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1. Journal of Sound and Vibration 488 (2020) 115625
Contents lists available at ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsv
Damper force characteristics of a separated dual-chamber
single-rod-type damper using an elastomer-particle
assemblage
Atsushi Toyouchia,b,∗
, Makoto Hanaib
, Yasushi Idoa
, Yuhiro Iwamotoa
a
Department of Electrical and Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan
b
KYB Corporation, 2548, Dota, Kani 509-0206, Japan
a r t i c l e i n f o
Article history:
Received 9 May 2019
Revised 6 August 2020
Accepted 7 August 2020
Available online 13 August 2020
Keywords:
Particle damper
Damper force
Elastomer
Particle assemblage
Discrete element method
Single-rod-type damper
a b s t r a c t
In the present paper, the damper force characteristics of dampers using a particle assem-
blage instead of oil have been reported. A separated dual-chamber single-rod-type damper
using an elastomer particle assemblage was produced. The effects of the packing ratio of
the particles, the vibration frequency, and the materials of the particles on the damper
force characteristics of the damper with elastomer particles only on one side chamber
were investigated experimentally. The maximum damper force was found to increase as
the packing ratio, the vibration frequency, or the Young’s modulus of the material of par-
ticles increased and the hysteresis of the damper force increased, i.e., the damping energy
increased. In order to visualize the behavior of particles inside the damper, numerical sim-
ulation using the discrete element method was carried out. The simulation results agreed
qualitatively and quantitatively with the damping characteristics of the experimental re-
sults, and large hysteresis was found to appear in the damper force due to deformation of
the elastomer particles.
© 2020 Published by Elsevier Ltd.
1. Introduction
Dampers are mechanical devices used to suppress vibration of structures, vehicles, and so on. Typical dampers include
viscous dampers using liquid, such as oil, and friction dampers using contact between metal surfaces. In viscous dampers,
influences on the damper force characteristics due to temperature and liquid leakage occur. Since the friction surface of
friction dampers deteriorates, their durability is not high and their application are limited. As a method by which to solve
the problems of oil leakage and temperature dependence, dampers using a particle assemblage have attracted attention in
recent years.
There are two types of damper that use a particle assemblage, i.e., a particle impact damper and a damper using a par-
ticle assemblage rather than oil in an oil damper. There have been a number of studies on particle impact dampers [1]. In
the case of using non-obstructive particle damping technology, which absorbs vibration energy using such as spherical or
irregularly shaped metal particles or nonmetal particles, Panossian proposed a design optimization procedure, reported ex-
perimental results [2], and indicated that the smallest response amplitude can be obtained using a honeycomb structure as
a container containing particles. Liu, et al. investigated the frequency response in a structure with particles in a cylindrical
∗
Corresponding author.
E-mail address: toyouchi-ats@kyb.co.jp (A. Toyouchi).
https://doi.org/10.1016/j.jsv.2020.115625
0022-460X/© 2020 Published by Elsevier Ltd.

2. 2 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
vessel, and showed that the attenuation strongly depends on the response level and that vibrational energy is dissipated
mainly due to the friction between particles and between particles and the wall [3]. Du et al. showed that a fine-particle
impact damper, which plastically deforms fine particles to attenuate vibration, works well in vibration control at frequencies
lower than 50 Hz [4]. Dehghan-Niri et al. reported the damping effect of a horizontally installed impact damper using one
particle. In this report, dampers designed to withstand large-amplitude vibrations were shown to attenuate vibration with
smaller amplitude well [5]. Sanchez et al. showed that there is no significant change in frequency-response characteristics
when the impact damper using multiple particles is vibrated in the direction of gravity, even if the friction characteristics
of the particles change due to a temperature change, for example [6]. Lu et al. investigated the damping effect when an
impact damper using multiple particles was used for a three-story steel frame and showed that the vibration can be re-
duced. In addition, the numerical results obtained using the discrete element method agreed well with their experimental
results [7]. They also investigated the damping properties of a horizontally mounted two-degree-of-freedom particle impact
damper and found that increasing the mass of the particles improves damping but is only effective up to a specific response
level. Furthermore, it was shown that, if the repulsion coefficient of the particle is increased, then the damping effect can
be obtained up to a wide response level [8]. Zahrai and Rod reported that when a sinusoidal vibration is applied to a hori-
zontally installed particle impact damper using one particle, increasing the stiffness increases the damping effect. However,
the damping effect does not become high in all vibration modes [9]. In addition, they showed that, regarding the damping
effect in the case of two different vibrations, the particle impact damper is an efficient device for reducing vibration of a
structure subjected to shock and harmonic vibrations [10]. Saeki performed a numerical analysis of the discrete element
method (DEM) in order to analyze the behavior of particles in the particle impact damper vibrated in the direction of grav-
ity [11]. They proposed a new energy dissipation model for estimating the efficiency of particle damping, and compared the
numerical results with the experimental results in order to verify their new model. Inoue et al. investigated the damping
effect of a multi-degree-of-freedom particle impact damper using multiple particles vibrated in the direction of gravity, and
they showed that a damper having an appropriate mass ratio and clearance effectively suppresses the resonance peak in
a wide frequency range [12]. Takahashi and Sekine examined the behavior of particles in the particle impact damper ex-
perimentally using piezoelectric elements and DEM simulations. In their paper, it was shown that proper choice of packing
fraction and particle size provides a good damping effect due to the resulting effective collisions [13].
