# Study on the Damping Effect of Particle Dampers considering Different Surface Properties

Authors: Xiaowei Li; Yue Yang; Weixing Shi
Journal: Shock and Vibration
 (2019)

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## Abstract

Particle dampers are nonlinear vibration control devices. The surface property has a great influence on the performance of the particle damper, but it is difficult to be considered and analyzed. This paper firstly gives a view of how to establish a theoretic model of the particle damper. The dynamic equation and energy dissipation coefficient of collision are revised from the Hertz contact theory in the proposed theoretic model, considering the friction of particles. Then, a contrastive collision model relying on the finite element method is established to verify the reasonability of the theoretic model. The effects of different factors which will have an influence on the performance of the particle damper are discussed, and several conclusions on how to optimize the particle damper are proposed. Except for the aforementioned dynamic analysis, this paper also presents a particle damping index to evaluate the capability of energy dissipation of different materials, in order to facilitate the material selection in the practical design. Finally, an experiment is developed to verify the character of the collision and energy dissipation. The feasibility of the proposed method to estimate the surface property of different particles is validated by the free vibration experiment.


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## Body

## 1. Introduction

Civil structures are suffering from earthquakes and wind-induced vibrations. To protect a civil structure, one choice is to strengthen them. However, it may cost a lot but little effect is obtained, which is not sustainable. A sensible choice is to use the structural control technology [1–4], which includes passive control, active control, and semiactive control [5, 6]. In real projects nowadays, passive control still has the dominant application [7, 8]. The particle damper is one of the passive controllers [9–11] which can dissipate the vibration energy via friction and impact among the particles filled in a body cavity [12–15]. Particle dampers can be used in a harsh environment because its performance is insensitive to temperature [16–20]. Furthermore, particle dampers can dissipate vibration energy in a relatively broader range of frequency [21–24].Momentum exchange, friction, and acoustic radiation are generally considered as the main damping mechanism of the particle damper [25–27]. However, there is no consensus yet. Some researchers thought that the particle damper can reduce the response of primary structure via momentum exchange [28–30]. However, others hold the opinion that the momentum exchange cannot dissipate the kinetic energy for transient and random disturbances. Cempel and Lotz [31] investigated the damping performance of particle dampers and found that the energy dissipation was related to the impacts among particles or between particles and the container. Kerwin [32] referred that the particle material consumed the energy of the main structure by three approaches: (a) the friction among the particles, (b) nonlinear deformation of the contacting points among the particles, and (c) the resonance of the particle material. Xu et al. [33] investigated an elastic beam and plate structures with drilled holes filled with particles. They proposed a numerical procedure to predict the damping effect and validated the model predictions by experimental results. Lu et al. [34] adopted an elastoplastic collision model and a simplified frictional-elastic collision model to analyze the energy dissipation due to impact and friction, respectively. They conducted the free vibration tests and aeroelastic wind tunnel tests of a benchmark model attached with particle dampers to validate the rationality of the theoretical results. Lu et al. [35] provided an overview of particle damping technology on theoretical analysis and numerical simulation, together with their pros and cons. They tended to hold the opinion that the impact damping is mainly achieved by a certain kind of energy loss that was produced on the basis of the nonperfect elastic collision. Lu et al. [36] supposed that the vibration of the structure was attenuated by nonlinear deformation occurred in collisions among particles and collisions between particles and the container wall. They proposed a simplified numerical simulation method of equating multiple particles to a single particle. The simplified method was used for the preliminary optimal design of a PTMD (particle-tuned mass damper) attached to a five-story steel structure through a shaking table test. Although these researchers have explored that the impact and friction are the main energy dissipation mechanism of particle dampers, they are generally qualitative and based on either experimental experiences or numerical simulations. There is little theoretical research.The most widely used theory for the particle damping is the discrete element method (DEM) [37, 38]. While the DEM regards the particle as a rigid body, the particles are simplified as elements and the contact between particles is represented by some springs and damper interfaces. DEM does not consider the process of collisions adequately, and it will ignore the influence of materials on the vibration energy dissipation. Consequently, conclusions obtained from DEM must be independent of the properties of the particle surface. In other words, the surface property cannot be considered in the analysis via DEM.Because the energy absorbed during the plastic deformation is irreversible and is not permanently consumed via momentum exchange, the plastic deformation should be considered as one of the main factors of particle damping. Correspondingly, it is important to study the role of particle surface properties during collision. Therefore, this paper uses the Hertz contact theory as the basis of the collision analysis. The effect of different surface properties on collision is qualitatively evaluated. A new index, named particle damping index, is proposed to identify the particle materials. This index can be applied to the initial design stage of the particle damper to maximize the damping performance from the material’s perspective. Finally, an experiment is designed and proposed to verify the theoretical analysis.


## 2. Mechanical Model and Theoretical Analysis

### 2.1. Hertz Contact Theory

Hertz contact theory, which assumes that periodic positive collisions will occur between two balls under an external excitation, is introduced briefly in this section. In the theory, a motion cycle is divided into three phases [31]: elastic compression, elastoplastic compression, and elastic recovery. Some assumptions are introduced in the collision model, e.g., friction among particles, and friction between particles and the air is ignored during the whole process. The geometric deformation relationship in the elastic phase is still applicable in the elastoplastic phase, and the relationship between the recovery force and relative displacement in the elastic recovery phase is equal to that in the elastic compression phase. The three phases are introduced in the following.


