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This is to certify that the

thesis entitled

Effects of Mistuning on the Performance of
Centrifugal Pendulum Vibration Absorbers

presented by

Vishal Garg

has been accepted towards fulﬁllment
of the requirements for

M. S . . Mechanical Engineering

degree in

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EFFECTS OF MISTUNING ON THE PERFORMANCE
OF CENTRIFUGAL PENDULUM VIBRATION
ABSORBERS

By

Vishal Garg

Thesis

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

MASTER OF SCIENCE

Department of Mechanical Engineering

Michigan State University
East Lansing, MI

1996

ABSTRACT

EFFECTS OF MISTUNING ON THE PERFORMANCE
OF CENTRIFUGAL PENDULUM VIBRATION
ABSORBERS

By
Vishal Gary

Torsional vibrations in IC engines are known to cause fatigue and NVH related
difﬁculties. In this thesis, the use of centrifugally driven pendulum absorbers for
the attenuation of torsional vibrations is investigated. In particular, the effects of
mistuning and nonlinearities on the performance of the absorbers is considered in
detail. The model employed is a simple two degree-of—freedom system that includes
a rotor and an absorber mass that rides along a quite general path relative to the
rotor. Analytical and computational approaches are undertaken to capture the model
dynamics for a wide range of path types, in order to evaluate their effectiveness in
reducing torsional vibrations of the rotor. The path types considered utilize a general
mistuning of their linearized behavior along with a parameter that allows one to vary
their large-amplitude character. The results show that overtuned paths generally
perform better than undertuned paths. Perfectly tuned cycloids are found to perform
best overall among perfectly tunued paths, whereas the perfectly tuned circular path
performed the worst. When mistuning is taken into account, it is found that slightly
overtuned circular path absorbers offer overall satisfactory performance, and they
have the advantage of being simple to implement in practice. The comparison between
the numerical and the analytical approaches Was generally very accurate. Hence, the
analytical results presented herein can be used to provide valuable guidelines for initial

design and evaluation of absorber systems.

ACKNOWLEDGEMENTS

I wish to express my sincere gratitude to my thesis advisor Dr. Steven W. Shaw.
His timely guidance, advice and suggestions through out the duration of this work is
greatly appreciated. I especially thank him for his endless help during the last few
months of this work. It has been a great privilege to work with him.

My special thanks to Prof. Alan G. Haddow and Prof. Brian Feeny for agreeing
to serve as my committee members and Dr. C. T. Lee , Mr. C. P. Chao for providing
many insightful discussions on various topics.

I would also like to thank Dr. S. Kudapa for his friendship and constant encour-
agement. I

Finally, I reserve my deepest and most sincere gratitude to my family members:
my parents (Dr. B. Kumar and Rajrani), my brother (Gaurav Garg) for their endless
sacriﬁces.

iii

TABLE OF CONTENTS

LIST OF FIGURES vi
1 Introduction 1
1.1 Background and History ......................... 2
1.2 Motivation for the Present Work ..................... 5
1.3 Organization of the Thesis ........................ 6

2 Mathematical Model
2.1 Simplified Model ............................. 7
2.2 Equations of Motion ........................... 9
2.2.1 Nondimensionalization ...................... 11
2.3 Generalized Absorber Path ........................ 12
2.3.1 Mistuning Parameter ....................... 17
3 Perturbation Analysis 20
A 3.1 Procedure ................................. 20
3.2 Absorber Paths .............................. 21
3.2.1 K44 Relationship ......................... 22
3.3 Steady-State Response .......................... 23
3.4 Jump Analysis .............................. 25
3.5 Numerical Examples . ........................... 28
3.5.1 Numerical Examples of the Steady-State Results ........ 28
3.5.2 Numerical Examples for the Jump Phenomenon ........ 34
4 Numerical Analysis 42
4.1 Introduction - Optimization Methodology ............... 42
4.1.1 Optimization Set-Up ....................... 44
4.2 State Equations .............................. 46
4.3 Statement of Optimization Problem (in Standard Form) ....... 48
4.4 Numerical Results ............................. 49

iv

4.4.1 General Observation of the Response .............. 49

4.4.2 Perfectly Tuned Dynamics .................... 51

4.4.3 Mistuned Dynamics ........................ 54

4.5 Perturbation -Numerical Comparisons .................. 63

5 Discussion and Future Direction for Research 68
5.1 Design Guidelines ............................. 68
5.1.1 Tuned Path Designs ....................... 68

5.1.2 Mistuned Path Designs ...................... 69

5.1.3 Dynamic Response ........................ 70

5.2 Limitations and Future Work ...................... 71

A Derivation of Linear Solution 74
A.1 Linearization in s and 0 ......................... 74
A.2 Linear Solution .............................. 75

B State equations for the model 77
BIBLIOGRAPHY 78

2.1
2.2

2.3

2.4

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

3.9

4.1

4.2

4.3“

4.4
4.5
4.6
4.7

LIST OF FIGURES

Schematic diagram for the CPVA and the rotor from the cross-section view. .
Pendulum path in a coordinate system relative to carrier center of
rotation .................................. 13
Generalized absorber paths in intrinsic one-parameter A form, for m =

2, R0 = 1. ................................. 15
Mistuned absorber paths for A = 0.6, for m = 2, R0 = 1 ........ 18

Performance of the zero-mistuned absorber paths; Peak angular accel-
erationvstorque..............L ............... 29

Performance of the mistuned A = 0,0.6 paths — acceleration versus

torque ................................... 31
Performance of the mistuned A = 0.8944, 1 paths- acceleration vs torque. 32
Mistuning performance at I‘,1 = 0.6, 0.4 ................. 35
Mistuning performance at R, = 0.2 ................... 36
Absorber amplitude .50 versus 1“,, at different path A values ...... 36
Performance of the mistuned A = 0, 0.6 paths. Absorber amplitude vs

torque level ................................. 39
Performance of the mistuned A = 0.8944,1 paths. Absorber amplitude

vs torque level. .............................. 40
Jump torque E; versus mistuning a at different A path values . . . . 41

General absorber response for cycloid and cicular paths- acceleration

vs 9 .................................... 50
General Absorber Response for Cycloid and Cicular paths- Absorber

Amplitude vs 0 .............................. 52
0 - Response of perfectly tuned circular paths ............. 55
0 - Response of perfectly tuned A = 0.6 paths ............. 56
0 - Response of perfectly tuned epicycloid paths ............ 57
0 - Response of tuned cycloid paths ................... 58
0 - Response of mistuned A = 0 (circle) paths ............. 59

vi

4.8 0 - Response of mistuned A = 0.8944 (epicycloid) paths ........ 60
4.9 0 - Response of mistuned A = 1 (cycloid) paths ............ 61
4.10 Maximum amplitude of rotor acceleration, max (5) versus 1‘”; solid
lines are from perturbation; ”o” are from numerical results ....... 64
4.11 Maximum absorber amplitude, max (3..) versus 1‘"; solid lines are from
perturbation; ”o” are from numerical results ............... 66
4.12 Acceleration, 5 vs 0; solid lines are from perturbation; ”o” are from
numerical results ............................. 67

vii

CHAPTER 1

Introduction

Centrifugal Pendulum Vibration Absorbers (CPVA’s) are used to reduce torsional
oscillations in rotating machinery. The reduction of torsional vibrations is important
in many applications because they are a source of noise and vibration and they also can
reduce the fatigue life of components. A C PVA is essentially a moveable counterweight
that is designed to dynamically counteract applied torque excitations of a given order
over a range of operating speeds and torque amplitudes. (The order of the excitation
referred to here is the number of excitation cycles encountered in each revolution of the
rotating shaft on which the absorbers are mounted.) CPVA’s have been successfully
employed in internal combustion engines and helicopter rotors. In this thesis we
choose a simple CPVA model problem in order to investigate certain aspects of the
nonlinear dynamic response of absorber systems. We describe the inﬂuence of various
system parameters, in particular, the parameters that ﬁx the path of the absorber
CG, on the overall system response. The results of this study offer an improved
understanding of the effectiveness of absorber systems in reducing torsional vibrations

and also provide specific information pertaining to the design of CPVA systems.

(O

1.1 Background and History

Background

A CPVA is essentially a mass which is restricted to move along a prescribed path
relative to the base rotating system. It is driven by the centrifugal field of the rotating
system and provides a torque which can be designed to counteract components of
disturbance/ applied torques acting on the system. Since the centrifugal force acting
on the mass is proportional to the square of the rotating speed, the natural frequency
of the CPVA, and thus the frequency which can be accessed for absorption, varies
linearly with the rotation speed. This means that once a CPVA is properly tuned
to a given harmonic of the nominal rotation speed, that is, for a given order, it will
remain effective over a continuous range of rotational speeds. This feature is very
useful since many sources of vibrations, such as those arising from inertia and gas
pressure effects in IC engines, are dominated by harmonics of the rotation speed.

Other devices that are used to control torsional vibrations include torsional fric-
tion dampers, ﬂywheels and conventional tuned vibration absorbers. Torsional fric-
tion dampers consume energy and generate heat; ﬂywheels increase the mass and
rotational inertial of the system, thereby reducing system responsiveness; and con-
ventional tuned vibration absorbers use elastic elements that can be tuned only to
single frequency and are therefore not useful except at a speciﬁc rotation rate (in
fact, they may have detrimental effects at other rotation rates). CPVAs offer many
advantages over these devices. They can often be implemented without increasing the
overall mass of the system, or its rotational inertia. When properly designed, they
dissipate an insigniﬁcant amount of energy in the form of heat and they are effective
over a wide range of rotational speeds and torque levels.

A CPVA is characterized by its mass, path shape, and location. Geometric con-

straints and balancing issues often limit the choice of mass and location for a CPVA.

In that case, the path shape becomes the most important design variable. The most
commonly used paths are circles. However, circles mistune at moderate torque am-
plitudes due to nonlinear effects and can undergo an undesireable jump behavior.
Other path types used are cycloids and epicycloids. Certain epicycloids have a spe-
cial tautochronic property, that is, they have a constant period of oscillation for all
amplitudes of motion when operating in a constant-speed centrifugal ﬁeld [1]. In this
thesis we systematically investigate a family of paths for CPVA’s that includes all of

the above as special cases.

History

Some parts of this section have been directly taken from the review chapter on CPVA’s
by Shaw and Lee [2].

A thorough account of the history of the CPVA up through the 1960’s, including
several applications, can be found in Ker Wilson [3] and DenHartog [4]. The ﬁrst
sound technical discussion on CPVA’s was published by Miessner at a conference at
Stockholm in 1930 (as referenced in Ker Wilson [3]). His experiments demonstrated
the effectiveness of such devices and opened the way for intensive development of
practical forms of CPVA’s. In 1938 Den Hartog published a paper regarding the
dynamics of the CPVA at moderate amplitudes of oscillation and pointed out the
shortcomings that can occur if one uses purely linear tuning to design the absorber
path [5]. Many forms of CPVAs were proposed and patented in Europe during the
1930’s and 1940’s. Taylor introduced the CPVA to the United States in 1936, and
Chilton was the ﬁrst to build a CPVA in the United States [6, 7]. The Sulzar brothers
adapted CPVA’s for use on trains and automobiles in the 1930’s and 1940’s, and
achieved very good results. Today, CPVA’s are applied in the aircraft / helicopter
industry [6, 8, 9] and recently have been tried in experimental automotive applications

[10, 11, 12].

