The 10th International Symposium on Transport Phenomena andDynamics of Rotating Machinery

Honolulu, Hawaii, March 07-11, 2004

ISROMAC10-2004-043

EXPERIMENTAL OBSERVATIONS OF CENTRIFUGAL PENDULUMVIBRATION ABSORBERS

Tyler M. Nester, Peter M. Schmitz, Alan G. Haddow, and Steven W. ShawMichigan State University

Department of Mechanical EngineeringMichigan State UniversityEast Lansing, MI 48824

Phone:Email: shawsw@msu.edu

ABSTRACTThis paper describes results from an experimental

investigation into the dynamic response of rotor systemsfitted with centrifugal pendulum vibration absorbers. Twotypes of absorbers are considered which exhibit differenttypes of nonlinear behavior. Systematic measurements ofthe rotor and absorber responses are taken for each type ofabsorber and compared against one another, and againstpreviously obtained theoretical predictions. The dramaticinfluence of the absorber nonlinearity is demonstrated, andthe results allow one to draw conclusions about theselection of absorber parameters. Results for systems withmultiple absorbers are also presented, showing how theabsorbers may behave in a complicated manner, evenwhen identically tuned.

INTRODUCTIONThe use of centrifugal pendulum vibration absorbers

(CPVA’s) is a proven method for reducing torsionalvibrations in rotating systems. They consist of masselements suspended from a rotor in such a manner thattheir centers of mass move along a prescribed path. Whenthe rotor is subjected to a fluctuating torque, the movementof the absorbers acts to counteract the applied torque,thereby reducing torsional vibration levels. The absorbersare tuned (using centrifugal effects) to a given order ofrotation, rather than to a set frequency, and are thereforeeffective over a continuous range of rotating speeds [DenHartog, 1985]. These devices have been in use for manyyears, most commonly in light aircraft engines [KerWilson, 1968] and helicopter rotors [Miao and Mouzakis,1981, Hamouda and Pierce, 1984]. More recent work hasproven them to be feasible for automotive applications,although they have not appeared yet in a productionvehicle [Borowski et al 1991, Nester et al., 2003].

These devices have some very appealing features forapplications, including: the order tuning mentioned above,

the fact that they dissipate very little energy, and they canoften be designed such that no mass or rotating inertia isadded to the rotor (for example, achieved in automotiveapplications by replacing existing counterweights byCPVA’s, which then serve dual use for balancing andtorsional vibration reduction [Nester et al., 2003]).

The basic operation of CPVA’s has long beenunderstood. Their tuning is achieved by selection of theplacement and curvature of the path of the absorber centerof mass, which is typically circular. This is most oftenachieved using a bifilar suspension, although many typesof physical arrangements are possible [KerWilson, 1968].The path parameters set the tuning order of the absorber,which is essentially the linear, small amplitude frequencyof free oscillation when the rotor runs at a constant speed.They also dictate the nonlinear behavior encounter atlarger amplitudes. The operating envelope of such adevice is dictated by the amount of absorber mass that oneemploys. Since added mass penalties are often very stiff,especially in aerospace applications, it is crucial thatdesigns minimize the amount of mass used. This leads tocomplications, however, since small absorber massnecessarily leads to large absorber responses, wherenonlinear effects become very important.

The potentially devastating effects due to nonlinearresponses are well known. As torque levels increase, thesystem can experience a jump, due to nonlinear effects,which results in absorber responses that actually amplifyvibration amplitudes. This phenomenon was investigatedby Newland [1964], who observed that absorber swingangles above 25o could cause a nonlinear jump in theabsorber amplitude, which was accompanied by a 180o

change in the absorber’s phase angle. This change in thephase causes the absorber to amplify the applied torque,rather than absorb it. This behavior, which will be shownboth experimentally and numerically, emphaticallydemonstrates that nonlinear effects cannot be ignored.