On the other hand, a damper using a particle assemblage instead of oil in the oil damper, as a structure other than a
particle impact damper, was proposed by Hayashi and Ido [14,15]. They reported the damping characteristics of the double-
rod-type linear damper using particles such as glass beads and steel particles, and found that the damper force for these
dampers shows hardening-type characteristics. In the case of a damper using a steel-particle assemblage, it has been re-
ported that the damper force characteristic can be changed by applying a magnetic field, and the damper force drastically
increases near the dead points of vibration, while the damper force becomes weak near the vibration center [16]. Ido et al.
experimentally investigated the influence of container size and stroke on the damper force of the double-rod-type linear
damper using a steel-particle assemblage [17]. The behavior of particles in the double-rod-type damper using a steel-particle
assemblage was simulated by Hanai et al. using the discrete element method [18]. Morishita et al. used elastomer particles
instead of steel particles or beads for the damper using a particle assemblage [19]. They reported the effects of the vi-
bration frequency, stroke, and packing fraction of the particles on the damping characteristics of a double-rod-type linear
damper. Kawamoto et al. examined the damper force characteristics of a double-rod-type linear damper using elastomer
particles (silicone rubber) mixed with micrometer-size silicon particles [20]. In the above mentioned reports, dampers us-
ing a particle assemblage instead of oil in an oil damper were double-rod-type linear dampers. However, since there are
many single-rod-type oil dampers, it is considered to be possible to produce a single-rod-type damper using an elastomer-
particle assemblage. However, the damper force characteristics of such a single-rod-type damper using an elastomer-particle
assemblage have not been clarified.
In the present paper, the experimental and simulation results for the damper force characteristics of the single-rod pro-
totype linear damper using an elastomer-particle assemblage are reported. We propose a separated dual-chamber single-
rod-type damper using an elastomer-particle assemblage, one of the basic structures of single-rod-type linear dampers, and
investigate the damper force characteristics of the damper when elastomer particles are placed in only one of the two cham-
bers. The effects of the packing fraction of particles, the vibration frequency, the hardness of particles, and the position of
the vibration center on the damper force characteristics of the proposed damper are examined. In order to verify the behav-
ior of the elastomer particles inside the damper, numerical simulations are carried out using the discrete element method,
and the simulation results are compared with the experimental results.
The reason why particles are placed in only one chamber is to make the damper force dependent on the direction of
travel of the piston. As a specific example, this technique can be used as an alternative to a shock-absorbing damper or a
silicone oil damper attached to casters, such as those on wheelchairs. These dampers have a one-sided damper force.
2. Experiments
2.1. Damper using a particle assemblage
Fig. 1 shows a schematic diagram of the damper prototyped in the present study. The damper consists of particles, a
piston, a rod, and a container consisting of a cylinder and end covers. The coordinate system is set as shown in Fig. 1, and

3. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 3
Fig. 1. Schematic diagram of the separated dual-chamber single-rod-type damper using an elastomer-particle assemblage. (1). Cylinder, (2). end cover, (3).
bearing, (4). rod, (5). piston, (6). elastomer particles. (Units: mm).
the columnar piston is displaceable in the z-axis direction. When an external force is applied to a columnar rod integrated
with the piston, the damper force due to elastomeric forces generated by compressing the particles and the frictional forces
generated between the particles or between the particles and the piston, cylinder, and rod, are obtained. The cylinder, piston,
and end caps are made of carbon steel S45C, and the rod is made of stainless steel SUS440C. The piston has a rod on one
side, and its outer diameter is of a size such that particles cannot enter the gap with the inner wall of the cylinder. The
origin of the coordinate system is the center point of the space surrounded by the cylinder and the end covers. The stroke
center position is defined as the distance from the origin to the vibration center of the central point of the piston.
