Research on Particle Assemblage Damper

Technical
ArticleResearch on Particle Assemblage Damper
TOYOUCHI Atsushi

Abstract
Dampers are devices which suppress vibrations thatcause harm to vehicles, buildings, and the like. There arevarious types of dampers, depending upon the application.
However, each type of damper comes with its own problems. Typical examples of such problems are that in thecase of oil dampers, liquid leakage may occur, and thedamping force may be dependent on temperature. Particle
dampers suppress vibrations by generating resistanceforce through the use of particles in place of oil or the like.
These dampers can be exemplified as a possible solution
to the abovementioned problems. In this study, an investigation and examination were carried out in order to clarify
the generative force characteristics and force generation
factors of a separated dual chamber single rod-type particle damper which employs a particulate elastomer*. This
type of damper holds the possibility of solving these
problems by means of a simple structure.
With regard to the damper used in this study, a particle
chamber within the cylinder was divided in two by apiston. It was found that when one chamber of the damper
was filled with particles, the generative force had thecharacteristic of being gradually hardening with hysteresis, with great influence exerted by elastic force in thenormal direction and frictional force in the tangential
1

Introduction

The particle assemblage damper is a damper filled withsmall steel or elastomer balls in place of a fluid such as oil
as in oil dampers. These spherical particles that fill thedamper are fluidized by the movement of a piston to yielda resistive force. (1, 2) The existing oil damper involves
problems including liquid leakage and temperaturedependent damping force. One of the possible solutions to
the problems is the use of particle dampers. A particle
damper using particles of a larger size than the clearanceof the sliding part of the damper is free from leakage. For
a particle damper using small elastomer balls (hereinafter"elastomer particles"), using elastomer material withexcellent heat/cold resistance whose glass transition temperature is low can reduce the temperature dependence ofthe generative force.

direction. The elastic force in this case was mainly
accounted for by repulsion force under compression generated by the compression of the particles, and the frictional force was generated by sliding friction between thecylinder and the particles. Furthermore, it was found thatin particular, the packing fraction and Young’s modulus,
which affects the material, exerted a large effect on the
maximum generative force and the hysteresis.
When both chambers were filled with particles, contrary
to the findings of single-chamber filling, it was found thatthe frictional force was generated by the sliding friction
between the cylinder and rod and the particles, whereby,
due to the friction of the rod, a waveform of the generative
force was asymmetrical with respect to the origin.
These results were demonstrated by means of simulations of generative force and particle behavior using thedistinct element method, as well as by experiments. The
simulation results and the experimental results were ingood agreement with one another both qualitatively andquantitatively, and it was confirmed that the simulations
were able to reliably reproduce the phenomena which
occurred in the experiments.
*“Elastomer” is a generic term for a polymer having elasticity. Rubber and the like correspond to this term.

This study used a separated dual chamber single rodtype particle damper which had two chambers divided bya piston and employed elastomer particles. An investigation was carried out to clarify the generative force characteristics and force generation factors or mechanism fortwo different cases: with particles filling only one of thetwo chambers and with particles filling both chambers.
Elastomer particles were selected with an aim at delivering a high generative force by densely filling them, which
was possible thanks to their high elastic deformation. The
investigation used two approaches: one is to develop a
damping theory based on the distinct element method andthe other to simulate generative force and particle behavior. The simulation results were compared with thoseobtained from experiments using an actual machine todetermine the generative force characteristics and mechanism and to verify the adequacy of the developed theory.

― 24 ―KAYABA TECHNICAL REVIEW No. 64 APR. 2022

Please note that this damper generates a combinedelastic and viscous force, although the term "damper"
might cause you to imagine a damper that can suppress
vibration by exerting a damping force attributable to apressure loss or viscous resistance of the fluid used. That's
why the force generated by this damper is called a "generative force", not a damping force, for clarification purposes.
2

Generative Force Characteristics of Particle
Damper with One of the Chambers Filled

2.1 Damper Structure
Fig. 1 shows the general structure of the damper used insimulations and experiments. The internal void of thedamper is divided into two chambers by a piston and only
the chamber without the rod is filled with spherical particles. The clearance between the piston perimeter and thecylinder bore is very small so that no particles can movefreely between the chambers. Friction in the sliding part isnegligible as it is considerably smaller than the generative
force of the damper. The piston displacement of thisdamper described in the following sections refers to the
movement of the piston in the direction of the z-axis
shown in Fig. 1.

d 2r i
(1)
m
＝Fi
i
dt 2
dΩi
Ii
＝Ti
(2)
dt
where:
i is particle number;t is time;
mi is particle mass;ri is position vector of the particle;
Fi is the sum total of contact forces;
Ii is inertia moment;
Ωi is angular velocity vector of the particle, and;Ti is the sum total of torques applied to the particle.
Fi, Ti and Ii can be expressed in Equations (3) to (5)below:
Fi＝Fcn＋Fct＋mi g(3)Ti＝ri×Fct(4)
8
(5)
Ii＝
ρπr 5
15
In the equations above, the subscript n indicates thenormal direction at the point of contact and the subscript t
the tangential direction at the point of contact. Fcn and Fct
are the contact force in the normal and tangential directions, respectively. mi is particle mass, g is gravitational
acceleration, p is particle density, and r is particle radius.
The concept of contact force was proposed by Cundall
and Strack (3). They used an analysis model consisting of
three elements shown in Fig. 2: a spring, a dashpot and afriction slider. The model can be expressed in Equations(6) and (7). Fig. 3 shows a conceptual rendering of theforce applied in the normal direction and the force appliedin the tangential direction.

