## 2 Theoretical Model  
### Fluid-Structure Interaction (FSI) 14-Equation Model of Hydraulic Pipeline  
The FSI dynamic 14-equation model of hydraulic pipeline is mainly based on the water hammer theory, using classical beam equations theory to describe the kinetic relationship between pipeline and fluid. The model is built based on the following basic assumptions: (1) the velocity of the sound wave is greater than the flow velocity; (2) the wavelength of the sound wave is longer than the pipeline diameter; (3) only elastic deformation is occurred in the pipeline wall; (4) the hydraulic oil is a kind of Newtonian fluid; (5) no separation and cavitation in the fluid during the movement process. Figure 1 shows the FSI dynamic 14-equation model of the bent pipeline microelement, which consists of two hydrodynamic equations and twelve pipeline equations [26]. (1) Axial dynamic model  
The axial dynamic model has two components: the water hammer equation of fluid and the axial dynamic equilibrium equation of pipeline, and it describes the transient behavior of the fluid and the pipeline. \[\frac{\partial P}{\partial z}=-\rho_{f}\frac{\partial V}{\partial t}-\frac{2 \tau_{0}}{R\rho_{f}} \tag{1}\]  
\[\frac{\partial V}{\partial z}=-\frac{1}{K^{*}}\frac{\partial P}{\partial t}-2v \frac{\partial i_{z}}{\partial z}-\frac{i_{y}}{R} \tag{2}\]  
\[\frac{\partial f_{z}}{\partial z}=-A_{p}\rho_{p}\frac{\partial i_{z}}{ \partial t}-\frac{f_{y}}{R} \tag{3}\]  
\[\frac{\partial i_{z}}{\partial z}=\frac{\partial f_{z}}{EA_{p}\partial t}- \frac{vr}{Ee}\frac{\partial P}{\partial t}-\frac{i_{y}}{R} \tag{4}\]  
(2) Transverse y-z dynamics model  
This model describes the vibration of the bent pipeline microelement by the force and moment balance equation in the y-z transverse direction. \[\frac{\partial f_{y}}{\partial z}=-\Big{(}\rho_{f}A_{f}+\rho_{p}A_{p}\Big{)} \frac{\partial i_{y}}{\partial t}+\frac{f_{z}}{R}+\frac{A_{f}}{R}p \tag{5}\]  
\[\frac{\partial i_{y}}{\partial z}=\frac{1}{k^{2}GA_{p}}\frac{\partial f_{y}}{ \partial t}-\dot{\theta}_{x}+\frac{i_{z}}{R},\;k^{2}=2\frac{1+v}{4+3v} \tag{6}\]  
\[\frac{\partial M_{x}}{\partial z}=-\Big{(}\rho_{p}I_{p}+\rho_{f}I_{f}\Big{)} \frac{\partial\dot{\theta}_{x}}{\partial t}+f_{y} \tag{7}\]  
\[\frac{\partial\dot{\theta}_{x}}{\partial z}=\frac{1}{El_{p}/ff}\frac{ \partial M_{x}}{\partial t} \tag{8}\]  
(3) Transverse x-z dynamics model  
This model describes the vibration of the bent pipeline microelement by the force and moment balance equations in the x-z transverse direction. \[\frac{\partial f_{x}}{\partial z}=-\Big{(}\rho_{f}A_{f}+\rho_{p}A_{p}\Big{)} \frac{\partial i_{x}}{\partial t} \tag{9}\]  
Figure 1: The schematic diagram of the bent pipeline element. \[\frac{\partial V}{\partial z}=-\frac{1}{K^{*}}\frac{\partial P}{\partial t}-2v \frac{\partial i_{z}}{\partial z}-\frac{i_{y}}{R} \tag{2}\]0  
\[\frac{\partial V}{\partial z}=-\frac{1}{K^{*}}\frac{\partial P}{\partial t}-2v \frac{\partial i_{z}}{\partial z}-\frac{i_{y}}{R} \tag{2}\]1  
\[\frac{\partial V}{\partial z}=-\frac{1}{K^{*}}\frac{\partial P}{\partial t}-2v \frac{\partial i_{z}}{\partial z}-\frac{i_{y}}{R} \tag{2}\]2  
(4) Torsion dynamics model  
This model describes the torsional vibration of the bent pipeline microelement rotate around the z axis by the axial torque balance equations and the connection between the torsion and its angle.