## 3 Mathematical Model Simplification and Distinct Features of Passive Dynamics
### Mathematical Model Simplification in Different Frequency Bands  
In the analysis of passive dynamics, the engine-side displacement \(y_{1}\) is the input excitation, and the engine-side reaction force \(f_{1}\) and transmitted force \(f_{5}\) to the chassis side are the outputs. Based on the mechanical model for the AHM-IT-DM-OCA shown in Figure 2, by setting \(m_{1}=0,y_{5}=0\), and \(f_{a}=0\), the nonlinear mathematical model for passive dynamics can be expressed as Equation (6). The equations are, in order, the Bernoulli equation for fluid flowing in the inertia track [24], differential equation for the mover, fluid continuity equation, equilibrium equation for engine-side reaction forces \(f_{1}\), and transmitted force \(f_{5}\) to the chassis side. As a significant distinct feature for parameter identification of passive hydraulic mounts, the fixed point related to the inertia track has been thoroughly studied. The studies [25] involve mid- and low-frequency bands, which refer to \(f<<f_{\rm n3}\). For the analysis in the mid- and low-frequency bands, simplification of the model can be carried out as follows: referring to the mechanical vibration theory [26], under excitation with frequency \(f<<f_{\rm n3}\) in the mid- and low-frequency bands, the mover displacement  
Figure 3: Low-, mid-, and high-frequency bands for the dynamic analysis of AHM-IT-DM-OCA [22]. \(y_{3}\) is almost in phase with the excitation, and the excitation force is mainly balanced by the elastic restoring force. In this case, the inertia and damping force [4; 16; 26; 27] can be neglected. Therefore, the nonlinear model in Equation (6) for passive dynamics in the mid- and low-frequency bands can be simplified by eliminating intermediate variables \(p_{1}\) and \(y_{3}\). The nonlinear model can be expressed using Equation (7). \[\left\{ \begin{array}{l}\rho_{2}\ddot{y}_{2}+\frac{1}{2}\rho\big{(}\xi_{l_{2} }+\xi_{d_{2}}\big{)}|\dot{y}_{2}|\dot{y}_{2}=-p_{1}\\ m_{3}\ddot{y}_{3}+c_{3}\dot{y}_{3}+k_{3}y_{3}+A_{3}p_{1}=0\\ A_{2}y_{2}+A_{3}y_{3}=A_{1}y_{1}+p_{1}/K_{1}\\ f_{1}=c_{1}\dot{y}_{1}+k_{1}y_{1}-A_{1}p_{1}\\ f_{5}=c_{1}\dot{y}_{1}+k_{1}y_{1}+c_{3}\dot{y}_{3}+k_{3}y_{3}-(A_{1}-A_{3})p_{1 }\\ =f_{1}-m_{3}\ddot{y}_{3}\end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{l}\ddot{y}_{2}+\left(\frac{32\mu}{\rho d_{2}^{2}}+\frac {\xi_{d_{2}}}{2\lambda_{2}}|\dot{y}_{2}|\right)\dot{y}_{2}+\frac{A_{2}K_{\rm ul }}{\rho_{2}^{2}}y_{2}=\frac{A_{1}K_{\rm ul}}{\rho_{2}^{2}}y_{1}\\ f_{5}=f_{1}=c_{1}\dot{y}_{1}+k_{1}y_{1}+A_{1}^{2}K_{\rm ul}y_{1}-A_{1}A_{2}K_{ \rm ul}y_{2}\end{array} \right. \tag{7}\]  
Experiments and numerical simulations [23] have shown that, under the excitation of harmonic displacement \(y_{1}\), the excitation and responses can be expressed as follows:  
\[y_{1}=Y_{1}e^{i\omega t},\,y_{2}=Y_{2}e^{i(\omega t-\varphi_{2})},\,f_{1}=f_{5 }=F_{1}e^{i\omega t}=F_{5}e^{i\omega t} \tag{8}\]  
where \(Y_{1}\) denotes the excitation displacement amplitude, \(Y_{2}\) is the response amplitude of the inertia fluid displacement, \(\varphi_{2}\) is the loss angle of \(y_{2}\) with respect to \(y_{1}\), and \(F_{1}\) and \(F_{5}\) are the drive point and cross-point response force amplitudes, respectively. For the analysis in the mid- and high-frequency bands, the simplification method can be carried out as follows: according to the principle of equivalent energy [26], the FRF of \(Y_{1}\)-\(Y_{2}\)[22] can be expressed as Equation (9). This indicates that the fluid motion in the inertia track can be ignored at frequencies above the fluid channel resonance frequency \(f_{\rm n2}\). Therefore, the nonlinear factors related to the inertia track are not considered to clearly study the distinct features related to the decoupler membrane in the mid- and high-frequency bands. When \(f>>f_{\rm n2}\), the linear model can be expressed as Equation (10) by setting \(y_{2}=0\) and simplifying Equation (6). \[