## 3 The Proposed MR Mount
### Mathematical Model of the Mount
#### 3.2.2 Mount Dynamic Model  
The proposed RSD inspection car is a single degree of freedom (SDOF) type with a base including a bogie-wheel cluster. We assumed that (1) the mass is uniformly distributed to four mounts, (2) the vertical vibration is the primary vibration of the car, and (3) the left and right rail surface profiles are the same. Figure 2(a) presents the dynamic model for the quarter-car suspension system. In this model, the sprung mass displacement and motion velocity are determined in the local coordinates of \(Z(t)\), with a upward positive direction. Similarly, those of the un-sprung mass determined in the local coordinates \(q(t)\)have the same positive direction. The model is built with the denotations as the following: \(m\), \(Z\), \(\dot{Z}\), and \(\ddot{Z}\) are the mass, the absolute displacement, the velocity, and the acceleration of the sprung mass, respectively; \(M\), \(q\), \(\dot{q}\), and \(\ddot{q}\) are the mass, the absolute displacement, the velocity, and the acceleration of the base mass, respectively. The rubber body is equivalent to the spring stiffness \(K_{r}\). The internal friction damping caused by the rubber is \(C_{r}\). From the model in Figure 3a, the active force on the sprung mass is derived in Figure 3b. Assuming that the sprung part moves upwards of the equilibrium position of the static state, the elastic force of the rubber body \(F_{Kr}\) tends to pull the sprung mass part back to the equilibrium position; therefore, the direction of \(F_{Kr}\) will point down. The damping force caused by the internal friction of rubber \(F_{Cr}\) and the damping force generated by the pressure \(P_{1}\) of the upper chamber \(A_{p}P_{1}\) are all in a downward direction. The inertia force of the sprung mass \(F_{n}\) also has a downward direction, opposite to the acceleration. The dynamic balance equation for the sprung mass is determined in Equation (6):  
\[F_{n}+F_{K_{r}}+F_{C_{r}}+A_{p}P_{1}\ =0,\ \text{with}\ F_{a}=m\ddot{Z},\ F_{K_{r}} =K_{r}(Z-q),\ F_{C_{r}}=C_{r}\Big{(}\dot{Z}-\dot{q}\Big{)}, \tag{6}\]  
Finally, the parameters determined in Equation (5) are substituted into Equation (6) to find the acceleration of the sprung mass as in Equation (7):  
\[\ddot{Z}=\ \frac{1}{m}\left[-K_{r}(Z-q)-C_{r}\Big{(}\dot{Z}-\dot{q}\Big{)}- \frac{A_{p}^{2}(Z-q)}{(C_{1}+C_{2})}-\frac{A_{p}^{2}C_{2}R_{\mu}\Big{(}\dot{Z} -\ \dot{q}\Big{)}}{(C_{1}+C_{2})}-\frac{A_{p}C_{2}c\tau(R_{o}-r_{i})k_{\tau}}{(C_ {1}+C_{2})g}sgn\Big{(}\dot{Z}-\ \dot{q}\Big{)}\right], \tag{7}\]  
Equation (7) is the general dynamic equation for the sprung mass. The acceleration of the sprung mass is inversely proportional to the \(m\) of the sprung mass. It is also associated with the relative displacement \((Z-q)\), the relative velocity \((\dot{Z}-\dot{q})\), and the structure of the mount, such as the effective working area of the upper chamber, the elastic stiffness, the damping coefficient of rubber by the inertia friction, the equivalent viscous resistance coefficient generated by MRF viscosity \(R_{\mu}\), the yield stress of MRF when the external magnetic field is applied, and the gap of MRF. Finally, the acceleration of the sprung mass is dependent on the compliances of the upper and lower chambers. The study of [PERSON] et al. [26] showed that the stiffness of rubber is linear in the displacement range less than or equal to 6 (mm). In this paper, the displacement of the rubber is 6 mm; thus, we can use the linearity of the rubber stiffness. The proposed mount has similar structures to the rubber and diaphragm, and its size is smaller than one-half of that of the mounts designed by [PERSON] et al. [17] and [PERSON] et al. [26]. So, the parameters \(K_{s}\), \(C_{s}\), \(C_{1}\), and \(C_{2}\) are referred to as one-half of the corresponding parameters in these studies.  
Figure 3: Dynamic model: (**a**) a quarter-car suspension system; (**b**) the active force on the sprung mass.