## 3 Numerical Simulation
### Dynamic Response of Wet Friction Clutch  
Figure 3a,b demonstrate the wet clutch rotation speed and drag torque dynamic response curves, respectively. As the control oil pressure increases, the wet friction clutch successively goes through the engagement process (I), the engagement state (II), and the disengagement process (III), and finally returns to the disengagement state (IV). When the wet friction clutch is in the engagement process, as the control oil pressure increases, the oil film thickness decreases, and the viscous torque first increases and then decays to zero, while the friction torque increases, the output shaft speed increases to 2388 rpm, and then remains constant. However, when the clutch is in the disengagement process, as the control oil pressure decreases, the frictional torque gradually decreases to zero, the oil film thickness increases, the viscous torque increases and then decays, and the output shaft speed decreases to zero.  
Figure 3c illustrates the friction element axial displacement curves, where the axial displacement of the friction components decrease to a stabilized value and then increase to a stabilized value with the control oil pressure. The oil film thickness can be obtained  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Outer radius of friction pairs \(R_{o}\) & 0.094 m & Inertia of output shaft \(I_{f}\) & 0.3 kg\(\cdot\)m\({}^{2}\) \\ Outer radius of friction pairs \(R_{l}\) & 0.06 m & Thickness of friction plate \(H_{fd}\) & 0.003 m \\ Number of friction pairs \(n\) & 8 & Thickness of steel plate \(H_{fd}\) & 0.002 m \\ Return spring stiffness \(R_{r}\) & \(1.52\times 10^{6}\) N\(\cdot\)m\({}^{-1}\) & Asperities density \(\lambda\) & \(7\times 10^{7}\) m\({}^{-2}\) \\ Input rotation speed \(\omega_{1}\) & 2388 rpm & Asperities radius \(R\) & \(8\times 10^{-4}\) m \\ Control oil pressure \(P_{app}\) & 1.6 MPa & RMS roughness \(\sigma\) & \(8.41\times 10^{-6}\) m \\ Initial gap of friction pairs \(h_{o}\) & \(5\times 10^{-4}\) m & Equivalent Young\({}^{\prime}\) modulus \(E^{\prime}\) & \(4.7\times 10^{9}\) Pa \\ Resistance torque \(T_{r}\) & 460 N\(\cdot\)m & Non-groove area coefficient \(A_{red}\) & 0.78 \\ Permeability \(\varphi\) & \(4\times 10^{-13}\) m\({}^{2}\) & Friction material thickness \(d_{m}\) & 0.001 m \\ Mass of friction plate \(m_{f}\) & 0.6 kg & Mass of piston \(m_{p}\) & 3 kg \\ Mass of steel plate \(m_{s}\) & 0.45 kg & Lubricant oil density \(\rho\) & 875 kg\(\cdot\)m\({}^{-3}\) \\ Spline friction coefficient \(\mu_{s}\) & 0.1 & Damping coefficient \(c_{s}\) & 0.08 N\(\cdot\)m\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation input parameters.  
Figure 3: The response curve of: (**a**) rotation speed, (**b**) drag torque, (**c**) axial displacement, and (**d**) oil film thickness in clutch operation.  
from Equation (1) and the corresponding result is shown in Figure 3d. In the friction clutch engagement process, the friction elements respond sequentially from left to right, while in the disengagement process, the friction elements respond sequentially from right to left, the first steel plate follows the piston movement, resulting in a gap of almost zero throughout the clutch disengagement process.