Powder Technology 142 (2004) 154 – 165
www.elsevier.com/locate/powtec
DEM simulation of particle damping
Kuanmin Mao a, Michael Yu Wang b,*, Zhiwei Xu c, Tianning Chen d
a
College of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, China
Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong, China
c
The State Key Laboratory for Smart Materials and Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China
d
College of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China
b
Received 9 July 2003; received in revised form 22 April 2004; accepted 22 April 2004
Available online
Abstract
Particle damping is a technique of providing damping with granular particles embedded within small holes in a vibrating structure.
Particle-to-wall and particle-to-particle collisions arise under the vibrating motion of the structure. As a result, the structure and the particles
will exchange momentum and thus dissipate kinetic energy due to frictional and in-elastic losses. The particle damping technology has been
used successfully in many fields for vibration reduction. However, it is difficult to predict the damping characteristics due to complex
collisions in the dense particle flow.
In this paper, we utilize the discrete element method (DEM) for computer simulation of particle damping. By considering thousands of
particles as Hertz balls, the discrete element model can describe the motions of these multi-bodies and determine the energy dissipation. We
describe a DEM modeling system with an efficient collision detection algorithm for large-scale particle problems. The simulation system is
validated by comparing with experimental results of a physical system. The DEM simulation system is further examined with examples for
its computational complexity and effectiveness for different density parameters. It is concluded that the particle damping is a mix of two
damping mechanisms of impact and friction. It is further shown that the relative significance of these damping mechanisms depends on a
particular arrangement of the damper.
D 2004 Elsevier B.V. All rights reserved.
Keywords: DEM; Impact; Friction; Particle damping; Vibration
1. Introduction
Passive vibration damping is often a preferable solution
in most industrial applications of vibration control for the
unconditional stability, robustness, and cost effectiveness
[1]. Conventional passive damping techniques include viscoelastic material applications, frictional devices, impact or
fluid dampers, tuned masses, and isolators. While viscoelastic materials are typically considered to be effective,
viscoelastic materials, however, have a tendency to degrade,
embrittle, and even disintegrate with time and in temperature changing environments.
Particle vibration damping (PVD) is a combination of
impact damping and friction damping [2,3]. It is a rela* Corresponding author. Department of Automation and ComputerAided Engineering, The Chinese University of Hong Kong, Shatin NT,
Hong Kong. Tel.: +852-2609-8487; fax: +852-2603-6002.
E-mail address: yuwang@acae.cuhk.edu.hk (M.Y. Wang).
0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.powtec.2004.04.031
tively simple concept where metal or ceramic particles or
powders of small size ( f 0.05 – 0.5 mm in diameter) are
placed inside cavities within or attached to the vibrating
structure (see a schematic of Fig. 1). In contrast to
viscoelastic materials which dissipate the stored elastic
energy, particle damping treatment focuses on dissipation
of the kinetic energy. Particle damping involves the potential of energy absorption and dissipation through momentum exchange between moving particles and vibrating
walls, friction, impact restitution, and shear deformations.
It is an attractive alternative in passive damping due to its
conceptual simplicity, potential effectiveness over broad
frequency range, temperature and degradation insensitivity,
and very low cost [2– 7].
Particle damping is a derivative of single-mass impact
damper that has been thoroughly studied on the influence of
mass ratio, particle size, particle/slot clearance, excitation
levels, and direction of excitation [8,9]. In the single-mass
case, direct analyses exist and reveal design criterion for
K. Mao et al. / Powder Technology 142 (2004) 154–165
155
Fig. 1. A particle vibration damper (left) and its body contacts (right).
optimal efficiency based on reduction in system response. It
is observed that a plastic ‘‘bean bag’’ filled with lead shot
exhibited much greater damping effectiveness and ‘‘softer’’
impacts than a single lead slug of equal mass [6,8]. There
have been some research efforts toward analytical or numerical models of the complicated phenomenon of multi-grain
particle damping [4,6]. However, this has been considered as
a specialty area and the literature is relatively scarce.
