## 2 Half-Vehicle Seat System Integrated with NESIs
\[z_{sf}=z_{s}-l_{f}\sin\varphi\approx z_{s}-l_{f}\varphi\quad z_{sr}=z_{s}+l_{r} \sin\varphi\approx z_{sr}+l_{r}\varphi\quad z_{sa}=z_{s}-l_{a}\sin\varphi\approx z _{s}-l_{a}\varphi \tag{9}\]  
Substituting Equation (9) into Equations (1)-(8) results in the following equations:  
\[m_{s}z^{\prime\prime}{}_{s}-b_{a}(z^{\prime\prime}{}_{ba}-z^{ \prime\prime}{}_{s}+l_{a}\varphi^{\prime\prime})+c_{f}(z^{\prime}{}_{s}-l_{f} \varphi^{\prime}-z^{\prime}{}_{uf})+c_{r}(z^{\prime}{}_{s}+l_{r}\varphi^{ \prime}-z^{\prime}{}_{ur})+c_{a}(z^{\prime}{}_{s}-l_{a}\varphi^{\prime}-z^{ \prime}{}_{a})\] \[+k_{f}(z_{s}-l_{f}\varphi-z_{uf})+k_{r}(z_{s}+l_{r}\varphi-z_{ur}) +k_{a}(z_{s}-l_{a}\varphi-z_{a})+c_{bf}(z^{\prime}{}_{s}-l_{f}\varphi^{\prime }-z^{\prime}{}_{bf})\] \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]0  
\[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]1 \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]2 \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]3 \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]4  
\[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]5 \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]6 \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]7 \[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]8  
\[({m_{ur}}+{b_{r}}){z^{\prime\prime}}_{ur}-{b_{r}}{z^{\prime\prime}}_{br}-{c_{ r}}{z^{\prime}}_{sr}+{c_{r}}{z^{\prime}}_{ur}-{k_{r}}{z_{sr}}+({k_{r}}+{k_{ tr}}){z_{ur}}={k_{tr}}{z_{rr}} \tag{4}\]9  
\[{m_{a}}{z^{\prime\prime}}_{a}+{c_{a}}({z^{\prime}}_{a}-{z^{\prime}}_{sa})+{k_ {a}}({z_{a}}-{z_{sa}})+{k_{bu}}({z_{a}}-{z_{bu}})^{3}+{c_{bu}}({z^{\prime}}_{a }-{z^{\prime}}_{bu})=0 \tag{5}\]0  
Furthermore, Equation (10) can be converted into a matrix form, which is provided as  
\[{m_{a}}{z^{\prime\prime}}_{a}+{c_{a}}({z^{\prime}}_{a}-{z^{\prime}}_{sa})+{k_ {a}}({z_{a}}-{z_{sa}})+{k_{bu}}({z_{a}}-{z_{bu}})^{3}+{c_{bu}}({z^{\prime}}_{a }-{z^{\prime}}_{bu})=0 \tag{5}\]1  
In Equation (11), \(\mathbf{M}\) is the mass matrix, \(\mathbf{C}\) is the damping matrix, \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{H}(\mathbf{Z})\) is the nonlinear term matrix, \(\mathbf{F}\) is the excitation matrix, and \(\mathbf{Z}\) is the displacement matrix, which is provided by  
\[\mathbf{M}=\left( \begin{array}{cccccccc}m_{s}+b_{a}&-b_{a}l_{a}&0&0&0&-b_{a} &0&0\\ -b_{a}l_{a}&l+b_{a}l_{a}{}^{2}&0&0&0&b_{a}l_{a}&0&0\\ 0&0&m_{uf}+b_{f}&0&0&0&-b_{f}&0\\ 0&0&0&m_{ur}+b_{r}&0&0&0&-b_{r}\\ 0&0&0&0&m_{at}&0&0&0\\ -b_{a}&b_{a}l_{a}&0&0&0&b_{a}&0&0\\ 0&0&-b_{f}&0&0&0&b_{f}&0\\ 0&0&0&-b_{r}&0&0&b_{r}\end{array} \right)\]  
\[\mathbf{C}=\left(