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Newland [1964] also describes a systematic means ofdealing with these effects, by intentionally detuning theabsorbers at small amplitudes in such a manner that theycome into proper tuning at moderate amplitudes. Thisworks, but at the expense of absorber performance, asshown below. In a major development, Madden [1980]proposed the use of noncircular (specifically, cycloidal)paths that allow for small amplitude tuning whilemaintaining, at least approximately, the desired tuningover a large range of amplitudes, thereby avoiding thejump. Subsequent studies considered the effectiveness ofabsorbers constructed with a range of paths, which includethe 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 absorbermass is divided up into a set of nominally identicalabsorbers that are arranged to satisfy balancing and spacerequirements. In such a case, the absorbers may notbehave in a synchronous manner with equal amplitudes.In fact, analytical results have predicted that systems ofnearly identical absorbers can undergo two basic types ofinstabilities. The first type is the jump described above,which maintains the absorbers in an equal amplitudemotion. The other type is a symmetry breaking bifurcationthat results in the absorbers oscillating at differentamplitudes and/or phases [Chao et al., 1997]. The formerinstability (which occurs for circular path absorbers, butnot cycloids or epicycloids) results in a catastrophic failureof the absorber system, while the latter instability (whichcan occur for all path types, depending on tuning) is morebenign, although it does limit the torque range over whichthe absorbers operate effectively [Alsuwaiyan and Shaw,2002]. Similarly, when the absorbers have smalldifferences among them, the response can becomelocalized, wherein a small subset of absorbers (maybe onlyone) does most of the work of canceling vibrations andtherefore oscillates at a much larger amplitude than thatpredicted by assuming identical absorbers [Alsuwaiyanand Shaw, 1999, 2003]. The method to avoid thislocalization and the instabilities is to intentionally overtuneall the absorbers, which reduces absorber effectiveness[Alsuwaiyan and Shaw, 2002, 2003].

Although much work has been done in this area, pastexperimental work has been limited primarily to specificimplementations. Various types of engines, helicopterrotors, etc., have been built and tested, but previousexperimental efforts have been limited to verifying thatCPVA’s lowered the vibration levels in the system ofinterest. In contrast, the present work describessystematic, controlled experiments that monitored both theresponse of the absorbers and the rotor for a variety ofoperating conditions.

In this work we report and compare results fromrecent experimental studies that consider two absorbersystems, one with circular path compound pendulums eachsuspended 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 briefoverview of the theoretical background, followed by ashort description of the experimental apparatus used. Themain results are then shown in the form of responsediagrams that depict experimental data compared againsttheoretical predictions. The paper is closed with adiscussion of the differences between the two differentabsorber systems and some conclusions.

THEORETICAL BACKGROUNDIn this section we first outline the basic tuning results

for the absorbers, and the nonlinear nature of the two pathsto be considered is then described. A brief summary of theanalysis is presented, along with some typical theoreticalresponse curves. These results are used as background andto compare with the experimental results, which form themain part of the paper.

An idealized representation of a rotor fitted with fourabsorbers is shown in Figure 1. For bifilar suspensions,such as the one shown in Figure 3 below, the absorbers canbe modeled as point masses riding along paths on the rotor(in this case their rotational inertias simply add to the rotorinertia), whereas for single point suspensions the rotationalinertia of the absorbers must be taken into account for thetuning.

Figure 1: Schematic of a rotor fitted with four general pathCPVA’s.

The absorber paths can be taken to be quite general,but two features are of importance: the linear tuning andthe nonlinearity of the path. According to small amplitudevibration theory, the linear tuning order of the CPVA’sshown in Figure 1 is given by:

( )22

~

r+=

r

rRn (1)

where R and r are the distances shown in Figure 1 and r isthe radius of gyration associated with the non-zeromoment of inertia of the absorber about its center of mass.Note that r=0 for the bifilar suspension, since the absorber

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rotates with the main rotor. By adjusting the radius of theabsorber path and the absorber’s center of rotation, it ispossible 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 W= n~ 0w (which reduces to

W rR / , the classical result, for r=0). For a system withan applied torque of order n, for example, of the form

)sin(0 qWnT , where q is the rotor angle, a CPVA is

referred to as being overtuned (undertuned) fornn )(~ <> , respectively. Absorbers are generally

overtuned, in order to avoid localization and nonlinearjumps [Alsuwaiyan and Shaw, 2002, 2003].

The nonlinearity in the path can be describedmathematically [Denman, 1992, Alsuwaiyan and Shaw,2002], but can also be conveniently depicted graphically,as shown in Figure 2. The circular path is easily realizedin experiments, in the present case by using the “T” shapedpendulums shown in Figure 3. The epicycloidal path isachieved by having the absorber mass connected to twostraps that wrap along cheeks shaped so that the absorbermass follows the desired curve, as shown in Figure 2 and 3[Schmitz, 2003]. From Figure 2 it is seen that thedifferences between circles and epicycloids that are tunedto the same linear order are nearly identical up to moderateamplitudes. However, as demonstrated below, this smalldifference has a large effect on the qualitative behavior ofthe system response at moderate amplitudes.

