## 1 Introduction  
The vibration problem of pipeline systems has a broad engineering application background, and its research results can be directly applied to various fields such as ocean engineering, bioengineering, chemical machinery, aerospace, and the nuclear industry. What's more, cross-type branch pipelines are widely used in hydraulic systems and pipelines [1]. However, during the transportation of gases and liquids, intermittent suction or discharge movements of fluids in pipelines may cause flow and pressure pulsations in the pipeline system, leading to severe pipeline vibration in discontinuous areas, especially at branch points. At the interface between the branch pipeline and the main pipeline, the pressure flow pulsation of hydraulic oil and the fluid-structure interaction phenomenon caused by the external vibration of the pipeline are relatively concentrated. Abnormal or excessive vibration can lead to fatigue failure, leakage, and other safety issues in pipeline systems. Therefore, the problems surrounding the branch pipeline have aroused the research interest of many scholars. In recent years, [PERSON] et al. [2] of the United States analyzed the nonlinear stability of incompressible flow in cylindrical thin shells, combining the linear model and the Donnell shallow shell nonlinear equation in the model, and discussed its impact on pipeline stability by changing the length diameter ratio and thickness diameter ratio. In 2008, [PERSON] [3] studied the plumbing composed of straight pipes and elbows. The theoretical model is based on the idea of Tentarelli, and the models of two pipe segments are established. The straight pipe is solved in the frequency domain, and the solution of the elbow is not divided into units. The elbow model is directly used for the overall solution, and the influence of the bending radius and angle of the elbow on the fluid-solid coupling vibration characteristics is analyzed. [PERSON] [4] conducted airflow pulsation and pipeline vibration tests on the primary intake pipeline of a certain hydrogen compressor under 60% flow regulation and variable operating conditions. Based on the actual pipeline size and direction on site, an airflow pulsation analysis model and a pipeline modal analysis model were established to conduct airflow pulsation analysis and pipeline modal analysis, respectively. And targeted pipeline vibration reduction measures were proposed to reduce the vibration value of the first-stage intake pipeline of the compressor to an acceptable range. [PERSON] [5] studied a local resonance-type periodic pipeline and showed a simulation model of the periodic pipeline system. He analyzed the influence of parameters such as local oscillator width, outer diameter, and material density on the vibration reduction effect of the periodic pipeline and verified it through experiments. The results show that the width and material density of the local oscillator have a significant impact on the vibration reduction effect of the pipeline system. The outer diameter of the local oscillator has little effect on the vibration level drop of the pipeline system. [PERSON] [6] proposed a semi-analytical model for the vibration characteristics of multi-clamp-supported liquid-filled pipelines and a method for optimizing clamp layout. In order to study the effects of parameters such as liquid filling state and flow rate on the optimization results of clamp layout, a general clamp layout optimization model was established with the maximum first-order natural frequency as the optimization objective without introducing an external excitation frequency. The particle swarm optimization algorithm was used to solve the optimization model. The article [7] is devoted to the analysis of the reasons for the high level of vibration of piping, in particular during resonance with the disturbing frequency of main line pumps. The causes of vibration in the piping of the main line pumps and the influence of frequency regulation on the level of vibration of the main line pump are analyzed.