It is generally understood that a dominant mechanism of
particle damping is in the form of momentum transfer which
occurs when the particles impact the cavity walls of the
hosting structure [3,6]. The non-elastic impacts (or restitution) result in energy dissipation. This is similar to a singlemass impact damper, and the analogy permits the use of
impact damper theory to roughly size the particle damper.
However, in addition, particle damping dissipates energy
due to inter-particle and particle – wall friction [6]. The
relative effectiveness of these mechanisms changes based
on a host of parameters such as particle size, density, and
shape, frequency and amplitude of vibration [6,11]. Although the initial test results of the early 1990s substantiated
the potential of particle damping [2,3,5], applications in the
literature have been largely based only on heuristic guidelines [3]. Due to the complex interactions involved in
particle damping, a comprehensive analysis methodology
is not yet available [6].
Because of the very large number of particles in a typical
particle damper, it appears that applications of concepts
from particulate technologies on a particle-to-particle basis
would offer the best analysis and design tools. In fact,
molecular dynamics simulation (MDS) methods have been
attempted to capture motions of particulates [10,14,15] and
the technique appear promising in characterizing the particle
damping behavior, although it typically is applied to rapid
granular flows (or gas– solid flows [16]) involving moderate
computational complexities.
Recently, a discrete element method (DEM) has been
used to perform limited studies of particle dampers [5,11].
This is a technique pioneered by Cundall and Strack [12] for
the simulation of behavior of granular materials. The procedure is an explicit process with small time step iterations
to determine resultant forces on any particle in the system.
Over the years, the discrete element method has found a
wide range of applications in various disciplines (see Refs.
[15,20 –22]). In this paper, we describe a simulation approach to modeling particle vibration dampers (PVDs) with
the discrete element method. We have developed an efficient
procedure for the dynamics of a large number of particles
without excessive computing complexity. This allows us to
perform large-scale simulations with realistic particle numbers (e.g., 1000) on highly reasonable time scales. This
capability is especially important for designing PVDs for
applications on a lightweight structural component such as
wire/die bonder head arm [13]. It will facilitate the development of minimally invasive damping treatment with the
use a minimal amount of damping particles and without any
significant impact on the strength, stiffness, weight, and
dynamic characteristics of the structure itself.
2. The discrete element simulation
Discrete element simulation is a numerical scheme that
involves following the motion of every particle and its host
structure of a system, where each particle – particle and
particle –structure contact is detected. Over small-step iterative computation cycles, complete loss of contacts, and
formation of new contacts between the bodies take place.
The contact forces resulting from each contact of a pair of
bodies are calculated based on a contact force model. DEM is
uniquely suited for the particle vibration damping problem.
2.1. The particle vibration damper
The physical system of a simple particle vibration
damper consists of a structural body with a vertical hole
partially filled with metal particles of small size as shown in
Fig. 2. The structural body is modeled as a single degree of
freedom (SDOF) system with mass M, stiffness K, and
intrinsic (e.g., material) damping coefficient C. The hole
has diameter of D and a depth of H. The particles are
typically made of a material of high density, for example,
tungsten steel, for a better damping performance. The
particles are often described as spheres for simplicity in
discrete element simulation. The diameter of the spherical
particles is typically less than 1/5 of the hole’s diameter and
is usually in the range of 0.05 – 5 mm. Thus, it is not
uncommon that a single hole may contain a large number
of particles in an order of 1000 or even 10,000.
