Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

Abstract

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.

Dynamic modeling of a non-ideal gyroscopic rotor system with nonlinear damping and nonlinear rigidity of an elastic support

“…Damped beats observed in Figures 8 to 12, 14, and 15 occur due to the superposition of forced nonstationary oscillations and damped natural oscillations with frequencies that closely coincide with the vicinity of the resonance. 22…”

“…Damped beats observed in Figures 8 to 12, 14, and 15 occur due to the superposition of forced nonstationary oscillations and damped natural oscillations with frequencies that closely coincide with the vicinity of the resonance. 22 According to equations (10), the direct modeling results can also be provided in the form of phase trajectories a 0 = a 0 a ð Þ and the dependence graph a = a b ð Þ of oscillating systems with different characteristics of nonlinear elasticity and different values of the coefficient of nonlinear cubic damping of the support, in Figures 16 and 17. Figure 16 shows that the value m 3 dampens not only angular misalignment a but also the angular velocity a 0 ; the sign change C 3 affects the size and density of phase trajectories.…”

“…He proposed a methodology is proposed for determining and identifying linear damping and nonlinear cubic damping of elastic support of a rigid gyroscopic rotor, in which surge effects in the frequency characteristics of a weakly nonlinear oscillating system are eliminated. 22 Al-Solihat and Behdinan 23 numerically investigated the nonlinear dynamic characteristics and characteristics of the transmitted force of a flexible rotor shaft-disk system supported by a suspension system with nonlinear stiffness and damping. The harmonic balance (HB) method, in combination with the continuation circuit, is used to determine the nonlinear frequency response and force transfer curves due to the disk unbalance force.…”

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

“…Experimental studies in [1] , [2] , [3] confirmed that the restoring and damping properties of viscoelastic materials should be described in nonlinear models and it was found that rubber insulators have both nonlinear stiffness and nonlinear damping. Subsequently, nonlinear shock absorbers began to be studied both by analytical and numerical methods, as well as by experimental methods [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] .…”

“…In it, the vibration of the rotor with a flexible support containing springs or rubber sheets with rotor vibration with a rigid support base is compared in the experimental single-disk rotor system supported by ball bearings at both ends. The works [11] , [12] , [13] , [14] , [15] are devoted to the study of the combined effect of linear and nonlinear cubic damping on resonance and behind it stationary and non-stationary oscillations, and on the stability of the movement of a gyroscopic rigid rotor with linear rigidity and nonlinear cubic rigidity of an elastic support. In the case of a rigid nonlinear characteristic of an elastic support, it is proved that if linear viscous damping merely shifts the lower boundary of the instability region towards high rotation speeds, then nonlinear cubic damping narrows this region on all sides until it is completely eliminated [12] , [13] , [14] .…”