The paper considers an impact of viscous linear and cubic nonlinear damping of the elastic support on nonlinear vibrations of a vertical hard gyroscopic unbalanced rotor, taking into account nonlinear stiffness of the support material. Analyzing the research results shows that linear and cubic nonlinear damping can significantly suppress the resonance peak of the fundamental harmonic, eliminate the jumping phenomena of the nonlinear system. In non-resonance areas where the velocity is higher than the critical one, cubic nonlinear damping, unlike linear one, can slightly suppress amplitude of the rotor vibration. Therefore, in the high-velocity area, only nonlinear damping can maintain performance of a vibration isolator. In resonance area, an increase in linear or cubic nonlinear damping significantly suppresses the ability to absolute displacement. In non-resonance area, where the rotational velocity is lower than the critical one, they have almost no impact on ability to absolute displacement. In high velocity area, an increase in nonlinear damping may slightly increase the moment of force transmissibility, but linear damping has almost no impact on it. The obtained results can be successfully used to produce passive vibration isolators used for damping the vibrations of rotary machines, including gyroscopic ones.

The article considers the effect of joint linear and nonlinear cubic damping on dynamics of a gyroscopic rigid rotor interacting with an electric motor with a rectilinear characteristic, taking into account the nonlinear rigidity of the support material. The method of regulating the control parameter (voltage on the motor), the amplitude of vibration, and the angular velocity of the shaft in the frequency equation, depending on the value of the coefficient of nonlinear cubic damping of the support, offers the most effective options for controlling resonant oscillations of large amplitudes. It is shown that the greater the value of the coefficient of nonlinear cubic damping, the easier it is to control these oscillations. Moreover, it is proved that the Sommerfeld effect (of the first kind) can also be weakened and eliminate with the help of joint linear and nonlinear damping. To do this, in the case of a rigid characteristic of the nonlinear elasticity of the support material, in a rotor system with a nonideal energy source to eliminate the bistability region, that is, jumping effects, more nonlinear damping of support or energy from a nonideal energy source will be required than in the case of an ideal rotor system.

This study analytically and numerically modeled the dynamics of a gyroscopic rigid rotor with linear and nonlinear cubic damping and nonlinear cubic stiffness of an elastic support. It has been shown that (i) joint linear and nonlinear cubic damping significantly suppresses the vibration amplitude (including the maximum) in the resonant velocity region and beyond it, and (ii) joint linear and nonlinear cubic damping more effectively affects the boundaries of the bistability region by its narrowing than linear damping. A methodology is proposed for determining and identifying the coefficients of nonlinear stiffness, linear damping, and nonlinear cubic damping of the support material, where jump-like effects are eliminated. Damping also affects the stability of motion; if linear damping shifts the left boundary of the instability region towards large amplitudes and speeds of rotation of the shaft, then nonlinear cubic damping can completely eliminate it. The varying amplitude (VAM) method is used to determine the nature of the system response, supplemented with the concept of “slow” time, which allows us to investigate and analyze the effect of nonlinear cubic damping and nonlinear rigidity of cubic order on the frequency response at a nonstationary resonant transition.