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' LIBRARY
7% Michigan State
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This is to certify that the
thesis entitled

Theoretical and Experimental Investigation Into the Transient
Behavior of Centrifugal Pendulum Vibration Absorbers

presented by
Daniel Palmer

has been accepted towards fulfillment
of the requirements for the

MS. degree in Mechanical Engineering

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2/05 clelRC/DateDue.indd—p.15

THEORETICAL AND EXPERIMENTAL
INVESTIGATION INTO THE TRANSIENT
BEHAVIOR OF CENTRIFUGAL PENDULUM
VIBRATION ABSORBERS

By

Daniel Palmer

A Thesis

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

Master of Science

Department of Mechanical Engineering

2005

ABSTRACT

THEORETICAL AND EXPERIMENTAL
INVESTIGATION INTO THE TRANSIENT
BEHAVIOR OF CENTRIFUGAL PENDULUM
VIBRATION ABSORBERS

By

Daniel Palmer

This work focuses on the dynamic behavior of a rotor with attached, circular—path,
centrifugal vibration absorbers. Starting from the full non-linear equations of motion,
a new non-dimensional procedure is introduced that allows the results to be applied
to a wider range of applications than has previously been possible. In particular,
the research has been motivated by applications in the automotive industry, and as
such, an important component of the research work deals with the transient behavior
of the system that results from sudden changes in the applied torque and excitation
order. Tests are run on a fully instrumented experimental apparatus and the results
are compared to numerical simulations and approximate analytical solutions. All
comparisons are in very close agreement. Throughout the work, special attention is
given to the physical interpretation of the results and, for example, parameter values
are chosen to closely correspond to applications of practical importance. Related to
this is a discussion of the important design parameters and the influence they have

on the overall performance of a system with attached centrifugal vibration absorbers.

To my dad

iii

ACKNOWLEDGMENTS

My greatest thanks go to my advisor, Dr. Alan Haddow. for giving me the op-
portunity to work on this project, and for his encouragement and guidance along the
way. I want to thank Dr. Steve Shaw for all of his advice and his contributions to
this work. I would also like to thank Bruce Geist for everything he has done for me.
Not only for his advice. but for being responsible for the money I had to support
myself over the past. four months. I consider myself extremely lucky to have had the
chance to work with the three of them. and I hope the quality of this work reflects
my appreciation and respect.

Special thanks to the National Science Foundation for supplying the grant that
helped construct the lab equipment, and to Roy Bailiff for making the modifications
to lab equipment that allowed me to run my tests. Thank you to everyone who

assisted me during my time at Michigan State University.

iv

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

1 Introduction

1.1 Background ................................
1.2 Purpose of This work ..........................
1.3 Thesis Organization ............................

2 Mathematical Modeling

2.1 Equations of Motion ...........................
2.2 Simplifications ...............................
2.2.1 Linearization ...........................
2.2.2 Non-Dimensionalization .....................

3 General Information

3.1 Tuning ...................................
3.2 Mass ....................................
3.3 Radius, R .................................
3.4 Damping ..................................
4 Experimental Work

4.1 Experimental Setup ............................
4.2 System Parameter Values ........................
4.3 Transient Tests ..............................

4.3.1 Simulations ............................

4.3.2 Excitation Order Step Change ..................

4.3.3 Fluctuating Torque Amplitude Step Change ..........

5 Conclusions

5.1 Summary of Findings ...........................
5.2 Recommendations for Future Work ...................
BIBLIOGRAPHY

vi

vii

hWMi-i

(0010‘

51

01
52

54

LIST OF TABLES

3.1 System parameter values used in sinmlations .

4.1 Damping ratios for absorbers . . . .
4.2 Measural;)le system parameter values used in simulations

vi

19

34
34

2.1

3.1

3.2

3.3
3.4

3.5

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

LIST OF FIGURES

Model of rotor/ absorber system .....................

Absorber motion as a function of fluctuating torque amplitude for var-
ious levels of mistuning; created using Equation 2.35 .........
Rotor acceleration as a function of absorber tuning; created from Equa-
tions 2.35 and 2.31 ............................
Vibration reduction as a function of radius, R .............
Absorber damping influence on absorber motion; created using Equa-
tion 2.35, damping ratio defined in Equation 2.11 ...........
Absorber damping influence on rotor acceleration; created using Equa-
tion 2.35, damping ratio defined in Equation 2.11 ...........

Experimental apparatus: (A) Rotor, (B) Motor, (C) Bearings

Circular path, T-shaped absorber assembly in “free” configuration . .
Circular path, T-shaped absorber assembly in “locked” configuration
Schematic of experimental setup (from [8]) ...............
Experimental Results: order step change from n. = 2.384 to n, = 1.192
at T9 : 6.2N.m, locked absorbers ....................
Experimental and Simulated Results: rotor speed during order step
change from n = 2.384 to n = 1.192, T 9 = 6.2N.m, locked absorbers .
Experimental and Simulated Results: absorber motion during order
step change from n = 2.384 to n = 1.192, T9 = 5.25N.m ........
Experimental and Simulated Results: rotor acceleration during order
step change from n = 2.384 to n = 1.192, To = 5.25N.m ........
Experimental and Simulated Results: absorber motion during order
step change from n = 1.192 to n = 2.384, T9 = 4.2N.m ........
Experimental and Simulated Results: rotor acceleration during order
step change from n = 1.192 to n : 2.384, To = 4.2N.m ........
Experimental Results: step change in fluctuating torque amplitude
from T9 = 2N m to T9 = 6.2N m, n = 2.384, locked absorbers .....
Experimental and Simulated Results: rotor speed during step change
in fluctuating torque amplitude from T9 : 2Nm to To = 6.2Nm,
n. = 1.192, locked absorbers .......................

27

29

30

30

32

36

38

39

40

42

42

43

4.13

4.14

4.15

4.16

4.17

4.18

Experimental and Simulated Results: al)sorl.)er motion during fluctu-
ating torque amplitude step change from T9 : 2N.m to T9 = 4.2N.m,
n. : 1.192 .................................
Experimental and Simulated Results: rotor acceleration (luring fluctu-
ating torque amplitude step change from T9 = 2i\"m to T0 : 4.2Nm,
n. = 1.192 .................................
Experimental and Simulated Results: absorber motion during fluctu—
ating torque amplitude step change from To = 4.2;‘Vm to T9 = 2Nm,
n = 1.192 .................................
Experimental and Sinmlated Results: rotor acceleration during fluctu-
ating torque amplitude step change from T9 : 4.2Nm to T9 : 2N m,
n = 1.192 .................................
Simulated Results: phase portrait for excitation order change from
n = 2.384 to n = 1.192, T9 = 4.2N.m ..................
Simulated Results: phase portrait for fluctuating torque amplitude
change from T9 = 2r\".1n to T0 = 4.2N.m, n = 1.192 ..........

viii

46

47

47

50

CHAPTER 1

Introduction

Torsional vibration is common in many dynamic systems. Typically, it is caused by
a harmonically varying force, or torque. There are many ways of reducing torsional
vibration, such as the use of a vibration absorber. In a linear (non-rotating) system, an
absorber's natural frequency is tuned to match the frequency of the applied force. The
same is true when dealing with torsional vibration absorbers, which will be discussed
here. However. in such systems, the frequency of the applied torque is often related
to the rotational speed itself. An example of this is an internal combustion engine
where cylinder firings occur at a specific crankshaft position causing a spike in the
torque. For a single cylinder 4-stroke engine, this would occur once per every other
revolution of the crankshaft. In these types of systems the vibration-inducing torque
is a function of position rather than time. This makes the use of centrifugal pendulum
vibration absorbers (C PVAs) very appealing, since the linear natural frequency of a
CPVA varies with the speed of rotation, discussed in detail in Section 2.2.

A simple example of a CPVA is a pendulum attached to a rotor. Similar to the
way gravity is the restoring force in a vertically hanging pendulum, the spinning rotor
creates a centrifugal restoring force on the pendulum absorber. When setup correctly.
the swinging motion of the pendulum creates a counter-torque opposite to the torque

applied to the rotor. reducing or eliminating the torsional viln'ation.

1 . 1 Background

Wilson [10] gives a detailed report of the early history of Pendulum Absorbers. Cen-
trifugal Pendulum Vibration Absorbers have been used as early as the 19203 [2]. They
have been used successfully in various applications, including diesel engine camshafts,
high-performance race engine applications, helicopter rotors and even radial aircraft
engines in WWII.

Throughout the history of CPVAs, multiple geometric configurations have been
used to implement this type of dynamics. One of the earliest concepts for these
types of absorbers involves liquid columns in U-shaped channels designed to affect.
the dynamics of rotating components. Other designs that have been utilized include
a bifilar suspension design used in radial aircraft engines [4] and a roll-form absorber
used in helicopter rotors [6]. Both of these are still used today in various applications.

