1W

WI

 
   

|_H..z‘ ‘H' ‘

Michigan State i
University i

This is to certify that the 3
thesis entitled

i
EXPERIMENTAL iNVESTlGATiON lNTO
EPICYCLOIDAL CEN

TRIFUGAL PENDULUM ‘
VIBRATION ABSORBERS

presented by

Peter M. Schmitz

has been accepted towards fulfiilment .
of the requirements for the ;
MS. degree in M
\

echanicai Engineering

A K t
( OLM 5
Major Professor’s Signature

6/1 3/2003
E

Date

MSU is an Affimrative Aerial/Equal Opportunity Institution

 

l
l
.
. .J
O P A!
I . _'

PLACE IN RETURN Box to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6’01 cJCIRC/Dateouepssosz

 

 

EXPERIMENTAL INVESTIGATION INTO
EPICYCLOIDAL CENTRIFUGAL PENDULUM
VIBRATION ABSORBERS

By

Peter M. Schmitz

A THESIS

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

MASTER OF SCIENCE

Department of Mechanical Engineering

2003

ABSTRACT

EXPERIMENTAL INVESTIGATION INTO
EPICYCLOIDAL CENTRIFUGAL PENDULUM
VIBRATION ABSORBERS

By

Peter M. Schmitz

The purpose of this work is to experimentally investigate the performance of a
centrifugal pendulum vibrational absorber system with the absorber constrained to
move along an epicycloid path. An existing experimental facility was modified so
that the behavior of either one or two absorbers could be studied. The displacements
of each absorber were measured, as was the resulting angular acceleration of the
rotor, for a range of torque amplitudes and orders. The results were found to be in
close agreement to theoretical predictions. In addition, some instances of non-unison

motion were observed and these are deserving of further theoretical investigation.

To Marcia.

iii

ACKNOWLEDGMENTS

I cannot begin to count the number of times I wanted to give up. Each time I
was convinced that there was no end in sight my advisor, Dr. Haddow, always had a
remarkable way of showing that if I stay the course there will be an end. Dr. Haddow
has a very contagious positive outlook on difficult problems and an overly optimistic
view about finding the solution. These traits are what made working with him very
enjoyable!

I would also like to thank the staff of the Physics Machine Shop for both their
personal and professional advice throughout my time at MSU. Firstly for giving me
the opportunity to develop a better understanding of machining practices. Secondly
for giving me a job to help sustain myself financially during my two years as a graduate
student. Most importantly I am deeply thankful to my boss Tom Palazzolo for setting
other jobs on hold so that my project could quickly be manufactured by utilizing
almost all of the human resources available. A special thanks is also extended to Jim
Muns and Tom Hudson for their time and effort put towards the completion of my
project.

The best kick in the butt to move on and not complain about the problems faced
on my research came from my soon to be wife Marcia. She reminded me of my favorite
saying that you should not complain about something if you actively do nothing about
it. She has been more than understanding and made many sacrifices so that I could

finish out my masters degree, and for that I am deeply grateful.

iv

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

1 Introduction
1.1 Background of CPVAs ..........................
1.2 Motivation .................................
1.3 Thesis Organization ............................

2 CPVA Mathematical Model
2.1 General Information ...........................
2.2 Equations of Motion ...........................
2.3 Change of Variable and Transformation to Dimensionless Form . . . .
2.4 Absorber Tuning and System Resonance ................
2.4.1 Absorber Tuning .........................
2.4.2 System Resonance ........................
2.5 General Path Function ..........................
2.6 Scaling ...................................
2.7 Approximate Equations of Motion ....................
2.8 Mistuning .................................
2.9 Method of Averaging ...........................
2.9.1 Approximate Steady State Solutions ..............
2.10 General Response for Various Path Types ...............
2.10.1 Effects of Varying Torque Order on Angular Acceleration . . .
2.10.2 Effects of Absorber Amplitude on Tuning Order ........
2.11 The Epicycloid Path ...........................
2.11.1 The Involute ...........................
2.11.2 The Evolute ............................

3 Experimental Setup and Physical Parameters
3.1 Experimental Setup ............................

vii

viii

AODr—‘H

m'xlCEO"

10
10
11
12
12
13
14
14
16
16
17
17
18
20
22

23
23

3.1.1 Motor Control ........................... 24

3.1.2 Absorber Measuring Devices ................... 25

3.1.3 Absorber and Rotor ....................... 27

3.1.4 Experimentally Obtained Values ................. 28

3.2 Absorber Lengths ............................. 30
3.3 Damping .................................. 31
3.3.1 Absorber Damping Ratio ..................... 31

3.3.2 Non-dimensional Damping Factor ................ 32

3.4 Absorber and Rotor Inertias ....................... 33
3.4.1 Total Inertia ............................ 33

3.4.2 Absorber Inertia ......................... 34

3.5 System Resonance ............................ 35
3.6 Parameter Summary ........................... 35

4 Experimental Results 37
4.1 Experimental Limitations ........................ 37
4.2 Percentage Mistuning ........................... 38
4.3 Experimental Absorber Tuning ..................... 38
4.4 Order Sweeps at Constant Values of T9 ................. 40
4.4.1 One Operational Absorber .................... 40

4.4.2 Two Operational Absorber .................... 40

4.4.3 Discussion ............................. 41

4.5 Torque Sweeps at Constant Values of Mistuning ............ 43
4.5.1 One Operational Absorber .................... 43

4.5.2 Two Operational Absorber .................... 44

4.5.3 Discussion ............................. 45

5 Conclusion 51
5.1 thure Work ................................ 52
BIBLIOGRAPHY 54

vi

LIST OF TABLES

3.1 Various inertias and 6 values for different N’s.

3.2 System resonant orders at various number of unlocked absorbers.

3.3 Summary of physical parameters obtained experimentally. ......

vii

2.1
2.2

2.3

2.4

2.5
2.6

3.1

3.2

3.3

3.4

3.5

3.6

3.7

4.1

4.2

LIST OF FIGURES

General CPVA Model, shown for N =3 absorbers. ...........
Single absorber, circular path CPVA defining the parameters used in
Equation 2.10. ..............................
Effects of varying torque order on IO] and absorber motion for (a) cir-
cular path absorbers (b) epicycloidal path absorbers, where T2=2T1;
T3=2T2. .................................
Effects of torque amplitude on absorber tuning and absorber amplitude
for (a) circular path absorbers (b) epicycloidal path absorbers, where
T2=2T1; T3=2T2. ............................
Epicycloid Path. .............................

Constrained path of CG following contour of the cheek with inexten—
sible flexible bands. ...........................

Block diagram of experimental setup showing basic components and
signal paths. ...............................
Close up of Absorber highlighting various parts. ...........
Physical experimental setup. ......................
Relationship between arc length 5' and absorber angle, 45, measured
from point c in Figure 2.6. .......................
Epicycloidal path compared to a circular path of radius 1", both paths
are tuned to same order it. .......................
Angular displacement of an absorber showing the amplitude decay as
a function of time. ............................
|d| versus T9 with all absorbers locked, n=l.40. ............

Rotor angular acceleration versus applied torque order, n, at three
different To values. ............................
Effect of increasing order, n, at T9 = 0.198 Nm and 6:0.07531 with
one active absorbers. (a) |6| of rotor, (b) absorber amplitude.

viii

19
20

39

4.3

4.4
4.5

4.6
4.7

4.8

4.9

Effect of increasing order, n, at T9 = 0.205 Nm and 6:0.16290 with
two active absorbers. (a) [0| of rotor, (b) absorber amplitude.

Iél versus T9 with one absorber unlocked at various levels of mistuning.

Absorber motion versus T9 with one absorber unlocked at various levels
of mistuning. ...............................

Iél versus T9 with two absorbers unlocked at various levels of mistuning.

Absorber amplitudes versus To for various levels of mistuning, two
absorbers unlocked. ...........................
Absorber response at 0% mistuning showing deviation from unison
response and the average response of both absorbers. .........

Time traces for two different T}; values from Figure 4.8 (0% mistuning)
(a) T9=L03 Nm (b) T9=1.92 Nm. ...................