The frictional force at the sliding part was reduced by a bearing between the rod and the cylinder and was measured
when a sinusoidal vibration was applied to the damper with no particles using the experimental apparatus mentioned later.
The frictional force at the sliding part was very small (several newtons). Since the damper force generated by the damper
in this experiment was approximately several hundred newtons, the frictional force at the sliding part could be neglected.
2.2. Experimental apparatus
Fig. 2 shows a schematic diagram of the experimental apparatus used in the present study. The damper was installed
horizontally, and gravity acted in a direction orthogonal to the z axis. The rotational motion of the motor (Mitsubishi Electric,
HA-LP801) was converted to reciprocating linear motion using the piston-crank mechanism, and sinusoidal forced vibration
with a constant frequency and constant amplitude was given to the damper. Regarding the measurement of the damper
force, the damper force generated by forced vibration was output by a load cell (Kyowa Electric Instruments, LUK-A-20kN),
and its output was amplified by a dynamic strain amplifier (Kyowa Electric Instruments, DPM-751A) and recorded with
an oscilloscope (Yokogawa Electric, DLM 2024). For the displacement, we measured the piston displacement with a laser
displacement meter (Keyence, Sensor Head LB-300, and Amplifier Unit LB-2000) and recorded it with an oscilloscope, in
the same manner as the damper force. Moreover, the vibration frequency could be set to an arbitrary value by the motor
controller (Mitsubishi Electric, MR- J3-11KA). In the experiments, the data was measured three times for each condition in
consideration of measurement error.
Fig. 2. Schematic diagram of the experimental apparatus. (1). Motor controller, (2). motor, (3). laser displacement sensor, (4). rod, (5). damper, (6). load
cell, (7). strain amplifier, (8). oscilloscope, (9). amplifier unit.

4. 4 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
Fig. 3. Photographs of particles.
2.3. Experimental conditions
In the present study, the packing fraction, the vibration frequency, the material of the particles, and the distance of the
stroke center from the origin are used as variable parameters. Two-component addition-type high-strength silicone rubber
(Momentive Performance Materials, TSE 3466) and nitrile rubber (NBR) (Inaba Rubber, scattered) were used as the materials
of the elastomer particles. Photographs of each particle are shown in Fig. 3.
Table 1 shows the experimental conditions. The packing fraction is an index of the amount of the particle assemblage to
be filled in the damper and is defined as follows:
Packing fraction =
Total mass of the particles
Volume of the space in the container × Density of the particle
(1)
Table 1
Experimental conditions.
Material of the elastomer particle Silicone elastomer (TSE3466), Nitrile rubber (NBR)
Young’s modulus [MPa] 4.11 (TSE3466), 17.6 (NBR)
Durometer A hardness 60
Diameter of particles [mm] 3, 4, 5
Packing fraction of particles [-] 0.60, 0.65, 0.70
Stroke of forced vibration [mm] 10
Frequency of forced vibration [Hz] 0.1, 1, 5
Stroke center distance from the origin [mm] 0, 10

5. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 5
The volume of the space in the container is the volume of the space to which particles can be filled in the container,
i.e., the volume of the space surrounded by the cylinder, the end cover, and the piston surface on the side without the rod.
However, due to the structure of the damper, the packing fraction changes as the piston moves. In the present study, the
packing fraction when the piston is at the stroke center position is treated as a representative value. Thus, the volume of
the space in the container in Eq. (1) is the volume when the piston center is at the center of the stroke.
In our experiments, the damper was filled with particles manually, so a non-uniform initial arrangement of the particles
was possible. For this reason, preliminary excitation was performed in order to eliminate the influence on the damper force
due to non-uniform initial disposition of the particles. Prior to starting the measurement, preliminary excitation with a
frequency of 1 Hz was performed for 200 seconds.
