## 1 Introduction  
Particle damping technology can better solve the problem of damping performance degradation caused by thermal aging, creep, and brittle cracking and suppress the vibration and noise of large structures under complex and harsh working conditions. Therefore, it has a broad application prospect in engineering [1, 2]. In vibration and noise reduction, the damping characteristics and energy dissipation law of particle dampers are affected by the vibration state of internal particles [3]. However, the damping characteristics of particle damping technology are highly nonlinear in the application process [4], which makes it difficult to form a relatively complete theory so far, and its internal energy dissipation mechanism also limits the better application of particle damping technology. The particle damping energy dissipation mechanism is mainly manifested as energy transfer and energy dissipation. From the energy analysis of the system, the total nonlinear kinetic energy, the total rotational kinetic energy, the total elastic energy stored in the deformation of the particle, the potential energy of the spring, and the kinetic energy of the moving structure are all involved in the energy transfer. The sum of this instantaneous energy is the residual energy of the particle system. The total energy of friction dissipation and acoustic energy dissipation in the inelastic collision process is the cumulative energy generated by energy dissipation. The dissipated energy is converted into thermal energy dissipation and acoustic energy dissipation, which cannot be transferred back to the system. After energy is transferred to the particles, it is dissipated by friction and inelastic collisions between the particles. The relative importance of friction and collision depends on the particles' physical properties and excitation conditions. Among them, the thermal energy dissipation between particles is considered the main factor in determining vibration energy dissipation. However, there is no effective energy loss relationship between vibration input and heat dissipation. During forced vibration, particles show a wide range of motion states and complex kinetic properties. However, energy conversion, consumption, and transfer during particle aggregate vibration have yet to be unified [5]. The various engineering applications of particle aggregates have promoted research on their energy dissipation mechanisms. [PERSON] studied the energy loss of particles under forced vibration conditions and established design principles for particle applications [6; 7; 8]. [PERSON] analyzed the collision and friction between particles using the discrete element method and discussed the energy consumption forms of particles [9; 10]. Using the discrete element method, [PERSON] explored the dissipation capabilities of particles with various motion states [11]. [PERSON] conducted experimental studies on the energy loss process of particle aggregates during forced vibration and obtained changes in the kinetic and potential energy of particles during vibration [12; 13]. The research on the energy dissipation of particle aggregates focuses on the changes in kinetic energy and potential energy caused by particle collision and friction. The recovery coefficient is an essential basis for establishing different dissipation models. Models are established based on the change in recovery coefficient caused by collisions between particles. These models make it difficult to describe the relationship between particle collisions and thermal energy consumption. [PERSON] et al. [14] divided the loss mechanism of the particle damper into an external mechanism and an internal mechanism. The external mechanism is mainly the friction and collision between the discrete body and the container wall, while the internal mechanism is mainly the friction and collision between the discrete bodies. [PERSON] et al. [15] believe that the energy dissipation mechanism of particle dampers is mainly related to the highly nonlinear phenomenon of friction and impact.