Abstract. 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.
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.
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