Section A-A

Fig. 1 Damper structure
1. Cylinder, 2. End cover, 3. Bearing, 4. Rod, 5. Piston,
6. Elastomer particles, Unit: mm

2.2 Simulations and Experiments
2.2.1 Distinct Element Method
Simulations were conducted by using the distinct
element method (hereinafter "DEM") to clarify the particle
behavior, generative force characteristics, and components of the force. DEM is a method for computing the
behavior of particles by sequentially calculating the equation of the motion of particles for their translationalmotion and rotation at individual time points, with consideration given to contact interactions of particles.
2.2.2 Primitive Equation
DEM can be used to determine the velocity and positionof a particle by solving the equation of motion expressedin Equation (1) and the equation of angular motion
expressed in Equation (2) by taking into account thecontact force of each particle.
― 25 ―（）

Fcn＝ －Knδn1.5－CnVij・ni ni(6)Fct＝－Ktδt－CtVfij(7)Friction slider

(a)

(b)

Fig. 2 Contact model(a) Normal direction, (b) Tangential direction

Research on Particle Assemblage Damper

elastic modulus Kn of the particles during their contact.
Kniw indicates the elastic modulus Kn of the particle when
it makes contact with the cylinder wall. Ei and Ew are themodulus of longitudinal elasticity of the particle and thewall, respectively. vi and vw are Poisson ratios of the particle and the wall, respectively. The elastic modulus in thetangential direction is based on an assumption that noslide occurs at the point of contact and can be expressed in
Equations (15) and (16) in accordance with the Mindlin (4)theory:➡Normal direction, ➡Tangential direction

Fig. 3

Conceptual rendering of forces in the normal andtangential directions

In the equations above, Kn and Kt indicate elastic
modulus in the normal and tangential directions, respectively, and Cn and Ct are viscosity coefficients in thenormal and tangential directions, respectively. δn is displacement in the normal direction at the point of contactand δt displacement vector in the tangential direction at
the point of contact between the particles i and j. ni is theunit vector in the normal direction from the particle i tothe particle j at their contact point. Vij is the relative velocity vector of the particle i against the particle j. vi and vj arevelocity vectors of the particles i and j, respectively. Vfij is
the relative velocity vector in the tangential direction of
the particle i against the particle j at their contact point.
Vij, Vfij and δt can be expressed in Equations (8) to (10):

2 2a Gi 0.5
(15)
Ktij＝
δn
2－νi
8 a Gi 0.5
(16)
Ktiw＝
δ
2－νj nIn the equations above, Ktij is the elastic modulus of theparticles during their contact. Ktiw is the elastic modulus of
the particle and the wall during their contact. Gi is themodulus of transverse elasticity of the particle and can be
expressed in Equation (17):Vij＝νi－νj(8)

Ei
G
(17)
i＝
2（1＋vi）

Vfij＝Vij－(Vij・ni)ni＋2a(ωi－ωj)×ni(9)Cn and Ct can be expressed in Equations (18) and (19):δt＝Δt－Vfij(10)(18)C
mi Kn δn0.25
n＝α

In the equations above, a is the particle radius, and ωiand ωj are angular velocity vectors of the particles i and j.
Elastic modulus in the tangential direction can be
expressed in Equations (11) to (14) using the Hertziancontact theory:

（ ）

aδn
4 1
(11)
Knij＝
2
3π 2δi

（

）

4
1
(12)
aδn
Kniw＝
3π δi＋δw
1－vi2
δi＝
(13)
Eiπ
1－vw2
(14)
δw＝
Ewπ

(19)
Ct＝α mi Kt δct0.25
2
ln（e）
(20)
α＝2.2
2
ln（e）＋π

α is a dimensionless constant that decides the magnitudeof viscous force and can be expressed in Equation (20). e
in Equation (20) is the repulsion coefficient of the particle.
If the relative velocity vector in the tangential direction of
the particle on the contact surface is more than zero (0) or
the contact force in the tangential direction is higher than
the frictional force, it is considered that particle sliding
has occurred along the contact surface. The phenomenon
can be expressed in Equations (21) to (23):In the equations above, the subscript w indicates that therelevant amount is related to the wall surface of the cylinder bore (hereinafter "the cylinder wall"). Knij indicates the― 26 ―

If ｜Vfij｜＝0 and Fct≦μf｜Fcn｜,
Fct＝Fct(21)KAYABA TECHNICAL REVIEW No. 64 APR. 2022

If Fct＞μf｜Fcn｜,

Table 2

Fct＝－μf｜Fcn｜ti(22)
If |Vfij｜＞0,
Fct＝－μf｜Fcn｜ti(23)
Vfij
ti＝
(24)
|Vfij|
ti is unit vector in the Vfij direction of the particle and canbe expressed in Equation (24). μf is the friction coefficientof the particle.
Thus, the analysis model was designed to consider
elastic and viscous forces in the normal direction and, in
addition to these forces, frictional force as well in thetangential direction. This analysis also applied the secondorder accurate Adams-Bashforth method to time-steppingof displacement, velocity and angular velocity.
2.2.3 Simulation and Experimental Conditions
The overview of the experiment equipment is shown in
Fig. 4. The simulation conditions are shown in Table 1
and the experimental conditions are shown in Table 2. The
particles used in the experiments are shown in Photo 1.
Table 1

Simulation conditions

Particle material

Silicone rubber(TSE 3466 made by
Momentive Performance
Materials),
nitrile rubber (NBR madeby Inaba Rubber)Particle size [mm]3, 5Packing fraction [-]0.60, 0.70

Number of particles

1,339, 1,562

Stroke [mm]10

Excitation frequency [Hz]1, 5Time increments [sec.]