Although a strong argument can be made in their favor, pendulum absorbers are
not the only method used to reduce torsional vibration. One commonly used method
is to increase the mass (inertia) of a rotating system, typically by using a flywheel. A
drawback to this is that the extra mass decreases performance by requiring more effort
to induce changes in motion. Another method is to increase the rotational damping.
Dissipating energy through increased damping is not always the best solution as it
may generate unwanted heat or noise, and the system will require an additional energy
input to maintain a constant motion. There are also active vibration canceling devices
that monitor vibration levels to deal with unwanted vibration. These devices can be
very complex and expensive, and often only mask the unwanted vibration instead of
actually removing its root cause.

CPVAs are extremely useful for reducing torsional vibrations, since little addi-
tional mass is needed and often much of the additional mass can be removed from
elsewhere in the original system. For example, the counter-weight or external damper

on an engine crankshaft. Little of the energy applied to the system is lost through

damping, and a pendulum absorber system can be simple and cheap to incorporate
into a rotating assembly.

Previous works have focused on various aspects of pendulum absorbers, including
stability, modes of operation in multiple absorber systems and non-circular absorber

paths [1. 3, 6, 7]. However, there are no reports on the transient behavior of CPVAs.

1.2 Purpose of This Work

A significant amount of work has already been done on pendulum absorbers, as
discussed in the previous section. However, a recent topic of inquiry involving CPVAS,
especially by the automotive industry, has been their transient response. The wide use
of cylinder deactivation systems in internal combustion engines to conserve fuel during
light-torque operation is a leading cause for this investigation. Engines are typically
well-balanced for their normal operating mode, however, when the number of active
cylinders are reduced, unwanted vibrations can occur. One method used to avoid this
problem is to map a specific range of operation where the vibration is acceptable, and
only use the reduced cylinder mode in this range. With CPVAs, the vibration could
be reduced and reduced-cylinder mode could be implemented over a wider range of
operating conditions, greatly improving fuel economy. The purpose of this thesis is to
explore the behavior of the absorbers when the system is subject to changing torque
conditions, which would be similar to an automotive engine with cylinder deactivation
capabilities. The experimental results will be compared to theoretical work to check
that they agree. It is important to be able to predict transient behavior to be able
to utilize CPVAs in various applications where torque conditions are not constant.
This work will also serve as a brief compilation of previous works on circular path

absorbers.

1 .3 Thesis Organization

The next chapter presents the mathematical modeling of the system to be studied
and theoretical approximations along with some transformations allowing results to
be applicable to a much wider range of scenarios. Chapter 3 is more practical for
application. It has general information that is useful for the application of pendulum
absorbers, and how results are influenced by various parameters. Chapter 4 reports
on and discusses, the experimental results. Tests include step changes in applied
torque to monitor the transient behavior of the system. Simulated results are also
given in this chapter. The final chapter summarizes the main contributions of this

work and includes recommendations for future research in this area.

CHAPTER 2

Mathematical Modeling

This chapter outlines some of the mathematical theory behind CPVAs. First, the
equations of motion are derived for a simplified model of a rotor and single pendu-
lum absorber, using Lagrange's method. The equations are then linearized to extract
certain information from the system, for example, tuning. Other mathematical ma-
nipulations are done, uncluding non-dimensionalization, so that results can easily be

compared from different systems.

2.1 Equations of Motion

A simple model was used to derive the equations of motion, see Figure 2.1. There is
rotor with inertia J, with a single pendulum attached at. a distance R from the fixed
center of rotation of the rotor, point 0. The pendulum is made up of an arbitrarily-
shaped mass, m with a center of gravity a distance :r from its own center of rotation,
point 00. Viscous damping is also considered to be part of the model. The coefficients
of the viscous damping are co and ca, for the rotor and absorber respectively. Finally,
a torque, T, is applied to the rotor.

It should be noted here, that this model represents absorbers that follow circular

paths as they swing. While this does limit the range of uses for the model, past re-

absorber

 

,
.. ii
.- 0.
R
o 9
A
rotor
T

Figure 2.1. Model of rotor/ absorber system

searchers such as Alsuwaiyan and Sclunitz [1, 7], have derived equations for absorbers
that follow general paths. However, the model used here does allow for an absorber
with a distributed mass as opposed to a point mass absorber. This is an important
characteristic to include, especially for practical applications.

Since the primary purpose of this thesis is to explore the transient behavior of
pendulum absorbers, a single absorber model is adequate. A system with multiple
absorbers will have an identical response providing there is synchronous motion of all
absorbers. This is true as long as the mass of the single absorber is equivalent to the

sum of the masses of the multiple absorbers, and as long as the multiple absorbers

are identical in almost every way. The multiple absorbers must. be tuned identically,
have the same damping, and the same mass. Slight variations between absorbers can
greatly influence the stability of the system in some circumstances. Alsuwaiyan and
Chao [1, 3] have researched the stability and unison response of multiple absorber
systems.

In order to use Lagrange’s method to get. the equations of motion, the energy of
the system must first be calculated. It is easy to see that as long as the rotor is
spinning in a horizontal plane, there is no potential energy. Note, the kinetic energy
in the system can be broken into several components. From the rotor, there is a
rotational energy term, of the form 3162. From the absorber, there is a rotational
component as well as a translational component, which are %mp2(6 + ¢)2 and é-m'vg,
respectively. Where an, is the magnitude of the velocity of the absorber’s center of

gravity, and p is the radius of gyration of the absorber about its center of gravity.

Equation 2.1 shows the total kinetic energy of the system.
, 1 '2 1 2 ' '.- 2 1 2
I\.E. 2 —J6 + —mp (6 + e) + —mvm
2 2 2
where (2.1)

c2 :(R2 + r2)62 + (26+ o)r2(.§ + (6 + c3)2'rR6 cosgfi

’ m

Using Lagrange’s method, equations for 6 and g6 can be derived

J + 711(132 + r2 + p2) + 2er coscb 6 + m.(p2 + TH 008635 + TQM
(2.2)

— 2n‘zer6d) sin ab — rrzrl‘Rdz sin (I) = T — (‘06

m(p2 + r2)c3 + m(p2 + r2 + TR cos (9)6 + 'mxrfi’62 sin c‘) = -—cac§ (2-3)

The torque applied to the rotor is assumed to be of the form

T 2 To + T; sin(n.6) ( )
2.4

’3 T0 + Tf 81110190

where Q is the mean speed of the rotor and n is the order of excitation, for small
changes about a constant mean speed.

The two forms are nearly identical, with the 6 dependent version being slightly
more accurate for modeling the torque in an internal combustion engine, or similar
system. It is useful to be able to express the torque as either a function of rotor
position, 6, or time, t, depending on the independent variable used in the equations
of motion.

A torque of this form (6 dependent) is used to model torque that may actually
be seen in practice. While a realistic torque would be a much more complicated
signal, Equation 2.4 is meant to model the most dominant harmonic. Part of this
harmonic, the order of excitation, n, is often used when refering to this type of forced
vibration. In the example mentioned in the previous chapter, a single-cylinder four-
stroke internal combustion engine would have an order of excitation of %. Since the
order of excitation is often fixed for a given application, this term is useful when
designing an absorber geometry. The reason for this is described in the next section.

From Equation 2.2, an expression for the counter-torque on the rotor caused by
the motion of the pendulum can be derived. By separating the equation of motion

for a rigid rotor with no absorbers attached,
J6 ‘f‘ (‘09 Z T

the the remaining terms in Equation 2.2 must be the result of the absorber. Therefore,

an expression for the absorber counter-torque can be written

Tcounter = — [mh’2 + m(r2 + ,0?) + ‘Zer cos 6)] 6

C)!
V

— 772(I'H cos 6 + r2 + p2)<;) + er(26 + O)o sin (3') (2-

: — 7nrR(6 + 0) cos o - 771326 + mrlt(6 + m2 sin e + (12.65
2.2 Simplifications

In order to simplify the equations of motion, certain assumptions can be made and
manipulations can be used. Linearization about a fixed absorber position makes it
easy to calculate the linearized natural frequency of the absorbers, used for tuning
the absorber to a specific order. To get the equations into a form comparable with
previous researchers, as well as a form to perform the method of averaging, a trans—
formation of the independent variable is used. This is followed by defining some
non—dimensional parameters, which are also substituted.

The first simplification that is made is shown in Equation 2.6. It is assumed that
the mean component of the torque, T 0, is equal in magnitude to the torque generated
from the rotor damping. c0. when the rotor is turning at a steady speed. At slower
speeds. the mean torque is greater than the rotor damping, and a net acceleration
occurs until an equilibrium is reached. This equilibrium speed, or the mean speed at
which the rotor turns for a given To is given the symbol Q. The mean rotor speed is
used significantly in the derivations in this chapter.
_ E

(“o

o (2.6)

As mentioned in the previous section, the order of excitation is used when de-
signing absorber geometry. The reason for this is to match the natural frequency
of the absorber to the exciting torque. This, like a linear vibration absorber, will

theoretically counter the fluctuating torque and eliminate (or significantly reduce for

the damped case) the vibration in the rotor. Equation 2.3 is used to find the natural

frequency of the absorber.

