i
PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS
Except where reference is made to the work of others, the work described in this
dissertation my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
________________________
Roland Horv?th
Certificate of approval:
________________________ ________________________
Subhash C. Sinha George T. Flowers, Chair
Professor Professor
Mechanical Engineering Mechanical Engineering
________________________ ________________________
John E. Cochran, Jr. Dan B. Marghitu
Professor Professor
Aerospace Engineering Mechanical Engineering
________________________
Stephen L. McFarland
Acting Dean
Graduate School
ii
PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS
Roland Horv?th
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 11, 2006
iii
PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS
Roland Horv?th
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense.
The author reserves all publication rights.
______________________________
Signature of Author
______________________________
Date of Graduation
iv
VITA
Roland Horv?th, son of J?nos Horv?th and Erzs?bet Fazekas, was born on August
16, 1972, in Kaposv?r, Hungary. He entered the Budapest University of Technology and
Economics (former Technical University of Budapest), September 1994, where he
received Master of Science degree in Mechanical Engineering. He joined the doctoral
program in Mechanical Engineering at Auburn University, Auburn, Alabama, in the fall
of 2000.
v
DISSERTATION ABSTRACT
PASSIVE PENDULUM BALANCER FOR ROTOR SYSTEMS
Roland Horv?th
Doctor of Philosophy, May 11, 2006
(M.E., the Budapest University of Technology and Economics, 2000)
152 Typed Pages
Directed by George T. Flowers
Automatic, passive selfbalancing systems are important tools for reducing the
effects of synchronous vibration in a variety of rotating machinery. Such systems are
ideally capable of precise balancing, subject to certain dynamic restrictions. There are a
number of designs that are used, but the most common type is the ball balancer system
that employs balls that move inside a cylindrical race or channel. However, such systems
may be subject to a variety of effects that arise due to rolling resistance. An alternative
approach uses pendulums rather than balls to provide the balancing. In the present work,
a passive pendulum balancer system is investigated from several aspects. A mathematical
model has been developed to discover the stability characteristic of the pendulum
balancer. Because of the obvious potential for practical application pendulum balancers
this system was investigated from engineering point of view. These investigations tried to
cover all the possible differences that could arise when the mathematical model would be
materialized as a real passive balancing device. The application of non identical
vi
pendulums was studied in detail and its advantages and disadvantages are discussed. The
influences of rolling resistance and shaft misalignment on the functional capability of
pendulum selfbalancing systems are specifically examined.
The study of a passive pendulum balancer with nonisotropic suspension is also
presented. The existence of two natural frequencies results in two distinct areas of
stability. These stable areas are determined by Floquet analysis and verified by numerical
simulations and experimental measurements.
vii
ACKNOWLEDGMENTS
The author would like to express his appreciation and thanks to his advisor
Professor George T. Flowers, Department of Mechanical Engineering for his guidance
and support toward the completion of this dissertation. The author also wishes to
acknowledge the following committee members: Dr. Subhash C. Sinha, Professor, Dr.
Dan Marghitu, Associate Professor, Department of Mechanical Engineering; and Dr.
John Cochran, Jr., Department Head and Professor of Department of Aerospace
Engineering.
The author is thankful for the invaluable initial guidance to his former professor Dr.
G?bor St?p?n, DSc, Department of Applied Mechanics Budapest University of
Technology and Economics. As well, the author would like to thank his friend, Dr.
Tam?s Insperger, Associate Professor, Department of Applied Mechanics Budapest
University of Technology and Economics, for his great advice and ideas.
Finally, the author would like to thank J?nos Torma and B?la Pallos, his former
teachers for supporting and believing in him.
viii
Journal used: Journal of Sound and Vibration
Computer software used: Microsoft Office Word 2003
ix
TABLE OF CONTENTS
LIST OF TABLES............................................................................................................ xii
LIST OF FIGURES .........................................................................................................xiii
1 INTRODUCTION ...................................................................................................... 1
1.1 Background......................................................................................................... 1
1.2 Motivation for research....................................................................................... 4
1.3 Organization of Dissertation............................................................................... 5
2 BASIC TWO DEGREES OF FREEDOM (DOF) MATHEMATICAL MODEL ..... 6
3 PASSIVE PENDULUM BALANCER WITH ISOTROPIC SUSPENSION .......... 12
3.1 Balancing boundaries (relative balancing areas) of a twopendulum balancer
with nonidentical pendulums....................................................................................... 14
Visualization of balancing capability: ?CG circles?............................................. 17
Singular Points and Stability Characteristics........................................................ 20
Stability Analysis.................................................................................................. 29
3.2 EXPERIMENTAL INVESTIGATION............................................................ 41
3.2.1 Pendulum Balancer Experimental Facility ............................................... 41
3.2.2 Experimental validation for nonidentical pendulums.............................. 45
Subcritical operation ............................................................................................. 45
x
Supercritical operation.......................................................................................... 47
3.2.3 Numerical and Experimental validation ................................................... 50
Special case: Identical pendulums ........................................................................ 52
Numerical and Experimental validation ............................................................... 54
3.3 Influence of Pendulum Shaft Misalignment ..................................................... 58
3.3.1 Experimental investigation of pendulum shaft misalignment .................. 62
Pendulum Balancer Experimental Facility ........................................................... 62
Implementation of shaft misalignment ................................................................. 62
Description of experimental procedure................................................................. 63
3.3.2 Experimental comparison of the sensitivity and consistency of ball and
pendulum balancers .................................................................................................. 67
Ball Balancer Experimental Facility..................................................................... 67
Ball Balancer Experimental Results ..................................................................... 68
Pendulum Balancer Experimental Results............................................................ 74
4 PASSIVE PENDULUM BALANCER WITH NONISOTROPIC SUSPENSION 78
4.1 Analytical investigation .................................................................................... 78
4.1.1 Stability of homogeneous linear system with time periodic coefficients . 84
4.1.2 Floquet analysis: piecewise approximation .............................................. 86
4.1.3 Floquet analysis with numerical integration: single pass scheme ............ 88
4.1.4 The result of Floquet analysis................................................................... 90
Floquet characteristic multipliers for the Type I singular point ........................... 93
xi
4.1.5 Stability changes for different damping coefficients................................ 96
4.2 Experimental facility with nonisotropic suspension........................................ 99
4.2.1 Description of experimental facility ......................................................... 99
4.2.2 Description of experimental procedure................................................... 105
4.3 Numerical and experimental investigation ..................................................... 106
4.4 Summary of validation.................................................................................... 112
5 CONCLUSIONS AND FUTURE WORK ............................................................. 113
BIBLIOGRAPHY........................................................................................................... 115
APPENDIX A PARAMETERS OF EXPERIMENTAL FASCILITY .......................... 119
APPENDIX B MATLAB SOURCE CODES ................................................................ 121
5.1 MatLab source code: Model40.m ................................................................... 122
5.2 MatLab source code: ODE01.m ..................................................................... 131
5.3 MatLab source code: wRunUp.m ................................................................... 132
xii
LIST OF TABLES
Table 3.1 Stability table of Type I singular point ............................................................ 32
Table 3.2 Stability table of Type II singular point........................................................... 34
Table 3.3 Stability table of Type III singular point ......................................................... 35
Table 3.4 Stable configuration for two operational and three constructional cases ........ 36
xiii
LIST OF FIGURES
Figure 2.1 Two DOF model with radial mass imbalance: M
P
........................................... 6
Figure 2.2 Magnitude and phase shift of the frequency response...................................... 9
Figure 2.3 Configurations of the frequency response and the three forces of the
pendulum as a function of phase shift............................................................................... 11
Figure 3.1 4 DOF model of rotor with twononidentical pendulum balancer in a rotating
coordinate system.............................................................................................................. 13
Figure 3.2 Balancing boundaries and relative balancing areas of twopendulum set...... 16
Figure 3.3 Balancing boundaries of twopendulum set for a series of mass imbalance S
P
........................................................................................................................................... 17
Figure 3.4 CG circles relative to a certain radial mass imbalance S
P
.............................. 18
Figure 3.5 Illustrations of the three types of singular point............................................. 26
Figure 3.6 Singular point loss as a function of relative damping and operational speed for
properly oversized pendulums ........................................................................................ 38
Figure 3.7 Singular point loss as a function of relative damping and operational speed for
improperly oversized pendulums.................................................................................... 39
Figure 3.8 The side view of pendulum balancer experimental facility............................ 43
Figure 3.9 Top view of pendulum balancer experimental facility................................... 44
Figure 3.10 Side view of pendulum assembly................................................................. 44
Figure 3.11 Analytical and experimental results for the amplitude of vibration
(subcritical operation) ....................................................................................................... 46
xiv
Figure 3.12 Analytical and experimental results for the position of pendulums
(subcritical operation) ....................................................................................................... 47
Figure 3.13 Analytical and experimental results for the amplitude of vibration
(supercritical operation).................................................................................................... 49
Figure 3.14 Analytical and experimental results for the position of pendulums
(supercritical operation).................................................................................................... 49
Figure 3.15 Results of numerical simulation and experimental measurement of twonon
identical pendulum............................................................................................................ 51
Figure 3.16 Analytical and experimental results for the amplitude of vibration
(supercritical operation).................................................................................................... 53
Figure 3.17 Analytical and experimental results for the position of pendulums
(supercritical operation).................................................................................................... 54
Figure 3.18 Results of numerical simulation and experimental measurement of two
identical pendulum............................................................................................................ 55
Figure 3.19 Possible design solutions for pendulum balancing systems......................... 57
Figure 3.20 Mathematical model of rotor system with noncentered pendulums ........... 58
Figure 3.21 Simulation results showing the nondimensionalized rotor vibration level for
the system with pendulum shaft misalignment................................................................. 60
Figure 3.22 Simulation results showing the absolute and relative positions of the
pendulums for the system with pendulum shaft misalignment......................................... 61
Figure 3.23 Experimental results showing the nondimensionalized rotor vibration level
for the system with pendulum shaft misalignment ........................................................... 64
Figure 3.24 Experimental results showing the absolute and relative positions of the
pendulums for the system with pendulum shaft misalignment......................................... 65
Figure 3.25 Side view of ball balancer experimental facility .......................................... 69
Figure 3.26 Top view of ball balancer experimental facility........................................... 70
xv
Figure 3.27 The final positions of balancing balls and vibration level for different
startups .............................................................................................................................. 71
Figure 3.28 Deformation of contact surfaces and force distribution of the balancing ball
and channel ....................................................................................................................... 73
Figure 3.29 The final positions of balancing pendulums and the level of vibration for
different startups ............................................................................................................... 75
Figure 3.30 The final positions of balancing pendulums and the level of vibration on the
zoomed plot for different startups..................................................................................... 76
Figure 3.31 Force distribution of pendulum and ball balancer........................................ 77
Figure 4.1 Coarse Floquet stability map of Type I singular point................................... 91
Figure 4.2 Fine Floquet stability map of Type I singular point with the points used for
numerical and experimental validation............................................................................. 92
Figure 4.3 The unit cylinder and the biggest Floquet characteristic multiplier as a
function of relative running speed. ................................................................................... 94
Figure 4.4 Three different views of unit cylinder and the Floquet characteristic
multipliers of Type I singular point .................................................................................. 95
Figure 4.5 Floquet stability map of Type I singular point for different relative damping
coefficients of rotor suspension ........................................................................................ 97
Figure 4.6 Floquet stability map of Type I singular point for different damping
coefficients of pendulums................................................................................................. 98
Figure 4.7 Side view of pendulum balancer experimental facility with nonisotropic
suspension....................................................................................................................... 102
Figure 4.8 Top view of rotor assembly.......................................................................... 103
Figure 4.9 Side view of rotor assembly ......................................................................... 104
Figure 4.10 Numerical and experimental validation of point A .................................... 107
Figure 4.11 Numerical and experimental validation of point B .................................... 108
xvi
Figure 4.12 Numerical and experimental validation of point C .................................... 109
Figure 4.13 Numerical and experimental validation of point D .................................... 110
Figure 4.14 Numerical and experimental validation of point E..................................... 111
xvii
NOMENCLATURE
t = Time [s]
T = Time period [s]
? = Phase angle [deg]
? = Angular displacement of disk [deg]
?(t) = Angular velocity of the disk [rad/s]
?
n
= Natural angular velocity [rad/s]
f
ni
= i
th
Natural frequency [Hz]
?
