Macromodel Characterisation and Application ofParticle Dampers to Vibrating Structures

Chian X. Wong1 , Jem A. Rongong2

University of Sheffield, Sheffield, UK, S1 3JD

In this paper, the nonlinear damping property of a particle damper ischaracterized using both a time domain analysis via a Multi-Layered-Perceptron(MLP) neural network and that of an energy dissipator method. Data on the particledamper were acquired using a structure-free method, which leads to easyapplication of the particle damper to other structures. The validation of the particledamper model is performed by a comparison between the measured experimentalvalues from a damped structure and the predicted values from the neural networkand the energy dissipator method. The analysis of the structure is facilitated via aFinite Element model coupled with the eigenvalue analysis.

I. Introduction

Particle Dampers (PD) are enclosures of particles made of a variety of materials (e.g. metallic beads,polymer spheres, ceramic particles etc.). When attached to a vibrating structure, PDs dissipate energy viatwo primary mechanisms. The most significant mechanism is that via friction between the particles with thewalls of the enclosure and with one another. The second primary mechanism is that of inelastic collisionsbetween the particles and the walls of the enclosure. Secondary mechanisms of energy dissipation can alsobe introduced by applying a field of force around the particles (such as filling the enclosures with liquid orapplying an electromagnetic field1). The attractiveness of particle dampers mainly stem from theirapplicability over a large range of frequencies and temperature.

The concept of PDs is not a product of recent research. In fact, the earliest published work on PDs isthought to have been that of the work of Lieber and Jensen2 in 1945. In their work, the concept of anacceleration damper was introduced (which in fact is an impact damper, a PD with a single particle) tosuppress the aircraft flutter. Other applications of particle damping include the popular Non-ObstructiveParticle Damper (NOPD)3, which introduce damping to a structure by filling particles into voids createdwithin the structure.

Despite the increasing uptake of particle damping technology, the modeling of a particle damper is stillcomplicated by a number of issues. The effectiveness of a particle damper depends on a number of designparameters; such as the geometry of the enclosure, the material of the particles, the shape of the particles,the packing configuration of the particles, and the application of the field of force around the particles. It isthe compounded effects of these parameters that have led to PDs with highly nonlinear behaviour whenoperated with different operating parameters (such as varying velocity and acceleration). There is nopanacea currently available to practitioners of PD modeling. However, there are three general routes takenby investigators.

In the first route, investigators opt to model the behaviour of the PDs up to the individual particle level4

(known as particle dynamics analysis). The equations of motion of each particle is considered andcalculated using the Discrete Element Modelling (DEM). Models created in this way can provide a strongvisual clue and ideas on how the PDs work (possibly leading to better designs). The same models can also

1 Research Associate, Department of Mechanical Engineering.2 Rolls-Royce Lecturer in Engineering, Department of Mechanical Engineering.

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be reused in different operating conditions, which would allow further analysis without recreating newmodels for different operating conditions. The drawback of this method however is that it iscomputationally expensive (especially if the number of particles are in the millions). One might also needto know a large number of parameters a priori to modeling (e.g. size and shape distribution of particles,hardness characteristics of particles etc.). This might prove to be difficult in a number of situations.

In the second route, modelling is performed via a simplification of the problem. An example of such amethod is that of modelling the bed of particles as a single effective particle and estimating theperformance of the PDs based on this single particle5. Other possible methods are to linearise the model fordifferent operating conditions. This was the method taken by Liu and colleagues6 in estimating anequivalent linear viscous damping coefficient for different levels of excitation. This is a much simplermethod to implement compared to the particle dynamics method, although it would not provide the level ofdetail that the particle dynamics analysis can. Furthermore, most of these analyses have been conductedwithin a restricted range of operating conditions, such as that for free decay to equilibrium5 and for specificsinusoidal frequency excitation6. Since the damping performance of PDs vary significantly from oneoperating condition to another, the use of these models are restricted in terms of application.