2.2. 1 Linearization

Assuming a small pendulum motion, a. Equation 2.3 can be linearized to

In.(/)2 + r2)o + cad) + 77’27‘R620 = —m.(,r)2 + r2 + ~rR)6 (2.7)
This resembles the standard form for a forced vibration system,
51% + 2(a;n;i: + air = f(t) (2.8)

and the linearized natural frequency is easily calculated to be

r R
,r'Z _+_ p2

or (2.9)

 

Equation 2.9 states that the natural frequency of the absorber varies linearly with
the speed of the rotor, 6. This is one of the primary reasons CPVA’s are so appealing.
The absorber can be tuned to the order of excitation and will stay tuned at any rotor
speed. Section 3.1 will discuss absorber tuning in more detail, and its affect on the
systems response.

It is convenient. to introduce the variable, ft, to represent the tuning order of the

absorber. Using Equation 2.9.

(2.10)

:1
lll

 

Equations 2.2 and 2.3 are. non-linear, however, in practice it is often desirable to

have the absorber operate in their linear range. The reason for this is discussed in

10

the next section. Research has been done into cycloidal and epicycloidal paths, for
the absorbers, such that the non-linearity is reduced or eliminated.

Equation 2.7 can also be used to determine an expression for the absorbers lin-
ear damping ratio, (0, again by comparing it to a standard linear oscillator. Using
Equation 2.9 this leads to the following relation:

2c C“
aw" _ m(p"2 +13)

0a
m (,02 + 7‘2)

 

using the approximation 6 = Q
_ C(l
2‘771f2fl(p'2 + r2)

 

Ca

2.2.2 Non-Dimensionalization

It is desirable to be able to compare results from different. systems that may have
a variety of geometries, physical quantities and orders of excitation. To make this
possible, non-dimensionalization of the equations of motion and parameters can be
done. While previous researchers have also done this, for both circular and non-
circular paths, it was only useful when comparing systems with equivalent tuning
and excitation orders. A new method is developed here in which the excitation order
is scaled out of the dimensionless equation of motion [9]. This method is outlined
below.

Beginning with the full non-linear equations of motion given in Equations 2.2
and 2.3, previous methods have converted from time to rotor position, 6, as the
independent variable. Since the rotor excitation, given by Equation 2.4, is dependent
on the rotor position and the excitation order it is more convenient to define a new
variable that will be used as the independent variable. Non-dimensional versions
of the absorber position, and rotor speed are then defined and substituted into the

original equations of motion using the chain rule. Next, a series expansion is used

11

to simplify sines and cosines, and to reduce the complexity of the non—linear terms.
A set of non-dimensional variables are defined for the damping, inertia and torque
parameters, followed by the introduction of a "small" scaling parameter. Finally, after
simplificatirms are made, the equations can be combined to uncouple the rotor motion
from the absorber motion. The result is a non-dimensional first-order approximation,
non-linear equation that contains absorber dynamics as a function of the applied
torque.

To begin, several definitions must be made. and the chain rule applied. The
definitions for the new independent variable, \I/, the non-dimensional absorber path

arc-length. s. and the non—dimensional rotor speed, V, are given in Equation 2.12.

m 6
_* = _ 2.12
a ‘ V e ( )

\II = 116. s =
(3 is a coefficient length scale to be defined later for convenience. Once the substitu-

tions are made. the chain rule is used to change the ii‘idependent variable from time

into \I’. The transformation uses the rules below:

 

 

 

 

 

 

 

 

: _ (1(*)
(*) _ (It
, (I *)
W Z is"
. .. .- L (2-13)
d(*) _ ()_\I_I()(*) _ n6d(*) _ ”GI/an.)
(It —6t6\II— at“ at
d2(*) 2 Iaf’k) 202f*)
(“,2 2 (HQ) l/l/ 0‘11 + (Ill/Q) 0‘1“,
for example. using the chain rule:
6 = thn/
-, (3 ,
o : —nQI/s (214)
'r
.. 1’ .
(j) 2 301(2)) (VI/'8' + V28”)
7‘

N ext, an expansion in s is used to eliminate sines and cosines, keeping only terms

to order 33. After making these substitutions and transformations, Equations 2.2 and

12

2.3 become

(_ III/QIIQQ R(s"1/ 2 +V’s’z/))_ 2 Irirzgflflfu2s'2.s) 33
(I

 

 

 

 

 

 

2,.‘2 ,.'2
+ ( 2171129211’ss'z/2 77712le Razz/V) 32
_ — ,l
T T 045)
(77212292 (r2 + Hr + [22) (<5”1/2 + u’ s 'V)) ,
+ ,6
7.
+ 12522 ("21:2 + 2er + mR2 + mp2 + J) 111/ = To + COQV + Tg sin(\I/)
and
_ 77192‘ltz/2.s3 133 _ 17211522 Rut/32:32 + anQ (r2 + RI“ + p2) W,
67‘2 2r2 (9 16)

 

II V2 2 , .I
+£3(IIIRI/28)QQ+"2]127(2 +p2s)( +u’s’u)Q +canusQ) :0

r ,.
Non-dimensional variables, F0 and F9, for the applied torque and, no and ya, for the

damping are then introduced, such that

 

 

 

 

 

 

F To ('0
: —‘ 1, : —'—
0 J9? ‘0 J93 (216
F T0 ca '
: ——~ [‘0 : . ‘
6 J92 I 1129(1‘2 + p2)
Substituting these definitions into Equations 2.15 and 2.16 yields:
_2mn29 R(.I/s” 2+ V’S’I/).S‘2 IIIII292RV2S '25 {33
27:2 r2
2an2 Rss’ 1/2 7711292 Rszi/u 2
+ __ — 13
-r 7‘
+ man2 (r2 + Rr + ,0?) (5"1/2 + u’ s 'I/) 3 (2.18)
7. it

+ 7in (mr2 + 271112]: + 112.122 + mp2 + J) VI/

= JQQ(F0 — [101/ + F9 sin(\I/))

13

and

 

77'2f22R1/233 IIZIIQQIfI/V’z s
— -—,—/’33 — I32 + 7711292 (7‘2 + Rr + ()2) 1/11'
6r2 2r
. 2 2 r 2 2 2 2 n 1/2 2
. . m (an r + . 1/3 Q 77m 1‘ + + l/"SI/ Q
+ H (mRz/Zs-Q‘2 + '1 ( f) ) + ( p )(8 )
,. ,.

 

 

) (2.19)
= 0

The next step is to introduce a “small" parameter, 6, which is used for scaling.
In this case. 6. like ,6, will be defined later for convenience. The purpose of using the
specific exponents of 6 below is to bring out the desired dynamic effects of interest at a
given order in 6 (see [1] for details). Several scaled dynamic variables and parameters

are given by
s:e%z+..., u=(1+£%ur)+...
F0 : eFo, F9 2 (gig (220)
#10 = ([10, pa = ([1,,
By making the above substitutions and expanding in 6, Equations 2.18 and 2.19
become:

[ch10 — f‘omz] e

(22(13—J(i;—:33—)112 ” + Jrnu" — FgJI sin(\Il)) .. ‘ (221)
63/2 + O (62) = 0

 

+
,.

 

 

 

and
F . r II
,afifisfl ((r +p )2 722 + rRz) 63,2
,.

l

l" r2 I . ~ I

+ nit-5.6%“? (6nr(r ('2 + Hr + p?) w + (ma (r2 + {22) 2 ) - (133323) 65/2

6,2 (222)

+ O ((3)
= 0

14

Looking at the first order 6 terms in Equation 2.21, the balance between the mean
torque and the mean speed can be used, and Equation 2.6 is recovered. Setting the
next highest order term. (63/2), to zero and solving for the non-dimensional rotor

acceleration term it". ives
3 ;

f0 1312.

I o . I,
u: = —s1n \I/ — —~. z
n ( ) r(1 + 72.2)

(2.23)
Substituting u:’ into Equation 2.22, effectively uncouples the the absorber dynamics
from the rotor dynamics, leaving a single equation for the absorber dynamics that
includes the forcing term from the applied torque.

It is convenient at this stage to define e as a ratio of absorber to rotor inertia. In

this way, the rotor/ absorber coupling can be systematically taken to be small; as is

the case in practical applications. This motivates a definition of e as

m(r2 + r]? + ,02)2
= 2.24
C J (I‘2 + p2) ( )

 

which conveniently sets a coefficient to unity in the following development.

For simplification. L3 and some other new parameters are defined. Many of the
following parameters are. defined this way such that the final equation is in a conve-
nient form and similar to previous versions. ,3 is defined by setting the coefficient of

the (23 term to be equal to 1.

