## 1 Introduction  
Particle damping (PD) is a passive means of vibration control in which small metallic or ceramic particles are placed inside a cavity that attached to the primary structure at the place of high vibration amplitudes. The kinetic energy of the primary structure is dissipated by non-elastic impact and friction between particles and walls. In the 1930s, a primary PD technology--impact damping--was first used by Pagat [1] to attenuate the vibration of the turbine blade. Considering this kind of damper produced large impact noise and was susceptible to the design parameters, some modified dampers were successively introduced. According to the number of units and particles, PD can be classified into four categories [2,3]: impact damper [4], multi-unit impact damper [5], non-obstructive particle damper [6,7] and multi-unit particle damper [8], as shown in Figure 1.  
[PERSON] [5] first proposed an improved method of a multi-unit single-particle damper, replacing one impact damper with multiple impact dampers to reduce the impact of the single-particle damper. Many researches have indicated that particle dampers are highly effective in suppressing resonant structural vibration over a wide range of frequencies: Several modes of vibration can be damped simultaneously by one damper design. Experiments conducted by [PERSON] [4] showed that PD could effectively suppress the wings flutter and general mechanical vibration. [PERSON] [9] also successfully applied PD to a GE-XTE45 fan blade. [PERSON] [6] used it to damp the vibration in the liquid oxygen (LOX) inlet tee of a Space Shuttle Main Engine (SSME) and found that PD technology had a good damping efficiency in the high frequency range between 3000 Hz to 6000 Hz. [PERSON] et al. [10] managed to reduce the vibration levels of a desk-top banknotes processing machine by about 40 dB in the range of 2000 Hz to 6000 Hz. Moreover, to illustrate the application of PD in cryogenic temperature environment, [PERSON] et al. [11] performed a series of experimental studies on an impact damper and applied this technique to damp the lateral shaft vibration in a rocket engine turbo pump, where most conventional dampers would invalid. To gain an appreciation of the important parameters that may influence the performance of particle damper, most PD experiments were conducted on a cantilever beam with particle dampers. The parameters concerned usually included the type, shape, and size of the particles, the volume fraction, damper location, and the level of excitation. It is found that the factors mentioned above have different effects on particle damping. Some of them, moreover, interconnect with each other. [PERSON] and [PERSON] [12] summarized their experimental findings that the damper was strongly dependent upon vibration amplitude, damper location, and mass of particles. [PERSON] and [PERSON] [13] further pointed out the key aspect for amplitude dependent behavior under gravity was the particles' acceleration characteristics. Although these studies have highlighted the basic rules of particle damping, the acceleration response range, given the cantilever beam structure, is relatively limited for practical application. [PERSON] and [PERSON] laid foundational analytical methods for mechanism research on a single-unit multi-particle damper [5; 14; 15; 16]. But the research on multi-unit particle dampers (MUPDs) is still in its infancy. In 2005, [PERSON] [8] first gave the definition of multi-unit particle damper, and studied the influence of the number of units and cavity size on particle damping with the single-degree-of-freedom system of horizontal vibration. Since the multi-unit particle damper in the literature is in the same response environment, [PERSON] assumed that the damping effect of the multi-body particle damper was actually a linear superposition of single-unit multi-particle damper. Through the discrete element method (DEM) and experimental approaches, the damping mechanism of MUPDs under different excitations was studied in detail by [PERSON] [17; 18; 19; 20; 21].