42
44

45
46

47

48

NOMENCLATURE

 

 

Symbol Description

r"). Linearized absorber tuning order
n2 Second resonance order
R,- Length from center of rotation to center of gravity

7‘ Length from absorber pivot to absorber center of gravity
1 Length from center of rotation to absorber pivot
10 Inertia of all unlocked absorbers, measured from rotor’s CR
J Inertia of rotor and all locked absorbers

J Locked Total inertia of system including all locked absorbers

N Number of absorbers unlocked

6 Percentage of mistuning from it
n Applied torque order
To Fluctuating Torque amplitude

0 Constant applied torque
52 Mean speed of rotor

Instantaneous speed of rotor

v Dimensionless rotational speed of rotor

e Inertia ratio equivalent to b,
b,- inertia ratio

C Absorber damping ratio

Q

‘68»0"

Absorber damping coefficient
Dimensionless absorber damping
Scaled dimensionless absorber damping
Rotor damping coeflicient
Dimensionless rotor damping

Scaled dimensionless rotor damping
Arc length of absorber

Dimensionless absorber arc length
General path function

General dimensionless path function
Mistuning parameter

Dimensionless fluctuating torque amplitude
Dimensionless constant torque

Total mass of all unlocked absorber
Natural frequency of absorber

Second natural frequency of system
Averaged value of absorber amplitude
Averaged value of absorber phase
Dummy parameter used for definition of involute
Radius of base circle for involute
Radius of revolving circle for involute
Angle measured by encoder

Phase angle between absorber and applied fluctuating torque

xi

CHAPTER 1

Introduction

Torsional vibrations often arise in rotating systems because of variations in applied
torques. In some situations these vibrations have detrimental effects throughout the
entire system. Therefore, it is of importance to minimize these vibrations. Currently
there exist a broad array of possible ways to reduce these vibrations. They range from
complex active vibration control utilizing sensors and actuators, torsional dampers,
increasing overall inertia, and passive type vibration absorbers. An innovative way of
reducing vibrations in rotating machinery has been through the application of tuned

centrifugal pendulum vibration absorbers (CPVA).

1.1 Background of CPVAs

The principle behind how a CPVA works is analogous to that of the linear trans-
lational vibration absorber. In the translational absorber it is possible to reduce
vibrations of a primary mass by the introduction of a mass-spring combination tuned
to the excitation frequency. This type of absorber is only effective at one particular
frequency. As will be shown, the CPVA is tuned to a given order of excitation which
is independent of rotational speed. A detailed analysis pertaining to the absorber

tuning is provided in a later section, but in summary, a CPVA is tuned to a given

order rather than a particular excitation frequency, making the tuning of the CPVA
independent of the rotational speed.

In most rotating systems, the applied fluctuating torques are generally periodic,
based on the angular position, 0, of the system. For example, consider a 2 cylinder 4
stroke IC engine. Regardless of the rotational speed, each piston fires once per every
other revolution of the crankshaft. Thus the system has an order, n, of 1, where the
order refers to the rate at which the applied fluctuating torque repeats per revolution.
It is this order that the absorbers are tuned to such that the angular acceleration is
minimized.

The simplest CPVA consists of a pendulum attached to a rotor. The idea is to
tune the absorber such that the absorber counteracts the applied torque by creating
an opposing torque on the rotor. A further refinement to CPVAs is to choose the
path that the absorber’s center of gravity (CG) must follow. The path type influences
how well the given CPVA performs, both in stability and general performance terms.
The stability of CPVAs is an important topic, but is not discussed in detail in this
dissertation (consult the works of Garg [l] and Chao [2] for further reading). Ideally,
what is sought is to maximize absorber performance by utilizing a path type whose
period remains constant over a large range of applied torque.

There are three path types mentioned in this dissertation, but only one will be
explored in detail. Each path type has its advantages and disadvantages. The first
path to consider is the circular path. This is the easiest to manufacture and imple-
ment. However, as the amplitude grows, the period of the absorber begins to change,
i.e., the period of the circular path absorber is amplitude dependent. This causes
the absorber to mistune at modest levels of torque due to nonlinear effects and an
undesirable jump bifurcation can occur [3, 4].

The next type is the cycloidal path. This has a more desirable large amplitude

performance. In a gravitational field absorbers moving along a cycloidal path have

the ideal property of constant period of motion, independent of absorber amplitude.
However, when operating in a rotational field, the period is also slightly nonlinear
(i.e. dependent on absorber amplitude).

The final path to consider, and the focus of this dissertation, is the epicycloidal
path. It is known that in a rotational potential field, absorbers riding on epicycloid
paths experience periods that are independent of absorber amplitude [5]. The dis-
advantage of the cycloid and epicycloid is that they are generally more complex to

implement physically.

1 .2 Motivation

The problem at hand is to reduce the torsional vibrations that arise in a rotating
system as a result of applied periodic torques. A CPVA operating on a prescribed
path has been suggested to rectify this problem. Extensive analysis has already been
performed on the theoretical aspects of CPVAs for all of the paths mentioned above
[1, 2, 3].

Most recently, circular path CPVAs have been studied experimentally [4]. They
displayed encouraging results and supported the theory favorably. It was therefore
decided to extend the investigation to study absorbers riding on epicycloid paths.
To this end, an existing experiment was modified to accommodate a new absorber
design.

The primary interest of this dissertation will be on the performance of the ab-
sorber system and how it varies as it is mistuned relative to the applied torque. The
performance of the absorber system will be measured by two different criteria. The
first criteria will be how effective the absorbers are at reducing the rotor’s torsional
vibration, as measured by its angular acceleration. The second criteria is the over-

all range of applied torque over which the absorbers can function. As will become

evident, these two criteria oppose one another.

1 .3 Thesis Organization

In the next chapter the equations that govern the motion of the rotor and absorbers
will be derived. These equations will then be arranged into a dimensionless form and
subjected to an independent variable transformation. The method of averaging will
be set up to obtain approximate solutions to the exact equations of motion. The
second chapter will conclude with a general comparison between the responses at
various levels of torque amplitude for circular and epicycloidal path absorbers and a
discussion into the epicycloidal path.

In the third chapter the experimental setup will be introduced along with a de-
scription of what is measured and how it is converted into physical units. The physical
parameters introduced in Chapter 2 will be found via experimental methods. These
parameters include the absorber and rotor inertias, absorber damping ratio, absorber
tuning, and system resonance order.

Chapter 4 will present the experimental results for both one and two absorber
systems. The effects of varying torque order and torque amplitude will be explored.
The conclusions of this dissertation will be presented in the last chapter, along with

recommendations for future works.

CHAPTER 2

CPVA Mathematical Model

This chapter will serve as the theoretical basis for the dissertation. It follows from
the works by Garg [1], Chao [2], and Alusuwain [3], where the general equations of
motion and approximate solutions have been reported for various absorber paths.

In the subsequent sections of this chapter some assumptions will be introduced
to simplify the mathematical model of the CPVA. The derivation of the equations of
motion for both the absorbers and rotor will be outlined and the resulting equations
will be put into a dimensionless form. The general path function for an epicycloid
will be discussed and a proper scaling procedure introduced to facilitate the use of the
method of averaging. Particular attention will be given to the approximate solutions
for the epicycloid case proposed by Chao [2], and the derivation will be extended
to allow for a mistuning between the applied torque order and the absorber tuning
order. This chapter will conclude with a theoretical comparison between CPVAs with
circular path absorbers and epicycloidal path absorbers and a detailed description of

the geometry of the epicycloidal path.

 

Figure 2.1. General CPVA Model, shown for N =3 absorbers.

2. 1 General Information

The system to be studied is sketched in Figure 2.1. It consists of a rigid rotor rotating
in the horizontal plane with inertia J. Attached to this rotor are N general path type
absorbers each of mass m,, whose position in the horizontal plane can be defined by
a path function R,- = R,(S,~) and an arc length variable 3,. The path function is
measured from the center of rotation of the rotor, 0, to the center of gravity of the
absorber. The arc length’s origin is based at the vertex of the path function (i.e.
12,0 = R,(0)). The arc length variable is also taken to be symmetrical about the
vertex (i.e. R,(—S'.-) = Ri(+S,-)). The rotor is subjected to an applied torque of the
form

m0) = To + 7‘}, sin (77.0). (2.1)

Where To is the constant applied torque necessary to counteract rotor damping, T9
is the amplitude of the fluctuating torque, and n is the order of the applied torque.

