Elimination of the vibration center shift in the nonlinear oscillations of a MAGLEV vehicle subjected to steady and unsteady aerodynamic forces: Second‐order multiple scales analysis

Source abstract

Abstract This study explores the dynamical behavior and stability of a magnetic levitation (MAGLEV) vehicle system controlled by a proportional‐derivative (PD) controller under steady and unsteady aerodynamic forces. The governing equation for the model is derived, showing that the displacement control gain affects the vehicle's linear stiffness, identifying conditions for positive, quasi‐zero, and negative stiffnesses. From previous studies, the model showed a shift in its vibration center, which might risk the safe operation of the vehicle as the suspension gap became smaller. In addition, this shift can generate deviations between the approximate‐analytical and numerical results. The main idea of this paper is to suggest an effective treatment for this shift, in the positive stiffness case, by implementing a convenient slight reduction in the value of the nominal current during its dynamical state under aerodynamic lift force. This could enhance the safety level during the vehicle's operation. Moreover, in the positive stiffness case, the multiple scales (MS) method is used to provide an approximate second‐order analytical solution to the model. The analysis includes the vehicle's motion stability utilizing the first method of Lyapunov. The effects of changing the vehicle mass, unsteady aerodynamic force amplitude, vehicle speed, the displacement, and speed control gains on vibration level and stability are examined. For the quasi‐zero stiffness case, the vibration characteristics of the MAGLEV vehicle are numerically simulated and investigated using the fourth‐order Runge–Kutta (RK4) method. The speed control gain should be properly chosen as it may delay reaching steady‐state conditions despite its effectiveness in mitigating large vibrations caused by low‐frequency external lift forces. Eventually, the negative stiffness case operation should not be allowed as the system exhibits dangerously high vibration amplitudes.