#### 2.1.1. Phase of Elastic Compression

The parameters used in the theoretic model of two collision balls are as follows: elastic moduliE1 and E2, yield strengths σs1 and σs2, Poisson’s ratios μ1andμ2, radii R1 and R2, masses m1 and m2, normal velocities before collision vn1− and vn2−, tangential velocities before collision vt1− and vt2−, normal velocities after collision vn1+ and vn2+, tangential velocities after collision vt1+ and vt2+, relative deformation during collision δ, contact force p, contact radius ra, and plastic zone radius in the contact surface rp. Assuming that vn1−<vn2−, the normal relative velocity before collision will be vrn−=vn1−−vn2−, while normal relative velocity after collision will be vrn+=vn2+−vn1+, tangential relative velocity before collision will be vrt−=vt1−−vt2−, and tangential relative velocity after collision will be vrt+=vt2+−vt1+.Contact radiusra can be written as [38](1)ra=3p1/Et1+1/Et241/R1+1/R21/3.In equation (1), Et1=E1/1−μ12 and Et2=E2/1−μ22. When Et=Et1Et2/Et1+Et2 and R=R1R2/R1+R2, ra can be simplified as(2)ra=3pR4Et1/3.The schematic picture of the two-ball collision is shown in Figure1.Figure 1
Diagram of two-ball collision.The expression of the relative deformation in the two-ball collision is(3)δ=3p1/Et1+1/Et24ra=3p4Etra.The following expressions can be obtained from equations (2) and (3):(4)ra2=Rδ,(5)p=43EtRδ3/2.Then, the equation of motion in the elastic phase can be written as(6)md2δdt2=−43EtRδ3/2.In equation (6), the equivalent mass m is m=m1m2/m1+m2.According to the Hertz contact theory, the distribution of compressive stress in the elastic contact zone is approximately considered to be spherical. Then, the compressive stress of the point with radiusr can be expressed as(7)σr=σcrara2−r2.In equation (7), the contact stress at the contact center σc can be described as σc=3p/2πra2.According to formulas (4) and (5), σr is(8)σr=2EtπRra2−r2.At the end of the elastic compression phase, the compressive stress at the center of the contact zone reachedσp=minσp1,σp2, namely, σc=σp. Because the contact area is relatively small compared to the radius of the sphere, σp1 and σp2 can be derived approximately based on the solution of the rigid flat punch pressing:(9)σp1=1+π2σs1,σp2=1+π2σs2.Contact radiusra and relative deformation δ1 can be expressed as(10)ra1=πσpR2Et,δ1=Rπσp2Et2.Boundary conditions are considered as follows: ifδ=0, then δ˙=vr. Expression of the speed in the elastic compression phase can be obtained by solving equation (6):(11)v=vr−2−16EtR15mδ5/2.According to the requirementδ=δ1, at the end of the elastic compression phase, the expression of relative velocity of two balls are(12)δ˙1=vr−2−16EtR15mδ15/2.Assuming that the initial condition ist = 0 and δ=0, the relationship of the relative displacement over time can be obtained by the reintegration of equation (11) via the separation variable method. The expression of the relationship of t−δ in the elastic compression phase is(13)t=∫0δvr−2−16EtR15mδ5/2−1/2dδ.At last, the end time of the elastic compression phase can be obtained by the numerical integration:(14)t1=∫0δ1vr−2−16EtR15mδ5/2−1/2dδ.


#### 2.1.2. Phase of Elastoplastic Compression

As the contact force increases, the plastic zone emerges in the contact center. While the two balls continue to moving mutually, the plastic contact range and elastic contact range gradually expand outward. Whenrp represents the radius of the plastic zone at certain moment, then the contact force in the plastic contact zone can be expressed as rp=πrp2σp. If the stress distribution in the elastic contact zone is still expressed by equation (8), then the total contact force in the elastic annular contact zone can be expressed as(15)p1=∫rpra4rEtRra2−r2dr=4Etra2−r23/23R=σp3π3R26Et2.Combining equations (4) and (8), the total contact force in the elastoplastic compression phase will be(16)p=σp3π3R26Et2+πrp2σp=πRσpδ−σp3π3R212Et2.Similarly, the motion equation in the elastoplastic compression phase using Newton’s law can be written as(17)md2δdt2=−πRσpδ−σp3π3R212Et2.Considering the boundary condition as follows: ifδ=δ1, then δ˙=δ˙1. The velocity expression in the elastoplastic compression phase is(18)v=δ˙12+σp5π5R348mEt2+σp3π3R26mEt2δ−πRσpmδ2.The relative speed of the ball reduces to 0, which means thatδ˙=0; when the elastoplastic phase ends, the relative deformation δ2 is(19)δ2=δmax=σp2π2R6Et22+mδ˙12πRσp+σp2π2R12Et2.Initial conditions aret=t1 andδ=δ1. The relationship of the relative displacement over time can be obtained by the reintegration of equation (18) via the separation variable method. The expression of the relationship of t−δ in the elastoplastic compression phase is(20)t=t1+∫δ1δδ˙12+σp5π5R348mEt4+σp3π3R26mEt2δ−πRσpmδ2−1/2dδ.At last, the end time of the elastoplastic compression phase can be obtained by the numerical integration:(21)t2=t1+∫δ1δ2δ˙12+σp5π5R348mEt4+σp3π3R26mEt2δ−πRσpmδ2−1/2dδ.


#### 2.1.3. Phase of Elastic Recovery

Two balls will move away from each other after the collision; in other words, the relative speed drops to 0. The contact force in this phase is considered to be elastic recovery. Moreover, the relationship between the elastic recovery force and relative deformation in this phase is assumed to be equal to that in the elastic compression phase, except the deformation in the elastic recovery phase, which contains an irreversible plastic deformationδp. The contact force in the elastic recovery phase can be expressed as(22)p=43EtRδ−δp3/2.The force between the two balls is identical at the end of the second phase (elastoplastic compression) and the beginning of the third phase (elastic recovery). An expression can be obtained as follows:(23)πRσpδ2−σp3π3R212Et2=43EtRδ−δp3/2.And then,δp can be written as(24)δp=δ2−34EtRπRσpδ2−σp3π3R212Et22/3.The motion equation in the elastic recovery phase is(25)md2δdt2=−43EtRδ−δp3/2.Considering the boundary conditions as follows: ifδ=δ2, then δ˙=0. The velocity expression in the elastic recovery phase is(26)v=−1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2.A negative value is taken in equation (26) because the direction of relative deformation in the elastic recovery phase is opposite to the relative velocity. Considering that δ=δp at the end of elastic recovery phase, the relative speed at the end of the whole collision of the two balls is(27)vr+=v2+−v1+=1615EtRmδ2−δp5/2.Initial conditions aret=t2 and δ=δ2. The relationship of the relative displacement over time can be obtained by the reintegration of equation (26) via the separation variable method. The expression of the relationship of t−δ in elastic recovery phase is(28)t=t2−∫δ2δ1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2−1/2dδ.At last, the end time of the elastic recovery phase can be obtained by the numerical integration:(29)t3=t2−∫δ2δp1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2−1/2dδ.


#### 2.1.4. Definition of the Energy Dissipation Factor

Figure2 shows the curve of collision duration and relative deformation of three phases. The detailed derivation can be found in Sections 2.1.1∼2.1.3.Figure 2
Curve of collision duration-relative deformation of three phases.A cycle of collision is composed of three phases, i.e., the beginning of contact, relative motion, and finally separation. Then, the recovery factor is defined as(30)e=v2+−v1+v1−−v2−=vr+vr−.The energy loss value is obtained by calculating the difference in kinetic energy before and after collision. Firstly, the conservation of momentum during collision is displayed:(31)m1v1−+m2v2−=m1v1++m2v2+.The speed expression of two balls after the collision can be obtained through equation (27):(32)v1+=v1−−1+em2v1−−v2−m1+m2,v2+=v2−−1+em1v1−−v2−m1+m2.The energy dissipation factorη is defined as the ratio of the kinetic energy loss after the collision to the kinetic energy before:(33)η=1/2m1v1−2+1/2m2v2−2−1/2m1v1+2+1/2m2v2+22π1/2m1v1−2+1/2m2v2−2.The energy dissipation factor can measure the energy consumption caused by the collision and compare the collision energy consumption effects of different particles.