 

.13 = \/6n.2%(r2 + p2) (2.25)

A new damping parameter, 110, is also defined to eliminate the use of n. in the final

equation.

it... = ’12 (2.26)
n

And finally, a torque amplitude parameter, A0, is defined as

_ f9?“ + 732)

_ 2.27
A6 71.213 ( )

By making these substitutions the equation of motion can be simplified to the form

shown in Equation 2.28.
~ 2

z" + (g) z = c [—[Laz' + z" + 33 — Ag sin(\IJ)] (2.28)

Since the purpose of this revised method is to eliminate the dependence on n. in the

final equation of motion, a parameter is needed to describe the tuning of the absorber

relative to the excitation order. By replacing the explicit dependence on excitation

order with a ratio of the tuning to excitation orders, the results are applicable to a

much wider range of systems. This detuning parameter, a, is defined as follows:

it = n(1 + 6.0) (229)

Using the definition of a given in 2.29, (:11)2 becomes 1 + 260' + 0(E2) and z” =

—z + 0(6) which can be used in 2.28 to get the final form of the equation:
2" + z : e [—[zaz' —- z — 20: + 23 — Ag sin(\Il)] (2.30)

Looking at the components of Equation 2.30, several comments can be made. The
equation is made up of a linear ODE at order 60 with order 6 corrections to account for
various effects. First, the order 60 coefficient describes the linear absorber response.
The order 6 corrections includes the :3 coefficient, which describes the non-linear
effects of the absorber motion. Also of order 5 is a z’ component, representing the
effects of damping on the solution. The 2 coefficient, which is derived from the z"
in Equation 2.28, accounts for the absorber/ rotor interaction. Finally, the A9 term,

represents the fluctuating torque applied to the rotor, i.e. the forcing term of the

system.

16

Equation 2.30 can be used to determine the scaled non-dimensional absorber mo-
tion, 2, which can be used to calculate the scaled non-dimensional rotor acceleration,
'w’, by using Equation 2.23. 111’ can then be used to calculate the dimensional form of

the rotor acceleration, (2’ using the following relationships:

0 = 7192(3/21/1/

and

VV' (1 + 63/2)c3/2w'

(23m
= 63/211." + 0(62)

which gives

ézzwflewau)
Method of Averaging

A useful way to describe the dynamics of the system is by approximating solutions
of Equation 2.30 using the method of averaging. This results in equations for the
amplitude and phase of the absorber motion. As previous researchers [1, 6, 7] have
done, a polar coordinate transformation from 2 and z’ to a. and ,9 is introduced, such

that

z = asin(\IJ + up) (2 32)

z' : acos(\I/ + up)

where a represents the slowly varying amplitude of the absorber motion in non-
dirnensional arc-length units, and «,0 is the phase angle between the applied torque

and absorber motion. Using the method of averaging results in the following approx-

imations:
I [In A -
a. = e [—?a + 39 sm(,:)] (2.33)
1 A
(1,9’ = e [(0 + 5) a — 30,3 + 32 cos( 0] (2.34)

17

Using a method similar to previous researchers [5, 6], a relationship between the
steady-state amplitude, (’1, and the non-dimensional scaled torque parameter, A9, can
be written, as in Equation 2.35. Under steady-state conditions. the slowly varying
amplitude and phase (a. and 9;) are no longer varying, meaning a.’ = 59' = 0. Making

this substitution, Equations 2.33 and 2.34 can be squared and added to obtain

1 2 3 1 '2
A0 = 2 (52—5) + (gr—:3 — (a + 5) a) (2-35)

where (1. represents the steady—state value of a.

 

These results are useful because they reduce the equations to first order. Explicit
solutions can be used to determine effects of various parameters on system response.

Many of the figures in the next chapter are created using the above results.

18

CHAPTER 3

General Information

This chapter discusses some general information about CPVAS that are particularly
useful for practical applications. Topics include absorber tuning, effects of mass and
geometry on performance, and the influence of damping. Various figures are used
to explain the effects of changing parameters, each created using equations derived
in Chapter 2. Unless otherwise specified, all parameter values used to generate the

figures in the chapter are given in Table 3. Chapter 4 discusses how these values are

 

 

 

 

 

 

 

 

 

 

detern‘iined.

l Parameter Value
J 0.063 kg.m2
r 0.039 m

0.118 m

p 0.0337 171
m 0.52 kg
f1, 1.316
12, 1.192
9 107rra.(l/S
ca 00004 leg-1722 / s

 

 

 

 

Table 3.1. System parameter values used in simulations

19

3. 1 Tuning

When using pendulum absorbers in practice, there are general guidelines that are
useful to know. One of the primary concerns should be the “tuning” of the pendu-
lum absorbers. As discussed previously, many applications for pendulum absorbers
operate at a specific order of excitation, n. A four-stroke internal combustion engine
has a dominant excitation order equal to half the number of cylinders.

Tuning an absorber is done by adjusting its geometry such that its tuned order,
ii, given in Equation 2.10, is an appropriate percentage from the order of excitation.
In practice, perfect tuning (f1. = n) is generally undesirable. The reason for this can
best be explained by looking at the absorber response as the amplitude of fluctuating

torque varies, see Figure 3.1.

Absorber Amplitude vs Fluctuating Torque Amplitude

 

 

 

 

 

 

 

 

 

110 T T I I I ,l I
$100~ __fl__..—-—-""' _:
93 ”,',,' ""___...
390- -—" ,,.-~-' .
‘D
.E
6 80"

3

2 70~

T3.

E 60f

<

m h-

e 50

O

B .

< 40

9

E 30~

"P

>‘ 2o~ ' .

E . perfect tuning

6}).3 10~ 1’:’ --—5°/oovertune .

/5-”’. ~- -‘10°/oovertune

0 x/ 1 l 1 1 1 1 1
0 1 2 3 4 5 6 7 8

Fluctuating Torque Amplitude, 1;), in Nm

Figure 3.1. Absorber motion as a function of fluctuating torque amplitude for various
levels of mistuning; created using Equation 2.35

20

Figure 3.1 contains a large amount of information. First, consider the curve
corresponding to 5% overtuning. There are three distinct. branches to this solution,
the first branch is the positively-sloped line from the origin to the vertex marked A. In
practice, this is the branch where operation is desirable. On this branch, the counter-
torque generated by the absorber is out of phase with the applied torque, meaning
it is absorbing vibration rather than amplifying it. According to the stability study
done by Alsuwaiyan [1], this lower branch is a stable solution for a single absorber,
or absorbers acting in unison up to a certain torque level. However, the next branch
is unstable. This would be the middle, negatively-sloped curve between points A and
B. The final, topmost branch, is also stable, but undesirable. When the absorber
operates on this branch, not only is it associated with very large absorber angles,
it also creates a torque in phase with the applied torque, causing an increase in the
rotors fluctuation, which in many cases can be dangerous.

The absorber motion follows a hysteretic behavior between the upper and lower
branches. On the lower branch, as the fluctuating torque is increased, the absorber’s
magnitude will continue to increase, almost linearly, until operating near point A.
At (or near, since these calculations are based on approximations) this point the
absorber motion goes through a “jump” to the upper branch and causes an increase
in the torsional vibrations. Once operating on this upper branch, the fluctuating
torque must be reduced to (or near) point B before the motion will again experience
this “jump” behavior back down to the lower branch.

To help stay on the lower branch where the absorbers are acting as intended,
the absorbers should be tuned for their specific application. When the system is
tuned exactly to the order of excitation (f1 = n), the range of fluctuating torque
that they are capable of counter-acting is extremely limited. However, as will be
shown subsequently, while in this region, they do a much better job of absorbing the

torsional vibration. As tuning is increased further from perfect tuning, the “safe”

21

range of operation increases. This is shown by the 5V and 10% overtuned cases
(ft = 1.05 - n and 1.10 - n). However, increasing the amount of detuning is not always

desirable, as is described in the following.

 

Rotor Acceleration vs Fluctuating Torque Amplitude
250 l l V I T }

  
    

 

 

A

perfect tuning
— — — 5% overtune

— — 10% overtune
— locked absorber
''''''' no absorber

M
O
O

_A

U1

0
r

 

 

  

 

\‘
\
\
\
k\
\

Steady—State Rotor Acceleration in rad/52

 

 

 

2 4 5 6
Fluctuating Torque Amplitude, 15, in Nm

Figure 3.2. Rotor acceleration as a function of absorber tuning; created from Equa-
tions 2.35 and 2.31

The purpose of adding absorbers is to reduce torsional vibration in the rotor, which
can be measured using the magnitude of the rotors acceleration, see Figure 3.2. As a
comparison, there are two straight lines plotted on Figure 3.2 that represent the rotor
acceleration with the absorber locked at its vertex and with the absorber completely
removed. Adding the absorber, but having it locked does reduce the vibration slightly,
but this is only due to the flywheel effect, i.e. the additional mass of the absorber
adds inertia to the system. By allowing the absorber to swing, it generates a counter-

torque on the rotor. which reduces the effect of the applied fluctuating torque. As

22

mentioned before, having a. perfectly tuned absorber results in the best vibration
reduction (lowest acceleration), but only over a very limited range of applied torque
amplitudes. By detuning the absorber, the useful range is increased, but with a slight
sacrifice in performance.

When tuning an absorber, it is necessary to find the balance between performance
and a range of operation that will avoid unwanted jumps to the amplifying branch
of the solution. Alsuwaiyan [1] calculated a formula for predicting the amplitude at
which absorbers will “jump”. He also calculated when a multi-absorber system will
lose its syncronous motion. Section 2.1 briefly discussed multiple absorber systems,
and that single absorber systems can be used to model multiple absorbers when they

are behaving in unison.

