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The 10th International Symposium on Transport Phenomena and
Dynamics of Rotating Machinery
Honolulu, Hawaii, March 07-11, 2004
ISROMAC10-2004-043
EXPERIMENTAL OBSERVATIONS OF CENTRIFUGAL PENDULUM
VIBRATION ABSORBERS
Tyler M. Nester, Peter M. Schmitz, Alan G. Haddow, and Steven W. Shaw
Michigan State University
Department of Mechanical Engineering
Michigan State University
East Lansing, MI 48824
Phone:
Email: shawsw@msu.edu
ABSTRACT
This paper describes results from an experimental
investigation into the dynamic response of rotor systems
fitted with centrifugal pendulum vibration absorbers. Two
types of absorbers are considered which exhibit different
types of nonlinear behavior. Systematic measurements of
the rotor and absorber responses are taken for each type of
absorber and compared against one another, and against
previously obtained theoretical predictions. The dramatic
influence of the absorber nonlinearity is demonstrated, and
the results allow one to draw conclusions about the
selection of absorber parameters. Results for systems with
multiple absorbers are also presented, showing how the
absorbers may behave in a complicated manner, even
when identically tuned.
INTRODUCTION
The use of centrifugal pendulum vibration absorbers
(CPVA’s) is a proven method for reducing torsional
vibrations in rotating systems. They consist of mass
elements suspended from a rotor in such a manner that
their centers of mass move along a prescribed path. When
the rotor is subjected to a fluctuating torque, the movement
of the absorbers acts to counteract the applied torque,
thereby reducing torsional vibration levels. The absorbers
are tuned (using centrifugal effects) to a given order of
rotation, rather than to a set frequency, and are therefore
effective over a continuous range of rotating speeds [Den
Hartog, 1985]. These devices have been in use for many
years, most commonly in light aircraft engines [Ker
Wilson, 1968] and helicopter rotors [Miao and Mouzakis,
1981, Hamouda and Pierce, 1984]. More recent work has
proven them to be feasible for automotive applications,
although they have not appeared yet in a production
vehicle [Borowski et al 1991, Nester et al., 2003].
These devices have some very appealing features for
applications, including: the order tuning mentioned above,
the fact that they dissipate very little energy, and they can
often be designed such that no mass or rotating inertia is
added to the rotor (for example, achieved in automotive
applications by replacing existing counterweights by
CPVA’s, which then serve dual use for balancing and
torsional vibration reduction [Nester et al., 2003]).
The basic operation of CPVA’s has long been
understood. Their tuning is achieved by selection of the
placement and curvature of the path of the absorber center
of mass, which is typically circular. This is most often
achieved using a bifilar suspension, although many types
of physical arrangements are possible [KerWilson, 1968].
The path parameters set the tuning order of the absorber,
which is essentially the linear, small amplitude frequency
of free oscillation when the rotor runs at a constant speed.
They also dictate the nonlinear behavior encounter at
larger amplitudes. The operating envelope of such a
device is dictated by the amount of absorber mass that one
employs. Since added mass penalties are often very stiff,
especially in aerospace applications, it is crucial that
designs minimize the amount of mass used. This leads to
complications, however, since small absorber mass
necessarily leads to large absorber responses, where
nonlinear effects become very important.
The potentially devastating effects due to nonlinear
responses are well known. As torque levels increase, the
system can experience a jump, due to nonlinear effects,
which results in absorber responses that actually amplify
vibration amplitudes. This phenomenon was investigated
by Newland [1964], who observed that absorber swing
angles above 25o could cause a nonlinear jump in the
absorber amplitude, which was accompanied by a 180o
change in the absorber’s phase angle. This change in the
phase causes the absorber to amplify the applied torque,
rather than absorb it. This behavior, which will be shown
both experimentally and numerically, emphatically
demonstrates that nonlinear effects cannot be ignored.
