[American Institute of Aeronautics and Astronautics 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference - Palm Springs, California ()] 50th AIAA/ASME/ASCE/AHS/ASC

Effects of Enclosure Geometry on Particle Damping Performance

C. X. Wong1, A.B. Spencer2 and J.A. Rongong3

Department of Mechanical Engineering, University of Sheffield, Mappin Street, United Kingdom, S1 3JD

The performance of particle dampers are related to the complex interdependencies between the material parameters of the particles, shape of particles, operating conditions, particle damper enclosure geometry and location within the structure. In order to investigate the effects of all these interdependencies, a number of experimental techniques combined with intricate and simplified models have been proposed. One of the more popular methods of the past few years is the application of the so called power input/ power flow method to identify the power dissipation properties of particle dampers. The coupling of this measurement method with the Discrete Element Method (DEM) has shown that DEM can be use as a good predictor of performance for particle dampers with low particle count. This allows the utilization of this modeling technique for the investigation of different designs rapidly. In this work, the effect of the damper enclosure shape on steady state performance is studied by interpolation between a number of prototype shapes in DEM simulations, whereby other parameters are held constant. It is shown that the enclosure shapes itself has a very significant effect on the performance of the dampers, indicating another useful parameter to tweak to improve performance.

Nomenclature )(int,

ierjn = Cartesian coordinates of the i-th nodal point of the j-th interpolated shape

)(,inormjn = the normalized column vector of the Cartesian coordinate of the i-th nodal point for the j-th shape

volj,norm = the specified normalized volume of the j-th shape volj,unnorm = the original volume of the j-th shape

)(,iunnormjn = the original column vector containing coordinates of the unnormalised j-th shape. )(iN = matrix that concatenates the nodal points of all the prototype shapes

w = column vector that contains the weighting factors for n prototype shapes A = set containing independent physically admissible weighting factors S = set of shapes simulated V(t) = wall velocity F(t) = wall force Vamp = velocity amplitude Aamp = acceleration amplitude ceq = equivalent viscous damping coefficient meq = equivalent mass f = frequency of excitation

American Institute of Aeronautics and Astronautics

1

1 Research Associate, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield U.K., S1 3JD, AIAA Member. 2 IT Support Officer, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield U.K., S1 3JD, Non-AIAA Member. 3 Senior Lecturer, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield U.K., S1 3JD, AIAA Member.

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2689

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction The basis of particle damping revolves around vibrated granular media enclosed in some container. It has

numerous advantages over other traditional treatments of damping. It can be designed so that it is largely invariant to temperature and is efficient over a broad range of frequencies. Particle dampers can also be implemented quite easily in a plethora of applications by just placing particles within voids in a structure. In cases where this is not practical, it is also of course possible to create attachments to place the particles in. There is now an increasing uptake in particle damping technology1,2, although there are still many challenges that remain to be tackled. Firstly, vibration energy from the main structure is partially transferred to the particles as kinetic and strain energy, which is also in turn reduced via dissipation due to friction between the particles and the plastic deformations from impacts. This results in a highly nonlinear relationship between the design parameters of the particle damper to that of the performance of the particle damper. The behavior of these dampers also varies significantly from one operating condition to the next, with no strong consensus on a unifying formulation at the moment to quantify the performance for these different regimes. Some attempts however, have been made with some degree of success3. To complicate matters further, the dimensional complexities of the design space is prohibitively large to explore via experiments and even to some extent, via simulations if the design space is not constrained appropriately.

There are a multitude of methods available to model granular material itself, ranging from continuum approximations and discrete elements4. A discrete element approach which allows the particles to overlap, aptly dubbed the Discrete Element Method (DEM)5, has risen in popularity in recent years. The method allows the modeling of particles via interaction mechanisms between individual particles, as opposed to fixed observation volumes with simplified constitutive laws of the media. Some work using this method (based on the commercial software PFC3D 3.16) was performed in the past to model particle damping, with reasonable correlation to the power dissipated measured in steady state vibration experiments7,8. The power dissipation from these experiments was based on the Fourier Transform based power input method, which measures the average power dissipated for periodic vibration9. More advanced methods like those based on the Hilbert-Huang method has been used for transient vibration of particle dampers10.

