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dissertation entitled

FINITE AMPLITUDE EFFECTS ON THE
DYNAMIC PERFORMANCE OF THE
CENTRIFUGAL PENDULUM VIBRATION
ABSORBER

presented by

Mehrnam Sharif-Bakhtiar

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FINITE ANIPLITUDE EFFECTS ON THE
DYNAlVIIC PERFORMANCE OF THE
CENTRIFUGAL PENDULUM VIBRATION
ABSORBER

By:

Mehrnam Sharif-Bakhtiar

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

1989

ABSTRACT

FINITE AMPLITUDE EFFECTS
ON THE DYNAMIC PERFORMANCE
’ OF THE CENTRIFUGAL
PENDULUM VIBRATION ABSORBER

By:

Mehmam Sharif-Bakhtiar

The dynamic response of a centrifugal pendulum vibration absorber is
studied. The thesis consists of two parts. First, the effects of the absorber’s
motion limiting stops on the overall response of the system is
considered. The results of the analysis provide results on the existence and
stability of the nonlinear periodic motions of the absorber mass based on
various system parameters. Several properties which are inherent to the
system are discovered in this analysis which have not been previously
considered, including the coexistence of impacting and non-impacting
motions at the anti-resonance frequency (the frequency for which the

pendulum is designed).

For ranges of system parameters the stability of the symmetric periodic
motions of the absorber mass break down due to pitchfork bifurcations and
successive period doubling bifurcations ensue, leading to chaotic

motions. The effect of detuning of the absorber is also considered.

In the second part of the study the nonlinear dynamic response of the
system with damping in both the primary system and the pendulum is
analyzed using the method of harmonic balance and Floquet
theory. Periodic solutions are approximated by the first harmonic of the
response; the resulting frequency response curves agree well with the
simulations of the full nonlinear equations of motion. Particular attention is
paid to the response at the anti-resonance frequency. Cases are
demonstrated for which there exists more than one stable steady-state
periodic motion of the system in the neighborhood of the anti-resonance
frequency; this particular property of the system is due to nonlinear effects
and cannot be captured through the traditional linear
analysis. Furthermore, it is shown that for ranges Of system parameters, the
only stable periodic response of the system at the anti-resonance frequency is
one of much larger amplitude than predicted by the linear analysis. The
effects of system parameters on the shifting of the anti-resonance frequency

and on the corresponding carrier amplitude are also considered.

The results obtained in the first part of the analysis confirm the
conclusion that motion limiting stops for the absorber can be effectively
employed when placed at amplitudes larger than the steady-state response
predicted from the linearized analysis of the system at the anti-resonance
frequency. The impact dynamics of the CPVA with motion limiting stops
can be quite complicated when the system is subject to excitation frequencies

above the anti-resonance frequency and can include chaotic motions. If the

system is driven out of the region of validity for linearization, nonlinear

effects can lead to catastrophic failures.

The results obtained in the second part of the analysis depict how the
true response of the system can be drastically different than what is
predicted from the linear analysis. Hence, as far as design applications of
the CPVA are concerned, the response of the CPVA obtained through a
linearized analysis of the system should by no means serve as the basis for
the design, but rather as a first trial which is to be modified further through
either experimental or nonlinear analysis of the system. Nonlinear
characteristics of the system such as the shifting of the anti-resonance
frequency and the jump phenomena must be considered in the design of the
CPVA since, if ignored, the absorber’s effectiveness can be reduced or result

in larger oscillatory amplitudes in the primary system.

This work is dedicated to my family.

ACKNOWLEDGEMENTS

The author wishes to extend his deepest gratitude and appreciation to his
advisor, Dr. Steven W. Shaw for his invaluable guidance and assistance

throughout the course of his study along with the preparation of this manuscript.
Sincere thanks are also extended to the other members of the author’s

Doctoral Guidance committee, Dr. Ronald C. Rosenberg, Dr. Brian S. Thompson,

Dr. Shui-Nee Chow, Dr. Alan Haddow, and Dr. Konstantin Mischaikow.

vi

TABLE OF CONTENTS

LIST OF FIGURES

CHAPTER 1- INTRODUCTION

1.1-History & Literature Survey

1 .2-Objectives

1.3-Full Nonlinear Equations

1.4-The Undamped, Linear System

1.5-Steady-State Dynamics of the CPVA for various
combinations of damping and nonoscillatory excitation

CHAPTER 2- EFFECTS OF MOTION LIMITING STOPS
ON THE DYNAMIC RESPONSE OF THE CPVA

2.1-Equations of Motion & Assumptions
2.2-lmpacting, Periodic Response
2.2.1-Methods of Analysis
2.2.2-Symmetric, Double-Impact Motions-Existence
2.2.3-Symmetric, Double-Impact Motions-Stability
2.3-Results
2.3.1 -Frequency Response
2.3.2-The Response at the Anti-Resonance
2.4-Other Nonlinear Responses and Chaotic Motions

vii

26

29
39
39
47
50
55
55
64
73

CHAPTER 3- EFFECTS OF NONLINEARITIES AND
DAMPING ON THE RESPONSE OF THE CPVA

3.1 Equations of Motion

3.2-Periodic Responses

3.3-Stability Analysis

3.4-Frequency Response

3.5-The Response at the Anti-resonance Frequency

CHAPTER 4- CONCLUSIONS

4.1-The Effect of Motion Limiting Stops
4.2-The Effect of nonlinearities and Damping
4.3-Suggestions for Future Work

LIST OF REFERENCES

viii

82

83
87

100
119

127
127

129
132

134

FIGURE 1.

FIGURE 28.
FIGURE 2b.

FIGURE 3a.
FIGURE 3b.

FIGURE 4.
FIGURE 5.
FIGURE 6.
FIGURE 7.
FIGURE 8.

FIGURE 98.
FIGURE 9b.
FIGURE 9c.
FIGURE 10a.

FIGURE 10b.

LIST OF FIGURES

Schematic view of a CPVA as used in practical

applications ‘ 8
Schematic view of a CPVA; Bifilar configuration 9
Schematic view of a CPVA; Simple pendulum

configuration ‘ 10
Frequency of the carrier 19
Frequency response of the pendulum 20
Schematic view of a CPVA with rigid constraints 27
Schematic view of the CPVA with the pendulum at impact 32
Phase trajectory of a double-impact motion 40
Frequency response of the absorber 56
Frequency response of the absorber; Subharmonic

orders and the primary response 57
Variation of the response of the absorber with respect to

the excitation amplitude 58
Variation of the response of the absorber with respect to
theabsorber damping 59
Variation of the response of the absorber with respect to

the coefficient of restitution 60
Phase trajectory of the absorber depicting an

SDIP motion 61
Phase trajectory of the carrier depicting an

SDIP motion 62

FIGURE 11.

FIGURE 12.

FIGURE 13a.

FIGURE 13b.

FIGURE 14a.

FIGURE 14b.

FIGURE 15.
FIGURE 16.

FIGURE 17a.
FIGURE 17b.
FIGURE 17c.

FIGURE 18a.
FIGURE 18b.
FIGURE 198.
FIGURE 19b.

FIGURE 19c.
FIGURE 20a.

FIGURE 20b.

FIGURE 21.

FIGURE 22a.
FIGURE 22b.

FIGURE 23a.
FIGURE 23b.

Coexistence of linear and nonlinear motions at the
anti-resonance frequency

The types of motions coexisting at the

anti-resonance frequency

Coexistence corresponding to + 5% deviation from the
anti-resonance frequency

Coexistence corresponding to -5% deviation from the
anti-resonance frequency

Coexistence corresponding to +10% deviation from

the anti-resonance frequency

Coexistence corresponding to -10% deviation from

the anti-resonance frequency

Frequency response of the absorber

Phase trajectory of the absorber

Time response of the unstable SDIP motion of Figure 16
Time response of the unstable SDIP motion of Figure 16.
Time response of the stable anti-symmetric period-one
double-impact motion of Figure 16.

Poincare\ map of a strange attractor

Time response of a chaotic motion

Frequency response of the absorber-

Analysis and simualtion

Frequency response of the carrier-

Analysis and simulation

Phase difference of the carrier and the absorber
Variation of the frequency response of the absorber with
respect to the excitation amplitude

Variation of the frequency response of the carrier with
respect to the excitation amplitude

Turning frequency and the anti-resonance frequency
Variation of the mean value of the carrier velocity
Variation of the mean value of the absorber

angular displacement

Phase trajectory of the CPVA in three-dimensional space
Phase trajectory of the CPVA in three-dimensional space

65

67

68

69

70

78
80
81

101

102
103

105

106
109
111

112
113
114

FIGURE 24.

FIGURE 258.
FIGURE 25b.
FIGURE 268.
FIGURE 26b.
FIGURE 26c.

FIGURE 278.
FIGURE 27b.

Effects of small symmetry deviations on a

pitchfork bifurcation

Frequency response of the carrier

Frequency response of the absorber

Variation of the carrier amplitude at the anti-resonance
frequency woth respect to the excitation amplitude
Variation of the carrier amplitude at the anti-resonance
frequency with respect to the absorber damping
Variation of the carrier amplitude at the anti-resonance
frequency with respect to the carrier damping
Frequency response of the absorber

Frequency response of the carrier

xi

116
117
118

120

121

122

125
126

CHAPTER 1
INTRODUCTION

1.1- History and Literature Survey

The centrifugal pendulum vibration absorber (CPVA for short), although
patented previously in France, was independently conceived and put into
practice by E. S. Taylor in 1935 in order to eliminate torsional vibrations of
geared radial aircraft-engine—prOpeller systems [1]. This was the first
presentation of pendulous absorbers in the United States. The device,
however, is useful in a broader range of applications for the reduction of
torsional oscillations in rotating shafts of large engines and machinery which
are subjected to torsional disturbances of frequencies proportional to the
nominal speed of the shaft. Due to the relatively short length of the
pendulum required to function as an effective absorber, the device was not
suitable for small engines such as automotive engines until 1942 when the
absorber was modified and incorporated into the design of internal
combustion engines in order to alleviate the torsional vibrations of the crank
shaft. This was done by integrating the absorber with the crankshaft
counterbalance masses [2]. The efiective radius of the pendulum in this case
is the difference between the bushing’s inner radius and the mating pin’s
radius (ie. , the clearance in the journal bearing) which can be made as

small as necessary.

In general, the reduction of torsional vibrations in rotating shafts such as
automotive crank shafts is achieved using one of three different classes of
dampers or absorbers [3]. These are the frictional dampers [4], tuned
absorbers [5], and pendulous absorbers, where the CPVA belongs to this

latter class.

In frictional dampers, the torsional vibration of the crank shaft is damped
uniformly throughout the whole range of operational frequency of the
system. Its effectiveness depends on its moment of inertia and the fluid
viscosity which is used as the damping element. The main disadvantage of
such a device is the substantial increase in the overall weight of the engine
and the need to dissipate the heat generated by the damper. On the other
hand, tuned absorbers, like linear spring-mass absorbers, are effective only
for the fixed, narrow bandwidth of the frequency for which they are designed
and, except in a few specialized cases, tuned absorbers find limited
applications in the design of rotating shafts due to the variation of the speed

of the crank shaft and consequently, the excitation frequency applied on the

shaft.

The CPVA, however, is essentially a tuned absorber whose natural
frequency varies in direct proportion to the rotational speed of the
crankshaft. It also introduces no additional weight increase to the design of
the system in many applications, and hence the CPVA becomes an
attractive alternative to the reduction of torsional vibrations in engines and

it makes the detailed study of its dynamic performance worthwhile.

The centrifugal pendulum vibration absorber has been successfully
employed to overcome induced vibrations on the cabin and cockpit of
helicopters which are due to the transmission of vibration from the main

rotor hub of the craft. W. F. Paul [6] has noted an experiment on

commercial and USAF versions of the Sikorsky S-61 helicopters where an
airframe vibratory stress reduction of as much as 4:1 was achieved, and
cockpit and cabin vibrations of i0.1g were recorded through the use of the
CPVA indicating a substantial improvement over the recorded data of the

operation of the rotor without the absorbers.

After the original idea of E. S. Taylor [1] to devise the CPVA, numerous
research studies have been undertaken in order to exploit the potential
advantages of the centrifugal absorber. A few instances are cited in the

papers discussed below.

Zdanowich [8] presents an alternative approach to describe the mechanics
of reduction of the carrier amplitude to zero at wAR through the notion of
the effective inertia of the carrier. It is shown in [8] that at the anti-
resonance frequency, the effective inertia of the carrier with respect to the
sinusoidal excitation tends to infinity and therefore there will be no

oscillation of the carrier.

Den Hartog [9] observed in 1947 that for a rotor, translational vibration
of the rotor mass center in two orthogonal directions and torsional
vibrations could be simultaneously reduced to zero by employing three
pendula fixed to the rotor and disposed about the circumference of the rotor
at 120 degrees with respect to one another. However, as noted by T. C. Lim
[10], Den Hartog’s analysis is not free of flaws in the sense that, although
the rotor mass center translational acceleration in the direction of the
applied force can be reduced to zero, the rotating pendula will always induce
vibrations transverse to the line of action of the applied force. This problem

may be solved by employing several such sets of pendula along the shaft.

Robert Plunkett [11] has carried out a rather detailed survey on the study
and application of the CPVA along with other modes of vibration
absorption through the appropriate use of the damping characteristics in
materials and vibration isolation. The author is indebted to Plunkett’s
work [11] for providing some of the resources on the subject. Zdanowich
and VV'Ilson [8] present a description of application of CPVA prior to
WWII. Harker [12] gives charts and design guidelines for the use of practical
design; Meyer and Saldin [13] show an application of the absorber to turbine
buckets and Reed [14], like Den Hartog [9], indicates the possibility of
applying the principle to nonrotating machinery.

Some of the credit for the novel application of the idea of the CPVA goes
to Sarazin [15] and to Chilton and Reed [16] who in 1930 proposed the idea
of bifilar pendula as absorbers. Butler [17] elaborates on the notion of
having the absorber’s center of mass in the bifilar construction move on a
noncircular trajectory. H. H. Denman in 1988 illustrated in [18] that such
modification in the design of the absorber can be beneficial to the overall
response of the system. Crossley studies the nonlinear response of the
CPVA due to the wide angle of swing of the absorber in free [19] and the
forced [20] modes of response, respectively. W. L. Miao and T. Mouzakis
[21] present an experimental study on the nonlinear characteristics of the
bifilar absorber mounted on the rotor hub of a helicopter. This idea
originates from Kelly’s work [22] who proposed the notion of application of
CPVA to helicopter rotors to overcome excessive torsional vibrations of the
mainframe of the helicopter. Mouzakis [23] discusses the interesting notion
of a mono/ilar absorber which inherently possesses two distinct frequencies

at which the absorber can be employed effectively.

1 .2- Objectives

One of the major difficulties in the design and application of the CPVA is
excessively large amplitudes of oscillation of the absorber which can occur
during operation [19, 20]. In other words, absorbers that are thought to be
properly designed can exhibit dynamics that are inconsistent with the
original analysis. An example is cited by Newland [7] where an engine was
designed with pendula calculated to swing through 45 degrees. During tests
it was found that the vibration absorbers were not functioning properly and
on dismantling the engine they were found to have been oscillating through

a much larger angle than expected, damaging the stOps which were set at 75

degrees amplitude.

Most of the literature on the subject of CPVA is based on the linear
analysis of the system (see references [1-5], [8], [10] and, [30], for
example). The studies of the nonlinear response are more limited (see
Newland [7], Crossley [19, 20], Paul [6] and, Den Hartog [9]), and there are
many dynamic properties of the system which remain to be explored. The
increasing use of mathematical tools such as bifurcation theory [26] and the
Poincare map [27], to study nonlinear systems, along with the advent of
high speed computing systems, makes it possible to undertake a more
thorough study of the response of the CPVA. This report aims at shedding
light on some of the nonlinear dynamic properties of the CPVA which have

not been considered before.

The study is composed of two parts. In the first, an attempt is made to
alleviate the large amplitude oscillation problem of the absorber by limiting
its maximum angle of swing to a prescribed value. This is carried out by
incorporating rigid constraints in the design of the CPVA. The resulting

dynamics are dealt with in detail in chapter two.

In the second part of the study the objective is to better understand the
dynamics of the CPVA without the constraints. To this effect, higher order
nonlinear and damping terms are retained in the equations of motion of the
system and their effects on the response of the system are studied. This is
covered in chapter three of the thesis. Chapter 4 contains an overview of

the results, some conclusions, and suggestions for further work.

In order to carry out the numerical computations of the analysis, the
following system parameters are used throughout the study:
Crank radius, R - 2.24 in.
Crank polar moment of inertia, J - 772.28 lb. in 2.
Absorber effective length, r = .561 in.
Absorber mass, m =-- 25.6 lbm.

