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.