In this work, the effect of the geometry of the damper wall enclosure on the performance will be assessed. This has actually been explored using simple disc geometries in the past11. In Ref. 11, simple disc particle dampers of varying aspect ratios (ratio of disc diameter to depth) are attached to a frame (where excitation direction is perpendicular to the gravity loading direction). This then becomes a simple single-degree-of-freedom system, with the frame represented by a spring; the mass of represented by the joint mass of the frame, the particle damper casing and the mass of the particles; and the damping represented by an equivalent viscous damping coefficient. Fig. 1 shows the equivalent viscous damping coefficient for different velocity amplitudes and disc aspect ratios. It indeed shows that the power dissipation properties do change quite significantly given different disc aspect ratios. The only setback of using this experimental method is that it is infeasible to test out a large number of configurations due to the manufacturing costs for the particle dampers. Nevertheless, it has provided the incentive to further investigate the effect of enclosure shape in a different manner.

The work reported in this paper forms the first step in providing a framework for different enclosure shapes without limiting the shapes to just disc geometries. This was done by creating a number of prototype shapes and interpolating between these prototype shapes for a desired shape specification. The casings are then modeled with particles using DEM and excited in steady state vibration. The power dissipation and the effective mass of the different geometries are then calculated and assessed. In Section II., the procedure used in creating the wall mesh of the enclosure is described. This is followed by a short description of how the system is modeled in DEM in Section III. The results of the simulations and some discussion are presented in Section IV. The paper then concludes in Section V.

Figure 1. Equivalent viscous damping coefficient for SDOF systems with different particle damper disc aspect ratios, α 9


2

II. Wall Meshing Procedure In order to define the geometry of the walls of the enclosure, it is useful to understand how varying the shape in a

continuous fashion would impact the performance of the damper. The aspect ratio of disc dampers can be thought of as a continuous parameter that is used to vary the shape of an object. It is however constrained to that of only disc dampers. A more general way of defining shapes must be used if shapes of different types are also part of that continuous spectrum. A simple way that was used in this paper is to define a number of prototype shapes. The final shape is a simple mapping between these shapes.

Each prototype shape is defined by a mesh on the surface of the wall. The walls are defined by number of wall elements with 4 nodal points each. The center of the overall wall mesh is defined at the point of origin. For ease of explanation, a spherical type wall mesh is constructed as shown in Fig. 2(a). This spherical wall has 8 rotational segments and 16 lengthwise segments. This results in 153 nodal points, with 8 nodal points sharing the same location at the “north pole” and the “south pole” of the sphere. All of the other created shapes would also have 153 nodal points that are each uniquely identified by a nodal point number across all the different shapes. Obviously increasing the meshing density would allow a more accurate approximation of an arbitrary shape. It must be kept in mind though that increasing the wall meshing density would increase the number of contact elements in the subsequent DEM simulations. The meshing density is kept at this level to reduce the computational time which can be quite demanding for a large number of configurations.

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(a)

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x 10-3

(b)

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x 10-3-50

5x 10-3

-5

0

5

x 10-3 (c)

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0.01

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05

x 10-3 (d)

Figure 2. Mesh of wall geometries (a) sphere, (b) disc, (c) cube, (d) interpolated shape with weighting factor vector [-0.25 0 0.75]T.

In order to ensure a smooth interpolation between the shapes, two more shapes were defined as prototype shapes. These are the cuboid and the disc. For simplicity, the cuboid is constrained to be a cube while the disc has an aspect ratio of 10/3. The mesh structure for both the disc and the cube with 8 rotational segments and 16 lengthwise segments are shown in Fig. 2(b) and Fig. 2(c) respectively. The mesh structure for both the shapes is similar in the sense that 8 of the lengthwise segments are constrained for the top face and the bottom face of the shapes equally.