Crank rotational speed, (I = 1000 RPM.

The above dimensions might are chosen to be in the range of practical

applications. ,

1.3- Full Nonlinear Equations of Motion

Figure (1) depicts a schematic view Of the CPVA as applied in
practice. The pendulous absorber is employed as a bifilar pendulum (Figure
2a) in which the absorber is mounted on the carrier through two pivoting
arms. In addition, the absorber itself has a finite moment of inertia which
can effect the dynamics of the system. However, in the following it is
demonstrated that such bifilar configuration can be modeled as a simple
pendulum (Figure 2b) provided certain geometric relations hold [7]. This is

done through a comparison of the equations of motions of the models.

Referring to Figure (2a), which is a schematic view of a bifilar pendulum
absorber, the expression for the velocity of the mass center of the absorber

mass can be written as:

1700 =[(b +h )IP ’ costb-I—l (45 ’ —1,b’ )cos(t/)+¢)]i-+
+l-(b +h )1!) ’ simb+l (<15 ' -t/J’ )sin(1b+¢)J-' . (1.1.1)

Denoting the inertia of the absorber with respect to its center of gravity by

I, , the kinetic energy of the bifilar system can be written as:

XE =%.(1+1, )¢ ' 2+
+—;-m [2p ' 2(1) +12 )2-Hrfi ' —¢' )212—22p ' (<6 ' —¢' )1 (b H )cos¢l (1-1-2)

Lagrange’s equation can be written as:

‘39” :[a—KE:+ +9554, i=1,2 (1.1.3)

_d_
d Bq, aq,

 

where T denotes time, q, and q2 represent the generalized coordinates 1,0 and

95:66:92 Suzette 5 com: an «See a co 32> 626.528 1 _ 83m:

 

 

Figure 28 - Schematic view of a CPVA; Bifilar configuration

10

 

Figure 2b - Schematic view of a CPVA; Simple Pendulum Configuration

ll

45, and Qa- , i=1,2 are generalized forces being —C'p¢' for the absorber and

—Cc1[1 ’ +T(1') for the carrier, T(T) represents the torsional excitation on the
carrier and (3'6 and 0,, are the damping values of the carrier and the
absorber, respectively. The effect of gravity on the system is negligible for
even moderate rotational speeds and hence the potential energy is zero,
PE =0. Hence, the equations of motion for the bifilar pendulum vibration

absorber system can be written as:
[(I+Ip )+m (b +h )2+m12+2m (b +h)lcos¢]1b’ '
—ml((b +h )cos¢+l )(b ' ' +
+m (b +h )l¢ ’ 2sin43-2m (b +h )II/J ’ qS ’ sin¢+Cc 1b ' =T(T) (1.1.4a)

—ml ((b +11 )cos¢+l)1b ' ' +m12¢> ' ' +0, d; ' +m (b-l-h )lzp ' 23in¢=0 (1.1.4b)

Referring to Figure (2b), the relation for the absolute velocity of the

absorber mass can be written as:
77c =51: +51) /c

where subscripts c and p stand for the carrier and the pendulum,

respectively. Since
.7, =(R 1/J’sin¢)i-—(R w ' saw);
and,

17p /c =l-r(-¢' +¢ ' )Sin(¢+¢)]7+lr(-¢ ’ +45 ' )COS(¢+¢)17

12

then,
.7, =18 «12' simb+rw ' -¢ ' )sin(z/»+¢)17+l—R .1 ' com/2+
+1“ (11) ’ -¢ ’ )COS(1/J+¢)]J—'. (1.1.5a)
and hence,
v,2=R2¢ ' 2+r2(z/2' —¢ ' ream/2' (11' —¢ ' )cosqs (1.1.511)

Thus, the kinetic energy of the system of Figure (2b) can be written as:

1 1
KB =5sz ' HEM},

01‘
KB =—;-(J+mR2)7,/J ' 2+%mr2(1/J’ —¢ ' )2+mRri,b ' (w ' —¢ ' )cosa5 (1.1.6)

where J is the polar moment of inertia of the carrier, m is the absorber
mass, R and r are the effective carrier and the pendulum radii, respectively,
and 1b and d) are the angular displacements of the carrier and the pendulum,

respectively, as denoted in Figure (2).

Neglecting the effects of gravity on the system, and applying Lagrange’s
equation, the equations of motion for the simple pendulum absorber of

Figure (2b) can be written as:
[J+mR 2-I-mr"’+2mRrcos<15]1,b " —mr(R cos¢+r)¢ ” +mRr¢ ’ 23in¢—

—2mRr1/J ’ 43 ’ singb+0cib ’ =T(T) (1.1.7a)

—mr (R cos¢+r )1!) ' ’ +mr2¢ ”+0qu ' +mRr1/) ’ 2sin§b=0 (1.1.7b)

where ( )’ denotes differentiation with respect to time, 1', and T('r)

represents the excitation on the carrier.

13

Comparison of equations (1.1.4) and (1.1.7) reveals that the bifilar
pendulum vibration absorber of Figure (2a) can be dynamically modeled as
the simple pendulum vibration absorber of Figure (2b) provided the

following geometric conditions hold:
J =1 +1?

1'. e. , the inertia of the carrier in Figure (2b) should be the total inertia of the

carrier and the pendulum. Also,
r=l
R =b +1:

1'. e. , the effective radius of the carrier in Figure (2b) should be the sum of
the distance from the center of the carrier to the line joining the pivot points
of the bifilar absorber, plus the distance from the center of gravity of the
bifilar absorber to the line joining its pivot points (see Figure 2a). This
equivalent modeling of the bifilar absorber by a simple pendulum absorber is
possible primarily due to the fact that there is no rotation of the bifilar

absorber about its center of gravity.

Considering the above analysis, from hereafter the simple pendulum
model of the CPVA will be used for obvious reasons of simplicity, while it is
understood that the necessary geometric conditions hold between the simple

pendulum model used with its corresponding bifilar type.

By referring to the equations of motion for the CPVA (equations 1.1.7)
one can observe that the angular displacement of the carrier, 1b, does not
appear in either of the equations (1.1.7). This renders the variable 1b
ignorable and consequently, when written in first order form, the equations
represent a third order system, ie. , one equation each for 1b, <13 and,

<15. Hence, the solution of the equations of motion yields 1b where upon

l4

integration the angular displacement of the carrier, 11), can be obtained up to

an arbitrary constant.

15

1.4- The Undamped, Linear System

As a first attempt to the solution of the problem under study, the
dynamic response of an undamped, linear system is studied in this
section. Obviously, a true model of the actual system should incorporate the
effects of damping and also the effects of nonlinearities. However, a study of
this simple model provides insight into the overall dynamic response of the
system and yields approximate results to the actual nonlinear, damped
system. Furthermore, a linear undamped model of the CPVA is often a
valid model of the actual system since for very low absorber and carrier
damping values (as is typically the case) and small amplitudes of the
oscillation of the absorber, the CPVAIcan be modeled as a linear, undamped
system with an acceptable degree of accuracy. Also, introduction of
damping and nonlinear effects into the model can be looked upon, in an

asymptotic sense, as perturbations on the linear system.

In order to carry out the linear analysis, certain assumptions are made to
simplify the equations of motion, they are as follows:

1. The angular displacement of the absorber, (b, is small enough that

singb=gb , cos¢=1

2. The dampings in the carrier, 0,, and the pendulum, C are negligible.

p,

3. The excitation is of the form
T(T)=Tosinwr (1.2.1)

4. The motion of the carrier, 11), is a steady-state, constant rotation plus a

small sinusoidal oscillation, i.e. ,

16

1b=flr+l9 (1.2.2)

5. Terms involving products Of (b, 6 and their time derivatives are negligible.

Substitution of assumptions 1 through 5 into the equations of motion

(1.1.7 ) results in the undamped, linearized equations of motion as follows:

[J+m (R +r )2]6 ’ ’ —mr (R +r)(b ’ ’ =Tosinwr (1.2.3a)

—mr(R +r )0 ’ ’ +mr2¢ ’ ' +mRr 02¢=0 (1.2.3b)

Now we assume a sinusoidal system response:
¢=¢osinwr (1 .2.4a)
0=i90$inwr (1.2.4b)

where the relative phase (0 or 71') is accounted for in the signs of 90 and
430. Using equations (1.2.2) and (1.2.4), equations for the amplitudes (1)0 and

1% can be obtained:
—[J +m (R +r )2]w21,bo+mr (R +r )w2¢o=T0 (1.2.5a)
mr (R +r)a121/Jo-mr2e12¢0+mRr fl2¢0=0 (1.2.5b)

Equations (1.2.5) are the amplitude relations for the linearized model of the
CPVA. The natural frequencies and mode shapes of the system are

determined to be:

 

 

2
an? 2 /
wl=0 , w2= Jr [J+m(R+r)] (1.2.6)
and
vl= [06°] , c arbitrary, (1.2.7a)

 

_. 1 _-lJ+m(R+r)“’l
122— [II] , Il— mr(R+r) (1.2.7b)

17

In words, the mode shape represented by equation (1.2.7a) which
correponds to the w1=0 natural frequency, indicates a rigid-body mode in
which the angular displacement of the carrier tends to infinity while the
absorber retains a finite angle with respect to the <b=0 reference line. The
second mode, equation (1.2.7b), is associated with the nontrivial natural
frequency of the system and corresponds to the case where the carrier and

the absorber are oscillating out of phase.

Note that from equation (1.2.5b), the steady-state amplitude ratio

equation can be written as:

 

 

62—025—
1b
¢° - ’ . (1.2.8)
r .

This implies that if the frequency of the torsional excitation, w, on the
carrier is jfl, j=1,2,3,..., then the pendulum can be tuned such that
j==(R/ r)1/ 2, thereby reducing the oscillatory amplitude of the carrier to
zero. In fact, such disturbances whose frequencies are multiples of the
rotational speed of the carrier, (I, are typical in internal combustion
engines. This particular frequency for which the absorber is designed is

called the anti —resonance frequency and is denoted by wAR . Hence,

/2
W34) [.11] , (1.2.9)

r

Note that WAR varies in direct proportion to Q, the nominal rotational speed
of the carrier. The term j=(R /r)1/2 is called the order of the engine torque
and in most practical applications, torques of order half the number of
cylinders of the engine are the most dominant ones. For instance, for a four

cylinder, four-stroke automotive engine (the type of engine considered in this

18

report), the order j =2 torque, i. e. , the second harmonic Of the excitation,

carries a substantial portion of the overall torque.

Figures (3a) and (3b) illustrate the frequency response of the carrier and
the absorber, respectively, as obtained based on the undamped linearized
CPVA model. Note that at the anti-resonance frequency, (UAR, while the
carrier amplitude of oscillation is reduced to zero, the absorber has a finite
amplitude of oscillation described by

To
mR r 02(R +r)

 

¢0 ICU-(JAR =

At the anti-resonance frequency the torque exerted by the absorber on the
carrier is equal in magnitude and opposite in direction to that of the

external excitation torque, hence resulting in a net zero torque on the carrier.

In order to better understand the dynamics of the CPVA under the given
excitation and damping conditions, and to grasp a general overview of the
mechanics of the system, the next section is devoted to the steady-state
response of the system under various simple but basic excitation and
damping conditions. The solutions presented therein will provide the basis
for further analysis of the system, as the various expansion processes
described in the following chapters will be done about the appropriate

steady-state solutions demonstrated in the next section.

19

 

 

 

 

 

101—(—
g__
4
6——
.- H
4
2——\
\(
mRIII[IIII‘fiIITF i]llli

 

 

91.0 01.5 1.9 1.5 2.0 ~ 2.5

1)

Figure 3a — Frequency Response of the Carrier

20

 

 

 

 

ill—F

¢o
g__._
5_.__
4....
2...-

7’1 l

@IIIIIIIIF£]IIIIIIIIIIII_I

mom MOS ['00 1.05 2.0 _ 205

Figure 3b - Frequency Response of the Pendulum

21

1.5- Steady-State Dynamics of the CPVA for Various

Combinations of Damping and Nonoscillatory Excitation

In this section the steady—state solutions Of the nonlinear equations of
motion are obtained for various damping and constant torque
conditions. The steady-state response of the system in the absence of any
oscillating component of the torque is presented in the following in order to
provide the nominal operating conditions for the case of no applied periodic
torques. These operating conditions are used as a basis for linearization and

for nonlinear expansions throughout the thesis.

By referring to the equations of motion (1.1.7) and letting T (T)=To, a

constant torque, the equations of motion can be rewritten as

[J +mR 2+mr2+2mRr cos¢]1,b ' ' —mr (R cos¢+r)qb ' ’ +mRr¢ ’ 2sincjb—
—2mthb ’ d) ’ sin¢+Cctb ’ =To (1.3.1a)

-mr(R cos¢+r )1/1 ’ ’ +mr2¢ ’ ’ +01, 923 ’ +mth/J ’ 2sin<15=0 (1.3.1b)

Depending on whether 0,, C, or, T0 are taken to be zero in equations

(1.3.1), the following eight different cases can be outlined.

1. c,=c,=o; T0950

This renders the equations of motion (1.3.1) as
[J+mR 2+mr2+2mchos¢]1/J ’ ’ —mr (R cos¢+r)<b " +mRr¢ ’Qsinqb-
—2mRr1/2' <15 ’ sin¢=T0 (1.3.2a)

—mr (R cos¢+r)1,b ’ ' +mr2¢5 "+mRr1b’Qsingb=O (1.3.2b)

IO
“'2

In order to find the steady-state solutions of the CPVA corresponding tO

equations (1.3.2), let
1b ” =a , a=constant
(:5 " =¢ ' =0
Hence, integrating these latter equations with respect to time, 1', yields,
tb ’ =ar+f2
and
=constant
Substitution of these equation into equations (1.3.2) yields,
[J +mR 2+mr 2+2mRr cosqb]a=To (1.3.3a)
—mr (R cos¢+r )a+mRr(ar+n)Qsin¢=0 (1.3.3b)

Assuming 43 is small enough that one can write cos¢=1, equation (1.3.3a)
yields an expression for a as

To
a=
J-I-m (R +r )‘2

 

Note that this case corresponds to the first mode of the CPVA obtained in
equation (1.2.6) and (1.2.7a) implying that the angular displacement of the
carrier, 1b, tends to infinity while the pendulum makes a small but finite
angle with the ¢=0 line (see Figure 2b) at 7'=0. This angle tends to zero as

1b tends to infinity. To be specific, from equation (1.3.3b) one can write
R arzsin¢=R +r

In other words (I) varies in time in such a way that the above relation holds
for all time 7' implying that (IS—t0 as r—boo.

2. Cc =01) =To=0

Under these conditions, by letting
t/J ' ' =0 , 1,!) ’ =fl , (I constant (1.3.4a)
¢ " =45 ’ =0 , ¢=constant (1.3.4b)
equation (1.3.1b) is identically zero and equation (1.3.1a) yields
mRr 023in¢=0
implying
fl=0 , (b arbitrary
or
gb=n7r , II arbitrary.

where the first of these two solutions is the trivial static configuration and
the second corresponds to the case where the carrier is rotating with a
constant angular velocity while the pendulum remains at ¢=0 (¢=7r is

unstable ).

3. CC #0, Cp=TO=O
Using the assumptions outlined in equations (1.3.4), equations (1.3.1a,b)

yield
Ccfl=0 => fl=0 ,
and
mRrflzsin¢=O => ¢arbitrary.

In other words, the carrier comes to a stop with the absorber at an arbitrary
angle.

4. c,,c,+o, T0=0

24

Letting
¢"=0 , 1,0 ’ =0 , Q=constant
(25"fl , (b’=gb’0 , ¢0'=constant ,
equation (1.3.1a) yields
II=O

and from (1.3.1b) one obtains

a, 45,, ' +mth1251n¢=o
implying

¢ ’ =0 , =arbitrary.

which is the same as case 3 above.
5. Cpfl) , CC=T0=O
Using the same assumptions as equations (1.3.3), equation (1.3.1a) is

identically zero and equation (1.3.1b) yields
“=0 or ¢=n7r

again implying the same result as that of case 3 above.

6. Cc=0 , 0,3040

Letting
1!) ’ ’ =0: , oz=constant
45 I r =¢ I =0

the solutions turn out to be identical to those considered in the case 1.

7. 0,, =0, 0,, ,Toaéo Using equations (1.3.3), equation (1.3.1a) yields

25

 

and from equation (1.3.1b),
§b=n7r

implying the carrier rotating with a constant angular velocity 0 fixed by the

 

0 while the absorber remains at ¢=0 or 7r (the ¢=7r solution is
C

relation fl=

inherently unstable).
8. C, ,0, ,ToséO

Substitution of assumptions (1.3.3) into equations (1.3.1) yields the

equations of motion identical to those of the previous case 7.

