Sup-Resonant Response of a Nonautonomous Maglev System With Delayed Acceleration Feedback Control

A Maglev system with delayed acceleration feedback control is disturbed by the deflection of flexible guideway, and resonant response may take place. We have investigated sup-resonant response of the Maglev system by employing center manifold reduction and the method of multiple scales. We present the dynamic model and expand it to a third-order Taylor series. Taking time delay as its bifurcation parameter, we discuss the condition for the occurring of Hopf bifurcation. We apply center manifold reduction to get the Poincare normal form of the nonlinear system and employ the perturbation technique to study sup-resonant response of the system. This yields the sup-resonant periodic solution of the normal form. We analyze the stability condition of the free oscillation in the solution and discuss the relationship between guideway excitation and periodic solution. Finally, numerical results show how time delay, control, and excitation parameters affect the system response. With the proper system parameter, the free oscillation may vanish and only the periodic solution plays a part. Time delay can control amplitude of the forced oscillation. The appearance of the chaos phenomenon can also be governed by regulating time delay. And judiciously selecting a control parameter makes it possible to suppress the response.

I. Introduction

The maglev system carries out suspending function and is the core part of the maglev train. Fig. 1 gives its working principle. The guideway is flexible in the engineering environment [1]–[4]; thus, the oscillation induced by Bernoulli–Euler (B–E) beam equation must be considered. Dynamic responses of maglev vehicle/guideway system have great influence on running safety, ride comfort, and system costs, which are crucial factors for a maglev train's commercial application [1]. Cai et al. [2] built a dynamic model of the maglev system with multiple degrees of freedom (DOF) and guideway interaction, and numerically studied the elastic deformation of the guideway and dynamic behavior of the vehicle. Zheng et al. [3] discussed the stable and dynamic properties of the system coupled with the vehicle movements, guideway vibration, and control subsystem on the basis of a second-order suspension body with 5 DOF. By investigating the effects of the nonlinear spring on the dynamic behavior of such systems, Zheng et al. noted that nonlinear terms of spring, velocity of the vehicle, and control parameters influence the stable region of the maglev system, and chaos response may appear [4]. But resonant response of the maglev system induced by the guideway does not receive enough attention.