The final interpolated shape is essentially defined by nodal coordinates which are themselves functions of the nodal coordinates of the 3 prototype shapes. The simplest interpolating function is to use a linear model. This is given as:

(1) wNn )()(int,

iierj =

[ ]T

)()()()( iiii

nwww L21=w (2)

(3) ][ 21 nnnnN L=

where n is the Cartesian coordinates of the i-th nodal point of the j-th interpolated shape, is a column

vector that contains the weighting factors for n prototype shapes, and is a matrix that concatenates the nodal points of all the prototype shapes. The 3 prototype shapes in this paper are defined such that w

)(int,

ierj w

)(iN1 is the spherical


3

weighting, w2 is the disc weighting and w3 is the cube weighting. A weighting factor of 1 for a particular prototype shape and 0 for the other shapes would create an interpolated shape that is same as the associated prototype shape. The weighting factors however, do not need to be constrained in the interval [0,1]. It is in fact possible to create non-convex type shapes by using negative weighting factors. An example of such a shape is given in Fig. 2(d). It must be noted though that using negative weightings may produce physically impossible shapes.

So far nothing has been mentioned about the volume of each shape. The dimensions of each prototype shape can initially be defined arbitrarily as long as the cylinder is fixed at the aspect ratio of 10/3. This is so because the prototype shapes are then subsequently volume normalized to a value of 6.36 x 10-7 m3. This volume value was chosen in order to ensure that all shapes are comparable in terms of volume and also to enable some form of comparison with physical experimental data and simulations that was done in the past7. The interpolation procedure

of Eqs. (1), (2) and (3) are then applied to the set of volume normalized prototype shapes. The generated interpolated shapes are then itself also volume normalized.

The volume for each shape can be easily calculated by exploiting symmetry and convexity. Even if the overall geometry of the shape is non-convex, the geometry can be quartered and then segmented for each lengthwise element into convex segments. An example of this is shown in Fig. 3. Each lengthwise segment would form a convex shape if two additional faces are created by projection to the x-z and y-z plane. By moving in a stepwise fashion from the 1st segment to the last, the total volume of the geometry can be found by summing the volume of each segment. If the height of subsequent nodal points for the next lengthwise segment is increasing, the volume is considered negative, otherwise it is positive (refer to Fig. 3(c)). Each convex volume segment can be found quickly using the popular Qhull algorithm12. Each shape is then volume normalized by:

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0

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x

(a)

y

z

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0

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0

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x

(b)

y

z

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0

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0

0.005

0.01

x

(c)

y

z 9th convexsegment (positive)

1st segment2nd segment

16th segment

1st to 3rdconvex segment

(negative)

Figure 3. Mesh of interpolated shape with weighting factor vector [-0.25 0 0.5]T, (a) Overall mesh, (b) quartered mesh, (c) quartered mesh with bounded lengthwise segments that are negative and positive involume.

( ) )(,

3/1,,

)(, / i

unnormjnormjunnormjinormj volvol nn = (4)

where is the normalized column vector of the Cartesian coordinate of the i-th nodal point for the j-th shape;

vol

)(,inormjn

j,norm is the specified normalized volume; volj,unnorm is the original volume of the shape; and being the original column vector coordinates of the original shape.

)(,iunnormjn

In this paper, the interpolated shapes that were explored were limited to those where the weighting factors belong to the set:

AIS ∩= (5)

where


4

( ) ( ){ }XwXwXwwwwIIII ∈∈∈=××= 321321321 ,,,, (6)

(7) { }175.05.025.0025.0−=X

A is itself is the set containing independent physically admissible weighting factors. The set I contains 6 x 6 x 6 = 216 elements, which results in 216 different shape configurations. Some weighting factor set elements are the same as others. For example a weighting factor vector [1 0 0]T is the same as [0.25 0 0]T due to volume normalization. Other shapes are automatically inadmissible such as those that have purely negative weighting factors or purely zero weighting factors. Therefore these configurations are not part of set A.


