# Control of Automotive Semi-Active MR Suspensions for In-Wheel Electric Vehicles
## 2 Methods
### F-Class Vehicle Model  
For simulation purposes, an F-class vehicle is considered (Figure 2). F-class vehicles are known to be vehicles with higher performance, better handling, comfort, more top quality equipment and an stupendous design. Examples of commercial vehicles classified as F-class are: _Mercedes S class_, _Audi  
Figure 1: (a) Advanced-Dynamic-Damper-Motor (ADM), (b) In-wheel quarter of vehicle (QoV) three masses. _A8, and Cadillac DTS_. The parameters of the F-class vehicle shown in Table 2 were extracted from the CarSim parameters library. The main parameters of the \(QoV\) are the sprung mass (\(m_{s}\)), which corresponds to the vehicle chassis (body) and components supported by the suspension, and the unsprung mass (\(m_{us}\)) is compounded by the wheel components (tire, rim, and links). From the sprung mass of the whole vehicle (\(m_{bs}\)), the (\(m_{s}\)) for the \(QoV\) must be estimated by making use of the weight distribution of the vehicle; which can be computed employing the dimensions shown in Figure 3. For the present case study, the sprung mass for the QoV is calculated as follows: \(m_{bs}*[1-CG/Wheel_{base}]=m_{b_{s}}*(1-1265/3165)=m_{bs}*0.6\). For that, it can be considered that the 60% of the body mass is located at the front part of the vehicle and 40% in the rear part. Knowing that and considering an uniform distribution of the weight at the front (both corners) with two front passengers as well as a mid-level fuel tank, it is possible to estimate the \(QoV\) parameters shown in Table 3 for one corner of the F-class vehicle at the front, where \(k_{t}\) represents the tire stiffness which is assumed to be linked to the road (\(z\)), and the damping effect due the tire is not considered. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Positive Velocity, \(\hat{z}>0\)** & \multicolumn{2}{c}{**Negative Velocity, \(\hat{z}<0\)**} \\ \hline \(C_{MRpost-yield}(NA^{-1})\) & 873.9 & \(C_{MRpost-yield}(NA^{-1})\) & 828.4 \\ \({}^{\triangleright}\!r_{pre-yield}(sm^{-1})\) & 7.7 & \({}^{\triangleright}\!r_{pre-yield}(sm^{-1})\) & 9.6 \\ \(x_{pre-yield}(m^{-1})\) & 12.2 & \(x_{pre-yield}(m^{-1})\) & 15.3 \\ \(c_{p}(Nsm^{-1})\) & 530 & \(c_{p}(Nsm^{-1})\) & 925.2 \\ \(k_{p}(Nm^{-1})\) & \(-7361.6\) & \(k_{p}(Nm^{-1})\) & \(-8941.1\) \\ \(f_{f}(N)\) & 115.3 & \(f_{f}(N)\) & 31.7 \\ \(v_{friction}(sm^{-1})\) & 32.5 & \(v_{friction}(sm^{-1})\) & 208.4 \\ \(x_{friction}m^{-1}\) & 66.6 & \(x_{friction}m^{-1}\) & 609.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Semi-Phenomenological (SP) asymmetric model parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Description** & **Value** & **Unit** \\ \hline \(m_{bs}\) & Sprung mass or body & 1823 & kg \\ \(m_{ss}\) & \(QoV\) unsprung mass & 50 & kg \\ \(k_{s_{f}}\) & Front spring stiffness & 83 & N/mm \\ \(k_{s_{r}}\) & Rear spring stiffness & 44 & N/mm \\ \([z_{f},\overline{z_{f}}]\) & Suspension front stroke (joince/rebound) & \(+80/-60\) & mm \\ \([\overline{z_{r}},\overline{z_{r}}]\) & Suspension rear stroke (joince/rebound) & \(+70/-50\) & mm \\ \(k_{t}\) & Tire stiffness & 230 & N/mm \\ \(Wh_{base}\) & Wheel base & 3165 & mm \\ \(CG\) & Center of gravity (horizontal) & 1265 & mm \\ \(\eta\) & Motion ratio & 0.614 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: F-class vehicle parameters in CarSim. Figure 2: F-class vehicle, adapted from Carsim. A modified F-class vehicle model is also studied, where an in-wheel electric motor is considered inside each wheel hub instead of a unique internal combustion motor. The parameters in Table 4 are estimated knowing the new mass values, battery bank, and the characteristics of the electric motors taken from Reference [28] (for a general), and from Reference [29] (characterized for a Brabus Electric 4 WD Mercedes). The modified QoV parameters are shown in Table 5. The \(f_{m_{s}}\) and \(f_{m_{us}}\) resonance frequencies were estimated according to Reference [20].