## 5 Conclusions  
In this paper, a finite element model of a multi-branch pipeline is established and verified by experiments. The influence of different factors on the vibration characteristics of a hydraulic multi-branch pipeline is studied. The obtained conclusions are as follows:  
1. Fluid passing through the four-way junction changes abruptly, and the branch shunt causes a significant drop in flow rate. After the connection there is a clear low-pressure area downstream of the two branches. The particle motion trajectory indicates that the fluid generates a vortex under the influence of high and low pressure, which is the main cause of fluid-induced pipeline vibration.
2. By comparing the experimental results with the finite element model, the maximum error of the first three orders of natural frequency of the pipeline is under 6.15%, which verifies the effectiveness of the model. The deformation and equivalent force  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wall Thicknesses/mm** & **1 st Mode/Hz** & **2 nd Mode/Hz** & **3 rd Mode/Hz** \\ \hline
0.5 & 148.32 & 670.04 & 1288.81 \\
1 & 194.94 & 756.93 & 1367.28 \\
1.5 & 230.51 & 836.65 & 1468.37 \\
2 & 260.21 & 900.97 & 1550.93 \\
2.5 & 286.92 & 959.19 & 1628.12 \\
3 & 311.94 & 1014.74 & 1703.65 \\ \hline \hline \end{tabular}
\end{table}
Table 12: First three natural frequencies of the pipeline at different wall thicknesses.  
Figure 21: First three natural frequencies of the pipeline at different wall thicknesses.  
of the multi-branch pipeline with a joint are reduced by more than 5%, the natural frequency and amplitude of dynamic response are smaller.
* The increasing values of flow rate and pressure reduce the natural frequency of the pipeline. The pressure has significant influence on the vibration characteristics of the pipeline. As the fluid density increases, the natural frequency of the pipeline decreases. Pipe diameter, restraint stiffness, straight pipe length, and wall thickness, etc., have large effects on the vibration characteristics of the pipeline. When the pipe diameter, elastic restraint stiffness, and wall thickness increase, the natural frequency of the pipeline increases. While the length of the straight pipe increases, the natural frequency decreases.  
The multi-branch pipeline model proposed in this paper will be applied in fuel, lubricating oil, air and other pipeline systems, such as aerospace, marine, rail transit, irrigation, and petroleum etc. The above results can provide a reference for design and fabrication in the above fields.  
Software, Z.Z.; Validation, D.Z.; Formal analysis, Z.Z. and M.J.; Resources, T.Y. and D.Z.; Data curation, M.J.; Writing--original draft, Z.Z.; Writing--review & editing, J.J.; Supervision, T.Y. and J.J.; Project administration, T.Y.; Funding acquisition, T.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Provincial Natural Science Foundation of China, grant number ZR2022 QA041 and Provincial College youth Innovation technology project support program, grant number 2019 KJB005.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare no conflict of interest.