In the third route, neither assumption nor simplification of the overall behaviour of the PDs is assumed.It is based merely on the observed performance of the PDs under a set of operating conditions. Thedamping force of the PDs are modelled based on its dependency on the observed acceleration, velocity andpossibly displacement, leading to a macromodel characterization of the PDs. This method is probably theeasiest to implement although it also provides the least information about the physical aspects of thedamper. An example of a nonparametric PD macromodel produced this way is the Restoring Force Surface(RFS) model7. The RFS model provides a phase plot of the damping force with respect to velocity anddisplacement of the system.

In this work, the macromodel approach is attempted via two methods. The first approach uses the PowerFlow (PF) concept to interpret the effective mass and loss factor of the PD. A similar approach has beensuccessfully attempted by Yang8 in the modeling of PDs excited in a sinusoidal fashion. The advantage ofYang’s method is that characterization of PDs can be performed without the need to mount it on a structure.This effectively enables the easy transfer of the PD model to any structure (provided that enough data hasbeen acquired). Yang’s method however, can be easily extended for a non-sinusoidal periodic excitation.This improvement allows the method to be used for a wider range of different excitation signals that maybe encountered in practice. This improvement is demonstrated here for a few simple examples.

Although the PF method is shown to provide a reasonably accurate interpretation of the data8, someinformation might have been diluted. This is due to the nature of the Fourier Transforms usually used infrequency domain based PT that do not include local time information (although it is possible to avoidFourier Transforms by using instantaneous powers in the time domain). The second method captures thelocal time information by introducing a time-domain based neural network model. This neural networkutilizes the measured velocity and force data for a particular PD. The neural network then is able to predictthe expected damping force given the response. This method would provide easy integration of the modelinto a control system (especially if the PD is tunable online) as most control systems are based in the timedomain. It must be noted however, that the use of this model neglects the frequency domain informationavailable in PF.

II. Particle Damper Characterisation

A. Power Flow Concept

The Power Flow (PF) concept is a general concept to allow one to calculate the flow of power into asystem. Despite the age of this subject, it is still being explored extensively, especially in the field ofelectrical engineering. This is due to the persistent significance of efficient power transmission for devices

of increasing complexity. In a nutshell, the electrical PF concept allows one to calculate the powertransmitted into a device based on the voltage and current of an electrical circuit.

Work done in a cycle (of a time period, T) of an electrical device is given as:

∫=T

cycle dttItVW0

)()( (1)

Where V(t) and I(t) are the voltage and current respectively at a time t. It should be noted that thevoltage and current are assumed to be periodic over this time period. This concept can be easily transferredfor a mechanical mass by substituting voltage and current with force and velocity:

∫=T

cycle dttvtfW0

)()( (2)

Where f(t) and v(t) are the force and velocity respectively. The term f(t)v(t) is denoted as the

instantaneous power. The average power transmitted is therefore given as:

∫==T

cycleav dttvtf

TT

WP

0

)()(1

(3)

Equation (3) can be easily interpreted as average of instantaneous power. If the force and velocity

dataset is represented as a complex Fourier series in a discrete manner (with the real values represented inthe real portion of the Fourier series), one arrives at the equation:

( ) ( )∫ ∑∑

=

∞

=

−∞

=

−T

m

tmjm

n

tnjncomplex dtevef

TP mn

0 00

1 βωαω(4)

Where complexP is the complex power; nf and mv are the magnitude of the force and velocity of the n-

th and m-th harmonic respectively; e is the Euler number; j is the imaginary number; ω is the fundamental

frequency in rad/s; and nα and mβ are the phase of the force and velocity of the n-th and m-th harmonic

respectively. Due to the orthogonality of harmonics, the integrals between all cross-terms cancel out:

( )∑∞

=

−=02

1

n

jnncomplex

nnevfP βα (5)

The average dissipated power (also known as active power in electrical engineering) can be extracted

from the real part of the complex power. The imaginary part of Equation (5) corresponds to a term usuallyknown as reactive power (also a term borrowed from electrical engineering). Reactive power isconventionally interpreted physically as the maximum power stored in the system. The derivation ofreactive power comes from the phasor difference between apparent power (the product of root-mean-squareof force with the root-mean-square of velocity) and active power.