2.2. Contact forces
As described in the Introduction, the fundamental
mechanisms of the particle damping directly involve phys-
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K. Mao et al. / Powder Technology 142 (2004) 154–165
Fig. 2. The model of a single degree of freedom system of a particle
vibration damper.
ical contacts between particles and between particles and
the cavity walls. In the DEM simulation, any contact
between a pair of bodies (spheres and walls) is considered
to be a distinct single-point contact (see Fig. 1). The
bodies are considered to be rigid, however, the contacts
are deformable. Every contact, either sphere – sphere or
sphere – wall contact, is modeled using a contact law
[16,17]. Selection of an appropriate contact model for
inter-particle and particle –cavity impact and friction interactions is an important aspect of the modeling technique
[20]. The simplest model is a linear model that uses a
combination of a spring and a dashpot in both the normal
and the shear directions. However, a linear model is
known to be inaccurate in predicting the correct force –
displacement relationship [20,21]. Various nonlinear models have been developed in the literature [16,17,20,21] and
it is found that the nonlinear impact theory of Hertz is
adequately suitable to metal – metal impact [16,21]. In this
paper, the Hertz theory is used to describe the particle
interaction.
The contact model used in this work is described by a set
of force – displacement relationships
fn ¼ kn ðdn Þ þ cn ḋn
ð1Þ
fs ¼ ks ðds Þ þ cs ḋs ðif stickingÞ ¼ lfn ðif slidingÞ
ð2Þ
for the normal and tangential contact forces fn and fs,
respectively, where dn and ds are the contact distances in
normal and tangential directions, respectively. Small damping coefficients cn and cs are used for energy loss in the
normal and tangential directions, respectively, while frictional dissipation is described by Coulomb’s friction coefficient l [15]. For metal – metal contact for spherical
particles, Hertz theory of elastic contact yields the following
normal force –displacement relationship
kðdn Þ ¼
4
1 3
ER 2 dn2
3
ð3Þ
Fig. 3. A schematic of a particle vibration damper and its experimental
setup.
where E is the equivalent elastic modulus and R is the
equivalent contact radius of curvature. For convenience, we
1
refer to kn ¼ 43 ER2 as the contact stiffness [17].
2.3. Particle and structure motions
In the DEM simulation, the motions of each particle and
the host structure are calculated through a cycling process of
small time steps Dt. As the structure vibrates causing the
walls of the hole to oscillate, each particle may collide with
many other particle as well as the walls. All of these collisions
will dissipate energy from the system, thus providing a
damping effect on the host structure. The DEM makes the
assumption that any body is affected only by its current
contacts. This simplification substantially reduces the number of interactions per body that has to be resolved in order to
update its motion. A calculation cycle proceeds as follows.
First, the moving walls of the structure develop contact
forces on each particle contact them. At each of the contacts,
the normal and tangential components of the contact force
are updated. The net force and moment on each particle
involved is determined from all forces applied on the
particle. Then, the incremental linear and angular accelerations of the particle are determined from the unbalanced
force and moment acting on it. Using a central difference
scheme, the linear and angular equations of motion of the
particle can be integrated to obtain the incremental linear and
Table 1
Parameters of the PVD system in DEM simulation and experiments
System parameter
Mass M
Stiffness K
Damping C
Hole diameter D
Hole height H
Simulation parameter
0.0376 kg
470 N/m
0.1 N s/m
25.4 mm
7.04 mm
Normal stiffness kn
Normal damping cn
Tangential stiffness ks
Tangential damping cs
Friction coefficient l
Sphere diameter d
Sphere number n
360.0 kN/m
0.01 N s/m
330.0 kN/m
0.015 N s/m
0.55
0.88 mm
1246
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K. Mao et al. / Powder Technology 142 (2004) 154–165
Table 2
Parameters of the PVD system in DEM simulation
System parameter
Mass M
Stiffness K
Damping C
Hole diameter D
Hole height H
Fig. 4. DEM simulation results of free vibration velocity responses with and
without particles.
angular displacements. Therefore, in the selected Dt step, the
structure walls as well as all of the particles of any contact are
moved accordingly. As a result, in each time step, new
contacts are generated and some of the existing contacts
are broken. The cycle of contact calculation is repeated for
the next time step. This incremental updating scheme was
fully described in Ref. [12], and it has been widely tested in
the literature [20 – 22], showing that it is reasonably accurate
and stable when the time step Dt is properly chosen. For
brevity, we shall omit the details of its formulation here.
