Dynamic analysis of a self-excited hysteretic system

The dynamic behaviour of a self-excited system with hysteretic non-linearity is investigated in this paper. The averaging method is applied to the autonomous system and the resulting bifurcation equation of the self-excited response is analyzed using the singularity theory. The study of the bifurcation diagrams reveals the multivalued and jumping phenomena due to the effect of the hysteretic non-linearity. Secondly, the steady state response of the averaged system of the non-autonomous oscillator in primary resonance is investigated. Due to the effect of the hysteretic non-linearity, the system exhibits softening spring behaviour. A stability analysis shows that the steady state periodic response exists over a limited excitation frequency range. It loses its stability outside the frequency range through Hopf bifurcation and then the system undergoes quasi-periodic motion. Finally, by using circle maps to get winding numbers, various orders of super- and subharmonic resonance and mode-locking are investigated. The mode-locking, alternating with the quasi-periodic responses, takes place according to the Farey number tree as revealed in many other systems. The increase of the hystereticity can improve the stability of subharmonic resonance.