# The Effect of Porous Media on Wave-Induced Sloshing in a Floating Tank
## 2 Methodology
### Equivalent Mechanical Model
\begin{array}{ll}M_{o}=\gamma\rho\int_{0}^{h}\int_{0}^{B}\left(\left.\frac{ \partial\phi}{\partial t}\right|_{x=\frac{h}{2}}-\left.\frac{\partial\phi}{ \partial t}\right|_{x=\frac{h}{2}}\right)zdydz+\gamma\rho\int_{-L/2}^{L/2} \int_{0}^{B}\left.\frac{\partial\phi}{\partial t}\right|_{x=0}xdydx\\ \\ =-\gamma\rho B\theta_{0}\omega^{2}sin\omega t\sum\limits_{n=0}^{\infty}8 \left[\frac{sinh_{n}\frac{h}{2}-b_{n}\frac{h}{2}cosh_{n}\frac{h}{2}}{b_{n}^{ 5}hcosh_{n}\frac{h}{2}}+\frac{2 cosh_{n}h-1}{a_{n}^{3}Lsinh_{n}h}\right]\\ \hskip 14.226378 pt+\gamma\rho B\theta_{0}\omega^{2}\sum\limits_{n=0}^{\infty} \frac{8\omega^{2}(p_{n}sin\omega t+q_{n}cos\omega t)}{(p_{n}^{2}+q_{n}^{2}) cosh_{n}h}\bigg{[}\frac{b_{n}\frac{h}{2}cosh_{n}\frac{h}{2}-sinh_{n}\frac{h}{2}}{b_{n}^{ 5}hcosh_{n}\frac{h}{2}}\\ \hskip 14.226378 pt+\frac{a_{n}sinh_{n}h-cosh_{n}h+2}{a_{n}^{3}Lsinh_{n}h}+ \frac{(a_{n}sinh_{n}h-cosh_{n}h+1)tanh_{n}\frac{L}{2}}{a_{n}^{2}b_{n}^{3}h} \bigg{]}\end{array}  \tag{43}\]  
Based on the dynamic equivalent, the mechanical model should give the same overturning moment under the same circumstances. Therefore, the motion equation of the mechanical system can be written as:  
\[m_{n}\ddot{x}_{n}+c_{n}\dot{x}_{n}+k_{n}x_{n}=m_{n}z_{n}\theta_{0}\omega^{2} sin\omega t \tag{44}\]  
where \(x_{n}\) is the equivalent displacement of the moving mass relative to the ground. The overturning moment can be obtained as:  
\[\overline{M}_{o}=I_{e}\ddot{\theta}+m_{f}z_{f}z_{f}\ddot{\theta}+\sum\limits_ {n=0}^{\infty}m_{n}z_{n}\ddot{\theta}+\sum\limits_{n=0}^{\infty}m_{n}z_{n} \ddot{x}_{n}\]  
\[=\bigg{(}I_{e}+m_{f}z_{f}^{2}+\sum\limits_{n=0}^{\infty}m_{n}z_{n}^{2}\bigg{)} \ddot{\theta}+\sum\limits_{n=0}^{\infty}m_{n}z_{n}\ddot{x}_{n} \tag{45}\]  
Using the same technique described in reference [24] to equalize the overturning moments due to rigid-body motion (regardless of the oscillatory component), i.e., the first term on the right-hand side of Equations (43) and (45), the moment of inertia \(I_{e}\) can be obtained as:  
\[I_{e}=8\gamma\rho B\sum_{n=0}^{\infty}\left(\frac{3 sinh_{n}\frac{I}{2}-b_{n}\frac {I}{2}cosh_{n}\frac{I}{2}}{b_{n}^{5}hcosh_{n}\frac{I}{2}}+\frac{2 cosh_{n}h-1}{ a_{n}^{5}Lsinh_{n}h}-m_{n}z_{n}^{2}\right)-m_{f}z_{f}^{2} \tag{46}\]  
When the mechanical model is applied to the floating body, its motion equation can be expressed as:  
\[m_{n}\ddot{x}_{n}+c_{n}\dot{x}_{n}+k_{n}x_{n}=-m_{n}\ddot{x}_{G}+m_{n}z_{n} \ddot{\theta}_{G} \tag{47}\]  
Therefore, Equation (25) for the surge motion of the floating body can be modified by:  
\[M_{b}\ddot{x}_{G}=\int_{S_{n2}}pn_{x}dS+m_{f}z_{f}\ddot{\theta}_{G}+\sum_{n=0} ^{\infty}\left(c_{n}\dot{x}_{n}+k_{n}x_{n}+m_{n}z_{n}\ddot{\theta}_{G}\right) \tag{48}\]  
For the heave motion, the mechanical model assumes no masses move vertically relative to the tank, so the entire system will be regarded as a dead load to the floating body. Therefore, the equilibrium equation can be simplified as:  
\[M_{b}\ddot{z}_{G}=\int_{S_{n2}}pn_{z}dS-m_{w}g \tag{49}\]  
For the pitch motion, the movement of the equivalent masses will produce an external moment on the floating body; therefore, the equilibrium equation can be modified by:  
\[I_{b}\ddot{\theta}_{G}=\int_{S_{n2}}p(\mathbf{r}\times\mathbf{n})\ dS-\left(I_{e}+m_{f} z_{f}^{2}\right)\ddot{\theta}_{G}-\sum_{n=0}^{\infty}\left(c_{n}\dot{x}_{n}+k_{n}x _{n}+m_{n}z_{n}\ddot{\theta}_{G}\right)z_{n} \tag{50}\]  
During the computation of Equations (48)-(50), the resultants from the sloshing behavior can be obtained once Equation (47) is solved. Since the sloshing computation does not go through the EL method, the numerical simulation is very efficient and inexpensive. Figure 2: The schematic diagram of the equivalent mechanical model of a water tank filled with porous media.