1,2
= Linear degrees of freedom [m]
?
3,4
= Angular degrees of freedom [deg]
Y
1,2E
= Linear coordinates of singular points, (linearized system) [m]
Y
3,4E
= Angular coordinates of singular points (linearized system) [deg]
M
D
= Mass of disk [kg]
M
P
= Mass imbalance [kg]
M
BA
= Mass of pendulum A [kg]
M
BB
= Mass of pendulum B [kg]
M
S
= Mass of the entire rotor system [kg]
I
D
= Mass moment of inertia of disk [kgm
2
]
P
3
= Radial perturbation variable [m]
R
D
= Radius of disk [m]
xviii
R
BA
= Length of pendulum A [m]
R
BB
= Length of pendulum B [m]
e = Linear distance of CG of imbalanced rotor [m]
k
1,2
= Linear spring stiffness [N/m]
c
1,2
= Linear damping coefficients [Ns/m]
c
3,4
= Angular damping coefficients [Nms/rad]
? = Relative damping coefficient []
?
C
= Nondimensionalized shaft misalignment []
R
C
= Shift of suspension of pendulums [m]
e = Linear distance of CG of imbalanced rotor [m]
? = Nondimensionalized amplitude of vibration []
A
PR
= Amplitude of vibration with released pendulums [m]
A
PL
= Amplitude of vibration with locked pendulums [m]
N = Amplification factor []
A = Amplitude of vibration [m]
A
Stat
= Static amplitude of vibration [m]
[I] = Identity matrix
[M] = Inertia matrix
[D] = Damping matrix
[G] = Gyroscopic matrix
[K] = Elastic matrix
[N] = Nonconservative force matrix
[A] = Coefficient matrix
xix
[?] = Fundamental matrix
[C] = Floquet Principal Matrix
? = Characteristic multiplier
? = Relative running speed []
K = Relative spring stiffness []
S
A,B
= First order moment of inertia of pendulums [kgm]
S
P
= First order moment of inertia of mass imbalance [kgm]
T = Sum of all kinetic energies [J]
V = Sum of all potential energies [J]
D = Sum of dissipation energies [J]
Q
1,2
= Generalized forces [N]
Q
3,4
= Generalized torques [Nm]
F
CF
= Centrifugal force [N]
F
C
= Constraining force [N]
F
F
= Friction force [N]
F
R
= Force of rolling resistance [N]
F
B
= Balancing force [N]
1
1 INTRODUCTION
1.1 Background
Rotating machinery is commonly used in civil, military and industrial applications
including vehicle wheels, machining tools, industrial rotating machinery aircraft gas
turbine engines and helicopter blades. One of the primary sources of vibration is mass
imbalance, which occurs when the principal axis of inertia of the rotor is not coincident
with its rotational axis.
There are two common balancing methods which are used to align the principal
inertia axis and rotational axis. One method is offline balancing in which the rotating
machine is stopped for the adjustment of mass distribution. The second method is online
balancing in which the mass distribution rearrangement happens continuously during
rotation. Such automatic balancing can be either active or passive. Active balancing
systems use sensors to measure the unbalance level and actuators to shift the mass
distribution. Passive balancing systems perform a similar task but without sensors,
control laws and external power supplies.
The simplicity, reliability and relatively low cost of passive balancing systems
make them a very attractive solution and thus, have been the subject of significant past
research.
The first documented appearance of an auto balancing device was by A. Fesca who
patented improved centrifugal machine equipped with three ring balancer in 1872 [1] The
second documented automatic balancer is also a ring balancer patented by G. W. Ledyard
in 1896. Ledyard used a series of rings around the outer diameter of his centrifugal
machine [2]. In the same year a new type of rotor balancing system was registered by
2
United States Patent Office. M. Leblanc patented his automatic balancer for rotating
bodies. In his design, the balancer consists of a simple cylindrical chamber field with a
heavy liquid [3]. Thearle [4] in 1932 published a detailed experimental study. Probably
this design was the first to use selfaligning balls to achieve passive balancing. The first
documentation of pendulum type balancer is also a patented invention by K. Clark 1946
[5]. Clark used four noncentrally attached pendulums to reduce the level of centrifugal
machines. Thearle [6] investigated in detail the Leblanc balancer and summarized the
requirements of ideal balancers. In the same journal, Thearle [7] compared several
different types of automatic dynamic balancers, such as a ring, pendulum and ball
balancers. This is the first appearance of an automatic pendulumbalancer in the literature.
In his paper Thearle concluded that by placing the pivot of pendulums at the center of
rotation the pendulum balancer become equivalent to the ring balancer, his analysis was
heuristic and did not include a rigorous and detailed analytical study. In addition, he
concluded that ball balancers were a superior system.
Since the 1950?s, the majority of researchers have concentrated their efforts on
investigations of ballbalancing systems. Sharp [8] provided a stability analysis of the
balanced condition for a twoball balancer on a planar rotor, and the presented the results
of a parametric study of that system. Conclusions were drawn regarding the satisfactory
operation of such a balancer. B?vic and H?gfors [10] in 1986 by using the method of
multiple scales showed that an automatic ballbalancer reduces vibration for planar and
nonplanar 6 DOF rotor systems. Their research is a detailed analytical study but it was
not verified by any experimental measurements.
3
Wettergren [11] investigated a ballbalancing system with one and two balls in a
cylindrical groove. He also examined the effect of oil viscosity and found a relationship
between the viscosity and the stability properties of the balancing system. However, his
research was analytical in nature and not verified experimentally. Huang and his
colleagues [12] in 2002 presented a combined analytical and experimental study of the
loss of balancing capability for a balltype balancing system due to runway eccentricity
and rolling resistance. Other investigators [13] have considered the effects of dry friction
on ball balancer systems. This is a specific concern if such units are operated in an un
lubricated condition, which is desirable for some applications (such as optical drives)
where lubricants can cause contamination and damage. It was demonstrated that, ?even
for very low friction coefficients? the balancing behavior can deteriorate considerably. K.
Green and his colleagues demonstrated a nonlinear bifurcation analysis of an automatic
ball balancer [14]. Applying only analytical methods they discovered large regions in the
parameter space where the ball balancer shows instability. They also investigated the
effect of perturbations and transient dynamics.
The only substantial study in the current literature on pendulumbased passive
balancing systems was conducted by Kubo and his colleagues, who presented their
research on an automatic balancer using pendulums [9]. This paper was concerned with
theoretical and experimental investigations on the dynamic behavior and stability of an
automatic balancer using centrifugal pendulums. Although they noted the most important
requirement that has to be satisfied for proper balancing such as the pivot of the
pendulums have to be placed to the center of rotor, the experimental facility was
equipped with noncentrally attached pendulums. Additionally this paper has a
4
fundamental flaw. In their study they model the suspension of disk as single DOF. This is
not an adequate model for a rotating disk especially when a centrifugal force has a
balancing effect.
1.2 Motivation for research
Providing reliable, on line balancing for wide range of applications is a great
challenge. The complexity of active on line balancing reduces its reliability. This
complexity does not suit zero tolerance applications and it requires tremendous research
hours to develop a somewhat reliable and robust active balancing system, which is
expensive production cost and this way rejects any application on mass production level.
Application of automatic balancers on a mass production level causes a focus of
efforts on implementation of the simplest passive balancing system: the ball balancer.
The extreme simplicity of ball balancer systems has generated a huge amount of research
effort but the widespread utilization of ball balancer systems has not taken place. The
effect of rolling resistance greatly reduces number of potential applications of automatic
ball balancers.
5
The desire to design and build a simple, reliable passive balancing device has not
been fulfilled. This research purposefully targeted this technological deficiency. The
discovery of such a system would initiate numerous potential applications:
 Providing artificial gravity in space
 Centrifuges for Gravitational biology research
 Helicopter blade balancing
 Centrifugal casting
 Handheld power tools
1.3 Organization of Dissertation
This research effort is a study of a passive pendulum balancer. The research work
was performed in the Vibration Analysis Laboratory of the Department of Mechanical
Engineering at Auburn University. Specifically, the work includes:
? Development of mathematical model
? Development of numerical procedures
? Development and testing of experimental facilities
6
2 BASIC TWO DEGREES OF FREEDOM (DOF) MATHEMATICAL MODEL
In order to provide an appropriate background for the current work, this discussion
begins by considering the behavior of a 2 DOF thin disk model rotating in a horizontal
plane with a radial mass imbalance represented by a point mass M
P
that is shifted from
the center of disk by P
3
in the 'I
null
direction, as shown in Figure 2.1.
?1
?2
?(t)
P3
MP
k1
k2
c1
c2
?(t)
I'
I
J
0
0
0I
0J
J'
Figure 2.1 Two DOF model with radial mass imbalance: M
P
The disk model has two degrees of freedom, ?
1
and ?
2
, which are mutually
orthogonal linear displacements in the same horizontal plane. The model is symmetric,
having the same spring stiffness k
1,2
and damping coefficient c
1,2
in both directions,
which yields coincident natural angular velocity ?
n
.
7
The differential equations of motion are:
2
1113
cos( )
P
M ckPM t? ??+?+?=
nullnull null
(21)
2
2223
sin( )
P
M ckPM t? ??+?+?=
nullnull null
(22)
D P
M MM= +
(23)
12
cc c= =
(24)
12
kk k= =
(25)
The solution of radially unbalanced system is:
()
2
1
22
2
cos( )
e
tt
kc
MM
?
??
?
?
?= ?
????
?+
????
????
(26)
()
2
2
22
2
sin( )
e
tt
kc
MM
?
? ?
?
?
?= ?
????
?+
????
????
(27)
3
:
P
D P
PM
Where e
M M
=
+
(28)
The phase angle of frequency response is:
1
2
() tan
c
k
M
M
?
??
?
?
? ?
? ?
? ?=
??
? ?
?
??
? ?
??
? ?
(29)
8
The magnitude of frequency response is:
2
22 2 2 4 2
()
2
M
N
ck kM M
?
?
???
=
+? +
(210)

()
Stat
A
N
A
? =
(211)
22
12
A = ?+? (212)
3
P
Stat
M
A P
M
= (213)
The rotating position vector, directed from the center of rotation to the center of
disk, has a lag angle (phase angle ?) relative to the forcing vector. The forms of the phase
angle and magnitude relations are shown in Equations (29) and (210), respectively, with
plots for several different relative damping coefficients shown in Figure 2.2. As is well
known, the response amplitude and phase are sensitive to damping level.