Not shown in Figure 2.1, but nonetheless important, is damping. The resistance

between the rotor and its pivot point at O is assumed to be viscous with a damping
coefl'icient of co. Likewise there also exists damping on each of the N absorbers. Again,
viscous damping is assumed with a damping coefficient for each absorber taken to be
cai. It should be noted that accurately accounting for damping is very difficult since
its mechanism is complex and may depend on many factors, for example material
damping, friction, windage, etc. As will be shown in Section 3.3, an equivalent
viscous damping coefficient is empirically found from the actual system by a standard

logarithmic decrement test.

2.2 Equations of Motion

The equations of motion for the system are derived using the Lagrangian approach.
The preliminary steps to derive the kinetic energy may be found in Alsuwaiyan [3],

where it is shown for general paths that the kinetic energy is

 

N
-1 '2 ._.-2 22
KE— 2 {J6 +2321, [X,(S.)o +5. +2G.(s.)6s,]} (2.2)
where
X.<s.->=R;+’<s.-) and 0.45.):(Also—flasks»?

Note that X;(S,-) describes the path of the absorbers CG relative to the rotor. Various
path types may be studied by specifying X,(S’,-). In this dissertation, X,(S,~) will be
specified in Section 2.5 such that the equations of motion will apply to a CPVA with
absorbers riding on epicycloidal paths.

It should be noted that the potential energy of the system is zero since there are
no energy storage devices and all motions are in the horizontal plane. However, even

if the rotor was rotating in the vertical plane, the potential energy may be considered

negligible since the centrifugal force is much greater than that of gravity for any
modest level of angular velocity [2]. Applying Lagrange’s equations, it can be shown
(see Chao [2] and Alsuwaiyan [3]) that the equations of motion for the absorbers and

rotor are, respectively,

 

as. + G.<S.-)é — £33062] = —c...-S'.-, 1 s z' s N (2.3)
.. N dX- . . .. .. dG- . 2
J6 + 2:; m,[E§:(s,)S.-0 + X,(S,-)6 + G.(S.-)s.- + 5§(S‘)S‘ ] (2.4)

= Z [CaiGi(S.-)S,-[ — cod + To + T9 Sin (n0)

i=1

2.3 Change of Variable and Transformation to Di-
mensionless Form

Equations 2.3 and 2.4 represent a set of autonomous equations since no terms explic-
itly contain the independent variable t. To make these equations more amendable to
further analysis, it is convenient to change the independent variable to 0, the rotor
angle.

In order to accomplish this, we first redefine the rotor angular velocity in terms

of a dimensionless variable U, such that

(2.5)

c:
III

{OICD-

where Q is the mean speed of the rotor, determined by setting the fluctuating torque

to zero and holding the absorbers fixed at their respective vertices. In this case

Equation 2.4 simplifies to
C To

Physically, this means that the constant torque To serves to counteract the dissipative
torque from the rotor damping, which in turn sets the mean speed, 52.
Using the new dimensionless variable, v, the derivatives with respect to t are now

expressed with respect to 0 using the chain rule as follows:

0=dw=nmr

dt
" d2 ' II I
(.) = if?) = Q2U2(.) + Q2117} (.),
I d.
where (.) represents £52.
-- d20 ,
e.g. 6 = Et-z— = 02222)

The equations of motion, Equations 2.3 and 2.4, with the above substitutions
become

vsfi’ + [31+ g,(s,-)]v’ — %-d—f%(s,-)v = —/’lm-8;, 1 S i S N (2-7)

3:

2:11),- [%f;f(s,-)s;v2 + xi(s,)vv’ + gi(s,-)s;vv’ + gi(s,-)s§’v2 + fi§f(si)s[2v2]

(2.8)
+vv’ = 2i, afraid-(30321) — [102) + I}, + F9 sin(n6)
where the following dimensionless parameters have been defined
=s ms unfit: Amt-Rio (29>
_ L, _ T A _ T
bi‘ J PO—J—r‘i’f I“til—772917

and,

 

 

 

 

Figure 2.2. Single absorber, circular path CPVA defining the parameters used in
Equation 2.10.

2.4 Absorber Tuning and System Resonance

In order to gain an appreciation for the subsequent work, it is convenient to look
into the linearized equations of motion. From these, it is possible to determine the
absorber tuning, it, and the applied torque order that will result in resonant conditions
of the complete system. Using the model shown in Figure 2.2, the linearized equations

of motion are:

(J + m,(l + r)2)fl + cod + m,(rl + r2)¢ = To + T3 sin n0
(1+ T‘)9 + Si + 53:55 + (£92)S,- = 0.

(2.10)

2.4.1 Absorber Tuning

Each of the N absorbers has a natural frequency, run, that can be determined from
the linearized equations of motion, Equation 2.10, by assuming the angular speed of

the rotor is constant , i.e. d = {2. Under this condition, can = 9ft where

(2.11)

3t
||
~3|“-

The ’77. represents a fixed quantity determined by the geometry of the absorber system
shown in Figure 2.2. It can be shown that if 7‘7. 2 n and the absorber damping is zero
(or small), the resulting angular acceleration of the rotor, 6, will be zero (or small)

even if To aé 0. Specific values for l and 1‘ will be presented in the next chapter.

2.4.2 System Resonance

The rotor/ absorber system of Figure 2.2 has two natural frequencies, which can be
easily found from Equation 2.10. In future analysis it will be convenient to know
the order at which resonance occurs. The first natural frequency occurs at 0, which

corresponds to a rigid body mode. The second natural frequency is

 

N
i=1

r

The resonant order is defined from the above equation as

 

n2 = 5(1 + l imh‘ + ()2). (2.13)
r J i=1 2

Using the definition of b,- (Equation 2.9), Equation 2.13 becomes

 

n2 (2.14)

 

which shows that the resonance is close to the absorber tuning, it, since typically

b, << 1.

11

2.5 General Path Function

In Section 2.2, X,(S,) was introduced such that any variety of absorber paths could
be studied by simply specifying this function.

In this dissertation the CG will follow an epicycloidal path tuned to order ft.
A detailed description of what is meant by an epicycloidal path is deferred until
Section 2.11. To continue with the current analysis we will follow the work of Chao [2]

who has shown that an epicycloidal path can be described by
a:,-(s,-) = 1 — 732.93. (2.15)

With the path function for an epicycloid specified, g,(s,~) and $309,) become

 

 

dgi( ) -(fi2 + 771,4)81'

. . — ..... ~2 ~4 2 —-— =
g,(s,) — \/1 (n + n )3, and ds, 3 \/1 _ (fig + 73.4%?

(2.16)

 

The absorber motion is restricted by g,(s,-) in the sense that it must be kept real during
absorber motion [2]. The physical reason for this will be presented in Section 3.1.4.
However, experimentally there are far greater restrictions imposed on the absorber

motion. These will be discussed in Section 4.1.

2.6 Scaling

In the sections to follow the method of averaging will be employed to develop approx-
imate steady state solutions for the angular acceleration of the rotor and absorber
motion. Therefore, there is a need to define a small parameter that can be used to
correctly scale the terms in the equations of motion. Recall that the ratio of the

inertia of an absorber to the inertia of the rotor is defined as b,. This is physically

12

likely to be small, therefore we will define the needed parameter, 6, to be

N 1 N

E :0; = 'j Emino. (2.17)
i=1 i=1

This is the inertia ratio of all unlocked absorbers to the rotor inertia. Using this
definition other system parameters can now be scaled such that the equations of
motion will be in the appropriate form for the method of averaging. The following
system parameters that appear in Equations 2.7 and 2.8 are therefore redefined to
recognize that they are small quantities. The necessary system parameters scaled by

6 become,
fiat = 6/1111, I10 = «Silo, I‘o = CFO, and Pa = ef‘g. (2.18)

These 0(6) terms can now be substituted back into Equations 2.7 and 2.8. For more

details about the scaling see Chao [2].