5.0×10-9

Excitation waveform

Sinusoidal

Particle density [kg/m3]

1.10×103

Poisson ratio of particle vi

0.5

Friction coefficient μf(wall-particle)

0.5

Friction coefficient μf(between particles)

0.5

Young's modulus Ew [GPa](wall)

210

Young's modulus Ei [MPa](particle)

4.08 (TSE3466),
17.6 (NBR)

Viscosity coefficient α

0.5311

Experimental conditions

Particle material

Silicone rubber (TSE 3466),
nitrile rubber (NBR)

Young's modulus Ei [MPa](particle)

4.08(TSE3466),
17.6(NBR)

Particle size [mm]3, 5Packing fraction [-]0.60, 0.70

Stroke [mm]10

Excitation frequency [Hz]1, 5Fig. 4 Experiment equipment
1. Controller, 2. Motor, 3. Laser displacement gauge, 4. Rod,5. Damper, 6. Load cell, 7. Dynamic distortion amplifier.
8, Oscilloscope, 9. Displacement gauge amplifier unit

(a)

(b)

(c)

Photo 1 Particles used(a) Silicone rubber, particle size 3 mm(b) Silicone rubber, particle size 5 mm(c) NBR, particle size 3 mm

For simulation purposes, the general coefficient offriction between rubbers of 0.5 was used as shown in
Table 1. (5) In a condition for which two or more parameters
are indicated in Table 1 or 2, the one in red is the reference
conditions. The packing fraction refers to the ratio ofparticles in the particle chamber and has been determinedby dividing the volume of all the particles present in thechamber by the volume of the chamber.
2.2.4 How to Calculate Generative ForceIn the simulations of the particle damper with only oneof the chambers filled, the total generative force can be
determined, as shown in Fig. 5, by adding the compres― 27 ―Research on Particle Assemblage Damper

Generative force

Load cell➡Direction of stroke➡Force applied to cylinder bottom by particle compression
and friction➡Force applied to cylinder wall by particles(a)Generative force

Load cell➡Direction of stroke➡Force applied to cylinder bottom by particle compression
and friction➡Force applied to cylinder wall by particles(b)Fig. 5 Generative force in the z-axis direction(a) Positive displacement, (b) Negative displacement

2.2.5 Generative Force and Its Normal andTangential Components
The results of the simulations are plotted in Fig. 6.
These curves show how the generative force and its component in the normal direction (hereinafter "the normalcomponent") and its component in the tangential direction(hereinafter "the tangential component") change withdisplacement under the reference conditions (i.e., forcedisplacement curves). Fig. 7 shows the force-velocity
curves under the same conditions. All the following
figures of these kinds plot a cycle of data. The displacement refers to the piston displacement. The time development of curves is clockwise for force-displacement curves
and counterclockwise for force-velocity curves.
According to Fig. 6, the generative force shows gradually hardening characteristics that the generative force
increases as the displacement advances in the direction ofparticle compression (hereinafter "the compression

process") and involves hysteresis as the piston travels inthe direction opposite to the direction of particle compression (hereinafter "the retraction process"). The graduallyhardening characteristics may be attributable to elastic
repulsion generated by particle compression. The hysteresis may be attributable to viscosity caused by deformation of elastomer particles and/or friction between
particles or between particles and the wall.
Now, let us move on to discussion about the normal andtangential components of the generative force. The normalcomponent is related to resistance generated primarily
during particle compression and the tangential component
is related to shear force generated between particles or
between particles and the wall as well as resistance
between particles or between particles and the wall generated when particles move around.
In the compression process, both the normal and tangential components show gradually hardening characteristics as shown in Fig. 6. The normal component has ahigher maximum value and lower hysteresis than for thetangential component. Since the normal component peaks
at around the point of 0 m/sec. velocity as shown in Fig. 7,
the effect of the viscosity attributable to compressive
deformation of particles in the normal direction should betrivial, which may result in the gradually hardening characteristics and the lower hysteresis. In other words, thenormal component consists predominantly of the elastic
force of particles due to their compression. The tangential
component also shows gradually hardening characteristics, but has higher hysteresis, resulting in a negativevalue that gradually becomes smaller as the displacement
advances during the retraction process. Since the viscosity
affecting the hysteresis depends on the velocity, the effectof viscosity on the tangential component is probably low
because it peaks at around 0 m/sec. velocity as shown in
Fig. 7. In addition, the force converges to 0 N at a displacement of -5 mm, which is the start point of the compression process. Therefore, the tangential component
consists predominantly of frictional force, to which thenormal component acts as vertical resistance. As the positive/negative sign to friction force is switched overaccording to the direction of displacement, the tangential
Generative force
Normal component
Tangential component
Force [N]sive force applied to the cylinder bottom end surface(hereinafter "the cylinder bottom") in the z-axis during
particle compression by the piston to the frictional forceof particles against the cylinder wall in the z-axis direction. The component of the generative force in the normal
direction is attributable to viscoelasticity of particles dueto their compressive deformation. The component of thegenerative force in the tangential direction is attributableto viscoelasticity of particles due to their shear deformation and sliding friction between particles and the cylinder
wall.

Displacement [mm]Fig. 6― 28 ―

Force-displacement curves of generative force andits normal and tangential components

Force [N]Force [N]KAYABA TECHNICAL REVIEW No. 64 APR. 2022

Generative force
Normal component
Tangential component
Velocity [mm/sec.]