9
0 0.5 1 1.5 2 2.5 3 3.5 4
0
1
2
3
4
? / ?
n
N
[]
Magnitude of the frequency response
?
= 0.05
?
= 0.2
? = 0.5
? = 1
0 0.5 1 1.5 2 2.5 3 3.5 4
0
50
100
150
? / ?
n
?
[d
eg
]
Phase angel of the frequency response
? = 0.05
? = 0.2
? = 0.5
? = 1
Figure 2.2 Magnitude and phase shift of the frequency response
A series of thought experiments using this 2 DOF model provides substantial
insight into selfbalancing using pendulum balancers. Figure 2.3 is a series of snapshots
of the basic system for three operating speeds. The center of the centrifugal force field is
the origin of the first two DOF, ?
1
and ?
2.
An imaginary (almost massless) pendulum is
also shown as a part of this discussion to illustrate the force distribution along the path
where a real pendulum would move.
10
For each case, the combined effects of centrifugal force and the constraining force
acting on the system will cause the position of the imaginary pendulum to change until
these two forces become parallel.
For subcritical operating speeds, the steadystate phase delay of the displacement
related to the exciting force is less than 90?. In this case, the stable pendulum position
will be between 0? and 90? as measured from the horizontal axis (as shown in Figure
2.3.a) which exacerbates the mass imbalance of the system.
For the resonance case, the phase delay is 90?, and the equilibrium position of the
pendulum is 90? behind the position of the imbalance mass, as shown in Figure 2.3.b.
Again, the effect of the pendulum is to shift the CG further from the origin though not
quite as far as in the subcritical case.
Figure 2.3.c shows the system operating at a supercritical running speed. The phase
delay is between 90? and 180?. The resultant effect of the centrifugal and the constraining
force acting on the pendulum will drive it toward a position on the opposite side of the
disk from the mass imbalance, partially compensating the unbalancing effect.
11
Mp
Mp
Mp
Pendulum
? ?
?
a. b. c.
?1?1 ?1
Figure 2.3 Configurations of the frequency response and the three forces of the
pendulum as a function of phase shift
12
3 PASSIVE PENDULUM BALANCER WITH ISOTROPIC SUSPENSION
In this chapter, the passive pendulum balancer is investigated with isotropic
suspension equipped with a pendulum balancing system consisting of two nonidentical
pendulums. From a practical pointofview, basic differences in the pendulums (mass or
length) may be incidental due to manufacturing variations or accidental damage. On the
other hand, differences may be an intentionally designed into the system if they are
determined to be advantageous to the operation and/or performance of the balancing
system.
In the following paragraphs, a model of rotor with a two balancing system with
nonidentical pendulums is developed and analyzed. The basic elements of the
mechanical model are shown in Figure 3.1. The first two degrees of freedom (?
1
, ?
2
) are
linear displacements that describe the position of the rotor center. The third and the fourth
are angular displacements that describe the positions of pendulum A and pendulum B,
respectively.
13
I'
?1
0J
J'
?(t)
?(t)
I0
Mp
P
3
?2
I'
J'
B
A
?3
?4
Figure 3.1 4 DOF model of rotor with twononidentical pendulum balancer in a rotating
coordinate system
14
3.1 Balancing boundaries (relative balancing areas) of a twopendulum balancer
with nonidentical pendulums
First, consider the balancing capability of a pendulum balancing system with non
identical pendulums. Because the pendulums are not identical, the ability of the system to
counter a given imbalance level depends very strongly on the relationship between the
parameters of the individual pendulums and the initial imbalance level. A useful
parametric description for pendulums is the first mass moment of inertia with unit: [kg
m]. It should be noted that two nonidentical pendulums may have the same first moment
of inertia. For each of the two pendulums, A and B, and for the mass imbalance, the
following relations define the first mass moments of inertia.
ABABA
SMR=
(31)
BBBB
SMR=
(32)
3PP
SMP=
(33)
The balancing boundaries of the pendulum system for a variety of S
A
and S
B
combinations and for a specified mass imbalance, S
P
, are shown in Figure 3.2. The
shaded area ABCD with the dark line border shows the parametric values which can
compensate for the given level of imbalance. If the parametric values are in this shaded
area (as described by Equation (34)) then the pendulums are able to counterbalance the
mass imbalance. For values outside this area, the pendulums are unable to completely
compensate for the mass imbalance because they are either undersized or improperly
oversized.
15

AB P AB P
SS S and SS S+? ??
(34)
For parametric combinations either above line AD or below line BC in Figure 3.2,
the pendulums are improperly oversized relative to the mass imbalance, S
P
. The
magnitude of the difference between the first mass moments of the pendulums is greater
than the mass imbalance, as described by Equation (35). If this is the case, there are no
pendulum locations where the net unbalancing forces resulting from the two pendulums
and the rotor mass imbalance can completely cancel and the rotor will remain unbalanced.
The final level of net imbalance may be higher or lower than the initial level without a
balancing system depending upon the relative differences between the two pendulums.

AB P
SS S? >
(35)
If the parametric combination is located in the OAB triangle (Equation (36)) then
the pendulums can only partially counterbalance the mass imbalance. In this case, the
pendulum set is undersized relative to the mass imbalance.
AB P
SS S+ <
(36)
A series of similar relative balancing areas for a variety of mass imbalances are
illustrated in Figure 3.3. If the two pendulums have the same first mass moment (S
A
= S
B
),
then the parametric configuration will be located on a line that starts from the origin with
16
a 45? slope, as shown in figure 6 with a dashed line. For this case, the pendulums are
either undersized or properly oversized relative to any mass imbalance S
P
.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
S
A
[kgm]
S
B
[
kgm
]
Pr
o
p
e
r
l
y
O
v
e
r
s
i
z
e
d
I
mp
r
o
p
er
l
y
O
ve
r
s
i
ze
d
Im
p
r
o
p
e
r
ly
O
v
e
r
s
i
z
ed
U.
S.
O
A
B
C
D
S
P
= 0.010 [kgm]
Figure 3.2 Balancing boundaries and relative balancing areas of twopendulum set
17
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
S
A
[kgm]
S
B
[
kgm
]
S
P
= 0.010 [kgm]
S
P
= 0.015 [kgm]
S
P
= 0.020 [kgm]
S
P
= 0.025 [kgm]
S
P
= 0.030 [kgm]
S
P
= 0.035 [kgm]
Figure 3.3 Balancing boundaries of twopendulum set for a series of mass imbalance S
P
Visualization of balancing capability: ?CG circles?
The balancing capability of a pendulum balancing system can also be characterized
by the resulting steadystate location of the center of gravity of the entire rotor system
(rotor, pendulums and mass imbalance). Figure 3.4 shows a relative balancing boundary
plot which uses a series of circular symbols with a darkened area, which are referred to
herein as CG circles where the pendulums are able to relocate the CG of the entire system.
If these dark CG circles include the center of the rotor, then balancing can be archived for
proper operational conditions.
Again, for the case of same first mass moment, the CG circles are located on the 45?
dashed line. Leaving this line the CG circles have an internal ?blind spot?. Having this
18
blind spot not necessarily means the pendulums cannot do the balancing. If the chosen
pendulum combination stays inside the ABCD area the pendulums would balance
regardless of the existence of blind spot. Under the BC or above the AD line the blind
spot is over the center of the rotor this means balancing cannot be archived because the
pendulums bother each other they are insufficiently oversized. In order to achieve
balancing the point of chosen pendulums combination has to stay in the ABCD area. This
condition is specified by Equations (34).
SA
SB
A
B
C
D
Figure 3.4 CG circles relative to a certain radial mass imbalance S
P
O
19
In general, all of the relativebalancingarea plots shown above (Figure 3.2  Figure 3.4)
contain three distinct areas:
Rectangular region ABCD
If the parametric combination is located in this area, the pendulum system is
capable of counterbalancing the entire mass imbalance. The system is properly
oversized relative to the mass imbalance.
Triangular region OAB
If the parametric combination is located in this area, the pendulum system is not
capable of counterbalancing the entire mass imbalance. They are undersized
relative to the mass imbalance.
Regions above the AD line and below the BC line
If the parametric configuration of the pendulum system is located in either of
these regions, the pendulums are not capable of counterbalancing the entire mass
imbalance. They are improperly oversized relative to the mass imbalance and may
partially reduce the level of vibration if:

AB P
SS S? <
(37)
Otherwise, such an improperly oversized system will exacerbate the synchronous
vibration resulting in a higher amplitude of vibration than if the balancer was not present.
20
Singular Points and Stability Characteristics
The above discussion has examined the fundamental capability of a balancing
system with two nonidentical pendulums. The actual dynamic behavior of such a system
depends on its equilibrium points and their associated stability characteristics. So, the
next step in this investigation considers the occurrence of singular points and the stability
of motion in the vicinity of such points. The governing equations of motion for a
symmetric rotor suspension (c = c
1
= c
2
and k = k
1
= k
2
) in a diskfixed coordinate system
can be represented as:
1, 2, 3, 4
i
iiii
dT T V D
Qi
dt
??????
?++= =
??
?? ?? ?? ??
??
nullnull
(38)
Where: T = Sum of all kinetic energies
V = Sum of all potential energies
D = Rayleigh dissipation function
i
? = Generalized coordinates
i
Q = Generalized forces
21
() ()()
() ()
()( )()
()(
()
()
2
1323
1424
22 2
31 12
21 2 3 3 21
1 4 4 3 21 21
2313 1
1
2cos 2sin
2
2cos 2sin
1
2
2
sin
cos
cos sin
ABA
BBB
P D BA BB P D D
BA A P
B A BA BB D
PA B
TSR
SR
SP M M M M MR
MS M
SSR M
SS S
?
?
?
=+??+??+
?
?
++??+??+
?
++?++++?+?+ +
?
?
+??+???+?+
+? ??+ ?+ ??+ ??+
+?+?? ???
nullnullnull
nullnullnullnull
nullnull null
()
() ()
()
() )
()()
() ()
4
244 122 3
2412 4 12
1312
22
21
22
34233244
14
sin cos
cos
sin
11
11
cos cos
22
si
BBAA
BDBB
AP
DBABBP DBABBP
ABA BBB A B
B
SM
SRM
SM
MM M M MM M M
SR SR S S
S
?+
+? ??? ??+? ?+
+? ??? ?+ ?? ???
?? ??? ?+
++++?++++?+
+ ?+ ?+?? ?+?? ?+
???
nullnull null
nullnull nullnull
nullnull
nullnull
null null nullnull nullnull
nullnull
() ()
4133
nsin
A
S???? ?
nullnull
(39)
()
22
12
1
2
Vk=?+? (310)
()( )()()
22 222 2 2
12 1221 12 3344
111
Dc c c c c??= ? +? ? ?? ??? + ? +? + ? + ?
nullnull null null null null
(311)
0
i
Q =
(312)
22
The resulting equations of motion are:
() ( ) ( ) ( )
() () () ()
()
()()()
22
33 33 3 33
44 44 44 4
22
11 2 3 1
2
12 1 2 1
2 cos sin cos cos
2cos cos sin cos
2
20
A
B
P D BA BB
DBAB
S
S
MPMM
ckM
??
