## 4 Dynamic Analysis
### Dynamic Modeling of the Proposed Seat Suspension
Especially when the construction machinery is excited by larger amplitude, the vibration peak transmissibility rate can be further reduced by adjusting the parameter of the nonlinear inerter \(\lambda\). For the QZS vibration isolation system in Figure 2, assuming that the base displacement excitation is:  
Figure 8:3  
By introducing the dimensionless parameters as:  
Figure 8:4  
Figure 8:5  
Then, the dimensionless dynamic equation of Equation (12) is given by:  
Figure 8:6  
where \((\cdot)^{\prime}=\frac{d^{2}(\cdot)}{dt^{2}},\ (\cdot)^{\prime}=\frac{d(\cdot)}{dt}\), the Equation (14) contains some complex expressions, which increases the difficulty of the analytical response calculation. And thus, the fractional and radical expressions can be replaced by using the Taylor series expansion [26]. Then, the approximate expression of the equation of motion (24) is given by:  
Figure 8:7  
where  
Figure 8:8  
Figure 8:9  
From the dimensionless dynamic Equation (24), it can be seen that there are square term and cubic term in the equation. Therefore, the solution of Equation of motion will contain constant term and harmonic series, which can reflect the vibration offset and vibration, respectively. Equation (24) can be solved in the following application of the harmonic balance method (HBM). By omitting the higher order harmonic terms, the response of the system is assumed in the form:  
\[v_{a}=\sqrt{\left[\frac{d(\lambda x)}{dt}\right]^{2}+\left[\frac{d\bigg{(}\left(1- \lambda\right)\sqrt{b^{2}-\left(\mathrm{H}-x\right)^{2}}-\left(1-\lambda\right) \left(a-L_{0}\right)\bigg{)}}{dt}\right]^{2}}=x\sqrt{\lambda^{2}+\left(1- \lambda\right)^{2}\frac{\left(x-\mathrm{H}\right)^{2}}{b^{2}-\left(\mathrm{H}- x\right)^{2}}} \tag{12}\]0  
where \(A_{0}\) represents the offset of vibration, \(A_{1}\) is the relative displacement amplitude, and \(\varphi\) is the phase between the excitation and the response. Substituting Equation (27) into (25), assuming the coefficients of the same sines and cosines terms are equal, one can obtain:  
\[\left\{ \begin{array}{c}\left(\frac{15}{128}-\frac{7}{6}\sigma_{2}\Omega^{2} \right)A_{1}{}^{5}+\\ \left[\frac{3}{4}(\frac{1}{4}+\sigma_{2})-\sigma_{2}\Omega^{2}\right]A_{1}{}^ {3}\\ +(2-\sigma_{1}\Omega^{2})A_{1}\end{array} \right\}\cos\varphi+(-2\zeta A_{1} \Omega)\sin\varphi=u_{0}\Omega^{2} \tag{29}\]  
transmissibility is defined as the ratio of the magnitude of the absolute displacement of the QZS vibration isolation system to the base as:  
\[T=20\log\frac{|u_{m}|}{|u_{e}|}=\frac{\sqrt{{A_{1}}^{2}+U_{L}{}^{2}+2A_{1}U_{L} \cos\varphi}}{U_{e}} \tag{46}\]  
Combining with Equations (28) and (29), substituting \(\cos\varphi\) into Equation (46), then, the absolute displacement transmissibility is expressed as:  
\[T=20\log\frac{|u_{m}|}{|u_{e}|}=20\log\sqrt{1+\frac{{A_{1}}^{2}}{{U_{L}}^{2}}+ \frac{{A_{1}}\Lambda}{{U_{e}}^{2}}} \tag{47}\]  
where  
\[\Lambda=(\frac{15}{64}-\frac{7}{4}\sigma_{2}\Omega^{2}){A_{1}}^{5}+\left[ \frac{3}{2}(\frac{1}{4}+\sigma_{2})-2\sigma_{2}\Omega^{2}\right]{A_{1}}^{3}+(4- 2\sigma_{1}\Omega^{2})A_{1} \tag{48}\]  
In the case of small amplitude excitation, the degree of nonlinearity of dynamic equation will be weak so that the dynamic Equation (22) can be linearized [28]. Thus, by the linear vibration theory the transmissibility can be given as:  
\[T=20\log\sqrt{\frac{1+(2\zeta\Omega)^{2}}{\left(1-\Omega^{2}\right)^{2}+\left( 2\zeta\Omega\right)^{2}}} \tag{49}\]