Resonant vibrations of a non-ideal gyroscopic rotary system with nonlinear damping and nonlinear stiffness of the elastic support.

When motor performance characteristic is unknown, non-linear differential equations of motion of nonideal gyroscopic rigid rotary system with nonlinear cubic damping and nonlinear stiffness of the elastic support turn out to be numerically unsolvable.•In this case, the method uses the motor performance characteristic expression found from the frequency equation of forced stationary oscillations based on assumption that the angular acceleration is many times less than the square of the angular speed of rotation, replacing the stationary rotation angular speed with the shaft rotation angle derivative.•The method correctness is evidenced by a good consistency between the rotor motion equation numerical solution results and the analytical solution results, and by the nonlinear cubic damping of the shaft angular coordinate oscillograms obtained by direct simulation, as well as by comparison with the results of numerical simulation for a straight-line DC motor performance characteristic.•The method limitations are that it is used for the first approximation and weak nonlinear oscillations in the resonance region, where the shaft rotation speed is of the order of the oscillating system natural frequency.