## 1 Introduction
On the basis of the model equations, the following researchers have investigated the fluid-structure interaction of pipelines using different calculation methods. [PERSON] et al. [21; 22; 23] used the characteristic line method to carry out research on pipeline system vibration for many years. They then put forward a mathematical model describing the fluid-structure interaction of a liquid-filled pipeline and carried out modal analysis of a liquid-filled pipeline system through the transfer matrix method. [PERSON] et al. [24] studied the influence of the combined force caused by fluid flow on the natural frequency of straight pipes and elbows by using the transfer matrix method. When analyzing the vibration of pipes containing flowing fluid, the stable resultant force generated by fluid flow must be considered. [PERSON] [25] realized accurate FSI calculation under the structural response dominated by axial, bending and shear motion through the ALE formula, and verified the accuracy of the method using existing commercial software. [PERSON] et al. [26] used the element-free Galerkin method to calculate the natural frequencies of the two ends of the straight pipe under different boundary conditions and obtained the natural frequency equations of the fluid transmission pipeline under different boundary conditions. [PERSON] et al. [27] used fluid dynamics (CFD) and fluid-structure interaction calculation methods to obtain the flow characteristics of the oil flow, and analyzed the effect of the diameter ratio of the branch pipe to the straight pipe in the T-pipe and the merging angle of the three branches on the flow and deformation of the pipe. [PERSON] [28] proposed a complete Lagrangian finite element method (FEM) with node integration to simulate the fluid-structure interaction problem. The accuracy of the method was proved through experimental tests, providing a reference for solving complex FSI problems. [PERSON], [PERSON] et al. [29; 30] used the finite element method combined with the characteristic line method (MOC-FEM) to solve the pipe equations based on the characteristic line method. The transient analysis of the fluid-filled pipeline was completed, which significantly improved the calculation efficiency of the coupled vibration of the hydraulic pipeline system. [PERSON] et al. [31] developed the transfer matrix method for vibration analysis of branch pipes to compare the calculated pressure frequency response of rigid pipes and flexible pipes. This method was easier to use and can be used to calculate branch pipes with different shapes. Based on the fluid-structure interaction, [PERSON] et al. [32; 33] used the structural and fluid-dynamic models to analyze the fluid-structure coupling vibration of the pipeline, as well as analyzing the pipeline vibration response under different working conditions, providing a reference for practical engineering application. The research on the fluid-structure interaction vibration characteristics of pipelines has already yielded a wealth of results. Different mathematical theoretical models have been established for straight pipes, elbows, T-shaped pipes, etc. The pipeline vibration has been analyzed using the characteristic line method, transfer matrix method, and finite element method, etc. However, there are few literature reports on the multi-branch pipeline, so it is necessary to explain the law of fluid in the multi-branch pipeline. Compared with other types of pipes, the fluid changes acutely when flowing in the multi-branch pipeline, especially at the junction. Where the fluid collides with the wall to generate eddy current, the influence of the fluid-structure interaction on the pipeline is significantly increased. Therefore, the finite element model of a multi-branch pipeline is established, and the influence of different parameters on the vibration characteristics of the pipeline is analyzed, providing a reference for the subsequent structural optimization design.