## 3 Mathematical Model Simplification and Distinct Features of Passive Dynamics
### Distinct Features of Passive Dynamics and Verification for Model Simplification Method  
In the mid-and low-frequency bands, the passive dynamics of the AHM-IT-DM-OCA have features similar to those of a conventional hydraulic engine mount, which is essentially caused by the inertia track. The significant distinct feature of the fixed point has been thoroughly studied and can be obtained through tests under excitation with different amplitudes. The feature of constant dynamics in the mid-frequency band is named as the 1 st horizontal segment. The **fixed point\(f_{R}\)** and **1 st horizontal segment\(k^{\prime}_{\infty,1}\)**, which can be inferred from the nonlinear model in Equation (7) can be expressed as Equation (11). These features can be applied to identify the equivalent piston area \(A_{1}\) of the main rubber spring and bulk stiffness \(K_{\rm u1}\) of the upper chamber. It should be noted that there were no differences between the drive-point and cross-point dynamics when \(f<<f_{\rm n3}\). \[f_{\rm n2}=f_{R}\,\ k^{\prime}{}_{\infty,1}=k_{1}+A_{1}^{2}K_{\rm u1} \tag{11}\]  
In the mid- and high-frequency bands, distinct features can be inferred based on the linear model shown in Equation (10). It should be noted that the inertial force of \(m_{3}\) cannot be ignored; thus, \(f_{1}\) and \(f_{5}\) are no longer the same. Therefore, the drive-point \(k^{*}_{11}\) and cross-point \(k^{*}_{51}\) passive dynamic stiffnesses can be expressed by Equation (12). Subsequently, the dynamic stiffness in-phase can be obtained. \[ \begin{array}{c}k^{*}_{11}=\frac{F_{1}}{T_{1}}=\left(k_{1}+A_{1}^{2}K_{1} \right)+j\omega c_{1}-\frac{A_{11}^{2}K_{1}^{2}}{K_{1}+K_{3}}\frac{1}{\left(1- A^{2}+j2\xi_{3}\lambda\right)}\\ k^{*}_{51}=\frac{F_{5}}{Y_{1}}=k_{1}+A_{1}^{2}K_{1}+A_{1}A_{3}K_{1}\frac{ \left(A^{2}-G\right)\left(1-\lambda^{2}\right)}{\left(1-\lambda^{2}\right)^{ 2}+\left(2\xi_{3}\lambda\right)^{2}}\\ \qquad\qquad+j\bigg{[}\omega c_{1}-A_{1}A_{3}K_{1}\frac{2\xi_{3}\lambda\left(A ^{2}-G\right)}{\left(1-\lambda^{2}\right)^{2}+\left(2\xi_{3}\lambda\right)^{2 }}\bigg{]}\end{array}  \tag{12}\]  
where \(\lambda\), \(\xi_{3}\) and \(G\) can be expressed as follow:  
\[\lambda=\lambda_{3}=\frac{\omega}{\omega_{\rm n3}}=\frac{f}{f_{\rm n3}}\,\ \xi_{3}=\frac{c_{3}}{2m_{3}\omega_{\rm n3}}=\frac{c_{3}}{2\sqrt{m_{3}A_{3}^{2} (K_{1}+K_{3})}}\,\ G=\frac{A_{1}K_{1}}{A_{3}(K_{1}+K_{3})}=\frac{A_{1}K_{ \rm u1}}{A_{3}K_{3}}\]  
In the mid-frequency band, the frequency ratio \(\lambda\) can be zero because \(f<<f_{\rm n3}\). Subsequently, drive point \(k^{\prime}_{11,0}\) and cross point \(k^{\prime}_{51,0}\) dynamic stiffness in-phase in the high-frequency band can be expressed by Equation (13). It should be noted that \(k^{\prime}_{11,0}\) and \(k^{\prime}_{51,0}\) are equal to the 1 st horizontal segment, which is obtained by the nonlinear model, even though \(f_{1}\) and \(f_{5}\) are different from each other in the linear mathematical model. The consistency of the 1 st horizontal segment obtained by nonlinear and linear models at mid-frequency indicates that the method to achieve a linear model by removing the nonlinear factors is feasible and rational. Therefore, the model simplification method can be verified and applied to study the distinct features of the active dynamics and parameter identification. \[k^{\prime}{}_{11,0}=k^{\prime}{}_{51,0}=k_{1}+A_{1}^{2}K_{\rm u1}=k^{\prime}{ }_{\infty,1} \tag{13}\]  
In the high-frequency band, the frequency ratio \(\lambda\) can be \(\infty\) because \(f>>f_{\rm n3}\). Furthermore, drive point \(k^{\prime}_{11,\infty}\) and cross point \(k^{\prime}_{51,\infty}\) dynamic stiffness in-phase in the high-frequency band can be expressed by Equation (14). \(k^{\prime}_{11,\infty}\) and \(k^{\prime}_{51,\infty}\) are denoted as \(k^{\prime}_{\infty,2}\) and \(k^{\prime}_{\infty,3}\), respectively, which are constant in the high-frequency band. Therefore, \(k^{\prime}_{\infty,2}\) is named the **2 nd horizontal segment**, and \(k^{\prime}_{\infty,3}\) is named the **3 rd horizontal segment**.