2.4. Contact detection and tracking
The DEM requires the determination of the contacts
made by a particle with another particle or wall at any step
in time. If a collision detection technique has to trace the
path of each moving particle to perform contact checks with
Fig. 5. The specific damping capacity versus the dimensionless acceleration, obtained with this simulation study and the experiments in Ref. [4],
respectively.
Simulation parameter
0.0376 kg
1410 N/m
0.1 N s/m
10 mm
24.64 mm
Normal stiffness kn
Normal damping cn
Tangential stiffness ks
Tangential damping cs
Friction coefficient l
Sphere diameter d
Sphere number n
360.0 kN/m
0.01 N s/m
330.0 kN/m
0.015 N s/m
0.55
0.88 mm
1246
all other bodies regardless of actual collisions, this would be
extremely time-consuming when the number of particles is
large. The success of the DEM simulation for a large-scale
problem such as the particle damping simulation depends on
an efficient contact detection scheme.
To keep the virtual collision check for all pairs of n
bodies in space, we need a calculation of computational
complexity O(n2) at each step. Fortunately, there exist a
number of efficient algorithms developed over the years in
various disciplines (see, e.g., Refs. [23,24]). For hard sphere
collision models, the most promising algorithm is the socalled ‘‘box’’ algorithm which has an O(log n) complexity
on average per time step and the overall simulation complexity of O(n log n) [23]. This is the algorithm used in our
implementation.
The basic idea of the box algorithm is a subdivision of
the space containing the sphere particles into a 3D grid of
cubic cells. Within each cell, one or more spheres can be
accommodated to move. The space subdivision lends itself
to a method of localized contact detection method to
determine nonempty cells intersecting one or more spheres.
By keeping track of the path of every moving sphere and the
list of the spheres intersecting each cell, the occurring times
of the contacts can be calculated. The approach requires
efficient data structures for keeping the nonempty cells and
for handling the collision events [24]. For the large-scale
problem of particle damping, this algorithm allows us to
Fig. 6. The displacement of the system mass obtained with the DEM
simulation.
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K. Mao et al. / Powder Technology 142 (2004) 154–165
avoid the contact detection process becoming the dominant
cost of the simulation.
3. Simulation of particle damping
In this section, we present the simulation results of a
single degree of freedom system with a particle vibration
damper by embedding metal particles in a closure attached to
the end of a beam. Experimental tests of this system were
reported in Ref. [4] with an experimental setup as illustrated
in Fig. 3. Both the enclosure and the beam are made of 3003
aluminum alloy with Young’s modulus E of 70 GPa and
density of 2.7 g/cm3. The mass of the enclosure is 15.2 g, and
the diameter of its interior hole is 2.54 cm. Lead particles of
4.0 g were filled in the hole, resulting in a filling volume ratio
of 25%. The beam has a length of 25.4 cm, width of 3.18 cm,
and height 0.23 cm. For the motion at the free end of the
beam, the continuous beam is reduced to an equivalent single
degree of freedom system with a natural frequency of 17.80
Hz when without the particles. The intrinsic material damping is measured as of a damping ratio of 0.12% [4]. The
parameters of the system for the experiments in Ref. [4] and
for our DEM simulation are listed in Table 1.
Fig. 4 shows a DEM simulation result of the free vibration
velocity of the mass enclosure for an initial condition of
displacement of 15.7 mm and zero velocity. Velocity history
for the case of the same system but without the particles is
also plotted. It clearly shows a dramatic attenuation of the
transient vibration due to the presence of particles.
Comparing this simulation result with the experimental
data of the same system presented in Ref. [4], we conclude
Fig. 7. Snapshots of the particles during the vibration of the damper.