One can readily see (for obvious reasons) that in all the cases considered
above, none admit an oscillatory solution to the problem. Furthermore, one
can see that the carrier damping, Cc, has a significant effect in determining
the rate of steady rotation of the carrier (cases 7 and 8), for in the absence
of 0, (cases 1, 2, 5 and, 6) the carrier’s velocity either tends to zero or
infinity as ‘r—too. Note in cases 3 and 4 that, although 0, is nonzero, the
other determining parameter, namely, To, is set to zero and “=0 results.
The pendulum damping, 0,, does not show any significant effect on the

steady-state response of the system.

It should be noted that in the chapter 2 the linearization of the equations
of motion will be carried out about the system parameters corresponding to
case 6 and the nonlinear expansion of chapter 3 will correspond to the

system parameters outlined in case 8 above.

CHAPTER 2

EFFECT OF MOTION LIMITING STOPS
ON THE DYNAMIC RESPONSE
OF THE CPVA

Figure (4) shows a schematic view of a CPVA with rigid constraints
which are employed in order to limit‘ the maximum allowable amplitude of
oscillation of the absorber to a prescribed value, denoted as H. The objective
of the design is to have the absorber oscillate freely between the two
constraints in normal operatiOn without coming into contact with the stops,

except possibly during transient motions.

In order to achieve this objective, the response of the absorber in relation
to the constraints and the range of parameters for which impacting and/or
non-impacting periodic motions exist is to be considered. In particular, the
possibility of coexistence of linear non-impacting and nonlinear, impacting
motions at the anti-resonance frequency is of interest. Such coexistence can,
in fact, occur and cannot be predicted without a detailed consideration of
impacting motions. An absorber which is thought to be properly designed
(i.e., using only linear analysis and the addition of stops at amplitudes
larger than the steady-state amplitudes predicted by the linear theory) may

encounter impacting steady-state dynamics.

26

27

T, cos(cuT)

 

Figure I» - Schematic View of a CPVA with Rigid Constraints

28

It is shown in the following that such coexistence occurs only for damping
characteristics of the absorber too large for practical applications of the
absorber. This result cannot be obtained without a detailed investigation to
determine the range of system parameters for which such multi-steady-state
dynamic behavior can occur. Also, should the system be subjected to a
higher frequency disturbance than predicted, the response of the absorber

with constraints can be quite complicated as shown in the following.

In addition, all previous studies of the CPVA have been carried out based
on the assumption that the absorber does not come into contact with
motion-limiting stops. Hence the analysis is done without considering any
possible vibrO-impact behavior in the system, even though such stops (Often

referred to as snubbers) are employed in all practical implementations of the

CPVA.

The aim of this part of the study is to achieve a better understanding of
these issues and to determine the range of system parameters for which a

linear absorber with stops can function satisfactorily.

29

2.1- Equations of Motion and Assumptions

The full nonlinear equations of motion, equations (1.1.7) are rewritten
here as

[J+mR 2+mr2+2mRr cos¢]1,b ’ ' -mr (R cos¢+r)¢ ” +

+mRr¢ ' Qsintb—2mRr1/J’ a ' sin¢+c, 1b ' =T(r) (2.1.1a)

—mr(R cos¢+r )zl) ' ' +mr2¢ ' ’ +0, (15 ' +mer/J ’ 2sin<f>=0 (2.1.1b)

where the definition of all of the terms in equations (2.1.1) above can be
found following equations (1.1.7). Equations (2.1.1) are valid for the case

when the pendulum is not in contact with either of the constraints,
i. e. , I¢I<fl.

At impact, [43 I=fi, use is made of the total angular momentum of the
system. The total angular momentum of the carrier is in the direction
perpendicular to the plane of Figure (4), i. e. , out of the paper, and can be

written as
c =—Jt/)’ . (2.1.2a)
The vector angular momentum of the pendulum mass is given by
555me ,
with
6. =IR 1b ' sun/mop ' -¢ ' )sin(¢+¢)l ?+
41—311" COSWTOD' -¢> ’ )COS(¢+¢)] i— .

being the velocity of the pendulum mass and

30
“545+?
=[R costb+r cos(1.b+¢)l Iii-[R Sim/1+7 sin(7,b+¢)] 5- ,

being the position vector of the pendulum mass. The angular momentum of
the absorber is also in the [Edirection (outwards, perpendicular to the plane

of the paper) is given by

p =—m (R 2+r 2-I-2Rr cos¢)t/2 ’ +mr (R cos¢+r )qS ’ . (2.1.2b)

Consequently, the total angular momentum of the CPVA in the Ic— direction

can be written as:
H =Hc +Hp ,
or

=_[J+m (R 2+r2+2Rr cosrbhb ' +mr (R COS¢+T )¢ ' - (2-1-3)

Assuming that at impact the contact time is small enough that the law of
the conservation of the total angular momentum of the system can be

instantaneously applied, one can write,
H +=H -
or
Imp’ ++k2¢ ’ +=k11/J’ "+k245 ' ' (2.1.4)

where superscripts - and + refer to the times just prior and after impact,

respectively, and,
k 1=—[J +m (R 2+r2+2Rr cosfl)] , (2.1.5a)

k2=mr (R cosfl+r) . (2.1.5b)

31

Another relation describing the dynamics of impact can be determined by
the usual definition of the coefficient of restitution, e. This can be written

e =—-;———3— (2.1.6)

where v denotes absolute velocity and subscripts p, c and, n refer to the
points of contact on the pendulum, the carrier and, the direction normal to

the line Of impact, respectively.

To find the expressions for up“ and v6“, reference is made to Figure (5)

which shows the pendulum at impact with one of the constraints. For the
sake of clarity, only one of the constraints is shown. Rewriting equation
(1.1.5a) which is in reference to the (t-, ;) coordinate system of Figure (5),

yields
17, =[R 10' swamp ' —¢ , )sin(1b+¢)] 7+
+[-R 1b ’ cos¢+r (t/J ' —¢ ’ )cos(1b+<b)] j— . (2.1.7)

Equation (2.1.7) can be written in terms Of the (71' , t.) coordinate system as

[2" [=E [:3 ] (2.1.8a)

where R: is the so-called rotation matrix having the form,

' R=[:S“‘ 3,3115, (2.1.8b)

where E=1b+a and where the angle a can be observed from Figure (5) and
its explicit form is presented below. From equations (2.1.7) and (2.1.8) one

Obtains,

vpu=—R it! ' cosa+r(-1/) ’ +45 ’ )cosw-Oz) (2.1.9)

32

 

 

 

Figure 5 - Schematic View of a CPVA with the Pendulum at Impact

33

similarly, since

5,456

=(pw ' sine) 7—(pz/2' cosE) 7 .
fi=[§cosd)+r cos(z/)+¢)] z'_+[R sin¢+TSin(¢+¢)] ;

E ' =—¢' I?
then

”C“ =—p 1p I

(2.1.10a)
(2.1.10b)

(2.1.10c)

(2.1.11)

where p is the magnitude of E and the angle H can be observed from Figure

(5) as well.

Substitution of equations (2.1.9) and (2.1.11) into equation (2.1.6) yields:

6: (k3+p)1,b' ++k4¢ ' +
‘(ks‘i'PW’ ' --k4¢ ' —

 

7

where
k3=—R cosa—r cos(fi—a) ,

k4=r cos(,8-a) ,

rsinfl
__ , d
R +r cosfl ] an

a=tan‘l

 

p=[(R +r cosfl)2+r 2sin2925] ”2

(2.1.12)

(2.1.1321)

(2.1.13b)

(2.1.13c)

(2.1.13d)

Equations (2.1.4) and (2.1.12) can be solved simultaneously, for (b ' "' (or

1b ' +) in terms of the variables prior to impact, i.e. , 1b ’ ‘ and ¢ ’ ‘, to yield,

¢I+=k8¢I-+k9¢l- ,

where

(2.1.14)

34

_ klk5(l+e)
8" kaS—m, ’

 

9_ kzks-k1k4

 

k5=k3+p

(2.1.15a)

(2.1.15b)

(2.1.15c)

One can readily observe that equations (2.1.14) in conjunction with

equations (2.1.15) are complicated enough so to obscure the true dynamics of

the absorber and the carrier at impact. An alternative,

and more

enlightening, approach employs the geometrical properties of the system to

simplify the equations of impact obtained above. Referring to Figure (5) it

can be seen that
k3+p=—[R cosa+r cos(a—fl)]+P
=—(R cosa+r cosacosfi+rsin015inm+P

Considering

R +r c036

cosaa ,
p

 

. rsinfl
sma— ,

 

and equation (2.1.l3d), one obtains
k3+p=0 ,

which renders equation (2.1.12) as

 

c=_¢r+ ,
¢”
01'
¢I+=_e¢l—

which is a familiar equation of impact for a single mass system.

(2.1.16)

This simple

35

result is a natural consequence of the fact that 45 is a relative coordinate.

Returning to the freeflight (i.e. , non-impacting) mode of the system, the
following assumptions are then made to simplify the free-flight equations of
motion (2.1.1) (see Figure 4):
l. The carrier runs at nearly constant speed 0 with a small time-dependent
variation, i.e. ,

1,!)(7)=flr+0(r) ,
2. The carrier damping is assumed negligible,
Cc=0

3. ¢ is small enough so that sin¢-¢, cos¢-1 are valid approximations.
4. Second and higher order terms of 0, 43 and their derivatives can be
neglected (i.e. , linearize the free-flight equations of motion).

5. The applied torque is sinusoidal in time.

Incorporation of the above assumptins renders the equations of motion

(2.1.1) as,
[J+m (R +r )2]0 ’ ’ —mr(R +r)¢ ’ ' =T(T)=Tlcoswr (2.1.17a)
—mr (R +r )0 ' ’ +mr2¢ "+0? (6 ' +mRrfl2¢=0 (2.1.17b)

Eliminating the 9’ ’ variable between equations (2.1.17a) and (2.1.17b)

one obtains,
er2¢ ’ ’ +0? [J+m (R +r )2]¢ ' +

+mRr 02[J+m (R +r )2]¢=mr (R +r)T1cosz (2.1.18)

Rescaling equation (2.1.18) by redefining time and angular displacement

as (see [28] ),

/2
t:— EJ—tjz—[J+m((R +r)2]I 7' (2.1.19a)

 

36

(2.1.19b)

equation (2.1.18) can be put into the following nondimensional form:
Ei+2k13 +3 =Kcosnt (2.1.20a)

where

x=c,

 

/2
fig—[Hm (R +r )2] I

is the nondimensional damping ratio,

_ (R '1" )T1
R 02,3[J+m (R +r )2]

 

is the nondimensional excitation amplitude,

1

RR2

2
=w T[J+m (R +1?)

 

is the nondimensional excitation frequency and, an over-dot denotes

differentiation with respect to the rescaled, dimensionless time, t.

As stated previously, equation (2.1.20a) is valid when the pendulum is not
in contact with the constraints, i.e. , Ia: I<1. For impacts, ( Ia: I=l),
equation (2.1.16) is also rescaled to yield equation (2.1.20b) as

+

:i: =—e:i:_ , Ix |=1 . (2.1.20b)

It should be noted that the rescaling (equations 2.1.19) that led to
equations (2.1.20) renders the anti-resonance frequency (equation 1.2.9) as

l

2

,7AR= 1+%(R +r)?) . (2.1.21)

 

This is the nondimensional frequency at which 6:0 for the undamped, linear

response, {.6 , it is the desired operating frequency.

37

The original equations of motion (2.1.1) were in coupled form. Use of the
assumptions (1) through (5) with the nonlinear equations of motion resulted
in a set of linear equations (2.1.17) describing the motion of the pendulum
between the constraints which were easily decoupled. In addition, the
impact equations can be expressed in an uncoupled form (equation 2.1.16)
and consequently equations (2.1.20a) and (2.1.20b) fully describe the motion
of the pendulum for a given set of initial conditions and parameter
values. The dynamics of the carrier can then be determined using equations
(2.1.17a) or (2.1.17b). In particular, equation (2.1.17a) yields,

mr(R +r )d) ’ ' +Tlsinwr
J+m (R +r)2

0 ” (2.1.22a)

 

and successive integrations with respect to time, 7', results in an expression

for 9’ and 6 as,

T
mr (R +r )¢ ’ —-ZJ-l—coswr

6’: (2 122b)
J-l-m(R+r)2 . .

 

and

(R + >45— T‘ °
mr r —smwr
52

6’: . (2.1.22c)
J +m (R +r )2

 

The constants of integration in equations (2.1.22b) and (2.1.22c) are taken to
be zero without any loss of generality since the addition of a nonzero
constant to equation (2.1.22b) would result in a term like or (c a nonzero
constant) whereas 9 in assumption (1) is assumed to be a small variation
about the steady-state rotation, and this component of w is already
accounted for by (I in 1/)=QT+0. Also, the addition of a nonzero constant to
equation (2.1.22c) would indicate a trivial phase shift in the value of 6 and a

simple change of variables would result in the same expression as (2.1.22c).

38

Equation (2.1.20a) can be solved for a set of initial conditions. Such
solutions are valid only for excitation amplitudes which result in amplitudes
of the response (i.e. , mm”) less than unity. In other words, any solution of
equation (2.1.20a) which yields zmax>1 is considered to be invalid since such
condition implies that the absorber is passing through the rigid
constraints. Given a set of initial conditions ($0,to,i:0), the steady-state
amplitude of the linear free-flight equations, X, obtained from the solution

of equation (2.1.20a), can be written as

X— K
[(l_n)2+4>\2n2]1/2

 

Thus, for this solution to be valid, the torque intensity, K must satisfy the

following inequality:
K <19,=[(1-17)2+4Vv?211/2

since Kc, renders X as unity. For K >Kc, the desired steady-state solution

will not exist.

39

2.2- Impacting, Periodic Response

2.2.1- Methods of Analysis

In the following, equations are derived for the existence and stability of
certain periodic motions. The type of impacting motions described here are
the symmetric, double-impact periodic motions (SDIP, for short) which have
one impact for everyehalf cycle of the excitation. A brief discussion will also
be presented on the existence of pairs of anti-symmetric double impact
periodic motions which are brought forth as a consequence of the breakdown
of the stability of certain SDIP motions. This will be discussed towards the

end of the chapter.

To study the existence and stability of SDIP motions, it is convenient to

write equations (2.1.20a) in first order form as
i=1]
' =—)\y +Kcos17t (2.2.1)
i=1
where the three variables (x,y,t) determine the state of the system for those
solutions restricted to Ix I<+l in the three-dimensional phase
space. Reference is made to Figure (6), which illustrates the phase trajectory
of a double-impact motion in the (a: , 3]) plane. Starting at point A
(corresponding to the absorber coming in contact with the constraint at

x=+1 with positive velocity), the impact equation (2.1.20b) is used to obtain

the time and velocity at point B in Figure (6), i.e. ,

X=y

 

 

 

 

 

 

 

 

Figure 6 - Phase Trajectory of a Double-Impact Motion

41

3’3 =—€3IA (2.2.2a)
g=q aazm

(in Figure 6 the trajectory from A to B is actually exactly on z=+l, but is
shown as is for clarity. Likewise for the C to D trajectory at z=—l). The
motion from point B to point C is governed by the free-flight equations
(2.2.1). The solution of (2.2.1) with initial conditions corresponding to point
B is valid until x=—1 again. Since double-impact orbits are of interest
here, it is then assumed that the next impact after point B occurs at point
C with a:=—l as shown in Figure (6). Excluded, for the present, are orbits
that leave the rigid constraints at x-+l and next impact at x=+l
again. For the motion of interest, the time and velocity at point B
(t3, 3<0) uniquely, although implicitly, determine the time and velocity at

point C (tc,yc <0) through the expressions,

$(tc;+1,t3,y3 )=—1 , (2.2.3a)

and,

i (tc;+l,t31y3)=yc (2-2-3b)

where x(t ;+1,t3,y3) is the explicit solution of equation (2.1.20a) with initial
conditions corresponding to the state of the absorber at point B, i.e. ,
x=+1, tatB, y-ByB. Equation (2.2.3a) can be inverted to yield to. as a

function of 03,313), i.e. ,

tc=tc(ta .113) , (2-2-4)

where to is the first root of equation (2.2.3a) for which
tC>tB. Substitution of this equation into equation (2.2.3b) yields an

expression for 310 as a function of (t3,y3 ), i.e. ,

3Ic=3/o(t3,3/3) - (2.2.4b)

42

By referring to Figure (6), the motion continues to point D via the

impact rule:

310 =—ey0 (2.2.5a)
t0 =tC . (2.2.5b)
The free-flight motion from point D to point E is again governed by
equation (2.1.20a), where the time and velocity at E are uniquely

determined from those at D, same as the case of the motion from point B

to point 0. Hence, the following conditions can be written:
x(tE;—1,tD ,yD )=+1 (2.2.6a)
and
93(‘1: 3-11tp 1w) )=!lc - (2-2-6b)
and when equation (2.2.6a) is inverted, one obtains,
tE =t1‘.‘(tD 1310) (23-73)

where t3 is the first root of equation (2.2.6a) for which t3 >tD. Then, from

equations (2.2.2a) and (2.2.7a) we have 315 as follows:
yea/£00,310) - (2-2-7b)

In reference to equations (2.2.2) through (2.2.7), one can observe that a
two-impact cycle has been completed taking the motion from one impact at
z=+1 to the next similar impact at $=+l. From these considerations it is
seen that by making use of the equations (2.2.2), (2.2.4), (2.2.5) and (2.2.7)
the time and velocity at point A in the cycle uniquely determines those at

point E, i.e. ,
2:20,. a.) , (2.2.82)

315 =yE(tA ,yA) . (2.2.8b)

43

In this sense each excursion from x=+l to z=—l and back is reduced to
the simple recursion relation of equation (2.2.8). This recursion is a natural
consequence of equations (2.1.20a) and (2.1.20b) and it will be used in the
following to study the motions of the absorber. One should note that
explicit expressions for above relations such as equations (2.2.8) are not
available in explicit form since such a task would imply inverting several
transcendental functions, making the process impossible. In spite of this,
explicit solutions for the existence of SDIP motions are obtained in the
following through certain conditions which govern the nature of the SDIP

motions.

