## Appendix A The mass matrix \(\mathbf{M}\) is  
\[\mathbf{M}=\begin{bmatrix}[M^{*}]_{N\times N}&\left\{A_{1n}^{*}\right\}_{N}^ {\text{T}}&[0]_{N\times N_{\text{h}}}&\left\{A_{1n}^{*}H_{1n}^{*}\right\}_{N} ^{\text{T}}&[0]_{N\times N_{\text{T}}}\\ \left\{A_{1n}^{*}\right\}_{N}&M^{h}&\left\{0\right\}_{N_{\text{h}}}&M^{hr}& \left\{0\right\}_{N_{\text{f}}}\\ \left[0]_{N_{\text{f}}\times N}&\left\{0\right\}_{N_{\text{h}}}^{\text{T}}& \left[0\right]_{N_{\text{h}}\times N_{\text{f}}}&\left\{0\right\}_{N_{\text{h }}}^{\text{T}}&[0]_{N_{\text{f}}\times N_{\text{f}}}\\ \left\{A_{1n}^{*}H_{1n}^{*}\right\}_{N}&M^{hr}&\left\{0\right\}_{N_{\text{h}}} &M^{r}&\left\{0\right\}_{N_{\text{f}}}\\ \left[0\right]_{N\times N}&\left\{0\right\}_{N_{\text{f}}}^{\text{T}}&\left[0 \right]_{N\times N_{\text{f}}}&\left\{0\right\}_{N_{\text{f}}}^{\text{T}}& \left[0\right]_{N_{\text{f}}\times N_{\text{f}}}\end{bmatrix},\]  
in which, \([M^{*}]_{N\times N}=diag(A_{11}^{*},\ \ldots,\ A_{1N}^{*}),\ M^{h}=\sum\limits_{n=1}^{N}A_{1n}^{*}+A_{10}^{*}+M_{ \text{t}},\)\(M^{hr}=\sum\limits_{n=1}^{N}A_{1n}^{*}H_{1n}^{*}+A_{10}^{*}H_{10}^{*}+M_{ \text{t}},\)\(M^{hr}=\sum\limits_{n=1}^{N}A_{1n}^{*}H_{1n}^{*}{}^{2}+A_{10}^{*}H_{10}^{*} {}^{2}+M_{\text{t}}^{2}+M_{\text{t}}.\)  
The damping matrix \(\mathbf{C}\) and stiffness matrix \(\mathbf{K}\) have  
\[\mathbf{C}=\begin{bmatrix}0&0\\ &\mathbf{C}^{h}\\ 0&\mathbf{C}^{r}\end{bmatrix},\mathbf{K}=\begin{bmatrix}\mathbf{K}^{*}&0\\ &\mathbf{K}^{h}\\ 0&\mathbf{K}^{r}\end{bmatrix},\mathbf{K}^{*}=\begin{bmatrix}k_{11}^{*}&0\\ &\ddots\\ 0&k_{1N}^{*}\end{bmatrix},\]\[\mathbf{C}^{h}=\begin{bmatrix}c_{0}^{h}&&&&\mathbf{0}\\ c_{1}^{h}&&&\\ &\ddots&&\\ &&&c_{l_{1}}^{h}&\\ &&&\ddots&\\ \mathbf{0}&&&&c_{N_{h}}^{h}\end{bmatrix},\mathbf{\cdot}\mathbf{C}^{\prime}=\begin{bmatrix}c_{0 }^{r}&&&&&\mathbf{0}\\ c_{1}^{r}&&&\\ &&\ddots&&\\ &&&&&c_{j_{2}}^{r}&\\ \mathbf{0}&&&&c_{N_{h}}^{r}\end{bmatrix},\] \[\mathbf{K}^{h}=\begin{bmatrix}k_{0}^{h}+k_{1}^{h}&-k_{1}^{h}&&&\\ -k_{1}^{h}&k_{1}^{h}+k_{2}^{h}&-k_{2}^{h}&&\\ &&\cdots&&&0\\ &&-k_{l_{1}-1}^{h}&k_{l_{1}-1}^{h}+k_{l_{1}}^{h}&-k_{l_{1}}^{h}\\ &0&&\cdots&\\ &&&-k_{N_{h}-1}^{h}&k_{N_{h}-1}^{h}+k_{N_{h}}^{h}&-k_{N_{h}}^{h}\\ &&&-k_{N_{h}}^{h}&k_{N_{h}}^{h}\end{bmatrix},\] \[\mathbf{K}^{\prime}=\begin{bmatrix}k_{0}^{r}+k_{1}^{r}&-k_{1}^{r}&&&&\\ -k_{1}^{r}&k_{1}^{r}+k_{2}^{r}&-k_{2}^{r}&&&\\ &\cdots&&0\\ &-k_{j_{2}-1}^{r}&k_{j_{2}-1}^{r}+k_{j_{2}}^{r}&-k_{j_{2}}^{r}&&\\ \mathbf{0}&&\cdots&\\ &&-k_{N_{h}-1}^{r}&k_{N_{r}-1}^{r}+k_{N_{r}}^{r}&-k_{N_{r}}^{r}\\ &&-k_{N_{h}}^{r}&k_{N_{h}}^{r}\end{bmatrix},\]  
where \(c_{j_{1}}^{h}=\frac{\delta_{j_{1}}\tau_{j_{2}}k_{0}^{h}}{\Psi_{s}}\), \(c_{j_{2}}^{r}=\frac{\delta_{j_{2}}\tau_{j_{2}}k_{0}^{r}}{\Psi_{s}}\), \(k_{j_{1}}^{h}=\chi_{j_{1}}k_{0}^{h}\), \(k_{j_{2}}^{r}=\chi_{j_{2}}k_{0}^{r}\), \(k_{0}^{h}=\frac{8\zeta_{j_{2}}\tau_{j_{2}}}{2-\
u}\), \(k_{0}^{r}=\frac{8\zeta_{j_{3}}\tau_{j_{3}}^{3}}{3(1-\
u)}\). \(k_{0}^{h}\) and \(k_{0}^{r}\) denote the static stiffnesses of horizontal and rocking nested LPMs, respectively. \(\chi_{j_{1}}\) (\(j_{1}=1,\ \cdots,\ N_{h}\)) and \(\delta_{j_{1}}\) (\(j_{1}=0,\ \cdots,\ N_{h}\)) represent horizontal stiffness and damping coefficients for \(j_{1}\)th degree of freedom for LPM, respectively. \(\chi_{j_{2}}\) (\(j_{2}=1,\ \cdots,\ N_{r}\)) and \(\delta_{j_{2}}\) (\(j_{2}=0,\ \cdots,\ N_{r}\)) represent the rocking stiffness and damping coefficients for \(j_{2}\)th degree of freedom for LPM, respectively. \(G_{8}\) represents the soil shear modulus.