Abstract A kinematic coupling model of the rotation-axial engagement process of friction elements is established based on the oil film dynamic pressure lubrication and an elastic contact model of the asperities. The engagement characteristics of the multi-disc wet friction clutch are analyzed. The equation for the transmission torque of the clutch is derived by considering the axial movement of the friction elements and the surface morphology of the friction pairs. The influences of the piston pressure and the number of friction pairs on the speed and transmission torque of the clutch during engagement are analyzed considering the viscous torque and the contact torque. Finally, the effects of engagement pressure on the torque and engagement time of the clutch are verified.