## 3 Static Analysis  
### Theoretical Model  
In Figure 2, \(K_{v}\) and \(K_{\rm H}\) represent vertical spring stiffness and horizontal spring stiffness, respectively; \(L_{0}\) represents the original length of horizontal spring; \(L_{\rm H}\) represents the length of horizontal spring at any position; \(F\) represents the elastic restoring force of the vibration-isolated object; \(b\) represents the length of connecting rod between horizontal spring and mass block; \(a\) represents the distance from the isolated equipment M to the fixed end of horizontal spring. The initial state of the mass block is the position of the dashed frame, and each spring of the suspension system is in a compression state during vehicle operation. The dynamic stiffness of the suspension system can be adjusted by changing the above parameters.  
Assuming that the spring in Figure 2 is in equilibrium, the deformation length of the horizontal spring is:  
\[\Delta_{l}=\Delta_{r}=\sqrt{b^{2}-\left(\mathrm{H}-x\right)^{2}}-\left(a-L_{0}\right) \tag{1}\]  
where H is the initial deformation of the vertical spring, obtained as:  
\[\mathrm{H}=\sqrt{b^{2}-\left(a-L_{0}\right)^{2}} \tag{2}\]  
Then, the horizontal force produced by the horizontal spring is:  
\[F_{\rm Hc}=2K_{\rm H}\bigg{(}\sqrt{b^{2}-\left(\mathrm{H}-x\right)^{2}}-\left( a-L_{0}\right)) \tag{3}\]  
Figure 1: Simplified model of the vehicle seat.  
Figure 2: Mathematical model of the seat isolation system with inerter elements.  
To convert it to the force \(F_{\rm{H}v}\) in the vertical direction:  
\[F_{\rm{H}v}=2K_{\rm{H}}\bigg{(}\sqrt{b^{2}-\left({\rm{H}-x}\right)^{2}}-\left(a- L_{0}\right)\bigg{)}\tan\alpha \tag{4}\]  
where \(\alpha\) is the angle between the rod and the horizontal plane, obtained as:  
\[\tan\alpha=\frac{{\rm{H}-x}}{\sqrt{b^{2}-\left({\rm{H}-x}\right)^{2}}} \tag{5}\]  
The force produced by the vertical spring in the vertical direction can be expressed as:  
\[F_{\rm{v}}=K_{\rm{v}}\cdot x+K_{c}\cdot x^{3} \tag{6}\]  
Thus, according to the principle of virtual work, the restoring force \(F\) of the system in the vertical direction is obtained as:  
\[F=F_{\rm{v}}+F_{\rm{H}v}=K_{\rm{v}}\cdot x+K_{c}\cdot x^{3}+2K_{\rm{H}}({\rm{H }-x})(1-\frac{a-L_{0}}{\sqrt{b^{2}-\left({\rm{H}-x}\right)^{2}}}) \tag{7}\]  
For dimensionless treatment, introducing the structural parameters of the system are, sequentially, as follow:  
\[F^{\prime} =\frac{F}{K_{\rm{v}}L_{0}},\alpha=\frac{K_{\rm{H}}}{K_{\rm{v}}}, \eta=\frac{K_{\rm{v}}L_{0}^{2}}{K_{\rm{v}}},x^{\prime}=\frac{x}{L_{0}}\] \[\gamma_{1} =\frac{b}{L_{0}},\gamma_{2}=\frac{a}{L_{0}}\] \[{\rm{H}}^{\prime} =\sqrt{\gamma_{1}{}^{2}-\left(\gamma_{2}-1\right)^{2}}\]  
where \(F^{\prime}\) is dimensionless restoring force, \(x^{\prime}\) is dimensionless displacement of polynomial. Thus, the dimensionless force-displacement relation of system is obtained, as follow:  
\[F^{\prime}=x^{\prime}+\eta{x^{\prime}}^{3}+2\alpha({\rm{H}}^{\prime}-x^{ \prime})(1-\frac{\gamma_{2}-1}{\sqrt{\gamma_{1}^{2}-\left({\rm{H}}^{\prime}-x ^{\prime}\right)^{2}}}) \tag{8}\]  
Then, by differentiating Equation (8) the relationship between the dimensionless dynamic stiffness \(K^{\prime}\) and displacement \(x^{\prime}\) of system is obtained as:  
\[K^{\prime}=1+3\eta{x^{\prime}}^{2}-2\alpha(1-\frac{{\rm{H}}^{\prime 2}(\gamma_{2}-1)-2{ \rm{H}}^{\prime}(\gamma_{2}-1)x^{\prime}+(\gamma_{2}-1)x^{\prime 2}}{\left( \gamma_{1}{}^{2}-\left({\rm{H}}^{\prime}-x^{\prime}\right)^{2}\right)^{3/2}}- \frac{\gamma_{2}-1}{\sqrt{\gamma_{1}{}^{2}-\left({\rm{H}}^{\prime}-x^{\prime} \right)^{2}}}) \tag{9}\]  
According to Equation (9), it can be seen that there are four design parameters \(\alpha\), \(\gamma_{1}\), \(\gamma_{2}\), \(\eta\), and their variation can have the influence on the dynamic stiffness of system.