Resonant Oscillations of a Nonideal Gyroscopic Rotor System with Nonlinear Restoring and Damping Characteristics

Abstract

This paper presents a method for numerical solution of differential equations of motion of a nonideal gyroscopic rigid rotor with nonlinear cubic damping and nonlinear cubic stiffness of an elastic support in the case of unknown characteristics of the energy source. The essence of the method is that the excitation source characteristic is replaced by its expression found from the frequency equation of forced stationary oscillations under the assumption that the angular acceleration is many times smaller than the square of the angular rotation speed, replacing the angular rate of stationary rotation by a derivative of the shaft rotation angle. The results of the numerical solution are in good agreement with the results of the analytical solution of the nonlinear differential equations of motion, also the results of direct simulation of the rotor system motion when the force source characteristic is unknown and the direct current motor rectilinear characteristic are close to each other. Numerical dynamic analysis confirms that the amplitude of near-resonance oscillations is suppressed by combined linear and nonlinear cubic damping. The method is designed for solution of the first approximation and for weak nonlinear oscillations in the resonance region, where the shaft rotation speed is of the order of natural frequency of the oscillating system.