???
?
???+?+?+??
???+?+?+??
????+?++++?+
+???+?? + + ?+?=
nullnullnull null
nullnullnull
nullnull null nullnull
nullnull
(313)
( ) ( ) ( ) ( )( )
() () () ()
()()()
22
33 33 3 33
44 44 44 4
2
22 1 2
2
21 2 1 2
2 sin cos sin sin
2 sin sin cos sin
2
20
A
B
P D BA BB
DBAB
S
S
MMM
ckM
??
??
?
?????+?+??
? ? ?+ ? ?? ? ?+ ? ?
??????+++?+
+?+?+?+ + + ???=
nullnullnull null
nullnullnull
nullnull null nullnull
nullnull
(314)
( ) ( )( ( )
() () ())
31 3 3 2 3 2
22
31 32 3133
sin cos 2 sin
2cos cos sin 0
ABA
SR
c
?
?? ?
???+?+??+ ??+
+??? ??+ ??+?=
nullnull nullnull nullnull null
nullnull
(315)
( ) ( ) ( )(
() () ())
41 4 4 2 4 2
22
41 42 4144
sin cos 2 sin
2cos cos sin 0
BBB
SR
c
?
?? ?
? ? ?+ ?+ ? ?+ ? ?+
+??? ??+ ??+?=
nullnull nullnull nullnull null
nullnull
(316)
23
The singular points are found by setting the time derivative terms in Equations (313) 
(316) to zero. This results in the following set of algebraic equations.
( ) ( )( )
2
13421
cos cos 0
ES A E B E E E
YMSYSY cYkY???+ + ?+=
(317)
( ) ( )( )
2
23412
sin sin 0
ES A E B E E E
Y M S Y S Y cY kY?+ + ++=
(318)
( ) ( )( )
2
132 3
sin cos 0
AE E E E
SY Y Y Y ?? =
(319)
( ) ( )( )
2
142 4
sin cos 0
BE E E E
SY Y Y Y ?? =
(320)
These equations are nonlinear and are not, in general, easy to solve in a closed form.
However, the possible solutions are of three distinct types. The different types of singular
points and the associated stability characteristics are detailed below.
24
Type I:
The Type I singular points represent a configuration where the rotor base motions
(Y
1E
and Y
2E
) are zero, as illustrated in Figure 3.5.a. The pendulums settle into the
positions where they counter the mass imbalance, M
P
. The resulting closed form
solutions for Y
3E
and Y
4E
, in terms of a four quadrant inverse tangent function, are shown
in Equations (321) and (322). Of course, this configuration only occurs when the
pendulums are physically capable of balancing the system, which means the parametric
configuration is in the ABCD region. When the parametric configuration is outside of
that region, the terms under the square roots in Equations (321) and (322) takes on a
negative value and there is no real solution.
From the perspective of balancing effectiveness, this singular point is the most
important of the three types. If this singular point is stable, the center of rotor is not
vibrating because the relative locations of the pendulums and mass imbalance result in a
balanced system.
222
34
arctan ,
22
ABP
E
PA PA
SSSAB
Y
SS SS
??
?+?
=??
??
??
(321)
222
44
arctan ,
ABP
E
PB PB
SSSAB
Y
SS SS
??
???
=
??
??
(322)
( )( )
()()
:
A BPABP
ABPABP
WhereASSSSSS
B SSSSSS
=+? ++
=?? ?+
25
Type II:
The Type II singular points (as shown in (324) and illustrated in Figure 3.5.b)
represent a configuration where Y
4E
= Y
3E
+ 180?. If this type of singular point is stable,
the rotor exhibits a steady state synchronous whirl, because Y
1E
and Y
2E
are not zero in
the rotor fixed coordinate system. Please note that, although closed form symbolic
solutions were developed for both the Type II and the Type III singular points, they are
not shown here because of the large size of those expressions.
26
Type III:
The Type III singular points (as shown in Equation (325) and illustrated in Figure
3.5.c). In the algebraic equation system Y
4E
= Y
3E
. The Type III singular point also
includes a steady state synchronous whirling of the rotor, but for this case the pendulums
are overlapping one other.
[ ]
1234
:00
EEEE
Type I Y Y Y Y==
(323)
0
1E 2E 3E 4E 3E
:00 18Type II Y Y Y Y Y???? =+
??
(324)
[ ]
1E 2E 3E 4E 3E
:00Type III Y Y Y Y Y?? ?
(325)
a.
?
?
b.
I'
Type I
J'
Type III
I'
Type II
J'
Mp
Mp
Mp
?
J'
I'
c.
A
B
B
AB
A
Y3E
Y4E
Y3E
Y1E
Y2E
Y4E
Y1E
Y2E
Y3E Y4E
Figure 3.5 Illustrations of the three types of singular point
27
In order to assess the stability characteristics of each type of singular point, the
differential equations of motions were linearized in the neighborhood of the singular
points.
The resulting equations in vectormatrix form are:
[] [ ] [ ]
M0YDGYKNY+ +++=
nullnullnull
nullnull null
(326)
The inertia matrix is:
[]
( ) ( )
() ()
() ()
() ()
34
33
34
0sinsin
0cosco
sin cos 0
sin cos 0
SAEBE
SAEBE
AEA E ABA
BEB E BB
MSYSY
MSYSY
M
SYS Y SR
SYS Y SR
????
??
=
?
?
??
(327)
:
S D BA BB P
Where M M M M M= +++
The damping and the gyroscopic matrix is:
[ ]
() ()
() ()
() ()
() ()
34
333
44 4
2 2 cos 2 cos
22sinin
2cos 2sin 0
2cos 2sin 0
SAEBE
SA
AE AE
BE BE
DG
c M SY SY
M c SY SY
SY SY c
SY SY c
?? ?
??
??
+=
?? ???
??
??
?
?
??
(328)
28
The elastic and the nonconservative force matrix is:
[]
( )()
() ()
() ()
() ()
222
34
22
22
343
44 4
sin sin
cos cos
sin cos 0
sin cos 0
SAEBE
SA
AE A E A
BE B E B
Mk c SY SY
cMk
KN
SY S Y SKN
SY S Y SKN
??? ?
??
???+ ?
??
?+? ?
+=
?
?
??
(329)
( ) ( )
() ()
33 2 3 1 3
44 2 4 1 4
:sincos
sin cos
EEE E
EEE E
Where KN Y Y Y Y
KN Y Y Y Y
=+
The Cauchy transformation of equation (326) results in a system of first order linear
differential equation system represented by 8x8 coefficient matrix [A]. Where:
X=
Y
Y
? ?
? ?
? ?
? ?
null
null
null
null
(330)
X=[A]X
null null
null
(331)
[] []
[][ ] [][ ]
11
0
[]
I
A
M KN M DG
??
??
??
=
?+?+
??
(332)
29
Stability Analysis
For the purposes of this stability investigation, all three types of singular points
were examined separately for subcritical and supercritical operation. Because of the
symmetric rotor suspension (k = k
1
= k
2
), the undamped first and the second natural
frequencies are the same, as indicated by the following expression:
n
D BA BB P
k
M MMM
? =
+++
(333)
The sub and supercritical stability investigation also had to be further divided for
the three relative balancing areas. This means for one certain singular point there is 2x3
cases to investigate.
Each case was also further examined separately for each of the three relative
balancing areas. So, six cases are considered for each type of singular point. The specific
cases that were investigated  subcritical or supercritical operation and each of the three
relative balancing areas are shown in a series of Tables (13) in the following pages. The
specific parametric combinations were chosen to encompass all three relative balancing
areas, with the specific location of each test case marked by a dot. The real part of the
calculated eigenvalues as a function of pendulum damping, c
34
, is also shown on the right
side of each table.
30
Stability investigation of Type I singular point
From a balancing perspective, this is the most important singular point. If it is stable
for a given operating condition, the synchronous vibration caused by mass imbalance will
be eliminated. The stability characteristics of this singular point are only investigated
inside the ABCD relative balancing area. The Type I singular point does not exist outside
of that region. The analysis results are shown in Table 3.1. They demonstrate that the
Type I singular point is stable for supercritical operation and unstable for subcritical
operation.
In addition, it is important to note that the stable position of the pendulums could be
mirrored about axis X. This operation has no effect on the location of the common CG of
the pendulums. Basically, Type I singular point consists of two sets that produce identical
balancing results.
Stability investigation of Type II singular point
For the Type II singular points, the steadystate locations of the pendulums are directly
opposite to one another. Solving the associated equations resulted in two sets of explicit
symbolic solutions.
Table 3.2 summarizes the stability analysis results for each of the six cases that
were considered (as described above). This singular point is only stable for supercritical
operation and when the parametric configuration is improperly oversized relative to the
mass imbalance. For the other five cases, this singular point is unstable.
31
Stability investigation of Type III singular point
For the Type III singular points, the steadystate locations of the pendulums are
directly on top of each other. Table 3.3 summarizes the stability analysis results for each
of the six cases that were considered, in a fashion similar to that done for the other two
types of singular points. This singular point is stable for subcritical operation regardless
of the parametric configuration with regard to the relative balancing areas. In addition, it
is also stable for supercritical operation when the parametric configuration is undersized
relative to the mass imbalance. For the other two cases, this singular point is unstable.
32
Type I
x
y
5 4 3 2 1 0 1
0
1
2
3
4
5
x 10
4
Real(?)
c
34
S
A
[kgm]
S
B
[
kgm]
UNSTABLE
S
A
[kgm]
S
B
[
kgm]
Subcritical operation
S
A
[kgm]
S
B
[kg
m
]
Singular point does not exist
x
y
20 15 10 5 0
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[
kgm]
STABLE
S
A
[kgm]
S
B
[
kgm]
Supercritical operation
S
A
[kgm]
S
B
[kg
m
]
Singular point does not exist
Table 3.1 Stability table of Type I singular point
33
Further investigation has demonstrated, that for certain cases, the stable singular point is
a function of the damping coefficient of the rotor suspension. Table 3.4 summarizes the
stable singular points for all of the six cases and their dependence on rotor suspension
damping.
