ﬁltering these signals. These harmonic
expressions would be used to generate the corresponding terms in equation (2.8), resulting in a second order ordinary di↵erential equation for the unknown S(t). The
resulting equation has many time-periodic terms, both additive and multiplicative to
41

Absorber Amplitude

Soln. from numerical simulation of EOM
0.04
Soln from diff. eqn. with initial condition from simulations

0.02
0
−0.02
−0.04
50

50.05

50.1
50.15
Time (sec)

50.2

50.25

Figure 3.1: Comparison of simulated absorber dynamics with the numerical solution of
di↵erential equation (2.8).

S and its derivatives, and the unknown S appears through the known path functions
X(S) and Y (S). The desired solution of this equation is time-periodic, and the initial
conditions for this solution are unknown. It would be convenient if direct simulations
of equation (2.8) with arbitrary initial conditions would settle onto this desired solution, but this is not the case. In fact, the desired solution is locally unstable. (Note
that the periodic response of interest is dynamically stable, but here we are solving a
purely kinematic equation.) Figure 3.1 shows two curves: the steady-state absorber
response taken from direct simulations of the systems dynamics equations of motion
(given in [7]), and a numerical solution of equation (2.8) obtained using initial conditions taken from the simulated response. In this case, the direct dynamic simulation
provides all features of the periodic response S(t) of interest to within small numerical
errors, including the required initial conditions.

By treating S(t) as unknown in the

corresponding kinematic di↵erential equation (2.8), and using the correct initial conditions (to some ﬁnite precision), the solution of the kinematic equation should track the
42

desired solution. Figure 3.1 shows that it does so for a short time, but it quite quickly
diverges from the desired periodic response, indicating its local instability. Therefore,
one must formulate the solution of equation (2.8) as a boundary value problem and
use numerical methods, for example, shooting methods, in order to obtain the desired
solution. The resulting periodic solution will provide the desired absorber response
relative to the rotor, from which its harmonic component at order n can be obtained.
This is the recommended approach for large amplitude absorber motions.
(vi) In fact, solutions of Equations (2.4) and (2.8) is central to all such studies, and a
fundamental study of its properties would be of interest.

43

APPENDIX

44

Appendix A

Full Coe cients

A.1

Full Coe cients: Hinged Absorbers

From equation (2.16) we can write the absorber dynamics in terms of harmonics with complex
coe cients P and P ⇤ , where
P = Pr + iPi

(A.1)

Similarly in equation (2.15), the individual coe cients Qj can be written as

Qj = Qjr + iQji , j = 1, 2, 3

(A.2)

We substitute these values into equation (2.15), and then into the expression for a in
equation (2.13) to obtain a Fourier series, as given in equation (2.18). Since we are only
interested in the

1

term, we extract the ﬁrst harmonic in complex form to obtain

1

= r+i i
45

(A.3)

A.1.1

Linear Approximation

If we truncate equation (2.13) at linear terms in
Q in the linear expression, we get the

r

and

i

and substitute the expressions of P and
as

r

= f 1 Pr + f 2 P i + c 1

i

= f 3 Pr + f 4 P i + c 2

(A.4)

where f1 , f2 , f3 , f4 , c1 and c2 are as follows:

f1 = R (2⌦Q2i + 2 (Q1i + Q3i ) Q1r

n⌦Q2r

2Q1i Q3r ) + 2n⌦ ( ⌦ + Q2r ) x

⇣
⌘
f2 =
Ln2 + R ⌦2 + 3RQ2 + 2RQ2
2RQ1i Q3i +
1i
2i
⇣
⌘
2 + Q2 + 2Q ( ⌦ + Q ) 2Q Q + 2Q2
R 2Q3i
2r
2r
1r 3r
1r
3r

f3 =

⇣

Ln2

⌘

n⌦Q2i (R

n 2 ⌦2 y
(A.6)

+ R ⌦2 + RQ2 + 2RQ2 + 2RQ1i Q3i +
1i
2i

⇣
⌘
R 2Q2 + 3Q2 + 2Q2r (⌦ + Q2r ) + 2Q1r Q3r + 2Q2 + n⌦Q2i (R
1r
3r
3i
f4 = R (2⌦Q2i + 2 (Q1i + Q3i ) Q1r

c1 = 2 ( ⌦Q1i + (Q1i
c2 =

2 x)

(A.5)

n⌦Q2r

46

n 2 ⌦2 y

2Q1i Q3r ) + 2n⌦ (⌦ + Q2r ) x

Q3i ) Q2r + Q2i Q3r ) x + Q1r

2 (Q2i Q3i + Q1r (⌦ + Q2r ) + Q2r Q3r ) x

2 x)

2Q2i x + n⌦ L + R + y

Q1i 2Q2i x + n⌦ L + R + y

(A.7)
(A.8)

(A.9)
(A.10)

Using equations (A.5) to (A.10) we can obtain the amplitude of absorber P in closed form
as shown in equation (2.20)