Circular paths have been widely used for the CPVA because of their simple im-
plementation and effectiveness at small amplitude oscillations. Meissner provided
the geometric relationships for achieving the correct linear tuning frequency using
circular paths. Den Hartog extended these results to include the effects of small
damping and nonlinearity [5]. Nonlinear mistuning has been observed in experiments
and simulations for moderate amplitude vibrations [13, 14, 15]. More complete non-
linear analysis of the undamped system, both for free oscillations [16] and for forced
oscillations [13, 17, 18], have been carried out in order to determine the source of the
failure of the linear analysis. Newland and Den Hartog [5, 13] suggested intentionally
mistuning the linearized system in order to produce favorable tuning at moderate
amplitudes of oscillation. It was not until quite recently that possibilities offered by
using non-circular paths were explored.

Absorber paths other than circles are possible due to the development of biﬁlar
constructions that use machined paths and rollers [3, 4, 11]. The idea behind these
paths is that they reduce the mistuning problems associated with circular paths. -
Madden [19] was awarded a patent for CPVA’s on helicopter rotors which employed
cycloidal paths for the absorbers. A cycloid was chosen by Madden, presumably,
since it is tautochronic (i.e., its period of motion is independent of amplitude) in
a gravitational ﬁeld. However, a certain epicycloid is known to be tautochronic in
a radial centrifugal ﬁeld [20]. This motivated a study, sponsored by Ford Motor
Company, which implemented epicycloidal paths on a four-cylinder, in-line, four-
stroke engine. The papers by Denman, Cronin and Borowski et a1. [10] describe
various aspects of this effort which showed a 92% reduction in the second order
oscillation amplitude at the rated torque of this 2.5 L engine, as compared to the

engine without absorbers.

1.2 Motivation for the Present Work

The basic desirable feature of the CPVA is that it remains effective, essentially by
staying in tune, over a continuous range of rotational speeds. This argument is based
on the dynamics of the linearized system, in which the absorber mass does not move
far from its equilibruim state. At moderate amplitudes of oscillation, absorbers are
known to have the potential to fail miserably [14, 13]. Possible causes for these failures
are:

(1) The absorber frequency is generally dependent on its amplitude of motion and
this mistunes the absorber signiﬁcantly at moderate amplitudes. This may even cause
a disastrous jump in the system response as the torque level is increased.

(2) Each absorber is designed to counteract only a single frequency disturbance
and higher harmonics may be ampliﬁed through nonlinear effects, thus leading to
reduced effectiveness of the absorber. (This can be addressed by employing several
absorbers which are tuned to selected harmonics [10, 11].)

Note that these causes are directly linked to nonlinear dynamic effects. In this
thesis we will study the effects of modifying the absorber path both at moderate
amplitudes, by using cycloids, etc., and at small amplitudes, by intentionally mis-
tuning the absorbers at the linear response level. In this way a direct comparison
can be made about the merits and drawbacks of the approaches currently used and,
furthermore, new parameter regions can be explored for potential designs.

Therefore, the purpose of this thesis is to investigate the overall system dynamics
as a function of the following primary design variables, referred to herein as the path

parameters:

e the level of intentional linear mistuning, described by a percentage of mistuning,

and,

e the large-amplitude path shape, speciﬁed herein by using a one-parameter fam-

ily of paths varying from a circle to a cycloid, as developed by Denman [11].

The dynamic analysis contained herein is carried out in two ways. First, a per-
turbation analysis is performed that captures the nonlinear effects in an analytical
manner. These results are then backed up by a systematic numerical study for a

given case.

1.3 Organization of the Thesis

This thesis is arranged as follows. The description of the model employed, the equa-
tions of motion describing the dynamics of the model and the path parametric equa-
tions are provided in Chapter 2. Chapter 3 describes the analytical method employed
to obtain approximate solutions of the model equations. Speciﬁc numerical examples
demonstrating the main results of the analytical study and describing the effects of
various important system parameters on the response are also presented. Chapter 4
describes the numerical methodology employed to solve the model equations, and nu-
merical simulation results are also given to describe the effects of system parameters
on the response. Comparisons with the analysis are also considered in Chapter 4. A
summary, a discussion of the limitations of the current study, and some directions for
future research are provided in Chapter 5. Some detailed derivations and a summary

of the simulation routine are provided in the appendices.

CHAPTER 2

Mathematical Model

In this chapter we describe the model employed and derive the equations which gov-
ern its dynamics. The description of the simpliﬁed model employed for the CPVA
is provided ﬁrst. Next, the differential equations which describe the dynamics of the
CPVA are derived by Lagrange’s method and then nondimensionalized. The equa-
tions used to describe the generalized path curve followed by the absorber center of
mass are then derived. Also, the deﬁnition of the mistuning and nonlinearity param-
eters associated with this family of paths is included. This provides a complete set

of equations which are studied in subsequent chapters.

2.1 Simpliﬁed Model

In order to provide some insight into the effects of absorber path curves on the steady
state motion of a rotational system, it is necessary to understand the basic operation
of the CPVA. A CPVA consists of an absorber mass which is driven by the centrifugal
ﬁeld of the rotational motion in such a manner that its center of mass is restricted to
move along a prescribed path relative to the rotor. The motion of this mass provides
a torque which can be designed to counteract the disturbance torque, thus smoothing

out torsional vibrations.

  
      
  

Id: The moment of inertia of the rotor
\
m.: Absorber’s mass

\3: Arc variable

T09)

3(5)

Absorber ’s path

Figure 2.1. Schematic diagram for the CPVA and the rotor from the cross-section view.

The simpliﬁed model employed is shown in Figure 2.1. It consists of a rigid disk
and a point mass ma for the absorber riding on a prescribed path relative to the disk.
The disk is called the carrier, and it represents the rotational inertia of the system
L; (for example a crankshaft or a helicopter rotor); its orientation is given by 0 .
The absorber mass rides on the prescribed path which is speciﬁed by a function of
- its position relative to the carrier. The disk is subjected to an external torque T(0),
which is assumed to be of the form T (9) = Tn sin n0, n representing the dominant
harmonic that occurs in the rotational system. Due to symmetry of the applied
torque, the absorber path is chosen to be symmetric about its vertex [21]. (The
vertex is deﬁned to be the point on the absorber path which is furthest from the
carrier center of rotation O). From the vertex, an arc length variable 3 is assigned
to specify the location of the absorber mass along the path, that is, S = 0 at the
vertex; see Figure 2.1. The distance from a point on the path to point 0 is denoted
as R, and R is expressed as a function of S, that is, R = R(S). The path is uniquely
determined once 12(5) is chosen. The value of R at the vertex is denoted as R0, that
is, R0=R(0). The absorber has a nominal moment of inertia of 11 = muff}, about

point 0. The damping for the disk and the absorber are ignored in this study. (This

assumption is signiﬁcant, but valid for initial design evaluations, as one must keep the
absorber damping small in order to achieve satisfactory performance). The nominal
speed of the disk is (I, while the oscillating torque is the source of speed ﬂuctuations

about 9.

2.2 Equations of Motion

The equations of motion are determined by Lagrange’s method. For the generalized
coordinates which describe the conﬁguration of the system we choose 5, the arc-
length parameter along the absorber CG path, and 0, the angular orientation of the
disk relative to an inertial frame of reference. The total kinetic energy of the system
can be divided into the kinetic energy of the disk and that of the absorber.

The kinetic energy of the disk is given by §Idéz where 0 is the rotation rate of
the disk. The kinetic energy of the absorber is given by §malz - 17; where 17:, is the

velocity of the CG of the absorber. The velocity, Va, is given by
v, = Rée'é + sag (2.1)

where e} is a unit vector along circumferential direction of the disk rotation at the
absorber location and 6'; is a unit vector tangent to the path direction at that point.
From the geometry of the path, the inner product 65 . c} is given by Rig where d5 is
the differential of S, and dd) is the angle expanded by d5 from the disk rotation center.
The parameters, d5, (IR and dd) are related by the following geometric constraint on
the arclength,

(d5)? = (dB)? + (my, (2.2)

d¢ _1 dB 2

which leads to

 

10

where the positive sign is chosen for the square root in the above equation since c5 is

assumed to increase with 5. Therefore,

8-; - e} = \Jl— (Cd—{92) (2.4)

The translational part of the kinetic energy of the absorber is then given by

 

“Incl/a . I/a :: gm“ (1220.2 + $2 + 2Rose-ﬁ-e-S)

= émJXéz + $2 + 24930) (2.5)

where X = X(S') = 122(5), and G = 0(5) = ‘/X— iX” where 0’ denotes the
derivative with respect to S.

We can now write down the total kinetic energy of the system, K E, which is given

by

, 1 .2 1 a ..
RE — §(Id)9 +§mVa-Va

= any}? + émuw'? + $2 + 263G) (2.6)
Assuming that gravitational effects are small, there are no forces derivable from po—
tential energy.
For determining the equations of motion, Lagrange’s equations of motion are
apphed:
d 0L 6L ,
_ __ .. — = . 2.7
dt (34') 5‘15 Q' ( )

where, for the assumptions stated above,

95 = {0,5}

11

Q: = {57(9), 0}
which results in the following equations of motion for the system

(1:05 + m. (X45 + ms + G5” + 0’5“?) = 7(0)

3 + 05 — gm)2 = o (2.8)
Note that ma simply divides out of the second equation.

2.2.1 Nondimensionalization

It is convenient to present the equations of motion in dimensionless form. By nondi-
mensionalizing, the number of parameters associated with the equation of motion
is reduced. In order to nondimensionalize the equations of motion, we rescale by

deﬁning the following dimensionless quantities:

 

 

 

 

H = R2(B308)
( ) 33
9(3) = \J$(S)-i(d::3))
Id
”° = E
I." = T. T.

12

Note that we have redeﬁned the overdot. The equations of motion in terms of these

parameters are given by

.. .. 1dx .2
3 + g(3)0 — §E(3)9 — 0, (2.9)
" d3 . ° '° .. d9 .2 .
1100 + Z;(s)30 + $(s)0 + g(s)s + HTS-(3)3 = I‘n Sin n0 (2.10)

If the disk never reverses its direction, 6.9., 0 > 0 always, then any function of 1' can
be expressed a as function of 0, which makes possible a switch of the independent
variable from r to 0. Note that with this change the angular speed of the disk is
treated as a variable dependent on 0, denoted as y = 0(0). Deﬁning ()’ as “-351, the

equations of motion become

iii
2 ds
9(8)y23” + (be + 33(8) + 9(8)8') 113/ + 7(03’3/

ys"+(3’+y(8))y’- (8)11 = 0,

-i-—(s)s’2y2 = I‘nsinno, (2.11).

