## 4 Soil-Tank Coupling Model  
The dynamic impedance \(K(\omega)\) describes the force displacement relation between foundation and soil. The dynamic flexibility \(F(\omega)\) is normalized associated with the static flexibility \(F_{\text{s}}\) in the form of  
\[F(\omega)=\frac{1}{K(\omega)}=F_{\text{s}}F_{\text{d}}(a_{0}) \tag{36}\]  
in which \(F_{\text{d}}(a_{0})\) denotes the dynamic flexibility coefficient. \(a_{0}=\omega r_{\text{s}}/V_{\text{s}}\) denotes the normalized frequency. \(r_{\text{s}}\) is the radius of the circular surface foundation. \(V_{\text{s}}\) represents the soil shear wave velocity. The dynamic flexibility coefficient for soil foundation is fitted in the form of a ratio of two Chebyshev polynomials [32]:  
\[F_{\text{d}}(a_{0})=F_{\text{d}}(\lambda)\approx\frac{Q^{(0)}(\lambda)}{P^{(0) }(\lambda)}=\frac{1+q_{1}^{(0)}\lambda+q_{2}^{(0)}\lambda^{2}+\ldots+q_{N_{ \text{s}}}^{(0)}\lambda^{N_{\text{s}}}}{1+p_{1}^{(0)}\lambda+p_{2}^{(0)} \lambda^{2}+\ldots+p_{N_{\text{s}}}^{(0)}\lambda^{N_{\text{s}}}+p_{N+1}^{(0)} \lambda^{N_{\text{s}}+1}} \tag{37}\]  
in which \(\lambda=ia_{0}/a_{0\text{max}}\), \(a_{0\text{max}}\) denotes the maximum in the approximate frequency range. \(N_{\text{s}}\) denotes the degree for the Chebyshev polynomials. The unknown coefficients in Equation (37) are obtained by the least squares technique [31]. The dynamic flexibility function \(F(\omega)\) is written in the nested form of  
\[ \begin{array}{l}F(\omega)=F_{\text{s}}F_{\text{d}}(a_{0})=\frac{F_{\text{s }}}{\frac{1}{1+p_{1}^{(0)}\lambda+p_{2}^{(0)}\lambda^{2}+\ldots+p_{N_{\text{s} }}^{(0)}\lambda^{N_{\text{s}}}+p_{N+1}^{(0)}\lambda^{N_{\text{s}}+1}}}\\ =\frac{1}{\frac{1}{N_{\text{s}}}+i\omega\frac{q_{N}}{q_{N}}+\frac{F_{\text{s} }}{\lambda^{2}}+\frac{1}{\lambda^{2}}+\frac{1}{\lambda^{2}}}{\frac{1}{N_{ \text{s}}}+i\omega\frac{q_{N}}{q_{N}}+\frac{1}{\lambda^{2}}+\frac{1}{\lambda^ {2}}}{\frac{1}{N_{\text{s}}}+i\omega\frac{q_{N}}{q_{N}}+\frac{1}{\lambda^{2}} +\frac{1}{\lambda^{2}}}{\frac{1}{N_{\text{s}}}+i\omega\frac{q_{N}}{q_{N}}+ \frac{1}{\lambda^{2}}}}\\ \ddots\\ +\frac{1}{\frac{F_{\text{s}}}{\lambda^{2}_{N_{\text{s}}}}+i\omega\frac{1}{ \lambda^{2}_{N_{\text{s}}}}}\end{array}  \tag{38}\]  
in which the dimensionless coefficients of the dashpot \(\delta_{j}\) and spring \(\chi_{j}\) for the nested LPM in Figure 4 are calculated from  
\[\delta_{0}=\frac{p_{N_{\text{s}}+1}^{(0)}}{q_{N_{\text{s}}}a_{0\text{max}}}, \chi_{j}=\frac{p_{N_{\text{s}}-j+1}^{(j)}}{q_{N_{\text{s}}-j+1}^{(j-1)}},\delta _{j}=\frac{p_{N_{\text{s}}-j+1}^{(j)}}{q_{N_{\text{s}}-j}^{(j)}a_{0\text{max}}} \text{ for }j=1,\cdots,\ N_{\text{s}} \tag{39}\]  
Figure 3: Equivalent analytical model for liquid sloshing. where \(q_{n}^{(j)}=q_{n}^{(j-1)}-\frac{q_{N_{\rm s}-j+1}^{(j-1)}}{p_{N_{\rm s}-j+1}^{(j)}} p_{n}^{(j)}\), \(p_{n}^{(j+1)}=p_{n}^{(j)}-\frac{p_{N_{\rm s}-j+1}^{(j)}}{q_{N_{\rm s}-j}^{(j)}}q_{n -1}^{(j)}\) for \(n=1,\cdots,\ N_{\rm s}-j\). In the following analysis, the Chebyshev polynomials of the degree \(N_{\rm s}=5\) are considered. Based on the substructure approach, the model for the sloshing of liquid is conveniently assembled with the LPM for soils, as shown in Figure 5. The governing equation for motion of the coupling model can be formulated based on the [PERSON]'s principle:  
\[\int_{0}^{t}[\delta(E_{K}-E_{P})+\delta W_{c}]\ dt=0 \tag{40}\]  
in which the kinematic energy \(E_{K}\) and potential energy \(E_{P}\) for the coupling system have  
\[ \begin{array}{l}E_{K}=\frac{1}{2}\sum\limits_{n=1}^{N}A_{1n}^{*}(\dot{q}_{n} ^{*}+\dot{u}_{0}+H_{1n}^{*}\dot{\varphi}_{0}+\dot{u}_{g})^{2}+\frac{1}{2}A_{10 }^{*}(\dot{u}_{0}+H_{10}^{*}\dot{\varphi}_{0}+\dot{u}_{g})^{2}\\ +\frac{1}{2}M_{\rm t}(\dot{u}_{0}+y_{t}\dot{\varphi}_{0}+\dot{u}_{g})2+\frac{1 }{2}I_{\rm t}(\dot{\varphi}_{0})^{2}\end{array}  \tag{41}\]  
\[