R

CENTER Of

ROTATION

CHEEK

r m i CG PATH

c

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 bifilarabsorber configuration with straps and cheeks.

The analysis for determining the system response canbe found in previous work by the authors [Alsuwaiyan andShaw, 2002, Nester 2002]. The results are obtained byperturbation methods that are based on a particular scalingof the system parameters. This scaling takes advantage ofrealistic ranges of physical parameters and renders thefollowing small quantities: the ratio of absorber inertia torotor inertia, the absorber damping, the applied torque, andthe nonlinearities in the response. It should be noted that

the scaling for the case of absorbers that use a single pointsupport requires some modification [Nester 2002], whichresults in equations of motion that are equivalent to thosewhich use a bifilar support [Schmitz 2003].

The basic model consists of N identical (or nearlyidentical) absorbers attached to a rigid rotor that issubjected to an applied torque of order n. In order toanalyze the equations of motion using perturbationtechniques, the equations are formulated for a general path

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absorber and then non-dimensionalized. The resultingequations are then transformed such that the rotor angleserves as the independent variable, instead of time[Alsuwaiyan and Shaw, 2002]. Using this change ofvariables 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 systemare given by:

( ) ( ) '21

''''~

svsdsdx

vsgsvs am-=-˙̊˘

ÍÎÈ ++ (2)

( ) ( )

( ) ( )

( ) ( )qmem

e

qG+G+-

=+

˙˙˙˙

˚

˘

ÍÍÍÍ

Î

È

++

++

00

~

22

~

2~

~2

'

'

''

''''

vvssg

vv

vssds

gdvssg

vvssgvvsxvsdsdx

a

(3)

where s is the normalized displacement along theabsorbers’ arc length (see Figure 1), v is the rotor’s angularvelocity normalized by the rotor’s mean rate of rotation,and primes denote differentiation with respect to q. Theabsorbers each have mass m and their paths are describedin Equations (2) and (3) by the function x(s), whichrepresents the square of the distance from the center ofrotation to the absorber’s center of mass as it moves along

the path. The function ( )sg~

is a path function related tox(s ) arising from the Lagrangian derivation of theequations of motion. The nondimensional damping termsin Equations (2) and (3) are given by ma for the absorbersand m0 for the rotor. The parameter e is a measure of theratio of the absorber inertia to the rotor inertia. For pointmass absorbers and bifilar absorbers, e =mR0

2/J, where J isthe rotor moment of inertia and R 0=R+r . When theabsorbers cannot be modeled as point masses, such as forthe “T” absorbers, additional scaling must be performed sothat the equations of motion match the form of Equations(2) and (3) [Nester 2002]. The non-dimensional torqueapplied to rotor consists of a constant term given by G0 anda fluctuating term given by Gq (q) (which are the actualtorques normalized by the kinetic energy of the rotor).Extensive analytical results have been obtained usingperturbation techniques for Equations (2) and (3)[Alsuwaiyan and Shaw, 2002, Chao et al., 1997]. Theanalysis first uncouples the rotor dynamics to leading orderusing the small inertia ratio. Averaging is then employedby assuming that the absorber response is harmonic with aslowly varying amplitude and phase. This leads to slowtime equations governing the amplitude and phase, whichare examined for steady state responses and their stability.

Here we present results from these analyses in theform of response curves, in order to compare experimentalresults to the analytical predictions. Response curves aregenerated for two types of parameter sweeps. In these, theamplitudes of the order n harmonic components of the

steady state responses are shown for both the absorbermotion and the rotor response. The measure used torepresent the rotor torsional vibration response is the ordern harmonic amplitude of its angular acceleration, whichrepresents its fluctuating component.

The first type of response curve, which is more usefulfrom a parameter selection/design point of view, is for thecase in which the torque amplitude and absorber order arefixed and the order of the applied torque is varied. Thisallows one to consider how the absorber should be tunedrelative to the torque to achieve good performance. Figure4 shows samples of these curves, indicating the basicdifferences in the responses for absorbers with circular andepicycloidal paths as the order of the applied torque isvaried. (Note that these theoretical results are equivalentto frequency response curves.) Response curves for threelevels of torque are shown. On the left are the results fromcircles, which show how the detuning of the absorber atmoderate amplitudes shifts the minimum point of the rotoracceleration and bends the absorber resonance curve (dueto the softening nonlinearity of the pendulum). Here theeffects of intentionally overtuning the absorber are clear,specifically, by designing the absorber so that the responseis to the left of the low torque amplitude minimum, onecan maintain a relatively small vibration level as the torqueis increased. These nonlinear features are notably absentfrom the epicycloidal case, shown on the right, even out tolarge amplitudes. This is due to the fact that the epicycloidis precisely the curve that is neither hardening norsoftening at linear order [Denman, 1992, Alsuwaiyan andShaw, 2002, Chao et al., 1997].