Note that in terms of these variables, the angular acceleration, 5, is equal to yy’.
Also, the unforced base operating point for the system is given by I‘n = 0, y = l and

3:0.

2.3 Generalized Absorber Path

In order to complete the equations of motion, the absorber path followed by the
pendulum mass must be speciﬁed. Figure 2.2 deﬁnes the geometric quantities, that
are needed in the development of equations for the generalized path.

In the ﬁgure,

e (p,q) are cartesian coordinates deﬁned at carrier centre of rotation.

13

q
GA Carrier center of rotation
P

K : Path Curve

 

Figure 2.2. Pendulum path in a coordinate system relative to carrier center of

rotation

14

e R is distance from the carrier center of rotation to the point on the path curve
where the absorber mass is located and is given by 12(5) = 192(5) + q2(5). Note

that the coordinates have been parameterized in arc length variable, 5.

e 0, deﬁnes the slope of the path curve with respect to the p axis, where a =

tan”1 dq/dp

e p is the local radius of curvature of the path curve, where p = (l + (112)3/2/qn,
here 0' = d()/dp.

e 5 = 0 is the vertex of the path curve, which is taken as symmetric about the q

axis; thus, ats=0z a=0,dq/dp=0andp=0.
e qz—Roat5=0.

The generalized curve represents a family of curves that in intrinsic one-parameter

form is given by,

p2 = p3 - A252, (2-12)

where p0 is the radius of curvature at the vertex and A is used to specify the nonlinear
character of the path. Note that this form is not the most general, but it contains the
most common paths employed. The constant p0 sets the frequency of small oscillations

for the system, and if set as

P0 = —Ro/(m2 +1) (2-13)

it provides the tuning required to cancel an order m torque in the linear range.

The A parameter values of interest are in the range A 6 [0,1] , where A = 0 gives
a circle, A = 1 is a cycloid, and A2 = mz/(m2 + 1) is an epicycloid with its base
circle of radius (R0 - p0) centered at the disk center. Other values of A correspond to
epicycloids scaled by a factor p0 / 123(1 — A2) and then translated vertically until their

vertices are at (0, —R0) [11].

15

A hypocycloid family of curves corresponds to A > 1. Only hypocycloids close to
the cycloid curves are considered, as those corresponding to large values of A result
in smaller arclength between the cusps, thus limiting the pendulum mass to very
small motions. Figure 2.3 shows the family of curves generated for different values of
A as represented by equation (2.2). Note that these curves correspond to the (p, q)
coordinate system mentioned earlier. Note also that all paths have a practical limit
imposed either by a cusp point or by the point on the path where the centrifugal ﬁeld
will no longer provide the contact force needed to maintain the mass on the path.

(Since absorbers are not designed like a ”bead on a wire”.)

 

-O 6 r 1 f is r 1 i
' 02 3
_o 7 )_ .................................................. O ........ \. ................... .4
0.6 i
-0.8 .. .................................................................................... ..

 

 

 

_o.9._ ..................... \ ........................................... / .......... 8 ........ .,
0.9;
_,_... ...................................... 1.. -
1.
_11........ ..........., ................................. 1.3.4 .................... . ..... .(
-o.s -02 —01 o 0.1 02 0.3

Path generated at diiierent values of Lamda

Absorber paths for different values of A.

Figure 2.3. Generalized absorber paths in intrinsic one-parameter A form, for m = 2,

120:].

16

We now derive some relationships between the various geometric path variables

with the ultimate goal of obtaining the function X (5), needed in the equations of

motion, in terms of po and A.

 

Notethat
a-fda— SdS—/S d5 —-1-sin'1-/-\§-
— —0 P_0 \/if’§"X§3i A 100,
or
Po .
5=TsmAa, A750; 5=poa, A=0.
Also, since

S a S
p = / cos adS =/ cos apda, q = R0 —/ sin ad5,
o o 0

we obtain , for A :,£ 1 ,

p 1—,\2

 

(sin 0 cos Aa - A sin Aa)

[20+ 1 p0” (cosacosAa + AsinAasina — 1),

 

Q
l

and, for A = 1 (cycloid),

- l (a+-l-s'n2a)

1
q = —Ro+ Epo(1—cos2a).

Therefore, we get

X=R$Y=ﬁ+f

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

which using equation (2.17) specifies X (5 ) as a functon of po and A, as required.

17

Also,

2

 

, _ dR_ p0 ,
X _ 2RdS—2(1_A2—Ro)sma 1423, A¢1,
X’ = po (0: cos a + sin a) — R0 sin a, A = 1 (2.20)

Similarly, expressions for G and G’ can also be obtained using the deﬁnitions in
equation (2.5). The functions X, X’, G, G’ can then be nondimensionalized to obtain

1:, g, g and 53% respectively.

2.3.1 Mist uning Parameter

It is interesting to examine the effect of altering the tuning of the pendulum relative
to the order of the dominant harmonic of the disturbing torque. If n is the dominant
harmonic of the disturbing torque and m is the order to which the absorber path is

designed, then one can deﬁne a mistuning parameter a which is given by the relation
n(l + a) = m. O (2.21)

Therefore, p0 from equation (2.13) becomes
p0 = —R0/ (n2 (1 + a)’2 + 1) . (2.22)

Figure 2.4 shows the family of curves generated for different values of mistuning
at A = 0.6 as represented by equation (2.2). The mistuning values are expressed in

percentage and are given by
a = {—6%, -4%, —2%, 0%, 2%, 4%,6%}. (2.23)

We now have a model by which one can investigate the effects of mistuning (a)

18

 

—o.75

 

 

 

l

-02 -0.15 -0.1 —0.05 0 0.05 0.1 0.15 0.2
Paths generatedatciﬂemﬁnietuningieveisatLambda=0£

 

Absorber paths for different mistuning values at A = 0.6.

Figure 2.4. Mistuned absorber paths for A = 0.6, for m = 2, R0 = l

19

and nonlinearity (A) on the performance of an absorber.

CHAPTER 3

Perturbation Analysis

In this chapter the analytical method for deriving approximate solutions for the
steady-state response to the model equations is described. First, the procedure for
developing perturbation solutions for the system is presented. Next, absorber path
functions that are suitable for perturbation analysis are formulated. The relation-
ship between these path functions and the general path functions is also presented.
Then the steady-state response of the system is described. The solution obtained
from linear theory is then derived and used for comparative purposes. The nonlinear
solution is obtained next. Finally, results that summarize the analysis are graphically

presented and explained for some speciﬁc cases of interest.

3.1 Procedure

The following approach is taken for performing the perturbation analysis. The torque
I), is assumed to be small, and the two variables .3 and y are expanded in power series

in terms of 1"", as follows:

3(0) = Fn81(0) + P:32(0) + F:S3(0) + ' ' ' , (3.1)

y(9) = 1 + I‘ny1(9) + Itit/2(9) + Fit/3(9) + ° " (3-2)

20

21

Note that the leading term for y is unity since y -> 1 as I), —-> 0. Also, all the yj’s,
where j = l, 2, - - - , should oscillate about zero.

This expansion will capture ﬁrst-order nonlinear effects, but can be quite accurate
out to moderate levels of torque. The results obtained are of importance in path
design considerations.

The approximate solution is achieved by substituting these expansions for s and
y into the equations of motion, (2.11), collecting terms of the same order, and solving

for the sj’s and yj’s forj = 1,2, 3,- -- [21].

3.2 Absorber Paths

Absorber paths are represented by their corresponding path functions. These path
functions must be represented as expansions in s in order to perform perturbation
analysis. Absorber path functions, in both exact forms and expansions up to order 3‘,
are given below for some absorber paths. Note that in each case the curve is chosen
so that the curvature at the vertex provides order m tuning in the linear range (recall
that the torque is of order n). This implies that each path has the same coefﬁcient
at order 32.

Circle:

_ 2m2{1— cos [(m2 +1)s]}

 

 

= 1
$(S) (m2+1)2
2 2 1 2 4
= 1—m232+ m (m1: ) 8 +O(36); (3.3)

Cycloid:

22

 

 

 

3 {sin'1 [(m2 + l) 6]}2
= _ 2 _ 2
23(5) 1 (m + 4) s + 4(m2+1)2
+sin"1 [(m2 + 1)s] sin [23212-1 (1122 + 1) s]
4(m2 +1)”
_ 22 (m2+1)234 0 6 , 34
—1—ms— 12 + (s), (.)
Tautochronic epicycloid:
x(s) = 1 — mzsz; (3.5)
General:
:1:(s) = 1 — mils2 + K434 + - ~ - , (3.6)

where K, is a constant such that this form accounts for the ﬁrst-order nonlinearities

in all possible paths [21].

3.2.1 K4-A Relationship

Before evaluating the steady-state response of the system, it is important to obtain
a relationship between the path parameters A and m and the path coefﬁcient K...
This relationship can be obtained by expanding X (5, A) in terms of 5 using equa-
tions (2.13) through (2.18), non-dimensionalization, and a direct comparison of the
resulting coefficient of s4 with equation (3.16). The relationship between K, and A

and m is found to be

= (m2+1)’(m2- 12 (1 + m2»

K“ 12

 

. (3.7)

The above relationship is important because it forms the basis for comparing results

obtained in this chapter with the numerical results obtained in the next chapter. The

23

reader should recall that hidden in m is both the order of the applied torque, n, and
the level of mistuning, 0. Since n is pre-determined, one can use either (0, K4) or

(m, A) as the path design variables, since either pair can be speciﬁed.

3.3 Steady-State Response

We take the absorber path to have the general form described by (3.16), and then
specialize to speciﬁc cases of interest by assigning different values of K4 and 0. Fol-
lowing the methodology described in the procedure section, the leading order (linear)

terms in I‘ﬂ are found to be

31(9) +m2s1(0) +y;(0) = 0 (3.8)

s’,’ (6) + (60 + 1) y;(6) = sin n0, (3.9)

which yield the following solution

 

sinn0
“(0) = ‘(m2(1+b.)_6...2)

311(0) = _(n($2(1_:2)cfzfn2))°

 

(3.10)

When the absorber is tuned at linear level such that m = n, the solution is given by

sin n0

31(0)=-— n2 , y1(0)=0. (3.11)

 

Note that this is the desired behavior since the rotor runs at a constant speed. If
one could insure that amplitudes remained small, this would be a good design. In
practice, however, one sees a range of torques and it is impractical to keep amplitudes

small in all cases, as this would require either large absorber mass ma or large moment

arm R0.