### 2.2. Revised Three-Phase Collision

It is unsuitable to analyze the collision of the particles using the Hertz contact theory because it ignores the friction generated during the collision although the specific contribution of friction to energy dissipation is not clear. Especially for the particles with surface properties close to linear elastomers, the main energy dissipation mechanism is the friction consumption, while the inelastic motion hardly occurs. In this paper, the friction mechanism is introduced into the Hertz contact model to revise the collision theory of particles. The friction between the balls is taken into consideration. The superposition of particles in contact is shown in Figure3.Figure 3
Superposition of particles in contact.For a particle, the tangential force is(34)Ft=−ktδt−ηtGct.Here,δt is a deformation in tangential direction caused by the tangential force; kt and ηt are the stiffness coefficient and damping coefficient in the tangential direction; and Gct is the relative speed of the contact point in the tangential direction, namely, slip speed. Taking the ball i as an example, Gct is(35)Gct=vti−vtj+riωti−ωtj.In the equation (35), ri is the radius of the ball, ωti and ωtj is the tangential acceleration of ball i and ball j. Stiffness coefficient kt in equation (34) can be determined according to the Hertz contact theory. The damping coefficient can be obtained through the following formula:(36)ηt=αmkn.Coefficientα is related to the recovery factor and usually set as 2 [35].If the following condition is detected during the collision, i.e., when the tangential frictional force is greater than the static frictional force, mutual movement between the two balls occurs:(37)Ft>usjFn.μ
s
j is the coefficient of static friction between the balls. The tangential force at this moment is revised as(38)Ft=−usdFn.μ
sd is the coefficient of dynamic friction between the balls. The existence of tangential velocity makes it possible for tangential mutual movement between the balls. The existence of tangential force and deformation makes it possible for the dissipation of energy. Therefore, the energy consumed by friction is(39)ΔEf=usdFnδt.The relative tangential velocity of two balls isvrt=v1t−v2t. At the moment the collision begins, take δt=0 and ignore the rolling of the ball, which means setting the ωt1 and ωt2 to be 0. Take kt=640N00.65/1+0.05N00.65, in which N0 is the normal positive pressure. Assuming that the normal action is a linear collision, p=4/3EtRδ. The tangential force is(40)Ft=−ktδt−ηtGct=αmknvt1−vt2=2×m×43EtRvt1−vt2.IfFt>μsj⋅p, that is, the mutual movement in the tangential direction occurs, Ft can be expressed as(41)Ft=usd43EtRδ.Exit form of collision of the balls can be described in two types: the tangential direction collision exit previously or the normal direction collision exist previously. That is, the normal collision can exit in the phase of elastic compression, or elastoplastic compression or elastic recovery, depending on the exit of tangential direction collision. The analysis process is shown in Figure4.Figure 4
Analysis process of the revised Hertz contact theory considering the tangential friction.


### 2.3. Revised Energy Dissipation Factor considering the Tangential Friction

Assumed that the normal motion can fully experience the three-phase collision (elastic compression, elastoplastic compression, and elastic recovery), Zhang [39] proposed a simplified formula which can be used to describe the energy dissipation factor:(42)Ek0−Ek=12m1−e2−1+e27μs22v01n−v02n2+21+eμsv01n−v02nv01t−v02t.In the formula,Ek0−Ek is the difference of kinetic energy before and after the collision; m is the effective mass of the two-ball system, m=m1⋅m2/m1+m2; e is the recovery factor; μs is the dynamic friction coefficient between the balls; and v01n, v02n, v01t, and v02t represent the initial normal velocity of two balls and initial tangential velocity of two balls, respectively. This formula can approximately describe the correlation between the normal and tangential collisions.The revised energy dissipation factor is obtained by referring to equation (42):(43)η=1/2m1v1−2+1/2m2v2−2−1/2m1vn1+2+1/2m2vn2+2+m1+eμsvn1−−vn2−vt1−−vt2−2π1/2m1vn1−2+vt1−2+1/2m2vn2−2+vt1−2.


## 2.1. Hertz Contact Theory

Hertz contact theory, which assumes that periodic positive collisions will occur between two balls under an external excitation, is introduced briefly in this section. In the theory, a motion cycle is divided into three phases [31]: elastic compression, elastoplastic compression, and elastic recovery. Some assumptions are introduced in the collision model, e.g., friction among particles, and friction between particles and the air is ignored during the whole process. The geometric deformation relationship in the elastic phase is still applicable in the elastoplastic phase, and the relationship between the recovery force and relative displacement in the elastic recovery phase is equal to that in the elastic compression phase. The three phases are introduced in the following.


### 2.1.1. Phase of Elastic Compression

The parameters used in the theoretic model of two collision balls are as follows: elastic moduliE1 and E2, yield strengths σs1 and σs2, Poisson’s ratios μ1andμ2, radii R1 and R2, masses m1 and m2, normal velocities before collision vn1− and vn2−, tangential velocities before collision vt1− and vt2−, normal velocities after collision vn1+ and vn2+, tangential velocities after collision vt1+ and vt2+, relative deformation during collision δ, contact force p, contact radius ra, and plastic zone radius in the contact surface rp. Assuming that vn1−<vn2−, the normal relative velocity before collision will be vrn−=vn1−−vn2−, while normal relative velocity after collision will be vrn+=vn2+−vn1+, tangential relative velocity before collision will be vrt−=vt1−−vt2−, and tangential relative velocity after collision will be vrt+=vt2+−vt1+.Contact radiusra can be written as [38](1)ra=3p1/Et1+1/Et241/R1+1/R21/3.In equation (1), Et1=E1/1−μ12 and Et2=E2/1−μ22. When Et=Et1Et2/Et1+Et2 and R=R1R2/R1+R2, ra can be simplified as(2)ra=3pR4Et1/3.The schematic picture of the two-ball collision is shown in Figure1.Figure 1
Diagram of two-ball collision.The expression of the relative deformation in the two-ball collision is(3)δ=3p1/Et1+1/Et24ra=3p4Etra.The following expressions can be obtained from equations (2) and (3):(4)ra2=Rδ,(5)p=43EtRδ3/2.Then, the equation of motion in the elastic phase can be written as(6)md2δdt2=−43EtRδ3/2.In equation (6), the equivalent mass m is m=m1m2/m1+m2.According to the Hertz contact theory, the distribution of compressive stress in the elastic contact zone is approximately considered to be spherical. Then, the compressive stress of the point with radiusr can be expressed as(7)σr=σcrara2−r2.In equation (7), the contact stress at the contact center σc can be described as σc=3p/2πra2.According to formulas (4) and (5), σr is(8)σr=2EtπRra2−r2.At the end of the elastic compression phase, the compressive stress at the center of the contact zone reachedσp=minσp1,σp2, namely, σc=σp. Because the contact area is relatively small compared to the radius of the sphere, σp1 and σp2 can be derived approximately based on the solution of the rigid flat punch pressing:(9)σp1=1+π2σs1,σp2=1+π2σs2.Contact radiusra and relative deformation δ1 can be expressed as(10)ra1=πσpR2Et,δ1=Rπσp2Et2.Boundary conditions are considered as follows: ifδ=0, then δ˙=vr. Expression of the speed in the elastic compression phase can be obtained by solving equation (6):(11)v=vr−2−16EtR15mδ5/2.According to the requirementδ=δ1, at the end of the elastic compression phase, the expression of relative velocity of two balls are(12)δ˙1=vr−2−16EtR15mδ15/2.Assuming that the initial condition ist = 0 and δ=0, the relationship of the relative displacement over time can be obtained by the reintegration of equation (11) via the separation variable method. The expression of the relationship of t−δ in the elastic compression phase is(13)t=∫0δvr−2−16EtR15mδ5/2−1/2dδ.At last, the end time of the elastic compression phase can be obtained by the numerical integration:(14)t1=∫0δ1vr−2−16EtR15mδ5/2−1/2dδ.


