## 2 Model of an Active Hydraulic Mount and Frequency Band Definition  
### Model of the Active Hydraulic Mount with Oscillating Coil Actuator  
A schematic of the AHM-IT-DM-OCA is shown in Figure 1. The upper end is connected to the engine, and the lower end is connected to the vehicle body or chassis. The coil bobbin was rigidly connected to the skeleton of the decoupler membrane, which was regarded as an actuator mover. Alternating current (AC) power is applied to the coil, and the coil oscillates under an alternating ampere force in the constant magnetic field of a permanent magnet. The ampere force is the active force controlled by the amplitude and frequency of the AC. The resultant force acting on the vehicle body consists of the active force transmitted to the vehicle body through the secondary path and the force transmitted from the engine to the vehicle body through the primary path, which can be controlled such that it remains zero under ideal conditions.  
The variables for the lumped parameter model of the AHM-IT-DM-OCA are shown as follows. Referring to the hydraulic mount mechanical models [4; 16], a lumped parameter mechanical model for the AHM-IT-DM-OCA was established, as shown in Figure 2. The displacement on the engine side is \(y_{1}\), and the corresponding reaction force \(f_{1}\) is the force acting on the mount. The displacement of the fluid in the horizontal fluid channel is \(y_{2}\), and the mover displacement is \(y_{3}\). The displacement on the chassis side is \(y_{5}\), and the force transmitted to the chassis side is \(f_{5}\). The pressure fluctuations of the upper and lower fluid chambers relative to the static state are \(p_{1}\) and \(p_{2}\), respectively. The active force of the actuator is \(f_{a}\).  
Figure 1: Configuration schematics for the AHM-IT-DM-OCA [22]: 1. rubber bellow; 2. lower chamber; 3. inertia track; 4. decoupler membrane; 5. upper chamber; 6. main rubber spring; 7. actuator mover; 8. coil; 9. permanent magnet.  
Figure 2: Lumped parameter mechanical model for the AHM-IT-DM-OCA [22].  
The parameters for the lumped parameter model of the AHM-IT-DM-OCA are shown as follows. The mass on the engine side is denoted by \(m_{1}\), the mass of fluid in the inertia track is denoted by \(m_{2}\) and the mass of the actuator mover with the attached fluid is denoted by \(m_{3}\). The in-phase vertical dynamic stiffness, viscous damping, and equivalent piston area of the main rubber spring are denoted by \(k_{1}\), \(c_{1}\), and \(A_{1}\), respectively. The linear stiffness, viscous damping, and equivalent piston area of the decoupler membrane are denoted by \(k_{3}\), \(c_{3}\), and \(A_{3}\), respectively. The bulk stiffnesses of the main rubber spring, rubber bellow, and decoupler membrane are \(K_{1}\), \(K_{2}\), and \(K_{3}\), respectively. The bulk stiffness \(K_{2}\) of the lower fluid chamber enclosed by the wrinkled rubber bellows was several orders of magnitude smaller than \(K_{1}\). Thus, \(K_{2}=0\), and the pressure fluctuation in the lower fluid chamber can be ignored by assuming \(p_{2}=0\)[22; 23]. The relationship between linear stiffness \(k_{3}\) and bulk stiffness \(K_{3}\) can be expressed by Equation (1).  
\[A_{3}^{2}K_{3}=k_{3} \tag{1}\]  
The length, cross-sectional area, and hydraulic diameter of the inertial track are denoted by \(l_{2}\), \(A_{2}\), and \(d_{2}\), respectively. The loss factor of the fluid flowing along the inertia track can be expressed as \(\xi_{l2}\)[24]. The local loss factor of the fluid flowing at the entrance and outlet is given as \(\xi_{d2}\).  
Eight key parameters of the main rubber spring, decoupler membrane and actuator mover need to be identified. Parameters \(k_{1}\) and \(c_{1}\) for the main rubber spring can be acquired directly from the test results. Therefore, six key parameters (\(K_{1}\), \(A_{1}\), \(K_{3}\), \(A_{3}\), \(m_{3}\), and \(c_{3}\)) that are independent of each other must be identified by distinct features of the dynamics.