The other type of response curve considers a fixedorder of the applied torque and a fixed tuning of theabsorber and an increasing amplitude of the fluctuatingtorque. These curves show how an absorber responds inactual implementations, since the tuning is fixed byhardware, while the applied torques vary depending onoperating conditions. Figure 5 shows theoretical responsecurves for the rotor vibration levels (on the left) and theabsorber amplitudes (on the right) for circular pathabsorbers, for various levels of absorber tuning. Theprimary features of these curves can be described byconsidering a situation in which the amplitude of thefluctuating torque is slowly increased from zero up tosome level and then decreased back to zero. It is seen thatthe absorber amplitudes increase up to a point at which ajump occurs, resulting in the response being on the upperpart of the curve, where the absorber amplifies thetorsional vibration. Further increases simply make thissituation worse. The attendant rotor vibrations, measuredby the amplitude of the rotor’s angular acceleration, alsoincreases as the torque increases, almost up to the jumppoint, although for some levels of mistuning it turns downslightly just before the jump. Note that the absorbers-locked reference line clearly shows the vibrationattenuation and amplification for responses before andafter the jump, respectively.

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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 theoperating range. In fact, one can see that for very lowtorques the perfectly tuned absorber works very well, butthat the jump occurs at small torque levels. Thus, it is seenthat there exists a compromise between absorberperformance and operating range. Typical designs that use

circular path absorbers use intentional detuning of about5%. Similar curves for epicycloidal path absorbers areshown below along with experimental data.

In addition to response curves predicted by theoreticalmethods, simulations have also been carried out on thefully nonlinear equations of motion. Therefore, in theresults shown below, experimental data are comparedagainst 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 circularpaths (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.

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EXPERIMENTAL SETUPThe experimental apparatus, described in detail in Nester[2002], consists of a pulse-width modulated electric motorthat drives a rotor to which up to four absorbers can beattached; see Figure 6. The motor is controlled such thatthe rotor maintains a constant average speed, upon whichis superimposed a fluctuating torque. This torque isgenerated using the angular position of the motor, and cantherefore be adjusted to any desired order – that is, theorder is not linked to the mean rotation rate. This allowsone to carry out order sweeps, something not possiblewhen the torques arise from the inertial effects of attachedcomponents, gas pressure loading, etc. Absorbermeasurements are taken by optical encoders on eachabsorber and fed out through a slip ring. Rotormeasurements are also taken by an encoder. The appliedtorque is measured by monitoring the armature current inthe motor, which was calibrated using a torque meter. Alldata is fed into a computer for storage and presentation.

Steady-state signals for the torque and system responsesare stored and processed using FFT software and a FFTanalyzer, which distills the amplitudes of the harmoniccomponents at the order of interest. Physical parameterswere obtained via a series of measurements. Some of theimportant parameters for the two absorber systems arelisted in Table 1. Results were obtained for each of theabsorber systems shown in Figure 3. For the circular pathabsorber, data was taken for one, two and four absorberconfigurations, while one and two absorber configurationswere used for the epicycloidal path absorbers.

Circular Path Epicycloidal PathDamping Ratio 0.00241 0.00344Absorber Order 1.31 1.465

Inertia Ratio 0.052 0.075Table 1: Parameter values for single absorber case.

Figure 6: A schematic diagram of the experimental apparatus and a photo of the actual setup.

EXPERIMENTAL RESULTS FOR A SINGLEABSORBER

We present and contrast results obtained fromexperiments for rotors with circular and epicycloidal pathabsorbers. In each case, the system with a single absorberis considered. Results are presented in the form ofresponse amplitudes verses the amplitude of the fluctuatingtorque, where amplitudes are taken to be the magnitude ofthe Fourier component at the order being considered.Theoretical response curves are shown for the absorberamplitudes (on the left) and the rotor vibration levels (onthe right), obtained using perturbation techniques. Solidlines represent stable responses while dashed linesrepresent unstable responses. Note that the referencestraight lines on the rotor vibration response plotscorrespond to the case in which the absorbers are locked attheir equilibria; this is used as a reference line on whichthe absorbers simply add inertia to the system, serving assimple flywheels. So, responses below this line

correspond to vibration reduction due to the absorberdynamics, while those above it represent vibrationamplification due to the absorber. We begin with resultsfor the circular path case, then turn to the epicycloidal pathcase, and finally provide a general comparison.