34
Type II
Set I Set II
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[k
g
m
]
UNSTABLE UNSTABLE
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[kgm
]
UNSTABLE UNSTABLE
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
x
y
8 6 4 2 0
0
0.5
1
1.5
2
x 10
3
Real(?)
c
34
Sub
c
ritical op
eratio
n
S
A
[kgm]
S
B
[k
gm
]
UNSTABLE UNSTABLE
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[k
g
m
]
UNSTABLE UNSTABLE
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[kgm
]
UNSTABLE UNSTABLE
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
Superc
ritical operation
S
A
[kgm]
S
B
[k
gm
]
UNSTABLE STABLE
Table 3.2 Stability table of Type II singular point
35
Type III
Set I Set II
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[k
g
m
]
UNSTABLE STABLE
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[kgm
]
UNSTABLE STABLE
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
x
y
8 6 4 2 0
0
0.2
0.4
0.6
0.8
1
x 10
3
Real(?)
c
34
Sub
c
ritical op
eratio
n
S
A
[kgm]
S
B
[k
gm
]
UNSTABLE STABLE
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[k
g
m
]
UNSTABLE UNSTABLE
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
S
A
[kgm]
S
B
[kgm
]
UNSTABLE STABLE
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
x
y
40 30 20 10 0 10
0
1
2
3
4
5
6
x 10
3
Real(?)
c
34
Superc
ritical operation
S
A
[kgm]
S
B
[k
gm
]
UNSTABLE UNSTABLE
Table 3.3 Stability table of Type III singular point
36
S
A
[kgm]
S
B
[k
g
m
]
Type III:
(function of
rotor
damping)
x
y
S
A
[kgm]
S
B
[k
g
m
]
Type III:
x
y
Subcritical operation
S
A
[kgm]
S
B
[k
gm
]
Type III:
(function of
rotor
damping)
x
y
S
A
[kgm]
S
B
[k
g
m
]
Type I: x
y
S
A
[kgm]
S
B
[k
g
m
]
Type III: x
y
Supercritical operation
S
A
[kgm]
S
B
[k
gm
]
Type II:
(function of
rotor
damping)
x
y
Table 3.4 Stable configuration for two operational and three constructional cases
37
A close examination of the characteristics of the various singular points shows that
for a certain level of rotor suspension damping there is a rotor speed domain where
certain singular points become complex. Such a result has no physical meaning and the
singular points do not exist for those cases.
Figure 3.6 shows a highlighted area in which the singular point is lost for properly
oversized pendulums. For supercritical operation, the stable singular point is Type I and it
is independent of damping. This configuration is represented by the horizontal upper
boundary line of this area. For subcritical operation, the stable singular point is Type III
and it is a function of damping. The Type III singular point does not exist when the
system is operating slightly below the critical speed and the damping is relatively high.
38
Figure 3.6 Singular point loss as a function of relative damping and operational speed for
properly oversized pendulums
Figure 3.7 shows the dark area where the singular point is lost for improperly oversized
pendulums. For supercritical operation, the stable singular point is Type II and for
subcritical operation the stable singular point is type III. Both are a function of damping.
The damping dependence results in a singular point loss near the critical speed for cases
with a relatively high rotor damping.
0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 25 50 75 100
c
12
[Ns/m]
?
?
/
?
n
Singular point does not exist
Together: Set II
Separated
x
y
x
y
Singular Point Type I
Singular Point Type III
39
Figure 3.7 Singular point loss as a function of relative damping and operational speed for
improperly oversized pendulums
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
0 25 50 75 100
c
12
[Ns/m]
?
?
/
?
n
Singular point does not exist
Together: Set II
Opposite: Set II
x
y
x
y
Singular Point Type II
Singular Point Type III
40
Table 4 shows that the stable singular points for relatively undersized pendulum
combinations have no damping dependence. This means there is no singular point loss for
any operational speed or rotor damping.
In summary, for each of the six cases, there is only one stable singular point. For
subcritical operation, only the Type II singular point is stable regardless of location
within the relative balancing areas. When the pendulums are oversized, either properly or
improperly, relative to the mass imbalance the singular point is a function of damping
when the system is operating subcritically. When the pendulums are undersized and the
rotor system is operating subcritically, there is no damping dependence.
For supercritical operation, if the pendulums are undersized relative to the mass
imbalance the stable singular point is damping independent. When the pendulums are
oversized, the stable singular point is Type III and is independent of damping. Improperly
oversized pendulums for supercritical operation have a stable Type II singular point
which is damping dependent.
41
3.2 EXPERIMENTAL INVESTIGATION
3.2.1 Pendulum Balancer Experimental Facility
In order to gain detailed insight into the dynamic characteristics and performance of
a pendulum balancing system, an experimental facility was designed, fabricated and used
for a series of tests. The design of this test rig is such that the weights of the pendulums
can be changed while maintaining the same lengths.
Figure 3.8 shows a side view of the entire experimental system. The disk rotation is
in the horizontal plane. It is supported by a base plate (5) which is mounted on the shaft
of a DC motor (11). The DC motor is supported by a column attached to the ground with
a flexible joint. This vertical column is supported from in the horizontal (x and y)
directions by springs attached at their opposite ends to fixed supports. This arrangement
allows the disk to move in nearly horizontal plane with minimal friction.
Figure 3.9 shows a topview and Figure 3.10 a side view of the pendulum assembly.
In these figures, the shaft of the pendulums (1) and the pendulums themselves (2) and (3)
can be seen. Each pendulum is attached to an aluminum disk and supported at the center
by two small ball bearings. The pendulums can be locked at the 0 and 180? positions by
plastic pins (6) to the frame of the pendulum assembly (4) when the system is stopped.
The motion of the pendulums is regulated by a damping system, which consists of two
magnet magazine rims (9) which can accommodate up to 12 small rear earth magnets
(10). One of the damping magazines is placed beneath the bottom pendulum and the
other is placed above the top pendulum. Between the pendulums there is a steel magnetic
field guide (12). The pendulums, the damping rims and the magnetic field guide comprise
a sandwich structure. This arrangement generates eddy currents inside the aluminum disk
42
of each pendulum, producing a velocity proportional damping force that can be
controlled by the number and the polarity of the magnets (10). The pendulum assembly
(4) is fixed to the base plate (5) by two bolts from underneath. The gross imbalance of the
system center of mass is set by adjusting the brass weights (7, 8), M
P
, and its radial
position, P
3
. Weights (7) are basically identical brass nuts on threaded radial rods. When
these weights (7) are twisted to the base (showed on Figure 3.9) and the auxiliary mass
imbalance (8) is removed from the system the rotor is balanced because the main brass
weights (7) are counterbalancing each other. By changing the location of one of these
brass weights the desired mass imbalance can be added to the rotor system.
43
Figure 3.8 The side view of pendulum balancer experimental facility
44
Figure 3.9 Top view of pendulum balancer experimental facility
Figure 3.10 Side view of pendulum assembly
45
Using this pendulum balancing facility, a series of experiments were conducted to
validate the analytical results obtained in the previous section. The details and results of
these experiments are presented in the following sections.
3.2.2 Experimental validation for nonidentical pendulums
For the first series of experimental measurements, the facility was equipped with
two nonidentical pendulums. The parameters of the pendulums were not changed. But,
the magnitude of the rotor imbalance was adjusted by changing the radial position of the
brass weight. In this way, the three relative balancing areas were investigated for both
subcritical and supercritical operation.
Subcritical operation
As shown in Table 4, for subcritical operation only the Type III singular point is
stable regardless of balancing area. For this configuration, the center of rotor will have a
steadystate whirling motion even for zero mass imbalance. The analysis results are
shown with a continuous line on Figure 3.11, along with the results form the
experimental measurements. The horizontal axis is the first moment of the mass
imbalance, S
P
, and the vertical axis is the vibration amplitude.
In similar fashion, Figure 3.12 also shows the analytical and experimental results
but the vertical axis represents the steadystate position of the pendulums. The ?circle?
symbols represent pendulum ?A? which has the higher first mass moment, S
A
, compared
to pendulum ?B? which is represented by the ?x? symbols. The experiments show that
the pendulums stayed together regardless of the location of the system within the relative
46
balancing areas, which tended to verify quite well the previously described analysis
results for subcritical operation.
0 1 2 3 4 5 6 7 8 9
x 10
3
0.8
1
1.2
1.4
1.6
1.8
2
Sp[kgm]
Vi
b
.Am
p
 [
m
m
]
Analysis
Experiment
Figure 3.11 Analytical and experimental results for the amplitude of vibration
(subcritical operation)
47
0 1 2 3 4 5 6 7 8 9
x 10
3
260
270
280
290
300
310
320
330
340
350
360
Sp[kgm]
?
3
,
?
4
[d
e
g
]
Analysis: Pend. A & B
Experiment: Pend. A
Experiment: Pend. B
Figure 3.12 Analytical and experimental results for the position of pendulums
(subcritical operation)
Supercritical operation
In contrast to the situation just described for subcritical operation, it is important to
distinguish the three relative balancing areas for supercritical operation. As summarized
in Table 3.4, for supercritical operation each relative balancing area has a different type
of stable singular point. Figure 3.13 and Figure 3.14 show the three relative balancing
areas and the transition regions using five small relative balancing area plots. For both the
analytical results and for the experimental results described, the pendulum parametric
combination is held constant and the mass imbalance was varied. This is why the point of
pendulum combination shown is at the same location on all of the relative balancing plots,
while the ABCD open area changes.
The first configuration that will be considered is for low mass imbalance, which
results in the pendulums being improperly oversized relative to the mass imbalance. This
48
situation is represented by the leftmost section of Figure 3.13 and Figure 3.14. The
pendulums do not have the same first mass moment, and cannot properly counterbalance
each other. This results in a stable Type II singular point, with the rotor engaging in a
steadystate whirling motion. As the mass imbalance is increased, the rotor vibration
amplitude decreases until the configuration transitions to a properly oversized condition
in which the steadystate rotor vibration ceases.
The middle section of Figure 3.13 and Figure 3.14 shows the results for pendulum
combinations that are properly oversized. The relative balancing area plot shows the
point of pendulum combination on the S
A
S
B
plane is inside the ABCD rectangular area.
For this configuration, the Type I singular point is stable.
As the mass imbalance is increased further, the pendulums do not have sufficient
balancing capability to properly compensate and the pendulum combination is undersized
relative to the mass imbalance. The pendulums come together opposite to the mass
imbalance, resulting in a stable Type III singular point. The pendulum system is able to
counterbalance only part of the mass imbalance. In this case, if the mass imbalance is
increased further, the rotor vibration amplitude increases also.
49
Figure 3.13 Analytical and experimental results for the amplitude of vibration
(supercritical operation)
Figure 3.14 Analytical and experimental results for the position of pendulums
(supercritical operation)
0 1 2 3 4 5 6 7 8 9
x 10
3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Sp[kgm]
V
i
b
.A
m
p
. [
m
m
]
Analysis
Experiment
0 1 2 3 4 5 6 7 8 9
x 10
3
100
50
0
50
100
150
200
Sp[kgm]
?
3
,
?
4
[d
e
g
]
Analysis: Pend. A
Analysis: Pend. B
Experiment: Pend. A
Experiment: Pend. B
50
3.2.3 Numerical and Experimental validation
In this chapter the result of numerical simulation and experimental measurements
are discussed and compared. After several repeated measurement the system parameters
were identified. These parameters were used as input parameters of simulations.
Both in the numerical simulation and the experimental measurement the after the
startup a proper amount of time was spent to let the transients die out. During this time
the pendulums were locked at their base position 0? and 180?. From a balancing point of
view, this initial position of pendulum would be neutral if the pendulums had the same
first order moment of inertial. This part of research investigated non identical pendulums
with different first order moment of inertia. In the numerical simulation and in the
experiment initially the pendulum, with higher first moment of inertia was locked at 0?
location. This initial configuration added extra imbalance to the system and resulted
higher amplitude of vibration. The results of numerical simulation and the experimental
measurement the logged coordinates ?
1..4
showed in time from prior pendulum release 3
seconds.
The comparison of numerical simulation and experiment had resulted in quite similar
system response for pendulum release proving the success of the model development and
system parameter identification.
51
0 1 2 3 4 5 6 7 8 9 10
1.5
1
0.5
0
0.5
1
1.5
Time[s]
?
1
[mm
]
Experiment
0 1 2 3 4 5 6 7 8 9 10
1.5
1
0.5
0
0.5
1
1.5
Time[s]
?
