## 4 Analysis Example and Methods
### Numerical Solution and Experimental Methods
#### 4.2.5 Middle Constraint  
The HB3-25 LB10B P-type clamp is used for analysis and verification, the P-type clamp is composed of a metal band and rubber washer, as shown in Figure 7. For the numerical solution of the FSI model by the discrete time transfer matrix method, the middle constraint mainly contains stiffness, damping and added mass for numerical solution. The clamp mainly provides elastic constraints of translation and rotation, the translational stiffness and rotational stiffness are obtained through finite element method simulation, while the added mass is offset by the force provided by the test bench. The fastening effect of the P-type clamp is achieved by the outer surface extrusion of the rubber washer and pipeline. The bolt has little effect on the stiffness of the clamp, therefore, the bolt is neglected in the modeling process. The Finite Element Analysis model of load setting is shown in Figure 7, and uses the Cartesian coordinate system to indicate the axial, circumferential and radial directions. Using ABAQUS 6.14 finite element software, the model is meshed in fully structured hexahedron cells as shown in Figure 7, which can effectively guarantee the simulation accuracy. The metal band material is 2A12 Aluminum Alloy and the constitutive model of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Work** & **Fluid Pressure** & **Flow Velocity** & **Flow Rate** & \(k_{q}\) \\
**Conditions** & **(MPa)** & **(m/s)** & **(L/min)** & **Pa/(m/s)** \\ \hline
1 & 5 & 2 & 6.82 & \(2.5\times 10^{6}\) \\
2 & 5 & 8 & 27.3 & \(6.25\times 10^{5}\) \\
3 & 10 & 2 & 6.82 & \(5\times 10^{6}\) \\
4 & 10 & 8 & 27.3 & \(1.25\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Throttle coefficient of four work conditions. clamp rubber washer material adopts the Mooney-Rivlin model; Table 6 lists the related material properties. In practice, the constraint between the metal band and rubber washer is set as binding, and there is no relative slide between them. The boundary condition of the bolt-hole in the metal band is set as fixed to limit the degrees of freedom of the clamp in six directions. First, coupling the inner surface of the metal band in contact with the pipe wall into a point and applying concentrated force or a moment at the coupling point, then, six-DOF equivalent stiffness of clamp can be obtained according to the linear displacement and angular displacement of the coupling point. The detailed solving methods and parameter settings are: the type of step is set to \"static and general\", the geometric nonlinearity is turned on, the time period is set to one second, the minimum incremental step is set to 1 and the maximum incremental step is set to 100. The applied concentrated force is 1 N and the moment is 1N-m. The six-DOF displacement results of the clamp are shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Material** & \begin{tabular}{c} **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} & \begin{tabular}{c} **Elastic Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s Ratio** \\ \end{tabular} \\ \hline \multirow{2}{*}{**Metal band**} & \begin{tabular}{c} **Aluminium alloy** \\ (2A12) \\ \end{tabular} & 2750 & 70 & 0.33 \\ \hline \multirow{2}{*}{**Rubber washer**} & \begin{tabular}{c} **Rubber** \\ (EPDM8370) \\ \end{tabular} & \begin{tabular}{c} **C10** \\ 0.774 \\ \end{tabular} & \
\begin{tabular}{c} **C01** \\ 0.193 \\ \end{tabular} &
\begin{tabular}{c} **D0** \\ 0.025 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material properties of the metal band and rubber washer. Figure 8: Six-DOF displacement results of the clamp. (**a**) Linear displacement in the \(x\)-direction; (**b**) Linear displacement in the \(y\)-direction; (**c**) Linear displacement in the \(z\)-direction; (**d**) Angular displacement in the \(x\)-direction; (**e**) Angular displacement in the \(y\)-direction; (**f**) Angular displacement in the \(z\)-direction. Figure 7: Finite element model of clamp and analysis settings.