## 1 Introduction  
Pipeline systems are mainly used for fluid energy transmission and generally consist of pumps, valves, pipelines, clamps, joints, etc. They are widely used in many engineering fields, such as aerospace, petrochemistry, nuclear power, etc. The excitation of the pump source fluid and the base of the mechanical equipment can cause severe vibration in the pipeline system. Therefore, effective methods are highly recommended to reduce the vibration of pipeline systems. In the past half-century, the vibration problems of pipeline systems have attracted great attention and many remarkable achievements have been made. According to the characteristics of the fluid, these problems are divided into ones dealing with the coupling of constant flow with pipelines and ones dealing with the coupling of non-constant flow with pipelines. In the area of coupled vibration problems between constant flow and pipelines, [PERSON] [1, 2] had long been devoted to the study of nonlinear dynamical behaviors such as instability, bifurcation, and chaos of the pipeline system caused by the fluid under the action of steady flow. [PERSON] et al. [3] studied flow-bend bifurcation and chaotic motion under nonlinear constraints using the cantilever flow convolution tube as a model, and their experimental results showed that in certain parameter regions, the path to chaos is through double-period bifurcation. [PERSON] et al. [4] made a detailed review of the pipeline vibration and stability due to flow around the infusion tube structure at macro, micro, and nano scales. [PERSON] et al. [5] established a nonlinear Timoshenko model for coupled vibration of flow transmission pipelines and concluded that the coupled Timoshenko theory is applicable to simulate fluid pipelines with high speed, high amplitude, or short pipeline conveyance. The coupling problem of unsteady flow and pipelines is mainly based on the classical water hammer theory. [PERSON] et al. [6] and [PERSON] [7] revealed the vibrational mechanism of coupling between unsteady fluid and pipeline systems generated by the water hammer effect, and proposed a 4-equation model, an 8-equation model, and a 14-equation model for fluid-structure interaction. [PERSON] et al. [8] provided a detailed review and discussion of the research content and progress of linear and nonlinear pipelines in the field of mechanics. [PERSON] et al. [9] proposed a frequency-domain transfer matrix method for fluid-solid coupling considering an elastically constrained liquid-filled pipeline based on pipeline vibration and verified the effectiveness of the method by means of measurement and simulation data. Based on the Tijsseling method, [PERSON] et al. [10] proposed to replace the recursively accelerated fluid-solid interacted quaternary model with time-line interpolation, and concluded that the quadratic linear interpolation scheme is the best. Many scholars have made rich research achievements on the fluid-solid coupling vibration of the pipeline system. Since the vibration problems of the pipeline system are prominent, more effective damping methods are urgently needed to suppress pipeline vibration. The particle impact damping technique is an effective damping method to achieve vibration reduction in structural systems, which is based on the principle of collision and frictional energy dissipation between particles. This technique has the advantages of wide temperature tolerance, wide frequency domain, radiation resistance, and high reliability [11; 12; 13]. In recent years, many scholars have conducted a lot of fruitful work on particle damping. [PERSON] et al. [14] designed a particle damper based on aircraft duct damping, explored the mechanism and the influencing factors of vibration damping, and verified the effectiveness and practicality of the aircraft duct particle damper through experiments and finite element simulations. [PERSON] et al.