Newland [1964] also describes a systematic means of
dealing with these effects, by intentionally detuning the
absorbers at small amplitudes in such a manner that they
come into proper tuning at moderate amplitudes. This
works, but at the expense of absorber performance, as
shown below. In a major development, Madden [1980]
proposed the use of noncircular (specifically, cycloidal)
paths that allow for small amplitude tuning while
maintaining, at least approximately, the desired tuning
over a large range of amplitudes, thereby avoiding the
jump. Subsequent studies considered the effectiveness of
absorbers constructed with a range of paths, which include
the traditional circles, the cycloids proposed by Madden,
as well as a class of epicycloids with special properties
[Denman, 1992, Shaw et al., 1997, Chao et al., 1997].
In typical applications the total amount of absorber
mass is divided up into a set of nominally identical
absorbers that are arranged to satisfy balancing and space
requirements. In such a case, the absorbers may not
behave in a synchronous manner with equal amplitudes.
In fact, analytical results have predicted that systems of
nearly identical absorbers can undergo two basic types of
instabilities. The first type is the jump described above,
which maintains the absorbers in an equal amplitude
motion. The other type is a symmetry breaking bifurcation
that results in the absorbers oscillating at different
amplitudes and/or phases [Chao et al., 1997]. The former
instability (which occurs for circular path absorbers, but
not cycloids or epicycloids) results in a catastrophic failure
of the absorber system, while the latter instability (which
can occur for all path types, depending on tuning) is more
benign, although it does limit the torque range over which
the absorbers operate effectively [Alsuwaiyan and Shaw,
2002]. Similarly, when the absorbers have small
differences among them, the response can become
localized, wherein a small subset of absorbers (maybe only
one) does most of the work of canceling vibrations and
therefore oscillates at a much larger amplitude than that
predicted by assuming identical absorbers [Alsuwaiyan
and Shaw, 1999, 2003]. The method to avoid this
localization and the instabilities is to intentionally overtune
all the absorbers, which reduces absorber effectiveness
[Alsuwaiyan and Shaw, 2002, 2003].
Although much work has been done in this area, past
experimental work has been limited primarily to specific
implementations. Various types of engines, helicopter
rotors, etc., have been built and tested, but previous
experimental efforts have been limited to verifying that
CPVA’s lowered the vibration levels in the system of
interest. In contrast, the present work describes
systematic, controlled experiments that monitored both the
response of the absorbers and the rotor for a variety of
operating conditions.
In this work we report and compare results from
recent experimental studies that consider two absorber
systems, one with circular path compound pendulums each
suspended from a single point, and the other with
epicycloidal paths suspended with a bifilar arrangement.
The paper is outlined as follows. We begin with a brief
overview of the theoretical background, followed by a
short description of the experimental apparatus used. The
main results are then shown in the form of response
diagrams that depict experimental data compared against
theoretical predictions. The paper is closed with a
discussion of the differences between the two different
absorber systems and some conclusions.
THEORETICAL BACKGROUND
In this section we first outline the basic tuning results
for the absorbers, and the nonlinear nature of the two paths
to be considered is then described. A brief summary of the
analysis is presented, along with some typical theoretical
response curves. These results are used as background and
to compare with the experimental results, which form the
main part of the paper.
An idealized representation of a rotor fitted with four
absorbers is shown in Figure 1. For bifilar suspensions,
such as the one shown in Figure 3 below, the absorbers can
be modeled as point masses riding along paths on the rotor
(in this case their rotational inertias simply add to the rotor
inertia), whereas for single point suspensions the rotational
inertia of the absorbers must be taken into account for the
tuning.
Figure 1: Schematic of a rotor fitted with four general path
CPVA’s.
The absorber paths can be taken to be quite general,
but two features are of importance: the linear tuning and
the nonlinearity of the path. According to small amplitude
vibration theory, the linear tuning order of the CPVA’s
shown in Figure 1 is given by:
~
n=
rR
r + r2
(
2
)
(1)
where R and r are the distances shown in Figure 1 and r is
the radius of gyration associated with the non-zero
moment of inertia of the absorber about its center of mass.