5

As mentioned previously, some of the shapes with negative weighting factors are also physically impossible. This is

due to the nature of the shape “folding” into itself. There are some shapes that are very difficult to create physically as well, such as those that appear to be highly “spiky”. Examples of physically impossible and “spiky” shapes are given in Fig. 4. These shapes tend to be those with negative weighting factors. Therefore, shapes with negative weighting factors are visually inspected and deleted from set S when deemed necessary.

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(a)

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0

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0

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0

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(b)

Figure 4. Mesh of infeasible interpolated shapes (a) w = [0.75 -0.25 -0.25]

III. DEM simulations

A. Simulation setup The 3-dimensional discrete element method used

here is based on the commercial software, Particle Flow Code in 3 Dimensions (PFC3D) 3.16. The reader is referred to Ref. 5 and Ref. 13 for a more detailed review of the method. A brief overview of the method and implementation is described in7,8. Basically, all the particles are assumed to be perfect spheres (although clumps can be bonded together to form irregular particles). The particles and walls are also assumed to be rigid (rigid in the sense the geometry does not warp for the purpose of calculations), and the particle displacement and contact area small relative to the particle sizes.

The equations of motion are applied for each particle, based on the resultant force and resultant moment on each particle. The Laws of motion are not applied to the walls in the simulation. The motion of walls however is explicitly controlled utilizing a specified wall velocity as an input.

The forces and moments acting on each element are based on the amount of overlap between elements (walls and particles) and the contact model used and also of any external field of force. The contact model chosen here was based on the simple combination of spring, viscous damper and Coulomb slider. This contact model is shown in Fig. 5. Each particle is also acted upon by a

T, “self-folding” (b) = [-0.25 -0.25 0.75]w T, overly “spiky”

Ball-Ball

Normal Direction

Ball-Ball

Shear Direction

m1

m2

m1

m1

m2

m 1

kn cn FCoulomb

cnkn

Ball-WallBall-Wall

ks

cs

FCoulomb

ks

cs

Figure 5. Contact models used for different collision cases.

gravitational force. The formulation of this model and the extraction of the parameter values from experiments have already been discussed in Ref. 14. The contact forces and equation of motion are applied iteratively until a specified number of time-steps.


6

The particles chosen to be simulated are that of 150 1.5 mm in diameter stainless steel bearings. This low particle count was chosen to reduce computational burden for a large number of simulations. The walls of the particle damper are set as those made with Perspex. The material

properties and the contact parameters are shown in Table 1.

Table 1. Contact models used for different collision cases.

Material and contact parameters

Having defined the material and contact parameters, the model for each shape configuration can be created. First of all, the Perspex walls are created. Since PFC 3D can only create flat walls that are coplanar, each of the 4 nodal point elements defined in the previous section was split into triangles. Particles are then randomly placed within the walls. In order to fill the casings with particles in a uniform manner, the particles are initially generated within a small section in the center of each shape with a significantly reduced size (refer to Fig. 6(a)). The walls are also

made rigid during this particle filling step. A gravitational setting for the model is also set so that the particles will settle in the negative x-axis (refer to Fig. 6(d)). This particle filling setup was done in order to exploit the repulsive forces between the particles so that particles can fit into the available volume within the casing as efficiently as possible. Position of the particles are updated using the equations of motion for 2000 time steps before increasing the size of the particle diameter by a factor of 20 percent. This process is repeated until the particles reach the specified diameter of 1.5mm. This stepwise process is shown in Figs. 6(b), 6(c) and 6(d). When the particles have reached the specified diameter, the particles are allowed to settle by gravity for 20000 time steps. The positions of

Figure 6. An example of a particle filling procedure (a) initial allocation of particle position and size, (b)

increased particle diameter as the simulation is running, (c) further diameter increase, (d) final configuration showing excitation and gravity direction

Wall yield stress 75.9 x 106 Pa Wall Poisson’s ratio 0.35

Particle count 150 Particle density 7800 kg/m3

Particle yield stress 345 x 106 Pa Particle Poisson’s ratio 0.3

Coefficient of restitution for all contacts 0.92 Coefficient of friction for all contacts 0.4

gravity excitation

all the particles are then recorded; it will be loaded up for all subsequent runs associated with each shape in order to speed up the simulation process.


