## 6 Conclusions  
In this paper, a fluid shear stress model of the pipe is presented to modify the friction model of the FSI 14-equation for high-speed and high-pressure hydraulic pipeline, then, the modal and axial vibration characteristics of an aircraft hydraulic pipeline is analyzed. Based on the research, the following conclusions can be drawn. The fluid shear stress model is proposed based on the Brunone empirical model and the Zielke weighting function after analyzing and comparing the applicability of the various friction coupling models, which is suitable for the friction coupling analysis of hydraulic pipeline for a wide range of Reynolds number. The frequency-domain transfer matrix method is used to solve the FSI 14-equation model and to obtain the numerical simulation results of the modal analysis and axial vibration velocity of the hydraulic pipeline. The research result verify the correctness of the friction model developed, and indicate that the frequency-domain transfer matrix method could accurately analyze the formant frequency and axial vibration characteristics of fluid-structure interaction vibration of the hydraulic pipeline. Of the simulation and experimental results in summary, when the pressure inside the pipeline is constant, the axial vibration velocity response amplitude increases with the increase of the flow \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Flow Rate** & **1 st-Formant** & **2 nd-Formant** & **3 rd-Formant** \\ \hline Simulation results & 2 m/s & 4 m/s & & \\
6 m/s & 8 m/s & 228 & 484 & 712 \\ Experimental results & 2 m/s & 220.89 & 487.47 & 720.73 \\ Error & 2 m/s & 3.32\% & 0.66\% & 1.22\% \\ Experimental results & 4 m/s & 226.42 & 496.20 & 712.44 \\ Error & 0.89\% & 2.47\% & 0.06\% \\ Experimental results & 6 m/s & 222.06 & 487.47 & 716.99 \\ Error & 6 m/s & 2.80\% & 0.66\% & 0.70\% \\ Experimental results & 8 m/s & 223.28 & 485.90 & 716.94 \\ Error & 2.27\% & 0.34\% & 0.69\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The axial velocity response of the measuring point 2.  
velocity, while the general trend of the frequency domain characteristic curve keeps unchanged. This indicates that the friction term will not affect the resonance frequency of the pipeline, but the response amplitude of each resonant frequency is impacted slightly. At each natural frequency of the pipeline, the corresponding resonance peak will appear in the axial vibration speed response of each position of the pipeline. Comparing the axial vibration velocity of different measuring points in the pipeline shows that the closer to the fixed end of the pipeline, the smaller influence of fluid-structure interaction on the pipeline. The general trend of simulation curves come closer to the experimental curves when considering friction term, this denotes that considering friction term can enhance the analytic accuracy of FSI dynamic characteristics of the hydraulic pipeline. Hence, it is of great practical meaning to consider the friction coupling effect when analyzing the FSI vibration characteristics of high-speed high-pressure hydraulic pipelines. However, the calculation error in the process of simulation, the manufacturing error and installation error of the experimental pipeline, the multiple damping and external signal interference of the system cause small error to the final detection results. Further research will be carried out in the mixing region and the hydraulic rough region with higher Reynolds number in the future to improve the solution accuracy of the FSI 14-equation model. Conceptualization, L.Q. and M.G. ; methodology, H.G. and S.C.; software, M.G. and H.G. ; validation, L.Q., M.G. and H.G. ; formal analysis, C.G. ; investigation, C.G. ; resources, L.Q. ; data curation, M.G. and H.G. ; writing--original draft preparation, H.G. ; writing--review and editing, H.G. ; visualization, M.G. ; supervision, C.G. ; project administration, L.Q. ; funding acquisition, L.Q. All authors have read and agreed to the published version of the manuscript.