Note that r=0 for the bifilar suspension, since the absorber
2
rotates with the main rotor. By adjusting the radius of the
absorber path and the absorber’s center of rotation, it is
possible to tune the CPVA to absorb any desired order
(within hardware constraints). For a constant rotor speed,
W, the linearized natural frequency of the tuned absorber in
Figure 1 is given by
The nonlinearity in the path can be described
mathematically [Denman, 1992, Alsuwaiyan and Shaw,
2002], but can also be conveniently depicted graphically,
as shown in Figure 2. The circular path is easily realized
in experiments, in the present case by using the “T” shaped
pendulums shown in Figure 3. The epicycloidal path is
achieved by having the absorber mass connected to two
straps that wrap along cheeks shaped so that the absorber
mass follows the desired curve, as shown in Figure 2 and 3
[Schmitz, 2003]. From Figure 2 it is seen that the
differences between circles and epicycloids that are tuned
to the same linear order are nearly identical up to moderate
amplitudes. However, as demonstrated below, this small
difference has a large effect on the qualitative behavior of
the system response at moderate amplitudes.
w 0 = n~W (which reduces to
W R / r , the classical result, for r=0). For a system with
an applied torque of order n, for example, of the form
T0 sin( nWq ) , where q is the rotor angle, a CPVA is
referred to as being overtuned (undertuned) for
n~ > (<)n , respectively. Absorbers are generally
overtuned, in order to avoid localization and nonlinear
jumps [Alsuwaiyan and Shaw, 2002, 2003].
CENTER
Of
ROTATION
R
c
m
i
CHEEK
r
CG
PATH
Figure 2: Left: Schematic showing circular and epicycloidal path geometries tuned to the same linear order. Right:
schematic showing how the epicycloidal path is realized using straps wrapping on cheeks.
Figure 3: Schematic diagrams: Left: rotor with two compound pendulum CPVA’s attached. Right: Close up of the bifilar
absorber configuration with straps and cheeks.
The analysis for determining the system response can
be found in previous work by the authors [Alsuwaiyan and
Shaw, 2002, Nester 2002]. The results are obtained by
perturbation methods that are based on a particular scaling
of the system parameters. This scaling takes advantage of
realistic ranges of physical parameters and renders the
following small quantities: the ratio of absorber inertia to
rotor inertia, the absorber damping, the applied torque, and
the nonlinearities in the response. It should be noted that
the scaling for the case of absorbers that use a single point
support requires some modification [Nester 2002], which
results in equations of motion that are equivalent to those
which use a bifilar support [Schmitz 2003].
The basic model consists of N identical (or nearly
identical) absorbers attached to a rigid rotor that is
subjected to an applied torque of order n. In order to
analyze the equations of motion using perturbation
techniques, the equations are formulated for a general path
3
steady state responses are shown for both the absorber
motion and the rotor response. The measure used to
represent the rotor torsional vibration response is the order
n harmonic amplitude of its angular acceleration, which
represents its fluctuating component.
The first type of response curve, which is more useful
from a parameter selection/design point of view, is for the
case in which the torque amplitude and absorber order are
fixed and the order of the applied torque is varied. This
allows one to consider how the absorber should be tuned
relative to the torque to achieve good performance. Figure
4 shows samples of these curves, indicating the basic
differences in the responses for absorbers with circular and
epicycloidal paths as the order of the applied torque is
varied. (Note that these theoretical results are equivalent
to frequency response curves.) Response curves for three
levels of torque are shown. On the left are the results from
circles, which show how the detuning of the absorber at
moderate amplitudes shifts the minimum point of the rotor
acceleration and bends the absorber resonance curve (due
to the softening nonlinearity of the pendulum). Here the
effects of intentionally overtuning the absorber are clear,
specifically, by designing the absorber so that the response
is to the left of the low torque amplitude minimum, one
can maintain a relatively small vibration level as the torque
is increased. These nonlinear features are notably absent
from the epicycloidal case, shown on the right, even out to
large amplitudes. This is due to the fact that the epicycloid
is precisely the curve that is neither hardening nor
softening at linear order [Denman, 1992, Alsuwaiyan and
Shaw, 2002, Chao et al., 1997].