7

It is noted that even if all the shapes have the same volume, the effective volume for particle movement is essentially a function of shape and particle size and distribution. For example, shapes that have sharp corners and is filled with large particles would have a much lower effective volume than if it was filled with smaller particles. In Fig. 7, it can be seen that for a “spiky” shaped wall, it is much harder to fit the same number of particles of the same size within. For shapes where the particles cannot fit without having high contact forces, these shapes were considered physically inadmissible and were not simulated. After this process of filtering, only 154 different shapes were considered for simulation.

B. Simulation implementation All the different shapes were simulated with a sinusoidal excitation

over a range of amplitudes and frequencies. The direction of excitation is perpendicular to the direction of gravity (refer to Fig. 6(d)). The excitation is implemented by a velocity input at the walls given by:

Figure 7. A spiky shaped

configuration. Particles can barely fit within the walls, as the space in

between the spikes are not filled.

)2sin()( ftVtV amp π= (8)

f

V ampamp π2

=A81.9

(9)

where t is the time, V is the velocity, V is the velocity amplitude, is the acceleration amplitude given in the unit measure of g-force (varied between 5 g to 17 g in steps of 4 g) , f is the frequency of excitation (varied between 25 Hz to 145 Hz in steps of 40 Hz). All simulations were executed for 20000 time steps to remove transient effects and data only collected for 40000 time steps, which is assumed in steady state. It is in fact possible to retrieve a whole range of data from simulation, which includes but not limited to the energy trace in terms of kinetic, potential, strain, friction and viscous losses. For the moment however, the data collected from all simulations has

been constrained to only the force acting on the particle assembly by the walls in the excitation direction. This is to

)(t amp ampA

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055

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0

0.1

Time trace of force and velocity

Time (s)

Vel

ocity

(m/s

)

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055

-5

0

5

Forc

e (N

)

Figure 8. The time trace for the force and velocity for a shape with w = [0.5 0.25 0.75]T, excited at 5g and

65 Hz.

replicate the data that would have been most easily extracted from most physical experiments. A typical plot of the force and velocity time trace for a given shape is shown in Fig. 8. It can be seen that the force is not continuous, but

rather a series of random impulses that follows a trend of a sine wave. Therefore, in reality the signal is never strictly periodic.

8American Institute of Aeronautics and Astronautics

Structures tend to be modeled in finite element with a high degree of complexity. In those cases, it is highly desirable to represent the particle damper within the finite element model in a simple manner. One of many possible representations for a particle damper when attached to a SDOF system can be seen in Fig. 9. It can be seen that the particle damper acts almost like a combination of a tuned mass damper (where energy is transferred away from the structure that is to be protected into the kinetic energy and strain energy of the mass damper) and a skyhook damper (where damping is dependent on absolute motion instead of relative motion). In the case of the particle damper, the strain and kinetic energy is assumed trapped as kinetic energy only. By using this representation, the equivalent mass and viscous damping coefficient can be extracted for a range of frequencies and amplitudes. It is noted that this representation is not physically correct as the real particle damper dissipates and stores energy based on random collisions that may significantly be out-of-phase with the excitation. Nevertheless, it allows easy interpretation of the performance of the damper.

The equivalent viscous damping and equivalent mass can be calculated based on a form of energy balance. From the equation of motion on the particle damper:

csky

m2

m1

c2k2

Particle Damper

F

Figure 9. Lumped parameter model of

particle damper attached to a SDOF system.