2.7 Approximate Equations of Motion

The following results are a brief summary from the work of Chao [2]. Chao has shown
that the first order approximation of the rotor angular acceleration and the absorber

equations of motion are,

N
1 d - ..
’U’U’(0) = —6{1—V' Z (—2fi28j33 — fi2gi(8j)8j + fi(8j)822) — F9 sin 716’} + 0(62)
i=1
(2.19)
and

32’ + r128,- = cf + 0(62) (2.20)

13

respectively, with

N
c - , ~ d - ~ ,

f = “Mast + (5: + 9431)) Z(—2n28j3j — n2gj(sj)sj + -g—J—(sj)s’-2) — F9 sm n6 .
i=1

. .7
d8]

ZIP

Now that Equation 2.20 is uncoupled from the rotor equation it is possible to solve
for the absorber motion, 3. This is then substituted back into Equation 2.19 which

allows for the angular acceleration of the rotor to be obtained.

2.8 Mistuning

It is of interest to examine the effects of mistuning between it and n. Mathematically
this is accomplished by introducing a mistuning parameter 0, which will be defined

as

it = n(1+ ea). (2.21)

Substituting the mistuning into Equation 2.20 yields a new equation for the absorber

motion:

32' + 77.23,- = €(f — 207128,) + 0(62). (2.22)

2.9 Method of Averaging

The method of averaging will now be employed to find the first order approximate
solutions to Equation 2.22. Following the usual procedure, (see Chao [2] for details),

a transformation is made to polar coordinates

s,- = a, cos (qfi, — 719) and s,’ = nai sin ((15,- — n6). (2-23)

14

Next the above transformation is substituted into Equation 2.22 and integrated over
one period. The averaged values of a,- and 43,- are now expressed as I“, and (pi, respec-
tively. The result from this operation may be found in Chao [2](Equation 3.12).
These averaged first order equations will be further simplified by assuming that
each absorber is identical (m,- = m, co,- = 0,.) and that the absorbers undergo unison

motion,

rizr and 90,-:go for 1gz'gN. (2.24)

With Equation 2.24, the resulting averaged equations are

1 f“
I" = e {—Eflar + 7:3 cos (<p)F1(r)} (2.25)
and
f9 1 n
I _ __ - _ 5 2 _ _ _
1p — e{ 117' sm(go)F2(’r) + (4n 7' 2) no} (2.26)
where,
F1(r) = 51;; 02" sin2 :1: [1 —— (ft2 + 73.4)7'2 cos2 2:]% da: (2 27)
F20") = 2-11; 02“ cos2 x [1 — (it2 + I'i“)7‘2 cos2 2:]% dz

With all absorbers identical, the following quantities have been defined such that

they inherently account for the number of absorbers present:

Cao = NCa mo = Nmi Io = NA. (2.28)

Where J is equal to the inertia of the rotor plus the inertias of all absorbers locked
at their vertices and Io is the total inertia of all unlocked absorbers, Io = moRo2.

Similarly the damping coefficient, [10,, has become

. Nca _ Coo
”“ _ Nmn “ mon' (2'29)

 

15

Further insight into the damping will be given in Section 3.3. In particular, experi-

mentally determined values for [to will be determined from measured damping ratios,

C.

2.9.1 Approximate Steady State Solutions

The steady state values of 7' and (p are now determined by setting I" =-- cp’ = 0 in

Equations 2.25 and 2.26, which results in

 

~ ham"
I‘ =
" COW 2F1(r)
.. 1 ~ ~
I}; sincp = ((Ef‘t‘r’r2 — Z23) — oft) FETT)’ (2.30)

where n 2 ft to order 6 has been employed. Squaring and adding both sides results

in an expression for absorber motion versus applied torque as

I<><<>>

2.10 General Response for Various Path Types

 

 

 

In Chapter 1 the short comings of circular and cycloidal paths were discussed. In
both cases the effects of nonlinearities could cause mistuning of the absorber at large
absorber amplitudes. This mistuning introduces the possibility of jump bifurcations
to large amplitude absorber motion. Since both the circular and cycloidal paths
cause amplitude dependence, the circular path will be chosen as the representative
case. Its influence on the absorber motion will be compared and contrasted to the
equivalent system moving along an epicycloidal path. Approximate steady solutions
will be obtained from Equations 2.30 and 2.31 (for epicycloid case) and the associated

equations for the circular case (see Equations 2.23 and 2.24 in Alsuwaiyan [3]).

16

2.10.1 Effects of Varying Torque Order on Angular Acceler-

ation

When the order of the applied torque is increased, the magnitude of the angular
acceleration begins to decrease as it approaches the tuning of the absorbers, see Fig—
ure 2.3. The magnitude of the angular acceleration will eventually reach a minimum
value when forced near ft, which corresponds to the small amplitude absorber tuning.
Once the applied torque order is increased past it and begins to approach the second
natural frequency of the system, 722, (see Equation 2.12), the influence of the path
type on angular acceleration becomes distinguishable. The circular path type exhibits
jump bifurcations to large amplitude values in both angular acceleration and absorber
motion. However, the epicycloid path does not exhibit any such jump bifurcations.
It is also interestingly to note that from Equation 2.14 that 712 becomes very close to
it as 6 decreases. This results in a very sudden rise in the rotor’s angular acceleration
as n > it.

Although not shown here the cycloidal path differs from the circular path near res-
onance. Near resonance the circular case bends to the left, the opposite is encountered

with the cycloidal path.

2.10.2 Effects of Absorber Amplitude on Tuning Order

In the circular case, as the absorber amplitude grows the tuning order shifts to the
left. To demonstrate this Figure 2.3 is replotted to an enlarged scale in Figure 2.4.
They clearly show the dependence of the tuning order versus the absorber amplitude
at various increasing levels of applied torque. The epicycloidal path does not share
this amplitude dependence. As seen from the figure, the minimum point occurs at

the same order for each level of applied torque.

17

 

 

 

 

 

 

 

 

 

 

 

 
  

 

 

 

 

 

 

80 80 f .......... ; .......... . . ......... ’ ......... ‘
60 ........................................ [ 6o [ ........................................

g I I '. . .' :
§ 40 .................... ........... ........... 4O .. .................. . . ....... ..........
0 Increasing T 3 Increasing T g '
20 ................. , .......... ........... 20 .................. , ..........
0 . . . .

1 1.5 2 2.5 3 1 1.5 2 2.5 3

n fl

(a) (b)

Figure 2.3. Effects of varying torque order on [9] and absorber motion for (a) circular
path absorbers (b) epicycloidal path absorbers, where T2=2T1; T3=2T2.

2.11 The Epicycloid Path

The epicycloid path is the path traced out by a point on the circumference of a circle
of radius b rolling without slipping on the outside of a base circle of radius a. This
point is labelled as m,- in Figure 2.5 since it physically represent the CG of the i—th
absorber. This path will be referred to as the involute of the epicycloid. Likewise,

the evolute of the epicycloid is a scaled version of the involute, scaled by a factor of

a

m, and shifted such that the minimum of the evolute corresponds to the maximum

of the involute, (as shown in Figure 2.5). The evolute physically corresponds to the

18

 

 

 

 

 

 

    
 

,5. ........ f ........ ; ........ j ........ g ....... ..
glncreasingT ; ;

 

 

 

 

 

 

t, ...................................
8.0
o
5 .............
o L l L ; 0 L i A ; 4
1.4 1.42 1.44 1.46 1.48 1.5 1.4 1.42 1.44 1.46 1.48 1.5
n n
(a) (b)

Figure 2.4. Effects of torque amplitude on absorber tuning and absorber amplitude for
(a) circular path absorbers (b) epicycloidal path absorbers, where T2=2T1; T3=2T2.

19

cheeks used in the experimental application of the epicycloid CPVA.

CENTER 0 Y
OF [
ROTATION
IV [\ ’
/ I
__l / /

/
CHEEK/
.— /

CG PATH

 

 

xl’

Figure 2.5. Epicycloid Path.

Physically the CG of the absorber is constrained to follow the prescribed path by
using a flexible and inextensible material connecting the absorber to the base of the
cheeks at point c. Assuming that the flexible material is always under tension, as m,-
moves along the CG path the flexible material begins to follow the contour of the
cheeks. This is shown in Figure 2.6. Experimentally the evolute of the epicycloid is

used to constrain the CG of the absorber to follow the path defined by Equations 2.32.

2.11.1 The Involute

The involute of the epicycloid is easily found for a point mg, as shown in Figure 2.5.