The other type of response curve considers a fixed
order of the applied torque and a fixed tuning of the
absorber and an increasing amplitude of the fluctuating
torque. These curves show how an absorber responds in
actual implementations, since the tuning is fixed by
hardware, while the applied torques vary depending on
operating conditions. Figure 5 shows theoretical response
curves for the rotor vibration levels (on the left) and the
absorber amplitudes (on the right) for circular path
absorbers, for various levels of absorber tuning. The
primary features of these curves can be described by
considering a situation in which the amplitude of the
fluctuating torque is slowly increased from zero up to
some level and then decreased back to zero. It is seen that
the absorber amplitudes increase up to a point at which a
jump occurs, resulting in the response being on the upper
part of the curve, where the absorber amplifies the
torsional vibration. Further increases simply make this
situation worse. The attendant rotor vibrations, measured
by the amplitude of the rotor’s angular acceleration, also
increases as the torque increases, almost up to the jump
point, although for some levels of mistuning it turns down
slightly just before the jump. Note that the absorberslocked reference line clearly shows the vibration
attenuation and amplification for responses before and
after the jump, respectively.
absorber and then non-dimensionalized. The resulting
equations are then transformed such that the rotor angle
serves as the independent variable, instead of time
[Alsuwaiyan and Shaw, 2002]. Using this change of
variables and assuming that the N absorbers, their paths,
and their responses are identical, Alsuwaiyan and Shaw
[2002] showed that the equations of motion for the system
are given by:
È ~ ˘ 1 dx
(s )v = - m a s'
vs' '+ Í s '+ g (s )˙ v'Î
˚ 2 ds
~
È dx 2
˘
Í ds s ' v + x(s )vv'+ g (s )s ' vv'˙
˙ + vv' =
eÍ
~
Í ~
˙
dg
2
2 2
(s )s v ˙
Í+ g (s )s ' ' v +
ds
Î
˚
(2)
(3)
~
em a g (s )s ' v - m 0 v + G0 + Gq (q )
where s is the normalized displacement along the
absorbers’ arc length (see Figure 1), v is the rotor’s angular
velocity normalized by the rotor’s mean rate of rotation,
and primes denote differentiation with respect to q . The
absorbers each have mass m and their paths are described
in Equations (2) and (3) by the function x(s), which
represents the square of the distance from the center of
rotation to the absorber’s center of mass as it moves along
~
the path. The function g (s ) is a path function related to
x ( s ) arising from the Lagrangian derivation of the
equations of motion. The nondimensional damping terms
in Equations (2) and (3) are given by m a for the absorbers
and m 0 for the rotor. The parameter e is a measure of the
ratio of the absorber inertia to the rotor inertia. For point
mass absorbers and bifilar absorbers, e =mR02/J, where J is
the rotor moment of inertia and R 0=R+r. When the
absorbers cannot be modeled as point masses, such as for
the “T” absorbers, additional scaling must be performed so
that the equations of motion match the form of Equations
(2) and (3) [Nester 2002]. The non-dimensional torque
applied to rotor consists of a constant term given by G 0 and
a fluctuating term given by G q (q ) (which are the actual
torques normalized by the kinetic energy of the rotor).
Extensive analytical results have been obtained using
perturbation techniques for Equations (2) and (3)
[Alsuwaiyan and Shaw, 2002, Chao et al., 1997]. The
analysis first uncouples the rotor dynamics to leading order
using the small inertia ratio. Averaging is then employed
by assuming that the absorber response is harmonic with a
slowly varying amplitude and phase. This leads to slow
time equations governing the amplitude and phase, which
are examined for steady state responses and their stability.
Here we present results from these analyses in the
form of response curves, in order to compare experimental
results to the analytical predictions. Response curves are
generated for two types of parameter sweeps. In these, the
amplitudes of the order n harmonic components of the
4
The effects of mistuning the absorber are clear here.
The jump point is delayed by increasing the mistuning,
although the absorber performance is degraded over the
operating range. In fact, one can see that for very low
torques the perfectly tuned absorber works very well, but
that the jump occurs at small torque levels. Thus, it is seen
that there exists a compromise between absorber
performance and operating range. Typical designs that use
circular path absorbers use intentional detuning of about
5%. Similar curves for epicycloidal path absorbers are
shown below along with experimental data.