The idea of studying dynamical systems using recursion relations, or
maps, has its origins in the works of Poincare‘ [27]. To this effect, the
mapping given by equations (2.2.8) will be referred to as the Poincare‘ Map
for the system being studied. By considering the phase space (z,y,t) with
z_<_l, it is seen that this method determines how points with z==+l, y>0
are eventually mapped back to tea-+1, 31 >0 under the governing equations
of motion. Formalizing this, one can define a Poincare‘Section for this case

as

E={(z,y,t) : x=+1, 31 >0} , (2.2.9)
and the attendant Poincarc‘ Map, P, which is a rule taking points in )3 back
into E, i.e. ,

P=Z-+E ,or equivalently (t,-+l , y,-+1)=P (t; , y,-) , (2.2.10)
where

(t,- , 31,- )62

44

Points in the set 2 correspond to those states in which the absorber is
just coming into contact with the constraint at :c=+1. For instance, points
A and E in Figure (6) are in Z . The mapping P is equivalent to equations
(2.2.8) for those points in E which result in excursions like the one shown in
Figure (6). It is noteworthy to mention that some points in 2 may be
mapped back to 2 without any encounter at x=-l, in which case rules
other than those given by equations (2.2.8) must be used. In addition, some
points in 2 may be mapped onto [2: [=1, y=0 in which case discontinuities

in P may arise (see [28,29] for details).

Periodic motions of the absorber mass may be studied using P as
follows. Each iterate of P corresponds to an impact of the absorber mass
with the constraint at x=+l and relates the condition (time and velocity) at
the previous impact to those of the subsequent one. A motion which repeats

itself after lc impacts at :c=+1 necessarily satisfies the condition

27m

1? ,F)=P *(7. 37') (2.2.11)

(7+

 

where P " indicates that P has been applied k times (i. e. , defining
PJ.+1( . )=P (P j( . )), P°( . )=( . ) is sufficient to define P"( . ) ). Equation

(2.2.11) corresponds to a 2k impact motion of period 11 in which the

 

absorber mass repeats its motion after k impacts at :r=+1, during which n
cycles of the forcing pass. Such a motion is referred to as a

subharmonic of order n. The point (t-J) is referred to as a periodic point

of P.

Such a periodic motion may be either stable or unstable. The stability of
periodic points can be investigated by tracing in time the dynamics of small
perturbations on the initial conditions for the periodic point (2,37). This

procedure can be discretized by observing the effects of perturbations using

45

the Poincare map. Consider a small disturbance (E,V), where
(t,y)=(;+E,§/'+V) and [E],]V]<<1, imposed on the periodic point. Its
state is observed after subsequent returns to the Poincaré section, 2. Since

local stability is of interest here, linearizing about (£37) one obtains,

(€,-+1,V,'+1)=DP(6,314) , [E [, [l/ I<<l (2.2.12)

where (E,u) is the perturbation and DP is the first derivative of the Poincaré
map, P, evaluated at the periodic point (7,37). Using the notation employed

in Figure (6), the matrix DP can be written as:

 

 

lag, at, ]
atA 831A
DP - 83,3 83,3 (2.2.133)
[31,, 8“ iii)
and the above matrix, for convenience, is denoted as
at ,
DP= M (2.2.13b)
30A 1301) t-ji)

 

Hence, the problem of stability of periodic points is reduced to the study
of the eigenvalues of the DP matrix. If both eigenvalues of the DP have
moduli less than one then (2237) is stable. If any of the eigenvalues has a
modulus greater than one, then (t-J) is unstable (see [26, 31, 32] for
example). As system parameters are varied, the periodic point changes
continuously as do its associated eigenvalues, )‘1 and X2. Local Bifurcations
occur as eigenvalues pass through the unit circle in the complex plane, i. e. ,
when [X5 [=1, i=-l,2 [26]. It is shown in the following that the DP matrix
has a determinant with magnitude less than one for motions of interest and
thus only X=il bifurcations are possible. No Hopf bifurcations occur in
which X1 and X2 pass through the unit circle as a complex conjugate

pair. Period doubling, or flip bifurcations occur for >\,-=—1. Bifurcations

46

corresponding to >\,'=+1, saddlenode and pitchfork bifurcations, also

occur. These are discussed in the following.

47

2.2.2- Symmetric, Double-Impact Motions- Existence

Necessary conditions for the existence of an SDIP motion can be obtained
by matching end conditions which require that a periodic, symmetric,
double-impact motion exist. The conditions to be solved use the known
linear solution for the motion of the absorber during the free-flight motion

and the symmetry of a SDIP motion and are given by

x(t-=--"7;-‘—;+1,'t',—ei)=—1 (2.2.14a)
2(t-+-7%n—;+l,t_,—e37)=—37 (2.2.14b)
]x(t ;+1,t-,—ei/‘) |< 1 for “(RI-+101) (2.2.14c)

where :1: is the explicit solution of equation (2.1.20a). For a SDIP motion
with a corresponding fixed point (t-J) on the Poincare?

section, 2, equation (2.2.14a) states that starting at x=+l and after a time—

lapse equal to exactly % (i. e. , half of the n number of periods of the

excitation), the absorber should be at z=—1 at which point the velocity, by
equation (2.2.l4b), must equal —37. Condition (2.2.14c) merely states that
the mathematical solution of the displacement of the absorber, 1', should
remain within the physical boundaries of the model, i. e., the absorber

cannot penetrate into the constraints.

It is worth mentioning that for very low frequencies of excitation, n, say
n<0.5, the results corresponding to solutions of conditions (2.2.14) are in
fact what are called penetrating motions which are consistent with the

mathematical modeling of the system but are physically not

48

realizable. These solutions satisfy conditions (2.2.14a) and (2.2.14b) but

violate (2.2.14c).

Also, note that due to the symmetric nature of the SDIP motion only odd
orders of the subharmonics can exist. The reason for this is that the
excitation that takes the absorber from x=+1 to z=—l should be equal and
opposite in direction to the one that brings the pendulum back to x=+1,
implying n=2i +1 (i integer) periods of the forcing are necessary to sustain

a SDIP motion for one full cycle.

For a typical set of initial conditions ($0,to,yo), such a solution can be
written, for the underdamped case, X< 1.0, as
—).(t-to

e )
a: (t :20, to,yo)=——A— [[xo—C’ cos(17t0—c—r)] Acos(A(t —t0))+

+AcosA(t —t0)+()\a:o+yo—C kcos(17to—Ei)+

 

 

+Cnsin(nto-5'))sin(A(t-—to)) ]+C'cos(17t —5) (2.2.15)
where
K
C: , 2.2.16
[(1-77)2 +4)\2,72]1 /2 ( a)
5=ta.n-l [ 2X02 ] , (2.2.16b)
1—77

A=V 1->\2 . (2.2.16c)

Equations (2.2.14a,b) can be solved for (ti?) as follows. Substitution of
equation (2.2.15) into (2.2.14a) yields

—i— [(l—Ccn)Ac-+()\—C kc "—837+C nsfl)§]—-Cc n=—1 (2.2.17a)

where

49

 

 

—Xnn
E =e ’7 (2.2.18a)
c-=cos Arm , s-=sin (2.2.18b)
cn=cos(170'—® , sn=sin(no—57) (2.2.18c)
- 271'

Differentiation of equation (2.2.15) with respect to time, t, yields the
expression for i:(t;zo,to,yo) as:

”Mt-to)
$(t ;Io,to,yo)= e A [[yo+CnASIn(flto—E)]AOOS(A(t —to))+

 

H—xo—xyowcosma-o—cmanna-wanna-to)>)—
—C’nsin(17t—3) (2.2.18e)
Application of condition (2.2.14b) to equation (2.2.18c) results in equation
(2.2.l7b) as
E .. _ _ _
I [[—ey +0178 n]Ac +[-1+>\ey +C'c n—C >073 n]s]+
+Cns,=—37 (2.2.17b)

Any solution (t-,y—) of equations (2.2.17), which is a fixed point of the
Poincare‘ map, P“, corresponds to a SDIP motion in the phase space if
condition (2.2.14c) holds. Equation (2.2.17) can be solved for (£37) by

solving for c" and 8,7, which appear linearly in these equations to yield:

 

cn=cos(n0—5)=%)iy-+IE (2.2.19a)
1
. J2 _
sn=sm(no—a=CnJ y (2.21%)
1

where

50

J,=-A(1+E'2+2c‘E'1) (2.2.202)
and
J2=AE"1(1—e)?:'+>\E—l(1+e)§'+AE'2—Ae . (2.2.20b)

Equations (2.2.19a) and (2.2.19b) can be squared and added together using
c3+sg =1 to yield the following quadratic equation which can be used to

solve for if in terms of the system parameters:

 

filers-Q, J22 +2]1+e]s—

 

 

 

 

 

 

, y+—-—=O 2.2.21a
(E07,)? (0171,)2 130%,02 ( )
The solutions for 37 are:
J1202 _[1+€)3iA1/2
_ 1,0213
y- ( 2)” J2 (2.2.21b)
1+e 3 J2
E2 ' 772
where
(1+e)9§‘2+122 (1— 1 )
2 2 2
A: E ’7 0 (2.2.22)

(110)2

The corresponding phase at impact for this motion can be obtained once

if is known using the definitions of c” and 3,, given in equations (2.2.19).

2.2.3- Symmetric, Double-Impact Motions- Stability

To obtain the stability characteristics of the fixed point (237), the first
derivative of the Poincare map, DP, must be determined. Since explicit,
closed-form expressions for the Poincaré map are not available, the following

implicit differentiation procedure is used to obtain an expression for the

51

matrix DP (see [28], for example).

Using the chain rule and the notation defined in equation (2.2.13), one

 

 

 

can write
8(tEiyE)
DP: Bums/1)];
8(tE’yE) a(tDvyD) 8009310) (tBin)
a 5(T073/D) 3(tc,yc)][a(t3.y3)“QM/1,“ )] (2.2.23)

 

 

By referring to equations (2.2.2) and (2.2.5) it is easily seen that

303,303) 300,310) 1 O
0 _ . (2.2.24)
aT—tx WA) 5—00 .yc) '3
The remaining two matrices are computed using implicit differentiation as

follows. Differentiating equation (2.2.3a) with respect to t3 one obtains

atc e-XUO-tfl)
BtB = A310 [3’8 A°°S(A( to "43 ))+(—1+C cos(nt3 -5[)—

 

—)\y3 -2 C anin(nt3 —E)—C' n2cos(nt3 —5))sin(A(tC -t3 ))] (2.2.25a)

Differentiating equation (2.2.3a) with respect to 313 yields

atc _e-x(t0"t8) .
ayB = Ayc Slfl(A(tC—t8) (2.2.2513)

 

Noting that,

aye _ 33/0 3‘0 4 dye
613 _ atc 'atB ' dtB

 

(2.2.25c)

and

Bye _ 33/0 3‘0 _|_ dye

. T , 2.2.25d
53/3 ate ByB dye ( )

 

the unknown terms on the right-hand-sides of equations (2.2.25c) and

52

(2.2.25d) can be computed using equation (2.2.3b) as follows:

 

 

33/0 ..
Btc =$(tc;+1,t3,y3)
=Kcosntc —2>\yC-—1 , (2.2.25c)
and
dyC e-k(t0_t3)
"' [—)\SII1(A(tC _tB ))+XCOS(A(tC —t8 ))] , (2.2.25f)
dyB A
and

dt - A [[9213 +20knsin(nt3 —o:_)+1_
B

 

—CCOS(7]tB -E)+C 772COS(77tB —C—Y)] ACOS(A( t0 —t8 ))+
+(—>\+C Xcos(77t3 —_)+(1—2)\2)y3 +(1—2)\supu 2)C’ nsin(77t3 -—m-

—C’ nsin(17tB —E)—C X112cos(77t3 —5))sin(A( t0 -tB ))] . (2.2.25g)

Furthermore, differentiation of equation (2.2.6a) with respect to t0 and

310 and differentiation of equation (2.2.6b) with respect to lg yields:

BtE e-Mlg-to)
BtD - A [3’0 A°°S(A( tE ‘tp ))+(l+Ccos(17tD —®_

 

—2 C anin(17t3 —3)—C n2cos(17tD —5)yD )xsin(A(tE "tD ))] (2.2.26a)
and

at, —e‘*(‘€“vlsin(A(tE—t,,)

3w) A1115

 

(2.2.26b)

respectively. In addition, we have

31/3
at,

 

=Ei(tE;_11tD vyD)

53

 

=K cosntE —2>\yE —1 . (2.2.26c)
employing
831;; _6yg . ate + dyE (2.2.26d)
at!) at); 61,, dtp
and
By, 33/13 at1? ,dyE , (2.2.26c)

 

33(1) 3‘13 .530) I dyo

equations (2.2.6a) and (2.2.6b) can be utilized to compute the remaining
unknown terms on the right-hand-sides of equations (2.2.26d) and (2.2.26c)

as

d "Mts-to)

 

dtD A
—C' cos(r]tD ——)+C n2cos(77tD —_))Acos(A(tE -tD ))+()\+
+311) (1—2X2)+(1—172)C>\cos(ntp —5)—2)\23in(ntp -—_—))sin(A( t5 —tD ))] (2.2.26f)

and

(13,5. e-Mts'tol
- [Acos(A(tD —tE))-)ssin(A(tD 43)] . (2.2.26g)
dyD A

 

Substitution of equations (2.2.24) through (2.2.26) into equation (2.2.23)

yields the final expression for matrix DP as,

DP‘ldq-l . i,j=1.2 , (2.2.27)

where
du=plp2-ep3p4 , (2.2.28a)
dl2=-ep1p5+62p3p6 . (2.2.28b)

(121:? 2107—5194198 (22380)

54

and
d22=-€P5P 7+€2P 6P8 9 (23-28(1)

where

p1.1%[3—Ccn§'+2C)\nsfl§+eg7AE-+C1726QE—eikg] ,
—E-c-x—2cx--A‘ 2
p2_A—37[—s+ ens+ eys- nsns-ey c-cncné‘Tl ,

P 3=_—3

Ay
_ E _ _
p 4=(1—2)\y —Kcos17t)p 2+1—[-)\s +C Xe "s —
—(1—2>\2)e§/?—237Ax3+(1-2>.2)0nsfl§+2cAxnsnc‘+
+AE—C 17803—0 Aan—CXnQCflF+C 11112ch ,

P “-Ei-P
5 AF 3 9

p6=(1+2)\§'—Kcosnt)p3+—i—(—>\§+AE’) ,
p7=(Kcosnt —2x§—1)p ,+%(x:—cxc,§+ey(1—2x2)s—+
+2 e37 AN:— —(1—2)\2)C 178,78-—2 C AknsnE—Ac-+
+Cnsflg+Cnsns—+CAcne-+Ckn26nE—0An2cflF) ,
and
_ E _ _
p 8=(K cosnt —2)\y —1)p3+K-(-)\s +Ac )

Consequently, the expressions for the eigenvalues of the DP matrix can be

written as

1
x12"; [(d11+d22)il(d11+d22)2+4(d11+d22)(d11d22+d12d21)ll/2i - (2-2-29)

55

These calculations involve no approximations in the mathematics.