24

Higher order terms for s and y are determined by straightforward application of

the perturbation procedure, yielding

 

 

 

 

sin n0 2 ,
s = —I‘,, (m2(1+ b0) _ bonz) + I‘naosm 2n0
+l‘f’, (a1 sin 110 + 02 sin 3120) + - - -, (3.12)
(m2 — n2) cos n0 2
= — 0
+1?1 ((14 cos 120 + 05 cos 3n0) + - - - , (3.13)
where
_ 4 _ 4 2 2 _ 4
00 = 2m 2bom -i- 560m n 30011 (3.14)
277.001
—m6 (1 + b0) + m“n2 (5 + 6120) — man4 (—2 + 9110) + 450126
03 = 2 (3.15)
4n 001
am = (m2 + (10m2 — 460712) (m2 + bom2 — bong!)2 (3.16)

a, = ((—4au) m8 + (—12K4an) m2n2

+ (7 + 3460 + 2763) m6n2 + (3011) msnz + (4860K. + 4863K.) n4

+ (—8 - 6360 — 7163,) m‘n" + (—4 -1960 — 1563) msn‘

+ (2860 + 6863) ".266 + (166., + 1263) mm6 — 20113128) /

(8122001 (m2 + bom2 — bon2)2) (3.18)
a; = ((—8ozn) m8 + (4K4an) mzn2 + (—1 + 66120 + 67b3) man2

+ (—an) m8n2 + (~16boK4 — 16b3K4) n4

+ (-5960 — 16763) m4n‘ + (4 + 960 + 563) m6n4

+ (--1260 + 16863) m2n6 (-le0 — 463) - 60b3n8) /

(8n2a01 (m2 + bom2 — bonz) (m2 + bom2 — 9bon2) (3.19)

25

a4 = ((—0111)m10 + (860 + 863) 64862 + (12K, + 12K,60) m2n4

+ (3 — 1260 — 2263) m6n4 + (—4 — 460) msn“ - 486.,K.,n6

+ (—10 + 2060 + 2863) m‘n“ + (4 + 206.) m6n6

+ (—14bo — 17b3) mzn8 - 16(10m4n8 + 4b3n10) /

(8123001 (m2 + born2 — 60712)?) (3.20)
a. = ((—a,,) m10 + (12 + 2860 + 1663) m8n2 + (—12K, — 1260K.) m2n4

+ (15 — 60bo — 78113) mf’n‘1 + (4 + 4120)m8n4 + 48boK4n6

+ (2 + 2460 + 14863) m“n6 + (—12 — 2860) m6n6

+ ((3460 — 12163) mi’n8 + 48bom‘n8 + 36631110) /

(8713001 (m2 + bomz — 50722) (7712 + bomz — 9120112)) (3.21)

Using these results, the expansion for the angular acceleration 0 is found to be

 

 

é _ , _ R. (m2 — n2) sin n0 F3, (—3 (m‘n + m2n3)) sin 2120
— yy — m2 + born2 — 0077.2 2301 _
—nI‘:,’, (a4 sin 120 + 305 sin 3110) + - - ~ . (3.22)

Note that the n and 2n harmonics in 0 are not affected by K4, and 0 is therefore path
independent upto order [‘31. However, as expected, all orders of 0 are affected by m.

The desired result is now in hand: the relationship between angular acceleration,
0, and the path parameters, (0, K4). These can be used for a variety of investigations

of system performance.

3.4 Jump Analysis

The jump phenomenon has been observed and documented in analytical studies and

in real applications for absorbers using circular paths (Newland, 1964) [13]. This

26

phenomenon occurs when the amplitude of the absorber motion grows rapidly (even
discontinuously) with respect to the torque amplitude P... This corresponds to the
points of vertical tangency in classic nonlinear resonance curves. Jumps can spell
disaster for a system, since the response on the upper (that is, large amplitude) solu-
tion branch results in a torque generated by the absorber that is generally in phase
with the applied torque, causing a dramatic increase in torsional vibration amplitude.
The jump analysis also provides additional important information relative to the per-
turbation results. Speciﬁcally, the regular perturbation results as derived above will
generally give nice smooth, single-valued response curves as a function of torque arn-
plitude. However, these have a limited range of validity, and the primary limitation
is the jump encountered, where the response curve can become multi-valued. There-
fore, a bound on the torque range over which the steady-state perturbation results are
valid can be approximated by determining the torque level at which a jump occurs.

Mathematically speaking, a jump occurs when £13.: —> 00, where 3., is the am-
plitude of the absorber motion. However jumps do not occur for all paths, as the
absorber amplitude is limited by the fact that g(s) must be real. (At points where
g(s) = 0, the absorber point mass reaches a singular point, speciﬁcally, a cusp, in the
path. In practice, absorbers can not generally achieve such operating levels). Let 3]
and 89 be the absorber amplitude at which the jump occurs and the amplitude at
which 9(8) = 0, respectively. Then a jump can occur only if s; < 39'. This condition
is very difﬁcult to check, as estimates of Sj,S are not easily obtained.

We can, however, estimate 3,, by noting that in this undamped case 8 is a series
of sine harmonics only and that, since the order n sine harmonic is dominant, 3
reaches its maximum (or minimum) at 0 = 5%. This assumption is reasonable as
the fundamental harmonic for 8“, which corresponds to the order n harmonic, is

quite close to the linear solution of the system, while the remaining harmonics are

generated by nonlinear terms, and these remain relatively small for the absorber.

27

With this assumption, the amplitude so is affected by the coefficients of odd order
sines only, since all even order sines vanish at 0 = ﬁ. With the order m absorber

paths, the amplitude so achieved in this way are polynomials in I'm with a leading

 

order term of 3,, z m, (1+Eonl-bo?’ which is the solution of the linearized system. In
order to calculate s; the relationship between. 8,, and P“ is inverted so that 1‘" is
approximated by a series expansion in terms of 30. Then, the value of SJ is taken
as the value of 3., at which $1: = 0. At this jump point, the corresponding torque
amplitude (I‘n)J is obtained by substituting 3] into the relationship between I), and
s... The details of these calculations follow.

Following the assumptions outlined above we obtain the approximation

_ P"
_m2(1+bo)—bon

 

3a

2 + a5I‘f; + . - -, (3.23)

where

05 = (12 — 01. (3.24)

The inverse relationship between 3., and 1‘" is obtained by expressing F“ as a Taylor

series expansion in terms of 3,, and matching coefﬁcients at each order, yielding
2 2 2 2 4 3
1‘" = (m (1+bo)—bon )sa— (m (1+bo)—bon) assa+~u (3.25)

The jump occurs when 4%: = 0, at which point 3] is the corresponding value of so,

that is, an (8]) = 0. It is thus determined that

d8“

1

s z . 3.26
J \/30;. (m2 (1 + b0) — bun?)3 ( )

28
The corresponding jump torque ( 1‘"), is estimated to be

(I‘) 2(m (1+bo)—bon)

,, z . (3.27)
J 3\/3015(m2(1+ b0) - ban?)3

Note that for jumps to occur, both 9 (SJ) > 0 and 0:5 > 0 must hold, since both 9 (SJ)

and SJ must be real, where

 

g(s) = \/1—m2(m2+1)32+K4(4m2+1)s4+~-. (3.28)

These results are useful in determining whether a given path is susceptible to the
jump behavior. This is crucial in design considerations since, as described above,

jumps can wreak havoc on system performance.

3.5 Numerical Examples

Steady-state solutions and the jump phenomenon are investigated for absorber paths
varying from a circle to a cycloid for different levels of mistuning. The example
considered is from a 2.5 liter four-cylinder, four-stroke engine considered by Denman
and Lee and Shaw [1, 21], for which n = 2 and bu = 6.017. Absorber performance is
determined by comparing maximum acceleration levels at different torque amplitudes.
The jump phenomenon results from section 3.4 are used to estimate the region of
validity of the approximate steady-state solutions obtained from the perturbation

solution. The parameters A and a will be used to determine the path.

3.5.1 Numerical Examples of the Steady-State Results

Equation (3.22) is used to obtain the angular acceleration, 0 of the rotor as a function

of 0. The angular displacement, 0, is varied over a one-period interval, ["' '], and

n’n

29

the absolute maximum acceleration is numerically obtained for given values of A, a

and 1‘... Note that the K4 value required in equation (3.22) can be obtained from

equation (3.7), which relates K; to A and m.

0.12 7 ﬁr r I I

 

Epicycioid .Tautochronic -

0.1

p
8

'8

max(aceieration)

'2

0.02

 

 

 

 

Figure 3.1. Performance of the zero-mistuned absorber paths; Peak angular accel-
eration vs torque. '

Figure 3.1 shows a plot of |max(0)| versus 1‘“ for the case of perfect tuning, m = n,

for the following A paths

A = [0 ,0.2 ,0.4 ,0.6 ,0.8 ,(/4/5 ,1 1. (3.29)

In order to make a meaningful assessment of. the performance of a given absorber

30

path, it is compared against that of the system with the absorber locked at its vertex,

which has a peak value, max(0) = Tiff? This relationship provides a baseline that
can be used as a reference to immediately interpret absorber effectiveness.

Figure 3.1 shows a distinct decrease in the maximum acceleration levels as A varies
from 0 (circle) to 1 (cycloid), over all torque levels. A signiﬁcant difference exists in
the acceleration levels obtained from the A- absorber paths and the locked absorber
condition for low to moderate torques, but this difference decreases at higher torques.
In fact, for torque levels greater than 0.6, paths with A values ranging from 0 to 0.4
have acceleration levels greater than the locked condition. The A paths from 0.8 to I
maintain a good difference in acceleration levels as compared to the locked condition
for torque levels up to (I‘n)= 0.8, indicating that paths close to the cycloid are good
candidates for absorbers, if they are perfectly tuned to the order of disturbance. The
path with A = (/4_/5 is the special epicycloid that has the tautochronic property in a
radial ﬁeld for n = 2; it is shown as a dashed curve in Figure 3.1. This path behaves
very close to the cycloid and gives acceleration levels slightly lower than the cycloid
for I), greater than 0.7.

Figure 3.2 and Figure 3.3 depict the peak acceleration versus torque curves for

different levels of mistuning for each of the following path types: A = 0, 0.6, 1, (/4 / 5.

The values taken for the mistuning parameter, a, are
a = [-6%, -4%, —2%, 0%, 2%, 4%, 6%]. (3.30)
The corresponding m values are calculated from equation (2.20) with n = 2 and are

m = [1.88, 1.92, 1.96, 2.00, 2.04, 2.08, 2.12]. (3.31)

Figure 3.2(a) depicts the acceleration-torque relationship for the circular paths

31

m-hu-emrmd)
“6‘2 1 I W v v r

 

 

 

 

A

 

00 0.7 0.0

coo ~ 1
m .
/ as
- c1 0.: as on u 0.0
Tenn

(a) Plot of acceleration versus torque for circular paths.

MIDJ-(UN-OD‘Zﬂ-a
0.12 I Y r ﬁr v v

 

0.1 '

 

 

 

0 1 J L
0 0.7 0.0

 

.2. ..

(b) Plot of acceleration versus torque for A = 0.6 Paths

Figure 3.2. Performance of the mistuned A = 0,0.6 paths — acceleration versus
torque

32

McMim-WN-tm7.nna

 

 

 

 

0.12
0.1 _ 4%
4% 4’. a
0.0
gone )- Alleclher —>
0.01 .
2%
0.1m-
. A A A A L L
0 0.1 0.2 0.0 0.4 0.0 0.0 0.7 0.0
Tenure

(a) Plot of acceleration versus torque for mistuned versions
of the tautochronic epicycloid paths

urea-1:o,¢nu-(uuo.emr.u.a

 

0.1 , 4%

 

 

 

0 0.1 0.2 0.8 0.4 0.0 0.0 0.7 0.0
T".

