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

Abstract

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.

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] .…”

“…Another advantage of nonlinear cubic damping is that it not only expands the area of vibration isolation, but is also used to control the resonant oscillations of the rotor with large amplitudes and to exit the area of instability with a jumping effect or to attenuate and eliminate this effect [15] .…”

“…One of the known models of nonlinear damping is phenomenological, i.e. proportional to the n-th degree of velocity, adopted in [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] and many other works, for vibrating systems with small nonlinearities, close to linear systems. The results of theoretical studies of the responses of these systems are satisfactorily consistent with the results of experimental work [9] , [10] , [11] , [12] , [13] .…”

“…To estimate the stiffness and damping parameters of a system with strong nonlinearities, a method [41] based on measurements of both jump frequencies and jump amplitudes of the system subjected to sinusoidal excitations is used. In many nonideal nonlinear oscillating systems, the relationship between amplitude and frequency is determined using the averaging method [ 13 , 15 , 42 , 43 ]. Numerical simulations are also presented to illustrate the results.…”