### 2.1.2. Phase of Elastoplastic Compression

As the contact force increases, the plastic zone emerges in the contact center. While the two balls continue to moving mutually, the plastic contact range and elastic contact range gradually expand outward. Whenrp represents the radius of the plastic zone at certain moment, then the contact force in the plastic contact zone can be expressed as rp=πrp2σp. If the stress distribution in the elastic contact zone is still expressed by equation (8), then the total contact force in the elastic annular contact zone can be expressed as(15)p1=∫rpra4rEtRra2−r2dr=4Etra2−r23/23R=σp3π3R26Et2.Combining equations (4) and (8), the total contact force in the elastoplastic compression phase will be(16)p=σp3π3R26Et2+πrp2σp=πRσpδ−σp3π3R212Et2.Similarly, the motion equation in the elastoplastic compression phase using Newton’s law can be written as(17)md2δdt2=−πRσpδ−σp3π3R212Et2.Considering the boundary condition as follows: ifδ=δ1, then δ˙=δ˙1. The velocity expression in the elastoplastic compression phase is(18)v=δ˙12+σp5π5R348mEt2+σp3π3R26mEt2δ−πRσpmδ2.The relative speed of the ball reduces to 0, which means thatδ˙=0; when the elastoplastic phase ends, the relative deformation δ2 is(19)δ2=δmax=σp2π2R6Et22+mδ˙12πRσp+σp2π2R12Et2.Initial conditions aret=t1 andδ=δ1. The relationship of the relative displacement over time can be obtained by the reintegration of equation (18) via the separation variable method. The expression of the relationship of t−δ in the elastoplastic compression phase is(20)t=t1+∫δ1δδ˙12+σp5π5R348mEt4+σp3π3R26mEt2δ−πRσpmδ2−1/2dδ.At last, the end time of the elastoplastic compression phase can be obtained by the numerical integration:(21)t2=t1+∫δ1δ2δ˙12+σp5π5R348mEt4+σp3π3R26mEt2δ−πRσpmδ2−1/2dδ.


### 2.1.3. Phase of Elastic Recovery

Two balls will move away from each other after the collision; in other words, the relative speed drops to 0. The contact force in this phase is considered to be elastic recovery. Moreover, the relationship between the elastic recovery force and relative deformation in this phase is assumed to be equal to that in the elastic compression phase, except the deformation in the elastic recovery phase, which contains an irreversible plastic deformationδp. The contact force in the elastic recovery phase can be expressed as(22)p=43EtRδ−δp3/2.The force between the two balls is identical at the end of the second phase (elastoplastic compression) and the beginning of the third phase (elastic recovery). An expression can be obtained as follows:(23)πRσpδ2−σp3π3R212Et2=43EtRδ−δp3/2.And then,δp can be written as(24)δp=δ2−34EtRπRσpδ2−σp3π3R212Et22/3.The motion equation in the elastic recovery phase is(25)md2δdt2=−43EtRδ−δp3/2.Considering the boundary conditions as follows: ifδ=δ2, then δ˙=0. The velocity expression in the elastic recovery phase is(26)v=−1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2.A negative value is taken in equation (26) because the direction of relative deformation in the elastic recovery phase is opposite to the relative velocity. Considering that δ=δp at the end of elastic recovery phase, the relative speed at the end of the whole collision of the two balls is(27)vr+=v2+−v1+=1615EtRmδ2−δp5/2.Initial conditions aret=t2 and δ=δ2. The relationship of the relative displacement over time can be obtained by the reintegration of equation (26) via the separation variable method. The expression of the relationship of t−δ in elastic recovery phase is(28)t=t2−∫δ2δ1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2−1/2dδ.At last, the end time of the elastic recovery phase can be obtained by the numerical integration:(29)t3=t2−∫δ2δp1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2−1/2dδ.


### 2.1.4. Definition of the Energy Dissipation Factor

Figure2 shows the curve of collision duration and relative deformation of three phases. The detailed derivation can be found in Sections 2.1.1∼2.1.3.Figure 2
Curve of collision duration-relative deformation of three phases.A cycle of collision is composed of three phases, i.e., the beginning of contact, relative motion, and finally separation. Then, the recovery factor is defined as(30)e=v2+−v1+v1−−v2−=vr+vr−.The energy loss value is obtained by calculating the difference in kinetic energy before and after collision. Firstly, the conservation of momentum during collision is displayed:(31)m1v1−+m2v2−=m1v1++m2v2+.The speed expression of two balls after the collision can be obtained through equation (27):(32)v1+=v1−−1+em2v1−−v2−m1+m2,v2+=v2−−1+em1v1−−v2−m1+m2.The energy dissipation factorη is defined as the ratio of the kinetic energy loss after the collision to the kinetic energy before:(33)η=1/2m1v1−2+1/2m2v2−2−1/2m1v1+2+1/2m2v2+22π1/2m1v1−2+1/2m2v2−2.The energy dissipation factor can measure the energy consumption caused by the collision and compare the collision energy consumption effects of different particles.