(10) )()()( tVmtVctF eqeq&+=

where F(t) is the force calculated from the DEM simulation. The instantaneous power is therefore:

( ) )()()()()( 2 tVtVmtVctVtF eqeq&+= (11)

Substitution from Eq. (8) into Eq. (11) yields:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−= )2sin(

22cos(1

2)()(

22

tVm

tVc

tVtF ampeqampeq ωω

ω (12)

where ω = 2πf. The first term on the right hand side of Eq. (12) is the instantaneous power dissipated which is always positive. The second term on the right hand side is the power that is stored in the system, where a negative value represents stored power that is returned back to the source of power. The average power provided to the particle damper over a time period can be found by integration of Eq. (12):

( ) dttVm

tVc

ttdttVtF

ttampeqt

t

ampeqt

t ⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−=

− ∫∫ )2sin(2

2cos(12

1)()(1 22

0101

1

0

1

0

ωω

ω (13)

[ ][ ]1001

,|)()()()(1 1

0

ttttVtFEdttVtFtt

t

t==

− ∫ (14)

where E is the expectation operator, and t0 and t1 are the starting and ending time for a set of data captured respectively. If (t1- t0) = 1/(2f), the equivalent damping coefficient can be found from the application of Eq. (13) and Eq. (14). It must be noted however, that due to the stochastic nature of the force, the average power would vary from one time period to the next within the dataset. In order to take account of this, the average power is calculated by averaging over multiple windows with the width of (t1- t0) that slides over the entire dataset:

[ ][ ] [ ][ ]2

10 ,|,|)()(2

amp

endstarteq V

tttttttVtFEEc === (15)

where tstart and tend are the starting and ending points of the dataset respectively. A plot of the power decomposition with the sliding window is shown in Fig. 10.


9

By replacing the equivalent damping from Eq. (15) into Eq. (10), the equivalent mass for every instance of time can be calculated. An example of this is shown in Fig. 11(a). The probability density of the mass for the entire dataset is given in Figs. 11(b), 11(c) and 11(d). It is noted that there are negative values for the mass, giving further evidence that the simplified model is not physically correct. From the probability density, it was found that the mode is close to zero, though the mean of the mass is much higher due to bias to the positive extreme values. The equivalent mass can be selected based on the mean of this distribution, although a different method is utilized here based on energy equalization. From Eq. (12):

0.035 0.04 0.045 0.05

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0

0.05

0.1

0.15

0.2

0.25

0.3

Time (s)

Pow

er (W

atts

)

Time trace of power

Stored powerDissipated powerInstantaneous power

Sliding window

Figure 10. The time trace for power for a shape with w = [0.5 0.25 0.75]T, excited at 5g and 65 Hz.

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= )2sin(

22cos(1

2)()()(

22

tVm

tVc

tVtFtP ampeqampeqstored ω

ωω (16)

( ) ( )2222

2 )2sin(2

2cos(12

)()()(⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= t

Vmt

VctVtFtP ampeqampeq

stored ωω

ω (17)

Taking expectation on both sides of Eq. (17) and utilizing the sliding window method mentioned previously, the equivalent mass can be calculated as:

[ ][ ] [ ][ ]

210

2 ,|,|))((8

amp

endstartstoredeq V

tttttttPEEm

ω==

= (18)


10

By replacing the equivalent mass and equivalent viscous damping calculated into Eq. (10), the force measured from the DEM simulation and the simplified model can be compared. This is shown in Fig. 12. It can be seen that the smoothed force from the model lags the visually identified sinusoidal trend of the DEM calculated force. Of course one could attempt to match the trend of the phase of DEM calculated force by a form of least squares of Eq. (10) (although there is no guarantee that the coefficients will be positive). This is also given in Fig. 12. Note that due to the random nature of the force changing directions within the period of half a cycle, the least squares method will also underestimate the maximum forces. Suffice to say, any continuous (in terms of force values over time) surrogate model built from the calculated DEM forces will never be physically valid for all cases and should be used with care when integrated to any larger model.

