## 4 Dynamic Analysis  
### Dynamic Modeling of the Proposed Seat Suspension  
With respect to the seat suspension system, \(u_{m}(t)\) is the absolute displacement of the isolated equipment in space. When the isolated equipment M experiences upward a vertical displacement \(z\) from its initial position, the kinetic energy \(T\) of the system is:  
\[T=\frac{1}{2}Mu_{m}^{2}+m_{a}v_{a}^{2} \tag{11}\]  
Figure 8: Dynamic stiffness curves of systems with the various values of \(\alpha\) (\(z=\mathrm{H}^{\prime}-x^{\prime}\)). where  
\[v_{a}=\sqrt{\left[\frac{d(\lambda x)}{dt}\right]^{2}+\left[\frac{d\bigg{(}\left(1- \lambda\right)\sqrt{b^{2}-\left(\mathrm{H}-x\right)^{2}}-\left(1-\lambda\right) \left(a-L_{0}\right)\bigg{)}}{dt}\right]^{2}}=x\sqrt{\lambda^{2}+\left(1- \lambda\right)^{2}\frac{\left(x-\mathrm{H}\right)^{2}}{b^{2}-\left(\mathrm{H}- x\right)^{2}}} \tag{12}\]  
The vertical upward external excitation from the base is \(u_{e}(t)\). The relative vertical motion is \(z(t)=u_{m}(t)-u_{e}(t)\). \(z\) is the displacement of the isolated equipment relative to the base of the cab:  
\[z=\mathrm{H}-x \tag{13}\]  
The relative velocity and acceleration of the isolated equipment are, respectively, defined as below:  
\[\dot{z}=\dot{u}_{m}-\dot{u}_{e} \tag{14}\]  
\[\ddot{z}=\ddot{u}_{m}-\ddot{u}_{e} \tag{15}\]  
The potential energy of the system contains the elastic potential energy of the vertical springs and the two springs in horizontal direction, which is as:  
\[V=\frac{1}{2}k_{l}\Delta_{l}^{2}+\frac{1}{2}k_{r}\Delta_{r}^{2}+\frac{1}{2}k_{ z}\Delta_{x}^{2}+\frac{1}{4}k_{c}\Delta_{x}^{4}+Mg\cdot x+2m_{g}\lambda \Delta_{x} \tag{16}\]  
The dissipation function \(D\) is given by:  
\[D=\frac{1}{2}C_{l}\dot{\Delta_{l}^{2}}+\frac{1}{2}C_{r}\dot{\Delta_{r}^{2}}+ \frac{1}{2}C_{v}\dot{\Delta_{x}^{2}}=\frac{1}{2}C_{v}(\dot{u}_{m}-\dot{u}_{e} )^{2} \tag{17}\]  
where \(C\) is the damping coefficient (\(N/m/s\)), \(k\) is the spring stiffness (\(N/m\)), \(\Delta\) is the variation of the spring length, \(\dot{\Delta}\) is the relative velocity at both ends of the spring (\(m/s\)), Subscripts \(l\), \(r\) and \(v\), respectively, represent the left, right, and vertical direction of the suspension system. \(\lambda(0\leq\lambda\leq 1)\) is the distance ratio. Since the system is symmetrical structure, it is considered that \(k_{l}=k_{r}=K_{\mathrm{H}}\), \(C_{l}=C_{r}\), \(\Delta_{l}=\Delta_{r}\). For the model of this suspension system, the damping in the horizontal direction is designed to be \(C_{l}=C_{r}=0\). Note that the weight of the isolated equipment M and the two attached inerter elements can be offset by the precompression of the vertical spring. So, we can have:  
\[K_{v}\Delta_{s}+K_{c}\Delta_{s}^{3}=Mg+2\lambda m_{g}g \tag{18}\]  
\[\Delta_{x}=x+\Delta_{s} \tag{19}\]  
where \(\Delta_{s}\) is the precompression of the vertical spring. Applying the Lagrangian Equation of the second kind, it can be expressed as:  
Figure 8:0  
where \(Q_{x}\) comprises of the excitation force \(F_{e}\) and the damping force introduced to account for energy dissipation, and it can be expressed as:  
Figure 8:1  
By substituting Equation (12) into Equation (11), substituting Equations (18) and (19) into Equation (16), and then substituting the expressions of potential energy, kinetic energy into the Lagrange Equation (20), the dynamic equation of the system can be derived as follows:Figure 8:2  
where \(\Delta_{s}\) and gravity terms no longer exist due to Equations (18) and (19). The left-hand side of the equation of motion includes damping term and restoring force term, respectively, which are the same as those of the equivalent quasi-zero-stiffness (QZS) system without the attached inerter elements. From Equation (22), it also can be seen that the nonlinear stiffness term and the inertia term have nonlinear characteristics. The structural parameters of the system can be adjusted to satisfy different engineering requirements.