## 2 Half-Vehicle Seat System Integrated with NESIs  
The schematic diagram of the half-vehicle seat system integrated with NESIs is shown in Figure 1. The three NESIs are mounted between the bodywork and the seat and between the bodywork and two tires, which are parallel with the seat and suspensions, respectively, and are designed to dampen the vibration of the half-vehicle seat system. The seat is connected to the bodywork through the seat suspension and NESI, which can move vertically with respect to the bodywork. The bodywork is linked to the vehicle tires at each corner through the front and rear vehicle suspensions and NESIs, which can move freely in the vertical and pitch directions. Both the vehicle's front and rear tires can move vertically with regard to the bodywork. In addition, taking into account the vertical motions of the bodywork, tires, and NESIs, the pitching motion of the bodywork is also considered, which indicates that the system has eight DOFs. As displayed in Figure 1, \(m_{a}\) and \(m_{s}\) are the vehicle seat and bodywork masses, respectively; \(m_{uf}\) (\(m_{ur}\)) is the front (rear) tire mass; \(l_{\varphi}\) is the inertial pitch motion of the bodywork; \(l_{a}\), \(l_{f}\), and \(l_{r}\) are, seriatim, the lengths from the vehicle seat and the front and rear axles to the middle of the bodywork; \(k_{a}\) and \(c_{a}\) are the vehicle seat suspension stiffness and damping, respectively; \(k_{f}\) (\(k_{r}\)) and \(c_{f}\) (\(c_{r}\)) are the front (rear) suspension stiffness and damping, respectively; \(k_{uf}\) (\(k_{tr}\)) is the front (rear) tire stiffness; \(k_{bat}\), \(c_{bat}\), and \(b_{a}\) are the cubic stiffness, damping, and inertrance of the NESI mounted between the vehicle seat and bodywork; \(k_{bf}\) (\(k_{br}\)), \(c_{bf}\) (\(c_{br}\)), and \(b_{f}\) (\(b_{r}\)) are the cubic stiffness, damping, and inertrance of the NESI installed between the bodywork and the front (rear) tire. \(z_{a}\) is the vehicle seat displacement; \(z_{s}\), \(z_{sf}\), and \(z_{sr}\) are the middle, front, and rear bodywork displacements, respectively; \(\varphi\) is the pitching angle of the bodywork; \(z_{bu}\), \(z_{bf}\), and \(z_{br}\) are the vertical displacements of the connection points for the three NESIs; \(z_{uf}\) (\(z_{ur}\)) is the front (rear) tire vertical displacement; \(z_{sf}\) (\(z_{rr}\)) is the front (rear) pavement vertical input excitation. Table 1 displays all of the above structural parameters of the half-vehicle seat system integrated with the NESIs [41; 42]. Assuming that the half-vehicle seat system integrated with NESIs has vertical displacement and pitching angles in the vertical and pitch directions, the deflection shape of the system is shown in Figure 2. The nonlinear dynamic equations of the half-vehicle seat system integrated with NESIs are acquired based on [PERSON]'s second law and are built on the static balance locations of the vehicle seat, bodywork, NESIs, and tires. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(m_{a}\) & \(m_{s}\) & \(I_{\varphi}\) & \(m_{uf}\) & \(m_{ur}\) & \(m_{sf}\) & \(m_{sr}\) \\
80 kg & 690 kg & 1222 kgm\({}^{2}\) & 40 kg & 45 kg & 340 kg & 350 kg \\ \hline \(l_{a}\) & \(l_{f}\) & \(l_{r}\) & \(k_{a}\) & \(c_{a}\) & \(k_{f}\) & \(k_{r}\) \\
1.4 m & 1.3 m & 1.5 m & 8000 Nm\({}^{-1}\) & 800 Nsm\({}^{-1}\) & 18,000 Nm\({}^{-1}\) & 22,000 Nm\({}^{-1}\) \\ \hline \(c_{f}\) & \(c_{r}\) & \(k_{tf}\) & \(k_{tr}\) & \(S_{mtx}\) & & \\
1000 Nsm\({}^{-1}\) & 1000 Nsm\({}^{-1}\) & 200,000 Nm\({}^{-1}\) & 200,000 Nm\({}^{-1}\) & 0.081 m & & \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of the half-vehicle seat system integrated with NESIs. Figure 1: Half-vehicle seat system integrated with NESIs. The dynamic equations established for the bodywork in the vertical and pitch directions are  
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