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Time (s)

Mas

s (k

g)

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Pro

babi

lity

dens

ity

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20

40

60

80(c)

Mass (kg)0 5 10

0

0.2

0.4

0.6

0.8(d)

Figure 11. Shape configuration with w = [0.5 0.25 0.75]T, excited at 5g and 65 Hz. (a) The time trace for

equivalent mass based on force (b) probability density of equivalent mass at negative extremes, (c) probability density of equivalent mass centered at 0 kg, (d) probability density of equivalent mass at positive extremes.

IV. Simulation Results Fig. 13 and Fig. 14 show the spread of the different equivalent damping coefficient and mass respectively for all

154 shapes over the range of frequencies and amplitudes. It seems to suggest that there is a significant spread in

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-2

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0

1

2

3

4

Time (s)

Forc

e (N

)

Force(N) vs Time(s)

Force from least squaresForce from DEMForce from energy balance

Figure 12. Force-time plot comparing the force from the different calculation methods to the force

extracted from DEM

data, indicating the shapes chosen has a strong role in the design of particle dampers. Nevertheless, it remains to be seen whether this spread is caused by the random nature of the data and whether the performance of each shape is qualitatively different. Qualitative changes are significant as it shows that different operating conditions can be targeted for vibration suppression based on the application. Even if there is some deviation from one shape to the next due to randomness, there should be some underlying structure for each shape due to these qualitative properties.

4060

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5

10

15

-2

0

2

4

6

8

Frequency (Hz)

Equivalent damping coefficient (N/ms-1) vs amplitude (g) vs frequency (Hz)

Amplitude (g)

Figure 13. Spread of equivalent damping coefficient for varying operating conditions and shape

configurations.


11

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120140

5

10

15

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)

Equivalent mass (kg) vs amplitude (g) vs frequency (Hz)

Amplitude (g)

meq

(kg)

Figure 14. Spread of equivalent mass for varying operating conditions and shape configurations.


12

By plotting the correlation coefficients (1 indicating strong positive linear dependency, 0 as no dependency and -1 strong negative linear dependency) between the shapes for equivalent damping and mass (Fig. 15 and Fig. 16 respectively), it can be seen that there is indeed some pattern. Since the equivalent mass is almost the same for all shapes (bar a few shapes which will be discussed), it is useful to actually classify the performance just by looking at the equivalent damping. By exploration of the “patch” areas on the upper triangle of the correlation coefficients, it can be surmised that there is in fact roughly five classes of behavior present by varying the shapes:

1. Damping peaks at low frequencies for a broad range of excitation amplitude and over a high frequency for low excitation amplitude (refer to Fig. 17)

2. Damping peaks at very frequencies and for the broad range of amplitudes. This mostly occurs for dampers with very large aspect ratios. (refer to Fig. 18)

3. Damping peaks at around frequencies of 105 Hz at high amplitudes. (refer to Fig. 19) 4. Damping peaks at around frequencies of 65 Hz at high amplitudes. (refer to Fig. 20)

Correlation coefficients of equivalent damping between shapes

Shape number

Sha

pe n

umbe

r

20 40 60 80 100 120 140

20

40

60

80

100

120

140

-1

-0.5

0

0.5

1Correlation coefficients of equivalent mass between shapes

Shape number

Sha

pe n

umbe

r

20 40 60 80 100 120 140

20

40

60

80

100

120

140

-1

-0.5

0

0.5

1

Figure 16. Correlation coefficients of equivalent

mass Figure 15. Correlation coefficients of equivalent

damping

5. Damping performance is roughly similar for all frequencies and amplitudes (refer to Fig. 21).


13

Figure 17. (a) Equivalent damping, (b) equivalent mass, (c) weighting factor and shape




14



The negatively correlated equivalent mass seen in Fig. 16 could have been caused by data insufficiency, leading

to equivalent masses that are averaged over too short a time period to smooth out the impulses. Such a shape configuration is shown in Fig. 22, where there is a huge sudden peak and dip in the equivalent mass and damping respectively. This would hopefully be alleviated in the future by increasing the simulation time for longer periods.

