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Particle damping technology can greatly reduce vibration of equipment and structure through friction and inelastic collisions of particles. An energy dissipation model for particle damper has been presented based on the powder mechanics and the collision theory. The energy dissipation equations of friction and collision motion are developed for the particle damper. The rationality of energy dissipation model has been verified by the experiment and the distributions for the energy dissipation of particles versus acceleration are nonlinear. As the experiment process includes lots of factors of energy dissipation, such as the noise and the air resistance, the experimental value is about 7% more than the simulation value. The simulation model can provide an effective method for the design of particle damper. And the particle parameters for damper have been investigated. The results have shown that choosing an appropriate particle density, particle size, and particle filling rate determined based on the simulation model will provide the optimal damping effect for the practical application of particle damping technology.

Particle damping technology can greatly reduce vibration of equipment and structure by friction and inelastic collisions of particles [

The mechanism of energy dissipation for particle damping is involved in the mechanical behavior and particulate matter dynamics. The high damping performance in vibration reduction has led to advances through research [^{7} [^{8}, and it hinders the development of DEM. Now, some new methods are introduced into the research [

When the main structure and the damper are excited, the kinetic energy is dissipated through friction and inelastic collision between particles and damper. On this occasion, the powder mechanics is effective for calculating frictional energy dissipation, and the theory of collision energy is effective for calculating collisional energy dissipation for particle damper, which need less computation.

For the frictional motion, we established a model for particle damper based on powder mechanics, and the frictional energy dissipation was calculated. And for the inelastic collision motion, we established a collision model based on the motion of particles and the damper boundary. Then, the total energy dissipation was calculated, and the simulation results were compared with experimental results. So the simulation model can provide a new analysis method for the practical application of particle damping technology.

The discrete element method can simulate the response of particle and damper with small numbers of particles. The large number of particles in the damper (in excess of 10^{8}) will make the DEM method computationally very demanding. A simple assessment method considering layer’s pressure is of important value in the design of damper with large number of particles.

In this paper, a parameter

If the total height of the rectangular particle damper is

The forces on the particle layer

The forces on the particle layer

Substituting (

The distance of adjacent particles from the thinnest to the thickest is given by

During the slippage of particles, on the basis of the energy conservation law, the kinetic energy for particle system will be dissipated due to friction work. As the particle system is under a harmonic excitation, the vibration of particle damper will be reduced.

Assuming the depth of particles is

In this paper, a model of collision energy dissipation for particle damper is established based on the collision theory.

In order to judge the particle position at every moment, the local coordinates of interaction for the particles are established in Figure

Coordinates of collision for adjacent particles.

Assuming that the mass of particle 1 is

Before the collision of particles, assuming that the velocity of particles 1 and 2 in the local coordinates is

After the collision of particles, based on the momentum theorem, the equation for particles 1 and 2 is given by

The symbolic function

Assuming

The equation of velocity before and after collision of adjacent particles in local coordinates is given by

When two particles collide, one particle will move to a new position. After a time step

As the particles keep moving in the damper, the particles collide with the inner damper. The contact between the damper and the particles can be done as the mass-spring-damping system.

Assume that the mass for particle

Model of collision between damper and particles.

During the compression phase, based on the theorem of impulse, the equation of impulse is written as

After the compression phase, the component

During the recovery phase, the equation of collision impulse is written as

The Newton recovery coefficient

After the recovery phase, the motion velocity

After a whole collision process

After the collision with the inner damper for particle

The model after collision with the inner damper.

Assuming that the motion velocity of particle 1 is

In Figure

Before and after the collision, assume that the motion velocity for the damper is

From (

As the motion of particle changes, it will access the next step size of time and return to the above process, which calculates the increment of motion for particle again and again. By the iterative calculation, we can achieve the real time motion tracing for every particle.

For the particle damper, the energy dissipation of collision motion is defined as

For the damper model, the total energy dissipation

Assuming that ^{3}. Figure

The variation of

Figure

Variation of

Figure

Effects of particle number

Figure

Effects of recovery coefficients

The energy dissipation experiment is performed to verify the correctness of the model for particle damping. The shape of experimental damper is a cube, and the damper material is carbon steel, with the dimensions of 150 mm × 150 mm × 400 mm. The schematic plot of experimental system is shown in Figure

The schematic plot of experimental system.

The test setup is mainly composed of test components, electromagnetic exciter of JZ-30, acceleration sensors, and signal acquisition analyzer of INV-3118. The electromagnetic exciter produces the excitation signal through two-force bar. The acceleration signal of the damper is acquired by a signal collection and transferred to a signal analyzer. The sampling number per cycle is more than 500 and the experimental process is repeated 4 times. The excitation signal of the electromagnetic vibrator JZ-30 is the sine wave.

In this experiment, the energy dissipation for particles is written as

The photograph of this experimental system is shown in Figure

The photograph of experimental system.

Firstly, the aluminum oxide, the stainless steel, and the tungsten carbide are used for experimental material. The densities of the three materials are, respectively, 3.4 g/cm^{3}, 7.9 g/cm^{3}, and 18.4 g/cm^{3} with 85% volume filling rate and 4 mm particle size. The results showing the energy dissipation versus acceleration distributions for the simulation and experiment are shown in Figure

The energy dissipation results for the simulation and experiment.

From Figure

Secondly, in order to investigate the effect of particle size and exciting acceleration on the energy dissipation, the stainless steel is used for experimental material with 85% volume filling rate. The experimental results showing the energy dissipation versus acceleration distributions for the different particle size are shown in Figure

The experimental results for the different particle size.

Thirdly, in order to investigate the effect of particle filling rate and exciting acceleration on the energy dissipation, the stainless steel is used for experimental material 4 mm particle size. The experimental results showing the energy dissipation versus acceleration distributions for the different particle filling rate are shown in Figure

The experimental results for the different particle filling rate.

As there is no particle in the damper, the energy dissipation value is nearly zero. And as the particle filling rate increases to 90%, the energy dissipation values rise. However, as the particle filling rate exceeds 90%, the energy dissipation values begin to drop. So we can see that there is an optimum particle filling rate for damping effect. In this experiment, the 90% particle filling rate is the best for stainless steel ball of 4 mm particle size. The main reason for this result is that there is less clearance of particle motion in the damper for too much particle, which will decrease the damping effect of an exciting structure. So, choosing an appropriate particle filling rate based on the simulation model will provide for the practical application of particle damping technology.

An energy dissipation model for particle damper has been established based on the powder mechanics and the collision theory. The equations of friction energy dissipation and collision energy dissipation are developed for the particle damper. Through the experiment, the rationality of energy dissipation model is verified and the important parameters of particle damping have been investigated, such as the particle density, the particle size, and the particle filling rate.

For the design process of particle damper, choosing an appropriate particle density, particle size, and particle filling rate determined based on the simulation model will provide the optimal damping effect for the practical application of particle damping technology.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge financial support from National Natural Science Foundation of China (no. 51205382) and the Fundamental Research Funds for the Central Universities, Xiamen University (no. 20720150094), and Collaborative Innovation Center of High-End Equipment Manufacturing in Fujian.