Dynamic analysis of a self-excited hysteretic system

The dynamic behaviour of a self-excited system with hysteretic non-linearit y is investigated in this paper. The averaging method is applied to the aut onomous system and the resulting bifurcation equation of the self-excited r esponse is analyzed using the singularity theory. The study of the bifurcat ion diagrams reveals the multivalued and jumping phenomena due to the effec t of the hysteretic non-linearity. Secondly, the steady state response of t he averaged system of the non-autonomous oscillator in primary resonance is investigated. Due to the effect of the hysteretic non-linearity, the syste m exhibits softening spring behaviour. A stability analysis shows that the steady state periodic response exists over a limited excitation frequency r ange. It loses its stability outside the frequency range through Hopf bifur cation and then the system undergoes quasi-periodic motion. Finally, by usi ng circle maps to get winding numbers, various orders of super- and subharm onic resonance and mode-locking are investigated. The mode-locking, alterna ting with the quasi-periodic responses, takes place according to the Farey number tree as revealed in many other systems. The increase of the hysteret icity can improve the stability of subharmonic resonance. (C) 2001 Academic Press.