The complex power can be easily acquired from experiments via the cross-spectrum calculation built-inin most signal analyzers. The complex power spectrum for the n-th harmonic is given as:

rmsnrmsnn vfP ,*

,= (6)

Where rmsnf , denote the Fourier transform of the force normalized to its root-mean-square level

(dividing the magnitude by 2 ); while rmsnv ,* is conjugate of the Fourier transform of the velocity

normalized to its root-mean-square level. The total complex power, complexP is the sum of the complex

power spectrum over all the harmonics:

∑=n

ncomplex PP (7)

The damping loss factor, η from a system is given as:

cycleainstoredenergyMaximum

radianpercycleaindissipatedEnergy=η

==

∑

∑

nn

nn

P

P

Im)2(

Re

power)reactive(2

poweractive

ππ

(8)

Assuming that the stored energy is in the form of kinetic energy, one could calculate an effective mass,

effm , that is associated with the particle damper:

powerreactive2

1 2

=

∑

dt

vmdn

neff

= ∑∑n

nn

neff Pvmn Imω

∑∑

=∴

nn

nn

effvn

P

mω

Im(9)

The effective mass and more importantly, the damping loss factors would provide us with glimpses into

how damping quality (and its mechanisms to a certain extent) is affected for a particle damper in a certainoperating condition.

B. Particle Damper Experimental Characterisation

A Perspex particle damper was chosen for the characterization procedure. This is the same particledamper that was used in the RFS analysis7 and is similar in construction to the particle damper used byYang8. The particle damper consists of a cylindrical casing with a screw top lid and a securing ring. Thescrew top lid allows the attenuation of the distance between the top layer of the particles and the bottom ofthe casing lid (it is however fixed at 6 mm in the subsequent experiments). The transparent casing alsoallows one to view the movement of the particles during the operation of the damper. A schematic of theparticle damper is shown in Figure 1 while a picture of the particle damper is shown in Figure 2.

The particle damper was filled with 68.67 grams of steel shots (used for shot blasting), giving acombined damper weight of 173.67 grams. These steel shots are near-spherical particles that wereoriginally acquired as a batch with a great degree of variance in size. The particles were then filteredthrough a series of sieves to acquire particles within the desired range of diameters. The probabilitydistribution of diameters of the particles used in subsequent experiments is shown in Figure 3.

1270.4

38 30

Screw threads

Securing ring

3032.5

3mm screw thread

34.5

5

Figure 1. Schematic of the particle damper.

Figure 2. The Perspex particle damper filled with steel shots

The histogram in Figure 3 is approximately normally distributed, giving an average particle diameter of0.6557 mm and a standard deviation of 0.0317 mm. The number of particles in the casing is estimated atabout 9500. Characterising this damper using particle dynamics is possible, though a little expensivecomputationally (as it is with almost all DEM simulations).

Four different sets of experiments were conducted to characterize the damper; a vertical stepped-sinetest (between 5 to 200 Hz, in intervals of 5 Hz for a range of amplitudes); a horizontal stepped-sine test(between 5 to 200 Hz, in intervals of 5 Hz for a range of amplitudes); a vertical random excitation test(band-limited white noise from 0-200 Hz); and a horizontal random excitation test (band-limited whitenoise from 0-200 Hz). This involves mounting the Perspex damper onto an electromagnetic shaker. Theconnection between the PD and the shaker is buffered by a force transducer, while the velocity of thedamper is measured with a laser vibrometer. The damper was supported with fishing lines and excited via astinger for all horizontal experiments to prevent damage to the shaker. The laser vibrometer used here has asmall range of measurement. This severely limits the range of operation that can be observed from theparticle damper. However, it is highly accurate (compared to an accelerometer), which is critical for timedomain modeling. All data were sampled at 1.28 kHz for all experiments conducted for this paper.

Figure 4 shows the installation of the damper on top of the shaker for the vertical excitation tests whileFigure 5 shows the setup for the horizontal tests.

Figure 3. Size distribution of the steel shots

Figure 4. Installation of the Perspex casing on the shaker for vertical excitations.

Plots of the damping force versus acceleration are shown for a range of frequencies and amplitudes. Theacceleration signal was acquired offline using the 5-point central-difference method. It should be notedthat the damping force term mentioned here does not differentiate between the force that is responsible forenergy dissipation and the inherent inertial force of the damper. Nevertheless, as an attachable damper, thedamping force can be considered as external force acting on the structure. Therefore for the modelingpurposes of this paper, this distinction is not needed.