(a) Plot of acceleration versus torque for cycloid paths

Figure 3.3. Performance of the mistuned A = 0.8944,l paths- acceleration vs torque.

33

at different levels of mistuning. The acceleration levels increase as the mistuning
is varied from zero to negative values. The steep slopes of the curves for negative
mistuning indicate that a small increase in torque value causes a large increase in
acceleration; this is poor system performance, and in fact the absorbers can actually
amplify vibration levels. On the other hand, positive mistuning offers excellent per-
formance, especially for slight mistuning. The accleration curves decrease from 0%
to 2% and then increase from 2% to 6% mistuning. The shape of the acceleration
curves indicates that the system response becomes nearly linear over a large torque
range at higher positive mistuning levels.

Figure 3.2(b) depicts the acceleration-torque curves for the A = 0.6 paths at
different levels of mistuning. This case shows the same character as the circular path
case, although there are some differences in the general levels of acceleration at each
level of mistuning.

Figure 3.3(a) depicts the acceleration-torque relationship for the A = W5 (tau-
tochronic epicycloid) paths at different levels of mistuning. Note that the acceleration
curves for mistuning a E (0%, 6%) are very close to one another, whereas for mis-
tuning a E (-6%, 0%), they are distinctly apart. Again, when considering the entire
torque range, a small level of positive mistuning offers slightly better performance
than the perfectly tuned case.

Figure 3.3(b) depicts the acceleration-torque relationship for the A = 1 (cycloid)
paths at different levels of mistuning. This case shows the same character as the
tautochronic epicycloid path, again with slight variations in amplitude for speciﬁc
mistuning levels.

Figures 3.4 - 3.5 show the effects of mistuning the absorbers at the following
selected torque levels: R, = 0.6,0.4,0.2, for paths with A = 0,0.6, 4/5,l. The
torques here represent the mistuning behavior of the system at low, medium and

high torque levels.

34

Figure 3.4(a) shows that for 1‘" = 0.6 (high torque) the rotor acceleration for all
the paths decreases steeply as we approach 0% mistuning from —6% mistuning. The
acceleration also initially decreases just beyond 0% mistuning for all paths except the
cycloid where 0% is the minimum. The minimum acceleration achieved by the circular
path is at 2% mistuning, while for the other two paths the minima lie between 0%
and 2%. Also, the circular path with 2% mistuning gives the least overall acceleration
for all paths considered at this torque level.

Figure 3.4(b) shows that for I‘ﬂ = 0.4 (moderate torque) the distinction between
the paths is not signiﬁcant, especially for mistuning levels from zero to 1%. The best
performance in this case is offered by the circular and A = 0.6 paths, both for about
1% mistuning.

Figure 3.5 shows that for R, = 0.2 (low torque) A has very little effect on the
response, as expected since the system will operate in a nearly linear manner at this
level of torque. Perfectly tuned paths with A near to unity offer the best performance,
by a very small margin. ‘

Next, we explain the jump phenomenon results that provide important informa-
tion about the validity of the results shown in above ﬁgures, since they are obtained

by the perturbation results.

3.5.2 Numerical Examples for the Jump Phenomenon

Equations (3.23),(3.26) and (3.27) are used for determining the relationship between
the peak amplitude of the absorber 3,, and the torque amplitude 1‘", as well as the
associated jump torque, for a given absorber path.

Figure 3.6 shows a plot of 8,, versus 1",, for the case of perfect tuning, m = n, for

the following A paths

A = [0, 0.2, 0.4, 0.6, 0.8, (/4/5, 1]. (3.32)

35

m Perionnence et Torque - 0.0
0.12

 

0.1 -

 

0.02 - ‘

 

 

.0100 -0.02 -0.01 o 0.01 0.02 000 0.04 0.05 0.00 0.07
W 1‘0"!)

(a) Plot of acceleration versus mistuning for 1‘" = 0.6

 

mmnrmws

 

muceiereﬂon)

 

 

 

-0’.03 -0.02 -0.01 0 0.01 0.02 0.00 0.04 0.05 0.00 0.07
mm (m)

 

(b) Plot of acceleration versus mistuning for I} = 0.4

Figure 3.4. Mistuning performance at I‘ﬂ = 0.6,0.4

36

mmuTm-oz

ﬁ— 1 r

 

0.02

  
   

0.010 r Luna-0.0—
0.010 )-
0.014 ~
270012 -
i 0101 -
E666. _

0.0001
0.004 -

0.002 P

 

 

A A

41.00 .000 -0.01 0 001 0.02 0.00 0.04 0.00 0.00 0.07
W (M)

 

Figure 3.5. Mistuning performance at I‘n = 0.2

as Versus Torque at diiierent Lambda values

 

 

 

 

002 W I i I I I
- 0.4 .
0.18 0.2 0.6 0.8
0.0

0.16 - .

0.14 - 1 q
8
i
.3 0.12 - 4
3
E 01 - -
<
a 0.08 - .
8
2

0.06 - -

0.04 - Epicycioid .

0.02 - .

O l l l l 1 EL 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Torque

Figure 3.6. Absorber amplitude 3,. versus 1‘" at different path A values

37

Figure 3.6 demonstrates that the absorber amplitude so generally increases with I‘o.
However, for paths A E [0, 0.6], the absorber amplitude reverses direction in the torque
range shown, indicating that there exist multivalued solutions for low levels of 1‘...
The reversal point is the absorber amplitude/ torque level at which the jump occurs.
The regular perturbation solution is invalid near to and beyond this torque level.
Also, note that the jump torque increases as A is increased, and there is no jump
beyond a certain value, indicating that solutions for these paths are valid within the
Po range [0,08]. Therefore, for perfect tuning, paths like cycloids and epicycloids,
i.e A 2 0.8, offer the best possibility for good performance.

Figure 3.7 and Figure 3.8 depict the absorber amplitude versus torque curves for
different levels of mistuning for each of the following path types: A = 0, 0.6, 1, ([175.

The values taken for the mistuning parameter, a, are
a = [—6% ,—4% ,—2% ,0% ,2% ,4% ,6% ,8%],
and the corresponding m values, calculated from equation (2.20) with n = 2, are
m = [1.88, 1.92, 1.96, 2.00, 2.04, 2.08, 2.12, 2.16]. (3.33)

Figure 3.7(a) depicts the absorber amplitude versus torque relationship for circu-
lar paths at different levels of mistuning. The jump torque for negative mistuning
a E [—6%,0%] decreases from 0% mistuning to —6% mistuning. Note that at —6%
mistuning, the jump occurs very close to To = 0. Also note that for the cases in
which a jump occurs, there is large amplitude response curve that exists over the en-
tire torque range. We have not investigated the system performance on this branch,
but it is expected to be poor. Also, we have not investigated the dynamic stability

of these branches. However, one can guess that the lowermost and uppermost of the

38

three branches are dynamically stable, while the middle branch is unstable. Thus,
as the torque level is increased, at the jump point the solution will suddenly move
from the lower branch to the upper one. Similarly, once on the upper branch, when
decreasing the torque level the response will follow the upper branch all the way to
zero torque. Obviously, such behavior is undesireable. It can be avoided, as is done in
practice, by using postive mistuning. As the mistuning is increased from 0% to 8%,
the jump torque continues to increase until there is no jump at 6%, 8% mistuning. In
fact, the absorber motion is very close to linear at 8% mistuning.

Figure 3.7(b) depicts the absorber amplitude versus torque relationship for A = 0.6
paths at different levels of mistuning. This case shows the same character as the
circular path case. Note, however, that there is no jump at 4% and the jump torques
in this case are higher than for the circular path case.

Figure 3.8(a) depicts the absorber amplitude versus torque relationship for A =
(#175 (tautochronic epicycloid) paths at different levels of mistuning. Jumps occur
at a = —6%, —4%, —2%. However, there is no jump at perfect tuning or any values
of positive mistuning. The absorber amplitude curves decrease as 0‘ values vary from
—6% to 8%. (However, recall that lower absorber amplitude is not the desired goal,
and that small mistuning levels minimize torsional vibration amplitudes).

Figure 3.8(b) depicts the absorber amplitude versus torque relationship for A = 1
(cycloid) paths at different levels of mistuning. A jump only occurs at a = -—6%.
Therefore, perturbation results are valid for a wide range of mistuning levels for
cycloid paths. Again, absorber amplitude curves decrease as a values vary from —6%
to 8%.

These ﬁndings are summarized in a single ﬁgure, in which the jump torque levels
can be graphically read for a range of mistuning and A values. Figure 3.9 shows a plot
of the jump torque I‘J verses the mistuning parameter 0’, obtained by using equation

(3.27) for the values of A used in this study. Note that the jump torque does not

39

 

o: I ' M I ﬁ

0.10

1:31..
1:2,

 

 

 

00 0.1 0.2 0.0 0.4 0.0 0.0 0.7 0.0
Tm

(a) Plot of absorber amplitude versus torque for circular paths.

 

 

 

 

 

tenth I 0.0
o: w I I
0.10 -
0.10 2*
0.14 r 1
$0.12
0.1 >
g .. .
0%
0.“ ~
0.04
an
o L A A A A A. J
0 0.1 0.2 0.3 0.4 0.0 0.0 0.7 0.0
Tame

(b) Plot of absorber versus torque for A = 0.6 paths

Figure 3.7. Performance of the mistuned A = 0, 0.6 paths. Absorber amplitude vs
torque level.

40

 

mammogram

 

 

 

04 00

 

(a) Plot of absorber acceleration versus torque for epicycloid paths.

Luca-1:051:10

004 I V Y PT r

 

0.30iv

03* 3

in“
021»

 

 

 

-05 -4%
0J5-
OA-
00‘
o l L A L L L A
0 01 02 03 04 00 00 OJ 03
‘i’crrpe

(b) Plot of absorber amplitude versus torque for cycloid paths

Figure 3.8. Performance of the mistuned A = 0.8944,1 paths. Absorber amplitude
vs torque level.

41

 

 

 

 

 

Figure 3.9. Jump torque F] versus mistuning a at different A path values

exist for cycloids over the range of mistuning levels shown. Using this figure, one can
immediately determine the range ofvalidity of the perturbation results for a given set

of parameters. Such information is very useful in evaluating prospective designs.

CHAPTER 4

Numerical Analysis

In this chapter, the numerical method used for solving for periodic solutions of the
model equations is described. This approach is rather unusual in that it uses a stan-
dard optimization package to find periodic solutions of nonlinear differential equa-
tions. The chapter is organized as follows. We first describe the optimization method
employed by providing an optimization problem statement with an objective function,
constraint functions, and design variables pertaining to the given set of equations.
Next, the ﬁrst-order state equations for the model under investigation are presented.
Finally, some numerical results obtained from this approach are used to describe the
effects of the system parameters on the response. These are graphically presented

and compared with the analytical results obtained in Chapter 3.