2
[m
m
]
0 1 2 3 4 5 6 7 8 9 10
0
60
120
180
240
Time[s]
?
3
,
?
4
[d
e
g
]
Pendulum A
Pendulum B
0 1 2 3 4 5 6 7 8 9 10
1.5
1
0.5
0
0.5
1
1.5
Time[s]
?
1
[mm
]
Numerical Simulation
0 1 2 3 4 5 6 7 8 9 10
1.5
1
0.5
0
0.5
1
1.5
Time[s]
?
2
[m
m
]
0 1 2 3 4 5 6 7 8 9 10
0
60
120
180
240
Time[s]
?
3
,
?
4
[d
e
g
]
Pendulum A
Pendulum B
Figure 3.15 Results of numerical simulation and experimental measurement of twonon
identical pendulum
52
Special case: Identical pendulums
Now, consider the special case in which the pendulums are identical. For such a
configuration, the pendulums can counterbalance each other even if there is no mass
imbalance in the rotor system. So, the improperly oversized condition cannot exist and
there are only two relative balancing areas ? (1) properly oversized, and (2) undersized
relative to the mass imbalance. Figure 3.16 and Figure 3.17 show these two domains
separated by a vertical dash line. On the leftmost side, the pendulums are properly
oversized. For this configuration, the Type I singular point is stable and the center of the
rotor is not vibrating. On the rightmost side, the rotor mass imbalance is beyond the
balancing capability of the two pendulums and the system is undersized. The Type III
singular point is stable and the center of the rotor moves on a steadystate circular path.
The pendulums overlap each other in an angular location opposite to that of the mass
imbalance and partially counterbalance the system. For the case of identical pendulums,
the ?wishbone? shaped section of the curve in Figure 3.17 is symmetric, as compared to
the nonsymmetric appearance of the similar region for the nonidentical pendulum
system, as shown in Figure 3.14.
An interesting phenomenon can be observed in the responses illustrated by Figure
3.16 and Figure 3.17. The first pendulum, indicated by an ?x?, generally settles in the
location described by the upper curve and the pendulum, indicated by an ?o?, generally
settles in the location represented by the lower curve of Figure 3.17 as the mass
imbalance is increased. However, the two pendulums occasionally switch positions, with
the ?x? pendulum associated with the lower curve and the ?o? pendulum associated with
the upper curve. Inspection of Figure 3.16 shows this switching behavior results in an
53
abrupt change in the magnitude of the rotor vibration. This is in spite of the fact that the
pendulums are theoretically identical and should be interchangeable. However, in
practice the pendulums are not identical and have some small differences due to the
fabrication process. These differences produce the observed sensitivity to the switching
behavior described above, which is itself a result of initial conditions for the experimental
system, which are random for every startup of the pendulums. From a balancing point of
view, these switched positions are not equivalent because the residual mass imbalance is
different for the two cases. However, as the mass imbalance of the system is increased to
a sufficiently high value (around S
P
= 0.005[kgm] for the experimental system) this
sensitivity to initial conditions disappears and the settled positions of the pendulums are
consistent and appear independent of the initial conditions.
Figure 3.16 Analytical and experimental results for the amplitude of vibration
(supercritical operation)
0 0.002 0.004 0.006 0.008 0.01 0.012
0
0.1
0.2
0.3
0.4
0.5
Sp[kgm]

V
i
b
.A
mp
 [
m
m]
Analysis
Experiment
54
Figure 3.17 Analytical and experimental results for the position of pendulums
(supercritical operation)
Numerical and Experimental validation
In this chapter, the result of numerical simulation and experimental measurements
are discussed and compared in similar fashion as it was showed for nonidentical
pendulums. Because of the application of identical pendulums, the initially locked
pendulums had no effect on the level of mass imbalance. They simple counterbalanced
each other. The results of numerical simulation and experimental measurement are
summarized in Figure 3.18.
0 0.002 0.004 0.006 0.008 0.01 0.012
100
150
200
250
300
Sp[kgm]
?
3
,
?
4
[d
e
g
]
Analysis: Pend. A
Analysis: Pend. B
Experiment: Pend. A
Experiment: Pend. B
55
0 1 2 3 4 5 6 7 8 9 10
1
0.5
0
0.5
1
Time[s]
?
1
[m
m
]
Experiment
0 1 2 3 4 5 6 7 8 9 10
1
0.5
0
0.5
1
Time[s]
?
2
[mm
]
0 1 2 3 4 5 6 7 8 9 10
0
60
120
180
240
Time[s]
?
3
,
?
4
[d
e
g
]
Pendulum A
Pendulum B
0 1 2 3 4 5 6 7 8 9 10
1
0.5
0
0.5
1
Time[s]
?
1
[m
m
]
Numerical Simulation
0 1 2 3 4 5 6 7 8 9 10
1
0.5
0
0.5
1
Time[s]
?
2
[mm
]
0 1 2 3 4 5 6 7 8 9 10
0
60
120
180
240
Time[s]
?
3
,
?
4
[d
e
g
]
Pendulum A
Pendulum B
Figure 3.18 Results of numerical simulation and experimental measurement of two
identical pendulum
56
It is important to notice the learned property of identical pendulums is also valid for
pendulum combinations where the pendulums are not physically identical but they have a
same first order moment of inertia. The value of this discovery is important from
engineering point of view.
Two pendulums can be manufactured to be quite identical, but the CG of these two
pendulums never will move on the same plane as it is shown by Figure 3.19.a. These
pendulums will produce a dynamic imbalance on the rotor as they are rotating around the
shaft. Figure 3.19.b. shows another (better) possible design solution based upon the above
results. In this case, each of the pendulums has a different length but they have the same
first order moment of inertia around the axis of the shaft. In this design, the CG of each
individual pendulum moves in the same plane and will not dynamically imbalance the
system. This also provides a better force distribution on the bearing system. The
experimental setup had similar design to the engineering design showed by Figure 3.19.a.
57
a.
b.
Figure 3.19 Possible design solutions for pendulum balancing systems
58
3.3 Influence of Pendulum Shaft Misalignment
It was demonstrated analytically above that a pendulum selfbalancing system is
ideally capable of exact radial balancing. However, imperfections in the fabrication and
assembly of such a system will compromise some of the modeling assumptions that
provided this result. One major imperfection is shaft misalignment, which can easily
occur due to improper design, fabrication and assembly. In the following sections, the
effect of misalignment between the center of rotation of the pendulums and rotor shaft is
examined in detail.
Figure 3.20 illustrates the basic configuration for a system with noncentered
pendulums. The position of the center of the shaft is described in a diskfixed rotating
coordinate system. The shaft of the pendulums is shifted from the center of the rotor by
an amount, R
C
, in the ?
1
direction.
Mp
0J
?4
?1
?2
?(t)
?3
J'
B
A
?(t)
I0
I'
R
c
P
3
J'
I'
e
Figure 3.20 Mathematical model of rotor system with noncentered pendulums
59
Using this model, a series of simulation studies were performed for a variety of R
C
.
In order to better generalize the results, the nondimensional parameter, ?
C
and ? are used.
3
,:
CP
C
D P
RPM
Where e
eM
? ==
+
(334)


PR
PL
A
A
? =
(335)
R
C
is the offset of the pendulum center of rotation and e is the imbalance eccentricity
of imbalanced rotor system.
Where A
PR
is the amplitude of the steady state vibration with the pendulums free to
rotate and A
PL
is the amplitude of vibration with the pendulums locked at the 0? and 180?
positions, respectively.
Figure 3.21 and Figure 3.22 summarize the results of these simulations. The angular
velocity of the rotor system is higher than the first critical speed of the system, which is a
basic requirement for this type of passive balancing system to work properly. In the
numerical simulations, the operational speed was set at almost 20 times higher than the
first critical speed and the damping ratio for the rotor suspension was small, which
resulted in the frequency transfer function having a magnitude close to one.
60
0 0.5 1 1.5 2
0
0.5
1
1.5
2
?
c
?
Numerical simulation
Figure 3.21 Simulation results showing the nondimensionalized rotor vibration level for
the system with pendulum shaft misalignment
The upper part of Figure 3.22 shows the angular positions of the pendulums (?
3
and
?
4
) as a function of ?
C
. The lower part of this figure shows the closest relative angular
distance between the two pendulums. Examination of this figure shows that when ?
C
is
zero, the pendulums are close to the 180? position and almost overlapping one other,
which indicates that the radial imbalance of the system is only slightly less than the
balancing limit of the pendulums. At ?
C
= 1.0, the two pendulums are the farthest apart at
180
o
. For higher values of ?
C
, they move closer together and for ?
C
= 2.0, the pendulums
converge to overlapping angular positions and stay in that configuration for further
increases of ?
C
. At this ?
C
value, there is a breakpoint in the vibration level of the center
61
of the rotor (as shown in Figure 3.21). The rate of increase for the vibration amplitude
changes abruptly and almost settles into a plateau.
0 0.5 1 1.5 2
100
200
300
400
?
c
?
3
[d
e
g
],
?
4
[d
e
g
]
Numerical simulation
?
3
?
4
0 0.5 1 1.5 2
0
50
100
150
200
?
c
?
4

?
3
[d
e
g
]
Figure 3.22 Simulation results showing the absolute and relative positions of the
pendulums for the system with pendulum shaft misalignment
62
3.3.1 Experimental investigation of pendulum shaft misalignment
Pendulum Balancer Experimental Facility
To gain detailed insight into the dynamic characteristics and performance of a
pendulum balancing system with misalignment shaft the same experimental facility was
used that was described in chapter 3.2.1. The design of this test rig is such that the
pendulum rotation center could be offset in the ?
1
direction from the center of the rotor
by an amount R
C
.
Implementation of shaft misalignment
The pendulum assembly (4) is fixed to the base plate (5) by two bolts from
underneath. When these bolts are loosened, the pendulum assembly can be shifted in the
?
1
direction by an amount, R
C
, while still maintaining the shaft of the pendulums parallel
to the shaft to the DC motor (11). However, shifting the pendulum assembly and the
structural parts surrounding the pendulums, itself also produces an additional mass
imbalance, which is undesirable for the purposes of the present study. This imbalance can
be offset by adjusting the brass weights (7) in the radial direction. The gross imbalance of
the system center of mass is set by adjusting the brass weight (8), M
P
, and its radial
position, P
3
.
63
Description of experimental procedure
Using the pendulum balancing facility described previously, a series of experiments
was conducted to validate the simulation results obtained in the previous section and to
gain further insight into the influence of specific imperfections and nonidealities on the
dynamic performance of such passive balancing system. The steps in the testing process
are described below:
1. Pendulums are locked at 0
o
and 180? respectively with plastic pins
2. The pendulum assembly is shifted by an amount R
C
relative to the base plate
3. The rotor was spunup to the test operating speed and balanced.
4. A specific gross mass imbalance was set and the pendulums were released
5. The rotor was spunup to the test operating speed and three sets of data were
measured and recorded:
 The magnitude of vibration [mm], measured by a laser displacement system
 The final angular position of each pendulum ?
3
and ?
4
[deg]
Figure 3.23 and Figure 3.24 summarize the results for a typical set of experiments,
which are quite similar to the predicted behavior from the simulation results (Figure 3.21
and Figure 3.22). Inspection of these figures shows that increasing the shift of the
pendulums, ?
