## 2 Problem Formulation with Multi-Staged Clutch Dampers  
Figure 1a shows the nonlinear vibratory system with 1 DOF affected by piecewise-type nonlinearities based on multi-staged clutch dampers. The nonlinear model for piecewise-type nonlinearities is depicted in Figure 1b. To investigate the dynamic characteristics, the employed parameters for the torsional system shown in Figure 1a are as follows [1,2]: inertia of flywheel, \(I_{f}=1.38\times 10^{-1}\) kg\(\cdot\)m\({}^{2}\); viscous damping, \(c_{f}=1.59\) N\(\cdot\)m\(\cdot\)s\(\cdot\)rad. Here, the employed natural frequency \(\omega_{n}\) is 115.6 (rad/s) from \(I_{f}\) and 4 th stage of stiffness value listed in Table 1. In addition, \(\theta_{f}\) is the absolute displacement of the flywheel (subscript\(f\)), as indicated in Figure 1a. The profiles of the clutch torque \(T_{C}\) are listed in Table 1, where \(k_{Ci}\) and \(H_{i}\) (\(i=1\), \(2\), \(3\), and \(4\)) are the torsional stiffness and hysteresis at _the it_ stage, respectively, and \(\phi_{pi}\) (or \(\phi_{pi}\)) is the transition angle at the positive (or negative) side [1,2]. Figure 1b illustrates the 3 rd and  
Figure 1: A single-degree-of-freedom system with piecewise type nonlinearities: (**a**) a nonlinear torsional system model with 1 DOF; (**b**) Torque \(T_{C}(\delta_{1})\) profile for a multi-staged clutch damper [1,2]. 4 th stages of stiffness and hysteresis areas. Based on the 1 DOF shown in Figure 1a, the equation of motion is derived as follows:  
\[I_{f}\ddot{\theta}_{f}(t)+c\dot{\theta}_{f}(t)+f_{n}\left(\theta_{f},\dot{\theta }_{f}\right)=T_{E}(t)-T_{D}. \tag{1}\]  
Here, \(f_{n}\left(\theta_{f},\dot{\theta}_{f}\right)\) is a nonlinear function that describes the nonlinear clutch forces affected by piecewise-type nonlinearities. \(T_{E}(t)\) and \(T_{D}\) are the sinusoidal input and drag torque, respectively. In general, the input torque can be calculated using the Fourier coefficients based on the measured data, as follows:  
\[T_{E}(t)=T_{m}+\sum_{i=1}^{N_{max}}\ T_{pi}cos\left(i\omega_{p}t+\varphi_{pi} \right). \tag{2}\]  
Here, \(T_{m}\) and \(T_{pi}\) are the mean and alternating parts of the input torque, respectively; \(\omega_{p}\) and \(\varphi_{pi}\) are the excitation frequency and phase angle, respectively; and \(N_{\max}\) is the maximum number of harmonics correlated with the harmonic index of the HBM. The input torque profiles employed are listed in Table 2. In this study, the drag torque is assumed as \(T_{D}=T_{m}\) under steady-state conditions. In addition to the input conditions, \(f_{n}\left(\theta_{f},\dot{\theta}_{f}\right)\) as illustrated in Figure 1, is the main nonlinearity affected by multiple components, such as piecewise linear stiffnesses, hysteresis, and pre-load. Here, \(T_{SP}\), is the total clutch torque induced by the pre-load, \(T_{Pr1}\) (or \(T_{Pr2}\)) is the positive (or negative) torque induced by the pre-load, and \(\dot{\varphi}_{Pr}\) is the angle located at the pre-load. Based on prior studies, the nonlinear function \(f_{n}\left(\theta_{f},\dot{\theta}_{f}\right)\) (or \(T_{C}\)) is derived as \begin{table}
\begin{tabular}{c c c} \hline
**Property** & **Stage** & **Value** \\ \hline Torsional stiffness, \(k_{Ci}\) (linearized in a piecewise manner) & 1 & 10.1 \\
3 & 2 & 61.8 \\
3 & 3 & 595.8 \\
4 & 4 & 1838.0 \\ \hline Hysteresis, \(H_{i}\) (Nm) & 1 & 0.98 \\
5 & 2 & 1.96 \\
6 & 3 & 19.6 \\
4 & 4 & 26.5 \\ \hline Transition angle at positive side & 1 & 0.05 \\
6\(\left(\theta_{f}>0\right)\), \(\dot{\varphi}_{pi}\) (rad) & 3 & 0.30 \\
4 & 4 & 0.39 \\ \hline Transition angle at negative side & 1 & \(-\)0.04 \\
6\(\left(\theta_{f}<0\right)\), \(\dot{\varphi}_{pi}\) (rad) & 2 & \(-\)0.05 \\
3 & 3 & \(-\)0.09 \\
4 & 4 & \(-\)0.15 \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of the real-life multi-staged clutch damper [1,2].