The sine excitation at 5 Hz for both orientations produced a linear damping force with no energydissipation. The gradient of this curve corresponds closely to the total mass of the damper (0.180 kg,slightly higher than the measured mass of 0.173 kg).

The vertical stepped-sine tests produced a skewed damping force curve for frequencies between 20 Hzto 200 Hz (Figures 7-8). This corresponds to increasing effective mass as the damper is acceleratingupwards and reduced effective mass as the damper is decelerating. It is also noted that damping forcefollows two different paths as the excitation level is increased. This is due to the particles impacting thebottom and the top of the damper casing. It can also be seen that the damping force profile is dependent onfrequency.

The damping force for the stepped-sine horizontal excitation however, seems to be largely independentof the excitation frequency (refer to Figures 10-11). This is probably because the particles are mostly incontact with one another and with the wall. The particles are rarely airborne as in the case with the verticalexcitation.

In Figure 12, random band-limited vertical damping force points clusters around two linear regimes.These two regimes correspond to the bounds of the effective mass of the damper (which is between 0.116kg and 0.180kg). Even though most of the damping force points are around these regions, the area coveredby the damping force points actually increases with excitation. Although this is also noticeable in therandom band-limited horizontal damping force (refer to Figure 13), a shift in damping force gradient ismuch more noticeable in the horizontal orientation as one increases the excitation amplitude.

Figure 5. Installation of the Perspex casing on the shaker for vertical excitations.

Figure 6. 5 Hz vertical sine excitation.

Figure 7. 45 Hz vertical sine excitation.

Figure 8. 125 Hz vertical sine excitation.

Figure 9. 5 Hz horizontal sine excitation.

Figure 10. 45 Hz horizontal sine excitation.

Figure 11. 125 Hz horizontal sine excitation.

Figure 12. Random vertical band-limited (200 Hz) excitation.

Figure 13. Random horizontal band-limited (200 Hz) excitation.

By utilizing the PF concept as stated in Section II. A., one can calculate the damping loss factors andthe effective mass for a general form of excitation. Samples of the damping loss factors and effective massfor a variety of excitations are shown in Figures 14-18. Figure 14 and Figure 16 shows the extent offrequency dependence of the two different orientations, with the damper in the horizontal orientationclearly showing a lower dependence. It is also noted that the randomly excited tests produced lowerdamping loss factors compared to its sinuisoidal counterparts for the same level of acceleration.

Plots of the dissipation power spectrum for the random excitation shows that most of the energy isdissipated within of 20-75 Hz (Figure 18 and Figure 19). However, the random excitation tests wereconducted without a control system to ensure a flat plateau in the force autospectrum (refer to Figure 19 orthe vertical random test force autospectrum, the horizontal test showed a similar characteristic). Therefore,it is not known to what extent this bias of energy dissipation in this region is caused by the shakercharacteristic. Nevertheless, it bears further scrutiny in the future.

Figure 14. Vertical stepped-sine tests of varying amplitude.

Figure 15. Random vertical band-limited (200 Hz) excitation.

Figure 16. Vertical stepped-sine tests of varying amplitude.

Figure 17. Random horizontal band-limited (200 Hz) excitation.

Figure 18. Dissipation power spectrum for vertical random excitation (bandwidth = 200 Hz).

Figure 19. Dissipation power spectrum for horizontal random excitation (bandwidth = 200 Hz).

Figure 20. Force autospectrum for vertical random excitation (bandwidth = 200 Hz).

III. Experimental Validation

In order to utilize the damper characteristic values acquired from the experiments conducted fromSection II, two different modeling techniques are used here. Namely, the Nonlinear AutoRegressive witheXogeneous input Multi-Layered-Perceptron (NARX MLP) network and the kinetic energy dissipator.

A. MLP network

The force and velocity data acquired from the previous experiments were fitted using a NonlinearAutoRegressive with eXogeneous input Multi-Layered-Perceptron (NARX MLP) network. The MLP is infact a feedforward network where the neurons are arranged in layers. An example of such a network isshown in Figure 21.