With the point 771,- oriented as shown, the following parametric equations define the

20

0
CENHHI
OF
ROTKHON
a; I

(XEEK

  

 

CGPATH

Figure 2.6. Constrained path of CG following contour of the cheek with inextensible
flexible bands.

path that the point will move through,

221...,(6) = (a + b) cosfi + bcos Sit-bid

(2.32)
ymw) = (a + b) sin a + bsin @213

where 6 is a dummy variable used to construct the geometry of the involute.
Design wise, the desired value of the absorber tuning, it, and R0 will be set. Using
the geometry described in Figure 2.5 and recalling Equation 2.11, one can easily find

the relationships

 

 

Raft
a - 2.33
712 + 1 ( )
b = fi (1— —" )
2 it? + 1
where we have used the geometric relationship
a
l = . 2.
(a + 2b) a ( 34)

21

Hence we now can solve for a and b, given It and R0.

2.11.2 The Evolute

Recognizing that the evolute (the cheeks) are simply scaled versions of the involute

we can immediately obtain their cartesian coordinates as

 

 

xeww) = a :21) ((a + b) cosfl — bcos (a :- b) ,6)
yevow) = E—E—Z—b ((a + b) sinfl — bsin (fig—(26) .

Noting that the signs have been changed to put point c at its minimum point, in

accordance with geometric relationships mentioned previously.

22

CHAPTER 3

Experimental Setup and Physical

Parameters

The experimental setup will be described in this chapter along with a discussion on
how the dynamic data is extracted and processed.

Details of the empirically measured system parameters will also be presented, i.e.,
the absorber tuning (ft), absorber damping ratio (C), inertia of the rotor and all locked
absorbers (J Locked), and the mass of each absorber m,- will all directly be measured for
the system. Knowing these numerical values allows for the following parameters to
be calculated: absorber inertia (Io), absorber damping (fiao), inertia ratio (5), rotor

inertia (J), and the resonant order (712).

3. 1 Experimental Setup

A schematic of the experimental setup is shown in Figure 3.1 indicating the major

components and the path of the signals used to control and measure the experiment.

23

 

 

 

 

Instantaneous Speed Frequency to Instantaneous Speed
Filter Converter

 

 

 

 

 

 

 

 

 

 

 

 

“(mating 100°11'11“" Motor Broader
Torque
Generator r /AC Servo Motor

l— ’ —

Rotor
X

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' Motor ——
Adder Wk, /
\
Analyzer \
[ Slip Ring
Motor Speed EA - R8 Am “‘3'“
Controller .
Flucmanng [
Tome Data Logger &
instantaneous Speed [3.3th

 

 

 

 

 

 

Figure 3.1. Block diagram of experimental setup showing basic components and signal
paths.

3. 1.1 Motor Control

An Allen Bradley AC Servo Motor along with an associated Motor Controller gener-
ates the torque, T9 (6), as defined in Equation 2.1. This is accomplished by measuring
the angular position, 6, of the rotor via a 1000 pulse/ rev encoder that is attached

directly to the motor shaft. This measurement is utilized in three distinct ways:

1. To measure and control the mean speed, 9, of the shaft. This is associated with

the To component of the torque.

2. To generate a sinusoidal control signal that is proportional to the angular po-

sition of the shaft. This is associated with the T9 sin (n6) component of the

24

torque.
3. To measure the instantaneous speed, 6(t), of the rotor.

The signal 6(t) mentioned in item three is obtained by passing the TTL pulses from
the motor encoder through a frequency to voltage converter (F2V). The output for
this device is an analog voltage that is proportional to 6(t). This signal is routed to
the Labview Data Logger, real-time FFT analyzer, and a 1 Hz low-pass filter.

The purpose of the low-pass filter is to extract the DC component from the 6(t)
signal, i.e. the mean speed, (I. Ideally, this should be constant but in practice it was
found to slowly drift, on a time scale of approximately 5-10 seconds. Therefore, a
secondary control system was established to ensure that this slow drift was negated.
This is marked as a dashed line in Figure 3.1. The net result is a very slowly varying
voltage that is proportional to the To component required to maintain the constant
mean rotor speed, 9. In all of the experiments to be reported in Chapter 4, this mean
speed was fixed at 300 rpm.

Returning to item two, the alternating component of the torque is combined with
To in the adder shown in Figure 3.1. It is important to note that a sinusoidal func-
tion of angular position is required (i.e., T9 sin (716) and not To sin (nflt)). This is
accomplished by generating, in a Labview program, a sine wave that is based on
the TTL signal from the motor encoder. Each pulse from the encoder represents an

angular change of fig rad. Hence, for the i-th pulse in the sequence, we need the

 

Labview program to generate an analog output proportional to To sin (1115530). This is

accomplished in real time and works for real values of n from 1 to 10.

3.1.2 Absorber Measuring Devices

The absorber position is measured using encoders (supplied by USDigital S5D-360—

LB). Differential optical encoders were chosen to eliminate noise that is generated

25

 

Figure 3.2. Close up of Absorber highlighting various parts.

along the signal path. The encoder shaft is centered at point c (see Figure 2.5) and
uses a lightweight arm to measure the angular position of the CG of the absorber
(Figure 3.2). The wiring of the encoders is passed down the shaft and connected
to a 20 channel slip—ring. From the slip-ring the signals from the encoders go into
a differential receiver (USDigital EA-R8), which converts them to single-ended TTL

Outputs. A specialized board (MicroStar MSXB 023) and a Labview program converts

these quadrature signals into degrees.

26

 

Figure 3.3. Physical experimental setup.

3.1.3 Absorber and Rotor

The frame, rotor, and absorber assembly are labelled in Figure 3.3. The rotor consists
of: the main shaft that is connected to the motor through a coupler; a steel flywheel
that can accommodate up to four absorber; and the base parts of the absorber as—
sembly that are attached to the flywheel. The absorbers are attached to the absorber
assembly in a bifilar fashion. Figure 3.2 shows is closeup of the actual implementa-
tion of the epicycloidal path CPVA. The absorber is attached to the flywheel through
0.002” blue tempered spring steal bands. Visible in this figure is the encoder used to
measure the angle of the absorber and the cheeks that are located either side of the

spring steal bands that constrain the absorber’s CG to follow an epicycloid.

27

3.1.4 Experimentally Obtained Values

Experimentally, the values of To, [6|, and [S [ need to be measured. Each of these will

be discussed in turn.

Obtaining To

The fluctuating torque, T9, is obtained by measuring a voltage that is proportional to
the applied torque taken directly from the motor controller. This voltage is made up
of the constant applied torque, To, and the fluctuating applied torque, T9. Of interest
is the magnitude of the fluctuating applied torque at the order, n. An FFT analyzer
is used to measure the peak corresponding to the applied order, It. To convert this
millivolt reading to Nm it must me multiplied by a calibration constant. That has

been found to be 0.01853 Nm/mV.

Obtaining [6|

The angular rotor acceleration, [6|, is not measured directly. Instead, the 6(t) signal
that leaves the F2V is feed into the FFT analyzer. The peak that corresponds to
the applied order, n, is recorded and then multiplied by 7152 to convert it into an

acceleration.

Obtaining S

The equation of motion of the absorber is written in terms of the arc length, 3.
However, experimentally this length is difficult to measure and so the angle the CG
moves through is measured using optical encoders. Now S needs to be expressed in
terms of the measured angle, (b.

Using the parametric equations for the involute (Equation 2.32), it is possible to

28

determine the arc length, S, by using the relationship

5_ fofi( l/(dfl L$invw )2 )+(d_-y:6nnw))2dfi

.-__._.. G?)

 

which simplifies to,

N ow, an expression relating the measured angle 45 to the dummy variable, 6, will be
obtained. To do this, the origin must be moved to coincide with the actual location
that d) is measured from, i.e., from point 0 to point c (see Figure 2.5). The angle (b
(see Figure 3.5), is then described by

n = tan—1 (%) . (3.2)

Using Mathematica and the physical parameters obtained in this chapter, it is
possible to show numerically the relationship between (15 and S (see Figure 3.4). Sur-
prisingly, the relationship seems to be linear, resulting in

S
= —. 3.3
at r < >

It is interestingly to note that the relationship found here is the same as the arc
length of a circle of radius 1‘ whose center is at point c. It is likely that this relationship

could be proved. To better visualize this, Figure 3.5 compares the circular path to

that of the epicycloid path. Both paths shown are tuned to the same order.