3.2 Mass

There are ways of improving absorber performance without changing the tuning,
which could result in unwanted behavior. One way is to increase the mass of the
absorbers. From Equation 2.5, the counter-torque generated by the absorber is pro-
portional to the mass of the absorber. Increasing absorber mass increases the inertia
of the absorbers, which increases their influence over the rotor, without having to
increase their swing angle. Effectively, this delays the onset of the jump to the (unde-
sirable) upper-branch solution, see Figures 3.2 and 3.1. Absorbers with greater inertia
are less excited by the vibration of the rotor, which means that for the same condi—
tions, absorbers with greater mass will have smaller swing angles. The drawback to
increasing the mass is similar to the flywheel effect, system response becomes slower,
possibly sluggish. This is often not acceptable. Figure 3.3 shows several examples of

the difference in vibration reduction between a 1 kg absorber and a 1.5 kg absorber.

23

3.3 Radius, R

Another way to improve absorber performance is to increase the distance of the
absorber from the center of rotation, or R from Figure 2.1. The dominant term (for
small absorber angles) in Equation 2.5 is 'mR'“), meaning the torque created by the
pendulum absorber is proportional to If). It can be shown that the reduction in rotor
vibration is increased proportionally with R2. This allows smaller absorbers to have
greater leverage on the rotor. Again. for the same conditions, the absorbers will not.
only swing less. but generate a larger amplitude counter—torque, improving vibration
reduction. As with the increase in mass, the benefits from increasing R also become
drawbacks when the absorber motion generates a counter-torque in phase with the

applied torque. In this case, the the effect of vibration amplification is also increased.

Reduction In Rotor Acceleration as a function of R

 

90%
80% ~
....... 4»
----v”‘
A o -. ........
i 704. . .......... '
c A """
.3 60% ~ ,A'"
g “A" e
.0”

g 50% " ”y’fl- . .
1,“.9 °
5 40% 1i" 0
E
2 30% "
o
o
2 20%- e 1.5 kg

100/ d 0 1kg

0 ------- quadratic fit
0% I I I 1 T T
55 65 75 85 95 105 115
R(mm)

 

 

 

 

 

 

Figure 3.3. Vibration reduction as a function

24

of radius. R

 

 

Several simulations were run using numerical integration of Equations 2.2 and
2.3 to show how the radius influenced vibration reduction. The results are shown in
Figure 3.3. The figure shows how increasing the radius from 70 mm to 90 mm had
an improvement similar to increasing the mass by 50%.

In general, increasing the distance between the center of rotation and the ab-
sorbers center of gravity will improve performance. This means that increasing 7“
will also give the absorber greater leverage without having to increase the swing

angle.

3.4 Damping

The absorber damping plays a similar role to the damping in a frequency tuned
absorber system. Theoretically, a system without damping has the ability to eliminate
vibration in the primary system completely. However, the resonant response, where
the absorber is amplifying instead of absorbing vibration can have infinite amplitude.
Increasing energy dissipation in the system reduces both these effects, having the
drawback that absorber performance is also reduced.

Damping, however, is not entirely undesirable. Absorber damping also has a
similar influence as overtuning, described in Section 3.1. Figures 3.4 and 3.5 Show
how absorber damping can increase the usable range for the absorber operation.
Damping can also be a significant factor in transient behavior, as discussed in the
next chapter.

Similar to the plots used to describe the benefits of tuning, plotting various levels
of damping demonstrate the advantages and disadvantages the absorber damping has
on the absorber performance. Focussing on absorber motion, Figure 3.4 shows how
increased damping can extend the absorbers usable range. A huge advantage possible

with increased damping is that the hysteretic behavior of the absorber motion is

Absorber Amplitude vs Fluctuating Torque Amplitude

 

40

30

 

20

 

1% damping ratio
- - - 5% damping ratio
- — ~ — . 10% damping ratio

100 T I V I
w ’
s /.
a) 90 ‘/‘ ‘
8 /
E 30 / e
9’ ,1
of
‘o 70 l ‘
a \ .
E 60 \ I
< \ _/
a) ,
e 50 / ,/ "
8 / ./
.c: /
< .
9
£9
"3
>
‘0
as
.9
(D

10

 

 

 

 

 

l L

8 10 12
Fluctuating Torque Amplitude, 1;), in Nm

 

 

Figure 3.4. Absorber damping influence on absorber motion; created using Equation
2.35, damping ratio defined in Equation 2.11

significantly improved. With less damping, a system may suffer catastrophic failure
if a jump to amplifying behavior occurs, where a system with more damping can
be brought back under control by a slight reduction in the fluctuating torque. In
terms of an automotive application, it could mean the difference between shutting off
the engine completely or simply closing the throttle slightly. As Figure 3.5 shows,
increased damping does have a small influence over the ability of the absorber to

reduce vibration, but not nearly as dramatic as the effect of overtuning.

26

Rotor Acceleration vs Fluctuating Torque Amplitude

 

 

 

 

 

 

 

 

 

250 T 1 T f r ,7
1% damping ratio / ./
. . /
— - — 5% damping ratio / ’

“g — — ~ 10% damping ratio / ’ ./

S 200 - / .’ .
s / ’

.5 ’ ’

E ’ l

.9.) 150 ' I ] "
O

o I

< \ /,

§ \ / .

o \

m 100 b \ / 7 / -
Q) /

3": ‘ \ , ’

CD ’\

'>. , ’ > -

'D I / ’ I "

5.3 50 ~ , , , , -

O L l l l 1
0 2 4 6 8 1O 12

Fluctuating Torque Amplitude. 1;, in Nm

Figure 3.5. Absorber damping influence on rotor acceleration; created using Equation
2.35, damping ratio defined in Equation 2.11

27

CHAPTER 4

Experimental Work

This chapter outlines the experimental apparatus and tests that were run. Before any
transient response testing can be done, certain parameter values had to be determined
in order to compare results to theoretical predictions. Once the system parameters
are known, various input variables, such as fluctuating torque amplitudes and the

order of excitation. are changed while the system is running.

4. 1 Experimental Setup

A more detailed description of the experimental apparatus is given in [8]. An overview
of the device is presented here. Past researchers have also used this particular appa-
ratus for pendulum absorber testing. Not only is this device the only known (as of
the writing of this thesis) test rig capable of measuring the response of the absorbers
it can also accomodate various parameter changes. It has the ability to generate fluc-
tuating torque over a broad range of amplitudes and excitation orders, as well as the
space for various types of absorbers. Up to four absorbers can be accommodated.
Figure 4.1 shows the actual test rig. The unit consists of a shaft, mounted ver—
tically in an aluminum frame. An electric motor (B) attached to the top end, spins

the shaft on two bearings (C). Near the lower end, between the bearings, is a ro-

28

I m. ~- 3‘

“.4: 3“.- “‘, _'. w’ -'
‘ iv” , 1““

 

Figure 4.1. Experimental apparatus: (A) Rotor, (B) Motor, (C) Bearings

tor / flywheel (A), machined to accept up to four absorbers or other attachments. The
absorbers used in this work are shown in Figure 4.2. Under the lower bearing, cables
pass through a slip ring into differential encoders mounted on the rotor/absorbers
that measure the position of the absorbers. The encoder, absorber and its base are
shown in Figures 4.2 and 4.3. There is also an encoder on the motor that emits 1000
pulses per revolution. This is used to record the instantaneous position and speed of

the rotor.

29

 

T-shaped
a absorber

Figure 4.2. Circular path, T-shaped absorber assembly in “free" configuration

locking bar

 

Figure 4.3. Circular path, T-shaped absorber assembly in "locked” configuration

30

Feedback controls are used to control the motor. Two computers are used, one for
the mean component of the torque (to control mean speed), and another generates a
sine wave signal (the fluctuating component of the torque). The speed of the rotor
is fed into a computer that slowly increases or decreases the output voltage to main-
tain the set speed. Unfortunately, supplying a constant voltage to the motor torque
controller does not produce an absolutely constant mean speed. Left unchecked, the
mean speed slowly drifts from its originally set value. This is easily corrected by this
computer, which is used to monitor the mean speed of the rotor and adjust the volt-
age supplied to the motor to keep the mean rotor speed at the set level. The result is
a very slow variation in the mean speed about the set value. For more precise testing,
this problem should be corrected, but for the purposes of this work, the variation
does not cause any major problems.

The angular position of the rotor is fed into the second computer where it is used
as a time base to generate the voltage signal for the fluctuating torque. This can be
altered on-the—fly to any desired order, n. The voltages from the two computers are
added together before being sent into the motor controller where the voltage signal
is used to control the applied torque. A third computer is used to log data including
applied torque, instantaneous rotor speed and absorber positions. A schematic of the

experimental setup is shown in Figure 4.4.

Note

For the experimental tests given in this chapter, the test apparatus is fitted with two
(2) equally sized absorbers. Absorber amplitude results are presented as the aver-
age amplitude of the two absorbers. All test results presented here have syncronous
absorber motion. Many of the results presented in this chapter include rotor accelera-
tion, which is not directly measured experimentally. The acceleration is calculated by

numerically differentiating the instantaneous rotor speed, which is directly measured.