3. Numerical simulation
3.1. Simulation method
Behavior analysis of elastomer particles using the discrete element method was carried out. The discrete element method
is a simulation method in which the motion of a particle at each time is sequentially calculated considering the contact
between particles and follows the particle behavior. In the discrete element method, by solving the equation of motion
[Eq. (2)] and the equation of angular momentum [Eq. (3)] by considering the contact force for individual particles, we can
find the velocity and the angular velocity:
mi
d2
ri
dt2
= Fi, (2)
Ii
di
dt
= Ti, (3)
where mi is the mass of the particle i, t is the time, ri is the position vector of the particle, Fi is the total contact force
vector, Ii is the moment of inertia, i is the angular velocity vector of the particle, and Ti is the total torque vector acting
on the particle. Here, Fi, Ti, and Ii are obtained using the following equations:
Fi = Fcn + Fct + mig, (4)
Ti = ri × Fct , (5)
Ii =
8
15
ρπr5
. (6)
In these equations, the normal direction at the contact point is denoted by the subscript n and the tangential direction is
denoted by the subscript t. Here, Fcn and Fct are the contact forces in the normal and tangential directions, respectively, g is
the gravitational acceleration vector, ρ is the density of the particle, and r is the radius of the particle. In order to calculate
the contact forces, the model composed of spring, dashpot, and friction slider elements (as shown in Fig. 4) proposed by
Cundall and Strack [21] is considered and can be expressed as follows:
Fcn =
−Knδn − CnVij · ni
ni, (7)
Fig. 4. Contact force model of particles.

6. 6 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
Fct = −Kt δt − CtVf ij, (8)
η = α
m · k, (9)
Vij = vi − vj, (10)
Vf ij = Vij −
Vij · ni
ni + 2a
ωi − ωj
× ni, (11)
where Kn and Kt are the elastic coefficients in the normal direction and tangential directions, respectively, Cn and Ct are the
viscosity coefficients in the normal and tangential directions, respectively, δn is the amount of displacement in the direction
of the normal to the contact point, δt is the displacement vector in the tangential direction at the contact point of the
particle i with the particle j, ni is a unit vector in the normal direction from particle i to particle j with respect to the
contact point, η is the damping coefficient, α is a dimensionless constant that determines the magnitude of viscous damper
force, k is the elastic coefficient, Vij is the relative velocity vector of particle i with respect to particle j, vi and vj are the
velocity vectors of particles i and j, respectively, Vfij is the tangential relative velocity vector of the particle i at the contact
point with respect to particle j, a is the particle radius, and ωi and ωj are the angular velocity vectors of particles i and j,
respectively. The viscosity coefficients Cn and Ct are expressed, respectively, as follows:
Cn = α
miKnδ0.25
n , (12)
Ct = α
miKt δ0.25
ct , (13)
where δct is the amount of displacement in the direction of the tangential to the contact point. The elastic coefficient in the
normal direction is obtained using Hertz’s contact theory as follows:
Knij =
4
3π
1
2δi
aδn
2
, (14)
Kniw =
4
3π
1
δi + δw
aδn, (15)
δi =
1 − v2
i
Eiπ
, (16)
δw =
1 − v2
w
Ewπ
. (17)
The subscript w in the above equations indicates that the physical quantity relates to the wall, Knij is the elastic coefficient
Kn at the time of contact between particles, Kniw is the elastic coefficient Kn when the particle and the wall surface are in
contact, Ei and Ew are the moduli of longitudinal elasticity of the particle and wall, respectively, and νi and νw are the
Poisson’s ratios of the particle and wall, respectively. Assuming that there is no slip at the contact point, the tangential
elastic coefficient is expressed by the following equations based on the Mindlin theory [22]:
Ktij =
2
√
2aGi
2 − νi
δ0.5
n , (18)
Ktiw =
8
√
aGi
2 − νi
δ0.5
n , (19)
where Ktij is the elastic coefficient Kt when particles are in contact with each other, Ktiw is the elastic coefficient Kt when
particles and the wall are in contact, and Gi is the transverse elastic modulus of the particle and is expressed as follows:
Gi =
Ei
2(1 + vi)
. (20)
Regarding the frictional force, if the tangential direction relative velocity vector on the surface of the contact particle is
larger than 0, or the contact force in the tangential direction is larger than the frictional force, slippage is assumed to have
occurred at the contact surface. The frictional force can be expressed as follows:
In the case of |Vfij| = 0, when Fct ≤ μf|Fcn|:
Fct = Fct, (21)
and when Fct μf | Fcn |,
Fct = −μf | Fcn | ti (22)
In the case of | Vfij | 0,
Fct = −μf | Fcn | ti (23)
where, ti = Vfij / | Vfij |, which is a unit vector in the same direction as Vfij of the particle, and μf is the coefficient of friction
of the particles.
In our simulations, the elastic repulsive force and viscous force are taken into consideration both in the normal direction
and the tangential direction, and the viscoelastic model including the frictional force is also taken into consideration. The
Adams-Bashforth method of second-order accuracy was used for the time progression of the velocity, displacement, and
angular velocity.

7. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 7
Table 2
Numerical conditions.
Material of the elastomer particles Silicone elastomer (TSE3466), Nitrile rubber (NBR)
Diameter of particles [mm] 3, 5
Packing fraction of the particles [-] 0.60 or 0.70
Number of particles 1339 or 1562
Stroke of forced vibration [mm] 10
Frequency of forced vibration [Hz] 1, 5
Stroke center distance from the origin [mm] 0
Table 3
Mechanical properties for calculation.
Density of particle [kg/m3
] 1.10×103
Poisson ratio vi 0.5
Friction coefficient μf (wall-particle) 0.5
Friction coefficient μf (particle-particle) 0.5
Young’s modulus of wall Ew [GPa] 210
Compressive modulus of particles Ei [MPa] 4.11 (TSE3466), 17.6 (NBR)
Parameter of attenuation coefficient α 0.5311
3.2. Analytical model and numerical conditions
The analytical model of the damper used for the numerical simulations was set up similarly to the damper used in the
experiments, as shown in Fig. 1. Physical properties such as Young’s modulus were also set to the same values as those in
the experiments. Table 2 shows the numerical conditions, and the physical properties of the elastomer particles and wall
used in our simulations are shown in Table 3. Since the friction coefficient was unknown, a general friction coefficient value
of 0.5 between rubber and rubber was adopted in our simulations [23]. In the numerical analysis, the variable parameters
were set as the packing fraction and vibration frequency. The packing fraction was 0.60 or 0.70 (vibration frequency: 1 Hz),
and the vibration frequency was 1 Hz or 5 Hz (packing fraction: 0.60). In addition, the material of the elastomer particles
was assumed to be silicone rubber. The distance from the origin of the stroke center was 0 mm, and the stroke was set to
10 mm, which was the same as in the experiments.
4. Results and discussions
4.1. Damper force generation mechanism of the damper
Fig. 5 shows the time history of the damper force with respect to the piston displacement. Although the experimental
apparatus uses a piston crank mechanism, Fig. 5 shows that the displacement roughly matches a sine wave. Fig. 5 also
shows that the phase delay of the damper force with respect to the displacement of the piston due to the influence of
the structural viscosity of the particles appears. Time zones in which there is almost no damper force with respect to the
Fig. 5. Time history of the damper force and displacement.

8. 8 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
Fig. 6. Damper force vs. piston displacement. Comparison of the simulation and experimental results. The stroke center distance is 0 mm.
displacement of the piston repeatedly appear because the restoration of the compressed and deformed particles does not
follow the displacement of the piston in the non-compression direction. Therefore, the damper force is considered to have
characteristics with hysteresis.
Fig. 6 shows the simulation results and the experimental results of the damper force-piston displacement curves for each
packing fractions. Fig. 6 also shows a hardening-type characteristic, in which the damper force increases as the displacement

9. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 9
Fig. 7. Normal and tangential damper forces vs. displacement of the piston.
progresses in the compression direction, regardless of the particle packing fraction. When the piston moves in the direction
opposite to the compression, hysteresis due to the deformation of the elastomer particles appears. The simulation results
are in good agreement qualitatively and quantitatively with the experimental results, regardless of the packing fraction of
the particles.
In order to confirm the normal and tangential components of the damper force, the decomposition of the normal and
tangential components was performed on the simulation results for a packing fraction of 0.60 and a vibration frequency of
1 Hz. The results are shown in Fig. 7. Since only a small force is generated in the tangential direction, as compared with the
normal direction, the damper force of the damper is dominated by the force in the normal direction.
