Vol 88, No 1 (2024)

The problem of stability of non-degenerate linear systems admitting a first integral in the form of a non-degenerate quadratic form is considered. New algebraic criteria for stability, as well as complete instability of such systems, have been established in the form of equality to zero of traces of products of matrices, which include an additional symmetric matrix. These conditions are closely related to the symplectic geometry of the phase space, which is determined by the matrix of the original linear system and the symmetric matrix defining the first integral. General results are applied to finding conditions for complete instability of linear gyroscopic systems.

The article describes the results obtained for the upper and lower bounds (estimates) for the apsidal angle (precession angle) in the theory of the motion of the heavy symmetrical solid body about fixed point (Lagrange’s case) for arbitrary initial conditions and parameters of the body. All regions of initial conditions is divided into two sets. In the first set there is a direct precession of the top, in the second set there is a retrograde precession of the top.

The problem of trajectories avoiding in nonlinear conflict-controlled processes (differential games) in L.S. Pontrjagin and E.F. Mishchenko statement is considered. Terminal sets have a particular rarefied structure. Unlike other works, they consist of countable points and may have a limit points. New sufficient conditions and evasion methods are obtained, which make it possible to solve a number of avoiding trajectory problems of oscillatory systems, including the swinging problem of the generalized mathematical pendulum.

The problem of acceleration from a state of rest of a shear flow in a viscoplastic half-plane is studied analytically when a tangential stress is specified at the boundary. It is assumed that the dynamic viscosity and density of the medium are constant, and the yield stress can change in a continuous or discontinuous manner depending on the depth. The entire half-plane at any moment of time consists of previously unknown layers where shear flow occurs and rigid zones. The latter can move as a rigid whole, or they can be motionless, such as, for example, a half-plane, to which disturbances caused by the action of tangential forces have not yet reached. To find the stress and velocity fields, a method is developed based on quasi-self-similar diffusion-vortex solutions of parabolic problems in areas with moving boundaries. The question of what conclusions about the depth distribution of the yield stress can be drawn from available measurements of the velocity of the half-plane boundary is discussed.

The distributions of residual and working stresses in hollow spheres pre-strengthened using a combination of hydraulic and thermal autofrettage are investigated. The analysis is based on the theory of infinitesimal elastoplastic strains, the Tresca or von Mises yield condition, the associated flow rule and the linear isotropic hardening law. During unloading, the sphere material may exhibit the Bauschinger effect. All mechanical and thermophysical parameters are assumed to be independent of temperature. Exact analytical solutions are found for both loading and unloading stages including secondary plastic flow. The values of technological parameters are established at which the strengthening effect is achieved near the inner surface of the sphere. Analysis of the results shows that the use of a positive temperature gradient makes it possible to increase the absolute value of residual stresses on the inner surface of the sphere. On the other hand, with the help of a negative gradient it is possible to reduce working stresses in the sphere.