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K. Mao et al. / Powder Technology 142 (2004) 154–165
that the DEM simulation agrees with the experiments very
well. Since we are not able to directly reproduce the results
of [4] for a direct comparison, the reader is referred to Fig. 4
in that paper for the experimental results. Specifically, the
experiments showed that the addition of particles would
change the vibrating frequency of the system as predicted
given their mass and the damping effect. The first natural
frequency of the beam system without particles is at 17.80
Hz as predicted by the theoretical calculation. With the
particles added into the damper attached to the end of the
beam, the frequency was measured at 17.71 Hz [4]. A change
in the frequency is also obtained by our DEM simulation as
clearly shown in Fig. 4 between the calculated responses for
the cases of with and without particles. The DEM prediction
of the damped vibrating frequency is 16.88 Hz.
While the effect of the particle damping is clearly shown
in Fig. 4, the particle damping is highly nonlinear. Especially it depends on the amplitude of the vibration. Therefore, it is more appropriate to use the specific damping
capacity to define the damping effect [4]. In the context of
particle damping, the specific damping capacity is defined
as
w ¼ DT =T
ð4Þ
where DT is the kinetic energy converted into heat during
one cycle of vibration, and T is the maximum kinetic energy
during the cycle. The nonlinear nature of the damping is
more evident when the specific damping capacity wi is
shown as a function of the effective acceleration for each
vibration cycle [4]. In Fig. 5, the computed results of
specific damping capacity are given along with the experimental results reproduced from [4]. It is also convenient to
define another dimensionless parameter, effective acceleration C, as
C ¼ Ax2 =g
ð5Þ
with amplitude A and frequency x of the periodic vibration
and gravity g [4].
Fig. 9. The computing time for two different contact checking algorithms:
with subdivision and without subdivision.
The DEM simulation reveals an important characteristic
of particle vibration damping. The specific damping capacity depends highly on the dimensionless acceleration amplitude. Roughly speaking, there are three regimes [7,25].
At higher effective acceleration C from the beginning of the
vibration, the specific damping is relatively low. The specific damping increases while the acceleration amplitude is
reduced. Following Fig. 5 to yet smaller amplitudes, we find
that a maximum in the specific damping capacity is reached.
Around this point, the particle damping is remarkably high,
reaching a highest level of specific damping capacity of
approximately 40%. This high damping effect then reduces
the acceleration amplitude rather quickly down to C c 1,
i.e., when the maximal acceleration of the structure becomes
comparable with gravity g. The particle damping effect
nearly ceases to exist for C V 1, at which the acceleration
of the structure is not sufficient enough for the particles to
overcome the gravity to have any collisions.
4. Simulation performance analysis
In this section, we present the performance of our DEM
simulation system under varying particle number and density conditions in the particle motion simulation. Our
numerical experiments focus on the computational time of
the DEM simulation in contact detection and overall computation. The analysis results for various simulations described in this section were obtained on a personal computer
with a 1.70-GHz Intel Pentium processor.
We first consider a similar single degree of freedom
system with parameters given in Table 2. The particle
Table 3
Computing (execution) time (h) for different cell size
Fig. 8. The number of contacts and the computing time for every time step.
Cell size
Computing time (h)
1.5 d
5.14
3d
14.1
Without subdivision
25.08
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K. Mao et al. / Powder Technology 142 (2004) 154–165
Fig. 10. The computing time versus the particle number.
damper has a total of 1246 particles with a volume packing
density of 50%, representing a system of dense flow inside
the container hole.
An initial condition of displacement of 15.7 mm and zero
velocity is given to the system, and the motion of the system
is simulated for 1.5 s as shown in Fig. 6. The displacement
plot of the structure mass is similar to that of the example in
the previous section of experimental validation. The damping effect of the particles is clearly shown to be significant.