In addition to response curves predicted by theoretical
methods, simulations have also been carried out on the
fully nonlinear equations of motion. Therefore, in the
results shown below, experimental data are compared
against both theoretical curves and simulation results.
Figure 4: Effect of varying torque order n on the angular acceleration (top) and the absorber motion (bottom) for circular
paths (left) and epicycloidal paths (right), near the tuning order. Three increasing torque levels shown.
Figure 5: Theoretical response curves versus fluctuating torque level for circular path absorbers, for various levels of
mistuning. On the left is the rotor angular acceleration while on the right is the corresponding absorber amplitude.
5
Steady-state signals for the torque and system responses
are stored and processed using FFT software and a FFT
analyzer, which distills the amplitudes of the harmonic
components at the order of interest. Physical parameters
were obtained via a series of measurements. Some of the
important parameters for the two absorber systems are
listed in Table 1. Results were obtained for each of the
absorber systems shown in Figure 3. For the circular path
absorber, data was taken for one, two and four absorber
configurations, while one and two absorber configurations
were used for the epicycloidal path absorbers.
EXPERIMENTAL SETUP
The experimental apparatus, described in detail in Nester
[2002], consists of a pulse-width modulated electric motor
that drives a rotor to which up to four absorbers can be
attached; see Figure 6. The motor is controlled such that
the rotor maintains a constant average speed, upon which
is superimposed a fluctuating torque. This torque is
generated using the angular position of the motor, and can
therefore be adjusted to any desired order – that is, the
order is not linked to the mean rotation rate. This allows
one to carry out order sweeps, something not possible
when the torques arise from the inertial effects of attached
components, gas pressure loading, etc. Absorber
measurements are taken by optical encoders on each
absorber and fed out through a slip ring. Rotor
measurements are also taken by an encoder. The applied
torque is measured by monitoring the armature current in
the motor, which was calibrated using a torque meter. All
data is fed into a computer for storage and presentation.
Circular Path Epicycloidal Path
Damping Ratio
0.00241
0.00344
Absorber Order
1.31
1.465
Inertia Ratio
0.052
0.075
Table 1: Parameter values for single absorber case.
Figure 6: A schematic diagram of the experimental apparatus and a photo of the actual setup.
correspond to vibration reduction due to the absorber
dynamics, while those above it represent vibration
amplification due to the absorber. We begin with results
for the circular path case, then turn to the epicycloidal path
case, and finally provide a general comparison.
Figure 7 shows a set of sample experimental
results, compared against theoretical curves and data
obtained from simulations of the full equations of motion.
The details of the experimentally determined system
parameters can be found in Nester [2002]. Excellent
agreement is found between the simulations and the
experiments, and there is very good agreement between
these data and the theoretical predictions. Note that the
theory predicts the jump point to be higher than is
observed; this is due, at least in part, to the fact that the
response is highly sensitive near the jump point and the
upper branch has a larger basin of attraction near this
instability. Also, the theory is less accurate on the upper
response branch, which stems from the fact that the
EXPERIMENTAL RESULTS FOR A SINGLE
ABSORBER
We present and contrast results obtained from
experiments for rotors with circular and epicycloidal path
absorbers. In each case, the system with a single absorber
is considered. Results are presented in the form of
response amplitudes verses the amplitude of the fluctuating
torque, where amplitudes are taken to be the magnitude of
the Fourier component at the order being considered.
Theoretical response curves are shown for the absorber
amplitudes (on the left) and the rotor vibration levels (on
the right), obtained using perturbation techniques. Solid
lines represent stable responses while dashed lines
represent unstable responses. Note that the reference
straight lines on the rotor vibration response plots
correspond to the case in which the absorbers are locked at
their equilibria; this is used as a reference line on which
the absorbers simply add inertia to the system, serving as
simple flywheels. So, responses below this line
6
approximations are based on perturbation expansions that
become less accurate at larger
amplitudes. Overall, the theory does an excellent job of
predicting the gross features of the response, the vibration
amplitudes, and the trends that occur as system parameters
are varied.
