## 4 Numerical Simulation and Discussions
### Axial Vibration Velocity Simulation  
In order to simulate the high-pressure condition of the pipeline, the pressure of the aluminum alloy pipeline is setting and keeping at 12 MPa. The critical speed \(V_{c}\) (5.90 m/s) of internal flow change is obtained according to the critical Reynolds number (2320) of the hydraulic pipeline. Hence, the flow velocity, flow rate and corresponding Reynolds number are listed in Table 3.  
The periodic pulsating fluid at the exit of an axial piston pump is regarded as the excitation, and the instantaneous pulsating flow is  
\[q_{sh}=\sum_{i=1}^{z_{0}}Av_{i}=AR\omega\tan\gamma\sum_{i=1}^{z_{0}}\sin\varphi _{i} \tag{32}\]  
In the formula, \(A\) is the cross-sectional area of the plunger; \(\gamma\) is the rotating speed of the cylinder; \(z_{0}\) is the plunger number in the oil pressure area at the same time; \(\varphi_{i}\) is the angle of the \(i\) plunger in the oil pressure area relative to the top dead center; \(v_{i}\) is the axial velocity of the \(i\) plunger in the oil pressure area. Without considering the effect of piston pump speed on pipeline vibration, the motor speed is chosen as 1000 r/min, the frequency domain curves of the flow rate of piston pump are shown in Figure 4.  
The first three order modals of the frequency domain flow pulsation curve are superimposed and the frequency domain flow excitation under each flow can be described as  
\[\overline{q}_{shi}=\sum_{j=1}^{3}q_{ij}\omega_{ij}/\sin(s^{2}+\omega_{ij}^{2}) ;i=1,2,3,4 \tag{33}\]  
Figure 4: Frequency domain curve of flow pulsation of piston pump. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \multicolumn{4}{c}{**Value**} \\ \hline Flow velocity/m/s & 2 & 4 & 6 & 8 \\ Flow rate/L/min & 5.66 & 11.32 & 16.97 & 22.63 \\ Reynolds number & 786.60 & 1573.20 & 2359.80 & 3146.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The flow velocity, flow rate and corresponding Reynolds number. \(q_{ij}\) is frequency domain fluid pulsating excitation coefficient as shown below  
\[\left[ \begin{array}{ccc}q_{11}&q_{12}&q_{13}\\ q_{21}&q_{22}&q_{23}\\ q_{31}&q_{32}&q_{33}\\ q_{41}&q_{42}&q_{43}\end{array} \right]\left[\begin{array}{ccc} \begin{array}{ ccc}\alpha_{11}&\alpha_{21}&\alpha_{31}\\ \omega_{12}&\omega_{22}&\omega_{32}\\ \omega_{13}&\omega_{23}&\omega_{33}\\ \omega_{14}&\omega_{24}&\omega_{34}\end{array} \right]^{\mathrm{T}}=\left[ \begin{array}{ccc}0.05474&0.00992&0.00693\\ 0.10933&0.01982&0.01383\\ 0.16409&0.02974&0.02076\\ 0.21842&0.03959&0.02763\end{array} \right]\left[ \begin{array}{ccc}1430.6&304 0.1&4470.7\\ 1430.6&3040.1&4470.7\\ 1430.6&3040.1&4470.7\end{array} \right]^{\mathrm{T}} \tag{34}\]  
The fluid pulsation of the piston pump is regarded as the boundary condition, the fluid pulsation excitation of the pipeline is \(\mathbf{V}=\mathbf{Q}/Af\). After the Laplace transformation, the corresponding excitation vector can be obtained as  
\[\left\{\begin{array}{c}\mathbf{Q}_{0}(s)=\left[ \begin{array}{cccccccc}V_{e }&0&0&0&0&0&0\\ \mathbf{Q}_{L}(s)=\left[\begin{array}{cccccccc}0&0&0&0&0&0&0\\ \end{array}\right]^{\mathrm{T}}\end{array} \right]\right. \tag{35}\]  
When the pressure is 12 MPa, the temperature is 20 \({}^{\circ}\)C, and the above fluid pulsation excitation is applied, the axial vibration velocity response of the pipeline is shown from Figures 5-8. Figure 5: Axial velocity response when the flow velocity is 2 m/s. Figure 6: Axial velocity response when the flow velocity is 4 m/s. Between the comparison from Figures 5-8, we can see that the axial vibration velocity response curves have the same trend with or without friction term. However, several points are worth noting. (1) There will be corresponding peak value of resonance in the axial vibration velocity response of each position of the pipeline at each natural frequency of the pipeline.