## 2.1.1. Phase of Elastic Compression

The parameters used in the theoretic model of two collision balls are as follows: elastic moduliE1 and E2, yield strengths σs1 and σs2, Poisson’s ratios μ1andμ2, radii R1 and R2, masses m1 and m2, normal velocities before collision vn1− and vn2−, tangential velocities before collision vt1− and vt2−, normal velocities after collision vn1+ and vn2+, tangential velocities after collision vt1+ and vt2+, relative deformation during collision δ, contact force p, contact radius ra, and plastic zone radius in the contact surface rp. Assuming that vn1−<vn2−, the normal relative velocity before collision will be vrn−=vn1−−vn2−, while normal relative velocity after collision will be vrn+=vn2+−vn1+, tangential relative velocity before collision will be vrt−=vt1−−vt2−, and tangential relative velocity after collision will be vrt+=vt2+−vt1+.Contact radiusra can be written as [38](1)ra=3p1/Et1+1/Et241/R1+1/R21/3.In equation (1), Et1=E1/1−μ12 and Et2=E2/1−μ22. When Et=Et1Et2/Et1+Et2 and R=R1R2/R1+R2, ra can be simplified as(2)ra=3pR4Et1/3.The schematic picture of the two-ball collision is shown in Figure1.Figure 1
Diagram of two-ball collision.The expression of the relative deformation in the two-ball collision is(3)δ=3p1/Et1+1/Et24ra=3p4Etra.The following expressions can be obtained from equations (2) and (3):(4)ra2=Rδ,(5)p=43EtRδ3/2.Then, the equation of motion in the elastic phase can be written as(6)md2δdt2=−43EtRδ3/2.In equation (6), the equivalent mass m is m=m1m2/m1+m2.According to the Hertz contact theory, the distribution of compressive stress in the elastic contact zone is approximately considered to be spherical. Then, the compressive stress of the point with radiusr can be expressed as(7)σr=σcrara2−r2.In equation (7), the contact stress at the contact center σc can be described as σc=3p/2πra2.According to formulas (4) and (5), σr is(8)σr=2EtπRra2−r2.At the end of the elastic compression phase, the compressive stress at the center of the contact zone reachedσp=minσp1,σp2, namely, σc=σp. Because the contact area is relatively small compared to the radius of the sphere, σp1 and σp2 can be derived approximately based on the solution of the rigid flat punch pressing:(9)σp1=1+π2σs1,σp2=1+π2σs2.Contact radiusra and relative deformation δ1 can be expressed as(10)ra1=πσpR2Et,δ1=Rπσp2Et2.Boundary conditions are considered as follows: ifδ=0, then δ˙=vr. Expression of the speed in the elastic compression phase can be obtained by solving equation (6):(11)v=vr−2−16EtR15mδ5/2.According to the requirementδ=δ1, at the end of the elastic compression phase, the expression of relative velocity of two balls are(12)δ˙1=vr−2−16EtR15mδ15/2.Assuming that the initial condition ist = 0 and δ=0, the relationship of the relative displacement over time can be obtained by the reintegration of equation (11) via the separation variable method. The expression of the relationship of t−δ in the elastic compression phase is(13)t=∫0δvr−2−16EtR15mδ5/2−1/2dδ.At last, the end time of the elastic compression phase can be obtained by the numerical integration:(14)t1=∫0δ1vr−2−16EtR15mδ5/2−1/2dδ.


## 2.1.2. Phase of Elastoplastic Compression

As the contact force increases, the plastic zone emerges in the contact center. While the two balls continue to moving mutually, the plastic contact range and elastic contact range gradually expand outward. Whenrp represents the radius of the plastic zone at certain moment, then the contact force in the plastic contact zone can be expressed as rp=πrp2σp. If the stress distribution in the elastic contact zone is still expressed by equation (8), then the total contact force in the elastic annular contact zone can be expressed as(15)p1=∫rpra4rEtRra2−r2dr=4Etra2−r23/23R=σp3π3R26Et2.Combining equations (4) and (8), the total contact force in the elastoplastic compression phase will be(16)p=σp3π3R26Et2+πrp2σp=πRσpδ−σp3π3R212Et2.Similarly, the motion equation in the elastoplastic compression phase using Newton’s law can be written as(17)md2δdt2=−πRσpδ−σp3π3R212Et2.Considering the boundary condition as follows: ifδ=δ1, then δ˙=δ˙1. The velocity expression in the elastoplastic compression phase is(18)v=δ˙12+σp5π5R348mEt2+σp3π3R26mEt2δ−πRσpmδ2.The relative speed of the ball reduces to 0, which means thatδ˙=0; when the elastoplastic phase ends, the relative deformation δ2 is(19)δ2=δmax=σp2π2R6Et22+mδ˙12πRσp+σp2π2R12Et2.Initial conditions aret=t1 andδ=δ1. The relationship of the relative displacement over time can be obtained by the reintegration of equation (18) via the separation variable method. The expression of the relationship of t−δ in the elastoplastic compression phase is(20)t=t1+∫δ1δδ˙12+σp5π5R348mEt4+σp3π3R26mEt2δ−πRσpmδ2−1/2dδ.At last, the end time of the elastoplastic compression phase can be obtained by the numerical integration:(21)t2=t1+∫δ1δ2δ˙12+σp5π5R348mEt4+σp3π3R26mEt2δ−πRσpmδ2−1/2dδ.


## 2.1.3. Phase of Elastic Recovery

Two balls will move away from each other after the collision; in other words, the relative speed drops to 0. The contact force in this phase is considered to be elastic recovery. Moreover, the relationship between the elastic recovery force and relative deformation in this phase is assumed to be equal to that in the elastic compression phase, except the deformation in the elastic recovery phase, which contains an irreversible plastic deformationδp. The contact force in the elastic recovery phase can be expressed as(22)p=43EtRδ−δp3/2.The force between the two balls is identical at the end of the second phase (elastoplastic compression) and the beginning of the third phase (elastic recovery). An expression can be obtained as follows:(23)πRσpδ2−σp3π3R212Et2=43EtRδ−δp3/2.And then,δp can be written as(24)δp=δ2−34EtRπRσpδ2−σp3π3R212Et22/3.The motion equation in the elastic recovery phase is(25)md2δdt2=−43EtRδ−δp3/2.Considering the boundary conditions as follows: ifδ=δ2, then δ˙=0. The velocity expression in the elastic recovery phase is(26)v=−1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2.A negative value is taken in equation (26) because the direction of relative deformation in the elastic recovery phase is opposite to the relative velocity. Considering that δ=δp at the end of elastic recovery phase, the relative speed at the end of the whole collision of the two balls is(27)vr+=v2+−v1+=1615EtRmδ2−δp5/2.Initial conditions aret=t2 and δ=δ2. The relationship of the relative displacement over time can be obtained by the reintegration of equation (26) via the separation variable method. The expression of the relationship of t−δ in elastic recovery phase is(28)t=t2−∫δ2δ1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2−1/2dδ.At last, the end time of the elastic recovery phase can be obtained by the numerical integration:(29)t3=t2−∫δ2δp1615EtRmδ2−δp5/2−1615EtRmδ−δp5/2−1/2dδ.


## 2.1.4. Definition of the Energy Dissipation Factor

Figure2 shows the curve of collision duration and relative deformation of three phases. The detailed derivation can be found in Sections 2.1.1∼2.1.3.Figure 2
Curve of collision duration-relative deformation of three phases.A cycle of collision is composed of three phases, i.e., the beginning of contact, relative motion, and finally separation. Then, the recovery factor is defined as(30)e=v2+−v1+v1−−v2−=vr+vr−.The energy loss value is obtained by calculating the difference in kinetic energy before and after collision. Firstly, the conservation of momentum during collision is displayed:(31)m1v1−+m2v2−=m1v1++m2v2+.The speed expression of two balls after the collision can be obtained through equation (27):(32)v1+=v1−−1+em2v1−−v2−m1+m2,v2+=v2−−1+em1v1−−v2−m1+m2.The energy dissipation factorη is defined as the ratio of the kinetic energy loss after the collision to the kinetic energy before:(33)η=1/2m1v1−2+1/2m2v2−2−1/2m1v1+2+1/2m2v2+22π1/2m1v1−2+1/2m2v2−2.The energy dissipation factor can measure the energy consumption caused by the collision and compare the collision energy consumption effects of different particles.