Pr =
Pi =

A.1.2

( r
( r

c1 ) f 4
f1 f4
c1 ) f 3
f2 f3

( i c2 ) f 2
f2 f3
( i c2 ) f 1
f4 f1

(A.11)

Cubic Approximation

Substituting equations (A.1) and (A.2) into the cubic expression of equation (2.13), we get
expression for

r

r

and

i

as follows:

2
3
= c00 + c10 Pr + c20 Pr + c30 Pr + c11 Pr Pi +
2
c12 Pr Pi2 + c21 Pr Pi + c01 Pi + c02 Pi2 + c03 Pi3 (A.12)

i

2
3
= d00 + d10 Pr + d20 Pr + d30 Pr + d11 Pr Pi +
2
d12 Pr Pi2 + d21 Pr Pi + d01 Pi + d02 Pi2 + d03 Pi3 (A.13)

where the values of coe cients cii and dii are as follows:

c00 = c1

(A.14)

c10 = f1

(A.15)

47

c20 = 0
c30 =

(A.16)

1
R (3Q3i Q1r + Q2i (2⌦ + Q2r ) + Q1i (2Q1r
3

Q3r ))

(A.17)

c11 = 0
c12 = R ( (2Q1i + Q3i ) Q1r + Q2i ( 2⌦ + Q2r ) + 3Q1i Q3r )
c21 =

(A.18)
(A.19)

⇣
⌘
⇣
1 ⇣ 2
R ⌦ + 3Q2 + 2 Q2 + Q1i Q3i + Q2 + Q2 + 2 Q2
2r
1r
2i
1i
3i
2

Q1r Q3r + Q2
3r

⌘⌘

(A.20)

c01 = f2
c02 = 0

c03 =

(A.21)
(A.22)

⌘
1 ⇣ 2
R 3⌦ + 10Q2 + 5Q2
10Q1i Q3i +
1i
2i
6
⇣
1
6Q2 + 2Q2
8⌦Q2r + 7Q2
1r
2r
3i
6

6Q1r Q3r + 6Q2
3r

⌘

(A.23)

d00 = c2
d10 = f3

(A.25)

d20 = 0

d30 =

(A.24)

(A.26)

⌘
1 ⇣ 2
R 3⌦ + 2Q2 + 5Q2 + 6Q1i Q3i +
1i
2i
6
⇣
⌘
1
2 + 10Q2 + 8⌦Q + 7Q2 + 10Q Q + 6Q2
6Q3i
(A.27)
2r
1r 3r
1r
2r
3r
6
d11 = 0

48

(A.28)

d12 =

⇣
⌘
⇣
1 ⇣ 2
R ⌦ + 3Q2 + 2 Q2 + Q1i Q3i + Q2 + Q2 + 2 Q2
2r
1r
2i
1i
3i
2
d21 =

R (3Q3i Q1r + Q2i (2⌦ + Q2r ) + Q1i (2Q1r

Q1r Q3r + Q2
3r

⌘⌘

(A.29)
Q3r ))

(A.30)

d01 = f4
d02 = 0
d03 =

A.2

(A.31)
(A.32)

1
R ( (2Q1i + Q3i ) Q1r + Q2i ( 2⌦ + Q2r ) + 3Q1i Q3r )
3

(A.33)

Full Coe cients: Biﬁlar Absorbers

Using a similar approach, as in section A.1.1 with hinged absorbers we can write the order
n harmonic coe cient of S as follows

S = Sr + iSi

(A.34)

The rotor harmonics are expressed as given in equation (A.2). We substitute these expressions into equation (2.14) and carry out a Fourier expansion. Since we are only interested in
the ⇥1 term, we retain the ﬁrst harmonic from the expression obtained by the substitution
of equation (A.34) and (A.2) into (2.19) to obtain

⇥ 1 = ⇥r + ⇥ i
49

(A.35)

A.2.1

Linear Approximation

If we truncate equation (2.14) to include only linear terms in ✓ and substitute the expressions
of S and Q into the linear expression, we get the ⇥r and ⇥i as

⇥ r = F 1 Sr + F 2 Si + C 1

(A.36)

⇥ i = F 3 Sr + F 4 Si + C 2
where F1 , F2 , F3 , F4 , C1 and C2 are given by

F1 =

⇣⇣

⌘
⌘
1 + n2 ⌦2 + Q2 + 2Q2 + 2Q1i Q3i X1
2i
1i
⇣
⌘
2 + 3Q2 + 2Q (⌦ + Q ) + 2Q Q + 2Q2 X
2Q3i
2r
2r
1r 3r
1 (A.37)
1r
3r
F2 =

2 (⌦Q2i + (Q1i + Q3i ) Q1r
F3 =

F4 =

C1 =

⇣⇣

Q1i Q3r ) X1

F2

(A.38)
(A.39)