2.3- Results

2.3.1- Frequency Response

Figure (7) is a typical plot of § vs. 1) depicting the stable and unstable
response branches for n=1, that is, motions with the same period as the
excitation, and Figure (8) depicts a similar plot for n=1 and the
subharmonic orders n=3 and 5. Superimposed on Figure (7) is the non-
impacting, or linear branch of the response curve. This branch depicts the
maximum velocity (at I=0) of motions for which there are no impacts and
the absorber simply oscillates between the two constraints. Figure (9a),
(9b), and (9c) depict the variation of if as system parameters 1?, 3: and, e,
i.e., the excitation amplitude, the damping ratio and, the coefficient of
restitution, respectively, are varied. Figure (10) illustrates the dynamic
behavior of the absorber and the carrier for a typical stable impacting

motion, specifically point A in Figure (7).

By referring to Figure (7) and moving horizontally from the right towards
decreasing values of the frequency, 77, one observes the following. For
excitation frequencies around n=2.25, the only possible motion is the linear
one (or possibly the n=3 subharmonic) up to a point where stable and
unstable SDIP motions with n=1 motions emerge in a saddle-node
bifurcation. This point corresponds to the excitation frequency for which

real roots appear for the quadratic equation in terms of 37', that is,
A=0

where A is as defined in equation (2.2.22). Equivalently, in terms of the

driving amplitude, these saddle-node bifurcations occur at

 

 

56

 

K—1.0
2.0-——-
.1 6=1.0
I >.=o.o
J
1.5- n=1
4 [ .
'l l °.
1 O
1. 0—-— [ '
:1 l """" LEAR
0.5-3— [ lNSTABLE
.1 .
:1 [in I“ '. n
.J 1 1 1 1 1' 1 1 1 1
I I I I I I I r] I I I I ‘ I I I II I I I I [ I I I I [ I I I I—l \

0.30 0.73 1.00 1.23 1.30 1.73 2.00 2.13

Figure 7 - Frequency Response of the Absorber

57

e =0.90

 
   

 

stable (=0,)
_ _ _ unstable

 

,7 l ‘ l
[1[I1 1[1IIij 1111]1111|1111[11II[
°°°7n11 2.5 5.0 7.5 10.0 12.5 15.0

forcing frequency , r)

Figure 8 - Frequency Response of the Absorber; Subharmonic Orders
and the Primary Response

58

 

 

‘ 0 __ 6 =0.90
_ y l _ stable )=0-0
8 __ _ _ unstable ”=1
5 __
4 __
2 ._ _.
0
0

 

forcing frequency , 7

'Figure 9a - Variation of the Response of the Absorber with Respect
to the Excitation Amplitude

59

 

 

K=1.0
4...—
-. g. e=0.90
j ___stable n=, /
J __-unstable , I
5-...
2-..
1..—
1
‘l
- 0.5
1/
0 IIIIIIW—(IVIIIII[IIIF[IIIIIIIII—]

 

°.7nJ 2 6

forcing frequency. 7

Figure 9b - Variation of the Response of the Absorber with Respect
to the Absorber Damping

K=1.0
T5 #10151:

- - — “table X=0.0

Lu
1
l

'11]!

N

 

-.
Illllllll
l

C.

 

 

IIAI[IIIIIIIIII[11.

1
0 71 2 3 L

I11.

0'!—

forcing frequency , 7

Figure 9c - Variation of the Response of the Absorber with Respect to
the Coefficient of Restitution

61

e =0.10
)x=0.0
1
n =1
-1
l
c. x

Figure 10a - Phase Trajectory of the Absorber Depicting an SDIP
Motion

62

q '6 K=1.0
d C #110

o o 05-— >\=0.0

 

Figure 10b - Phase Trajectory of the Carrier Depicting an SDIP Motion

63

 

(1-772)2+4>\2772 JV?

(”manage

As the frequency is decreased further, the steady-state amplitude of the
linear branch, ymax, increases to a point where the motion just touches the
constraints at [2: [=1. This occurs in Figure (7) at a frequency denoted by
117‘' at which the linear motion with annual coexists along with an unstable
SDIP motion having i7 =0. That these two motions are in fact coincident at
n=n' can be shown as follows. For an SDIP motion with n=l to satisfy
37-0, equation (2.2.21b) yields, using the minus branch,

A1/2+ (1+8 )8-=0
1,02E

or, equivalently,
—(l+e )s_+J102E Al/2=0

If the steady-state amplitude of the linear branch is unity then from
equation (2.2.15) it follows that C'=1. It can be demonstrated that the
above condition holds only for C=1 by using equations (2.2.20a) and
‘ (2.2.22) for J 1 and A, respectively, as follows:

J, 0213 Al/Q- [CE [E'2(l+e )252+11‘2Jr§ (1--C‘2)l‘/2 [0-1

=—(l+e )3—

Any further decrease of the frequency results in the merging and
annihilation of the stable linear and the unstable SDIP motions in a

degenerate saddle-node bifurcation at n=17‘.

A point of interest about the transformation of linear motions into
impacting SDIP orbits is that as the amplitude of the oscillation of the

absorber is increased in the linear (non-impacting) range, eventually the

64

amplitude, I2: I, will equal 1.0, i. e. , the motion just grazes the stops while
still retaining its linear character. An attempt to further increase the
amplitude of the linear motion of the absorber by changing a parameter will
destroy the non-impacting motion. .However, contrary to what one might
intuitively expect, the transformation from linear to nonlinear orbits will not
be a smooth one and there will be jump discontinuity (see Figure 7). In
other words, a stable SDIP motion with i=6, 0<€<< 1, will not emerge
from the linear motion as n is decreased past 17'. This occurs since the
stable linear motion with zmax=l and an unstable SDIP motion with
37-0 annihilate each other in a saddle-node bifurcation. The overall result is
that as parameters are varied, a linear motion does not smoothly transform

into an SDIP motion.

It is possible for the stable SDIP response to become unstable in a
symmetry-breaking pitchfork bifurcation which results in an anti-symmetric
pair of periodic motions. These motions can, as 17 is decreased further,
undergo period doubling bifurcations which result in chaotic dynamics for

the system. This is described in more detail towards the end of the chapter.

2.3.2- The Response at the Anti-resonance

An extensive study was carried out to determine the possibility of
coexistence of impacting and non-impacting motions at the anti-resonance
frequency. To this effect, a fine grid of points in the parameter (K ,X,e)
space was searched to detect the occurrence of such coexistence. Figure (11)
is a plot showing the coexistence points for different values of K in the (>\,C)

space.

 

e K.i.3 214% 1515] 1:; '
0.0q l l l 1 X]

 

 

l ' I ' l r ' V '
0.0 0.2 0.4 Oifi 0i8 Tie

Figure 11 - Coexistence of Linear and Nonlinear Motions at the
Anti-resonance Frequency

66

The occurrence of coexistence at the anti-resonance frequency is crucial
from a design point of view since the dynamic behavior of the pendulum,
and consequently the carrier, depend on the intial conditions and slight
disturbances could take the motion from non-impacting to impacting
motions and vice versa. As can be observed from Figure (11), coexistence at
the anti-resonance frequency occurs only at unreasonably large values of the
damping ratio, )\>0.7. This places the issue outside the realm of practical
applications. The upper bound for K can be obtained by requiring that a
linear, non-impacting motion exist at the anti-resonance frequency, i.e. ,
031 at 17217,”. Employing the definition of C from equation (2.2.16a),

this implies that the excitation amplitude should be limited as follows:
2 2 2 2 /2
KSKcr(>‘)= (l-nAR) +4>\ 1IAR

for the existence of a linear motion. The function Kc, (k) is bounded above

by 0.50 over the range of realistic damping ratios, 0_<_>\SO.25.

Figure (12) depicts a case revealing the coexistence of all three types of
motions, namely, the stable and unstable SDIP motions and the linear, non-

impacting motion at X=0.80, as a function of the coefficient of restitution, e .

The effect of detuning can also be studied in this context. Detuning refers
to the presence of disturbances with slightly different excitation frequencies
than the frequency for which the absorber is designed. The primary focus of
the study of the detuning in this report is to investigate the sensitivity of
the occurrence of coexistence at the operating frequency. In other words, if
the excitation frequency is not exactly at the anti-resonance frequency, 17,“; ,

will this significantly affect the likelihood of coexistence? Figures (13) and

67

‘00-—

 

 

 

" mimeth g
0.0-.— $1.3
1.0.72
11.:
0.4
0.!

side at; '

 

.0. .11 0.4 0.. 0..

i - esonatce
Fi re 12 - The Types of Motions Coexisting at the Anti r
9‘ Frequency

1.0——-

 

 

°-a--

‘ K-10§ 12$ 14 $15 17:
°-4-—
0.2- - i j g i
0.0 , ] 1 [_ 1; l 1 l 1

l
0.0 0.2 0.4 . Oifi 0,3
damning ratio , )1

- Figure 13a - Coexistence Corresponding to +51 Deviation from the
Anti-resonance Frequency

69

 

 

e 5 5 : : i 3
K=12§13§ 14% 15% 18? IT?

5___ '

‘__

.2- -

.0 . I :% ‘ F I F Er§—%

 

0.0 0.2 0.4 0.6 0.8 I.
damping ratio . A

Figure 13b - Coexistence Corresponding to -5% Deviation from the
Anti-resonance Frequency

70

K=0.8§0.9;10; 1.2% 15 14: 1.5 15 171819

 

 

0.0 . ’- r',} .1555
0.0 0.2 0.4 0.5 0.3

damping ratio , A

Figure 14a - Coexistence Corresponding to +10% Deviation from the
Anti-resonance Frequency

71

0.0 L I I 1
0.0 0.2 0.4 0.5 050

damping _ ratio , .\

 

Figure Mb - Coexistence Corresponding to 40% Deviation from the
Anti-resonance Frequency

7.4...

72

(14), which are similar to Figure (11), were obtained in the following
manner. The operating excitation frequency in Figure (13) has a :l:5%
variation from the anti-resonance frequency and that of Figure (14) has a
i10% variation. As can be observed from these Figures, the effect of
detuning has a slight tendency to increase the possibility of
coexistence. However, it still does not lead to coexistence at anti-resonance
frequency for reasonable values of the damping ratio. This lack of
sensitivity implies that tuned CPVA systems should have no problem with

steady-state impacting motions.

73

2.4- Other Nonlinear Responses and Chaotic Motions

As mentioned earlier, in addition to the saddle-node bifurcations that
result in the jump phenomena (see Figures 7 to 9), another form of
bifurcation can occur which results in a change of stability of the SDIP
response. For instance, for K =2.0, e=0.95 and X=0.01, the frequency
response is as shown in Figure (15) and it is observed that as the excitation
frequency is decreased from n=2.0 the upper SDIP branch becomes unstable
at about 1721.60; this is indicated as point B in Figure (15). For excitation
frequencies between n=l.60 and n=l.0, where the latter frequency is close to
the anti-resonance frequency (17,”; =0.89), there exists no stable SDIP or
linear, non-impacting preioidic motions. However, a stability study reveals
that as the excitation frequency is decreased past point B, an eigenvalue of
the DP matrix passes out of the unit circle through +1 which renders the
response unstable. At this frequency a super-critical pitchfork bifurcation
takes place in which the stable SDIP motion becomes unstable and a pair of
stable, anti-symmetric, double-impact, periodic orbits are generated. These
motions then can each undergo a succession of period doubling, or flip,
bifurcations, which eventually result in nonperiodic, or chaotic motions. See
Shaw [28] for a more detailed bifurcation analysis for a similar, but simpler

system.

Figure (16) depicts the phase trajectory of an unstable SDIP motion along
with a pair of stable anti-symmetric period-one double-impact motions and
Figure (17 ) illustrates their corresponding time responses. Plots showing the
succession of period-doublings of similar anti-symmetric motions can be

found in Shaw [28]. Figure (18a) is a Poincare map indicating the existence

74

Sdflx

mad" at

9N" t<

noncomplx 05 do omCOQmom augment“... mp 95m:

 

mqmfirnmzz | l I l

 

Bamfikm l

 

la.

 

75

 

 

 

 

 

 

K=2.0
4__ STABLE 6:035
5
fl ———— UNSTABLE x=0.01
2....
.4
0....
_2__
a:
_4 lll[ITIIIIiillliiITETlilllilT]
-|.5 -l.0 -O.5 0.0 0.5 1.0 1.5

Figure 16 - Phase Trajectory of the Absorber

76

0— 9590. Co c0302 dam 03395 of co omcoamom we: 1 a: 95m:

ow on . cm 3 o

 

 

 

77

UmoETo—osoo oce..uofoa oucuoEE>m12c< 033m 9: ~o omcoamoz 0.5... 1 a: mesa:

 

~O.OHK

mad" 9

o.m.ll.v~

 

 

 

 

 

 

 

 

 

 

 

 

I

 

78

5:02
Hoods—103:8 0.501839; umcuoEEzmrzc< 033m of he omcoamom 9:: 1 u: 9.3mm”.

 

 

o v o m o N o _ o
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ N l
_ _ _ _ 1
g I
ll F l
I... o
F
21": 1
—" .2 ’ Tl
Sdflx _ ll —
no.0" e l
Qantas - a r.

 

79

of a strange attractor which corresponds to a chaotic motion; this occurs
after the completion of the period-doubling sequence. Figure (18b) depicts a
portion of a typical time response of the absorber mass for an initial
condition within the strange attractor. It is noteworthy to mention that the
succession of period doubling bifurcations leading to chaos occurred in such
a short interval in the parameter space that actual observation of the period
doubling of the trajectories turned out to be quite difficult [29]. In other
words, such transition in this case does not take place as orderly as the

problem considered by Shaw [28].

While chaotic motions may exist at frequencies close to the anti-resonance
frequency, no chaos was found at anti-resonance for absorbers designed to
operate within the linear range, i.e. , for K<Kcr and k small. However,
other unmodeled disturbances may lead to chaotic responses. In particular,
inputs may lead to chaotic motions near a response curve of subharmonic
order n in frequency ranges from n=n to a point analogous to point B in
Figure (15).

The system is nonlinear and one cannot rule out these types of effects
from disturbances, even if they are small, since superposition does not apply

when impacts occur.

     

 

 

 

O O I: O 3
O O 0 O O
0 Ln <-- O N
O 0‘: C O N
N O O '— '—
ll ll ll ll H
x ..
.1-
l.
—-—N
!
."'-‘ ‘
_ N
.— l
.>< _
l I l m
I 1 1 . 1 l 1 1 i 1 l 1 1 l 1 I 1 1 I I l

 

Figure 18a - Poincare Map of a Strange Attractor

 

 

81

 

 

 

 

 

 

K=2.0
2___
‘ e=-0.95
‘ k=0.01
I ‘ n=l
_ n=1.22
O
_] ___..
-2 ‘ 1 1 1 1 1
l l I i l I l ] V l l I I l i l l l l l l T—]
O 10 20 30 40 50

Figure 18b - Time Response of a Chaotic Motion

CHAPTER 3

EFFECTS OF NONLINEARITIES
AND DAMPING ON THE DYNAlVflC
RESPONSE OF THE CPVA

The objective in this chapter is to gain a better understanding of the
nonlinear dynamic response of the CPVA and the effect of damping on the
system. The results enable one to obtain a broader view of the different
nonlinear aspects of the response of the system along with more accurate
guidelines in terms of the limitations of the linear analysis. To this effect,
the nonlinear dynamic response of a centrifugal pendulum vibration absorber
with damping in both the primary system and the pendulum is analyzed
using the methods of harmonic balance and Floquet theory. In section 3.1
the full nonlinear equations of motion are rescaled and put into the proper
form for further analysis and section 3.2 deals with the existence of periodic
solutions of the system and the method of analysis. Section 3.3 is devoted
to the stability analysis of the periodic solutions obtained in section 3.2 and
frequency response results are presented in section 3.4. The final section of
the chapter, section 3.5, describes a study of the response of the system at

the anti-resonance frequency.

82

83

3.1- Equations of Motion

Rewriting the full nonlinear equations of motion (1.1.7):

[J+mR 2+mr2+2mchosqb]1/)’ ’ —mr (R cos¢+r )aS ” +mRr¢~ ’ 25inch—
—2mRr1!J’¢ ’ sin¢+Cc1b’ =T(T) (3.1.1a)
—mr(R cosaS-l-r )1D ' ’ +mr2¢ " +0? 4) ' +mRr¢ ’ 2sinr,15=0 (3.1.1b)

where the terms in equations (3.1.1) are defined following equations

(1.1.7). For simplicity, the following rescaling of the parameters is

defined. Let:

 

 

 

 

 

J r

H mR2 , ’Y—E (3.1.23)

.. C .. C
>\ =- ‘ , =- ’ 3.1.2b
° mR2 " 111122 ( )

- T0 .. T1
0-11132 . Kl- mR2 , (3.1.2c)

This will render the equations of motion (3.1.1) as
(1+u+32+2vcos¢)1/2 ' ' —3(cos ¢+r1)¢" +019 “tines—2311'

(fl sin¢+ic1b ’ =Ko+chos(w) (3.1.3a)
-’7(cos¢+’7)1,b ’ ' +7209 ’ +33, 05' +710’ 2sin<75=0 . (3.1.3b)

Note that ib and ¢ are the angular displacements of the carrier and the
absorber, respectively. In addition, as mentioned previously, primes denote
differentiation with respect to time, 1'. We have taken only the first term in

the Fourier series of a general periodic disturbance.