## 2.2. Revised Three-Phase Collision

It is unsuitable to analyze the collision of the particles using the Hertz contact theory because it ignores the friction generated during the collision although the specific contribution of friction to energy dissipation is not clear. Especially for the particles with surface properties close to linear elastomers, the main energy dissipation mechanism is the friction consumption, while the inelastic motion hardly occurs. In this paper, the friction mechanism is introduced into the Hertz contact model to revise the collision theory of particles. The friction between the balls is taken into consideration. The superposition of particles in contact is shown in Figure3.Figure 3
Superposition of particles in contact.For a particle, the tangential force is(34)Ft=−ktδt−ηtGct.Here,δt is a deformation in tangential direction caused by the tangential force; kt and ηt are the stiffness coefficient and damping coefficient in the tangential direction; and Gct is the relative speed of the contact point in the tangential direction, namely, slip speed. Taking the ball i as an example, Gct is(35)Gct=vti−vtj+riωti−ωtj.In the equation (35), ri is the radius of the ball, ωti and ωtj is the tangential acceleration of ball i and ball j. Stiffness coefficient kt in equation (34) can be determined according to the Hertz contact theory. The damping coefficient can be obtained through the following formula:(36)ηt=αmkn.Coefficientα is related to the recovery factor and usually set as 2 [35].If the following condition is detected during the collision, i.e., when the tangential frictional force is greater than the static frictional force, mutual movement between the two balls occurs:(37)Ft>usjFn.μ
s
j is the coefficient of static friction between the balls. The tangential force at this moment is revised as(38)Ft=−usdFn.μ
sd is the coefficient of dynamic friction between the balls. The existence of tangential velocity makes it possible for tangential mutual movement between the balls. The existence of tangential force and deformation makes it possible for the dissipation of energy. Therefore, the energy consumed by friction is(39)ΔEf=usdFnδt.The relative tangential velocity of two balls isvrt=v1t−v2t. At the moment the collision begins, take δt=0 and ignore the rolling of the ball, which means setting the ωt1 and ωt2 to be 0. Take kt=640N00.65/1+0.05N00.65, in which N0 is the normal positive pressure. Assuming that the normal action is a linear collision, p=4/3EtRδ. The tangential force is(40)Ft=−ktδt−ηtGct=αmknvt1−vt2=2×m×43EtRvt1−vt2.IfFt>μsj⋅p, that is, the mutual movement in the tangential direction occurs, Ft can be expressed as(41)Ft=usd43EtRδ.Exit form of collision of the balls can be described in two types: the tangential direction collision exit previously or the normal direction collision exist previously. That is, the normal collision can exit in the phase of elastic compression, or elastoplastic compression or elastic recovery, depending on the exit of tangential direction collision. The analysis process is shown in Figure4.Figure 4
Analysis process of the revised Hertz contact theory considering the tangential friction.


## 2.3. Revised Energy Dissipation Factor considering the Tangential Friction

Assumed that the normal motion can fully experience the three-phase collision (elastic compression, elastoplastic compression, and elastic recovery), Zhang [39] proposed a simplified formula which can be used to describe the energy dissipation factor:(42)Ek0−Ek=12m1−e2−1+e27μs22v01n−v02n2+21+eμsv01n−v02nv01t−v02t.In the formula,Ek0−Ek is the difference of kinetic energy before and after the collision; m is the effective mass of the two-ball system, m=m1⋅m2/m1+m2; e is the recovery factor; μs is the dynamic friction coefficient between the balls; and v01n, v02n, v01t, and v02t represent the initial normal velocity of two balls and initial tangential velocity of two balls, respectively. This formula can approximately describe the correlation between the normal and tangential collisions.The revised energy dissipation factor is obtained by referring to equation (42):(43)η=1/2m1v1−2+1/2m2v2−2−1/2m1vn1+2+1/2m2vn2+2+m1+eμsvn1−−vn2−vt1−−vt2−2π1/2m1vn1−2+vt1−2+1/2m2vn2−2+vt1−2.


## 3. Verification of Finite Element Analysis

Because the speed before and after the collision is directly related to the kinetic energy and then affects the energy dissipation, accurate estimation of the speed is the key to particle damping. The finite element collision model of two copper balls, with the same material and diameter, is established in ANSYS. Hexahedral solid element (solid 185) is used to simulate the behavior of balls with large strain during collision. The bilinear isotropic model is applied to consider the nonlinear property of the material, and it is suitable for large plastic strain analysis. Material parameters of the balls are elastic modulusE = 120 GPa, yield stress σs=100MPa, density ρ=7.9×103Kg/m3, and Poisson’s ratio μ=0.33. Tangential elastic modulus is calculated according to the formula: G=E/21+μ. Hex/wedge-sweep method is used in dividing the grid, and the grid size is 2 mm. Face-to-face contact pairs are defined via appointing one ball as a target unit (TARGE 170) and appointing another as the contact unit (CONTA 174). The contact type is chosen as “frictional” to consider the effects of friction, and the coefficient of friction is assigned as μs=0.2. ANSYS utilizes the contact stiffness (or penalty stiffness, FKN) to ensure the coordination of the contact interface. Small contact stiffness may lead to excessive intrusion in the contact surface, which means that a target surface penetrates the contact surface seriously. Large contact stiffness may cause the jump of the balls apart from each other, as well as the oscillation of calculation results. The contact stiffness is initially set as 0.1E, where E is the elastic modulus. Reducing the contact tolerance (FTOLN in ANSYS) also prevents an excessive intrusion of two contact surfaces. If the tolerance is too small, it will make it difficult to be converged. The FTOLN value is set to be 0.1. The normal and tangential displacement constraints of the balls are released, as well as their rotational constraints along the vertical axis, respectively. Initial speed of each ball is set to be in the normal and tangential direction. Large displacement transient is chosen as the analysis method, and the time at the end of a loadstep is set to be 1.8×10−4s, and the number of substeps is 100. The results of two methods (theoretic model and finite element model) are compared with different initial velocity and diameter of the balls. Initial velocity in the normal direction of the right ball before collision is −2.5, −1.5, −1.0, 0.0, 0.5, and 1.0 m/s respectively, while the initial velocity in the normal direction of the left ball is 1 m/s. The initial velocity in the tangential direction of two balls (left ball and right ball) is 0.0 and 0.0, 0.1 and −0.1, and 0.5 and −0.5 m/s. The diameter of the balls D is 5, 10, 20, and 30 mm, respectively. The finite element model in ANSYS is shown in Figure 5.Figure 5
Finite element model in ANSYS.Curves of the before-collision speed and the after-collision speed of the balls are presented in Figures6 and 7.Figure 6
Relationship between the after-collision speed of the left ball and the before-collision speed of right ball.Figure 7
Relationship between the after-collision and before-collision speed of the right ball.The difference between two models (theoretic model and finite element model) can also be observed by comparing the relative deformation of the collision process. Figures8–10 are curves of collision duration-relative deformation, with different tangential velocities. The X-axis is time, and the Y-axis is relative deformation. The relative deformation in the finite element model is calculated from the center to center of two balls.Figure 8
Curve of collision duration-relative deformation (tangential velocity of two balls is 0 m/s).Figure 9
Curve of collision duration-relative deformation (tangential velocities of two balls are 0.1 m/s and −0.1 m/s).Figure 10
Curve of collision duration-relative deformation (tangential velocities of two balls are 0.5 m/s and −0.5 m/s).The conclusion can be obtained that the result of the theoretic model coincides well with the finite element model, which can be seen in Figure8. Results of the theoretic model will have a larger deviation compared with the finite element if considering the tangential speed of the ball. It can be seen that the tendency deviates from Figures 9 and 10. This is owing to the hypothesis of theoretic model that the collision will carry through the whole three-phase collision (elastic compression, elastoplastic compression, and elastic recovery). Actually, if the tangential speed exceeds the limit, collision process will exit prematurely in the elastic compression phase or elastoplastic compression phase. Therefore, if the collision has a large tangential velocity, the theoretic model proposed in this paper cannot be accurately analyzed.The trends of the curves of collision duration-relative deformation obtained by two models are identical in spite of the discrepancy caused by the tangential movement. Therefore, the theoretic model has a sufficient accuracy in the factor analysis of damping effect with different surface properties.