Force-velocity curves of generative force and its normal and tangential components

component takes a negative value with higher hysteresis
during the retraction process. The frictional force generated here refers to the friction of sliding particles against
the cylinder wall.
2.2.6 Elastic and Viscous Forces in Normal/
Tangential Direction
To verify the assumption discussed in the previous
section that the normal component of the generative force
consists predominantly of elastic force and the tangential
component consists predominantly of frictional force, thissection further discusses these components by dividingthem into elastic and viscous forces. Fig. 8 shows theforce-displacement curves of the normal component andits breakdown into elastic and viscous forces. Fig. 9 is anenlarged view of the viscosity curve in Fig. 8. Fig. 10shows the force-displacement curves of the tangential
component and its breakdown into elastic and viscous
forces.
In Fig. 8, the normal component curve generally agrees
with the elasticity curve, which implies that the elastic
repulsion of particles due to their compressive deformation is predominant. However, the viscosity never takes avalue of zero (0). The enlarged view in Fig. 9 reveals thata minute force is always generated. Fig. 10 shows that thetangential component involves no elastic or viscous force
attributable to shear deformation of particles along thecylinder wall. Rather, the tangential component only con-

Force [N]Generative force
Normal component
Tangential component

Displacement [mm]Fig. 8

Force-displacement curves of normal component andits elasticity and viscosity

Fig. 9

Enlarged view of force-displacement curve ofviscosity of normal component

Tangential component
Elastic force
Viscosity
Force [N]Fig. 7

Displacement [mm]Displacement [mm]Fig. 10

Force-displacement curves of tangential component
and its elasticity and viscosity

sists of friction of particles against the cylinder wall,
which implies that particles are always sliding along thewall.
Now, the discussion above has revealed that the generative force is greatly affected by the elastic force of particles
in the normal direction due to their compressive deformation and the friction of particles against the cylinder wallin the tangential direction.
2.2.7 Distribution of Particle Compression
Since the generative force heavily depends on theelastic force in the normal direction, we tried to determine
the effect of the compressive force of particles filled in thedamper under the reference conditions. Fig. 11 (a) to (d)show where the piston and particles are located in thedamper when viewed from the section A-A in Fig. 1 andthe distribution of particle compression. The compressive
force on particles in these diagrams is due to the elasticity
in the normal direction.
In Fig. 11 (a), many of the particles close to the piston
are highly compressed during the compression process,resulting in higher particle repulsion against the piston.
Near the cylinder bottom in turn many particles are not sostrongly compressed. This means that the compression of
the particles close to the piston has not been transferredwell to the cylinder bottom. In Fig. 11 (c) for the retraction
process, many of the particles close to the piston are not sostrongly compressed, resulting in lower particle repulsion
against the piston. However, many of the highly com-― 29 ―

Research on Particle Assemblage Damperpressed particles still remain near the cylinder bottom.
This means that the compression of the particles close to
the cylinder bottom has not been transferred well to thepiston. It can be derived from these facts that there is atime lag between the forced excitation by the piston andthe transfer of forces caused by particle deformation.
In the compression process, the particle-particle compressive force, the force pressing particles against thecylinder wall, and the friction against which these twoforces act as vertical resistance increase, resulting in ahigher force applied to the cylinder bottom. In the retraction process, the particle-particle compressive force andthe force pressing particles against the wall becomeRod

Piston(a)Rod

Piston(b)

Rod

Pistonsmaller, resulting in lower friction. In this case, the force
applied to the cylinder bottom is the sum of the compressive repulsion of particles and the friction. Since the friction is switched from a positive value to a negative one orvice versa between the compression and retraction processes, the generative force at any given displacement
during the compression process is different from that at
the same displacement during the retraction process. The
discussion above has revealed that the delay in the transfer
of forces and the friction are one of the causes of the hysteresis of the generative force.
2.2.8 Comparison Between Experimental and
Simulation Results
To verify the adequacy of the simulations and determine
the generative force characteristics under different conditions, we conducted experiments under different conditions and compared the results with the simulation results.
Figs. 12 to 16 show the generative force-displacement
curves obtained from the experiments under the reference
conditions as well as under another set of conditions
where the packing fraction, excitation frequency, particle
material and particle size were changed from the reference conditions, along with the results of the simulations.
The particle material was changed from the reference
conditions in that only the Young's modulus of particles
was changed.
According to Figs. 12 to 16, the simulation results are ingood agreement with the experimental results both qualitatively and quantitatively. Like under the reference conditions, the generative force shows gradually hardening
characteristics during the compression process or hysteresis after the transition from the compression process to
the retraction process, regardless of the packing fraction,
excitation frequency, particle material or particle size.
Figs. 12 to 16 also show how the generative forcechanges under different conditions. Its peak and hysteresis
substantially change with the packing fraction and theYoung's modulus which affects the particle material. They
do not change so much with the excitation velocity or(c)Force [N]Rod

Experiment
Analysis

Piston(d)Fig. 11 Distribution of particle compression(a) Compression process, z = 0(b) z = 5(c) Retraction process, z = 0(d) z = -5

Displacement [mm]Generative force-displacement curves;
Experiment-simulation comparison
Under reference conditions (packing fraction 0.60, excitation
frequency 1 Hz, Young’s modulus 4.08 MPa, particle size
φ3 mm)
Fig. 12― 30 ―KAYABA TECHNICAL REVIEW No. 64 APR. 2022

Experiment
Analysis

Force [N]Force [N]Experiment
Analysis

Displacement [mm]Displacement [mm]Fig. 13G enerative force-displacement curves; Experimentsimulation comparison
With packing fraction changed (0.70)

Fig. 16Since the generative force characteristics are found to
be qualitatively the same under the different conditions,the particles in the damper should generally behave in asame way. It can now be concluded that the simulations
reliably reproduce the phenomena which occurred in theexperiments.

