## 3 The Proposed MR Mount
### Mathematical Model of the Mount  
#### 3.2.1 A Damping Force Model  
The damping force of the mount is constructed with the assumption of the Bingham plastic behavior of the MRF. This paper used MRF 140-CG, which generated high stress up to 58 (kPa) at a magnetic field intensity of 200 (kA/m) [23]. We used the experimental data (in graph form) showing the relationship between the yield shear stress \(\tau\) and the magnetic field intensity \(H_{m}\) for MRF 140-CG, which was published by the manufacturer, Lord Corporation. Based on these data, we approximated the polynomial function to represent the yield shear stress \(\tau\), using the least square curve-fitting method (LSM) to generate Equation (1):  
\[\tau=-2\times 10^{-12}H_{m}^{3}-2\times 10^{-6}H_{m}^{2}+0.6226H_{m}-389.85, \tag{1}\]  
The difference between the pressure of the upper chamber \(P_{1}\) and the pressure of the lower chamber \(P_{2}\) equals the sum of the pressure drops generated by inertia of fluid flow \(\Delta P_{I}\), the viscosity of fluid \(\Delta P_{\mu}\), and the yield stress of fluid \(\Delta P_{\tau}\). Each pressure drop for the MR mount with a valve structure using a flow mode is derived as in Equation (2). Further, the compliances of the upper chamber \(C_{1}\) and the lower chamber \(C_{2}\) also impact \(P_{1}\) and \(P_{2}\). They are determined based on the continuity equation of fluid [17]. Finally, the damping force equation of the mount is derived by the product of the upper chamber pressure \(P_{1}\) and the effective cross-sectional area \(A_{p}\) in Equation (3) [15,17,24,25]:  
\[P_{1}-P_{2}=\Delta P_{I}\ +\ \Delta P_{\mu}\ +\ \Delta P_{\tau}=I_{i}A_{p} \left(\ddot{Z}-\ddot{q}\right)+\frac{12k_{\mu}\mu QL_{m}}{\pi g^{3}Rn}+\frac{ 12k_{\mu}\mu Q}{\pi g^{3}}ln\frac{R_{ud}}{r_{i}}+\frac{128\mu QL_{ho}}{\pi \mathcal{D}{{}_{ho}}^{4}}+\frac{c\tau L_{\mu}k_{\tau}}{g}sgn\left(\dot{Z}- \dot{q}\right), \tag{2}\]  
\[F_{m}=P_{1}A_{p}=\left[\frac{C_{2}}{C_{1}+C_{2}}\left(\frac{c\tau L_{\mu}k_{ \tau}}{g}sgn\left(\dot{Z}-\dot{q}\right)+\frac{12k_{\mu}\mu QL_{m}}{\pi g^{3} R_{\mu m}}+\frac{128\mu QL_{ho}}{\pi g^{3}}ln\frac{R_{ud}}{\tau_{i}}\right)+ \frac{A_{p}(Z-\dot{q})}{(C_{1}+C_{2})}+iL_{\mu}\left(\ddot{Z}-\ddot{q}\right) \right]A_{p}, \tag{3}\]  
where \(c\) is the coefficient which depends on the flow velocity profile, \(c\) = [2.07-3.07] [24]; \(\mu\) is the viscosity of MRF; \(k_{\mu},k_{\tau}\) are the effective area ratio of the viscous fluid and the field fluid, respectively; \(Q\) is the flow rate (\(Q=A_{p}(\dot{Z}-\dot{q})\)); \(I_{i}\) is the inertia coefficient of the fluid (\(I_{i}\ \cong\ \pi\rho R_{od}^{4}/8g\)) [15]. Structural parameters of the magnetic part include the outer radius \(R_{od}\), the annular radius \(R_{an}\), the inner radius \(r_{i}\), the inner diameter \(D_{ho}\), the gap \(g\), the working radial and annular length \(L_{m}\), \(L_{an}\) of MRF; the height of core \(L_{ho}\). Dynamic parameters of the car will be explained in the next section. The values of all key parameters of the proposed mount are listed in Table 1.  
For convenience, a variable \(R_{\mu}\) is set as equivalent viscous resistance coefficient of the MRF flow (Equation (4)), and it depends only on structures and size of the proposed MR mount  
\[R_{\mu}=\frac{12k_{\mu}\mu L_{an}}{\pi g^{3}R_{an}}+\frac{128\mu L_{ho}}{\pi D _{ho}}^{4}\ +\frac{12k_{\mu}\mu}{\pi g^{3}}ln\frac{R_{od}}{r_{i}}, \tag{4}\]  
Further, the value of the term of inertial force of the MRF, \(I_{i}A_{p}(\ddot{Z}-\ddot{q})A_{p}\), is relatively small compared to the other terms. So, it is reasonable to neglect this term in Equation (3). Therefore, based on the formula of \(R_{\mu}\) and \(Q\), Equation (3) can be rewritten as the following:  
\[F_{m}=P_{1}A_{p}=\frac{A_{p}C_{2}c\tau(R_{o}-r_{i})k_{\tau}}{(C_{1}+C_{2})g} sgn\left(\dot{Z}-\dot{q}\right)+\frac{A_{p}^{2}C_{2}R_{\mu}\left(\dot{Z}- \dot{q}\right)}{(C_{1}+C_{2})}+\frac{A_{p}^{2}(Z-q)}{(C_{1}+C_{2})}, \tag{5}\]