29

 

 

 

 

1.5 , , f , T I
1,_. ....................................................... .1
8
m
‘6
tr:
0:
o i i i i i i
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
ArcLengtMm)

Figure 3.4. Relationship between arc length S and absorber angle, (25, measured from
point c in Figure 2.6.

3.2 Absorber Lengths

The quantity 1" can easily be measured from the experimental apparatus. It is simply
the length of the flexible steal bands and is found to be r =0.0576m. Rather than
experimentally finding the exact position of the absorber’s CG, the distance I is found
indirectly from the measured absorber tuning, it = 1.465 (this value will be reported
in Section 4.3). Hence recognizing that 62 = f; we find I =0.1236m. If required, R0

can be calculated from the relationship R0 = l + r.

30

EPICYCLOID

 

 

Figure 3.5. Epicycloidal path compared to a circular path of radius r, both paths are
tuned to same order it.

3.3 Damping

The damping mechanisms in the system are likely to be quite complicated. However
in this investigation it is assumed to be viscous throughout. The damping on the
rotor was discussed in the previous chapter and serves only to balance the constant
applied torque, To, thus fixing the mean speed, 0. The absorber damping will now

be discussed.

3.3.1 Absorber Damping Ratio

The following derivation of the damping ratio, C, is conducted on only one absorber.
The damping ratio will be found experimentally by running the rotor at a constant
mean speed and applying sufficient fluctuating torque, To, to cause the absorber to
exhibit large amplitude motion. The fluctuating torque will then be immediately
cut off to allow the absorber to experience free vibration. Shown in Figure 3.6 is the
resulting angular displacement of the absorber from this test. Next the log decrement

method is utilized yielding a damping ratio of C=.00344.

31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15 ......I ......... I ........ II ....... I ....... II ......... I ........ I ........ _
,0 . ... _.. . .. _. . .. ... . . ._ .. . ..... _
5.1.
....................................... _
........ ......... . ......... ,. ........ ........ , ......... . ........ _, ________ , ...... -
10 105 11 115 12 12.5 13 135 14 145 15
Tlme(s)

Figure 3.6. Angular displacement of an absorber showing the amplitude decay as a
function of time.

3.3.2 Non-dimensional Damping Factor

Assuming a constant rotor speed (i.e. 6 = 0), Equation 2.3 reduces to

.. ldX 2 _ .
mo [3 — 515009 [ _ was. (3.4)

Using Equation 2.15, noting that

dX _ -2

32

and using Equation 3.3, e.g. S = m5, and some dimensionless substitutions introduced

in Chapter 2.3, Equation 3.4 reduces to the following form,
3+ fin + (items = 0. (3.5)

Comparing this to the desired form of a viscously damped linear oscillator

as + 252113 + (mm = 0, (3.6)
we have
9m— : 2452a (3.7)
and so, from Equation 2.29
pa = 205. (3 8)

3.4 Absorber and Rotor Inertias

Determining the respective inertias will serve to fix the small parameter, c. The
total inertia, JLocked, will first be determined by locking all absorbers and recording
the magnitude of the angular acceleration at various levels of increasing T9 for an
arbitrary applied torque order. Then the rotor inertia, J, will be found by subtracting

10 from J Locked -

3.4.1 Total Inertia

Figure 3.7 shows the experimental results of a test completed with all absorber locked,
running at n=1.40. Since Torque = Inertia =1: AngularAccerle'ratt'on, determining
the slope of this graph will allow one to determine the total inertia. The total inertia

for the system is found to be JLocked=0.0811 kgm2.

33

 

30 I I I T I I T I T

 

 

 

o i i i i i i i i i
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Torque (Nm)

Figure 3.7. [6| versus T9 with all absorbers locked, n=1.40.

3.4.2 Absorber Inertia

The absorber inertia is not obtained directly. Instead the absorber’s mass is measured
and then multiplied by R: to obtain Io. Recalling that 1,, describes the inertia of all

unlocked absorbers, and that the measured mass of one absorber is 0.173 kg, we have

I, = mo(Ro)2 = N 4 017302,)?

The inertia of the rotor, J, is then obtained by,

J = JLocked _ I0-

34

Physical values for various numbers of unlocked absorbers are shown in Table 3.1.

Included in this table are the inertias of the absorbers, rotor, and 6.

Table 3.1. Various inertias and 6 values for different N’s.

N Io J 6
(kg :1: m2) (kg =1: m2)

1 .00568 .07542 .07531
2 .01136 .06974 .16290

 

 

 

 

 

 

 

 

 

 

3.5 System Resonance

The system resonance may be found in accordance with Equation 2.13. Table 3.2

shows the associated values for the resonant order, n2.

Table 3.2. System resonant orders at various number of unlocked absorbers.

 

 

 

N 77.2
1 1.519
2 1.579

 

 

 

 

 

 

3.6 Parameter Summary

A summary of the physical parameters obtained in this chapter are shown in Table 3.2.

35

Table 3.3. Summary of physical parameters obtained experimentally.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Values Description
Z .1236 m Length 0 to c from Figure 2.5
r .0576 m Length 0 to m in orientation shown, Figure 2.5
R0 .1812 m Length 0 to m in orientation shown, Figure 2.5
ft 1.465 Tuning of each absorber
C .00344 Absorber damping ratio
10 (N=1) .00568 kg*m2 Absorber inertia for one active absorber
Io (N =2) .01136 kg"‘m2 Absorber inertia for two active absorbers
e (N=1) .075314 Inertia ratio for one active absorber
c (N=2) .162890 Inertia ratio for two active absorber
71.2 (N =1) 1.519 Resonant order for one active absorber
712 (N =2) 1.579 Resonant order for two active absorbers
J (N=1) .07542 kg"‘m2 Inertia of rotor with one active absorber
J (N=2) .06974 kg*m2 Inertia of rotor with two active absorbers
J Locked .0811 kg*m2 Inertia of rotor with all absorbers locked
f2 107r rad/s2 Mean speed of rotor

 

36

 

CHAPTER 4

Experimental Results

In the following sections the results from a series of tests on one and two absorber
systems will be reported. The CPVA’S performance will be evaluated using two
criteria. Firstly, how effective the absorbers are at minimizing the angular acceleration
of the rotor and secondly, the range of torque the system can operate over.

The tests will be conducted at various levels of mistuning and torque amplitudes.
Recordings of both the magnitude of the first order angular acceleration and the
absorber amplitude will be reported and compared to the approximate solutions that

were derived in Chapter 2.

4.1 Experimental Limitations

It was not feasible to run the experiment at orders near 1 or 2, because the Allen
Bradley AC servo motor used in this experiment generated small speed fluctuations
at these orders. This interfered with measuring the true magnitude of angular accel-
eration. Hence the absorbers were designed to be tuned to an order close to it = 1.50,
thus avoiding the problem orders.

There also exists a limit to the amplitude the absorber may swing through on

account of the cusp point that exist along the epicycloid paths. This can be seen in

37

Figure 2.5 and is marked as point d. It corresponds to an angular displacement of ap-
proximately i70° measured either side of point c in the same figure. Mathematically,
this point is associated with the value of s,- where gi(s,-) of Equation 2.16 becomes
imaginary.

There are further physical limitations imposed on the absorber amplitude. Snub—
bers fixed to the rotor prevent the absorber from travelling any greater than :l:40°.
In addition, amplitudes greater than 25° fatigue the metal bands used to constrain
the absorber to the desired path resulting in premature failure of the metal bands.

Therefore, continuous operation at high absorber angles is avoided.

4.2 Percentage Mistuning

Since experiments will be conducted for one and two absorber systems it will be
convenient to quantify the mistuning using a parameter that is independent of e, the
inertia ratio of the system. Theoretically the mistuning is introduced using a, but
this may be misleading since it is a function of 6. That is, for a fixed torque order, n,
the value of o for one absorber will be different than for a two absorber case. This

issue can be alleviated by introducing a percent mistuning parameter, 6, such that

n = (1— 1’66)fi‘ (4.1)

This now makes the mistuning independent of e and will make direct comparisons for

both one and two absorber cases easier to make.

