## 1 Introduction  
A multi-branch pipeline is a typical structure, consisting of a straight pipe, nut, rear ferrule, front ferrule and four parts of the joint body. It is widely used in aerospace, shipbuilding and other fields, with the advantages of small size and convenient operation. The pipeline suffers serious vibration from fluid-structure interaction, and long-term vibration will reduce the life of the pipeline and hydraulic components, and in serious cases may even cause wall rupture and damage to the pipeline support structure. It causes loss of support stiffness, piping system failure, and joint leakage [1; 2; 3]. Therefore, it is urgent to study the vibration behavior of pipeline to reveal the influence mechanisms of various parameters on its vibration behavior. The fluid-structure interaction mechanism is defined as three kinds of coupling [4; 5; 6]. Poisson coupling is due to the Poisson effect, in which the oscillating pressure causes the radial pipe wall to expand, resulting in axial strain and movement. Due to the unbalanced pressure acting on a certain area, the coupling of the joint occurs at the boundary of change, such as elbow, valve, joint and pipe end. Friction coupling is caused by the shear stress on the pipe wall and is generally considered less important than the other two couplings. To describe the vibration characteristics of pipelines, several models are proposed according to basic equations. [PERSON] et al. [7; 8] considered the stress distribution around the section perpendicular to the axis of the hydraulic pipe according to the two-dimensional model, taking into account the effects of curvature and stress concentration. On the basis of the characteristic line method, the vibration characteristics of a simple straight pipe were accurately predicted by using the eight-equation model, and on this basis, the effects of curvature and friction on the resonance frequency of the pipe section were analyzed. [PERSON] et al. [9] developed a quasi-two-dimensional flow model for fluid transport in pipes, analyzed the energy transfer and dissipation effects in fluid piping systems, effectively described frictional coupling, and accurately captured energy dissipation in fluids. [PERSON], [PERSON] et al. [10; 11] used the 14-equation fluid-structure interaction vibration model of hydraulic pipe. The accuracy of the FSI frequency domain solution at high Reynolds number is improved and the accuracy of the friction coupling model analysis is verified. [PERSON] et al. [12] established a new dynamic model for the cantilever pipe conveying variable density fluid in combination with the Bernoulli-Euler beam model. The results showed that the pipeline system is prone to losing stability when the fluid density increases. [PERSON] et al. [13; 14; 15] analyzed the research status of aviation hydraulic pipeline vibration and control technology in the field. A model reduction method was developed, and artificial springs were introduced to simulate boundary conditions, which significantly reduced the computational cost of the vibration analysis of long-distance multi-elastically supported hydraulic pipelines. [PERSON], [PERSON] et al. [16; 17; 18] established a nonlinear coupling dynamic model of the flow conveying pipeline based on the Timoshenko beam and verified the accuracy of the model through the finite difference method (FDM). The results showed that the Timoshenko model was more accurate in analyzing large flow rate, amplitude or short pipeline length. [PERSON] et al. [19; 20] developed a novel finite element model based on the absolute nodal coordinate formula (ANCF) and a dynamic mechanical model based on the Timoshenko beam theory for curved pipes conveying fluid, and analyzed the vibration characteristics of curved pipes under the influence of the fluid flow. Then, through the finite element numerical simulation, they verified the accuracy of the model.