## 4. Parametric Study on the Particle Damping

The particle damping has a complex correlation with the density, elastic modulus, yield strength, friction coefficient, etc. In order to study the role of a certain factor, we can use the theoretic model of the particle collision established previously and the control variable method to analyze the impact of different factors on energy dissipation.The discussion on density is shown in Figure11.Figure 11
Curve of the energy dissipation factor under different densities.As can be seen from Figure11, the energy dissipation increases with the increase of density. This owes to that the higher the density is, the larger mass the particles have. Under the same conditions, the inertial force and the plastic deformation generated by the collision increase if the mass increases. Thereby, the energy dissipation factor increases as well. But the energy dissipation factor does not change linearly with density. When the density reaches a certain value, the increase of the energy dissipation factor is very small.The discussion on dynamic friction coefficient is proposed in Figure12.Figure 12
Curve of the energy dissipation factor under different dynamic coefficients of friction.As can be seen from Figure12, the energy dissipation factor increases with the increase of the dynamic friction coefficient. And in the case of tangential velocity increases, the trend of the increase of energy dissipation will be more significant.The discussion on elastic modulus is presented in Figure13.Figure 13
Curve of energy dissipation factors under different elastic modulus ratios.As can be seen from Figure13, the energy dissipation factor increases with the increase of the elastic modulus. But the rise of the factor is not great except for the curve with a modulus ratio of 0.1. Modulus ratio can be expressed as E2/E1. E1, E2 is the elastic modulus of the balls. The energy dissipation factor rises when the elastic modulus ratio increases. The rise trend is larger when the elastic modulus ratio is relatively small (0.1∼10), and the trend is smaller when the elastic modulus ratio is relatively large (10∼100). Overall, the increase in the elastic modulus ratio and the increase in the elastic modulus value are not particularly significant for the contribution of the energy dissipation factor.The discussion on yield strength is shown in Figure14.Figure 14
Curve of the energy dissipation factor under different yield strengths.As can be seen from Figure14, the energy dissipation factor decreases when the yield strength increases. Plastic deformation has a significant influence on the energy dissipation factor, while the yield strength reflects the ability of plastic deformation of the material. As long as the ball has a small yield strength, which means the material is more susceptible to plastic deformation, the more the vibration energy stored in the plastic deformation, the larger the energy dissipation factor. Yield ratio can be expressed as σs2/σs1. σs1 and σs2 are the yield stress of the balls. When the yield ratio is less than 1, the energy dissipation factor has different decrease trends. When the yield ratio is greater than 1, all the trends of decrease coincide. This is caused by the hypothesis of the Hertz contact theory: σp=minσp1,σp2.Some factors, e.g., fill ratio, fill position, and excitation, cannot be taken into account because the theoretic model is proposed aiming at the analysis of particle collision process. The discussions focus on the properties of the material. These factors can be studied experimentally. Some conclusions have been drawn by studies as follows. The optimal fill ratio of the particle damper is between 20%∼40% [33]. The vibration response of the main structure decreases when the particle mass ratio increases. But the reduction of vibration and increase of mass ratio are not linear [34]. The damping effect of the cylindrical container is better than that of a cuboid, which is probably because the cylindrical container has a good symmetry and the damping effect is not affected by the external excitation direction [34]. The damping effect becomes better with a high relative position to the ground because the high floor will have a larger displacement and more kinetic energy. The vibration of the floor will be attenuated via the dissipation of kinetic energy transferred from the structure to the particles [35]. The damping effect of the particle damper will be better if the main structure has a lower inherent damping. The particle damper can reduce the response of the main structure in a wide frequency band, no matter whether the external excitation frequency is close to the natural frequency of the structure. However, the particle damper will have an amplification effect when the external excitation frequency is much smaller than the natural frequency of the main structure [35]. As the intensity of the excitation increases, the particles in the container become more active and the momentum exchange between the particles and main structure also increases, as well as the energy dissipation. But for a certain size of container, the damping effect is no longer affected when the excitation is large enough to provoke all particles to collision [36]. If the particles have different materials, the greater the difference in hardness, that is, the greater the difference in elastic modulus and yield strength is, the more significant the damping effect would be.


## 5. Evaluation of Particle Materials

The collision balls used in particle dampers generally have the same material, meaning the same size, mass, mechanical properties, etc. According to the derivation of the theoretic model above, a dimensionless coefficient named particle damping indexζ is proposed to indicate the energy-consuming capacity of different particles during the collision:(44)ζ=υEρσs2vn2+μsvt2=3mυEvn2+μsvt24πR3σs2.The particle damping index makes it convenient for the selection and comparison of different materials. Physical properties of several materials which are commonly used are listed in Table1, as well as the particle damping index and energy dissipation factor. The normal and tangential velocity used in the formulas is set as a fixed value of 1 m/s and 0.1 m/s, respectively. As we can see from Table 1, the trend of particle damping index practically coincides with energy dissipation factor. In conclusion, it can be judged that the material has a better performance in energy dissipation if its particle damping index is larger.Table 1
Physical properties and energy consumption of various materials.
Material
Elastic modulus,E (GPa)
Yield strength,σs (MPa)
Density,ρe (kg/m3)
Dynamic friction coefficient,μs
Poisson’s ratio,μ
Particle damping index,ζ
Energy dissipation factor,η
Copper
120
100
8500
0.20
0.34
34.749
0.147
Steel
210
210
7800
0.10
0.27
14.038
0.139
Concrete
17
25
2400
0.60
0.20
13.134
0.132
Cast iron
160
250
7200
0.15
0.30
5.537
0.129
Alumina
70
100
3100
0.50
0.25
4.748
0.131
Glass
80
70
2700
0.30
0.20
4.342
0.129
Wood (pine)
14
40
600
0.40
0.25
1.317
0.102
Aluminum alloy
75
300
2700
0.32
0.33
0.7448
0.087
Rubber
0.0061
7
1000
0.45
0.45
0.056
0.004
The yield strength of concrete, glass, and alumina in this table uses their compressive strength. The energy dissipation factor is calculated according to the theoretic model established above.