4.3 Experimental Absorber Tuning

In this first test the applied torque order is gradually increased to observe where the

angular acceleration reaches a minimum value. Three different levels of T9 are used

38

 

 

+ 0.203 Nm
x 0.405 Nm

 

 

 

 

 

 

 

0
1.4 1 .41 1.42 1 .43 1.44 1.45 1.46 1.47 1.48 1.49

Figure 4.1. Rotor angular acceleration versus applied torque order, n, at three differ-
ent T9 values.

to verify that the absorber tuning does not exhibit any amplitude dependence.

As can be seen from Figure 4.1 the magnitude of the angular acceleration reaches
a minimum value at an order of 1.465. This did not change appreciably for the three
different levels of T9. It is concluded that the absorber tuning is not affected by the
absorber amplitude over the range of torque tested.

Additional tests were completed on the second absorber and it too was found to
have the same tuning order. Therefore, for the remainder of the study we will take

it = 1.465 for both absorbers.

39

4.4 Order Sweeps at Constant Values of T9

In this section the effect that torque order has on [6| and the absorber motion will
be investigated. While holding the physical value of To constant, It will be increased
and the corresponding values of [6| and absorber motion will be recorded. The values
will be compared to the theoretical predictions. Of particular interest is the response
close to perfect tuning, it, which has been found to be 1.465 for all absorbers. The
system resonance, n2, which is given by Equation 2.13 will also be studied. This

depends on the inertia ratio, 5, and so depends on the number of unlocked absorbers.

4.4.1 One Operational Absorber

With one absorber unlocked, and I}, held constant at 0.198 Nm an order sweep was
conducted and the results are shown in Figure 4.2. The experimental values for [6|
and absorber amplitude correlated quite well with the theoretical predictions. As
the order increases and approaches an 1‘2 of 1.465, [6] gradually decreases and reaches
a minimum value at perfect tuning. [6] begins a sharp rise in the vicinity of the
resonance, 11.2. This resonance was found experimentally to be at 1.52. This compared

well with the theoretical value of 712:1.519 given by Equation 2.14.

4.4.2 Two Operational Absorber

With two absorbers unlocked, and Ta held constant at 0.205 N m an order sweep
was conducted and the results are shown in Figure 4.3. As for the one absorber
system, the theoretical predictions matched the experimental results quite well. As
n is increased the absorber motion follows the unison motion assumption up to the
area near perfect tuning, see Figure 4.3(b). It is observed that the unison motion
assumption is not valid in the vicinity of ft to 712. However, as the order increases

past 712 the absorber amplitudes again match one another and the unison motion

40

 

 

25 r a .
1;. x Experimental
1 . — Theory

 

 

 

 

 

 

 

.........

 

 

 

 

 

 

 

 

 

 

   

(a) (b)

Figure 4.2. Effect of increasing order, n, at T9 = 0.198 Nm and e=0.07531 with one
active absorbers. (a) [6| of rotor, (b) absorber amplitude.

assumption becomes valid. Experimentally the resonance, 712, was found to be 1.58

which is in agreement with the theoretical prediction of 1.579 predicted in Section 3.5.

4.4.3 Discussion

Before discussing any of the results of Section 4.4, it should be emphasized that the
type of experiment conducted here, i.e. order sweeps, are very unlikely to occur in
practical applications. In almost all real-world usage, the forcing order, n, will be
fixed by the type of system. For example, a single cylinder 4 stroke IC will have a
dominant n = % order in its torque fluctuation. However, the experimental facility
has been designed to allow this order parameter to be varied. Hence, the influence of
the variation between 11 and it can be easily investigated.

To this end, the order sweeps shown in Figures 4.2 and 4.3 give a very good
overview of the general behavior of the complete system. The tuning order, fr, can
be clearly seen in Figures 4.2(a) and 4.3(a) (and in more detail in Figure 4.1). The

value of 71:1.465 agrees closely with the theoretical predictions (see Figure 4.1) and

41

 

 

 

 

x Experimental
7— “99'!

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

   

(a) (b)

Figure 4.3. Effect of increasing order, n, at T9 = 0.205 Nm and 6:0.16290 with two
active absorbers. (a) [BI of rotor, (b) absorber amplitude.

the resonant peak associated with 77.2 (see Equation 2.14) is measured to be 1.52 and
1.58 for the one and two absorber cases, respectively.

There are other interesting points to note from the order sweep experiments.
Theoretically, as n << ii the absorber motions are expected to become very small,
since the system is now responding well below it’s resonance. In this limit, as n
becomes small, the absorbers act as if they were locked and so the limiting case of
lél = To / J Locked should be reached. This is indeed the case and gives an independent
check on the empirical value of J Locked found in Section 3.4.1.

The counterpart of this is for the case of 72. >> ft. Now the absorbers are effec-
tively floating in space and not responding to the applied alternating torque, i.e., the
effective inertia of the system is now J = hooked — IO and thus the limiting value of
lél should be |6| = T9 / J. Again this theoretical limit is found to agree very closely to
the experimental findings.

One final point worthy of comment, and discussed more fully in Section 4.5.3, is

the issue of non-unison response. In the two absorber case (Figure 4.3) both absorbers

42

move as one over a wide range of n. However, close to n = ft, an instability occurs
and they no longer have the same amplitudes. With reference to Figure 4.3(b) there
are ranges of n over which each absorber has a different amplitude. As stated, this
will be discussed in more detail in Section 4.5.3 where torque sweep experiments are

undertaken.

4.5 Torque Sweeps at Constant Values of Mistun-
ing

In this section the effect that torque amplitude has on |6| and the absorber motion
will be investigated. While holding n constant, To will be increased and the corre-
sponding values of |0| and absorber motion will be recorded so that these values may
be compared to the theoretical responses. Of interest here is how closely the experi-
mental results match the theory and to study the possibility of bifurcations from the

unison response.

4.5.1 One Operational Absorber

With one absorber unlocked a series of torque sweeps were performed at five different
levels of mistuning. The results of these torque sweeps are shown in Figures 4.4
and 4.5. It was found experimentally that mistuning values of 5% and 3% best
matched the theoretical predictions. As the mistuning approached perfect tuning,
deviations from the theory became apparent.

The deviation from the theoretical predictions at and near perfect tuning may be
attributed to a combination of factors. It should be recalled that both the damping
ratio, C, and inertia ratio, 6, are extremely small (0.004 and 0.07531 respectively).

Consequently the angular acceleration of the rotor, IF], the absorber motion, S , and

43

 

 

 

 

 

 

 

Torque (Nm)

Figure 4.4. |0| versus T9 with one absorber unlocked at various levels of mistuning.

the phase relationship between the absorber and the applied torque all become very
sensitive to small changes in n, close to perfect tuning (see for example Figure 4.2).

This will be discussed more fully in Section 4.5.3.

4.5.2 Two Operational Absorber

A second set of torque sweep experiments was completed for the case of two absorbers.
Physically this meant that the second absorber, which was locked in place for the
previous experiment, was freed. Consequently, the inertia ratio, 6, changes from
0.07531 to 0.16290. The test results are shown in Figures 4.6 and 4.7. Figure 4.6 is
the counterpart of Figure 4.4, where the rotor’s acceleration is plotted as a function
of the torque, for various % of mistuning.

It should be noted that the % mistuning is independent of the number of active

44

 

 

 

 

 

 

 

 

3° ' w r Y F F
It
. u . - x . - u
25-...... ..... ........... ......... ii. ........... 3 ............ g .......... :, ......... ..
' ' *‘ ' 3 + 5 3
X . . .
+ I
20,. ............ .......... ........... ..
: . a 2
§ . 3 .‘ 2 2
515.. ........................... x ............... 3..........‘. ............ .... ......... q
8 I 0 ' i 3
0 : . ; :
.‘ 2 ;
‘ : : ,1 ‘ E O SSS-Experiment
10.. .......... .. ........ ......... .......... + SSS-Experiment .......... _,
: 4., : "’ : x vat-Experiment
' . f e (HS-Experiment
: ; ASS-Experiment
. .' » —Theory
5...... .. .5. . .......................... - ............ . ......... .4
' : :
0° 3 2 3
f.” I 1 i
. A l l l l l l
0 0.5 1 15 2 25 3.5 4
Torque(Nm)

Figure 4.5. Absorber motion versus T9 with one absorber unlocked at various levels
of mistuning.

absorbers, see Equation 4.1. That is, a mistuning of 5% corresponds to the same
n of 1.392, regardless of the number of absorbers active. Hence, the figures from

Section 4.5.1 can be compared directly with those in this section.