⌘
⌘
1 + n2 ⌦2 + 3Q2 + 2Q2
2Q1i Q3i X1 +
1i
2i
⇣
2Q2 + Q2 + 2Q2r ( ⌦ + Q2r )
1r
3i

⌘
2Q1r Q3r + 2Q2 X1 (A.40)
3r

2 (Q2i Q3i + Q1r (⌦ + Q2r ) + Q2r Q3r ) X +
Q1i (2Q2i X + n⌦ (Y0 + Y )) (A.41)

50

C2 = 2 ( ⌦Q1i + (Q1i

Q3i ) Q2r + Q2i Q3r ) X +
Q1r ( 2Q2i X + n⌦ (Y0 + Y )) (A.42)

where X1 = 1, X3 =

1 ,
6⇢2
0

1
2⇢0

Y0 = c, Y2 =

and ⇢0 is the radius of curvature of the

absorber COM path at the vertex.

Using equations (A.37) to (A.42) we can obtain the amplitude of absorber P in closed
form as shown in equation (2.21).

Sr =
Si =

A.2.2

(⇥r

C1 ) F 4
F1 F4
C1 ) F 3
F2 F3

(⇥r

(⇥i C2 ) F2
F2 F3
(⇥i C2 ) F1
F4 F1

(A.43)

Cubic Approximation

Similar to the cubic expressions developed for hinged absorbers in Appendix A.1.2 , we can
develop a cubic approximation of equation (2.14), resulting in expressions for ⇥r and ⇥i as

2
3
⇥r = a00 + a10 Sr + a20 Sr + a30 Sr + a11 Sr Si +
2
2
2
3
a21 Sr Si + a12 Sr Si + a01 Si + a02 Si + a03 Si (A.44)

2
3
⇥i = b00 + b10 Sr + b20 Sr + b30 Sr + b11 Sr Si +
2
2
2
3
b21 Sr Si + b12 Sr Si + b01 Si + b02 Si + b03 Si (A.45)

51

where the values of coe cient aii and bii are as follows:

a00 = C1

a10 = F1

(A.47)

a20 = n⌦ (Q1i + Q3i ) Y2

a30 =

(A.46)

(A.48)

⇣ ⇣
⌘
⌘
3 1 + n2 ⌦2 + 2Q2 + 5Q2 + 6Q1i Q3i X3
1i
2i
⇣
⌘
6Q2 + 10Q2 + Q2r (8⌦ + 7Q2r ) X3
1r
3i
⇣
⌘
10Q1r Q3r + 6Q2 X3 (A.49)
3r
a11 =

a12 =

3

⇣⇣

2n⌦ (3Q1r + Q3r ) Y2

(A.50)

⌘
⌘
1 + n2 ⌦2 + 3Q2 X3
2i
⇣ ⇣
⌘
⌘
2 + Q Q + Q2 + Q2 X
3 2 Q1i
3
1i 3i
2r
3i
⇣ ⇣
⌘⌘
3 2 Q2
Q1r Q3r + Q2
X3 (A.51)
1r
3r
a21 =

6 (3Q3i Q1r + Q2i (2⌦ + Q2r ) + Q1i (2Q1r

Q3r )) X3

a01 = F2
a02 =
a03 =

(A.53)

n⌦ (5Q1i + Q3i ) Y2

2 ((2Q1i + Q3i ) Q1r + Q2i (2⌦

52

(A.52)

Q2r )

(A.54)
3Q1i Q3r ) X3

(A.55)

b00 = C2

(A.56)

b10 = F3

(A.57)

b20 = n⌦ (5Q1r
b30 =

b21 =

3

⇣⇣

(A.58)

2 (3Q3i Q1r + Q2i (2⌦ + Q2r ) + Q1i (2Q1r
b11 = 2n⌦ (3Q1i

b12 =

Q3r ) Y2
Q3r )) X3

Q3i ) Y2

6 ((2Q1i + Q3i ) Q1r + Q2i (2⌦

Q2r )

(A.59)
(A.60)

3Q1i Q3r ) X3

(A.61)

⌘
⌘
1 + n2 ⌦2 + 3Q2 X3
2i
⇣ ⇣
⌘
⌘
3 2 Q2 + Q1i Q3i + Q2 + Q2 X3
2r
1i
3i
⇣ ⇣
⌘⌘
2
2
3 2 Q1r Q1r Q3r + Q3r X3 (A.62)
b01 = F4
b02 = n⌦ ( Q1r + Q3r ) Y2

b03 =

(A.63)
(A.64)

⌘
⇣ ⇣
⌘
2 ⌦2 + 10Q2 X
3 1+n
3
1i
⇣
⌘
5Q2
10Q1i Q3i + 6Q2 + 2Q2 + Q2r ( 8⌦ + 7Q2r ) X3 +
1r
2i
3i
⇣
⌘
6Q1r Q3r 6Q2 X3 (A.65)
3r

53

BIBLIOGRAPHY

54

BIBLIOGRAPHY
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55