31

 

 

Instantaneous Speed

 

 

 

 

 

 

Frequency to

Instantaneous Speed

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Pass Voltage
Filter Converter
__fi Alternating 1000 TI'L'rcv Motor Encoder
Torque
Generator [— /AC‘ Servo Norm.
‘ K
/ Rotor
Motor —--7 /
Adder Controller /
a
‘ t .2
0k
' FFT A U T Absorbers
i Analyzer
I _] ‘\ ~
Slip Ring
‘1
Motor Speed EA _ R8 Absorber angles
Controller _
F [urinating
Tom” Data Logger &
Instantaneous Speed Display

 

 

 

 

 

Figure 4.4. Schematic of experimental setup (from [8])

4.2 System Parameter Values

32

Before any experimental results can be analyzed and compared to theory, certain
parameters must be determined from the test apparatus. These fall into two classes:
the first are directly measurable. i.e. mass, radius, etc. The second have to obtained
experimentally and include the rotational inertia of the rotor, and the amount of
damping in the absorbers. It is also interesting to know the damping in the rotor.
However, the current test apparatus has such a small amount. of friction in the rotor
that the torque required (mean torque) to spin the rotor is so small that it is diflicult

to measure. Since the rotors damping has little to do with the steady-state dynamics

 

of the system, other than the mean speed of the rotor, testing to determine it. was
not done, although it can be estimated from the ratio TO/Q.

The inertia of the rotor was found by using the simple principle that. the fluctuating
torque applied to a rigid rotor is proportional to its rotational acceleration. The

constant of proportionality, J, is the rotors rotational inertia.
To = Jl|9ll (4.1)

To calculate the inertia, a torque with a constant fluctuating amplitude (of the form
given in Equation 2.4) is applied to the rotor and its motion is recorded to be used
with Equation 4.1. This was done using two different fluctuating torque amplitudes.
The inertia of the system without the absorbers present is found to be approximately
0.063 to 0.067 139.7722.

To find the absorber’s damping. a log decrement method was used. With a single
absorber free to swing and the other absorber locked, the system is run with a constant
fluctuating torque applied until it reached a steady state, then the fluctuation torque
was suddenly stopped and the decay in the free absorber’s swing was recorded. Using
Equations 4.2, the damping ratio for each absorber can be calculated. As with the
rotor inertia, the tests were performed twice for each of the two absorbers, using

different torque amplitudes.

Ca ("(2%)

, /1 _ Cf _ 27min

where (1.0 and an are the amplitudes at. the first and nth. peak of the absorber’s decay.

 

(4.2)

In is the integer number of peaks used in the calculation. The results are given in
Table 4.2. It should be noted that the damping ratio is expected to be a function
of the mean rotor speed, 52, as shown in Figure 2.11. The values given below were

obtained at a mean speed of 107? rad/s.

33

 

Absorber Damping Ratio, (0 ]

1 0.0042, 0.0043
2 0.0030, 0.0036

 

 

 

 

 

 

 

Table 4.1. Damping ratios for absorbers

Measurable parameter values. used in the simulations in this chapter, are given in

Table 4.2. Parameters marked with an * are estimated from other measured quanti-

 

 

 

 

 

 

 

 

ties.
Parameter Value

1" 0.039 m

H 0.118 m

p 0.0337 m

m 0.26 kg each 0.52 kg total
TO* 0.25 N.m

cui‘ 0.0004 kg.n22/s

 

 

 

 

Table 4.2. Measurable system parameter values used in simulations

A small amount of research was done to investigate certain parameter values for ac-
tual automotive applications, namely, the powertrain rotational inertia and damping.
It has been a topic of discussion as to how much of the powertrain should be included
when calculating rotational inertia. For a vehicle equipped with an automatic trans-
mission, components downstream of a torque converter may be less significant than
others. Certainly, the crankshaft, connecting rods and pistons in the engine should
be included in a calculation of this type. While sizes of engines will vary, a common
range of realistic. rotational inertias for an average-sized engine is 0.05 kgm2 to 0.15
kg.m2.

Friction in an internal combustion engine is also a function of many different
variables including size, type of motor oil, configuration, etc., and it is likely to not

be entirely viscous as modeled in this work. As an approximation, a viscous damping

34

coefficient for an average-sized internal combustion engine is likely to fall in the range

of 0.05 59—— to 0.2 52—.

4.3 Transient Tests

Two basic tests are performed to investigate the behavior of the absorbers during
changes in the system. The tests consist of step changes in the order of excitation
and the fluctuating torque amplitude. In each case, the system is allowed to reach a
steady state before the step change is implemented.

W hen changing excitation order, specific values of 1.192 and 2.384 are chosen.
These values are used to attempt to model a situation that might be seen in practice.
As mentioned in the introduction, vehicles equipped with cylinder deactivation tech-
nology are excellent candidates for pendulum absorbers. To use absorbers in practice,
a certain amount of overtuning is desirable to avoid jumps to amplifying solutions.
Since the test rig has a fixed tuning order of ft m 1.316, n = 1.192 makes the system
roughly 10.4% overtuned. When the vehicle is in its normal mode, the excitation
order would be twice the value as when it is in reduced-cylinder mode. Meaning that
n = 1.192 will represent operation in reduced-cylinder mode and n = 2.384 represents

operation in normal mode.

4.3. 1 Simulations

Many of the experimental results in the following section are accompanied by sim-
ulation results. Simulations refer to a numerical solution to Equations 2.2 and 2.3.
For each simulation, the parameters used are those determined from the experimental
apparatus, with the exception of the mean torque and rotor damping. Because these
values are difficult to determine, they are fixed to give the appropriate measured mean

speed.

35

4.3.2 Excitation Order Step Change
High Frequency to Low Frequency: Locked Absorbers

To establish a baseline, a test is run with the absorbers locked. In this configuration,
the system should behave as a single rotating inertia and have an equation of motion

given by Equation 4.3, where 122 is the combined inertia of the rotor and locked

 

 

 

 

 

 

 

 

 

 

 

 

absorbers.
T(t) = 1,,9' + cor) (4.3)
10 Applied Torque
e l]]”ill.]]]l]ll]l]l ilWl [ll
2 ° [1]] ]]]]]]]l [[1 l [[1]] [[1]
“105 5 7 ti é 1‘0 111 12
200 RotorS ibggliesration
o liillllllllill‘illlluli [1111111111111]
‘2005 5 7 13 5 1b 111 12
seconds
350 . . Rotor Speed ‘ I
11111111111me111111
5 E. 5 3 d 5 110 1‘1 12

Figure 4.5. Experimental Results: order step change from n = 2.384 to n. = 1.192 at
T9 = 6.2N.m, locked absorbers

Figure 4.5 shows the acceleration, 0, and speed, 6, of the rotor during a step change

36

in the excitation order, n. This is what might be seen in an engine when switching
to reduced-cylinder mode. While it is not transient behavior, it should be noted here
that the steady-state amplitude of the acceleration is the same for both excitation
orders, but the steady amplitude of the speed is doubled for half the excitation order.

This is expected, and can be explained by using Equation 4.4.

6,, 2 Qt + 60 sin(n§2t)

688

Q + 6071.9 cos(th) (4-4)
633 = —60(nQ)2 sin(th)

It can be assumed that the steady-state rotor motion can be approximated as in
Equation 4.4 when a torque given by Equation 2.4 is applied. There is a constant
increasing term as the rotor spins at some mean speed, 0, along with a small per-
turbation in that motion with amplitude 60. It can also be assumed that amplitude
of the rotor acceleration is dependent on the fluctuating torque, as in Equation 4.3.
Algebraically, if the fluctuating torque amplitude is constant and the excitation order
is halved, the steady-state magnitudes of the acceleration and speed are constant and
doubled, respectively.

Another point to notice is the rotors speed transition. Due to the conditions of
the step change (the order is set to decrease when the torque is at a positively sloped
zero-crossing), the applied torque is positive for a slightly longer period of time. This
causes the mean speed of the rotor to temporarily increase. This is shown in Figure
4.6 in both the experimental data and simulation. If the step change occurred when
the torque is decreasing, the opposite is true. The mean speed would temporarily
decrease. The transition back to the steady mean speed (300 RPM) is approximately
the same in both instances and is dictated by the homogeneous solution of equation

4.3, which is a first order ODE in the rotor speed, 6. It is easy to show that the

37

Experimental Rotor Speed

 

 

 

340 1

320
s 1 1 . 1 1 1i 1
(I 300 l‘ 1 ‘1 , . ‘ agility] l]; [ [1‘3 ,[hllgll ["1]l'|‘]vi1,,‘i] I» 1]]‘11 V

280

5 10 15 20 25
seconds
Simulated Rotor Speed
340 . ,

 

 

 

seconds

Figure 4.6. Experimental and Simulated Results: rotor speed during order step
change from n = 2.384 to n = 1.192, T9 = 6.2N.m, locked absorbers

associated time constant, 7', is
T = —— (4.5)

This dictates the rate for the transient motion to decay.