Although the performance has been classified, nothing conclusive has been said on how specific shapes itself relate to the performance. The only clear thing from the results so far was that having a very large aspect ratio tends to produce particle dampers that work well over high frequencies. This needs to be investigated in the future by measuring the energies of each particle directly in the time domain, for both dissipative and stored energy. This is easily done in a simulation, although measuring it in physical experiments might prove difficult. It would however give us further insight on how energy is dissipated in the assembly locally, rather than just looking at the energy trace of the whole assembly as a single value in time. Nevertheless, it has been shown that tuning the shapes does play a strong effect in affecting the qualitative performance of particle dampers.

Although the surrogate model leads to easy interpretation, it can only be used for model integration as a submodel in a larger more complicated model. It does not give us much information on how to create better and new particle dampers, as the local interactions between particles is not captured just from the force on the walls. Even


15

approximating the force continuously is not accurate as shown in the previous section. A better approach for surrogate models in the future is to include some probabilistic treatment of the forces. A possible approach is to use a Gaussian process


15 with a periodic mean and covariance function to predict the forces, although this approach does not lead to any physical interpretation.

V. Conclusion The methods described in this paper can be considered part of a framework to select a particular shape for a

particle damper. It is shown that it works, albeit more work needs to be done to improve some of the shortcomings mentioned previously. In terms of a physical application, the volume that one can work with is probably constrained within a certain dimension. This can be explored in the future by some form of a constrained optimization. It is also possible to control the performance in real time by morphing the walls, possibly by the application of shape memory alloys16. A more general framework still needs to be decided upon on how to combine particle size, distribution, material, volume and placement to create the optimal particle dampers. The approach mentioned in this paper also deals with only one frequency of excitation. Particle dampers however are proven to damp well over a broad range of frequencies. This will be explored in the future by coupling particle dampers with multiple-degree-of-freedom systems. Needless to say, the field of particle dampers remains an exciting and challenging field of research with plenty of potential.

Acknowledgments C.X. Wong is funded by EPSRC contract EP/D078601/1.

References

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2Simonian, S., Camelo, V., Brennan, S., Abbruzzese, N., and Gualta, B. "Particle damping applications for shock and acoustic environment attenuation," 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. Schaumburg, IL, 2008.

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7Wong, C. X., Daniel, M. C., and Rongong, J. A. "Prediction of the amplitude dependent behaviour of particle dampers," 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference. Hawaii, 2007.

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10Fang, X., Luo, H., and Tang, J. "Investigation of Granular Damping in Transient Vibrations Using Hilbert Transform Based Technique," Journal of Vibration and Acoustics Vol. 130, No. 3, 2008, p. 11 pages.

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12Barber, C. B., Dobkin, D. P., and Huhdanpaa, H. T. "The Quickhull algorithm for convex hulls " ACM Trans. on Mathematical Software Vol. 22, No. 4, 1996, pp. 469-483.

13Itasca Consulting Group, I. UDEC (Universal Distinct Element Code) Version 4.0. . Minneapolis:ICG, 2004. 14Wong, C. X., Daniel, M. C., and A., R. J. "Energy dissipation prediction of particle dampers," Journal of Sound and

Vibration Vol. 319, No. 1-2, 2009, pp. 91-118. 15Rasmussen, C. E., and Williams, C. K. I. Gaussian Processes for Machine Learning: The MIT Press, 2006. 16Hartl, D. J., and Lagoudas, D. C. "Aerospace applications of shape memory alloys " Proceedings of the Institution of

Mechanical Engineers, Part G: Journal of Aerospace Engineering Vol. 221, No. 4, 2007, pp. 535-552.

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