## 2 Overview of the TMM  
### The 14-Equation Model for FSI in Hydraulic Pipelines  
The 14-equation model, based on the Timoshenko beam, comprehensively characterizes the dynamic relationship between potential variables (force, moment) and flow variables (velocity, angular velocity) across each degree of freedom in the hydraulic pipeline. This model currently represents the most complete framework for describing the FSI characteristics inherent to pipelines. It comprises four equations that address axial vibrations of the pipeline, eight equations pertaining to transverse vibrations, and two equations related to torsional vibrations. Figure 1 illustrates both the force analysis diagram of a microelement within the elbow section and a schematic representation of axial friction coupling between the pipe and fluid. By considering the force conditions acting on the microelement of both the pipe and fluid, a 14-equation model that incorporates friction coupling can be established, while the detailed derivation process of the model can be found in reference [10], which is not described in detail in this paper due to space reasons. The equations of axial motion are as follows:  
\[\frac{\partial p}{\partial l}=-\rho_{f}\frac{\partial v}{\partial t}-\frac{2\tau} {r_{i}\rho_{f}} \tag{1}\]  
\[\frac{\partial v}{\partial l}=(-\frac{1}{K}-\frac{(1-\mu^{2})(2r_{i}+e)}{Ee}) \frac{\partial p}{\partial t}+\frac{2\mu}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{u_{y}}{R_{w}} \tag{2}\]  
\[\frac{\partial f_{z}}{\partial l}=A_{p}\rho_{p}\frac{\partial u_{z}}{\partial t }-2\pi r_{i}\tau-\frac{f_{y}}{R_{w}} \tag{3}\]  
\[\frac{\partial u_{z}}{\partial z}=\frac{1}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{\mu r_{i}}{Ee}\frac{\partial p}{\partial t}-\frac{u_{y}}{R} \tag{4}\]  
The equations of lateral motion in the y-z plane are as follows:  
\[\frac{\partial f_{y}}{\partial l}=(\rho_{f}A_{f}+\rho_{p}A_{p})\frac{\partial u _{y}}{\partial t}+\frac{f_{z}}{R_{w}}-\frac{A_{f}}{R_{w}}p \tag{5}\]  
\[\frac{\partial u_{y}}{\partial l}=\frac{1}{k^{2}GA_{p}}\frac{\partial f_{y}}{ \partial t}-\varphi_{x}+\frac{u_{z}}{R_{w}} \tag{6}\]  
\[\frac{\partial m_{x}}{\partial l}=(\rho_{p}I_{p}+\rho_{f}I_{f})\frac{\partial \varphi_{x}}{\partial t}+f_{y} \tag{7}\]  
Figure 1: A schematic representation of force for a liquid-filled pipeline microelement. \[\frac{\partial\varphi_{x}}{\partial l}=E_{A}\frac{\partial m_{x}}{\partial t} \tag{8}\]  
The equations of lateral motion in the x-z plane are as follows:  
\[\frac{\partial f_{x}}{\partial l}=(\rho_{f}A_{f}+\rho_{p}A_{p})\frac{\partial u _{x}}{\partial t} \tag{9}\]  
\[\frac{\partial v}{\partial l}=(-\frac{1}{K}-\frac{(1-\mu^{2})(2r_{i}+e)}{Ee}) \frac{\partial p}{\partial t}+\frac{2\mu}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{u_{y}}{R_{w}} \tag{2}\]0  
\[\frac{\partial v}{\partial l}=(-\frac{1}{K}-\frac{(1-\mu^{2})(2r_{i}+e)}{Ee}) \frac{\partial p}{\partial t}+\frac{2\mu}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{u_{y}}{R_{w}} \tag{2}\]1  
\[\frac{\partial v}{\partial l}=(-\frac{1}{K}-\frac{(1-\mu^{2})(2r_{i}+e)}{Ee}) \frac{\partial p}{\partial t}+\frac{2\mu}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{u_{y}}{R_{w}} \tag{2}\]2  
The equations of motion about the z axis are as follows:  
\[\frac{\partial v}{\partial l}=(-\frac{1}{K}-\frac{(1-\mu^{2})(2r_{i}+e)}{Ee}) \frac{\partial p}{\partial t}+\frac{2\mu}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{u_{y}}{R_{w}} \tag{2}\]3  
\[\frac{\partial v}{\partial l}=(-\frac{1}{K}-\frac{(1-\mu^{2})(2r_{i}+e)}{Ee}) \frac{\partial p}{\partial t}+\frac{2\mu}{EA_{p}}\frac{\partial f_{z}}{ \partial t}-\frac{u_{y}}{R_{w}} \tag{2}\]4  
where \(k^{2}=2\frac{1+\mu}{4+3\mu}\) is identified as the shear constant. \(E_{A}\) denotes the bending rigidity of the pipe. \(E_{A}=\frac{1}{EJ_{p}}\) applies when the object is a straight pipe, while \(E_{A}=\frac{ff}{EJ_{p}}\) pertains to cases involving a curved pipe. \(ff\) represents the rigidity correction factor, which is stated in Equation (15).