High Frequency to Low Frequency: Free Absorbers

The previous test is now repeated with the absorbers free. This time, the torque level
is slightly lower (5.25 Nm) to keep the absorbers from swinging too far, possibly caus—
ing damage. At 71 = 2.384, the excitation is far enough away from the tuning of the
absorbers that they are not significantly excited. However, when the excitation order

is halved, the absorbers are now only 10% away from perfect tuning, which means

38

the absorber amplitude should be much larger. Figure 4.7 shows the experimental

and simulated absorber motion during the step change.

 

 

 

 

 

 

 

 

[l l] ill 11 a 1'

seconds

Figure 4.7. Experimental and Simulated Results: absorber motion during order step
change from n = 2.384 to n = 1.192, T9 = 5.25N.m

The general trend of the absorber amplitude shown in Figure 4.7 is the same
in both experimental and simulated results; an initial overshoot followed by settling
to a new larger steady-state amplitude. The results from the method of averaging
(Equations 2.33 and 2.34) were also used to predict transient behavior, but because
this torque level is close to the jump point, the method of averaging simulation
resulted in a jump in the absorber’s motion. This is likely a consequence of the step
change initial conditions not lying close enough to the appropriate basic of attraction.
Those results are not included here, however, similar results are presented in a later

section of this chapter.

39

The beating that occurs after the step change is expected. Typically, a beating
response comes from two frequencies that are close to one another, which is the case
in this system. At steady state, the transient (free) vibrations have decayed away, so
the response is dominated by the forcing frequency, 11, in this case. Immediately after
the step occurs, the system once again has both a free and forced response until the
free response component decays. The corresponding frequencies are 71 and n, which
are only different by approximately 10%.

Although the general trend between the simulation and the experiment are correct,
the amplitude and the resulting frequencies are slightly off. This discrepancy is likely
due to the variation in the experimental rotor’s mean speed, which determines the

excitation frequency and as a result, the response frequency. As the rotor spins with

Rotor Acceleration
(T6 = 5.25 Nm. n = 2.384 to 1.192)

150 T ‘1’ I l

 

 

I
experimental .

simulated
100 | amplitude

_1 50 I I l l I
14 16 18 20 22 24 26

seconds

 

      

 

 

 

2

U)

E
E

1]] ,
-5o [1]],111‘

 

Figure 4.8. Experimental and Simulated Results: rotor acceleration during order step
change from n = 2.384 to n = 1.192. T9 = 5.25N.m

40

a slightly different mean speed, the two results slowly drift out of phase with each
other. A possible source for the amplitude difference is likely to be the form of the
clamping. It may be much more complicated than simple viscous damping, which
would certainly influence the amplitude of the response.

The absorber amplitude is also extremely dependent on the absorber tuning. The
variation in the rotor’s mean speed changes the frequency at which the excitation
occurs. This change translates into a slight change in tuning, which can dramatically
affect response amplitude, as shown in Figures 3.1 and 3.2.

Figure 4.8 demonstrates that when the excitation order is suddenly halved to be
close to 72, that the torsional vibration is reduced. Although the reduction in 6 is
only about 40% it should be noted that there is a considerably large amount (‘76) of

overtuning in the system (7‘1. 2: 77 + 10% 71).

Low Frequency to High Frequency: Free Absorbers

Figure 4.9 shows how the absorbers behave as the excitation order is changed from
a lower order (1.192) to a higher order (2.384). As expected, the absorber steady-
state amplitude is much smaller when excited by the 2.384 order torque because
of the dramatic amount of mistuning (ft is nearly 50% below 71, or undertuned). A
smaller torque value is used in this experiment again to avoid large absorber amplitude
responses at the lower excitation order.

Figure 4.10 shows the associated vibration (or acceleration) that results in the
rotor due to the increase in the excitation order. Again, the initial overshoot is
followed by a beating exponential decay to the new higher steady—state value. As
mentioned in Chapter 1, in practice, the high level of vibration seen at 77 = 2.384,
which corresponds to an engine operating with all cylinders, is controlled by other
means. Automotive engines are typically very well balanced in this configuration.

Experimental and simulation differences in the amplitude and phase can be ex-

41

Absorber Motion
(T9 = 4.2 Nm, n = 1.192 to 2.384)

[[le

 

0 ll1

  
   

o in degrees

, I

iW]]‘]l‘1li]‘]"'l‘]“ w
i [11]”111‘1 .‘
]

 

‘1 5 ] experimental ‘
_ simulation
20 amplitude

 

OR

5.:

 

18 20 22 24 26 28 30 32
seconds

Figure 4.9. Experimental and Simulated Results: absorber motion during order step
change from n = 1.192 to n = 2.384, T9 = 4.2N.m

 

 

 

 

Rotor Acceleration
(Te = 4.2 Nm, n = 1.192 to 2.384)
1 DU ‘F f 1 T .
100 -
50- . ‘ ] [1‘1] p ‘ .
o: l i] i ] i [[l] H] l 1
€ 0 L ‘ ] I ] I 1 “‘1‘ l ‘1‘
E ] W] l] ‘ 1 , ‘
-50 1 ]
‘1 00 - ‘ -xperlmental
simulated
-150 - -mp|itude
1.8 1‘9 2'0 2'1 22 23 2.4

 

 

seconds

Figure 4.10. Experimental and Simulated Results: rotor acceleration during order
step change from n = 1.192 to n = 2.384, T9 = 4.2N.m

42

plained using the same arguments as the step change from a high to a low excitation
order. The general trend, however, is very close between simulation and experimental

results.

4.3.3 Fluctuating Torque Amplitude Step Change
Increase in Fluctuating Torque: Locked Absorbers

As in the previous section, the first test. is run with the absorbers locked. In this
case, the behavior is again exactly as expected. An increase in the amplitude of the
fluctuating torque increases the amplitude of the acceleration, which results in an
increase in the fluctuation in the speed, this time due to the 60 term in Equation 4.4
since 71 and Q are constant. Figure 4.11 shows how the rotor responds during a step

change in the fluctuating torque amplitude.

Applied Torque

 

 

 

 

 

 

 

 

 

 

 

 

i o -
:z
-5 -
10 11 12 13 14 15
seconds
Rotor Acceleration
1 00 I F T I
(\l
(I)
:5 0 r
S
—100 1 '
10 11 12 13 14 15
seconds
Rotor Speed
350 l I I I
12
a.
c 300 VWWNVWWWNWMWW
10 11 12 13 14 15
seconds

Figure 4.11. Experimental Results: step change in fluctuating torque amplitude from
T9 : 2N m to T9 = 6.2N m, 71. = 2.384, locked absorbers

43

Again, looking at the disturbance in the mean rotor speed shows that it is slightly
increased due to the increase in fluctuating torque. The step occurs while the torque is
increasing, which accelerates the rotor, increasing the mean speed. The decay in this
transient behavior is exponential. Figure 4.12 gives a. closer look at the experimental
and simulated rotor speed. As is mathematically predicted, the steady-state ampli-

tude of the fluctuation in speed varies linearly with the fluctuating torque amplitude.

 

 

 

 

Experimental
320 1
2 [l
& I
300 1 1, .
l 1
280
10 15
seconds
Simulation
320 v
2 i ‘ ] l l l ‘ i
D. l y l ] ] ]
(I 300 i l 1
1 i l l ‘ ]]
280
56 58 60 62 64 66 68 70
seconds

Figure 4.12. Experimental and Simulated Results: rotor speed during step change
in fluctuating torque amplitude from Ta = 2N m to T9 : 6.2Nm, 71 = 1.192, locked
absorbers

Increase in Fluctuating Torque: Free Absorbers

For the following tests, not only are experimental and simulated results presented, but
results from numerical integration of the averaged equations, Equations 2.33 and 2.34,

are used to predict transient behavior. While the method of averaging is only valid

44

for slowly varying amplitudes and phases, all three results are in close agreement.
It should be noted that this slowly varying requirement can be relaxed if the step
changes are considered to be only an expression of a change in the intitial conditions.
This is explained further at the end of this chapter, see Figures 4.17 and 4.18.

Figure 4.13 shows the behavior of the absorber when the fluctuating torque am-
plitude is increased from 2 N .m to 4.2 N .m. Both the simulation and the averaged
amplitudes match closely with the experimental results. Discrepancies in amplitude
and frequency can be explained as previously. All three results show an amplitude
overshoot followed by the beating behavior. As expected, the steady-state amplitude
after the step change is larger than before.

Absorber Motion
(To = 2 Nm to 4.2 Nm, n =1.192)

 

v I I

30- ‘

20-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(I)
o
9
§ 0 -
.E
e
—10
experimental
simulated
—20 - amplitude
method of
averaging
'30 ’ 1 1 I n I d
10 11 12 13 14 15 16

Figure 4.13. Experimental and Simulated Results: absorber motion during fluctuat-
ing torque amplitude step change from T9 = 2N.m to T9 = 4.2N.m, n = 1.192

Figure 4.14 demonstrates how the rotor behaves during the step change in fluc-

tuating torque. In this case, Equation 2.23 is used with Equations 2.33 and 2.34

45

to calculate the amplitude of the rotor acceleration. The same general trends are
observed.