The different signal values, iI are first passed through the input layer. For each subsequent layer,

weighted sums of the signals with a bias element are passed into each individual neuron. These neurons,also called activation functions, are nonlinear differentiable functions. A common choice for the activationfunction is the hyperbolic tangent function and is adopted in this work. Differentiability of the activationfunction is a prerequisite for the application of the optimisation routine, which is a gradient descentalgorithm called the backpropagation algorithm9. The output layer activation function however, is usually alinear function.

Since damping is a function of acceleration, velocity and possibly displacement. The inputs to the MLPnetwork must be augmented from the virgin velocity signals. Numerical integration of the velocity signalscan sometimes lead to the introduction of low frequency distortions10. Removing these low frequencycomponents can be troublesome as simple filtering is actually an incorrect approach for nonlinear systems.Removing the low frequency trends using least-squares fitted polynomials may also distort the signalsfurther. An alternative approach is used here instead. The inputs to the MLP are encoded using the NARXapproach10, which actually uses the lagged versions of the inputs (velocity signals) and the outputs (forcesignals) of the system. This indirect manner allows the acceleration and displacement to be encodedindirectly into the MLP network. It also has the added advantage of modelling any complex hystereticbehaviour of the PDs due to the inclusion of the lagged outputs. The previous RFS model7 only attemptedto estimate dependencies based on both velocity and displacement. This model however is much moregeneral. It is also simple to implement as NARX MLP algorithms are widely available.

Layer 0(Input)

Layer 1(Hidden)

Layer 2(Hidden)

Layer l(Output)

I i yi

)1(0jw

j

m

n

)2(nmw

BiasElement

Figure 21. A multi-layer perceptron network.

The MLP network was used to fit a range of data acquired from the randomly excited horizontalorientation tests for a series of amplitudes. The network was then combined using an extra linear model.The resultant MLP network can hopefully be used as a general model to model the damping force given thevelocity. An example of the performance of the neural network in predicting the damping force is given inFigure 22 (note that the data predicted is not the same set of data used for optimizing the network).

Figure 22. Comparison of neural network model output and measured damping force.

The second macromodel approach utilises the same experimental data as the first method. It assumesthat the particles are equivalent to a solid kinetic energy dissipating material. The energy dissipated isrelated to an energy dissipation coefficient to the equivalent solid mass. Utilising the concept of modal lossfactor (which is a ratio of energy dissipated per radian to the total apparent kinetic energy), one can relatethe structural modal loss factor to the mass of the particles, the mode shape of the structure, the modal massand the dissipation coefficient. This approach has already been taken by Rongong and Tomlinson1. Thedifference however, is that the dissipation coefficient is calculated here based on the damper loss factorscalculated in Section II without the need to measure the dissipation coefficient from a structure.

B. Kinetic Energy Dissipator

For a vibrating system, the modal loss factor, mη can be related to the dissipated energy, dW using:

total

pD

total

dm

W

Τ

Τ== ηη

T(10)

Note that Ttotal is the apparent kinetic energy, and is the sum of Tp (the kinetic energy trapped in theparticles) in the damper and the kinetic energy of the host structure and damper casing. The so calleddissipation coefficient, Dη is actually the damper loss factor calculated in Section II. In a linear, flexible

structure, velocities at different positions vary according to:

)()( tqxx T && φ= (11)

where φ is the mode shape and q the vector of time varying coefficients. The apparent kinetic energy of thesystem is then given by:

221 qMtotal &=Τ (12)

where M is the modal mass, calculated using the equivalent solid mass for the granular medium. Theapparent kinetic energy for the particles is given by:

2221 qm ppp &φ=Τ (13)

where mp and φp are the mass of particles and the mode shape at the damper connection point. The modalloss factor can then be obtained by substituting Eq. (12) and Eq. (13) into Eq. (10) to give:

M

m ppDm

2φηη = (14)

It is important to note that the dissipation coefficient ηD is amplitude-dependent. Thus a predictionmade using Eq. (14) has to employ the correct value for the damping coefficient based on the accelerationlevel at the damper.