C
, produces a proportional increase in rotor vibration level until the
pendulums are overlapping one other. At ?
C
= 2.0, there is again a breakpoint, after
which further shifting of the pendulum axis has only a slight effect on the rotor vibration,
in a fashion similar to that observed in the numerical simulations.
64
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
?
c
?
Measurement
Figure 3.23 Experimental results showing the nondimensionalized rotor vibration level
for the system with pendulum shaft misalignment
65
0 0.5 1 1.5 2 2.5
100
200
300
400
?
c
?
3
[d
e
g
],
?
4
[d
e
g
]
Measurement
?
3
?
4
0 0.5 1 1.5 2 2.5
0
50
100
150
200
?
c
?
4

?
3
[d
e
g
]
Figure 3.24 Experimental results showing the absolute and relative positions of the
pendulums for the system with pendulum shaft misalignment
Top part of Figure 3.24 shows the absolute position of the pendulums and the
bottom part of this figure shows the relative position to each other. Again, obvious
similarity can be found between these experimental results and the numerical simulation
results shown in Figure 3.24. In particular, the lower graphs (showing the relative
position of pendulums) are almost identical. The upper graphs are also similar. However,
for values of ?
C
near 1.0, the experimental and simulation results differed for the absolute
position of the pendulums. This area is marked by dashed lines on the upper part of
Figure 3.24. Since the relative angular velocity (?/?
n
) is nearly 20 and the damping ratio
66
is quite low, the transfer function is close to 1. Thus, the center of rotation is very close to
the CG of the imbalanced rotor system, which is also the location of the pendulum when
?
C
is near 1.0. This means the center of pendulums is near the source of the centrifugal
force field. So at this location, the relative position of the pendulums will be 180? but the
absolute position of the pendulum configuration is indeterminate from the perspective of
balancing the system. When the suspension point of the pendulums is moved slightly
from the CG of the unbalanced rotor, the relative distance of the pendulums slightly
decreases and this neutral state will change. Thus, small effects that are not included in
the simulation model but that are present in the experimental facility, such as rolling
resistance and friction, will tend to produce a different absolute position of the pendulums
from that predicted by the simulations. This explains the differences between the
numerical simulations and the experimental measurements in the region where ?
C
is near
1.0.
67
3.3.2 Experimental comparison of the sensitivity and consistency of ball and
pendulum balancers
This investigation begins by considering the effects of rolling resistance. Previous
investigators have noted that rollingresistance and dry friction are significant problems
that tend to degrade the consistent performance of ball balancer systems [12], [13].
However, previous work has not considered the influence of such effects on pendulum
balancers. A basic question is: Are pendulum balancers less susceptible to such effects
(rolling resistance and dry friction) than ball balancer systems? Accordingly, in the
following sections, a comparison is made between the performance of these two types of
systems with the goal of providing some insight into this question. Two comparable test
rigs are developed, one with a ballbalancer system and the other with a pendulum
balancer system, and experimental results from each are compared. The observed results
are discussed and some insights into the expected relative performance of such systems
are presented.
Ball Balancer Experimental Facility
Photographs of the ball balancer test facility are shown in Figure 3.25 (side view)
and Figure 3.26 (top view). The same central suspension was used as for the pendulum
balancer described earlier. This suspension allows the center of the disk to move in a
nearly horizontal plane without significant friction.
Distinct from the pendulum balancer system, the moving elements are two steel
balls (2, 3) guided by a cylindrical channel (4) (also a secondary channel (5) can be seen
for later investigations) machined in an aluminum disk (1). The channels are covered by a
68
plexiglass cover (7) for safety reasons and to support the dial plate. The rotating disk and
the plexiglass cover together are well balanced. The imbalance is generated by
component (6) which is a brass block whose position in the radial direction can be
adjusted.
Ball Balancer Experimental Results
Using this facility, a series of experiments was performed to evaluate the
performance and consistency of this system. The experimental test procedure consisted of
the following steps.
First, the balance state of the rotor was set using the following procedure. The steel
balls were locked at ?30? (as shown in Figure 3.26) and the system was balanced by
adjusting the radial position of the brass block (6). Accordingly, the magnitude of the
imbalance and the desired settling position of the balancing balls (when released) are
precisely known.
Next, the balancing balls were released and repeated startups were performed
without changing any of the other physical properties of the system. For each startup, the
system was started from rest and was driven until the rotor disk had reached 1500 rpm.
The amplitude of the steadystate vibration and the position of the balancing balls were
then recorded.
69
Figure 3.25 Side view of ball balancer experimental facility
70
Figure 3.26 Top view of ball balancer experimental facility
Figure 3.27 serves to graphically summarize the results of these experiments. The
horizontal axis shows the positions of the balancing balls. Each pair is shown by a
different symbol and is connected by a dashed horizontal line. The vertical axis shows the
measured amplitude of the center of disk in the ?
1
direction. The smallest measured
vibration level was recorded when the balancing balls were locked at ?30?. The highest
vibration level was recorded when the rotating disk was not equipped with the balancing
balls. This level of vibration (0.80475 [mm]) would be the amplitude of the steady state
motion of the rotating system without any passive balancing mechanism. For all of the
other test cases, the balancing balls tended to reduce the overall vibration but they settled
71
to different locations each time with a relatively large scattering. The above experiment
was repeated with a variety of different sizes and numbers of balls, with similar results.
60 40 20 0 20 40 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 Vibration amplitude without balancing balls: 0.80475[mm]
30 30
Position of balancing balls [deg]
R
e
la
tiv
e
a
m
p
litu
d
e
o
f
v
i
b
r
a
tio
n
[
]
Figure 3.27 The final positions of balancing balls and vibration level for different
startups
These results clearly indicated that there are some physical effects that prevent the
balancing balls from settling into the proper positions and achieving more consistent
results. This effect can be traced to the deformation of the contact point of the ball and
channel surface due to the high normal forces generated by the centripetal acceleration
and the resulting dry friction. For the specific experimental setup used for the present
tests, the mass of each 15[mm] diameter chrome steel ball was 13.8[gram] moving in
114[mm] radius and the material of the channel was alloy 6061 aluminum. This material
72
and geometrical combination resulted in an approximately 16 [?m] deformations at the
ball contact points when the disk was rotating at 1500 rpm.
Figure 3.28 illustrates a graphical explanation of rolling resistance. The deformation
of both channel and ball forms a pocket at contact point A and the idealized pointto
point contact becomes a surface as a result of high centrifugal force F
CF
. This elastically
formed pocket changes the direction of constraining force, F
CS
, that was directed toward
the center of rotation in the idealized model. So, the constraining force has a component
that tends to oppose the balancing force, F
B
. In the ball balancing system that was
investigated, the centrifugal force, F
C
, is much greater than the gravitational force, mg.
Because of this high force ratio, the ball contact motion will primarily be rolling at point
A and slipping at point B. Relative motion at the contacting surfaces at point B will result
in a friction force, F
F
, which also tends to oppose the balancing force, F
B.
The action of
these combined forces tends to stop the motion of the balancing balls before they reach
the complete balancing position, resulting in a nonzero balance state for this system. This
remaining mass imbalance produces synchronous vibration, which is thus the indirect
result of rolling resistance.
73
A
B
mg
FC
FC
FCS
FB
FF
C
A
C
FCB
FCS
S
L
O
P
E
Figure 3.28 Deformation of contact surfaces and force distribution of the balancing ball
and channel
74
Pendulum Balancer Experimental Results
A similar series of experiments were performed for the pendulum balancing system.
The rotor was repeatedly started without changing any parameters of the system. The
rotor vibration level and the position of the pendulums were captured when the system
reached the final speed and exhibited steadystate behavior. The results of these
experiments are summarized in Figure 3.29 and Figure 3.30, in a fashion similar to that
which was used for the foregoing ball balancer experiments.
In Figure 3.29, the amplitude of the steadystate periodic motion is also shown to
illustrate the relationship of the different counterbalancing levels. When the pendulums
are locked at the 0
o
and 180? positions respectively, the overall vibration level of the
system is 1.75[mm]. Figure 3.30 shows a closer (zoomed) view of the settled pendulum
positions for better insight. A comparison of the results for the ball balancer system and
the pendulum balancer system shows that the pendulum balancing system has a much
greater consistency.
75
60 40 20 0 20 40 60
0
0.2
0.4
0.6
0.8
1 Vibration amplitude without balancing pendulums: 1.75[mm]
30 30
Position of balancing pendulums [deg]
R
e
l
a
t
i
v
e
a
m
pl
i
t
ude
of
v
i
br
a
t
i
on [

]
Figure 3.29 The final positions of balancing pendulums and the level of vibration for
different startups
76
32 31 30 29 28
0.0105
0.011
0.0115
0.012
0.0125
Position of balancing pendulums [deg]
R
e
la
tiv
e
a
m
p
litu
d
e
o
f
v
i
b
r
a
tio
n
[

]
27 28 29 30 31
0.0105
0.011
0.0115
0.012
0.0125
Figure 3.30 The final positions of balancing pendulums and the level of vibration on the
zoomed plot for different startups
Figure 3.31 illustrates the force distribution of a pendulum balancer system and of a
ball balancer system. F
RR
is the rolling resistance generated by the deformation of the
contact surface, friction and other less significant forces. F
CF
is the centrifugal force. F
C
is
the constraint force which guides the moving masses, pendulums or balls, in a circular
path. F
B
is the balancing force. Inspection of these diagrams show that the pendulum
balancing system has a better overall force distribution in the sense that (in general) a
larger percentage of the forces are directed toward achieving relocation of the balancing
mass. This serves to dramatically improve the overall sensitivity of the pendulum
balancer system with respect to a similar ball balancer system. Thus, pendulum balancer
77
systems tend to be inherently less susceptible to inconsistencies and errors as a result of
rolling resistance.
?1
MP
?
FCF
FB
FC
FRR
FCF
?1
FB
FRR
MP
FC
?
b.a.
Figure 3.31 Force distribution of pendulum and ball balancer
78
4 PASSIVE PENDULUM BALANCER WITH NONISOTROPIC SUSPENSION
4.1 Analytical investigation
The previous studies have targeted isotopic suspension of rotor system where the
spring stiffness and the damping were identical. Identical suspension results two identical
natural frequencies. It was demonstrated that the Type I singular point of an isotropic
pendulum system showed stability when the system was operating supercritically.
This part of the research has investigated a rotor system equipped with a pendulum
balancer and supported from ?
1
and ?
2
directions by nonidentical springs and dampers.
The application of two different supports results in two distinct natural frequencies. The
analytical investigation starts with the development of the equations of motion.
1, 2, 3, 4
i
i iii
dT T V D
Qi
dt
??????
?++= =
??
?? ?? ?? ??
??
nullnull
(41)
Where: T = Sum of all kinetic energies
V = Sum of all potential energies
D = Rayleigh dissipation function
i
? = Generalized coordinates
i
Q = Generalized forces
79
() ()()
() ()
()( )()
()(
()
()
2
1323
1424
22 2
31 12
21 2 3 3 21
1 4 4 3 21 21
2313 1
1
2cos 2sin
2
2cos 2sin
1
2
2
sin
cos
cos sin
ABA
BBB
P D BA BB P D D
BA A P
B A BA BB D
PA B
TSR
SR
SP M M M M MR
MS M
SSR M
SS S
?
