## 2 Fluid-Structure Interaction Vibration Analysis of Single Pipeline  
### Theoretical Model of Pipeline Vibration  
Figure 1 shows the geometric modeling of a single pipeline in the hydraulic system. Due to the fact that the inner diameter of the actual pipeline is much smaller than its length, the calculation process assumes the pipeline model as a beam model and solves it with fixed support at both ends. Based on the fluid motion control equation and the pipeline motion control equation, the motion coupling of the two media is realized by the contact compatibility condition of the fluid and solid structures on the boundary [23; 24]:  
Fluid motion control equation:  
The flow in the pipe can be regarded as incompressible flow, and the fluid continuity equation and momentum equation are:  
\[\frac{\partial(\rho\mathrm{u}_{i})}{\partial\chi_{i}}=0 \tag{1}\]  
\[\frac{\partial}{\partial\chi_{i}}\big{(}\rho\mathrm{u}_{i}\mathrm{u}_{j}\big{)} =-\frac{\partial\mathrm{P}}{\partial\chi_{i}}+\frac{\partial}{\partial\chi_{i }}\left[\mu\left(\frac{\partial\mathrm{u}_{i}}{\partial\chi_{j}}+\frac{ \partial\mathrm{u}_{j}}{\partial\chi_{i}}\right)-\frac{2}{3}\mu\delta_{ij} \frac{\partial\mathrm{u}_{k}}{\partial\chi_{k}}\right]+\frac{\partial}{ \partial\chi_{j}}\Big{(}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Big{)} \tag{2}\]  
where \(\rho\) is the density of the fluid; \(u_{i}\), \(u_{j}\) are the average velocity component for each time; \(\mu\) is the dynamic viscosity of the fluid; \(\delta_{ij}\) is \"Kronecker delta\" symbol (when i = j, \(\delta_{ij}\) = 0; k is the turbulent kinetic energy; and \(\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is Reynolds stress component. There is a vortex flow in the water flow inside the pipe, and the streamline is highly curved. RNG k-\(\varepsilon\) is chosen here as the turbulence model:  
\[ \begin{array}{c}\frac{\partial(\rho\mathrm{k})}{\partial t}+\frac{\partial( \rho\mathrm{k}\mathrm{u}_{i})}{\partial\chi_{i}}=\frac{\partial}{\partial \chi_{i}}\left[\alpha_{k}\mu_{\mathrm{eff}}\frac{\partial\mathrm{k}}{\partial \chi_{j}}\right]+\mathrm{G}_{k}+\rho\varepsilon\frac{\partial(\rho\varepsilon )}{\partial t}+\\ \frac{\partial(\rho\mathrm{u}_{i})}{\partial\chi_{j}}=\frac{\partial}{ \partial\chi_{j}}\left[\alpha_{\varepsilon}\mu_{\mathrm{eff}}\frac{\partial \varepsilon}{\partial\chi_{j}}\right]+\frac{\mathrm{C}_{1k}}{k}G_{k}-C_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k}\end{array}  \tag{3}\]  
where \(k\) is turbulent kinetic energy; \(\mu_{eff}\) is corrected turbulent viscosity; \(\varepsilon\) is the turbulent dissipation rate; \(\alpha_{k}\), \(\alpha_{\varepsilon}\) are turbulent kinetic energy k and dissipation rate \(\varepsilon\) corresponding Prandtl number, respectively, and its values are all 1.39; t is the time; \(G_{k}\) is the production term of turbulent kinetic energy k caused by the average velocity gradient, \(G_{k}=\mu_{t}\left(\frac{\partial u_{i}}{\partial u_{j}}+\frac{\partial u_{j} }{\partial u_{i}}\right)\frac{\partial u_{i}}{\partial u_{j}}\), \(\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon}\), \(C_{\mu}=0.0845\); \(C_{1k}^{*}\) is the correction factor; and \(C_{2\varepsilon}\) is an empirical constant, \(C_{2\varepsilon}=1.68\). Pipeline motion control equation:  
For a pipeline system with n degrees of freedom, the finite element equation under the state of unit motion is  
\[\mathrm{M}_{\mathrm{s}}^{\varepsilon,\mathrm{s}^{T}}+\mathrm{C}_{\mathrm{s}} ^{\varepsilon,\mathrm{e}^{\mathrm{e}^{\mathrm{e}}}}+\mathrm{K}_{\mathrm{s}}^{ \mathrm{e}}\mathrm{e}^{\mathrm{e}}=\mathrm{F}^{\mathrm{e}} \tag{4}\]  
where \(M_{\mathrm{s}}^{\varepsilon}\) is quality matrix for pipeline units; \(\mathrm{C}_{\mathrm{s}}^{\varepsilon}\) is damping matrix for pipeline elements; \(K_{\mathrm{s}}^{\varepsilon}\) is the stiffness matrix of the pipeline element; \(\delta^{\varepsilon}\) is the displacement vector of the pipeline unit; and \(F^{\mathrm{e}}\) is the external load vector of the unit.