In this paper, a new method for investigation of dynamics of fluid-filled rotor systems is presented. The method consists of development of finite degrees-of-freedom (discrete) models for the rotor systems. The discrete models are physically justified and demonstrative. Being described by the system of ordinary differential equations, they allow one to employ powerful tools of the theoretical mechanics and oscillation theory. The method is applied to the case of the plane model of the rotor system partly filled with incompressible liquid. Both the continuous and discrete models are considered. The main attention is paid to the latter model. The discrete model consists of a disk symmetrically fixed on the shaft (Laval scheme), the ends of which are in viscoelastic bearings, and a ring sliding over the disk with friction. The centers of the disk and ring are elastically connected. The disk models the rotor, while the ring describes the liquid filling. When the ring is sliding over the disk surface, an interaction force arises that is diverted from the direction of the relative velocity at the contact points. It is demonstrated that an appropriate choice of the parameters of the discrete model allows one to determine the stability domain of the steady-state rotation of the rotor in the plane of the parameters of the shaft bearings with an excellent accuracy. It is found out that when the parameters overstep the limits of the stability domain, the Andronov-Hopf bifurcation occurs: a periodic motion of a kind of a circular precession arises from the steady-state rotation regime either “softly” or “hardly.”

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The dispersion characteristics of gyroscopic waves in an incompressible liquid layer in a cavity of a rapidly rotating cylinder are studied. It is shown that in a viscous incompressible liquid layer, an inertial wave can be represented as the sum of six helical harmonics. The effects of the liquid viscosity and the ratio of the wave frequency to the angular velocity of rotation of the cylinder on the real and imaginary parts of the wavenumber are studied. Introduction.Wave processes in a liquid with the effect of the Earth's rotation taken into account were considered in papers dealing with waves in the ocean (see, for example, [1-4] and the bibliography therein). The strong effect of rotation on the liquid dynamics is exemplified by rotary systems. Rotors with a liquid-containing cavity, in which the centrifugal force can be several hundred thousand or even several millions of times the gravity. Disturbance of the balance between the pressure gradient and the Coriolis force can lead to the generation of wave motions in the rotating liquid, which are called inertial or gyroscopic waves. These waves play an important role in the problems of dynamics of rotors, turbines, separators, centrifuges, and rotating aircraft containing a liquid, and in some geophysical problems (flows in the Earth's liquid core [5]). Wave phenomena in a rotating liquid layer can have a significant effect on a number of technological processes (in particular, sedimentation processes), phase equilibria in multicomponent liquids, and on the aircraft flight trajectory. The dynamics of inertial waves in a liquid filling a rotating cylinder has been experimentally studied in a number of papers (see [6,7] and references therein). Stationary gyroscopic waves in a circular cylindrical layer of an ideal liquid bounded by solid walls were studied in [8,9]. Dispersion relations were obtained and dependences of dimensionless wavelengths on dimensionless frequencies were constructed. The behavior of a low-viscosity liquid in a rotating horizontal cylinder as a function of the rotation velocity and the degree of filling was studied in [10]. The resonant generation of waves in a liquid filling a rotor cavity, which is the main factor responsible for instability of steady-state rotation of rotary systems was investigated in [11][12][13][14][15][16][17][18][19].The present investigation of the properties of gyroscopic waves in a rotating liquid layer is motivated by interest in the problem of the stability of rotary systems with a fixed point at which the angular velocity of rotation of a rotor is maintained constant by a drive. A stability analysis method for similar rotary systems was proposed in [15,17]. In this method, one of the main stages involves calculation of the moments of the forces exerted by the liquid on the rotor walls during steady-state conical precession of the rotor. It is easy to show that, in the case of conical precession, the steady-state hydrodynamic problem is directly related to the problem of generation of inertial waves in a liq...

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Институт гидродинамики им. М А. Лаврентьева СО РАН, Новосибирск, 630090, Россия Рассматривается задача о распространении зон пластического состояния в безграничной среде от границы выпуклой поверхности, на которой действуют нормальное давление, касательные усилия и заданные скорости перемещений. В случае полной пластичности система квазистатических уравнений идеальной пластичности Треска, описывающих напряженно-деформированное состояние среды, является гиперболической. Для численного решения этой системы предложена разностная схема, применяемая для гиперболических систем законов сохранения.The problem of propagation of plastic zones in an unbounded medium from the boundary of a convex surface under normal pressure, tangential forces and given velocities is studied. When the medium is in the state of full plasticity, the system of quasistatic ideal plasticity Tresca equations describing the stress-strain state is hyperbolic. The difference scheme applied to hyperbolic systems of equations is proposed for the numerical solution of this system.