Figure 8 shows similar results for the epicycloidal
path absorbers with bifilar suspension, with experimental
data and theoretical curves [Schmitz, 2003] shown on one
set of plots for several levels of mistuning (simulation data
are not shown here, but they match the experiments quite
well). Note the excellent agreement between theory and
experiments, for a range of absorber tuning values. The
key feature of these results are that the jump does not
occur and that one can therefore use the small amplitude
tuning of the absorbers and still avoid the jump. However,
it should be noted that the experimental data extends out to
higher torque levels for absorbers that are more mistuned;
this is due to the fact that the absorber amplitude grows
less rapidly in these cases, and therefore the absorber
amplitude limitations imposed by hardware constraints are
encountered at higher torques.
Figure 7: Sample results for the circular path system response, amplitude versus torque level. On the left is the rotor
angular acceleration while on the right is the corresponding absorber amplitude.
Figure 8. Theoretical and experimental results for the epicycloidal path system response, amplitude versus torque level.
On the left is the rotor angular acceleration while on the right is the corresponding absorber amplitude.
EXPERIMENTAL RESULTS FOR MULTIPLE
ABSORBERS
Figures 9 and 10 show the absorber amplitudes and rotor
vibration amplitude versus non-dimensional torque level
for the case when four absorbers are active.
7
Figure 9: Absorber amplitudes versus fluctuating torque
amplitude for order 1.2987 torques.
Figure 10: Experimental and theoretical rotor angular
acceleration versus torque amplitude for order 1.2987
torques.
In Figure 9, it is observed that initially only two absorbers
are active, and that as the torque level is increased the
other two absorbers became active one at a time, until all
four are active and acting in unison. This trend was typical
of the experimental results obtained for four circular path
absorbers. This behavior, while intriguing, does not agree
with the theoretical predictions [Alsuwaiyan and Shaw,
2002], which indicate that the absorbers will move in
unison at low torque levels and that their amplitudes would
grow as the applied torque level was increased, until an
instability is reached. The outcome of the instability is a
nonunison response, many types of which are possible.
In addition to the experimental absorber
amplitudes, analytical predictions [Alsuwaiyan and Shaw,
2002] are shown for the cases when two, three, and four
absorbers are active (obtained by assuming that the
inactive absorbers are locked). These curves are shown for
reference, and it is seen that when only a subset of
absorbers are active, the responses follow these predictions
quite well.
The corresponding angular acceleration for the
absorber data shown in Figure 9 is shown in Figure 10.
This data shows that the experimental angular acceleration
follows the theoretical prediction for the case when four
absorbers are moving in unison reasonably well. The
scallops in the experimental angular acceleration at low
torque levels correspond with the activation of additional
absorbers as the angular acceleration approaches the
theoretical curve. This behavior is currently not well
understood and is being investigated analytically.
path types, the lack of a jump in the response of the
epicycloidal path absorber makes it more attractive, since
jumps lead to very undesirable system responses. The data
also confirm the utility of the perturbation analysis, which
accurately predicts the system response. Such results
allow designers of absorber systems to make intelligent
estimates for the selection of absorber parameters. The
experimental results for the multi-absorber case are not
well understood and are the subject of current analysis.
CONCLUSIONS
The experimental data presented clearly show that
advantages of using epicycloidal path absorbers over the
commonly used circular paths. While one faces the
tradeoff between performance and operating range for both
V. J. Borowski, H. H. Denman, D. L. Cronin, S. W. Shaw,
J. P. Hanisko, L. T. Brooks, D. A. Milulec, W. B. Crum,
and M. P. Anderson. Reducing vibration of reciprocating
engines with crankshaft pendulum vibration absorbers.
1991. SAE Technical Paper Series 911876.
ACKNOWLEDGEMENTS
This work was supported by the National Science
Foundation under grant CMS 0084947.
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A. S. Alsuwaiyan and S.W. Shaw. Steady-State Responses
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