In order to test the validity of these techniques, an experiment is conducted on a steel ring with the PDattached (refer to Figure 23). This steel ring is representative of a fan casing. The steel ring is 756 mm indiameter, 150 mm deep with a wall thickness of 6.35 mm. The ring was suspended, with its axis in thevertical direction, with fishing lines to mimic a free boundary. Excitation of the structure and responsemeasurements were carried out using the same equipment as that used for the SDOF test rig. The ring wasexcited radially for a range of fixed sinusoidal velocity amplitude over its first three in-plane bendingmodes. The modal loss factors were then obtained using curve-fitting software. The excitation andmeasurement points were located exactly half way along the axial depth of the ring in order to avoidexciting out-of-plane modes. The damper is attached on in the horizontal manner, where excitation sourceis located 180 degrees away.

Figure 23. Setup of the steel ring.

A finite element model (of 728 quadrilateral elements) was also generated to utilize the eigenvalueanalysis (refer to Figure 24-27). The natural frequencies of the finite element model was compared to theexperimental rig to ensure accurate modeling (refer to Table 1).

Table 1. Comparison of natural frequencies between the finite element model and experiment.Finite element Experiment

1st bending mode naturalfrequency (Hz)

28.41 28.61

2nd bending mode naturalfrequency (Hz)

80.35 80.71

3rd bending mode naturalfrequency (Hz)

154.09 154.75

Figure 24. Finite Elementrepresentation of fan casing.

Figure 25. 1st flexural mode.

Figure 26. 2nd flexural mode. Figure 27. 3rd flexural mode.

By assuming the attachment of the damper as a point mass in the FE model, the flexural modal massand mode shapes for flexural modes from 1 to 3 can be calculated via a real eigenvalue analysis. By usingthe dissipation coefficients calculated with the power measurement technique, the modal loss factor for arange of vibration amplitude can be calculated using the kinetic energy dissipator assumption.

To validate the MLP model, the velocity at the damper connection point of the FE model is calculatedfrom the modal analysis (with the mode shapes acquired from the finite element model). The damping lossfactor (which is also the dissipation coefficient of the kinetic energy dissipation method) can then beextracted for comparison purposes by utilizing the power flow equations of Section II. A. and the modalkinetic energy (to calculate the total trapped energy in the structure).

The comparison between the simulation and the experiment is shown in Table 2.

Table 2. Comparison of modal loss factors. Note that a very small background damping has beenremoved from the calculations.

Bending modes ofvarious amplitudes

Kinetic EnergyDissipator

(modal loss factor)

Neural Network(modal loss factor)

Experimental(modal loss factor)

1st mode (Level 1) 6.9650-4 2.4468e-4 8.6969e-4 (Level 2) 8.0236e-4 2.6765e-4 9.4166e-4 (Level 3) 8.5498e-4 2.8225e-4 0.0012(Level 4) 8.7120e-4 3.0012e-4 0.0013(Level 5) 9.3582e-4 3.1064e-4 0.0015

2nd mode (Level 1) 0.0011 2.4399e-4 0.0017(Level 2) 0.0010 2.5091e-4 0.0015(Level 3) 9.0192e-4 2.7239e-4 0.0014(Level 4) 7.2198e-4 2.9068e-4 0.0013(Level 5) 7.1203e-4 3.1047e-4 0.0012

3rd mode (Level 1) 6.1238e-4 1.5341e-4 8.3971e-4 (Level 2) 5.4210e-4 1.4210e-4 7.6369e-4 (Level 3) 4.4011e-4 1.4011e-4 6.9584e-4 (Level 4) 4.4910e-4 1.4910e-4 6.372e-4 (Level 5) 3.4001e-4 1.4001e-4 5.9061e-4

ConclusionsThe work reported in this paper aimed at finding more ways to characterize a particle damper. The

techniques used found moderate correlation with the measured modal loss factor from experiment.However, it must be stated that the levels of excitation were curtailed by the laser vibrometer used. It ishoped that more experiments can be conducted in the future to further refine and verify these techniques.

AcknowledgmentsC.X. Wong is funded by EPSRC contract GR/T28393/01 and is grateful for the help extended by Dave

Webster and Leslie Morton in setting up the experiment.

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