Forced Torsional Vibration of a Two-Degree-of-Freedom System Including a Clearance and a Two-Step-Hardening Spring

The engine-clutch-gearbox train of an automobile is modeled by a forced torsional vibration system including two nonlinear terms, namely a clearance and a two-step-hardening spring. Analytical solutions of the equations of motion of the system are obtained in recurrent form. Based on the solutions, the effect of the torsional stiffness of the clutch disk on the idle rattle of the gearbox is investigated. The results are as follows : (1) The sum of torque impulses per cycle of the forced motion increases with the increase in the ratio of the second-step stiffness to the first-step stiffness. Therefore, there is an upper limit to the second-step stiffness for the purpose of the prevention of idle rattle. (2) Both the sum of torque impulses and the number of collisions decrease with the increase in the range of torsional displacement covered by the first-step stiffness.