4.1 Introduction - Optimization Methodology

The following approach is taken to obtain accurate numerical versions of the periodic
solutions of the equations of motion given in equation (2.11). These equations describe

the dynamics of a disk with a single absorber. These are reproduced here as they

42

43

form the basis of this chapter:

II I I ldx _
313 +(3 +9(3))3/ ‘55“)?! — 0,
2 II I I dx I 2
9(3):; 3 + (be + ac(8) +9(S)s )yy + d—3(8)sy
+g‘-:-(.s)s’2y2 = F" sin 120, (4.1)

As is evident from the equations, they represent a non-linear, 0-dependent, conser-
vative set of equations with two degrees—of-freedom, s and y, subjected to a periodic
forcing, I‘ﬂ sin 720. Such non-linear conservative systems with more than one degree of
freedom do not generally yield periodic solutions for a given set of initial conditions.
In fact, the most generic types of response encountered are quasi-periodic (containing
several discrete frequencies) and chaotic. Therefore, in order to obtain the periodic
solutions to this set of nonlinear equations for a given set of system and path vari-
ables, and for a given torque level, a systematic, iterative procedure is required for
identifying the initial conditions that result in periodic solutions. (These solutions
are the ones that generally become dynamically stable when damping is added to the
system, and thus these are the ones of engineering importance.)

The approach employed in this thesis is one adopted from the solution of design
optimization problems. Optimization, as the name suggests, is concerned with the
minimization or maximization of functions. These functions can be multivariable,
non-linear, time-dependent functions and can be subjected to a set of non-linear or
linear, time dependent / independent constraint functions in their variables. There
are many optimization algorithms in use today. The algorithm used herein is a
MATLAB implementation of Sequential Quadratic Programming Method (SQP). A
full description of this algorithm can be found in MATLAB Optimization Toolbox
User’s Guide [22]. The method essentially involves solving a linear quadratic sub-

problem (QP) to obtain a descent direction from the initial starting solution. A line

44

search algorithm, a merit function and the descent direction are then used to obtain
a new solution point. A new QP problem is formulated and the process is repeated

until the descent direction is zero, which corresponds to the optimum solution.

4.1.1 Optimization Set-Up

In order to obtain periodic solutions to the above set of equations, one imposes the

following constraints on the state variables (3, y, and y’),

3(0) 3(0 + T)

s'(0) = s’(0+T)

31(9)

3109 + T), (4.2)

where T represents the O-period for the solution being sought: T = 21r/n. The
solutions for these dynamic variables comes from the numerical solution from a state-
variable representation of the equations of motion, which is given in the next section.

The choice of objective function in this problem is a secondary matter, since the
satisfaction of the above constraints is the main goal. Thus, there are many ways
one can select an objective function. Some of the possible choices are: the maximum
rotor acceleration (or some other norm of the rotor acceleration), or the maximum
absorber amplitude (or some other norm of the absorber amplitude). Since the rotor
acceleration is generally well-behaved, we choose the maximum rotor acceleration over
one 0 period as our objective function.

Design variables here refer to the variables that form the solution set of the op-
timization routine — that is, its output. The solution set is defined as the set of
quantities that we are interested in obtaining as a result of the iterative Optimization
procedure. In the present setting, this set must obviously contain the initial condition

of the state equations that results in the desired periodic solution.

45

Since we are interested in obtaining periodic solutions over a range of torque
amplitudes, we have included F" in the solution set, for the reason described here.
Because of the highly non-linear components in the state equations, especially at
large torque amplitudes, the design variables are highly sensitive to the objective
function and attempts to ﬁnd periodic solutions at a ﬁxed P" value are troublesome
(in terms of convergence) and time consuming. For this reason, a modified approach
was adopted in which the torque amplitude is included as one of the design variables,
that is, in the solution set, with some very specific bounds imposed to achieve our

desired goal. This is described in the next section.

Bounds on the Design Variable

With the set of design variables deﬁned, we now set the upper and lower bounds on
the design variables. Such bounds are set in order to constrain the design set to a
reasonable range and, in the present case, to facilitate the solution. When one has
some knowledge about the general location of the solution in design space, one can
impose bounds on the design variables that improves the efficiency of the process and
the probability of finding a solution. In this spirit, we often utilized the initial states
obtained from the perturbation analysis described in Chapter 3 to set the bounds on
the initial condition of the state equations.

The selection of bounds on the torque amplitude serves a special purpose here.
Since the objective function is generally a monotonically increasing function with
respect to torque amplitude, we know that the solution will nearly always be at the
upper bound of the torque amplitude. By using a set range of torque amplitudes,
one obtains a more robust optimization process and yet can essentially fix the torque

level for the solution point by setting the desired value as the upper bound.

46

Obtaining a Response Curve

By continually incrementing the upper torque bound upward, and with the lower
bound set by the previous solution, we can obtain a response curve that can be com-
pared with the perturbation results. This process is initiated by starting at low torque
levels where the linearized solution is known to be accurate, and then ratcheting the
process up in torque amplitude. The increments in torque level used depend on the
resolution desired in the response curve, balanced against convergence considerations
(since large increments will lead to convergence problems). This process requires the
solution of a well-defined optimization problem for each response point, and it can be
carried out for any set of system and/ or absorber path parameters.

In order to carry out this procedure, one must have in hand the following: the
state equations, the objective function, the constraints, and the bounds deﬁned. To
this end, we now turn attention to the formulation of the standard form of the state

equations of motion.

4.2 State Equations

Equation (4.1) can be expressed in first order standard form of
A2' = F (4.3)
where
y 8’ + 9(3)

9(3):;2 31 (be + a=(6) + 9(8)3’)

z’ = (4.5)

47

122.121,,

F = 2 do (4.6)
1“,, sin n9 - éﬁ-Eﬂs’zyz — d—Egﬂs’yz

Defining the states x1, 13;, $3 in terms of s, s’ and y as follows,

8 = $1
3’ = 132
y = (1+1‘3) (4-7)

the explicit state equations are given by

\

= A‘lF (4.8)

Note that 1:3 is defined as y — 1 so that all states (2:1, 13,33) are zero in the base -
operating condition. A complete representation of the state equations in terms of the
states 31, 9:2 and 2:3 can be found in Appendix B.

They have the general form

2’ = f(z). (4-9)

Recall that a prime denotes a derivative with respect to 9.
Equations obtained in this form can now be numerically integrated over any in-
terval of 0 to obtain solution traces of the state variables, or combinations of them,

as functions of 0. For example, the rotor acceleration is given by

48

3131,09) = —2I‘,, sin n0 + 2x§(0)z§(0)%%1 + 2x2x§é§fld + $§g($l)%£1ﬂ . (4.10)
2 ("be + 90502 — $($1))

 

4.3 Statement of Optimization Problem (in Stan-

dard Form)

With the state equations in hand, we now write the optimization problem statement

that mathematically summarizes the optimization procedure explained above.

Findx = {p1,p2,p3,I‘,,} ;XER4,
th t ' ' ' f = — '
a mmmuzes 9333513”, (0),
subject to the constraints :

h1 = (z(7r) — p)2 = O (periodic constraints),

91 = 1‘" S I‘m-9h (upper bound on 1‘"),

g2 = —I‘,, 5 Flow (lower bound on P"),
g3 = p g plow (lower bound on initial conditions),
g, = —p g phigh (upper bound on initial condition),
where
P1
P = P2 l I
p3 J
l
111
Z = 32 l ,
$3
J

 

49

and z is determined from the state equations, (4.9)

Note that I‘M-9;, is the torque at which the solution is desired and Flow is the torque
at which the solution was obtained in the previous iteration. plow and phigh are the
bounds set around the desired initial states, obtained from perturbation results. The
first initial guesses for p and I‘n are taken from the linear solution at low torque
amplitude and the subsequent initial guesses are obtained from the solution obtained

in the previous iteration.

4.4 Numerical Results

The optimization procedure was employed to produce numerical results which de-
scribe the effects of system parameters on the response and also to check the validity
of the perturbation results. We return to the example system that was used for
the perturbation results in Chapter 3, with n = 2 and b0 = 6.017. We begin with
some general observations of the system response and then turn to a more detailed

investigation of the system dynamics for a range of m, A and I‘n values.

4.4.1 General Observation of the Response
Typical Acceleration Response

Figure 4.1 represents a 0 trace of the rotor acceleration over one period for param-
eter values A = 0 (circle) and A = 1 (cycloid), for m = n (that is 0' = 0) and P = 0.2
(low torque). Positive acceleration corresponds to the forward 0 direction and nega-
tive acceleration corresponds to the backward 0 direction. Note that the solution is

periodic and includes higher order harmonics. In fact, there is a very strong second-

50

 

 

 

 

x 104 General Acceleration Response
6 l T T l a
3 Circle
4 ................................................................................. d
A; CYCIOCd
-\ /\
/ f \ , \
I \ I \
I \ I \
2....I ........... \ .......................... I ............ ‘ .................................... d
I \ , \
I \ , \
.5 I \ , \
g I \ \
g o ....................... \ ........................................ \. ........... u
\ \ '
1 I
§ \ . .,
. \; I . ‘ . /
-2. ...,\ ........... I. .............. E ........... .\..:. ....... ,3 ............ ..
. .\ I. . \I I I
\ I \: I
\ i‘ I
\II \VI
_4._ ................................................................................................. u
.6 l l l l L
0 OS 1 15 2 25 3 35
Theta

Figure 4.1. General absorber response for cycloid and cicular paths- acceleration vs

0

51

order harmonic, as expected from the perturbation results. Also note that, due to the
zero damping assumption, the acceleration is zero at the beginning and mid-points of
the period, exactly where the torque is zero (recall that the applied torque is a sine
function [2]). Furthermore, the response is reﬂection- symmetric about the period
mid-point [2]. Therefore, the peak magnitude can be taken from either half-period,
using the absolute value. Finally, note that the cycloidal path offers more favorable

characteristics as compared to the circular path. These effects are considered in more

detail subsequently.

General Absorber-Amplitude Response
0.06 I I l l

 

0.04

0.02 :

Absorber Amplitude
O

-0.02 ‘

-0.04

 

 

 

 

Figure 4.2. General Absorber Response for Cycloid and Cicular paths- Absorber
Amplitude vs 0

52

Absorber Amplitude Response of the System

Figure 4.2 represents a 0 trace for the absorber motion over one period for parameter
values A = 0 (circle) and A = 1 (cycloid), for zero mistuning (m = n or a = 0)
and I‘n = 0.2 (low torque). The response is periodic and is out of phase with the
excitation torque, which is a sine function in n. Note that the absorber response is
nearly a pure harmonic and that higher harmonics are quite small. Also note that the
amplitudes for these two paths are nearly the same at this torque level.

We next consider the system dynamics for different A and m parameters when

subjected to a range of torque disturbance amplitudes.

4.4.2 Perfectly Tuned Dynamics

In this section we consider the system response for paths that are perfectly tuned to
the excitation torque, that is, m = n (= 2 here) or, equivalently, 0‘ = 0. Simulations
are carried out with A = 0,0.6, 4/5,1; corresponding to a circle, an intermediate
case, the tautochronic epicycloid, and a cycloid, respectively. For each path type, the
response was determined for the torque amplitude I‘n varying from 0.0 to 0.7; the
results are shown in Figures 4.3-4.6 in the form of rotor acceleration and absorber
response versus 0.

