## 2 Problem Formulation with Multi-Staged Clutch Dampers
\begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Torque Component**} \\ \hline \multicolumn{3}{c}{**Magnitude (Nm)**} \\ \hline \(T_{M}\) & \(T_{pi}\) & **Phase (rad)** \\ \hline
168.9 & 251.5 & \(-\)1.93 \\ \hline \end{tabular}
\end{table} 
Table 2: List of the input torque profiles by assuming the WOT condition on the vehicle [1,2]. follows [1;2]: First, the clutch torque induced by the stiffness \(T_{S}\Big{(}\theta_{f}\Big{)}\) was defined using the smoothing factor \(\sigma_{C}\). \[T_{S}\Big{(}\theta_{f}\Big{)}=k_{C1}\theta_{f}+\frac{1}{2}\sum_{i=2}^{N}\Big{(} k_{C(i)}-k_{C(i-1)}\Big{)}\Big{(}T_{sp(i-1)}-T_{sn(i-1)}\Big{)}, \tag{3a}\] \[T_{sp(i)}=\Big{(}\theta_{f}-\phi_{p(i)}\Big{)}\,\Big{[}tanh\Big{\{}\sigma_{C} \Big{(}\theta_{f}-\phi_{p(i)}\Big{)}\Big{\}}+1\Big{]},\] (3b) \[T_{sn(i)}=\Big{(}\theta_{f}+\phi_{n(i)}\Big{)}\,\Big{[}tanh\Big{\{}\sigma_{C} \Big{(}\theta_{f}+\phi_{n(i)}\Big{)}\Big{\}}-1\Big{]}. \tag{3c}\]  
Here, \(k_{C(N)}\) (or \(k_{C(i)}\)) is _the \(N^{th}\)_ (or \(i^{th}\)) stage of the clutch stiffness (with subscript \(N\) or \(i\)), \(T_{sp(i)}\) (or \(T_{sn(i)}\)) is the positive (or negative) direction of the clutch torque induced by the stiffness at _the \(i^{th}\)_ stage (with subscript \(p\) or \(n\)), and \(\phi_{p(i)}\) (or \(-\phi_{n(i)}\)) is _the \(i^{th}\)_ transition angle of the positive (or negative) side [3]. Second, \(T_{H}\) induced by hysteresis was derived using the smoothing factor \(\sigma_{H}\)[1;2]. \[T_{H}\Big{(}\theta_{f},\dot{\theta}_{f}\Big{)}=\frac{H_{(N)}}{2}tanh\Big{(} \sigma_{H}\dot{\theta}_{f}\Big{)}+\sum_{i=2}^{N}\Big{(}\frac{H_{(i)}}{4}- \frac{H_{(i-1)}}{4}\Big{)}\Big{[}T_{Hp(i-1)}+T_{Hn(i-1)}\Big{]}, \tag{4a}\] \[T_{Hp(i)}=tanh\Big{\{}\sigma_{C}\Big{(}\theta_{f}-\phi_{p(i)}\Big{)}\Big{\}} \Big{[}1+tanh\Big{(}\sigma_{H}\dot{\theta}_{f}\Big{)}\Big{]},\] (4b) \[T_{Hn(i)}=tanh\Big{\{}\sigma_{C}\Big{(}\theta_{f}+\phi_{n(i)}\Big{)}\Big{\}} \Big{[}1-tanh\Big{(}\sigma_{H}\dot{\theta}_{f}\Big{)}\Big{]}. \tag{4c}\]  
Here, \(H_{N}\) (or \(H_{(i)}\)) is _the \(N^{th}\)_ (or \(i^{th}\)) stage of hysteresis (with subscript \(N\) or \(i\)), and \(T_{Hp(i)}\) (or \(T_{Hn(i)}\)) is the positive (or negative) side of the clutch torque induced by hysteresis at _the \(i^{th}\)_ stage (with subscript \(p\) or \(n\)). The pre-load \(T_{SPr}\) was calculated as a function of \(\theta_{1 pr}\). \[T_{SPr}\big{(}\theta_{1 pr}\big{)}=\frac{1}{2}T_{Pr1}\big{[}tanh\big{(}\sigma_{ C}\theta_{1 pr}\big{)}+1\big{]}+\frac{1}{2}T_{Pr2}\big{[}-tanh\big{(}\sigma_{C} \theta_{1 pr}\big{)}+1\big{]}, \tag{5a}\] \[\theta_{1 pr}=\theta_{f}-\phi_{Pr}. \tag{5b}\]  
Overall, the total clutch torque is estimated by the summation of, \(T_{S}\Big{(}\theta_{f}\Big{)}\), \(T_{H}\Big{(}\theta_{f},\dot{\theta}_{f}\Big{)}\), and \(T_{SPr}\big{(}\theta_{1 pr}\big{)}\) from Equations (3)-(5), as follows:  
\[f_{n}\Big{(}\theta_{f},\dot{\theta}_{f}\Big{)}=T_{C}\Big{(}\theta_{1 pr},\dot {\theta}_{1 pr}\Big{)}=T_{S}\big{(}\theta_{1 pr}\big{)}+T_{H}\Big{(}\theta_{1 pr},\dot{\theta}_{1 pr}\Big{)}+T_{SPr}\big{(}\theta_{1 pr}\big{)}. \tag{6}\]  
The employed values for \(\sigma_{C}\) and \(\sigma_{H}\) are \(1\times 10^{3}\) and \(0.1\), respectively.