?
?
=+??+??+
?
?
++??+??+
?
++?++++?+?+ +
?
?
+??+???+?+
+? ??+ ?+ ??+ ??+
+?+?? ???
nullnullnull
nullnullnullnull
nullnull null
()
() ()
()
() )
()()
() ()
4
244 122 3
2412 4 12
1312
22
21
22
34233244
14
sin cos
cos
sin
11
11
cos cos
22
si
BBAA
BDBB
AP
DBABBP DBABBP
ABA BBB A B
B
SM
SRM
SM
MM M M MM M M
SR SR S S
S
?+
+? ??? ??+? ?+
+? ??? ?+ ?? ???
?? ??? ?+
++++?++++?+
+ ?+ ?+?? ?+?? ?+
???
nullnull null
nullnull nullnull
nullnull
nullnull
null null nullnull nullnull
nullnull
() ()
4133
nsin
A
S???? ?
nullnull
(42)
() () () ()
22
11 221 2
cos sin sin cos
22
Vk t t k t t?? ??=???+?+? (43)
() ()
( )
() ()()()
(
() ()()()
)
() ()()
() ()()
22
2
11 2
11 2 2 1
2221
2
2
12213
2
2
22214
1
cos sin
2
cos sin cos
sin cos sin
cos sin
11
sin cos
Dctct
ct t t
ct t t
ct t c
ct t c
??
?? ? ? ?
?? ? ? ?
?? ? ?
?? ? ?
=? + +
+? ? ? ? ? +? +
+ ? ?+ ?+? +
+ ? ?? ?+? + ? +
+ ? ?+ ?+? + ?
null
nullnull
null
nullnull
nullnull
(44)
0
i
Q =
(45)
80
The most general form of the equations of motion are:
() ( ) ( ) ( )
() () () ()
() () ()()
() () ()()
() ()
22
33 33 3 33
44 44 44 4
22
11 2 3 1
122
221
11
2cos sin cos cos
2cos cos sin cos
2
cos cos sin
sin sin cos
cos cos
A
B
P D BA BB
S
S
MPMM
ct t t
ct t t
kt t
??
???
?????
?????
??
???+?+?+??
???+?+?+??
????+?++++?+
+????+?+
+ ? ?+ ?+? +
+?
nullnullnull null
nullnullnull
nullnull null nullnull
null
null
()
() () ()
()
2
21
2
21
sin
sin sin cos
20
DBAB
t
kt t t
MM M
?
?? ?
??
??+
+??
?++ ?+?=
null
(46)
() ( ) ( ) ( )
() () () ()
() () ()()
() () ()()
() ()
22
33 33 3 33
44 44 44 4
2
22 1 2
121
222
11
2 sin cos sin sin
2 sin sin cos sin
2
sin sin cos
cos cos cos
sin sin cos
A
B
P D BA BB
S
S
MMM
ct t t
ct t t
kt t
??
??
?????
?????
?? ?
???+?+?+??
? ? ?+ ? ?? ? ?+ ? ?
????+?+++?+
+????+?+
+ ? ?+ ?+? +
+??
nullnullnull null
nullnullnull
nullnull null nullnull
null
null
()
() () ()
()
2
21
2
12
cos cos cos
20
DBAB
t
kt t t
MM M
?? ?
??
?+
+?+
+++ ???=
null
(47)
( ) ( )(
() ()
() ())
31 3 3 2
32 31
22
32 31 33
sin cos
2sin 2cos
cos sin 0
ABA
SR
c
??
???+?+??+
??+ ???
???+??+?=
nullnull nullnull nullnull
nullnull
null
(48)
( ) ( )(
() ()
() ())
41 4 4 2
42 41
22
42 41 44
sin cos
2sin 2cos
cos sin 0
BBB
SR
c
??
???+?+??+
+??+???
???+??+?=
nullnull nullnull nullnull
nullnull
null
(49)
81
In order to verify that the Type I singular point exists for the nonisotropic system
the, time derivative terms are set to zero in Equations (46)(49). This results in the
following set of algebraic equations.
( ) ( )( )
() () ()
() () ()
() () ()
() () ()
2
134
21 2
12 1
11 2
21 2
cos cos
cos sin sin
cos sin cos
cos sin cos
sin cos sin 0
ES A E B E
EE
EE
EE
YMSYSY
ctY tY t
ctY tY t
ktY tY t
ktY tY t
?
?? ? ?
?? ? ?
???
?+ + +
+? ?
?+ +
+?
++ =
(410)
( ) ( )( )
() () ()
() () ()
() () ()
() () ()
2
134
11 2
21 2
12 1
22 2
sin sin
sin cos sin
cos sin cos
cos sin cos
sin cos sin 0
ES A E B E
EE
EE
EE
YMSYSY
ctY tY t
ctY tY t
ktY tY t
ktY tY t
?
?? ? ?
?? ? ?
???
?+ + +
++
+
+?
++ =
(411)
( ) ( )( )
2
132 3
sin cos 0
AE E E E
SY Y Y Y ?? =
(412)
( ) ( )( )
2
142 4
sin cos 0
BE E E E
SY Y Y Y ?? =
(413)
It is easy to prove that the Type I singular point, found earlier for the isotropic
system, is also a singular point for the nonisotropic system, especially when the
pendulums have the same first moment of inertia S
A
= S
B
. Further investigation showed
Type II and Type III singular points also satisfy Equations (410)  (413) for an arbitrary
chosen nonidentical pendulum combination.
82
In order to assess the stability characteristics of the Type I singular point, the
differential equations of motions were linearized around this point. The resulting
equations, in a matrix vector form, are:
[] () ()
M0YDtGYKtNY++++=????
????
nullnullnull
nullnull null
(414)
The inertia matrix is:
[]
( ) ( )
() ()
() ()
() ()
34
33
34
0sinsin
0cosco
sin cos 0
sin cos 0
SAEBE
SAEBE
AEA E ABA
BEB E BB
MSYSY
MSYSY
M
SYS Y SR
SYS Y SR
??
??
=
?
?
??
(415)
:
S D BA BB P
Where M M M M M= +++
The time periodic damping and the gyroscopic matrix is:
( )
() ()
() ()
() ()
() ()
22134
21 1 3 4
33
44 4
2()2cos 2cos
2 ( ) 2 sin 2 sin
2cos 2sin 0
2cos 2sin 0
SAEBE
SA
AE AE
BE BE
Dt G
Mcc SY SY
Mcc SY SY
SY SY c
SY SY c
?????
?? ?
??
+=??
??
?+?? ?
??
+? ? ?
?
?
(416)
( ) ( )
() ()
() ()
22
11 2
22
21 2
:sincos
cos sin
sin cos
Where t t
ctct
ct t
???
?? ?
? ??
=
=+
=+
83
The time periodic elastic and the nonconservative force matrix is:
( )
() () () ()
() () () ()
() ()
() ()
2 22
121 112 3 4
222
212 212 3 4
22
34
44
sin sin
cos cos
sin cos 0
sin cos 0
AE BE
AE BE
AE A E A
BE B E B
Kt N
Mcc kk SYSY
kk M cc S Y S Y
SY S Y SKN
SY S Y SKN
?? ? ?? ? ? ?
?? ? ? ? ? ? ?
??
+=??
??
? ??++? ???
? ?
?? ? ++? ? ?
? ?
? ?
?
? ?
?
? ?
? ?
(417)
( ) ( )
() ()
() ()
() ()
() ()
() ()
() ()
22
11 2
22
21 2
22
11 2
22
21 2
33 2 3 1 3
44 2 4 1 4
:sincos
cos sin
sin cos
cos sin
sin cos
sin cos
sin cos
EEE E
EEE E
Where t t
ktkt
ktk t
ctct
ct t
KN Y Y Y Y
KN Y Y Y Y
???
?? ?
?? ?
?? ?
?? ?
=
=+
=+
=+
=+
=+
84
The Cauchy transformation of Equation (414) results in a system of first order
differential equations with an 8x8 coefficient matrix ( )At? ?
? ?
.
()
=tX AX? ?
? ?
null null
null
(418)
Where:
=
Y
X
Y
? ?
? ?
? ?
? ?
null
null
null
null
(419)
The coefficient matrix of the linearized differential equations with time periodic
coefficients is:
()
[] []
[] () [] ()
11
0
[]
I
At
M Kt N M Dt G
??
??
??
=
?+?+? ???
? ???
??
(420)
4.1.1 Stability of homogeneous linear system with time periodic coefficients
This section describes the mathematical background that was applied for the
stability analysis. The general time periodic linear system is defined by equation (418)
where the coefficient matrix
[ ]
()A t is periodic with period 0>T , that is
[ ] [ ]
() ()At T At+= . Of interest is the stability of the equilibrium point 0X ?
nullnull
. To solve
the problem, we apply the Floquet theorem [28[29].
85
The fundamental matrix of system of the linearized (418) is
[ ]
()t? , if the
[ ]
[][]
()
() ()
dt
At t
dt
?
=? (421)
matrix differential equation is satisfied. The following statements can be proved:
 All solutions of equation (418) can be written in the form ( )tc?? ?
? ?
null
, where c
null
is a
constant vector.
 There exists a fundamental matrix ( )
0
t?? ?
? ?
, that all solutions of (418) come up in the
form ()
00
tX???
??
null
, where ( )
0
0XX=
nullnull
is the initial condition, that is ()
[ ]
0
tI?=??
??
, where
[ ]
I is the identity matrix.
 All fundamental matrix can be written in the form ( )tC
? ?
???
??
? ?
null
, where C
??
??
null
is a constant
matrix.
 For any fundamental matrix ( )t?? ?
? ?
, ( )tT?+? ?
? ?
is also a fundamental matrix.
 There exists constant matrix C
? ?
? ?
null
for which ( ) ( )tT t C
? ?
?+ =?????
????
? ?
null
, where C
??
??
null
is called
the principal matrix of (418), () ( )
1
CttT
?
??=? ?+
? ?? ?
? ?? ?
??
null
.
 The principal matrix belonging to the fundamental matrix
()
0
t???
??
assumes the
form
[] () ( ) () ( ) ()
11
00 0 0 0
00Ct tT T
??
=? ? + =? ? + =???? ???? ???
??? ???? ???
.
 All principal matrices are similar to each other, consequently the eigenvalues of the
principal matrix  called the characteristic multipliers (notation:
n
??? ?,,
21
)  are
invariant, and determined by the system.
86
 System (418) is asymptotically stable if and only if 1<
i
? , ni ?,2,1= .
 System (418) is stable in the Liapunov sense if and only if 1?
i
? , ni ?,2,1= , and
if 1=
i
? , than
i
? is simple in the minimal polynomial of the system.
In general the principal matrix can not be determined in an analytic way, but there are
several methods to approximate it [16].
4.1.2 Floquet analysis: piecewise approximation
If the coefficient matrix
[ ]
()At is piecewise constant, then  by the coupling of
solutions  the complete solution at time Tt = is obtained in the form:
[ ]() [ ]( ) [ ]( )
11 110
() exp exp exp
nn n n
X TtAtA tAX
??
=
nullnull
?
(422)
()
[ ]
[]
[]
11
2112
12 1 12
if 0
if
if
nnn
Att
At
A tttttttT
?
? ??
?