## 4 Governing Equations of Motion MF-Tank System  
The simplified mass-spring model of an \(N\)-layer MF-tank system is depicted in Figure 5.  
The whole system comprises \(2N\) degrees of freedom representing the MetaFoundation and two degrees of freedom demonstrating the convective and impulsive masses, respectively. The governing equations of motion of the MF tank system can be written in matrix form:  
\[M_{T}\ddot{u}+C_{T}\dot{u}+K_{T}u=-M_{T}\dot{m_{S}} \tag{21}\]  
where \(M_{T}\), \(K_{T}\), and \(C_{T}\) are the mass, stiffness, and damping matrices of the coupled tank system mounted on the MetaFoundation, respectively expressed by  
\[M_{T}=\left[ \begin{array}{cc}M_{MF}&0\\ 0&M_{Tank}\end{array} \right] \tag{22}\]  
Figure 5: Simplified mass–spring model of the liquid storage tank mounted on an N-layer MF. \[C_{T}=\left[ \begin{array}{cc}C_{MF}+C_{Tank}&-C_{Tank}\\ -C_{Tank}&C_{Tank}\end{array} \right] \tag{23}\]  
\[K_{T}=\left[ \begin{array}{cc}K_{MF}+K_{Tank}&-K_{Tank}\\ -K_{Tank}&K_{Tank}\end{array} \right] \tag{24}\]  
\[M_{MF}=\text{diagonal ($m_{1}$, $m_{2}$\,\ldots\, $m_{1}$, $m_{2}$)_{N \times 2N;}$}M_{Tank}=\text{diagonal ($m_{i}$, $m_{c}$)_{2\times 2}$}; \tag{25}\]  
\[C_{MF}=\left[ \begin{array}{cccc}2c_{1}+c_{2}&-c_{2}&-c_{1}&\cdots&0\\ -c_{2}&c_{2}&0&\cdots&0\\ -c_{1}&0&\ddots&\cdots&\vdots\\ \vdots&\vdots&\vdots&c_{1}+c_{2}&-c_{2}\\ 0&0&\cdots&-c_{2}&c_{2}\end{array} \right] \tag{26}\]  
\[C_{Tank}=\text{diagonal ($c_{i}$, $c_{c}$)_{2\times 2}$} \tag{27}\]  
\[K_{MF}=\left[ \begin{array}{cccc}2k_{1}+k_{2}&-k_{2}&-k_{1}&\cdots&0\\ -k_{2}&k_{2}&0&\cdots&0\\ -k_{1}&0&\ddots&\cdots&\vdots\\ \vdots&\vdots&\vdots&k_{1}+k_{2}&-k_{2}\\ 0&0&\cdots&-k_{2}&k_{2}\end{array} \right] \tag{28}\]  
\[K_{Tank}=\text{diagonal ($k_{i}$, $k_{c}$)_{2\times 2}$} \tag{29}\]  
\[u=[u_{1},\,u_{2}\,\ldots\,\,u_{1},\,u_{2},\,u_{i},\,u_{c}]^{T} \tag{30}\]  
The superscript \(MF\) and \(Tank\) stand for MetaFoundation and tank system, respectively; \(u\) is the displacement of the system relative to the ground; \(x_{2}=(u_{2}-u_{1})\), \(x_{i}=(u_{i}-u_{1})\), \(x_{c}=(u_{c}-u_{1})\) are displacement of internal, impulsive, and convective masses relative to the external mass, respectively. In Equation (21), the earthquake acceleration is shown by \(iig\), and \(r\) is a column vector of one. To obtain the time history of responses of the liquid storage tanks with MF, the governing equations of motion are derived and transferred to the first-order differential equations. Then, these equations are solved in each time step using the state-space representation and ODE15s in MATLAB programming language. The response quantities of interest are vertical displacement of the surface of the fluid (\(d_{v}\)), impulsive displacement (\(x_{i}\)), overturning moment (\(M\)) at the top of the MF, and structural base shear of the tank (\(F_{s}\)). While the impulsive displacement can be obtained directly through the solving of equations of motion, the vertical displacement of the free surface, the overturning moment at the top of the foundation, and structural base shear are calculated according to Equations (31)-(33), respectively [8]. The overturning moment determines the generated hydrodynamic forces in the tank wall, which is proportional to the axial compressive force. The axial compressive force is the main reason of buckling of the tank walls, either in the form of elephant foot buckling or diamond shape buckling. Conversely, the vertical displacement of the free surface caused by the convective component controls the required freeboard. In the case of lack of sufficient freeboard, the tank may experience a leak of fluid, tear of the shell or break of the shell-roof connection. \[d_{v}=0.837R\frac{\omega_{c}^{2}x_{c}}{g} \tag{31}\]  
\[M=-\{m_{c}l_{c}(\ddot{u}_{c}+\ddot{u}_{g})+m_{i}h_{i}(\ddot{u}_{i}+\ddot{u}_{ g})\} \tag{32}\]  
\[F_{s}=-\{m_{c}(\ddot{u}_{c}+\ddot{u}_{g})+m_{i}(\ddot{u}_{i}+\ddot{u}_{g})\} \tag{33}\]  
where \(\omega_{c}\) is the frequency of the convective mass (\(\omega_{c}=2\pi/T_{c}\)).