84

The variable 10 does not appear in the equations of motion and is
completely arbitrary, i.e., it is an ignorable coordinate. The system has
only one—and—a-half degrees of freedom and if equations (3.1.3) are
written in first order form, only three equations will be required, 1'. e. , one
each for the derivatives of d), d) ' and, 11)’ . Here the system is not reducible
to a second order system since we retain the constant torque and carrier
damping.

As one can see, the method of rescaling of the parameters in this chapter
(equations 3.1.2) is different than the one used in the previous chapter
(relations following equation 2.1.20a). The scaling of chapter 2 resulted in
nondimensional system parameters, while the scaling in equations (3.1.2)
does not yield all the parameters as dimensionless (see equation 3.1.2b
above). However, the latter scaling is employed in order to simplify the
terms in the equations of motion for further

analysis. Nondimensionalization is carried out below.

It is also important to note the lack of a certain symmetry in equations
(3.1.3), which arises due to the damping and constant torque terms. When
these terms are omitted, that is, he =33, =1? 0=0, the equations admit

solutions with the following symmetry:
(11,2111) -—> (—¢,—1/1,r+-:—) .

Such solutions represent motions which have zero mean and contain only
odd order harmonics. When K0 is nonzero, such symmetric solutions are
not possible and in general motions will have nonzero mean and contain all
harmonic orders. The physics behind this lack of symmetry is that the
constant torque, K0, and the counteracting damping, Xe, bias the rotation

of the carrier to a preferred direction.

85

To account for the fact that the system generally undergoes a gross

rotational motion, we shall break '10 up as follows:
¢(T)=QT+9(T) , ~ =constant ; (3.1.4)

that is, the carrier angular displacement is composed of a nominal steady
rotation and an oscillating part. The average rotational speed, fl, will be
solved for in the course of the analysis. Substitution of equation (3.1.4)

into (3.1.3) and rescaling time yields:

(1 +u-t-’Y2+2’7cos ¢)9—’y( cos¢-l#7)05-+-’7<132$in d)—

—2yfl¢sin¢—2'y.95¢sin¢+)\c 8=K1cos(17t)+Ko—)\c (I (3.1.5a)
—q(cos¢4q)8h2$+kp Mflzsin¢+
qézsin¢+2rynésin¢=o . (3.1.510)

The rescaled time is defined by

t=wn r

11),I = [—(—+—+*7+2)T

is the nontrivial natural frequency of the carrier-absorber system. This

where

results in the following nondimensional parameters which appear in

Equations (3.1.5):

>’
It
y:
'e
>’
II
VI
('5
:3
ll
31

 

 

 

8
E

3
a
3

 

 

 

Also, overdots denote differentiation with respect to the rescaled time t. In
equation (3.1.5a), one might be tempted to immediately cancel the K o and
-m,, terms by setting the average speed to be Q=Ko/)\c. This will be true
to first order, but it is not immediately obvious that there will be no
contribution to n from the terms on the left hand side of the equality in

equation (3.1.5a), due to the inherent asymmetric nature of the oscillations.

Two remarks about the rescaling are in order at this point. First, the
numerical values of the damping ratios kc and X? used in this study are all
between zero and 0.10, and are not as relatively small as one might think.
In fact, if the pendulum damping ratio is greater than 0.01, the system
response is drastically different when) compared to that for lower damping
values. This is demonstrated in section 3.4. Secondly, the numerical values
of the excitation amplitude K 1 used in this analysis (typically K1301)
represent reasonable values for the input torque. For instance, for the
particular geometrical dimensions of the system used in this analysis,
K1=O.l corresponds to an excitation amplitude in the system of about

5.0x104 ft. lb.

87

3.2- Periodic Response

The method of harmonic balance [32, 33] is employed here in order to
approximate the periodic responses of the CPVA. The harmonic balance
method is chosen since it provides results with relatively uniform accuracy
over a wide range of frequency and is not dependent on small parameter
assumptions for the equations of motion (although such an assumption is
made here in order to limit the number of terms used in predicting the
response; see equation 3.2.1 and the explanation following). This wide
bandwidth uniformity of the results of the harmonic balance method makes
it an attractive tool for analyzing nonlinear systems such as the one
presented here. In fact, the limitations of the method are primarily dictated
by the number of terms that one wishes to include in the analysis and also
by the degree of sophistication of the computing systems at hand. In an
asymptotic sense, the results of the harmonic balance obtained for problems
such as the one presented here converge to the exact result as more terms

are included in the analysis.

The reader should note that the method of averaging, which in its own
right is a powerful method in dealing with nonlinear systems, was employed
first. However, difficulties arose as one tried to obtain periodic solutions for
excitation frequencies that were outside of a small neighborhood of the
resonance frequency. The result was that, although the frequency response
of the pendulum was predicted satisfactorily, inaccuracies in the carrier
response prevented the results of the averaging to be acceptable for further

analysis.

88

To apply the method of harmonic balance [32,33], we assume

6=A lsinnt +Blcosnt (3.2.1a)
=a+A zsimyt +82cosnt (3.2.1b)

i.e. , the solution is approximated by its first harmonic plus an offset in (b
which accounts, to the first order, for the asymmetry in the solutions (the

oflset in 1P is given by 01', a constant in 9 would be meaningless).

Substitution of equations (3.1.4) and (3.2.1) into equations (3.1.5) results

in two nonlinear equations in terms of 07, Q, A 1, B 1, A 2, and B 2 as follows:
—172[1+u-l-/)12+2'71fc (t )](A lsinnt +B lcosnt )+
+7772[’7+f c (t )] (A 25innt +B2cosnt )+’m2(A 2cosnt —B 2sin17t )2 f 8 (t )—
—2’yfln(A 2cosnt —B 23mm )1, (t )—
—2’7772(A lcosnt —Blsin17t)(A 2cosnt —B 25in17t )f, (t )—
—)\c n(A1cosnt—Blsinnt)+>\c 0=K1cosnt +Ko , (3.2.2a)
and
”7172h1+/ c (t )] (A lsinnt +B lcosnt )—’72772(A 231nm +Bgcosnt )+
+)\p ”(A 2cosnt --BQSin17t)+ryQ"’f,3 (t )+
44777201 lcosnt —Blsin17t )2], (t )+2'7077(A lcosryt —Blsin17t )f, (t )=0 . (3.2.2b)
where
f, (t )=sin(a+A gsimyt +Bzcosnt) (3.2.3a)
[C(t)=cos(a+A 28innt+Bgcosnt) . (3.2.3b)

As one can see, the above substitution results in composite trigonometric

functions [C(t) and f 8(t). Expansion of equations (3.2.3) yields

89
f c (t )=sina[cos(A 28mm )cos(Bgcosnt )—sin(A 25mm )sin(Bgcosnt )]+
+cosa[sin(A 25innt )cos(82cosnt )+cos(A 28innt )sin(Bgcosnt) , (3.2.4a)
and
f c (t )=—sina[sin(A 251nm )cos(Bzcosnt )—cos(A 23in77t )sin(82cosnt )] +
+cosa[cos(A gsinnt )cos(82cos77t )—sin(A 2sinnt )sin(Bgcosnt )] . (3.2.4b)

The composite trignometric functions appearing on the right-hand sides of
equations (3.2.4) can be expressed in terms of infinite series of Bessel

functions as follows:

cos(:rsin17t)=-Jo(:t)+2 [J2(:c)cos217t+J4(z)cos4nt+. . . ] , (3.2.5a)
sin(xsin77t )=2 [J1(:r)sinnt +J3(2‘)sin3nt +. . .] , (3.2.5b)
cos(:rcosnt )=Jo($)—2 [J2(x)cos2nt—J4(x)cos417t+. . .] (3.2.5c)
and,
sin(:ccos17t)=2 [1,(z)cosm—J,(z)cos3m+. . .] , (3.2.50)

where $=A2 or B2, and where Jn(:1:) is a Bessel function of order n defined

 

0° -1 k a: +2]:
J"(z)=kz_30k!(n+k)! ET . (3.2.6)

(The composite terms such as cos(A,-sinnt), 1' a-l,2 can be expressed as a
power series in A2 and 82. The nonlinear equations will eventually be
truncated at third order, i.e. , only linear, quadratic, and cubic terms in
a, Q, A; and B, are retained; this will capture first order nonlinear
effects.) Utilization of equations (3.2.3), (3.2.5) in equations (3.1.5) yields

two nonlinear equations in terms of the aforementioned variables as follows:

90

x, 0+ {—1211 1l1+MH2+23cosaJoM 2)JO(BQ)1+
41/717271 2[’7+cosaJ0(A 2)JO(B,,)]+2myB,sinaJo(A ,)JO(B._,)—
—X. 1181 lsimwt + l—fl2Blll+/1+’72+2’YCOSOJO(A 2)J0(32)l+
Hn232[’y+cosaJ0(A 2ljolBell-2777A 2531104004 2)J0(B2)+
+>\c 17A 1 }osnt =K0+K1cosnt , (3.2.7a)
and
asinaJoM 2)Jo(32)+ “1112A llv+cosafo(A 2)J 0032)] -’1"’712A 2-

-)‘p n82+21003aJo(B 2)J 1(A 2)—2’7nB lsinaJ0(A 2)JO(BQ) )sinnt +

+ [7772B 1 [741303051004 2)J0(Bz)]""’277282+xp 77A 2+
+2’YCOSQJ0(A 2)J1(B2)+2’Y77A lSinaJ0(A 2)J0(BQ) }:os17t =0 . (3.2.713)

Matching the coefficients of terms multiplied by sinknt and cos/ant for Ic==0
and 1 in equations (3.2.7) yields six equations in terms of 0:, Q, A; and,

B,- , i=1, 2.
The constant order terms result in the following conditions:
kc fl=Ko , (3.2.8a)
and
7013111011 (A ,B)=0 , (3.2.8b)
where
f (A ’3 )=J0(A 2)JO(BQ)

Note that f (A ,B) is a function of A2 and 82 which is nonzero (except for

values of A2 and B? representing large amplitude motions,

91

7r . . . . . .
e.g., ¢max>-2—). These conditions immediately give the mean rotational

speed as Q=Ko/>\c and a zero offset in pendulum oscillations, a=0 (the oz=7r
case is inherently unstable and not of interest here). This implies that the
pendulum and the carrier oscillations can be well approximated by a zero

mean, symmetric (in the sense described in section 3.1) oscillation.

The four remaining equations are in terms of the A, and B, and are given

as follows, where we have employed conditions (3.2.8):
—112A 111+u+12+27J01A 2)Jo(32)l+

H0244 2l’7+Jo(A 2)Jo(B2)l->‘c "31:0

”702A il7+JolA 2W 0(3 2)l #7277214 2—

—>‘p ’73 2+2?” 0(3 2V 1(A 2)"'0 (3.2.9b)

—772B 1 ll +fl+’72 +271 0(A 2)Jo(B 2)] +7023 2l’7+

+1004 2)Jo(B2)l+)\c "A 1=Ki
77723 1 “+1004 zljolell—WQWQB 2+)‘p 77A 2+

+2710(A2)J1(82)=0 . (3.2.10b)

Noting that from equation (3.2.6),

and

92

3
x .1:

All the terms in equations (3.2.9) and (3.2.10) are expanded and we keep up

to cubic orders in A, and 31- This results in the following equations:

 

 

 

 

 

 

A 2 B 2 A 2 B 2
-112[1+u+72+2'7(1 2 42 )]A1+n2['12+*1(1 42 42 )lA2-
—)\c nB1=0 (3.2.11a)
2 B 2 ' A 2 B 2
—n211+11+q2+2»1(1 2 ,2 )]Bi+n2h2+'v(1 ,2 42 >132—
—)\c "A 1_Klfi) (3.2.11b)
A} B 2
A A B2 A 3
2 02 2 2 2 2 . .
+3 (2 8 10 )=o (3212a)

2 A22 B22 2
7? l’YzWU— 4 _ 4 ”Bi-”1277 32")»712424'

 

32A? 323
8 16

 

2 B2
+2111( 2 )=0 (3.2.12a)

At this stage it is convenient to express 6 and 45 in terms of amplitude-phase
variables as follows:
=alcos(nt—fll) (3.2.13a)

¢=a 2cos(77t —,82) (3.2.13b)

where

93
A,~2+B,-2=a,-2 , A,-=a,-sinfi,~ , B,-=a,-cosfl,~ , i=1,2.

Substitution of equations (3.2.11) into equations (3.2.9) and (3.2.10) (with
’ Q=K0/)\c and a=0) results in the following four equations for al, 31, 02,

and ,82:

 

 

 

 

22
—7]2 l+u+’722’7[1- a4 ]]alsinfil+
2
2 a2 .
+772 ’7 “Pill-T]LQSinfig—chalcosfilfl , (3.2.14a)
a22
—172 1+it-l-’)122’)1[1——4 ] lcosfll-l-
2
2 a2 .
+77 7244““? l]4200562+)\c’7“ismfli‘K1=0 1 (3°2'14b)

 

and

a 2
772 [724’7l1—'4il }" iSiDfii‘JY2772a25infl2-

3

2 a2 . a2 . 2
—)\p no 2cos,32+2’70 ——2—sm,32——8—sm,52cos 32—

023
_.—516 in3flz , (3.2.153)

 

a 2
172 [724“7l1-jf—l [10050147277211 goosfi2+

02 a; . 2
Ycosfig——8 cosflzsm ,32-

 

+>\p 770 28ln32+2702

3
a
_ 11: c03352]=o , (3.2.150)

 

94

Of these four equations, it turns out that equations (3.2.14a) and
(3.2.14b) can be solved for al and ,3, in terms of a2 and 162; that is, the

equations uncouple. In terms of 02 and fig, a1 and E, are given by:

a lsinfll=%[1112agsinflz+)\c n(K1—12a2cosflg)] (3.2.16a)
and
a lcos,81=—TI[II(K1-I2a 2cosflQ)—)\c nIQagsinfig] (3.2.16b)
where
2
2 2 “2
11=77 [1+u-l-q1 +27(1——4—)] , (3.2.17a)
. 022
I2=7l‘l'12‘l"‘f(1"T)l 1 (3-2-17b)
and
D =1,2 +0., 17)2 . (3.2.17c)

When the two expressions (3.2.16a,b) are substituted into the two equations

(3.2.15a,b), the result is a pair of equations involving only 02 and ,82:

I, , .
—D-[I 112a 281n32+kc 17(K l—I 202cosfl2)]-/721720 QSInflQ—Xp nagcosfl2

3 3
a a a
+2’yfl2( —22—sinfl2-Tzsin52c032182—T2—sin3flg)=0 (3.2.18a)

and,

I
-%[II(I(1_I20 200532)+>\c nIQa2sinfl2]-c112n2a 2cosfig+

3 3
. a a a 2 .
+>\p na 251nfi2+2702( T2cosBQ—ié—cos3fiQ—T51ngfigcosfig)=0 (3.2.18b)

95

This uncoupling simplifies the equations considerably. Equations
(3.2.18a) and (3.2.18b) can be solved for the pendulum amplitude and phase
( 02 and ’62 ) and the latter substituted into equations (3.2.16a) and
(3.2.16b) to yield those of the carrier, namely, 01 and 31. This is carried

out numerically.

The case of zero damping and zero constant torque, )\p =)\c =K0=0, is

useful for certain analyses and is presented here. For this case equations

(3.2.16a,b) and (3.2.18a,b) reduce to:

 

 

I a sin
alsinfll- 2 2 32 (3.2.19a)
Ii
K —I a cosfi
alcosBI=—( l 2 2 2) (3.2.19b)
11
I? . 2 2 .
TlgaQSinfig—q 77 a251n32+
1
3 3
2 a2 . 02 . 2 a2 . 3
+270 (TSinflz—TSinfl2cos 32—Fsm flz)=0 (3.2.19c)

I2 2 2
“‘I—(K1"‘12‘12‘:‘)Sfl2)'—’71 7? 0200532?
1

3 3

:2 3 -32-2
cos £2 8 Sin flgcosfl2)=0 (3.2.19d)

2 a2
+27“ (.2—cos02— 16

96

3 .3- Stability Analysis

In this section we consider the dynamic stability of the approximate
solutions which are obtained. This analysis uses Floquet theory (see [32] or
[34]). The idea is to determine the dynamics (actually only the growth or
decay) of small perturbations to a periodic solution. This is accomplished
by linearizing the full equations of motion about the periodic solution and
studying the resulting linear, time varying differential equations.

