## 3 Model Verification
Furthermore, it should be noted that the resonance peak values calculated via the FEM tend to be generally higher than those obtained through the TMM and the  
Figure 9: The experiment platform. MVTMM due to neglecting viscous friction effects within fluid modeling. Figure 11b further demonstrates that the TMM remains unstable at frequencies exceeding 500 Hz, making accurate predictions regarding pipeline flow-induced vibrations infeasible. Conversely, the MVTMM continues to exhibit stability throughout all frequency bands; its predicted natural frequencies align closely with those derived from the FEM analysis. A comparative assessment between theoretical predictions and simulation outcomes clearly indicates that the MVTMM possesses significant advantages over the TMM concerning stability. In order to further validate the accuracy of the proposed method for predicting the FSI frequency domain vibration characteristics of the pipeline, we compared the calculated results with the experimental results, as illustrated in Figure 12. The comparison of modal data across various orders is presented in Table 6. As shown in Figure 12a, the trend of fluid vibration characteristics predicted by our method aligns closely with the experimental  
Figure 11: The comparison of theory and simulation. (**a**) Comparison of H-P2/P1, (**b**) Comparison of H-A1/P1. Figure 10: A time–frequency diagram of the pressure pulsation at point P1. \begin{table}
\begin{tabular}{c c c} \hline
**Transfer Function** & **Mathematical Definition** & **Physical Significance** \\ \hline H-P2/P1 & Pipeline outlet pulsating pressure/inlet & \\ H-A1/P1 & Pipeline vibration acceleration/pipeline & \\  & inlet pulsating pressure & \\ \hline \end{tabular}
\end{table} 
Table 5: The verification indicators. results, with a maximum error of only 3.83% between the theoretical and experimental outcomes. In addition, it can be seen from Figure 12a that there is an anti-resonance peak at 760 Hz, which is caused by the FSI, but the anti-resonance peak obtained by the experiment is significantly higher than that obtained by the theoretical calculation. The reason for the difference may be that the pipe boundary condition adopted by the theoretical method is the ideal anchored boundary, which cannot be achieved completely in practice. Similarly, Figure 12b indicates that the pipeline flow-induced vibration characteristics predicted by our method are largely consistent with those observed in the experiment, exhibiting a maximum error of 4.35%. Notably, a formant at 460 Hz is detected in the experimental results for H-A1/P1, which is not anticipated by our theoretical model. It is speculated that this discrepancy may arise from boundary conditions within the actual pipeline failing to achieve absolute fixed support and rigid boundaries. To illustrate the necessity of considering FSI in the analysis of hydraulic pipeline vibration characteristics, the H-P2/P1 index was utilized as a case to compare the outcomes of the classical water hammer model with those of the FSI model. The results are presented in Figure 13. It is evident that the classical water hammer model fails to account for  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **The** \\ **Experimental** \\ **Result (Hz)** \\ \end{tabular}  } & \multicolumn{2}{c}{**MVTMM**} & \multicolumn{2}{c}{**FEM**} \\ \cline{2-5}  & **Natural** & **Relative Error to** & **Natural** & **Relative Error to** \\  & **Frequency (Hz)** & **Experiment (\%)** & **Frequency (Hz)** & **Experiment (\%)** \\ \hline
115 & 116 & 0.87 & 110 & 4.35 \\
155 & 158 & 1.94 & 156 & 0.65 \\
195 & 200 & 2.56 & 198 & 1.54 \\
235 & 2  
Poisson coupling and junction coupling; consequently, its predictions encompass only fluid vibration modes characterized by specific periodic patterns, neglecting pipeline vibration modes induced by FSI effects. Furthermore, the Poisson effect leads to a shift in fluid vibration modes towards lower frequencies.