Zharilkassin Iskakov

In this paper, the dynamic modeling of the anisotropic non-ideal weakly gyroscopic rotor system is considered. Equations of nonstationary transitions are derived from motion differential equations; then, the control equation, stationary frequency dependencies, and force and energy relations are obtained. When the rigidity of the elastic support is anisotropic in orthogonal directions, two critical velocities and, accordingly, two resonance regions are found. Because of the strong interaction of the rotor system with a non-ideal DC motor, slopes of the resonance curves are observed in the regions of critical speeds even in the absence of a nonlinear component of the reference stiffness, and loops are also recorded. It is proven that, compared with linear damping, the cubic nonlinearity of damping strongly suppresses the resonant amplitudes of the rotor, reduces the size of the loops even more, and strongly attenuates the Sommerfeld effects until they are completely eliminated. It is shown that an increase in the magnitude of the cubic nonlinearity of damping greatly facilitates the passage of the resonance region and expands the range of operating speeds. This proves that the amplification of linear damping with cubic nonlinearity is an effective method for controlling resonant passages and an effective damping model.

In this paper, the dynamic modeling of the anisotropic non-ideal gyroscopic rotor system is considered. The equations of nonstationary transitions are derived from the motion differential equations, from there the control equation and stationary frequency dependencies, force and energy relations. When the rigidity of the elastic support is anisotropic in orthogonal directions, two critical velocities and, accordingly, two resonance regions are found. Because of the strong interaction of the rotor system with a non-ideal DC motor, slopes of the resonance curves are observed in the regions of critical speeds even in the absence of a non-linear component of the reference stiffness, and loops. It has been proven that the cubic nonlinearity of damping strongly suppresses the resonant amplitudes of the rotor, reduces the size of the loops even more, and strongly attenuates the Sommerfeld effects until they are completely eliminated than linear damping. It is shown that an increase in the magnitude of the cubic nonlinearity of damping greatly facilitates the passage of the resonance region and expands the range of operating speeds. This proves that the amplification of linear damping with cubic nonlinearity of damping is one of the methods for controlling resonant passages and an effective damping model.

The dynamics of the vibrational mode of motion of the driving member of a nonideal system, a mixing–whipping device based on a simple slide-crank mechanism, was studied. The highly nonlinear differential equations of motion were solved numerically by the Runge–Kutta method. The interaction of the mixing–whipping device with the nonideal excitation source causes the rotational speed of the engine shaft and the rotation angle of the driving member to fluctuate, accomplishing a damped process. The parameters of the device and the nonideal energy source have an effect on the kinematic, vibrational and energy characteristics of the system. An increase in the engine’s torque, crank length, number and radius of piston holes, and piston mass, as well as a decrease in the fluid’s density, leads to a reduction in the oscillation range of the crank angle, amplitude and period of angular velocity oscillations of the engine shaft and the mixing–whipping force power. The effects of a nonideal energy source may be used in designing a mixing–whipping device based on a slider-crank mechanism to select effective system parameters and an energy-saving motor in accordance with the requirements of technological processes and products.

The dynamics of the librational mode of motion of the driving member of a nonideal system, a mixing-whipping device based on a simple slide-crank mechanism, is studied. The highly nonlinear differential equations of motion were solved numerically by the Runge-Kutta method. The dependences of the rotation angle of the driving member, rotational speed of the actuating motor shaft on time and the dependences of the mixing-whipping force power on the rotation angle and rotational speed of the force source shaft, and the parameters of the nonideal system are obtained. The interaction of the mixing-whipping device with the nonideal excitation source causes the rotational speed of the engine shaft and the rotation angle of the driving member to fluctuate, accomplishing a damped process. The parameters of the device and the nonideal energy source have an effect on the kinematic, vibrational and energy characteristics of the system. The effects of a non-ideal energy source may be used in designing a mixing-whipping device based on a slider-crank mechanism to select effective system parameters and an energy-saving motor in accordance with the requirements of technological processes and products.

In the work, the differential equations of motion of the gyroscopic rotor, built taking into account the anisotropy of stiffness and damping of the flexible support, are solved analytically, by the method of harmonic balance, convenient for obtaining separately amplitude-frequency and phase-frequency characteristics in the direction of oscillations. The equations of the non-stationary process are obtained by the method of changing amplitudes. It has been found that when the linear stiffness of the elastic support is different in two orthogonal directions, two critical velocities and the corresponding resonant regions arise. In the area of each critical speed, there are two amplitude-frequency curves of oscillations of the main direction and the direction perpendicular to it, respectively. The geometric nonlinearity of damping suppresses the elevations of these amplitude-frequency curves more significantly than linear damping. If only one of the two directions has a damping nonlinearity, then its effect is on the amplitude-frequency curves of the corresponding critical speed. It is more efficient to control resonant amplitudes for smooth resonant transitions by enhancing the linear damping with geometrically nonlinear damping. The results of the analytical solution of the equations of motion agree well with the results of direct modeling and experimental studies.