We denote the periodic solution as (5,3), and let ‘9 and £6 represent small
perturbations of 5 and 3. Substitution of 9=E+9 and ¢=$+€b into equations
of motion (3.1.3a,b), and noting that Sim} and coszb can be approximated by

97> and 1, respectively, one obtains
Il+u+e2+2cos$-&ssin$1(3+5)-[cos$—esin$+e1($+$)+
m<$+3s)2(sin$+ecos$)—2vm$+3>(sin$+$cos$)—
—27(§+3)($+3)(sin$+<i>cos$)+xc(E+E)=K,eesnt , (3.3.1a)
flicosg—éfisinEH](3+3)-Ifl”($+3)+>\p (5+33)+
Mn?(sin$+$cos$)+y(5+§)2(sin$+éseos$)+
+2en(5+§)(sin$+$eos$)=o . (3.3.1b)

Note that, by definition, 5 and a satisfy the equations of motion (3.1.5),
and also keeping only terms linear in the tilde variables in equations (3.3.1),

we obtain:

—273{>sin5§+[1+uH2+27cosgl§H$sin3$-

97
—i(i+cos$)25+2v$<issin$—2vnébsin5—
—2qn$&5cos$—2q(?o+$é)sin$+xc§=K1cosnt ,
véfisin33-7h+cos$]5+723+>xp 2b+
H02<3cos$+275~95in5+2705~¢cos$+
+2qnfisin$=o
Solving equations (3.3.2) for '2'} and 23 yields

3=%[72L.H(7+C)L.l

$=%.[L ¢(1+M+’72+2’YC )+’7(7+C )L a]

where
L o=+2v$sinfi—qisin?¢—2v$35sin$+2vn&sin$+
+2vn$ecos5+2v<3e+$é)sin$—x.‘o ,
L ¢=~7§$sin$5->\p (E-q0263cosg—2qb‘fisina—
—27Q:9-¢cos$-2flflgsin$
and
17=72(1+n-cos"’$)

(3.3.2a)

(3.3.2b)

(3.3.3a)

(3.3.3b)

(3.3.43)

(3.3.4b)

(3.3.4c)

Equations (3.3.3a) and (3.3.3b) can be put into first order form to obtain

four first order linear differential equations for 9, 9, 55, and 53 with time-

periodic coefficients. These equations are then numerically integrated

through one period of the forcing with initial conditions equal to successive

columns of the 4X4 identity matrix. The four resulting vectors can be

98

assembled to form the 4X4 monodromy matrix. According to the F loquet
theory, a periodic solution (5,.) is stable if all the eigenvalues of its
monodromy matrix, i.e., the Floquet multipliers, have moduli less than
unity; otherwise it is unstable. Bifurcations occur as eigenvalues pass
through the unit circle in the complex plane, and we shall predict post-
stability behavior wherever possible using concepts from bifurcation
theory. We shall use as 5 and 5 our approximate solutions obtained using

harmonic balance.

It must be re—emphasized that the variable 9 appears nowhere in the
governing equations. The system under consideration has actually only
one —and —a —half degrees of freedom and can be represented by three first
order differential equations. The effect of this on the stability analysis is
that one of the four eigenvalues will always be identically one, due to the
inherently neutral nature of 9. Thus, all stability considerations are based
on the remaining three eigenvalues. Another way to see this is to note that
we only need to consider the differential equations for 46, (i3, and 52>, and after
solving them '9 can be obtained by direct integration, i.e. , the Z) dynamics

are uncoupled.

In regards to symmetries, it must be noted that the linearized equations

(3.3.3a,b) are symmetric in the following sense:
($9bat )_’(_$7-99t +%)

leaves the equations unchanged only if the periodic solutions (iii-0.) have the

following symmetry:
(gravt )_*(—$2-§9t +'E;_) -

The original equations of motion do not admit such solutions, but it is

99

important to note that our approximate solutions have such a
symmetry. This will be a crucial point used in interpreting the stability

results.

100

3.4- Frequency Response

Figures (19a) and (19b) show typical frequency response curves for the
pendulum and the carrier, respectively, and Figure (19c) depicts the
corresponding phase difference of the carrier and the absorber. The solid
lines indicate stable periodic motions and the dashed lines indicate unstable
ones. Superimposed on these plots are the results from simulations of the

full nonlinear equations of motion (3.1.3a) and (3.1.3b), shown as circles.

As can be observed, the results of the harmonic balance agree well with
the simulation results and only when the absorber amplitude response
(Figure 19a) becomes quite large does one see a difference in the analytical
and simulation results. In particular, the results of the analysis start to
diverge from those of the simulation of the equations of motion when the
absorber sweeps through about 90 degrees in each direction during the
steady-state operation, well beyond the range of practical applications.
This is primarily due to the fact that in the harmonic balance analysis, the
solution was assumed to consist of only its first harmonic while the actual
response contains many harmonics, and as the amplitude of the oscillation
grows, the effect of these harmonics becomes more pronounced. Also, the
effects of terms ignored in the series expansions of the Bessel functions may
become non-negligible. As a test of the validity of our approximate
solutions, the Fourier spectrum of the response of the carrier obtained
through the simulation of the full nonlinear equations of motion was
analyzed over a range of frequencies. It was found that at low amplitude,
oscillation energy is primarily stored in the first harmonic, while as the

amplitude grows, the higher harmonics (in fact up to the fourth) acquire

O .

 

101

_ll
“2
_ O
. 5—— ‘.
-— 1 ‘\
--0 \O
0 —— ~
"‘ '.
5 __ \
-l
\
_y. r
d» ' .1 ' .
-———-—‘:
0 r I

K,=0.05
x, =0.01

x,=o.01

 

STABLE
_ _. _ — UNSTABLE

Q SIMULA TION

 

 

 

0.4 0.6

| 7‘ l
078 My!)

 

Figure 19a - Frequency Response of the Absorber - Analysis and

Simulation

O .

O .

O .

102

 

 

  

 

 

 

 

O 6—P-
K1=0.05
“1 O
' >\p =0.01
.. u
, - - xc=o.01
0 4 \ STABLE
— — - - UNSTABLE
1 ’ Q SIMULATION
0 2 O .
O \
_ ‘51
‘I
aQQo )3 O\‘7“o~
0 0 I I \91 I I I I
I I I I 'I I I I I I I I
11 O 6 O 8 AR I 0 l 2

Figire 19b - Frequency Response of the Carrier - Analysis and
Simulation

103

 

 

 

 

 

 

 

 

 

 

 

4—r
J lfie-Bil K,=o.05
— x,=o.01
—I.
.5—— >.C=o.01
Z
—I
I __
O I I I T I r I. I
O I- ’l H (l . 8 l I] I Z l 4-
17

Figure 19c - Phase Difference of the Carrier and the Absorber

104

enough energy to significantly affect the motion of the carrier.

The numerical value of the anti-resonance frequency for the example
system used in this analysis is determined from the linearized system to be
17 A R =0.89, which is close to the resonance frequency of the system, n=1.0.
As is shown in section 3.5, the nonlinear anti-resonance frequency for the
damped, nonlinear system will differ from "AR- Figures (20a) and (20b)
illustrate the frequency response of the absorber and the carrier, respectively,

for several values of the excitation amplitude, K1.

The frequency at which the slope of the response curves of the pendulum
and the carrier are vertical is referred to as the turning frequency, 17, . From
Figure (19a) one can observe that the turning frequency should not be too
close to the anti-resonance frequency for a feasible design. Obviously, if the
frequency of the excitation is slightly increased above 17,”; the resulting
response of the system will be drastically different than expected and the

absorber will not function properly.

The initial conditions play a major role in the response of the CPVA in
certain cases. For instance, in Figure (19) one can see that there exists more
than one steady-state periodic solution for the system at the anti-resonance
frequency, mm. In other words, depending on the initial conditions, the
system either responds as predicted by the linear analysis (the lower branch
on Figure 19) or oscillates with much larger amplitude (the upper branch in
Figure 19). The middle branch in Figure (19) represents unstable, and

unobservable, periodic solutions of the CPVA.

The numerical value of the turning frequency can be obtained as
described by Stoker [34] and outlined here. As mentioned earlier, at m the

slope of the frequency response of the pendulum is vertical,

i.e. ,Tgn—=O at n=17t. This property can be exploited to find 17, by
“2

2 .

O .

O .

105

 

 

 

 

 

 

 

 

 

xp=o.01
O >.C=o.01
_ a?
_ , STABLE
_. — — — — UNSTABLE
. 5——
.J.
0..—
.J
I
5
—I
(l
O . 4

 

Figure 20a -Variation of the Frequency Response of the Absorber
With Respect to the Excitation Amplitude

106

 

 

 

 

 

 

 

 

 

 

 

O 5 X? =0.01
. —-l'—
“1 : xc—om
L. K‘ ‘ ‘°°
' # STABLE
-II-
0 ' 4:— _ - _ — UNSTABLE
O . .I
:I
o . 2 x
: ~.\ . 5H0
\ " .
O I——— \\\o.5 ‘“~-.‘~
E -
n ~Q ' _ _ __'_E_:EE 3:3 :12:‘-~‘0-I \--
o o — iuL-Ifi‘ilgfin i ‘‘‘‘ ‘ 2% F i i
O 4 O 5 (l. 8 7&1 l lll l 7. I 4

Figure 20b -Variation of the Frequency Response of the Carrier
_ With Respect to the Excitation Amplitude

107

differentiating equations (3.2.18a) and (3.2.18b) with respect to a2. Two

as follows:

 

equations result in terms of 17,, a2, fi2, and

6a2
—1—2— (w I22a2sin52+2111212’ a,sinB,+I,I,2 sinfl2+1112202fi2’ eosB2+
+)\c 1712' (K l—I2a 2cosfl2)+)\c 171 2(—I 2’ a 2cosfi2-I2cosfl2+
+I2a 2fl2’ sinfl2)]D —D’ [1112211 2sinB2+I2kc n(K1-I2a 2cosB2)] ]—

#721728ln32—727720 2fl2’ 00832—Xp "C03fl2+)\p 770 2,82, Sin,82+

3022
+2702[—sinfi2+—fl2' cosfl2-—8—sinfl2cos2 E2—

 

023 3 “23 . 2 3a22
-—fi2' COS flew—32’ sm 5200352“ 3511352-
8 4 16
3a23 . 2
-n-fle'sn aces/as , (3.4.121)

and
I}; [D I I 1’ [2(K1—I2a 2cosfl2)+1112' (K 1‘12“ 2C03fl2l+
+1 1 I 2(—I 2’ a 2cosfl2-I2cosfl2-I-I2a 262’ sinfl2)+
+2)» ”1212' a 2sinfl2+bc 77122 sinfl2+kc 77122 a 2:32, cosfl2] —
—D ' [1112(K1—12a200532)+)\e 17122a esiDflel )-

-’72772cos,32«|-’72772a2fl2’ sin,32+>\p nsinfl2+>xp 17a 2fl2’ cosfl2+

a2

 

 

+27fl2[——cosfl2——-,B2' sin,B2-—3 Hoos3fl2+ 63"82 sinfl2cos2 B2—
3a22a231123 3
-—sinzfi2cosfi2-—fl2’ sinfl2cos2 fl2+——fl2’ sin B2]=O . (3.4.1b)

8

where

 

11’=-’7172a2 .
II’
I '=
2 2
and
D, =2IlIl’

Equations (3.4.1a,b) along with equations (3.2.18a) and (3.2.18b) form a

system of four nonlinear equations in terms of four unknowns, namely,

(9
(a2, 52, 5-51, 17, ). Here, 02 and ,32 are the absorber’s amplitude and phase,
2

2 is the rate of change of the relative phase of the

602

respectively, at 77, and

 

absorber with respect to the absorber amplitude. Given a set of system
parameters such as damping ratios and excitation amplitudes, these
equations can be solved numerically to yield the corresponding turning

frequency, 17, .

Figure (21) shows a plot of the turning frequency versus the amplitude of
the oscillating component of the excitation, K1 (the 77 curve in Figure (21) is
explained in section 3.5). It was found through numerous simulations and
analysis that the damping ratios of the carrier and the pendulum have little
effect on the turning frequency compared to the effect of varying K1. Hence
Figure (21) depicts the result for zero damping with the fact in mind that
similar curves for other sets of damping values (within practical range) all lie
in a small neighborhood of the curve shown in Figure (21). As described in
the next section, the turning frequency 17, can cross over the anti-resonance

frequency, f7, resulting in a drastic change in response.

109

 

20-——-Kl
-I
\\
—I> \\
\\
_ \\ 7?:
.. \\ \\
\ \
f5_— \ \ n
\ \
.. \ \
\ \
\
4 \\\ \
\ \
_. \\ \
\
.H}1——
as . I . I . I . I .

 

0.80 0.82 0.84 0.86 .I 0.88

Figure 21 - Turning Frequency aid the Anti-resonance Frequency

110

There are two dynamic instabilities observed in Figure (19). There exists
one at the turning frequency; this is a straightforward, simple saddle-node
bifurcation (Guckenheimer and Holmes [26]). The more interesting one
occurs on the upper branch of the response curve. It corresponds to an
eigenvalue passing through +1 in an increasing manner as 17 is
decreased. Such a transition corresponds in the generic case to a simple
saddle-node bifurcation, but this does not occur here. The instability results
in a symmetric saddle-node, or pitchfork, bifurcation which results in a pair
of anti-symmetric motions [26]. This bifurcation is predicted from our
analysis, which yields symmetric approximate solutions, which cannot occur
in the actual, nonsymmetric system. By considering a small, unsymmetric
variation in the predicted instability, we predict that near this point a large
deviation from symmetry will occur. That is, solutions will be nearly
symmetric up to the instability, and above the instability the asymmetry of
the solutions will be amplified quite sharply (see section 7.1 of Guckenheimer
and Holmes [26] for a more thorough explanation). Figures (22a,b) show the

K0
kc

variation of the mean values of 9 and (b, that is, deviations from Q: and

 

oz=0, as computed from direct simulations of the steady-state solutions of
equations (3.1.5) while following the upper branch of the response
curve. The sharp rise in the mean value of 0 near the instability indicates
that our approximation for Q is breaking down. In Figures (23a,b) we show
a series of periodic motions as limit cycles in the three dimensional phase
space (¢,<I),0) for 17 varying near the point of loss in symmetry; it clearly

shows the breakdown in symmetry as 17 is decreased near the instability.

Another point of interest is that there should exist another branch of
solutions arising via a saddle-node bifurcation near the predicted pitchfork

bifurcation point. We were unable to find any evidence of these motions,

A
v

C)

.001

.000

111

 

 

 

 

A mean of 9 K1=0,05
_, \/\ x,=o.01
'I’ xc=0.01
_I
‘I
I I I W1. - . n L."
I I I I I I l I I I I
0 4 0 5 0 . 8 I . 0 I 2 I 4

Figure 22a - Variation of the Mean Value of the Carrier Velocity

 

0 .

0 .

0.

O .

0 .

.10

112

 

_ mean of 45

08

 

06

 

04

 

 

02

K,==0.os
x, =0.01

>2 =0.01

 

 

 

00 I I
0.4 0.

Figure 22b — Variation of the Mean Value of the Absorber Angular

Displacement

 

 

 

113

Son .4
Son 24

modui

ouoom 353585181... h 5 (Ph.D 9: no 338.235. conca— I can 959...

 

1

114

ouoam 3835551822.- 5 <>QU 05 yo >Louuonoch omega I saw 0.5m:

 

Son .4

Son 2

 

Song +

115

although they may well, and should, exist. Figure (24) depicts the effects of

small symmetry deviations on a pitchfork bifurcation.

Finally, we remind the reader that the rescaling of time and other system
parameters that led to the equations of motion (3.1.5) results in a scaling
down of the damping values. In Figure (25) we show a set of response
curves corresponding to )\c=>\p =0.1 and K1=0.1. Note that for these
apparently moderate damping values the response has no usual resonance
peak or anti-resonance point. Such response curves were found to occur for
A? greater than about .01 over a wide range of K1 and )2 values. A reason

for this is provided in the following section.

116

8.50.535 xcohzuzm o co mcofiogoo xcuoEExm :mEm .0 muootw

 

I on 95m:

 

 

0 .

0

_I al K1=O.1
.. xp=o.1
' 1 xc=0.1
_I
5——
I I I I I I I I 7 I I I I I I
0.0 0 S 1.0 I 5 2.