Figure 4.3 depicts the response for the circular path absorber. Responses beyond
1‘" = 0.3177 were not obtained, as a jump occurs just above this torque level. (In
fact, below this value the system possesses two stable solutions, only one of which
is shown.) Note that the slope of the rotor acceleration response curve at 0 = 0 is
negative just before the jump torque level, whereas it is positive away from the jump.
The absorber amplitude response shows a steady increase in the amplitude levels from
P7, = 0.1 to I‘n = 0.3177. In fact, the absorber response grows nearly exactly linearly

with respect to 1‘7, until just before the jump occurs.

53

Figure 4.4 depicts the response for a system with an absorber with a A = 0.6
path. This case shows the same character as the circular path case, although the
jump occurs at the higher torque level of l"n = 0.4052.

Figure 4.5 depicts the response of for a system with an absorber with the tau-
tochronic epicycloidal path. No jump occurs in this case (up to 1‘" = 0.7). The
acceleration and absorber amplitudes show a steady rise from I‘n = 0.1 to I‘n = 0.7.
In this case, the absorber response is very closely approximated by its linearized
solution over the entire range of F... values shown.

Figure 4.6 depicts the response for a system with an absorber with a cycloidal path.
Again, no jump occurs in this case (up to I‘n = 0.7), and the acceleration and absorber
amplitudes show a steady rise from F" = 0.1 to I‘n = 0.7. However, in contrast to the
cases shown before, the first peak of the rotor acceleration response is now greater in
amplitude than that of the second peak (during the ﬁrst half-period). A quick visual
comparison at the maximum values-of the rotor acceleration for the epicycloid and
cycloid cases shows that cycloid gives slightly lower peak values. This shift in the
location of the maximum peak value indicates that there exists a path where the two
peaks match. Such a condition will offer the optimal path. However, this optimal
path will depend on the torque level, and therefore is not of much practical interest.
This path is not considered in this thesis, but it can be obtained by investigating

paths lying in between cycloid and epicycloid; see [21] for such a study.

4.4.3 Mistuned Dynamics

In this section, absorber paths with A = 0.0, 4/5,l.0 are considered. The paths
are mistuned over a range of values of the parameter 0. These values for a and the

corresponding m values are given below

 

 

 

 

 

0.5 1 1.5 2 2.5 3

(a) Plot of rotor acceleration versus 0 for circular paths for
I"" = {0.1, 0.2, 0.28, 0.3177}

 

 

 

 

 

o 0.5 1 a} Q is 5
M
(b) Plot of absorber amplitude versus 0 for circular paths for
I‘n = {0.1, 0.2, 0.28, 0.3177}

Figure 4.3. 0 - Response of perfectly tuned circular paths

55

men-panama”

 

 

 

 

 

(a) Plot of rotor acceleration versus 0 for A = 0.6 paths for
I‘n = {0.1, 0.2, 0.3, 0.4052}

 

Mammoth-0.0m
o.:: T . . r v

 

 

 

 

2 2.5 3 3.5

(b) Plot of absorber amplitude versus 9 for A = 0.6 paths for
I‘n = {0.1, 0.2, 0.3, 0.4052}

Figure 4.4. 0 - Response of perfectly tuned A = 0.6 paths

56

 

 

 

 

 

 

 

 

 

(b) Plot of absorber amplitude versus 0 for epicycloid paths for I‘n = [0.1, 0.7]

Figure 4.5. 0 - Response of perfectly tuned epicycloid paths

57

 

 

 

 

 

 

 

 

 

0 0.5 1 1.5 2 2.5 5

(b) Plot of absorber amplitude versus 9 for circular paths for I‘n = [0, 0.7]

Figure 4.6. 9 - Response of tuned cycloid paths

58

a = {0%, 2%, 4%, 6%, 8%}

m = {0,2.04,2.08,2.12, 2.16}. (4.11)

The reason for considering only positive values of mistuning is that they gener-
ally offer improved performance, whereas negative mistuning nearly always degrades
performance; this fact is further demonstrated in section 4.5 wherein the simulation
results are compared with the perturbation results. Three levels of torque amplitude
are considered for each path: I} = 0.2,0.4,0.6. Figures 4.7-4.9 show the system
responses in the form of rotor acceleration for these cases. In all cases the absorber
response is nearly a pure harmonic with an amplitude that varies signiﬁcantly from
that predicted by linear theory only when approaching a jump condition.

Figure 4.7 depicts the system performance in terms of rotor acceleration for mis-
tuned circular paths. Figure 4.7(a) shows that for I‘n = 0.2 (low torque) the maximum
acceleration increases as the mistuning is increased. Again the response characteris-
tics change from a dominant second-order harmonic at 0% mistuning to a dominant
primary harmonic at 8% mistuning. The shift in the location of the response peak oc-
curs again between 0% and 2% mistuning, suggesting that the best path conﬁguration
lies in between these two paths.

Figure 4.7(b) shows that for I‘n = 0.4 (moderate torque) there is no solution for
0% mistuning. Recall that this is expected since the jump for 0% mistuning occurs
at I‘n = 0.3177. However, for the other levels of mistuning shown, the response exists
and the maximum acceleration increases with the level of mistuning, from 2% to 8%.
Again, the response becomes dominated by the primary harmonic as the mistuning
level is increased, although the secondary harmonic persists for larger mistuning levels.

Figure 4.7(c) shows that for I‘ﬂ = 0.6 (high torque), no solution exists for 0% and

59

 

 

 

 

 

 

 

 

 

 

3
‘
._ :~
3,. ..
-
O
I

-1:

 

 

 

 

 

 

(c) Plot of rotor acceleration versus 0 for A = 0 (circle) paths for I‘n = 0.6

Figure 4.7. 0 - Response of mistuned A = 0 (circle) paths

60

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
..
;.
ﬂ
5
u

(c) Plot of rotor acceleration versus 9 for A = 0.8944 (epicycloid) paths for I}, = 0.6

Figure 4.8. 9 - Response of mistuned A = 0.8944 (epicycloid) paths

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15 I u 2

=
g»
a

(c) Plot of rotor acceleration versus 9 for A = 1 (cycloid) paths for I‘n = 0.6

Figure 4.9. 9 - Response of mistuned A = 1 (cycloid) paths

62

2% mistuning. The jump torque for 2% mistuning is I‘n = 0.4501. Responses for the
other mistuning levels are qualitatively similar to those for the moderate torque case.
However, as expected, the secondary harmonic is even more persistent in this case.

Figure 4.8 depicts the system performance in terms of rotor acceleration for mis-
tuned epicycloidal paths. Figure 4.8(a) shows that for I‘n = 0.2 (low torque) the peak
acceleration value increases with mistuning and the response characteristics change
from having two peaks at (0% mistuning) to just one at (8% mistuning) and from
nonlinear (many harmonics) to almost linear (a single harmonic). This is due to the
fact that at low amplitudes the system is nearly linear and perfect tuning offers good
performance in which the rotor acceleration is dominated by a small second harmonic
term. As the level of mistuning is raised, the primary harmonic grows and quickly
overcomes the second harmonic.

Figure 4.8(b) shows that for 1",, = 0.4 (moderate torque), the maximum accelera-
tion again shows a steady rise as the mistuning is increased. The maximum acceler-
ation at 0% mistuning is actually little less than that of the 2% mistuning. However,
it is interesting to note that there is a shift between 0% and 2% mistuning in the
peak at which the greatest acceleration occurs. This indicates that there exists a
mistuning level between these values which offers the minimum acceleration for this
torque level.

Figure 4.8(c) shows that for I‘n = 0.6 (high torque), the maximum acceleration
is lowest for 2% mistuning and then rises from 2% to 8% mistuning. The maximum
acceleration at 8% and 6% mistuning is actually greater than that with zero mis-
tuning. The shift in the largest acceleration peak again occurs between 0% and 2%
mistuning, suggesting that the best path lies between these two conﬁgurations.

Figure 4.9 depicts the system performance in terms of rotor acceleration for mis-
tuned cycloidal paths. No jump occurs here for the levels of mistuning and torques

considered. The maximum acceleration for all the torque levels increases with greater

63

mistuning. Importantly, for 0% mistuning the difference in magnitude for the two
acceleration peaks is insigniﬁcant at all torque levels, suggesting that this is close to

the minimum acceleration path conﬁguration.

4.5 Perturbation -Numerical Comparisons

The analytical results produced by solving the perturbation equations in Chapter 3
and the results produced in this chapter are compared here. These comparisons es-
tablish the accuracy of the perturbation procedure which can be used as an important
initial design assessment tool for absorber systems.

Figure 4.10 shows the maximum rotor acceleration as a function of torque ampli-
tude for three different paths and a variety of mistuning values. In this figure, results
from the analysis are shown as solid curves while the numerically obtained data points
are shown as open circles. Note that the analytical results are generally very good,
except when the response curves turn past a jump point. The perturbation analysis
fails beyond the jump torque value. It is important to note that for cases where the
response is nearly a straight line, the linear response provides a satisfactory approxi-
mation. However, it is not possible using linear theory to determine where the linear
result breaks down.

Figure 4.11 shows numerical and analytical results for the absorber amplitude
versus F" at different a values. The analytical results are obtained from section
3.5.2, which allows one to obtain accurate results beyond the jump. This alternative
perturbation approach ﬁxes the large discrepancies between the numerical and the
analytical results that are observed in Figure 4.10 when I‘n 2 I). These analytical
results compare very well with the numerical results and can be used to accurately
determine the jump torque (PI) for general absorber path conﬁgurations.

Figure 4.12 depicts system responses obtained numerically and analytically in

64

 

 

milky“.
3
o
3

\

 

 

 

4‘- ‘ ~A 4 A A
o M a: a; as 0.0
”You.“
Hm
0.1: . v
0
0.1L ..
e . e

T’

 

with“.
e
e
O
Q
\ e

5

i
i
a
- 3».

 

 

 

 

 

 

 

Figure 4.10. Maximum amplitude of rotor acceleration, max (9) versus I‘n; solid
lines are from perturbation; ”o” are from numerical results.

65

 

 

 

 

o m m an m m an an are are «1
”GM“!

 

Ina-- - - -
e. '0’ 0‘ j
e e e
0 .6

M r ’ O I J

_ o I

~ e

5 ~ 0

 

 

 

o 0.3 m an ace 0.! 0.12 an 0.10 0.90 o:
«mun-n

 

0.5 >

 

an I“ 0.“ nos 0.1 Q12 0.16 Q15 0.15 0.2
WOW“

 

 

Figure 4.11. Maximum absorber amplitude, max (3“) versus I‘n; solid lines are from
perturbation; ”o” are from numerical results.

66

 

 

 

 

 

 

 

 

 

Figure 4.12. Acceleration, 9 vs 9; solid lines are from perturbation; ”o” are from
numerical results

 

67

terms of rotor acceleration versus 9 for 1‘” = 0.5. The responses are for a circular
path with mistuning of a = 4%, and for a perfectly tuned cycloidal path. This further

demonstrates the accuracy of analytical approach.