Force [N]Experiment
Analysis3

Displacement [mm]Fig. 14

Generative force-displacement curves; Experimentsimulation comparison
With particle size changed (φ5 mm)

Generative force-displacement curves; Experimentsimulation comparison
With excitation frequency changed (5 Hz)

Characteristics of Generative Force ofParticle Damper with Both Chambers Filled

3.1 Damper Structure
Fig. 17 shows the general structure of the damper used
for the simulations and experiments. The only difference
from the damper described in Chapter 2 is that both chambers divided by the piston are filled with particles. In thecase where both chambers are filled with particles, thechamber without the rod is called Chamber A and thatwith the rod Chamber B.

Force [N]Experiment
Analysis

Section A-A

Displacement [mm]Generative force-displacement curves; Experimentsimulation comparison
With material changed (Young’s modulus 17.6 MPa)

Fig. 15

particle size. The reason for the minor effect of excitation
velocity is the very small effect of viscosity according to
the simulation results. The reason why the particle size
does not influence so much is that, if larger particles areused, the particles would have larger deformation as thedisplacement advances but would have lower friction ofparticles against the cylinder due to the lower number ofparticles. This lower friction was a probable cause of why
the generative force did not increase so much.

Fig. 17 Damper structure
1. Cylinder, 2. End cover, 3. Bearing, 4. Rod, 5. Piston, 6.
Elastomer particles
Unit: mm

3.2 Simulations and Experiments
3.2.1 Simulation and Experimental Conditions
Table 3 shows the simulation conditions. Table 4 showsthe experimental conditions. The simulation methods,
primitive equations and experimental conditions describedin Chapter 2 were also used in these simulations andexperiments. The simulations and experiments in this
Chapter were conducted only under the reference conditions described in Chapter 2 because the generative force
characteristics have been generally verified in Chapter 2.

― 31 ―

Research on Particle Assemblage Damper

Simulation conditions

Table 3

Load cell

Particle material

Silicone rubber (TSE 3466)Particle size [mm]3

Packing fraction [-]0.60

Number of particles

2533

Stroke [mm]10

Excitation frequency [Hz]1Time increments [sec.]

5.0×10-9

Excitation waveform

Sinusoidal

Particle density [kg/m3]

1.10×103

Poisson ratio of particle vi

0.5

Friction coefficient μf(wall-particles)

0.5

Friction coefficient μf(between particles)

0.5

Young's modulus Ew [GPa](wall)

210

Young's modulus Ei [MPa](particle)

4.08(TSE3466)

Viscosity coefficient α

0.5311

Table 4

➡Direction of stroke➡Force applied to cylinder bottom by particle compression
and friction➡Force applied to cylinder wall by particles(a)Load cell➡Direction of stroke➡Force applied to cylinder bottom by particle compression
and friction➡Force applied to cylinder wall by particles(b)Force applied to cylinder bottom in the z-axis
direction(a) Positive displacement, (b) Negative displacement

Fig. 18

Experimental conditions

Particle material

Silicone rubber (TSE 3466)

Young's modulus Ei [MPa](particles)

4.08(TSE3466)Particle size [mm]3

Packing fraction [-]0.60

Stroke [mm]10

Excitation frequency [Hz]1

3.2.2 How to Calculate Generative ForceIn the simulations of the particle damper with both of its
chambers filled, the force applied to the cylinder bottom
Dz shown in Fig. 18 can be determined by adding thecompressive force applied to the cylinder bottom in thez-axis during particle compression by the piston to thefrictional force of particles against the cylinder wall in thez-axis direction. For simulations of a damper with bothchambers filled with particles, the force applied to the rod
end Rz shown in Fig. 19 can be determined by adding thecompressive repulsion applied to the rod end in the z-axis
during particle compression by the piston to the frictional
force of particles against the rod wall in the z-axis direction. There is a balance between Dz and Rz.
The normal component of the generative force is attributable to Viscoelasticity generated by compressive deformation of particles. The tangential component of the― 32 ―

Load cell➡Direction of stroke➡Force applied to piston by particle compression and friction
as well as friction of particles against cylinder wall
➡Force applied to rod wall by particles(a)Load cell➡Direction of stroke➡Force applied to piston by particle compression and friction
as well as friction of particles against cylinder wall
➡Force applied to rod wall by particles(b)Fig. 19 Force applied to rod end in the z-axis direction(a) Positive displacement, (b) Negative displacement

KAYABA TECHNICAL REVIEW No. 64 APR. 2022

friction of particles against the rod wall, and the tangential
friction of particles against the cylinder wall. The asymmetry of the generative force curve to the origin is attributable to the contact force of particles against the rod
located in Chamber B. This issue will be verified later inthe section discussing the results related to the generative
force Rz.
Fig. 21 shows that the generative force and its normal
and tangential components' peak at a point slightly shiftedfrom the 0 m/sec. velocity. Without the influence of viscosity, the force-displacement curve should peak at the
point of 0 m/sec. velocity. According to Equations (6) and(7), the velocity only affects the normal/tangential viscosity. Therefore, the maximum generative force occurs at apoint slightly shifted from 0 m/sec. probably because of
the effect of viscosity.
Then, another simulation was conducted with a focus
on the force applied to the rod wall as shown in Fig. 19.