In Fig. 7, a number of snapshots are given to show the
motion of the structural mass and the particles contained in
its hole. These snapshots illustrate the approximate damping
regimes during the course of free vibration [7]. In the first
regime approximately between 0.0 and 0.05 s, the damper
container walls impact the particles transferring momentum
to the particles. The second regime immediately follows the
first region and lasts until 0.24 s. This regime has the most
significant damping effect. The DEM simulation reveals that
Fig. 11. The computing time versus the hole’s size.
K. Mao et al. / Powder Technology 142 (2004) 154–165
Table 4
Parameters of the particle damping system for damping analysis
System parameter
Mass M
Stiffness K
Damping C
Hole diameter D
Hole height H
Packing ratio p
Simulation parameter
0.0134 kg
Mx2n
0.0875 N s/m
10 mm
12 mm
95%
Normal stiffness kn
Normal damping cn
Tangential stiffness ks
Tangential damping cs
Friction coefficient l
Sphere diameter d
Sphere number n
300.0 kN/m
0.02 N s/m
270.0 kN/m
0.018 N s/m
0.55
1.2 mm
386
the particles undergo a large number of collisions with
significant relative speed, thus dissipating mechanical energy through both impact and dry friction. After this regime,
the particles are nearly at rest relative to the container mass
and little damping effect is observed.
A time step Dt = 2.5 10 6 (s) is used in the DEM
discrete integration scheme. This example represents a
system of dense particle flow and the flow is highly
transient. The motions of the particle relative to the structural mass last for less than 1 s. Within each time step of
iteration, there are often substantially a large number of
collisions between particles and between particles and the
walls. Fig. 8 depicts the number of contacts for every time
step. Within the period of 1.5 s, there are 1571 contacts on
average. This phenomenon of dense particle flow is very
different from the result in a dilute granular – gas flow [18].
Furthermore, after around t = 1.0 s when the particles nearly
come to rest with respect to their container hole, the
computational situation becomes extremely intense. The
state of stationary contacts for nearly all bodies is represented in the DEM scheme by a resolution of repeated
contacts at a high concentration in time.
Fig. 8 also shows the total computing time for updating
the motions of all bodies of the vibration system for every
time step. The total computing time is directly proportional
to the number of contacts detected within the time step.
Therefore, the localized contact checking algorithm described in the previous section is an important element for
the efficient implementation of the DEM simulation.
The benefit of the localized contact detection algorithm is
further illustrated in Fig. 9. For the simulation results shown
in Fig. 7, the space of the container hole is divided into
multiple cubic cells in the ‘‘box’’ algorithm with the side
length of the cell equal to three times the sphere radius (i.e.,
l = 1.5d). As expected, without using the cell subdivision,
the computing time of each time step would be much longer.
In Fig. 9, the computing time is compared with that of the
contact detection algorithm without using the cell subdivision, plotted for every time step of the DEM simulation.
The computational efficiency is directly proportional to
the size of the subdivision cell. As discussed in Refs.
[23,24], the primary benefit of the subdivision cells is that
they limit the number of collision calculations required for
any given particle. A specific particle will be checked only
with other particles in the same cell and in the cells adjacent
(27 in total for three dimensions). All particles in nonadja-
161
cent cells are ignored [24]. A drawback of this time-saving
method is that collisions from particles or walls more than
one cell away are not accounted for, thus resulting approximation in contact detection and motion calculation. Therefore, a trade-off has to be balanced. In our experience, the
cell length l = 1.5d gives an acceptable level of accuracy
while yields a practically reasonable computational complexity. Table 3 shows the total execution time for the above
example for three different sizes of subdivision cells including without subdivision.
Finally, we illustrate the computational complexity of the
DEM simulation with respect to ‘‘particle density’’ of the
damper. The density is simply defined as n/Vh, where n is
the number of particles and Vh is the volume of the container
hole. The same example of Table 2 is used here. In Fig. 10,
we keep the hole’s size and the total mass of the particles the
same while changing the number of particles and the sphere
diameter. The total execution time and the mean number of
collisions per time step are plotted for a number of cases of
300 – 1300 particles. As the number of particles increases,
the computational effort clearly increases approximately
linearly. This indicates that our algorithm behaves with a
computational complexity of O(log n) per time step as
predicted. Next, we keep the number of particles at 936,
and vary the diameter of the container hole between 10 and
25 mm. As shown in Fig. 11, the execution time and the
mean number of contacts would decrease as an increase in
the hole’s diameter decreases the particle density in a
quadratic inverse relation. This illustrates the said computational complexity of our simulation.