Figure 7 shows a set of sample experimentalresults, compared against theoretical curves and dataobtained from simulations of the full equations of motion.The details of the experimentally determined systemparameters can be found in Nester [2002]. Excellentagreement is found between the simulations and theexperiments, and there is very good agreement betweenthese data and the theoretical predictions. Note that thetheory predicts the jump point to be higher than isobserved; this is due, at least in part, to the fact that theresponse is highly sensitive near the jump point and theupper branch has a larger basin of attraction near thisinstability. Also, the theory is less accurate on the upperresponse branch, which stems from the fact that the

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approximations are based on perturbation expansions thatbecome less accurate at largeramplitudes. Overall, the theory does an excellent job ofpredicting the gross features of the response, the vibrationamplitudes, and the trends that occur as system parametersare varied.

Figure 8 shows similar results for the epicycloidalpath absorbers with bifilar suspension, with experimentaldata and theoretical curves [Schmitz, 2003] shown on oneset of plots for several levels of mistuning (simulation dataare not shown here, but they match the experiments quite

well). Note the excellent agreement between theory andexperiments, for a range of absorber tuning values. Thekey feature of these results are that the jump does notoccur and that one can therefore use the small amplitudetuning of the absorbers and still avoid the jump. However,it should be noted that the experimental data extends out tohigher torque levels for absorbers that are more mistuned;this is due to the fact that the absorber amplitude growsless rapidly in these cases, and therefore the absorberamplitude limitations imposed by hardware constraints areencountered at higher torques.

Figure 7: Sample results for the circular path system response, amplitude versus torque level. On the left is the rotorangular 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 MULTIPLEABSORBERSFigures 9 and 10 show the absorber amplitudes and rotorvibration amplitude versus non-dimensional torque levelfor the case when four absorbers are active.

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Figure 9: Absorber amplitudes versus fluctuating torqueamplitude for order 1.2987 torques.

Figure 10: Experimental and theoretical rotor angularacceleration versus torque amplitude for order 1.2987

torques.

In Figure 9, it is observed that initially only two absorbersare active, and that as the torque level is increased theother two absorbers became active one at a time, until allfour are active and acting in unison. This trend was typicalof the experimental results obtained for four circular pathabsorbers. This behavior, while intriguing, does not agreewith the theoretical predictions [Alsuwaiyan and Shaw,2002], which indicate that the absorbers will move inunison at low torque levels and that their amplitudes wouldgrow as the applied torque level was increased, until aninstability is reached. The outcome of the instability is anonunison response, many types of which are possible.

In addition to the experimental absorberamplitudes, analytical predictions [Alsuwaiyan and Shaw,2002] are shown for the cases when two, three, and fourabsorbers are active (obtained by assuming that theinactive absorbers are locked). These curves are shown forreference, and it is seen that when only a subset ofabsorbers are active, the responses follow these predictionsquite well.

The corresponding angular acceleration for theabsorber data shown in Figure 9 is shown in Figure 10.This data shows that the experimental angular accelerationfollows the theoretical prediction for the case when fourabsorbers are moving in unison reasonably well. Thescallops in the experimental angular acceleration at lowtorque levels correspond with the activation of additionalabsorbers as the angular acceleration approaches thetheoretical curve. This behavior is currently not wellunderstood and is being investigated analytically.

CONCLUSIONSThe experimental data presented clearly show thatadvantages of using epicycloidal path absorbers over thecommonly used circular paths. While one faces thetradeoff between performance and operating range for both

path types, the lack of a jump in the response of theepicycloidal path absorber makes it more attractive, sincejumps lead to very undesirable system responses. The dataalso confirm the utility of the perturbation analysis, whichaccurately predicts the system response. Such resultsallow designers of absorber systems to make intelligentestimates for the selection of absorber parameters. Theexperimental results for the multi-absorber case are notwell understood and are the subject of current analysis.

ACKNOWLEDGEMENTSThis work was supported by the National Science

Foundation under grant CMS 0084947.

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