Force [N]Generative force
Force applied to cylinder wall
Remaining component

Displacement [mm]Fig. 20

Force-displacement curves of generative force Dz,
force applied to cylinder wall and remaining
component
Generative force
Force applied to cylinder wall
Remaining component

Force [N]generative force is attributable to Viscoelasticity generated by shear deformation of particles and the sliding
friction of particles against the cylinder or rod wall.
Therefore, determining Dz will identify the friction ofparticles against the cylinder wall and determining Rz will
identify the friction of particles against the rod wall.
3.2.3 Generative Force and its Breakdown
As shown in Fig. 18, simulations were conducted with afocus on the force applied to the cylinder wall. The simulations produced the results shown in Figs. 20 and 21. Fig.
20 shows the generative force Dz that changes with displacement under the reference conditions, the tangential
component of Dz applied to the cylinder wall, and theremaining component. Fig. 21 shows the generative force
Dz that changes with velocity under the same conditions.
The remaining component was determined by subtracting
the tangential component applied to the cylinder wallfrom Dz. According to Fig. 20, the generative force shows
hardening characteristics and involves hysteresis after the
direction of displacement changes. In general, themaximum generative force during the negative displacement in which the particles in Chamber B are predominantly compressed is larger than the maximum generative
force during the positive displacement in which the particles in Chamber A are predominantly compressed. The
curve of the generative force is asymmetrical with respect
to the origin. Like the damper with one of the chambers
filled with particles, this damper also delivers the generative force with gradually hardening characteristics probably because of the effect of the elastic repulsion due toparticle compression. The hysteresis is attributable to theviscosity due to deformation of the elastomer particles
and/or the effect of the particle-particle or particle-wall
friction. The generative force has been plotted to be a nonlinear curve probably because, as the piston moves toward
Chamber B where the rod exists to increase the displacement, the particle-rod friction increases.
The next step is to discuss the generative force characteristics based on the force applied to the cylinder walland the remaining component. Fig. 20 indicates that, as
the displacement advances in each direction, both theforce applied to the cylinder wall and the remaining component increase. In this process, the force applied to thecylinder wall abruptly jumps when the direction of displacement changes. After a certain point of displacement,
the force mildly increases to show gradually hardening
characteristics. On the other hand, the remaining component involves this gradually hardening characteristics all
the time. So, the hysteresis is considered to be greatly
influenced by the force applied to the cylinder wall. The
maximum generative force is attributable to the force
applied to the cylinder wall and the remaining component
to almost the same extent. Here again, as in Fig. 18, thegenerative force applied to the cylinder bottom on the side
without the rod consists of the normal compression generated when the piston compresses particles, the tangential

Velocity [mm/sec.]

Fig. 21

Force-velocity curves of generative force Dz, force
applied to cylinder wall, and remaining component

Fig. 22 shows the generative force Rz that changes withdisplacement under the reference conditions, its tangential
component applied to the rod wall, and the remaining
component. The remaining component was determined bysubtracting the tangential component of the force applied
to rod wall from Rz. The only difference from Dz is that theforce applied to the rod wall shows gradually hardening― 33 ―

Research on Particle Assemblage Damper

characteristics during the negative displacement, andincreases until the displacement increases to a certain
positive level and then gradually decreases until it gets
down to about 0 N. The remaining component shows
almost the same characteristics as those of the generative
force however considerably differs from the generative
force in the negative displacement region. The force
applied to the rod wall shows gradually hardening characteristics as the friction of particles against the rod wall,
which acts as vertical resistance, increases due to theincrease in particle compression in Chamber B caused bynegative displacement. During the positive displacement,
the friction of particles against the rod wall decreases
because the particle compression in Chamber B decreases.
Furthermore, this force applied to the rod wall makes thegenerative force asymmetrical with respect to the origin.
The gradually hardening characteristics of the remaining
component is attributable to the compressive repulsion ofparticles and the friction of particles against the cylinderwall, which acts as vertical resistance. The hysteresis is
attributable to the friction of particles against the cylinderwall.
Now the force applied to the cylinder wall shown in Fig.
20 can be compared with the force applied to the rod wallshown in Fig. 22. It is obvious that the force applied to therod wall is smaller. This is because the rod has a smaller
outside diameter than the cylinder bore. In other words,
since the rod perimeter has a smaller surface area, the rod
is affected by particle friction to a smaller degree than thecylinder is affected by the same.
From the above, the discussion about Dz and Rz can besummarized as follows. The generative force in Chamber
A mainly consists of the normal repulsion generated
during particle compression by the piston and the tangential friction of particles against the cylinder. In Chamber
B, the tangential friction of particles against the rod is
added to the forces stated above to constitute the generative force. Furthermore, the generative force, the force
applied to the cylinder wall, and the remaining component
peak at a point shifted from the 0 m/sec. velocity as shownin Fig. 21. This implies that these forces are affected by
the viscous force of elastomer particles.