5. Distribution of energy dissipation
In this section, we further examine the relative importance of the two energy dissipation mechanisms under a
Fig. 12. Various cumulative energy dissipations for 95% packing ratio. (A)
Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D)
total energy dissipation; (E) remaining energy in the system.
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K. Mao et al. / Powder Technology 142 (2004) 154–165
Fig. 13. Various specific damping w for 95% packing ratio during t = 0.00 –
0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
Fig. 15. Various specific damping w for 50% packing ratio during t = 0.00 –
0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
different set of arrangements of the major parameters of
the particle damper. This represents our effort towards a
better and more complete understanding of the energy
dissipation mechanisms in particle damping [11]. We
utilize our DEM simulation system to calculate the dissipated energy by collisions between particles and between
particle and container wall interfaces. The dissipated energy is divided into two classes due to impact and dryfriction respectively for each collision. When summarizing
all the particles in collision, we obtain the system energy,
the frictional energy dissipation, and the impact energy
dissipation over the course of vibration attenuation of the
free-vibrating system. The parameters of the particle damping system are listed in Table 4. In all the cases to be
discussed, the particle mass remains the same of 10% of
the structural mass M.
5.1. Energy dissipation versus packing ratio
We first examine the effect of packing ratio of the
particles. Under free vibration condition, the undamped
natural frequency of the system is set at xn = 250 Hz, for
three different packing ratios with H = 12, 24, and 48 mm,
respectively, for three different packing ratios p = 95%,
50%, and 25%. In all cases, the initial displacement is 5
mm with zero initial velocity. The energy dissipations of
the system due to impact, friction, and viscous damping
are shown in Figs. 12 –17, respectively, for the three cases.
In all three cases, the DEM simulation shows that both
friction damping and impact damping play significant role.
However, at different levels of height of the hole, each
damping mechanism may have different relative contribution. At the high packing ratio of 95%, the container hole is
Fig. 14. Various cumulative energy dissipations for 50% packing ratio. (A)
Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D)
total energy dissipation; (E) remaining energy in the system.
Fig. 16. Various cumulative energy dissipations for 25% packing ratio. (A)
Viscous dissipation; (B) friction dissipation; (C) impact dissipation; (D)
total energy dissipation; (E) remaining energy in the system.
K. Mao et al. / Powder Technology 142 (2004) 154–165
Fig. 17. Various specific damping w for 25% packing ratio during t = 0.00 –
0.02 s. (A) Impact damping; (B) friction damping; (C) viscous damping.
163
Fig. 19. Various cumulative energy dissipations at xn = 100 Hz. (A) Viscous
dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy
dissipation; (E) remaining energy in the system.
nearly packed full of the particles. The friction damping
becomes more significant than the impact damping (Figs. 12
and 13). Intuitively, the small clearance between the particles
and the container walls reduces the potential of strong
collisions between the particle – particle and particle – wall
contacts. For a higher clearance at p = 50%, the total energy
dissipation remains nearly at the same level with that at
p = 95%. In this case, however, the friction and impact
damping mechanisms have reversed their roles with energy
dissipated by impact being about 30% more than energy
dissipated by friction (Figs. 14 and 15). For the even higher
clearance at p = 25%, the impact energy dissipation is even
more pronounced while the friction energy dissipation
remains nearly the same (Figs. 16 and 17). Therefore, the
total level of energy dissipation is higher. At this clearance
level, the particle damper has the best performance for the
particles have a best chance to shake and tumble inside the
hole, thus dissipating energy through the impact mechanism
[25]. It should be noted from Figs. 12 – 17 that the bulk of the
vibration energy is dissipated within a very short period of
time approximately between t = 0.00 – 0.18 s for all the three
cases. This again is a reflection of the nonlinear characteristics of the particle damping as discussed in Section 3.