CHAPTER 5

Discussion and Future Direction

for Research

The objective of this investigation was to improve our understanding of the dynamic
performance of CPVA systems in terms of the absorber path. The model representing
absorber systems was chosen to facilitate the analysis and simulations in order that
some general observations could be made regarding the inﬂuence of various parameters
on the system response. Of particular interest were the nonlinear effects produced by
general mistuned paths on absorber system dynamics when subjected to a range of

applied torques.

5.1 Design Guidelines

5.1.1 'Ihned Path Designs

It has been shown that nonlinear effects are signiﬁcant over a range of applied torque
amplitudes except at very low torques (I‘" S 0.1) where absorber motions are small.
In applications, it is common to design circular paths for the absorbers, as they can

provide the required small-amplitude curvature and are easy to manufacture. How-

68

69

ever, the nonlinear jump occurring at large absorber amplitudes, even at moderate
torque levels, defeats the purpose of the CPVA. Newland [13] provides details about
this behavior. The results presented here indicate that for other path conﬁgurations
(represented here by varying A from zero [circles] to unity [cycloids]), the jump torque
level keeps increasing as we move away from the circle to virtually no jump occurring
for paths closer to the cycloid. This is expected and is due to the fact that an increase
in the absorber amplitude causes a change in period and this change in its period, if
signiﬁcant, will not maintain the right phasing to absorb the disturbing torque, and
will in some cases amplify the disturbing torque. As the path is shifted towards the
tautochronic epicycloid (which is quite close to the cycloid for n = 2 values of two
[where A = \/4/—5] and larger), the absorber period becomes less and less sensitive
to the absorber amplitude, thus pushing the jump out to larger and larger torque

amplitudes.

5.1.2 Mistuned Path Designs

As has been long known to designers of CPVA systems, the results presented in this
thesis indicate that intentional overtuning (positive mistuning) produces great per-
formance enhancements in absorber systems. It was observed that by making tuning
adjustments in circular and other paths near the circle, many undesirable nonlinear
behaviors (especially the jump) can be completely avoided. The avoidance of the
nonlinear jump with positive mistuning is due to the simple reason that absorbers’
preferred frequency of motion decreases as their amplitude is increased, and by inten-
tionally overtuning the absorbers, they come into more favorable tuning as amplitudes
become large. Both numerical and perturbation approaches also indicate that there
is an optimum level of mistuning for a path that will give the best performance for a
particular level of disturbing torque. However, it should be understood that this ad-

vantage is at the cost of some performance degradation (in terms of rotor acceleration)

70

at lower torques where a perfectly tuned absorber system offers better performance.
This is not crucial, however, since vibration levels at low torque levels are not of
importance.

On the other hand, intentional undertuning (negative mistuning) promotes jumps
and results in large rotor accelerations for all path conﬁgurations. This is expected
and is due to the fact that signiﬁcant frequency discrepancies occur at even smaller
absorber amplitudes as compared to the case of zero mistuning.

The net conclusion is that with a slight tuning adjustment, a circular path offers
performance comparable to the tautochronic epicycloid and to the cycloid, and it
has the added beneﬁt of being easier to implement. One must be careful about the
ﬁrmness of this conclusion however, as there are limitations in the model employed
and ignored effects may be important. These results should be used as rough guide-
lines and further investigated in light of other signiﬁcant dynamic effects which are

discussed subsequently.

5.1.3 Dynamic Response

The dynamic response of the absorber to the harmonic applied torque is nearly a pure
harmonic of order n and is always out of phase with the disturbing torque. This is true
over a wide variety of paths that include mistuning. In fact, the absorber amplitude is
' very well approximated by the linear response (given in equation (3.11)) over a range
of torque levels from zero up to nearly the jump torque. This simply follows from the
basic kinematics of the system, which do not promote the generation of higher order
harmonics in the absorber motion. Due to the kinematics of the system, however,
such a harmonic absorber motion does not translate into a harmonic torque on the
rotor, and signiﬁcant harmonics are generated in the rotor acceleration.

The dynamic response of the rotor acceleration has a primary harmonic only when

the system is signiﬁcantly mistuned, but possesses a sizable secondary harmonic in all

71

cases. Also, it is observed that the rotor acceleration is more sensitive to mistuning

than to the nonlinear path parameter. Therefore, when developing absorber design

speciﬁcations, mistuning should always be taken into account.

5.2 Limitations and Future Work

In this section we list a few limitations of the model employed and possible directions

for improving the model for future studies. Some known facts about these limitations

from concurrent studies are brieﬂy described. Also, additional comments are offered

about another absorber system and some speculations about added beneﬁts of using

such absorbers.

0 Consideration of dissipation. Damping is ignored in this study. It generally
has the effect of reducing the performance of absorbers in terms of rotor ac-
celeration, but it also causes a slightly increased torque range for acceptable
performance. An improved model including damping effects should be investi-
gated. Also the dynamic stability of the steady state response of the system
should be determined, and this result must be obtained from a model with some

dissipation.

Generalized N absorber dynamic system. The dynamic system should be gener-
alized to include the dynamics of a set of N absorbers, even if they are identical.
Recent studies have shown that multi-absorber systems can behave quite dif-
ferently than the present idealized model. Absorbers do not always move in a
synchronous manner and studies have shown that beyond certain torque level,
the amplitude of one of the absorbers in the set may become quite large. Multi-
absorber systems are mostly used for balancing and/ or due to restricted space

around the rotor. See recent papers for detail [23, 24].

72

0 Consideration of rolling and slipping effects for biﬁlar absorbers. Biﬁlar ab-
sorbers employ rollers and they may have rolling/ slipping effects depending
upon the operating and lubrication conditions. However, these effects will be
small if their inertia is small relative to that of the absorber mass. Denman

includes a detailed analysis of roller dynamics in his study [11].

0 Consideration of rotor rigidity. An improved model should include rotor ﬂexibil-
ity. Although rotors are generally quite stiff as compared to the absorbers, their
effects on the relative motion of the absorbers may be signiﬁcant. This effect
is especially important when more than one absorber is used in the absorber

system.

0 Inclusion of a complex applied torque model. The applied torque is much more
complex than the simple single-harmonic model considered here. It generally
includes multiple harmonics and is generated by complex interaction of the
inertia forces of engine components, the bearing frictional forces, the gas pres-
sure forces, etc. The paper by Borrowski et a1. [10], describes using multiple
absorber paths to counteract each individual harmonic of the torque and satis-
factory results were obtained using this approach. Similarly, the paper of Lee
and Shaw [25] offers a more comprehensive engine dynamics model, incorporat-
ing the aforementioned effects. One could (and should) test proposed absorber

paths using such a model before building hardware.

0 Use of ﬂywheels. Flywheels, owing to their simpler design, can be used in com-
bination with absorber systems. Flywheels also can smooth out irregularities
in the torsional system response, but at the expense of reduced system respon-
siveness and increased weight. The use of absorbers in place of ﬂywheels may

be critical for some high-performance applications.

73

e The subharmonic absorber system. This system, as described in Lee et al. [26]
offers absolutelyperfectperformance by the measures used in this study. It
renders zero rotor acceleration over a large torque range using only two ab-
sorbers. Its main limitation is that it requires slightly more space for absorber

movement. Also, this system is yet to be tested experimentally.

By implementing the above improvements, one can gain a more accurate predic-
tion of the dynamics involved in CPVA systems and have greater insight in solving

torsional vibrations problems using these devices.

APPENDICES

APPENDIX A

Derivation of Linear Solution

A.1 Linearization in s and 9

Note that from equations (2.9) and (2.10), a: and g are functions of s, and since 3 will

be restricted to small values, the following approximations can be used for 2(3) and

9(8),

:c(s) = 1_m232

(m2 + m4) 32

2

 

9(8) = 1 (A.1)

Note that here m is given by equation (2.20). These approximations are obtained by
expanding the general path function :r(s) and keeping terms up to order 32. A further
account of these approximations can be found in Chapter 3. Note that the above
equations are independent of the path parameter A. By using these assumptions,
retaining only those terms linear in s and 9, equations (2.9) and (2.10) can be reduced

to

,. ~ 1dr
“(a—53(3) _ o, (A.2)
§+(1+bo)9 = I‘nsinn9 (A.3)

74

75

The above equations represent the linearized, dimensionless equations of motion for
the system under investigation. The solution for the above equations is developed in

the next section.

A.2 Linear Solution

Equations (2.22) and (2.23) can be expressed in the second order standard form of
M :2 + K x = F (A.4)

where

M is the mass matrix,
K is the stiffness matrix,
F is the force vector and

:c is the displacement vector, resulting in

(Him) -1 ('9' + o o o Fusinn9

1 1 5 0m2 s 0

(A.5)

Note, that there is no damping in the system and the stiffness matrix is only
positive semi-deﬁnite, indicating the presence of a rigid body mode.

For small amplitude motion, 9 can be approximated by,

9 = r+e (A.6)

where, 7' = Qt and e is a small deviation in 9.

76

The steady state solution of the system is given by

688 Al

x,, = = sin nr = A sin nr (A.7)
3” A2

where A = (K — nzM)_1 F0 and F0 represents the constant part of the forcing
vector.
Substituting equation (A.7) into the equations of motion equation (A5) and solv-

ing for A yeilds

 

1 m2 — n2 n2 F”
n2 —n2 (1 + b0) 0
or
[‘n m2 — n2
A = — (A.9)
A 2
n
where A = [K - n2M| = n2 (nzbo — m2 (1 + bo))
Therefore,
- Pa 2 2 -
6,, = A1 srn nr = K (m — n )sm nr, (A.10)
and the steady-state linearized acceleration, 9,,, is given by
9,, = 2,, = — 2E": (m2 — n2) sin nr
2 __ 2
PM (m n ) sinnr. (A.11)

m2 + born2 — ban2

Note that this linearized result corresponds to the ﬁrst term of the non-linear
acceleration as obtained by the perturbation method given in equation (3.22). Also,

note that it is independent of K. (or, equivalently, of A).

APPENDIX B

State equations for the model

I (1 +133)2 (bo+$2g($1)+$($1))$'($1)
2 (be (1 + .3)? — (1 + x3)2g(:c1)2 +(1+ warm»)
(—$2 — g (2:1)) (I‘n sin n9 - mg (1 + 2:3)2 9’ (21) — 32(1-l- 3:3)2 :c’ ($1))
50 (1 + 1‘3)2 "' (1+ $3l29($1)2 + (1 + $3)2$ (1'1)
, — ((1 + 23)3g(x1)z'(x1))
2(bo(1+ x3)? — (1+ 33)29($1)2+(1+ arm.»
(1+ 2:3) (I‘n sin n9 — $3 (1 + x3)2g' (1:1) — 32(1-1- 2:3)2 3’ (2:1))

B.I
bo(1+1'3)2-(1+$3)29(1‘1)2+(1+$3)2$($1) ( )

Note that g,g’,:r and :r’ are functions of 2:1.

77

 

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