Force [N]Generative force
Force applied to rod wall
Remaining component

Displacement [mm]Fig. 22 F orce-displacement curves of generative force Rz,
force applied to rod wall, and remaining component

3.2.4 Elasticity and Viscosity of Force Applied
to Cylinder or Rod Wall and of Remaining
Component of Generative Force
To verify the effect of viscosity described in the previous section, the force applied to the cylinder or rod walland the remaining component of the generative force weredivided into elasticity and viscosity.
Fig. 23 shows the force-displacement curves of theforce of Dz applied to the cylinder wall and its elasticity
and viscosity. Fig. 24 shows the force-displacement
curves of the remaining component of Dz and its elasticity
and viscosity. Fig. 25 is an enlarged view of Fig. 24
showing the viscosity.
According to Fig. 23, the force applied to the cylinderwall only consists of the friction of particles against thecylinder. No elastic or viscous force is generated due toshear deformation of particles. Like the discussion about
the tangential component of the generative force in theparticle damper with one of the chambers filled, thisdamper also has sliding of particles against the cylinder inthe tangential direction all the time. According to Fig. 24,
the remaining component consists predominantly of elasticity with gradually hardening characteristics and hysteresis. The remaining component is generally in good
agreement with the elasticity. Therefore, like the normalcomponent of the particle damper with one of the chambers filled, this damper also delivers the generative force
whose remaining component consists predominantly ofthe elastic repulsion due to compressive deformation ofparticles. The effect of the viscous force generated during
particle deformation is very small. Unlike the normalcomponent of the particle damper with one of the chambers filled, this damper delivers the generative force
whose elasticity involves hysteresis. This is because theelastic modulus finally formed by the particle groups
filled in chamber A and chamber B are different in eachchambers. Another reason is, according to Fig. 18, that theelastic force of this damper includes the frictional force ofparticles against the rod. Therefore, properly speaking,this is not an elastic force or elasticity, but it is called here
an elastic force or elasticity for convenience purposes.
The viscous force never takes a value of zero (0). Rather,a minute viscous force is generated as shown in Fig. 25.
Fig. 26 shows the force-displacement curves of theforce of Rz applied to the rod wall and its elasticity andviscosity. Fig. 27 shows the remaining component of Rz
and its elasticity and viscosity. Fig. 28 shows an enlarged
view of the viscosity curve in Fig. 27.
According to Fig. 26, like Dz, Rz also comes with theforce applied to the rod wall only consisting of the frictionof particles against the rod. No elastic or viscous force isgenerated by particle deformation. This means that particles around the rod are sliding all the time. According to
Fig. 27, like Dz, Rz also comes with the remaining component consisting predominantly of elasticity with graduallyhardening characteristics and hysteresis. The remaining― 34 ―KAYABA TECHNICAL REVIEW No. 64 APR. 2022

reason is, according to Fig. 19, that the generative force of
this damper includes the friction between the cylinder andparticles. Like Dz, Rz also includes a minute viscous force
as shown in Fig. 28.
Now it may be useful to compare the generative force
characteristics of this elastomer particle damper with therepulsion characteristics of cylindrical or other-shapedelastomer blocks during compression. When elastomer
cylinders are compressed, many of them produce thegenerative force that depends on the displacement andinvolves viscoelasticity with hysteresis. In this damper inturn, friction of elastomer particles against the cylinder orForce [N]Force applied to cylinder wall
Elasticity
Viscosity

Displacement [mm]Fig. 23 F orce-displacement curves of component ofgenerative force Dz applied to cylinder wall andits elasticity and viscosity

Force [N]Force applied to rod wall
Elasticity
Viscosity

Force [N]Remaining component
Elasticity
Viscosity

Displacement [mm]Fig. 26 F orce-displacement curves of component ofgenerative force Rz applied to rod wall and its
elasticity and viscosity

Displacement [mm]Fig. 24 F
 orce-displacement curves of remaining
component of generative force Dz and its
elasticity and viscosity

Force [N]Force [N]Remaining component
Elasticity
Viscosity

Displacement [mm]Fig. 27 F orce-displacement curves of remaining
component of generative force Rz and its
elasticity and viscosity

Displacement [mm]Enlarged view of force-displacement curve ofviscosity of remaining component (in Fig. 24)component is generally in good agreement with the elasticity. Properly speaking, according to Fig. 19, this is not
an elastic force because the elasticity in Fig. 27 consists ofthe elastic repulsion of particles during compressive
deformation and the friction between the cylinder andparticles, but it is called here an elastic force or elasticity
as in the case of the force applied to the cylinder wall andthe remaining component. The reason why the elastic
force involves hysteresis is that, like Dz, the chambers
show different non-linear Young's modulus values
depending on the presence/absence of the rod. Another

Force [N]Fig. 25

Displacement [mm]Fig. 28― 35 ―

Enlarged view of force-displacement curve ofviscosity of remaining component (in Fig. 27)

Research on Particle Assemblage Damper

Rod

Piston(a)Rod

Piston(b)

Rod

Piston(c)

Rod

Piston(d)Fig. 29 Distribution of particle compression(a) Compression process, z = 0(b) z = 5(c) Retraction process, z = 0(d) z = -5