Fig. 18. Various cumulative energy dissipations at xn = 20 Hz. (A) Viscous
dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy
dissipation; (E) remaining energy in the system.
Fig. 20. Various cumulative energy dissipations at xn = 250 Hz. (A) Viscous
dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy
dissipation; (E) remaining energy in the system.
5.2. Energy dissipation versus frequency
Next, we examine the effect of vibrating frequency on
the energy dissipation. The particle volumetric packing
ratio is set at 50%, and the initial displacement is taken
at 10 mm. The natural frequency of the system is set at
four different levels, xn = 20, 100, 250, and 500 Hz. As
shown in Figs. 18 –21, respectively, the friction dissipation
is greater than the impact dissipation for the lower frequency cases of xn = 20 and 100 Hz. But when system
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K. Mao et al. / Powder Technology 142 (2004) 154–165
numerical simulations of different physical and numerical
parameters, including the number of particles, the hole’s
size, the particle density, and the size of subdivision cells.
The particle damping is a combination of the impact and the
friction damping. It is further shown that the relative
significance of these damping mechanisms depends on a
particular arrangement of the damper. It is shown that our
implementation is particularly efficient with a near linear
dependency on the number of particles per time step. Thus,
the DEM simulation can be readily applied as a practical
tool for designing a PVD. It should complement experiments to furnish a deeper understanding of particle damping
and to provide a comprehensive methodology for analysis
and design of PVDs.
Fig. 21. Various cumulative energy dissipations at xn = 500 Hz. (A) Viscous
dissipation; (B) friction dissipation; (C) impact dissipation; (D) total energy
dissipation; (E) remaining energy in the system.
frequency is higher (at 250 and 500 Hz), the impact
dissipation is greater than the friction dissipation. At
xn = 20 Hz and x0 = 10 mm, the dimensionless acceleration
of the system is initially C = x0xn2/g c 400. We found that
when C < 400, the friction dissipation is greater than
impact dissipation. This conclusion is consistent with the
results of [10,25]. In the low frequency case, the friction
dissipation is nearly twice greater than impact dissipation
as shown in Fig. 18. However, at the much higher
frequency of 500 Hz, the situation is nearly reversed with
impact damping to be much more significant than the
friction damping.
6. Conclusions
We have presented a 3D DEM to simulate the motion of
particles in a PVD. The metal particles are represented as
spheres. For particle – particle and particle – wall contacts, a
normal contact model and a tangential contact model are
used to account for the coefficient of restitution and the dry
friction. Both impact and friction are known to play an
important role in the damping effect provided by the
particles on their container structure. An efficient ‘‘box’’
algorithm is used for the detection of collisions in the dense
particle flow during the course of vibration.
With the DEM modeling tool, we performed numerical
simulation for a number of examples of a single degree of
freedom system. For a physical particle vibration damper
with experimental results of a previous investigation, simulation results of the DEM modeling are found to have a
good agreement with the experimental data. Furthermore,
simulations provide information of particle motions within
the container hole during three different regions and help
explain their associated damping characteristics. The computational complexity of the DEM scheme is analyzed with
Acknowledgements
The research work reported in this paper is sponsored in
part by the Hong Kong Research Grants Council (project No.
CUHK4196/01E), ASM Assembly Automation, a Croucher
Chinese Visitorship of the Croucher Foundation of Hong
Kong for Professor T. N. Chen, and the National Science
Foundation of China (NSFC) under grant No. 59775019.
The authors wish to express their sincere thanks to C.K. Liu,
P.K. Choy, and G.P. Widdowson of ASM Assembly
Automation for their valuable insight and support.
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