Vol 54, No 2 (2019)

For a conflict-controlled dynamical system described by neutral-type functional differential equations in Hale’s form, a differential game is considered in the classes of control strategies with a guide for a minimax-maximin of the quality index, which evaluates the system’s motion history implemented by the terminal time moment. The differential game is associated with the Cauchy problem for a functional Hamilton-Jacobi type equation in coinvariant derivatives. It has been proven that the game value functional coincides with the minimax solution of this problem. A method of constructing the optimal strategies of players is given. The approximation by ordinary Hamilton-Jacobi equations in partial derivatives is proposed for this functional Hamilton-Jacobi equation in coinvariant derivatives.

Quaternion equations are proposed for the ideal operation of spatial inertial navigation systems with an azimuthally stabilized platform and a gyrostabilized platform that keeps its orientation invariant in inertial space, quaternion equations for the ideal operation of strapdown inertial navigation systems in the regular four-dimensional Kustaanheimo-Stiefel variables with consideration of zonal, tesseral, and sectorial harmonics of the Earth's gravitational field. The equations are dynamically similar to the regular equations of perturbed spatial two-body problem in the Kustaanheimo-Stiefel variables, which enables to use the results obtained in regular celestial mechanics and astrodynamics theory in inertial astronavigation. The development of operational algorithms for these navigation systems using these equations is considered.

In this paper, we studied the effect of internal dissipation on the rotational motion of a satellite in a central gravitational field using the Lavrent'ev model. Evolution equations are derived, and the results of an evolution analysis of the rotational motion of a dynamically symmetric satellite moving in a Keplerian circular orbit depending on the parameter values and initial conditions are presented.

The present work is aimed at studying the rotational motion of a satellite in a central force field in an elliptical orbit. A satellite is simulated by a dynamically symmetrical solid body with flexible viscoelastic rods, firmly fastened along the symmetry axis. In the absence of strain in the rods, the central ellipsoid of inertia of the satellite has a spherical shape. The averaged set of equations of perturbed motion near a 1 : 1 resonance at small eccentricities is obtained. Capture in a 1 : 1 spin-orbital resonance is substantiated.

The problem of the motion of a rigid body having a fixed point in a potential field of forces is considered. The existence conditions of three invariant relations of a special type, the choice of which is due to the integration of the Poisson equations by quadrature, are investigated. A new solution of the equations of motion of a dynamically symmetric body is found. A dynamically symmetric body is characterized by one arbitrary function of the vertical vector component. The case where the angular momentum modulus is constant is studied.

A discrete numerical tracking of the trajectories of dynamic systems is applied to demonstrate the existence of the closed trajectory in a 3D model of cyclic reactions that are associated with a Prigozhin brusselator.

The quasi-static plane-parallel motion of an infinite elastic cylinder along a horizontal foundation made of the same material is studied (the cylinder axis is horizontal). The distribution of normal and tangential stresses in the region of contact interaction between the cylinder and plane is derived from the solution to the problem of the theory of elasticity, where the Amontons-Coulomb dry friction model is used in the slip area and the presence of a stick area is taken into account. The analytic solution to the problem is obtained and the phase portrait of the system is drawn. We compare this with the problem of the motion of a perfectly rigid cylinder along a rigid surface with dry friction and without slip.

Based on the Kelvin model of a viscoelastic medium, the steady-state motion of a viscous lubricant thin layer between a viscoelastic cylinder and a viscoelastic base is considered. The effect of a dimensionless parameter proportional to the relaxation time of viscoelastic bodies exerted on the characteristics of the lubricant layer is studied. It is shown that for a high value of the mentioned parameter, the solution of the problem concerning the contact between viscoelastic bodies separated by a thin layer of lubricant is close to the solution of the elastohydrodynamic problem in the case of the elastic modulus equal to the instantaneous elasticity modulus inherent in viscoelastic bodies. As the value of the parameter decreases, the high-pressure area exhibits an expansion, whereas the maximum pressure exhibits a decrease, and the second pressure maximum decreases faster than the first one and then disappears. It is shown that the coefficient of rolling friction as a function of the relaxation time has a maximum. At small relaxation time values, the gap at the entry takes such a shape wherethrough the mode of abundant lubrication is impossible. As a result, the thickness of the lubricant layer rapidly decreases with decreasing relaxation time.

The interdependent dynamic behavior of a two-dimensional elastic system in the form of a one-dimensional mechanical object moving on a band is considered. The Lagrangian density of the two-dimensional system depends on the generalized coordinates and their derivatives up to and including the second order, and the Lagrangian of the moving object as one of the generalized coordinates contains the motion law, which is an unknown function of this problem. The physically and mathematically correct conditions at the moving boundary have been found as a result of formulating the self-consistent boundary-value problem based on the Hamilton variational principle. The problem of the unseparated motion of a rod, which performs bending and torsional vibrations, along a plate with consideration for the rotational inertia of its components is formulated as an example. The differential and integral laws of the change in energy and wave momentum are derived for both the entire complex system and its isolated parts. The relationships true at the moving boundary are established between the components of the energy flux density vector and the wave momentum flux density tensor.

Analytical solutions are obtained for the problem of torsion of incompressible cylinders, whose materials are described by the constitutive relations of viscoelastic media, generalizing the relations of the elementary Maxwell model for the case of finite deformations. Constitutive relations differ from each other in the value of the parameter specifying the type of an objective derivative used from the Gordon-Showalter family, including the Oldroyd, Cotter-Rivlin, and Jaumann derivatives.

It is shown that a longitudinal compressive force (the Poynting effect) is generated when twisting a cylinder with a constant length, whose material is described by any constitutive relation from the one-parameter family under consideration. For stepwise deformation processes, a qualitative description of a number of available experimental data are also obtained.

The transverse distributed load on an elastic thin plate in a gas environment is determined. Different values of the pressure of the medium on both surfaces form both the pressure difference and the transverse unbalanced force depending on the average pressure and curvature of the mid-surface. It is shown that in the general case both of these components of the transverse force should be taken into account. With a small ratio of the average pressure to the elastic modulus of the plate material and a large relative thickness, the influence of the second load component is small. With a small relative thickness of the plate and a large ratio of the average medium pressure to the elastic modulus of the material, the influence of the second component of the transverse bending load becomes comparable to the bending stiffness of the plate. Linear bending and plate stability are considered.