Rotor Acceleration
(T9 = 2 Nm to 4.2 Nm. n =1.192)

 

0’)
C)

I I I 1*

A ., .
7 112“"

 

 

 

experimental
simulated
amplitude
method of ‘
averaging

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—
I

I

 

11 12 13 14 15 16
seconds

Figure 4.14. Experimental and Simulated Results: rotor acceleration during fluctu-
ating torque amplitude step change from T9 = 2N m to T9 = 4.2N m, n = 1.192

Decrease in Fluctuating Torque: Free Absorbers

Figures 4.15 and 4.16 give the results of the step change from a higher fluctuating
torque amplitude to a lower fluctuating torque amplitude. Similar comments can be
made about the agreement between experimental, simulated and averaged approxi-
mation results. The behavior of the absorber and rotor acceleration follows the same
type of pattern as before. The decay of transient motion to the new steady amplitude
is exponential, and it has the same beating behavior seen in many of the previous

tests.

46

Absorber Motion
(T9 = 4.2 Nm to 2 Nm, n =1.192)

20 v v 7 . f

 

/

.41
0| 0
‘v—
\
‘—

 

 

 

 

 

 

 

 

 

 

 

|
\.‘ /\‘l
N." l
1
a \ll
9 0
a)
«8 ~5
.E
e —10
-15 experimental
simulated
—20 - amplitude
method of
'25 averaging
10 11 12 13 14 15

seconds

Figure 4.15. Experimental and Simulated Results: absorber motion during fluctuat-
ing torque amplitude step change from T9 = 4.2N m to T9 = 2N m, n = 1.192

Rotor Acceleration
(To = 4.2 Nm to 2 Nm, n =1.192)

“ii i i i ii 1

experiments

rad/$2

i

——

 

simulated
amplitude

  
 

-30 method of
averaging

 

14
seconds

Figure 4.16. Experimental and Simulated Results: rotor acceleration during fluctu-
ating torque amplitude step change from To = 4.2Nm to T9 = 2Nm, n = 1.192

47

Discussion

These results appear similar to the results obtained when the excitation order was
changed from low frequency to high frequency. The resulting amplitude may look
similar, but the actual behavior of the absorber relative to the rest of the system
is very different. This may be better explained using Figures 4.17 and 4.18, which
show simulated phase portraits for two cases of an excitation order change and a
fluctuating torque amplitude change. They are obtained by numerical integration of
the averaged equations 2.33 and 2.34. In both cases, the final state is the same.

Before the change in excitation order, indicated as the initial state in Figure
4.17. the absorber is only excited to a relatively small amplitude. This is because the
absorber is so far out of tune from the excitation order. Notice also, that the absorber
motion is in phase with the applied torque, i.e. it is amplifying the vibration in the
rotor. Once the order is changed, the absorber is tuned much closer to the excitation,
which results in an increase in amplitude. The torque level remains constant, but.
now the excitation is more in tune with the absorber. After the change, the absorber
motion is 180° out of phase with the applied torque, so the counter-torque is opposite
the applied torque, and a vibration reduction in the rotor occurs.

Considering a similar situation, but now the excitation order remains constant
as the torque level is increased. The results from the numerical intergration of the
averaged equations are shown in Figure 4.18. Again, the amplitude of the absorber is
only excited to a relatively small amplitude until the step change. Then the amplitude
increases to the same steady value as in Figure 4.17. This time, however, both the
initial state and the final state are 180° out of phase (absorbing rather than amplifying
vibration). Since, the tuning has not changed, the increase in amplitude comes from
being excited by a larger torque amplitude. While the two cases appear similar, there
are significant differences between a step change in excitation order and a step change

in fluctuating torque amplitude.

48

The use of the averaged equations here indicates a possible way to efficiently find
the bounds on the overshoots, beating frequency and the decay rates. This is left for

future work.

49

Phase Portrait for Excitation Order Change
(To =4.2Nm n= 2.t384 01.192)

 

     
   

overshoot

SM

'0 M
O 01
v v

absorber amplitude, ¢, in degrees
6‘.

 

 

 

o 50 100 150 200 250
phase (degrees)

Figure 4.17. Simulated Results: phase portrait for excitation order change from
n. = 2.384 to n = 1.192, T9 = 4.2N.m

Phase Portrait for Torque Amplitude Change
Nm t.o42Nm n=1.19 9)

 

no I I l

final state

absorber amplitude. q: (degrees)
a;

8 - initial state

 

 

so 160 170 130 190 200 210
phase (degrees)

 

Figure 4.18. Simulated Results: phase portrait for fluctuating torque amplitude
change from T9 = 2N.m to T9 = 4.2N.m, n = 1.192

50

CHAPTER 5

Conclusions

Over the past years many researchers have worked on various aspects of the design
and performance of CPVAS. To-date, none have studied transient behavior of such
devices and in particular how the absorbers and the rotor respond to sudden changes
in the excitation order and the amplitude of the fluctuating torque component.

The purpose of this work was to explore the behavior of circular-path centrifugal
pendulum absorbers, as they pertain to real-world applications, specifically the au-
tomotive industry. This has been accomplished through both theoretical work and
experimental testing. The following is a sunnnary of the main contributions of this

work.

5.1 Summary of Findings

0 The inclusion of a general information chapter, useful for real-world applications

of CPVAS.

0 Outline of a new procedure for non-dinlensionalizing the equations of motion,
better suited to unifying results from different applications and allowing one
to compare the performance from seemingly disparate systems. Results from

different systems are able to be mapped into a non-dimensional parameter space

for direct comparison. Previously, this was not possible with systems operating

at different excitation and tuning orders.

0 The choice of parameters used in simulations, approximate solutions and ex-
periments were all guided by values from real automotive applications, with

specific regard to cylinder deactivation technology.

0 As predicted by the mathematics. when operating at orders well above the tuned
order (71 >> ft) the influence of the absorbers on the rotor dynamics is very small.
Typically, absorber swing angles are an order of magnitude less than when the
system is operating closer to perfect tuning. For this set of parameter values.
if transients are introduced, they decay to 10% of their initial overshoot value

within approximately 50 revolutions of the rotor.

0 Although the results presented in Chapter 4 are not exhaustive, there is a very
good indication that the amplitudes of the overshoots in the system response
are within tolerable limits. For example, 50% overshoot and decay takes ap-
proximately 50 revolutions. This has been verified experimentally and through
numerical simulation for cases where the absorber damping is extremely small
(very conservative case). In practice the damping is likely to be larger and

therefore overshoots and decay rate smaller.

0 Important to note is that during the transient testing. no cases were observed
where the absorber motion became unstable or “jumped" to the undesirable,

large amplitude solution.

5.2 Recommendations for Future Work

Pendulum absorbers have been in use, successfully, for a significant. amount of tune,

but there are still questions regarding various aspects of their behavior. C ertain toplcs

Cit
[\D

have been explored in this work that lead to new questions and possible directions

for future work. These are:

e A more rigorous analytical study of overshoots and decay rates. The averaged
equations derived in Chapter 2 could be used to explore the basins of attraction
and the form of the dynamics in the vicinity of the steady-state solutions. A
sudden change in a system parameter, e.g. the excitation order, 77., could be

viewed as a change in the initial conditions in the averaged “state-space”.

0 Multiple absorbers and their non-syncronous response. Although not reported
here, there is experimental evidence that this can become more problematic the
smaller the detuning value is, i.e. the closer n is to ft. In an attempt to optimize
CPVA performance it would be advantageous to be able to operate closer to
perfect tuning. However, the robustness of the performance in this region would

have to be fully explored.

0 Experimental and theoretical work could be expanded to non—circular path ab-

sorbers.

BIBLIOGRAPHY

[1] AS. Alsuwaiyan. Performance. Stability. and Localization of Systems of Vibra-
tion Absorbers. PhD thesis, Michigan State University, 1999.

[2] BC. Carter. British patent no. 337446, 1929.

[3] CF. Chao. The Performance of Multiple Pendulum Vibration Absorbers Applied
to Rotating Systems. PhD thesis, Michigan State University, 1997.

[4] The Chilton and Reed Propeller Company. British patent no. 460088, 1935.

[5] Alan G. Haddow and Steven W. Shaw. Centrifugal pendulum vibration ab—
sorbers: An experimental and theoretical investigation. Nonlinear Dynamics,
2003.

[6] Tyler Nester. Experimental investigation of circular path centrifugal pendulum
vibration absorbers. Master’s thesis, Michigan State University, 2002.

[7] Peter M. Schmitz. Experimental investigation into epicycloidal centrifugal pen-
dulum vibration absorbers. Master’s thesis, Michigan State University, 2003.

[8] Peter M. Schmitz. Technical operation manual. CPVA Operation: Technical
Operation Manual for Michigan State University Test Apparatus, July 2003.

[9] Steven “7'. Shaw. Revised circular path non-dimensionalization method. Math-
ematica work and hand-written notes (unpublished), September 2005.

[10] W. Ker Wilson. Practical Solution of Torsional Vibration Problems. Chapman
and Hall Ltd, 3rd edition, 1968.

   

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