## 2 Active HVS System Coupled with NESI  
Figure 1 presents the schematic diagram of an active HVS system coupled with NESI. \(m_{s}\), \(m_{uf}\), \(m_{ur}\) and \(m_{a}\) are the masses of the vehicle body, front tire, rear tire, and seat, respectively; \(I_{g}\) is the body pitch moment of inertia; \(k_{f}\), \(k_{r}\), \(k_{bf}\), \(k_{tr}\) and \(k_{a}\) are the stiffnesses of the front and rear suspensions, front and rear tires, and seat suspension, respectively; \(c_{f}\), \(c_{r}\), \(c_{tf}\), \(c_{tr}\) and \(c_{a}\) represent the damping coefficients of the front and rear suspensions, front and rear tires, and seat suspension, respectively. \(l_{f}\), \(l_{r}\) and \(l_{a}\) are the distances from the center of mass to the front axle, rear axle, and seat, respectively; \(b_{f}\)(\(b_{r}\), \(b_{a}\)), \(k_{bf}\)(\(k_{br}\), \(k_{ba}\)) and \(c_{bf}\)(\(c_{br}\), \(c_{hu}\)) represent the inertance, cubic stiffness, and damping coefficients of the NESI installed between the vehicle body and the front and rear tires, and between the vehicle body and seat, respectively. \(\varphi\) is the body pitch angle; \(z_{s}\), \(z_{a}\), \(z_{uf}\), \(z_{ur}\), \(z_{bu}\), \(z_{bf}\) and \(z_{br}\) denote the displacements of the vehicle body center of mass, seat center of mass, front and rear tires, the inerter of the NESI installed between the vehicle body and seat, and the inerters of the NESIs installed between the vehicle body and the front and rear tires, respectively. \(z_{rf}\) and  
\(z_{\pi r}\) represent the front and rear pavement excitations, respectively; \(u_{a}\), \(u_{f}\) and \(u_{r}\) are the active forces acting on the seat suspension and front and rear suspensions, respectively. Table 1 presents the structural parameters of the active HVS system. The structural parameters of the NESIs are obtained in [27], which are the optimized structural parameters using the genetic algorithm. For the NESI installed between the vehicle body and seat, the structural parameters are \(b_{a}\) = 299.662 kg, \(k_{ba}\) = 3.25 \(\times\) 10\({}^{5}\) N/m, and \(c_{ba}\) = 634.83 Ns/m. For the NESIs installed between the vehicle body and the front and rear tires, the structural parameters are \(b_{f}\) = 86.7921 kg, \(b_{r}\) = 97.9515 kg, \(k_{bf}\) = 2.76414 \(\times\) 10\({}^{6}\) N/m, \(k_{br}\) = 1.25991 \(\times\) 10\({}^{6}\) N/m, \(c_{bf}\) = 504.536 Ns/m, and \(c_{br}\) = 616.126 Ns/m. Utilizing [PERSON]'s second law, the dynamic equations for the active HVS system coupled with NESI are derived. These equations are formulated based on the static equilibrium positions of the vehicle body, seat, front and rear suspensions, NESI, and front and rear tires. The dynamic equation describing the vertical motion of the vehicle body is  
\[ \begin{array}{l}m_{s}z_{s}^{\prime\prime}-b_{a}(z_{ba}^{\prime\prime}-z_{sa} ^{\prime\prime})+c_{f}(z_{sf}^{\prime}-z_{uf}^{\prime})+c_{r}(z_{sr}^{\prime} -z_{ur}^{\prime})+c_{a}(z_{sa}^{\prime}-z_{a}^{\prime})+k_{f}(z_{sf}-z_{uf})+k_ {r}(z_{sr}-z_{uf})\theta\\ +k_{a}(z_{sa}-z_{a})+c_{bf}(z_{sf}^{\prime}-z_{bf}^{\prime})+c_{br}(z_{sr}^{ \prime}-z_{br}^{\prime})+k_{bf}(z_{sf}-z_{bf})^{3}+k_{br}(z_{sr}-z_{br})^{3}- u_{f}-u_{r}-u_{a}=0\end{array}  \tag{1}\]  
The dynamic equation describing the pitch motion of the vehicle body is  
\[ \begin{array}{l}I_{q}\varphi^{\prime\prime}+c_{f}l_{r}(z_{sr}^{\prime}-z_{ ur}^{\prime\prime})-c_{f}l_{f}(z_{sf}^{\prime}-z_{uf}^{\prime})-c_{d}l_{a}(z_{ sa}^{\prime}-z_{a}^{\prime})+k_{r}l_{r}(z_{sr}-z_{ur})-k_{f}l_{f}(z_{sf}-z_{uf})- \\ k_{d}l_{a}(z_{sa}-z_{a})+b_{a}(z_{ba}^{\prime\prime}-z_{sa}^{\prime\prime})l_{ a}+c_{br}l_{r}(z_{sr}^{\prime}-z_{br}^{\prime})-c_{bf}l_{f}(z_{sf}^{\prime}-z_{ bf}^{\prime})+k_{br}l_{r}(z_{sr}-z_{br})^{3}\\ -k_{bf}l_{f}(z_{sf}-z_{bf})^{3}+l_{f}u_{f}-l_{r}u_{r}+l_{a}u_{a}=0\end{array}  \tag{2}\]  
The dynamic equation describing the front tire is  
\[