117

 

 

 

 

Figure 25a - Frequency Response of .the Carrier

 

118

 

 

I.O———
02 K1=O.1
. xp=o.1
0.8——
xc=o.1
— \
\\
0.6——
0.4——
0 2—-- \‘\\
7)
NC I I I I
V! IIFITTIIIIIIIIIIIII—I
0.0 0.5 I.U 1.5 2.0

Figure 25b - Frequency Response of the Absorber

119

3.5- The Response at the Anti-resonance Frequency

A study was done on the behavior of the carrier near the predicted anti-
resonance frequency for a range of damping and excitation amplitude
values. The variations of the carrier amplitude at 17,”; as a function of
K1, )‘p and, )\c are shown in Figures (26a), (26b) and, (26c), respectively.
From these Figures one can observe that the carrier amplitude at the anti-
resonance frequency, 17,43, varies almost linearly as a function of the
excitation amplitude, K 1- It depends strongly on the pendulum damping
)xp for 03%,, <0.02 and is relatively insensitive for )‘p >0.02. The carrier
amplitude at 1743 remains essentially constant as )\c is varied. The
sensitivity of the response to changes in )\p as shown in Figures (25) and

(26b) indicate that the pendulum damping should be kept as low as possible.

As shown immediately below, the actual desired operating point may vary
away from 17,“; due to nonlinear and/or damping effects. This is crucial
from a design standpoint. Thus, the sensitivity of the value of the anti-
resonance frequency itself to the variation of the system parameters was also

investigated.

A first order approximation of nonlinear effects on the anti-resonance
frequency can be obtained by assuming that the carrier and the pendulum
dampings are small enough so as to be neglected. The absence of damping
implies that the carrier amplitude can be identically zero at a single value of
17, the true anti-resonance frequency, denoted by 57 . This value is exactly at

17 A It only for the linear, undamped system.

The condition for anti-resonance in the undamped system is a1=0, 1'. e. ,
the carrier has no oscillatory response. Imposing a1=0 on the undamped

response conditions (3.2.19) yields the following conclusions. From equation

0.

(D

120

o 2 ('I—I—-~

)xp =0.001

_ x3001
.OISf~—-

 

 

 

0I0-
GUS-
K1
~00.OIIIIIILTIIIIITIIIIIII—IIIITIIIIIIIV‘I'

0.000 0.025 0.050- 0.075 0.100 0.125 0.150

Figure 26a - Variation of the Carrier Amplitude at the Anti-resonate
Frequency with Respect to the Excitation Amplitude

121

 

 

 

 

 

_ al
.015—— / K,=0.1

— xc=o.01
010——
.005—

- X.

I I L,__ _= I

.000 I I I I f I I T I

0.00 0.07. 0.04 0.06’ 0.08‘ 010

Figure 26b - Variation of the Carrier Amplitude at the Anti-resonate
Frequency with Respect to the Absorber Damping

122

 

 

 

 

xto-Z
0.20--~
.. a!
:I—— —
0.15—L K,=0.1
:. )\p=0.001
_I
0.10 _.
II
0.05——
I X.
III . I . I=II
0.00 0.07. 0.04 0.06 0.08 0.10

Figure 26c - Variation of the Carrier Amplitude at the Anti-resonate

Frequency with Respect to the Carrier Damping

123

(3.2.19a) it is concluded that either a2=0 or ,B2=O or 11’ (I2=0 corresponds to
a2=2 which is beyond the range of our approximation, 1'. e. , ¢max>%)° The

a2=0 solution is unfeasible with al=0. Note that due to the particular
configuration of the directions of the excitation and the rotation of the
carrier and the motion of the absorber (Figure 2b), the fi2=7r solution
represents the upper branch of the response curve and the fl2=0 solution
represents the desired solution on the lower branch of the response curve;
this, in fact, corresponds to the pendulum moving out of phase with respect
to the forcing, as it must in order to act as an absorber. Equation (3.2.19c)

is automatically satisfied for B2=0. Equation (3.2.19b) reduces to
K1_12&2=0 (3.5.13)

where I 1 and 12 are as in equatiOns (3.2.17a) and (3.2.l7b), respectively,
with n=f7 and a2=fz2 where (12 is the pendulum amplitude at the anti-
resonance frequency, 17. Equation (3.2.19d) simplifies to the following
expression by imposing al-O, 762:0 and equation (3.5.1a):

a 2
a
222_n2(1——82—)=o (3.5.1b)

Note that if 62 is small, i.e. , if we linearize, then f7=17AR is recovered
from equation (3.5.1b) and (3.5.1a) yields the linear pendulum response

amplitude at 17243 .

Equations (3.5.1) were solved numerically for the unknowns (1‘2 and 17 as a
function of the excitation amplitude K1. Figure (21) illustrates the result of
this analysis for 1"]. As can be observed, the anti-resonance frequency, 17,
decreases in value as the excitation amplitude is increased. According to
Figure (21), for K1>0.12 the anti-resonance frequency is predicted to be

larger than the turning frequency, 77,. When 17, <17 is predicted from our

124

approximate analysis, no 17 can actually occur since a1=0 cannot exist at a
frequency above 17, . In such a case no low amplitude carrier motion can
exist, and a steady-state response at n=f7 will occur which corresponds to the
upper branch of the response curve. This effect renders the absorber useless,
and it cannot be captured from a linearized system analysis. Figure (27)
depicts the predicted and simulated carrier response for K =0.5. Note that
in this situation no range of 17 values exists for which the system acts as an
absorber. The conclusion is that these devices will have a limited torque

range.

When damping is taken into account the amplitude of the carrier at the
anti-resonance frequency can be no longer set identically to zero. Instead,
the derivative of the carrier amplitude with respect to the excitation
frequency at the anti-resonance frequency can be set to zero since the carrier
response curve, by definition of the anti-resonance frequency, has a minimum
at that frequency. Accordingly, equations (3.2.16) and (3.2.18) can be
differentiated with respect to the excitation frequency, 17, resulting in four
coupled partial differential equations. The subsequent eight unknowns

(i.e. , a1, a2, 31, 32, and their partial derivatives with respect to 17, with
801
317

using the four equations (3.2.16) and (3.2.18) and the four partial differential

 

set to zero, and 17 which is not known) can be solved (in principle)

equations mentioned above. However, the resulting equations are
complicated enough (and are not shown here) that results obtained directly
from the simulation of the equations of motion (3.1.5) proved more
practical. It was found as a general rule that increasing the absorber
damping tends to shift the 17 curve in Figure (21) slightly to the left, while
the carrier damping has no detectable effect on the shifting of the anti-

resonance frequency (for )\c <0.1).

125

 

 

 

 

. O'T— K l=0.5
“2 x, =0.001
_. o xc=o.01
. 5.—-—
— Q SIMULATION
. 0———
. 5——
- \s\
" I I 701‘ I I I
I I l I 'f I T I F I I
0 . 4 0 6 0 . 8' I 0 I 2 I 4

Figure 27a - Frequency Response of the Absorber

126

 

A, (11 XIII-10.5
I x, =0.001
. 4 --I .. -
._Ir kc £001
-3 *— o SIMULATION

0
d

   

 

 

 

I—O
LL.
o— . II I «I . I .
0.4 0.5 0.8-IMI.II 1.2 1.4..

Figure 27b - Frequency Response of the Carrier

CHAPTER 4

CONCLUSIONS

The dynamic response of a centrifugal pendulum vibration absorber was
investigated in terms of the effectiveness and range of applications of the
device. The nonlinear methods of 4 analysis undertaken in this work have
revealed some aspects of the dynamics of the CPVA which have not been

understood before.

4.1- The Effects of Motion Limiting Stops

The main conclusion drawn from the first part of the thesis is that
motion limiting stops can be effectively employed when placed at amplitudes
larger than the steady—state response predicted from the linear system at the
anti-resonance frequency. This is so due to the fact that steady-state
impacting motions can occur at the anti-resonance frequency only if the
damping between the pendulum and the carrier is unreasonably large. The
impact dynamics of the system can be very complicated when it is subjected
to frequencies above anti-resonance; this can include chaotic motions and/or

a variety of periodic responses. The satisfactory performance of the

127

128

centrifugal absorber with motion limiting stOps is dependent on the
amplitudes of the excitation that is inducing the undesirable vibration on
the primary system (carrier). If the excitation amplitude becomes larger
than the free —flight threshold, the absorber will interact with the stops

resulting in poor, if not damaging, performance.

Recent experiments have been carried out on a simple, non-rotating,
impacting pendulum which is governed by the same equations of motion as
the linearized system with stops (equations 2.1.20a,b). These demonstrated
the existence of SDIP motions, pitchfork bifurcations and chaotic dynamics
[36]. Frequencies near the corresponding anti-resonance frequency were not,
however, attainable in that system since 17 was restricted to be 1.0 or higher
by equipment limitations.

The SDIP motions are a very specific type of periodic motion. An
infinite number of other types of periodic motions exist for this system.
This is known since the presence of chaotic dynamics indicates the existence
of horseshoe sets for the map P which in turn contain this infinity of
periodic motions along with nonperiodic motions [37]. Thus a complete
study of the impacting dynamics is out of the question. However, SDIP
motions are the most common; this has been observed in experiments [36]
and simulations. In addition, they are a good indicator of where other
impacting motions exist since many (although not all) of the other types

arise out of bifurcations which are directly tied to an SDIP motion.

If stops are not employed at all, and the pendulum is allowed to swing
over the top (an extreme situation which is rarely, if ever, seen in
applications), the system can undergo chaotic motions in which the
pendulum undergoes chaotic sequences of clockwise and counter-clockwise

rotations [35].

129

4.2- The Effects of Nonlinearities and Damping

In the second part of the study, the nonlinear response of a centrifugal
pendulum vibration absorber was studied using harmonic balance and
Floquet theory. Although specific geometrical dimensions for the model
were used in this report, some general conclusions can be drawn from the

results.

The method of analysis applied to the basic model resulted in sufficiently
accurate results and their agreement with the simulation of the full
nonlinear equations of motion was satisfactory. The results depict how the
true response of the system can be drastically different than what is
predicted from the linear analysis. Hence, it is well advised that when
designing a CPVA that the undamped, linear analysis should be used as a
means of first trial of the required system dimensions and geometric
properties. However, once designed, the model must be analyzed using a
nonlinear method such as the one outlined in this study to account for the
nonlinear and damping characteristics of the system, which affect the
dynamic response of the CPVA. This can be true even in some cases where
one might think that the mere presence of low amplitude motions justifies

using the linear analysis results.

Nonlinear characteristics of the system such as the shifting of the anti-
resonance frequency and the jump phenomena must be considered in the
design of the CPVA since if ignored, the absorber’s efiectiveness can be
reduced, or it might even result in larger oscillatory amplitudes in the

primary system.

130

It is often assumed that carrier damping has negligible effect on the
response of the CPVA since, when properly working, the carrier runs at
constant speed, and hence the carrier damping is completely counteracted by
the mean component of the torque. Based on the results indicated in Figure
(23c), it appears that such an assumption is reasonable since the response at
the anti-resonance is quite insensitive to the values of )xc. However, the
anti-resonance response is extremely sensitive to the magnitude of pendulum
damping. It may be possible to lump the carrier damping in with the
pendulum damping, resulting in some increased value for the effective
pendulum damping, due to the system’s insensitivity to )‘c at anti-

resonance.

Several other researchers have investigated the nonlinear dynamic
response of the CPVA. Den Hartog [38], Crossley [20], and Newland [7 , 25]
seem to have contributed the most. A brief comparison is presented here
between the results obtained in this report and those of these
references. The model used in this report is a generalization of the one used
by Crossley [20] except that damping is neglected in [20]. Although the
equations of motion in both reports are equivalent (with the exception of
damping terms), the final expressions for the nonlinear dynamic response of
the pendulum and the carrier are different. Since Crossley models the
system without damping, the equations of motion are integrable and the
result is an expression for the angular displacement of the pendulum in
terms of hyperelliptic functions. Also, no stability analysis was carried out
by Crossley and the subjects of jump phenomena and the turning frequency
were not considered. The conclusion drawn from Crossley’s analysis is that,
due to the wide angle of swing of the absorber, the designed aborber should

never be too long and they should be of a length somewhat shorter than the

131

value given by the linear analysis. The results in this report show in a more
detailed manner as to the reason that the pendulum should be designed with
a shorter length than predicted by the linear analysis. This is performed by
the computation of the shifting in the anti-resonance frequency and as
noted, 17 AR has a tendency to decrease as K1,, and consequently the
absorber amplitude, is increased. Keeping in mind that the excitation

frequency is fixed at n=jfl, and since the minimum amplitude of oscillation
R /2 R
of the carrier occurs at a value below j= T , then the ratio 7 should

be adjusted in such a manner as to increase the frequency at which the
minimum oscillation amplitude of the carrier takes place, i.e. , the frequency
at which the absorber is most effective. By the virtue of the fact that the
carrier effective radius, R , is essentially fixed, the only parameter over which
the designer can have effective control is the pendulum length, 1'. An
increase in the effective frequency of the absorber (a right-ward shift of the
anti-resonance frequency) then necessarily implies a decrease in the effective
radius of the absorber. This fact, as mentioned earlier, is in agreement with

Crossley’s results.

Den Hartog [38] uses a model equivalent to Crossley’s. The equations are
linearized and the concept of anti-resonance frequency for small angle
displacement is discussed through the definition of the equivalent inertia of
the system. The effects of pendulum damping and nonlinearities are briefly
touched upon and expressions are presented for the dynamic response of the
absorber in terms of hyperelliptic functions. As far as the shift in the anti-
resonance frequency is concerned, remarks similar to those given by Crossley
[20] are presented, indicating an increase in the anti-resonance frequency due

to the wide angle of swing of the pendulum. No stability analysis is

132

presented by Den Hartog [38] or Crossley [19, 20].

Newland [7, 25] has presented the most comprehensive study on this
subject. However, his model does not consider the effects of damping.
Newland obtains an approximate solution using the Ritz minimization
method. Some stability analysis is presented in Newland [7] which explains
the unstable nature of the middle branch of the frequency response. The
stability results presented in this report extend those of [7] in the sense that
in addition it has been shown that there can be breakdown in the stability
of the periodic motions on the upper branch of the frequency response

curve. The efi'ect of jump phenomena is dealt with extensively in [25].

In this work it is shown that the method of harmonic balance with
nonlinear terms of up to cubic order yields quite accurate and satisfactory
results when compared to the results obtained from simulations of the full
nonlinear equations of motion (Equations 3.1.1). The method of averaging
[33] was also applied to the present system. This efiort was not fruitful,
however, due to the fact that the low-amplitude carrier response near the

anti-resonance frequency could not be captured using first-order averaging.

4.3- Suggestions for Future Work

A limiting factor in many designs of the CPVA is that the steady-state
pendulum angle amplitudes must become large when the disturbing torque
amplitudes are large. If the system is driven out of the region of validity for
linearization, then nonlinear effects can lead to catastrophic failures [7].
This is currently dealt with by making the effective pendulum path non-
circular. Cycloidal paths are common in helicopter applications; more

optimal paths are currently being worked on in the automotive industry

133

[18]. It is found, through research and experiments, that many of the
undesirable nonlinear properties of the conventionaly circular-path CPVAs
(for example, the jump phenomena) can be eliminated by incorporating
non-circular paths in the design of the CPVA. Also worthy of note is the
work of Mouzakis [23] on a monofilar pendulous absorber in conjunction

with helicopter rotor torsional vibrations.

An obvious study complementary to the one presented in this thesis is an
experimental investigation of the response of the CPVA. Further research
can also be carried out on other aspects of the dynamics of the CPVA. For
instance, more detailed analysis of the response of the absorber with respect
to non-circular paths deserves more attention. Also, the use of multiple-
pendula sets on shafts that are subject to excitations with more than one
dominant frequency needs to be investigated. From a design point of view,
one can study the range of system parameters for which an absorber with
optimal performance characteristics can be designed. In particular, one can
study the effects of combining the results of the two parts of this thesis to
obtain guidelines and data for the response of the CPVA with motion
limiting stops with respect to the turning frequency, anti-resonance
frequency and possible interactions with the stops. Also, the dynamic
interactions which result when multiple pendula are used on a shaft is of
interest, as are the efi'ects of the coupling between vibrations due to shaft
flexibility and/or translations of the entire structure. Furthermore, during
simulations of the full nonlinear equations of motion of the CPVA it was
observed that a subharmonic of order two occured near n=0.5 (chapter 3 of
the thesis) and has a significant effect on the response of the system at that
frequency. This characteristic of the CPVA is one which deserves further

study.

10.

11.

12.

13.

14.

15.

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