As is clear from Fig. 7, since the force in the normal direction is dominant, the compressive forces acting on the particles
were investigated using the simulation results. Both the experimental results and the simulation results showed that the
damper force curve has a similar shape under any conditions. As such, there was considered to be no large difference
in compressive force distribution during forced vibration. Thus, the simulation of the packing fraction of 0.60 and of the
vibration frequency of 1 Hz was performed as a typical example. The results are shown in Fig. 8, which shows the particle
distribution in the section including the z axis and the intensity of the compressive force acting on the particles. The time t
is non-dimensionalized with the period T of the forced vibration. The darker the color of the particle in Fig. 8, the greater the
compressive force acting on the particle. From Fig. 8(a), in the compression process when the piston moves toward the left-
hand side, there are many particles with a strong compressive force near the piston, so the repulsive force from the particles
to the piston becomes strong, i.e., the damper force applied to the piston becomes strong. However, the compressive force
of the particles on the bottom side of the cylinder is small, and the compressive force of the piston is not sufficiently
transmitted to the bottom side of the cylinder. From Fig. 8(c), in the return process when the piston moves toward the right-
hand side, most of the particles have a small compressive force near the piston, so the repulsive force from the particles
to the piston becomes small, and the damper force applied to the piston also becomes small. However, since the particles
on the bottom side of the cylinder have a high compressive force, the compressive force on the bottom side of the cylinder
is not sufficiently transmitted to the piston. The reason for this phenomenon is that the restoring speed of the compressed
particles does not follow the moving speed of the piston, i.e., response delay of particle deformation to the piston motion
is considered to occur. Even at the same piston position (z = 0 mm), the damper force is large in the compression process
when the piston moves toward the left-hand side and the damper force is small in the non-compression process when the
piston moves toward the right-hand side, so that a difference occurs in the damper force between the compression process
and the non-compression process. As the compression of particles progresses, the compressive force between particles, the
force of pressing the particles against the wall surface, and the frictional force increase, so the repulsive force applied as
a damper force to the piston increases. In contrast, as the particles move in the non-compression process, the compressive
force between the particles and the force of the particles pressed against the wall surface become smaller, and the frictional
force becomes smaller. At this time, since the repulsive force applied to the piston is the frictional force subtracted from the
compression repulsive force of the particles, there is a difference in the damper force between the compression process and
the non-compression process. As a result, the damper force has characteristics with large hysteresis. As a result, the damper
force has characteristics with large hysteresis.
Next, in order to confirm the behavior of the particles in the damper during the forced vibration, the results of computing
the velocity vectors of the particles are shown in Fig. 9. The condition is the same as the calculation of the compressive force
of particles. In Fig. 9, only the velocity vector of particles in the plane including the z-axis is displayed. Fig. 9(a) and (c) show
that the particles near the piston in both the compression process and the non-compression process have velocities in the
same direction as the direction of the piston progression. Furthermore, with regard to the particles other than the particles

10. 10 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
Fig. 8. Distributions of the compressive force acting on the particles. The position of the piston center and time are (a) z = 0 mm, t/T = 0 (compression
process), (b) z = 5 mm, t/T = 0.25, (c) z = 0 mm, t/T = 0.50 (non-compression process), and (d) z = -5 mm, t/T = 0.75. The arrow on the piston indicates
the direction of the piston velocity.
near the piston, the velocity is generally in various directions at every time in Fig. 9. When the velocity of the particles is in
the same direction, the interparticle friction force increases according to the particle velocity difference. However, when the
particles have velocities in different directions, particles with opposite velocities often collide with and contact each other.
As a result, more compression and friction occur between particles and between particles and the wall as compared to the
case in which the velocity is directed in a uniform direction.
4.2. Influence of the packing fraction on the damper force
Figs. 10 and 11 show, respectively, the damper force-displacement curves and the damping energy in the cases of packing
fractions of 0.60, 0.65, and 0.70. The damping energy is obtained by calculating the area of the region surrounded by the
damper force-piston displacement curve. The subsequent damping energy is the average value of the data measured three
times. Moreover, from Fig. 5, the force generated in the experiment is not a pure damper force because the displacement
and the phase are not 90° out of phase. Since the force includes both stiffness and damping related aspects, the force is
referred to as the damper force. From Figs. 10 and 11, the damper force shows the hardening-type characteristics in which
the damper force increases as the displacement of the piston progresses in the compression direction. When the piston
moves in the restoring direction, the damper force has a characteristic with hysteresis due to the restoration delay of the
elastomer particles. Since the damper force exhibits the hardening-type characteristics, the compressive force of each par-
ticle increases as the piston displacement progresses in the compression direction from the distribution of the compressive
force acting on the elastomer particles shown in Fig. 8. The amount of elastic deformation of each particle is considered to
increases with the piston displacement in the compression direction, i.e., the elastic repulsive force increases. The damper
force has hysteresis because, as discussed in Section 4.1, since the restoration from the compressive deformation of the par-
ticles cannot follow the piston displacement in the restoring direction, even for the same piston displacement position, the
damper force in the restoring process is smaller than that in the compression process. At this time, the energy given to the
elastomer particles by compression is considered to be converted into thermal energy due to the structural viscosity of the
particles. From the velocity vector diagram shown in Fig. 9, since the particles have velocities in various directions at any
displacement, the elastic repulsive force of the particles generated by the particle motion and the frictional force between

11. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 11
Fig. 9. Selected velocity vectors of the particles inside the damper. The position of the piston center and time are (a) z = 0 mm, t/T = 0 (compression
process), (b) z = 5 mm, t/T = 0.25, (c) z = 0 mm, t/T = 0.50 (non-compression process), and (d) z = -5 mm, t/T = 0.75. The arrow on the piston indicates
the direction of the piston velocity.