4.5.3 Discussion
One Operational Absorber

Comparing Figure 4.4 with 4.5 one may make the following observations for one un-
locked absorber. Higher levels of positive mistuning (i.e. n < it) allow one to operate
over a larger range of torque. However the effectiveness at reducing lfll becomes worse
the further one moves from n = 77.. At 0% mistuning the effectiveness at minimizing

[0| is the greatest, but the associated absorber amplitude is relatively large. The worst

45

 

 

 

 

 

 

 

 

 

0 Mounted-Experiment E

O 5%-Experlment 1 A.-.'r3- ‘ ‘.

+ 3%-Experimem g Locked- ; 5

X 1%-Experiment ; j : :

25.. Q 0%-Experlmgm ............ .............

- -1%-—Exper|mem : : : : 596'

—Th°0fy E E E E E

s 3 i s i 9 o i

20... ........... ........... ..................... ......... o.............

2 § : 3 z 3 3%?

: : : : :’ :

“g 2 3 3 2 . z ‘ .

E15.. ........... ........... _ .......... ......... .,

: i 3 ‘. I 1+ 1

E ‘ E 1 i :+ H5 3

f : f +§ f f

i E E e 1 3 1%3

1o............. ....... ’1................ .......... I ............................................. ‘

r f g 3 0%;

5.. ........ I, ........ ’.-... ............ .................................. -/

."v : . " x x :1T‘

, ‘ . / E ' : 3 ”1% E

//-/ 3 I I: *‘i *3 E E E

. {4“ l L 1 L 1 1 1

. 0.5 1 1.5 2 2.5 3 3.5 4
Torque(Nm)

Figure 4.6. [6| versus T9 with two absorbers unlocked at various levels of mistuning.

overall performance occurs at -1% mistuning. This is a consequence of the resonant
order, 712, being very close to the absorber tuning, ii. With reference to Figure 4.2,
-1% mistuning (n=1.48), is located just to the right of perfect tuning and very close
to 722. Hence, small changes in n have a large effect on the response.

The most likely reason for the discrepancy between theory and experimental re-
sults from the influence of the phase, d), as defined in Equation 2.23. The value of (b is
critical to the evaluation of |0|. The phase is very sensitive to the absorber damping in
the region of n2 and it is also likely to be affected by slow drifts in Q. As a result, one

cannot expect high correlation between theory and experimental results for n 2 72,2.

46

5% Mistuning 3% Mistuning
, , 25 ........... ‘ .......... : .......... . .......... .

Degrees

 

 

 

 

 

 

 

 

 

 

+ Absorber 1
at Absorber 2
— Theory

 

 

 

 

 

 

Torque (N'm)

Figure 4.7. Absorber amplitudes versus T9 for various levels of mistuning, two ab-
sorbers unlocked.

Two Operational Absorber

The experimental results support the general theoretical trends. Since there are now
two absorbers active, there is twice the inertia available to absorb IOI. Hence, the
absorber amplitudes are smaller than in the one absorber case. Consequently higher
levels of applied torque can be accommodated.

Of more interest is the absorber response near perfect tuning. By inspection of
Figure 4.7 it is clear that the absorber amplitudes are experiencing non-unison mo-

tion. However, when the experimentally obtained time traces for the two absorbers

47

 

 

 

 

 

Degrees

 

 

 

l l l J 1 L l

o 0.5 1 1.5 2 2.5 3 3.5 4
Torque (Nm)

Figure 4.8. Absorber response at 0% mistuning showing deviation from unison re-
sponse and the average response of both absorbers.

are averaged the resulting amplitudes fit the theoretical predictions, see Figure 4.8.
The details of this are worthy of further comment. Consider the time histories of
the absorber motion shown in Figure 4.9. Figure 4.9(a) shows the time trace for
each absorber at Tg=1.03 Nm. The relative phase difference between the two ab-
sorbers is found to be 61° and so the resulting average amplitude is less than either
of the individual amplitudes. Indeed, it is very close to the theoretical prediction, see
Figure 4.8.

Viewing Figures 4.9 more carefully, it is clear that a true steady-state response
has not been reached. Although only 0.5 seconds of data are shown, the test was
run for five minutes at each torque level prior to collecting data. Yet there is still

a slight amplitude variation present. There are a number of possible reasons for

48

 

 

 

 

 

 

 

 

 

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
time (s) time (s)

 

 

 

 

(a) (b)

Figure 4.9. Time traces for two different To values from Figure 4.8 (0% mistuning)
(a) To=1.03 Nm (b) T9=1.92 Nm.

this. Clearly a non-unison response is present and so from the analysis presented
here, we cannot predict what the true response will be. However, it is likely to be
very sensitive to small changes in the input parameters of the system, for example
the mean speed, 9. Very small changes in this may add transients to the responses
that take a considerable time to decay. Alternatively, the responses observed may be
the true post bifurcation responses that theory could predict. This is deserving of
attention in future work.

Returning to Figure 4.9, a second pair of time responses are shown for a torque
level of Tg=1.92 N m (Figure 4.9(b)). Here the relative phase difference between the
two absorbers is found to be only 6" and so the average motion is very close to the
average of the individual absorber amplitudes, i.e. mid-way between the values shown
in Figure 4.8.

Finally, as shown in Figure 4.6, although there is close agreement between theo-
retical and experimental results in many cases the agreement is not so good as one
approaches n 2 n2. The most likely reason for the discrepancy results is from the

phase, a5, as defined in Equation 2.23. In addition to the reasons stated in the one

49

absorber case, the introduction of two absorbers is likely to make the system even

more sensitive to small changes in the input parameters.

50

CHAPTER 5

Conclusion

It was the goal of this dissertation to explore the effects of mistuning on the perfor-
mance of absorbers in a CPVA system riding on epicycloidal paths. The following

will summarize the conclusions that are based on the findings of this dissertaion.

0 Introduction of more absorbers increases the effectiveness of the absorber sys-
tem. As the number of absorbers increases, more inertia is made available to
help absorber the fluctuating torque. Therefore, larger torque amplitudes may

be applied to the system.

0 Bifurcations to non-unison response were observed. In the two absorber case,
small amounts of fluctuating torque would often result in the absorbers under-
going non-unison motion. This is especially true if operating close to perfect

tuning.

0 Generally there was excellent agreement between the theory and experimental

results.

51

5. 1 Future Work

Based on the successes encountered from the experiments conducted here, many pos-

sibilities are now open for exploration.

0 Since bifurcations from the unison response were observed for all mistuning

values, theoretical work should extended on this behavior.

0 Future tests should also measure the phases of each absorber. This would be
useful for more detailed comparison with the theory - both for the unison and

non-unison responses.

0 The effects of subharmonic absorbers should be explored, i.e., absorbers tuned

to % the applied torque order.

0 Effects of additional absorbers should also be explored to see their influence on

reducing |6| and study the bifurcations to the non-unison response.

52

BIBLIOGRAPHY

53

BIBLIOGRAPHY

[1] V. Garg, Effects of Mistuning on the Performance of Centrifugal Pendulum Vi-
bration Absorbers. MS Thesis, Michigan State University, 1996.

[2] C.-P. Chao, The Performance of Multiple Pendulum Vibration Absorbers Applied
to Rotating Systems. PhD Thesis, Michigan State University, 1997.

[3] A. S. Alsuwaiyan, Performance, Stability, and Localization of Systems of Vibration
Absorbers. PhD Thesis, Michigan State University, 1999.

[4] T.Nester, Experimental Aproach to Centrifugal Pendulum VIbration Absorbers.
MS Thesis, Michigan State University, 2002.

[5] H. H. Denman, “Remarks on brachistochrone-tautochrone problems,” American
Journal of Physics, vol. 53, pp. 781—782, 1985.

54

   

    

   
 

  

EFISI

G‘AleT‘AT‘E LlN[I\[I ‘ ‘ TY LIBRARIES
[[[l[Willi
[w ‘[v[[.‘.[[[ ll[1[[ [ «‘l[.‘[*[
1293 02470 0027

‘ [MIC‘HI
[1]]
l l [ l

3