Experiment
Analysis

Force [N]rod wall occurs to greatly affect the generative force.
Thus, the generative force of this damper shows higher
hysteresis in comparison with the characteristics of theelastomer block of the same volume when it is compressed.
3.2.5 Distribution of Particle Compression
Like the particle damper with one of the chambers
filled, this damper with both chambers filled also produces
the generative force Dz heavily dependent on the othercomponents than that applied to the cylinder wall, that is,particularly the elastic force in the normal direction. Then,
we tried to determine the effect of the compressive force
applied to particles filled in the damper. Fig. 29 (a) to (d)show where the piston and particles are located in thedamper when viewed from section A-A in Fig. 17 and the
distribution of particle compression. The compressive
force applied to particles in these diagrams is due to theelasticity in the normal direction.
In Fig. 29 (a), as the piston travels in the direction of
positive displacement, many of the particles close to thepiston in Chamber A are highly compressed while manyof the particles close to the cylinder bottom are weakly
compressed. In Chamber B, many of the particles close to
the piston are weakly compressed while many of the particles close to the cylinder bottom are highly compressed.
This means that, as the piston travels in the direction of
positive displacement, the force applied to the piston by
the particles in the Chamber A increases while the force
applied to the piston in Chamber B decreases. This phenomenon also takes place when the piston travels in the
opposite direction as shown in Fig. 29 (c). However, acomparison between Figs. 29 (a) and 29 (c) reveals that
Chamber B in Fig. 29 (c) has a greater number of highly
compressed particles than in Chamber A in Fig. 29 (a)while Chamber A in Fig. 29 (c) has a greater number of
weakly compressed particles than in Chamber B in Fig. 29(a). This means that the compressive force applied toparticles in Chamber B during displacement in the negative direction has greater influence. This is probably
because Chamber B has a smaller volume than Chamber
A by the volume of the rod, thereby giving particles
smaller space to escape in Chamber B. As a result, particle
compression in Chamber B is started at a smaller displacement than in Chamber A. In addition, a delay in the transfer of the force due to particle deformation can be foundin Fig. 29 as in the case of the damper with one of thechambers filled.
3.2.6 Comparison Between Experimental and
Simulation Results
To verify the adequacy of the simulations, we conducted
experiments and compared the results with the simulation
results. Fig. 30 shows the generative force-displacement
curves obtained from the experiments and simulations.
According to Fig. 30, the simulation results are in good
agreement with the experimental results both qualitatively
and quantitatively. It can now be concluded that the simu-

Displacement [mm]Fig. 30 G enerative force-displacement curves; Experimentsimulation comparison― 36 ―KAYABA TECHNICAL REVIEW No. 64 APR. 2022

lations reliably reproduce the phenomena which occurredin the experiments, as in the case of the damper with oneof the chambers filled.

4

Concluding Remarks

In this study, an investigation and discussions wereconducted to clarify the generative force characteristics
and force generation factors of a separated dual chamber
single rod-type damper using elastomer particles for thetwo cases of one of the chambers filled and both chambers
filled.
When only one of the chambers is filled with particles,
the generative force shows gradually hardening characteristics along with hysteresis and is greatly affected by theelasticity in the normal direction and the friction in thetangential direction. This elasticity is mainly attributable
to the compressive repulsion of particles during compression. The friction here indicates the sliding friction ofparticles against the cylinder. Increasing the packing fraction, the excitation frequency, the Young's modulus which
affects the particle material, and the particle size will raise
the maximum generative force and hysteresis. In particular, the packing fraction and Young's modulus strongly
affect the generative force.
With both chambers filled with particles, the generative
force shows gradually hardening characteristics alongwith hysteresis and is affected by the frictional force
applied to the cylinder and rod walls in the tangential
direction and the elasticity included in the remaining
component, as in the case of the damper with one of thechambers filled. The force-displacement curve of thegenerative force is asymmetrical with respect to the origin
because of the frictional force applied to the rod wall.
The results of DEM-based simulations of a damper with
one of the chambers filled and a damper with both chambers filled were in good agreement with the experimental
results both qualitatively and quantitatively. This has
proved that the simulations reliably reproduce the phenomena which occurred in the experiments.

This report has now successfully clarified the generative force characteristics and force generation factors.
A future challenge is to enhance the accuracy of discussions and analyses. This report has mainly discussed thegenerative force based on its force-displacement curves
and distribution of particle compression. Rather, weshould determine the variations in elastic modulus ofparticles and the amount of particle deformation in theanalysis process and then verify them in connection withthe equations used for analysis. Furthermore, when considering to commercialize the damper, we also need todetermine how the generative force changes with the sizeof the piston or the length of the particle chambers and toclarify advantages and disadvantages of this damper compared with other types of dampers.
Note) This report has been prepared by re-editing the FY2020
doctoral dissertation thesis in Nagoya Institute of Technology: Generative Force Characteristics of Separated Dual
Chamber Single Rod-type Particle Damper Using Elastomer Particles.

References
1) IDO Y, HAYASHI K: Damping Force of Damper Utilizing a
Spherical Particle Assemblage, Proceedings of 15th International
Conference on Experimental Mechanics, 2012, Paper ref: 2714.
2) MORISHITA Y, IDO Y, MAEKAWA K, TOYOUCHI A: Basic
Damping Property of a Double Rod Type Damper Utilizing an
Elastomer Particle Assemblage, Advanced Experimental
Mechanics, Vol. 1, 2016, pp. 93-98.
3) P.A. Cundall, O.D.L. Strack: A Discrete Numerical Model for
Granular Assemblies, Géotechnique, Vol. 29, Issue I, 1979, pp.
47-65.
4) R.D. Mindlin: Compliance of Elastic Bodies in Contact,
Transaction of ASME, Series E, Journal of Applied Mechanics,
Vol. 16, 1949, pp. 259-268.
5) The Japan Society of Mechanical Engineers, JSME MechanicalEngineers' Handbook, Vol. 6, 1977, pp. 3-34, Maruzen
Publishing Co., Ltd.

Author 
TOYOUCHI Atsushi
Joined the company in 2009
New Products Development Sect.,
Developmental Center, Engineering
Headquarters, Automotive
Components Operations
Engaged in research anddevelopment of particle assemblage
dampers― 37 ―