Fig. 10. Damper force vs. displacement (influence of packing fraction). The vibration frequency is 1 Hz, and the material of the particles is silicone elas-
tomer TSE3466. The stroke center distance is 0 mm.
the particles and between the particle and the wall surface causes the hysteresis. Figs. 10 and 11 show that the maximum
damper force, hysteresis, and damping energy increase as the packing fraction increases. This phenomenon is explained as
follows. The number of particles increases as the packing fraction increases, and the gap in the cylinder becomes small in
the initial state when the piston is located at the stroke center before vibration starts (hereinafter referred to as the initial
state). When the packing fraction is low, the particles move to some extent with the displacement of the piston. However,
when the packing fraction is high, the particles are held by other particles, so that the amount of compression of the parti-
cles due to the piston displacement increases. In other words, the damper force increases from the small displacement. For

12. 12 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
Fig. 11. Damping energy (influence of packing fraction). The vibration frequency is 1 Hz, and the material of the particles is silicone elastomer TSE3466.
The stroke center distance is 0 mm.
Fig. 12. Damper force vs. displacement (influence of frequency of forced vibration). The packing fraction is 0.60, and the material of the particles is silicone
elastomer TSE3466. The stroke center distance is 0 mm.
the hysteresis, the higher the packing fraction and the larger the number of particles, the larger the frictional force and the
heat energy converted inside the particle.
4.3. Influence of the vibration frequency on the damper force
Fig. 12 shows the damper force-displacement curves for the cases of the vibration frequencies of 0.1 Hz, 1 Hz, and 5 Hz.
From Fig. 12, the damping energy can be obtained as 148.3 mJ, 163.0 mJ, and 208.6 mJ at vibration frequencies of 0.1 Hz, 1
Hz, and 5 Hz, respectively. Fig. 12 shows that the maximum damper force, hysteresis, and damping energy increase as the
vibration frequency increases. This is because, as the speed of compressing the particles increases, the viscous resistance
force of the viscoelastic elastomer particles increases, and the elastic repulsive force also increases because the deformation
proceeds promptly.
4.4. Influence of the material of the particles on the damper force
Fig. 13 shows the damper force-displacement curves when the materials of the particles are silicone elastomer TSE3466
and NBR. The damping energies obtained from the damper force-piston displacement curves in Fig. 13 are 163.0 mJ for
TSE3466 and 547.1 mJ for NBR, indicating that the influence of the material of the particles on the damping energy is large.
Fig. 13 shows that the maximum damper force, hysteresis, and damping energy depend on the particle material. This is clear
also from the comparison of Fig. 5(c) and (d). The larger the larger the Young’s modulus of the particle material, the larger

13. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 13
Fig. 13. Damper force vs. displacement (influence of material of the particles). The vibration frequency is 1 Hz, and the packing fraction is 0.60. The stroke
center distance is 0 mm.
Fig. 14. Damper force vs. displacement (influence of the stroke center distance from the origin). The vibration frequency is 1 Hz, and the packing fraction
is 0.60. The material of the particles is silicone elastomer TSE3466.
the damper force, hysteresis, and damping energy. Thus, changing the particle material is very effective when changing the
strength of the damper force of the damper using an elastomer-particle assemblage.
4.5. Influence of the stroke center distance on the damper force
Fig. 14 shows the damper force-displacement curves when the stroke center distance from the origin is 0 mm or 10 mm.
For stroke center distances from the origin of 0 mm and 10 mm, the damping energies are 163.0 mJ and 174.7 mJ, respec-
tively, and the amount of increase is small but the damping energy is increasing. From Fig. 14, the maximum damper force
becomes larger as the distance from the origin of the stroke center is larger. This is because even when the packing fraction
is the same, as the stroke center distance from the origin increases, the volume of the particle packing space decreases, so
that the packing fraction based on the vibration center increases, and therefore the damper force becomes stronger. In other
words, the apparent filling rate increases from a low displacement, and the damper force increases. When the elastic repul-
sive force increases, the normal force increases, and the frictional force also increases. Thus, the hysteresis and the damping
energy should be large. However, the hysteresis and the damping energy do not increase in the experimental results. Since
the force in the normal direction is dominant and the force in the tangential direction is very small, the hysteresis does not
increase because the increase in the frictional force due to the increase in the normal force is also small. Furthermore, even
if the stroke center distance from the origin is changed, the number of particles is the same, and so it is conceivable that
there is no significant difference in frictional force and thermal energy converted inside the particles Figs. 4 and 16.

