## 3 Theoretical Modeling  
### Dynamic Model Considering Friction Coupling of the BHP  
#### 3.1.1 The FSI Dynamic Model  
The force schematic diagram of the BHP is established according to the right-hand rule shown in Figure 4. To establish the dynamic model, we assume that the pipeline is installed parallel to the horizontal plane. Considering the influence of elastic material deformation, fluid viscous friction, pipeline, and fluid inertia [27], this paper establishes the FSI 14-equation for a single straight pipe as follows [20]:  
Axial dynamic model:  
\[\frac{1}{\rho_{1}}\frac{\partial P}{\partial z}+\frac{\partial V}{\partial t}+ \frac{f_{\rm f}}{A_{1}\rho_{1}}=0 \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Name**} & **Electron Beam Acceleration** & **Amplification (Times)** & **Resolution (nm)** & **Maximum Sample Size (mm)** \\  & **Voltage (kV)** & **Amplification (Times)** & **Resolution (nm)** & **Maximum Sample Size (mm)** \\ \hline value & 0.5–30 & 30–800,000 & 1 (15 kV) & \(\Phi\)100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Technical parameters of the scanning electron microscopy. Figure 4: Force schematic diagram of the BHP. \[\frac{\partial\dot{w}_{x}}{\partial t}+\frac{1}{\rho_{t}A_{t}+\rho_{r}A_{r}}\frac{ \partial f_{x}}{\partial z}+\frac{f_{t}}{\rho_{t}A_{t}+\rho_{r}A_{r}}=0 \tag{7}\]  
\[\left[\frac{1}{K}+\frac{1}{3(C_{01}+C_{10})}\frac{r_{r}}{e_{t}+e_{r}}\right] \frac{\partial P}{\partial t}+\frac{\partial V}{\partial z}=0 \tag{8}\]  
\[\frac{\partial\dot{w}_{x}}{\partial z}+\frac{1+v_{r}}{6C_{10}+6C_{01}}\frac{1 }{A_{t}+A_{r}}\frac{\partial f_{x}}{\partial t}+\frac{v_{r}}{6C_{10}+6C_{01}} \frac{r_{r}}{e_{t}+e_{r}}\frac{\partial P}{\partial t}=0 \tag{9}\]  
\(X\)-\(z\) plane vibration equation for the BHP:  
\[\frac{\partial f_{x}}{\partial z}+(\rho_{t}A_{t}+\rho_{r}A_{r}+\rho_{1}A_{1}) \frac{\partial\dot{w}_{x}}{\partial t}+(\rho_{t}A_{t}+\rho_{r}A_{r}+\rho_{1}A _{1})g=0 \tag{10}\]  
\[\frac{\partial f_{x}}{\partial t}+k(A_{t}G_{t}+A_{r}G_{r})\left(\frac{ \partial\dot{w}_{z}}{\partial z}-\dot{\theta}_{y}\right)=0 \tag{11}\]  
\[\frac{\partial m_{y}}{\partial z}+f_{x}+(\rho_{t}I_{t}+\rho_{r}I_{r}+\rho_{1} I_{1})\frac{\partial\dot{w}_{y}}{\partial t}=0 \tag{12}\]  
\[\frac{\partial m_{y}}{\partial t}+(E_{t}I_{t}+E_{r}I_{r})\frac{\partial\dot{ w}_{y}}{\partial z}=0 \tag{13}\]  
\(Y\)-\(z\) plane vibration equation for the BHP:  
\[\frac{\partial f_{y}}{\partial z}+(\rho_{t}A_{t}+\rho_{r}A_{r}+\rho_{1}A_{1}) \frac{\partial\dot{w}_{y}}{\partial t}=0 \tag{14}\]  
Figure 4:0  
Figure 4:1  
Figure 4:2  
Torsional vibration equation for the BHP:  
Figure 4:3  
Figure 4:4  
where \(V\) is the liquid flow rate (m/s), \(K\) is the liquid elastic modulus (MPa), \(P\) is the liquid pressure (MPa), \(\rho\) is the density (kg/m\({}^{3}\)), \(g\) is the acceleration of gravity (m/s\({}^{2}\)), \(E\) is the elastic modulus (MPa), \(\upsilon\) is the material Poisson's ratio, \(G\) is the shear modulus (MPa), \(k\) is the shear distribution coefficient, \(w\) is the straight-line displacement (m) of the BHP, \(A\) is the cross-sectional area of each layer (m\({}^{2}\)), \(\theta\) is the angle of rotation (rad) of the BHP, \(f\) is the cross-section force of the BHP (N), \(m\) is the bending moment of the BHP (N\(\cdot\)m), \(J\) is the polar moment of inertia (kg/m\({}^{2}\)), \(r\) is the pipeline radius (m), \(e\) is the wall thickness for each layer (m), and \(I\) is the moment of inertia (kg/m\({}^{2}\)). The subscripts x, y, and z are the coordinate axis direction, \(t\) is the quantity related to the stainless steel material on the outer layer of the pipeline, \(r\) is the quantity related to the elastic material in the pipeline, l is the quantity related to the fluid, and \(C_{10}\) and \(C_{01}\) are temperature-related parameters for the elastic materials, which can be obtained through experimental tests.