PRIMARY AND SUBHARMONIC SIMULTANEOUS RESONANCE OF DUFFING OSCILLATOR 1)

In this paper, the dynamics and stability of the Duffing oscillator subjected to the primary resonance together with the 1/3 subharmonic resonance are studied. At first, the approximate analytical solution and amplitude-frequency equation are obtained through the method of multiple scales, and the correctness and satisfactory precision of the approximate solution are verified by simulation. Then, the amplitude-frequency equation and phase-frequency equation of steady-state response are derived from the approximate analytical solution, and it can be found there are at most seven different periodic solutions, which are called multi-value characteristics and can be used to switch the state of the system. Moreover, the stability condition of steady-state response is derived based on Lyapunov theory, and the amplitude-frequency curves of steady-state response are compared with the cases where the primary or 1/3 subharmonic resonance exists alone, and it is found that the system contains both resonance characteristics. At last, the effects of nonlinear factor and excitations on the system response are analyzed by simulation. The particular phenomena in this system are revealed, i.e., the nonlinear factor affects the response amplitude, multi-value characteristics and stability of the system with stiffness softening. However, for the stiffness hardening system, the nonlinear factor only affects the response amplitude, which is similar to the cases of single-frequency excitation. These results are important for the study on the Duffing system or other similar systems.