## 6. Experiment of Particle Dampers

An experiment is designed to analyze the damping effects of particle dampers filled with different particles. The experimental model is a single-degree-of-freedom (SDOF) frame structure. The bottom of the frame is fixed to the support, and the top is a beam with a relatively large rigidity. After giving an initial displacement in the horizontal direction at the top of the frame, it will swing freely in this direction. The natural frequency of the frame is adjusted to 1∼2 Hz with reasonable material, size, and top weight, approximately simulating the vibration of tall buildings. The container of the particle damper is glued on the beam. The shape and dimensions of the experimental model are shown in Figure15. The measured natural frequency of the frame is 1.69 Hz, and the total weight is 5.76 kg.Figure 15
Experimental model of the SDOF frame structure (Unit: mm). (a) Photo. (b) Size. (c) Particle damper.
(a)
(b)
(c)The frame is fixed on the experimental site, and the accelerometer is installed on the side wall near to the top. Different kinds (divided into 7 conditions) of particles are fill in the damper container. Types of particles in each case can be seen in Table2. And a horizontal initial displacement of 30 mm is placed to the top beam. The frame for free vibration is released and observed, and the vibration attenuation is recorded at the same time. Table 2 also lists the values of the natural frequency and the revised damping ratio of the structure in each case.Table 2
Natural frequency and revised damping ratio of each case.
Case
Particles
Natural frequency (Hz)
Revised damping ratio (%)
Fill ratio (%)
Mass ratio (%)
Remarks
1
Empty
1.69
0.45
–
0.00
Free vibration
2
9 steel balls,D = 20 mm
1.65
1.41
70
10.99
Same fill ratio
3
9 alumina balls,D = 20 mm
1.68
0.95
70
4.72
4
14 glass balls,D = 15 mm
1.67
0.69
70
2.75
5
3 steel balls,D = 20 mm
1.68
0.79
24
3.66
Same mass ratio
6
7 alumina balls,D = 20 mm
1.67
0.61
55
3.66
7
19 glass balls,D = 15 mm
1.68
0.63
95
3.66
Fill ratio: projecting from the top to bottom, the ratio of the balls’ area to the bottom area of the container. Mass ratio: ratio of particle mass to frame mass.Figures16–22 show the acceleration-time history curves of the frame in 7 cases, as well as the identified frequency via applying Fourier transform and removing interference frequency. The damping ratio of part (b) in Figures 16–22 is a synthesis calculation of multiple vibration periods, and it is related to the initial state of the frame and the length of the intercepted time. The revised damping ratio is calculated again using the logarithmic decay rate method to eliminate the interference of monitoring data, as shown in Table 2.Figure 16
Monitoring results without the particle damper. (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Figure 17
Monitoring results with the particle damper (9 alumina balls in container). (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Figure 18
Monitoring results with the particle damper (9 steel balls in container). (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Figure 19
Monitoring results with the particle damper (14 glass balls in container). (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Figure 20
Monitoring results with the particle damper (19 glass balls in container). (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Figure 21
Monitoring results with the particle damper (7 alumina balls in container). (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Figure 22
Monitoring results with the particle damper (3 steel balls in container). (a) Acceleration-time history curve of the frame. (b) Identified frequency and damping ratio.
(a)
(b)Conclusions can be obtained from Table2 and Figures 16–22 as follows:(1)
Vibration of the frame is significantly reduced by the additional particle damper. While the initial damping ratio is 0.45%, the minimum damping ratio of the damper cases is 0.63% and the maximum damping ratio of the damper cases is 1.41%. The acceleration curves of the frames with dampers attenuate faster, indicating a significant damping effect.(2)
In the cases of the same fill ratio (cases 2, 3, and 4), the damping effect of steel balls is better than the alumina balls and the damping effect of alumina balls is better than the glass balls. In the cases of the same mass ratio (cases 5, 6, and 7), the steel balls still have the best damping effect and the damping effect of 19 glass balls and 7 alumina balls is nearly the same. The regulation of the experiment is identical with that shown in Table1. The particle damping index ζ in Table 1 of steel, glass, and alumina is 14.04, 4.34, and 4.75, respectively. The conclusion obtained from both experimental and theoretical analysis is that the damping effect of some “flexible” materials (such as copper and lead) is better than that of the “brittle” materials (such as stone and glass).(3)
In the cases of the same fill ratio (cases 2, 3, and 4), the difference of damping effect in material selection is very significant, while the damping ratio changes from 1.41% to 0.98% to 0.68%. In the cases of the same mass ratio (cases 5, 6, and 7), the difference of damping effect in material selection is not significant, while the damping ratio changes from 0.79% to 0.61% to 0.63%. In summary, when the mass ratio of the particles is adjusted properly, it is possible to choose particles which are inexpensive and easy to access.(4)
In the cases of the same particle material (cases 2 and 5, 3 and 6, and 4 and 7), the mass ratio of “flexible” materials will have a more significant impact on damping than that of the “brittle” materials. For example, when the mass ratio’s increase is same, the damping effect becomes better for the steel balls than the glass balls. But the relationship of mass ratio and damping ratio is not linear. For example, the damping ratio of 14 glass balls is 0.68%, but it drops to 0.63% when filled with 19 glass balls. The mass ratio increases while the damping ratio decreases. This is because when the fill ratio is relatively large, the movement space for the particles in the container is reduced. The kinetic energy of each ball reduces relatively. So the energy dissipation by the collision decreases. Even when the fill ratio is too large, the balls pile up and are difficult to move and the particle damper becomes invalid. In design, a balance between the high mass ratio and fill ratio is extremely important.


## 7. Conclusions

Particle dampers can increase the damping effect and effectively consume the vibration energy of the main structure. While metal particles have advantages in energy dissipation relatively, common building materials such as stone and concrete can also be applied as particles in the damper, deduced form particle damping indexζ. However, their practical effects on damping need to be verified by further theories and experiments. The mechanism of the particle damper is very complicated, because of various uncertainties and nonlinearities during the collision, as well as numerous mutual coupling factors which will affect the results. Effects of some factors can be derived from the theoretic model, while others require a lot of experimental researches.This paper firstly gives a view of how to establish a theoretic model of the particle damper. The dynamic equation and energy dissipation coefficient of collision are revised from the Hertz contact theory, considering the friction of particles. Then, another collision model established by the finite element method verifies the reasonability of the theoretic model. The effects of different factors which will have an influence on the particle damping are discussed, and several conditional conclusions on optimized design are proposed. In spite of the aforementioned theoretical analysis, this paper also presents a particle damping index to evaluate the capability of energy dissipation of different particle materials, in order to facilitate the material selection. Finally, an experiment is developed to verify the character of the collision and energy dissipation. The feasibility of the proposed method to estimate the surface properties of different particles is proved by the free vibration experiment.It is found that the properties of the particles have a significant impact on particle damping. The damping effect increases especially for materials with high elastic modulus and low yield strength, as well as some secondary factors, like high density, large normal velocity, and large dynamic friction coefficient.



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*Source: 8293654-2019-07-21.xml*