## 5 Conclusions  
In light of the coupling vibration problem caused by the fluid pressure beat and external mechanical excitation of the pump at the junction of the branch of the cross-type hydraulic pipeline, a method combining finite element simulation and fluid-structure interaction vibration experiments is proposed for research. And the different pressures of the fluid in the pipeline and the elastic modulus of the pipeline are considered variables, so complex systems can be calculated using this method. The biggest advantage of the present method is that the influence of internal fluid beats and external excitation can be comprehensively considered. It means that there is no need to replace methods and models when solving the working conditions of internal fluid coupling and external excitation  
Figure 11: A 70 GPa harmonic response analysis curve of cross-type pipe.  
changes, in contrast to most existing methods. Thus, this method is simple, reduces measurement time and cost, and has practical engineering value.  
Experimental measurements of a single pipeline in a simple laboratory apparatus are presented. The results calculated by the presented method are in good agreement with the experimental data, which verified the accuracy of the proposed method.  
The fluid-structure interaction analysis and vibration characteristics of a cross-type hydraulic pipeline were carried out using the finite element method. As the fluid pressure increases, the maximum deformation of the cross-type pipeline decreases in a small range, and the maximum stress shows a decreasing trend. But only lower-order modes are significantly affected by modal analysis. As the elastic modulus of the pipeline increases, the maximum response value decreases, and the maximum stress shows a linear trend of increase. While the higher-order modes are significantly affected in modal analysis. With the increase in fluid pressure and solid elastic modulus, the pipeline's stiffness will go up. So the self-excited vibration period of the vibration system decreases, and the natural frequency increases.  
Under external excitation, the first, second, and fifth-order modes of the cross-type tube are sensitive to external excitation forces. And the vibration direction is in the X direction. The analysis results have a certain reference value for the vibration reduction design of liquid pipelines in practical engineering. When designing and installing the hydraulic pipeline system, adjust the stiffness parameters of the components reasonably based on the differences in the load frequency band of the area where the pipeline is located to avoid resonance frequencies. In the operation and maintenance stages, if there is a pipeline failure caused by vibration, it can be focused on the easily excited modes, so that maintenance is more targeted.  
In actual hydraulic pipelines, the fluid flow rate is often variable, so in future research, the changing fluid flow rate can be considered the research object.  
Conceptualization, Z.L. and H.L.; methodology, Z.L. and B.L.; software, J.C.; validation, Z.L., H.L. and Y.L.; formal analysis, B.L.; investigation, H.L.; resources, H.L.; data curation, Y.L.; writing--original draft preparation, Z.L.; writing--review and editing, H.L.; visualization, J.C.; supervision, B.L.; project administration, Y.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for Central Universities at the Civil Aviation University of China, grant number \"3122022062\".  
Not applicable.  
The data presented in this study are available from the corresponding.  
The authors declare no conflict of interest.