Sub-harmonic resonances of the non-autonomous system with delayed position feedback control

The response of the Maglev system with delayed position feedback control under the sub-harmonic excitation of the flexible guideway is investigated. The dynamical model is linearized at the equilibrium. Employing time delay as its bifurcation parameter, the condition under which the Hopf bifurcation may occur is investigated. Center manifold reduction is applied to get the Poincare normal form of the nonlinear system with guideway disturbance so that we can study the relation between periodic solution and system parameter. The sub-harmonic resonant periodic solution of the normal form is calculated based on the method of multiple scales, and we get the bifurcation equation of the free oscillation. The existence condition of the free oscillation in the solution is analyzed. Relationship between periodic solution and control and excitation parameters is also investigated. Finally numerical method is applied to study how system and excitation parameters affect the system response. It was shown that the critical time delay to keep the response of the system stable is less than that without perturbation. Time delay can not only suppress sub-harmonic resonance, but also control the appearance of the chaos. Control parameter can govern the emergence of the free oscillation and affect the amplitude of the forced oscillation. So carefully selecting the system parameters can restrain the oscillation effectively.