Limit cycle oscillations in a mechanical system under fractional-order liénard type nonlinear feedback

Abstract

Anti-control of self-excited oscillation in mechanical and micro-mechanical systems is an important research problem due to its potential applications. In this paper, a novel fractional-order Liénard type nonlinear feedback is proposed to generate a limit cycle of desired frequency and amplitude in a single-degree-of-freedom spring-mass-damper mechanical oscillator. The feedback comprises two different fractional-order terms which are associated with both linear and nonlinear parts of the feedback and thus making it more general, with the van der Pol and Rayleigh type feedback as special cases. The analytical relations for steady-state amplitude and frequency of oscillation with the system and controller parameters are obtained by performing the nonlinear analysis with the method of two-time scale. Bifurcations of amplitude and frequency of oscillation with the fractional orders are studied in details. For any desired frequency and amplitude of oscillation, the controller parameters are obtained for minimum control cost. The analytical results are verified by numerical simulations performed in MATLAB SIMULINK and experiment. An equivalent integer-order model of the proposed fractional-order feedback system is developed to decipher the dynamics behind the generation of the limit cycle at different desired frequencies and amplitudes.