## 2 Modelling of ACM  
The ACM model, including the electromagnetic actuator model, is build based on the passive hydraulic mount model. In order to simplify the model, the following assumptions are made: (1) the fluid in the mount is incompressible, (2) the pressure distribution in the upper and lower fluid chambers is uniform, (3) the fluid in the upper and lower chambers is leak-free, (4) the fluid does laminar flow in the inertia track.  
The output force of the electromagnetic actuator in the ACM is linear with the input current between 25 Hz and 250 Hz, while the excitation frequency of the four-cylinder four-stroke in-line engine is generally 25-200 Hz [13]. In this way, the actuator is suitable for the application in ACM.  
The ACM mechanical model is established as shown in Figure 1, it is stipulated that upward is the positive direction. The meaning of each parameter and the corresponding values are shown in Table 1. The excitation displacement at engine side is \(y_{1}(t)\) and it is assumed that the chassis side degree of freedom \(y_{5}(t)\) is fixed, that is \(y_{5}(t)=0\). The forces applied to the engine side and transmitted to the chassis side are\(f_{1}(t)\) and\(f_{5}(t)\), respectively. The displacement of the fluid in inertia track and the DM are \(y_{2}(t)\) and \(y_{3}(t)\), respectively.  
The relative pressures of the upper and lower chambers are \(p_{1}(t)\) and \(p_{2}(t)\), respectively. Since the bulk stiffness of the rubber bellow, \(K_{2}\), is about zero due to the crimpled lower rubber bellow, set \(p_{2}(t)=0\). The \(f_{a}(t)\) is the active force of electromagnetic actuator. According to the ACM mechanical model shown in Figure 1, the ACM mathematical model can be derived as Equation (1) [3,14,15].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Para.** & **Name** & **Value** \\ \hline \(m_{1}\) & mass of engine side/kg & 60.00 \\ \(m_{1}\) & mass of piston of engine/kg & 0.40 \\ \(m_{2}\) & mass of fluid in inertia track (\(m_{2}=\rho_{2}A_{2}\))/g & 30.41 \\ \(m_{3}\) & mass of actuator mover including attached fluid in fluid-filled state/g & 113.93 \\ \(k_{1}\) & dynamic stiffness in-phase of main rubber spring in vertical & 522.04 \\ \(k_{3}\) & dynamic stiffness in-phase of decoupler membrane (\(k_{3}=A_{3}^{2}K_{3}\))/N\(\cdot\)mm\({}^{-1}\) & 61.95 \\ \(c_{1}\) & viscous damping of main rubber spring in vertical direction/N\(\cdot\)s\(\cdot\)m\({}^{-1}\) & 17.90 \\ \(c_{3}\) & viscous damping of decoupler membrane/mover in fluid-filled & 44.40 \\ \(A_{1}\) & equivalent piston area of main rubber spring/mm\({}^{2}\) & 3584.40 \\ \(A_{2}\) & cross-sectional area of inertia track/mm\({}^{2}\) & 92.50 \\ \(A_{3}\) & equivalent piston area of decoupler membrane/mm\({}^{2}\) & 1320.03 \\ \(l_{2}\) & length of inertia track (fluid channel)/mm & 328.80 \\ \(\rho\) & fluid density/kg\(\cdot\)m\({}^{-3}\) & 1000 \\ \(K_{1}\) & dynamic bulk stiffness of main rubber spring/GN\(\cdot\)m\({}^{-5}\) & 121.53 \\ \(K_{2}\) & dynamic bulk stiffness of lower fluid chamber/GN\(\cdot\)m\({}^{-5}\) & 0 \\ \(K_{3}\) & dynamic bulk stiffness of decoupler membrane/GN\(\cdot\)m\({}^{-5}\) & 35.55 \\ \(k_{\text{M}}\) & voice coil constant \(B\)-\(l\)/T\(\cdot\)m & 14 \\ \(R\) & radius of the crank/m & 0.04 \\ \(L\) & length of the connecting rod/m & 0.15 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the ACM with oscillating coil actuator (OCA).  
Figure 1: ACM mechanical model.  
\[\left\{\begin{array}{l}\rho l_{2}\dot{y}_{2}+\frac{1}{2}\rho(\xi_{2}+\xi_{2})| \dot{y}_{2}|\dot{y}_{2}=-p_{1}\\ m_{3}\dot{y}_{3}+c_{3}\dot{y}_{3}+k_{3}y_{3}+A_{3}p_{1}=f_{a}\\ A_{2}y_{2}+A_{3}y_{3}=A_{1}y_{1}+p_{1}/K_{1}\\ m_{1}\dot{y}_{1}+c_{1}\dot{y}_{1}+k_{1}y_{1}-A_{1}P_{1}=f_{1}\\ f_{a}(t)=k_{\rm M}i(t)\\ f_{5}=c_{1}\dot{y}_{1}+k_{1}y_{1}+c_{2}\dot{y}_{3}+k_{3}y_{3}-(A_{1}-A_{3})p_{1 }-f_{a}\\ =f_{1}-\left(m_{1}\dot{y}_{1}+m_{3}\dot{y}_{3}\right)\end{array}\right. \tag{1}\]