## 4 Controller for the MR Mount  
### Analysis of a Skyhook Control Strategy  
The desired damping force of a conventional damper is calculated following the damping force of an ideal Skyhook damper. The Skyhook damping force is determined by Equation (8), where \(C_{sky}\) is the damping coefficient of the Skyhook damper and \(\dot{Z}\) is the velocity of the piston in the Skyhook damper. The desired damping force of a conventional damper was formulated by [PERSON] et al. (2010) as in Equation (9), in which \(C_{sa}\) is the desired damping coefficient and \((\dot{Z}-\dot{q})\) is the relative velocity of the damper piston (equivalent to the relative velocity of the mount piston). To create the desired damping force, we must change the desired damping coefficient value according to the Skyhook damping coefficient at the velocity of the sprung and un-sprung mass, using the following formula:  
\[F_{d}=C_{sky}\dot{Z},\text{ and }F_{sa}=C_{sa}\left(\dot{Z}-\dot{q}\right), \tag{8}\]  
\[F_{d}=F_{sa}\ \leftrightarrow\ \ C_{sa}=\frac{C_{sky}\dot{Z}}{\left(\dot{Z}-\dot{q} \right)}, \tag{9}\]  
In the actual control of damping in the suspension system, a high damping force is required in the case of rebound (extension) motion \(\dot{Z}(\dot{Z}-\dot{q})\ \geq\ 0\). In contrast, it requires a low damping force in the case of jounce (compression) motion \(\dot{Z}(\dot{Z}-\dot{q})<0\). Therefore, two types of control change the desired damping coefficient, including the on-off type and continuous variable type, which are introduced in Figure 4a,b. For on-off type, there are two states, i.e., with and without the magnetic field applied, leading to the generation of the force or not, respectively. This type is simple to control; however, the range of damping coefficient is discrete. For the continuous variable, the damping coefficient varies on a continuous scale following a specific function, which will help the device produce the desired force range as well as to avoid fast state transitions that reduce the jerk phenomenon. Figure 4c,d illustrate the desired damping force control domain. The damping coefficient control of continuous variable type can provide a larger controllable force domain which enhances the effectiveness of the controller. The damping coefficient control following the ideal Skyhook damping coefficient of continuous variable type is given as follows:  
\[C_{sa}=\left\{ \begin{array}{ll}C_{sky}\frac{\dot{Z}}{\dot{Z}-\dot{q}}&\dot{Z }\left(\dot{Z}-\dot{q}\right)\ \geq\ 0\\ C_{min}&\dot{Z}\left(\dot{Z}-\dot{q}\right)<0\end{array} \right., \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Stiffness of the rubber, \(K_{s}\) (kN/m) & 50 \\ \hline Damping coefficient of the rubber, \(C_{s}\) (N\(\cdot\)m/s) & 50 \\ \hline Compliance of the upper chamber, \(C_{1}\) (m\({}^{5}\)/N) & \(2\times 10^{-10}\) \\ \hline Compliance of the lower chamber, \(C_{2}\) (m\({}^{5}\)/N) & \(2\times 10^{-8}\) \\ \hline Effective area of the upper chamber cross-section, Ap (m2) & \(7.065\times 10^{-4}\) \\ \hline Gap of MRF, \(g\) (m) & 0.002 \\ \hline Outer radius of the magnetic part \(R_{mp}\) (m) & 0.03 \\ \hline Equivalent viscous coefficient caused by MRF, \(R_{\mu}\) (Ns/m5) & \(9.91\times 10^{7}\) \\ \hline Ratio of the effective area caused by the magnetic field, \(k_{t}\) & 0.58 \\ \hline Weight of the sprung mass per each mount, \(m\) (kg) & 15 \\ \hline Density of MRF 140-CG (kg/m\({}^{-3}\)) & 3540 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of the RSD inspection car’s components. In the control Equation (10), the value of the desired damping coefficient will increase to infinity when the denominator \((\dot{Z}-\dot{q})\) is too small or close to zero. To overcome this problem, the upper value of the damping coefficient is limited by Equation (11) [27]:  
\[C_{sa}=\left\{ \begin{array}{ccc}max[C_{min},\min[C_{sky}\frac{\dot{Z}}{Z-\dot{q }},&C_{max}]]&&\dot{Z}\Big{(}\dot{Z}-\dot{q}\Big{)}\geq&0\\ C_{min}&&\dot{Z}\Big{(}\dot{Z}-\dot{q}\Big{)}<0\end{array}