Based on the mechanism of linear and nonlinear forces on the rotor system

the linear scale factor

nonlinear scale factor and coupling ratio was introduced as characteristic parameters

the general expressions of damping and stiffness forces were established

so a general form of linear and nolinear coupling vibration model was formulated. Considering a single-disc rotor system excited only by mass unbalance force

the steady-state and transient-state solutions were derived through multi-scale method. The analysis on the analytical solutions showed the vibration mechanism of linear and nonlinear coupling effects and responses. The influence of transient time scale factor on responses was analyzed through numerical calculation. When transient time scale factor was larger

the transient-state solution decayed faster

approaching the steady-state solution closer. The amplitude-frequency characteristics were also analyzed by the short time Fourier transform of the steady-state solution. It can be seen that the first harmonic generation has nonlinear characteristics and the third harmonic generation has double-peak characteristics

further elaborating the influences of nonlinear scale factor on the coupling vibration of the rotor system.