14. 14 A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625
Fig. 15. Damper force vs. displacement (influence of the diameter of particles). The vibration frequency is 1 Hz, and the packing fraction is 0.60. The
material of the particles is silicone elastomer TSE3466.
4.6. Influence of the diameter of the particles on the damper force
Fig. 15 shows the damper force-displacement curves for the cases of particle diameters of 3 mm, 4 mm, and 5 mm. From
Fig. 15, the damping energy can be obtained as 163.0 mJ, 222.7 mJ, and 261.2 mJ at diameters of 3 mm, 4 mm, and 5 mm,
respectively. From Fig. 15, the maximum damper force, hysteresis, and damping energy increase as the diameter of particles
increases. This is considered to be because the bulk density of the particles increases as the particle size increases, and the
larger particles have a larger amount of deformation due to compression for the same displacement.
Next, the damper force versus velocity curves for a packing fraction of 0.60, a vibration frequency of 1 Hz, silicone
elastomer particles, and a stroke center distance from the origin of 0 mm are shown in Fig. 16. Fig. 16 shows that the
maximum damper force is obtained at a velocity of approximately 0 m/s, and the damper force when the piston velocity
becomes maximum is small. As for the damping characteristics, the damper force does not increase according to the ve-
locity, in the manner of oil dampers, due to the viscous force of the oil, but the damper force becomes larger according to
the displacement, in the same manner as a spring or rubber. However, since the velocity at which the maximum damper
force is obtained deviates from 0 m/s, the velocity dependency due to the viscous force also appears in the damper force
characteristics although the degree of influence is small. A region in which the damper force is negative during the restoring
process is shown in Fig. 16, because, after the piston returns to the position where the damper force is zero, the piston is
further pushed in a direction in which the damper force becomes negative due to the inertia force.
Fig. 16. Damper force vs. velocity. The vibration frequency is 1 Hz, and the packing fraction is 0.60. In addition, the material of the particles is silicone
elastomer, and the stroke center distance is 0 mm.

15. A. Toyouchi, M. Hanai and Y. Ido et al. / Journal of Sound and Vibration 488 (2020) 115625 15
5. Conclusions
In the present study, the separated dual-chamber single-rod-type damper using the elastomer-particle assemblage were
produced experimentally, and when the elastomer particles were placed in only one chamber, the damper force charac-
teristics were examined by varying the particle packing fraction, vibration frequency, material of the particles, and stroke
center position. Numerical simulations using the discrete element method were carried out in order to investigate the be-
havior of particles in the damper, and it was confirmed that the simulation results when changing the packing fraction and
vibration frequency agree well qualitatively and quantitatively with the damper force characteristics obtained experimen-
tally. The damper force was decomposed into normal and tangential components, and the force in the normal direction was
found to be dominant. Based on the simulation results of the distribution of the compressive force acting on the particles,
the compressive force of the particles increases from the periphery of the piston during the compression process, whereas
the compressive force of the particles from the periphery of the piston decreases in the restoring process. In both the com-
pression and restoring processes, the particles near the piston have velocities in the moving direction of the piston, and
the particles, other than those near the piston, have velocities in various directions, except when the piston is near dead
center. The damper force shows a hardness-type damping characteristic in which the damper force increases as the piston
displacement progresses in the compression direction and has a hysteresis characteristic when the piston moves in the non-
compression direction. In addition, the force generated by this damper does not deviate in phase from the displacement by
90° and includes both rigidity and side surfaces related to damping. Regarding the influence of the parameters, the maxi-
mum damper force and the damping energy increase as the packing fraction or the vibration frequency increases. However,
the influence of the vibration frequency on the damper force is small with respect to the influence of the packing fraction
on the damper force. Furthermore, the damper force characteristics change when the particle material is changed, and the
maximum damper force increases when the stroke center distance increases. However, there is no large difference in the
hysteresis.
Declaration of Competing Interest
The authors declare that have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
CRediT authorship contribution statement
Atsushi Toyouchi: Methodology, Investigation, Software, Writing - original draft. Makoto Hanai: Methodology, Software.
Yasushi Ido: Supervision, Software, Conceptualization. Yuhiro Iwamoto: Software, Supervision.
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