Vol 54, No 7 (2019)

The dynamic problem of the rotation of a rigid body (for example, a spacecraft) from an arbitrary initial to the required final angular position in the presence of control restrictions is considered and solved. The end time of the maneuver is known. To optimize the rotation control program, a quadratic quality criterion is used, the minimized functional characterizes energy costs. The construction of optimal turn control is based on quaternion variables and the L. S. Pontryagin maximum principle. The features of optimal motion are studied in detail. Key properties of the optimal solution are formulated in an analytical form. It is shown that in the case of limited control, the moment of forces in the process of optimal rotation is parallel to a straight line that is stationary in inertial space, and during rotation of a rigid body (spacecraft) the direction of the kinetic moment is constant relative to the inertial coordinate system. Optimal control is presented in the form of synthesis—the synthesizing function is found and the dependence of the control variables on the phase coordinates is given. Formalized equations and calculation expressions are obtained to determine the optimal rotation program. The constructive scheme for solving the boundary value problem of the maximum principle for arbitrary rotation conditions (initial and final positions and moments of inertia of a solid body) is also described. An example and results of mathematical modeling of the motion of a spacecraft as a solid with optimal control are presented, demonstrating the practical feasibility of the developed method for controlling the spatial orientation of the spacecraft. For a dynamically symmetric solid, a complete solution of the reorientation problem in closed form is given, control variables and the optimal trajectory of motion as functions of time are presented in an analytical form.

The problem of forced oscillations of a heavy inhomogeneous ball placed on a horizontal base is considered. It is assumed that the base moves in a horizontal direction according to harmonic law and at the point of contact between the surface of body and the base, the force and moment of viscous friction act. The equations of motion for a mechanical system are derived. Their solution is obtained by the averaging method. The dependences of the oscillation amplitudes for the excitation frequencies close to the eigenfrequency of the oscillations of the inhomogeneous ball on an absolutely smooth base are constructed. Both plane and spatial modes of oscillations have been found.

The long-term strength of rods under tension in an aggressive environment is investigated. We consider rods of various shapes of a single-cell cross-section (circle, square, and rectangles with different aspect ratios), provided that the areas of these cross-sections are equal. To determine the level of inclusions from aggressive environment into the rod at different time moments, approximate diffusion equations based on the motion of diffusion fronts from the rod surface are used. The high accuracy of the obtained approximation is shown. To assess the influence of an aggressive environment on long-term strength, the Yu. N. Rabotnov kinetic theory with two structural parameters (material damage and concentration of environment elements in the rod material) is used. The dependence of the time to rupture for the rods with various cross-section shapes under the same level of tensile stress is obtained. It is shown that the time to rupture for a rod with square cross-section under the indicated conditions exceeds the time to rupture for a rod with circular cross-section. In the case of rods with the same cross-section shape, the relation of the cross-section perimeter to time to rupture for the rod is considered. The minimum time to rupture for the considered rods is observed for rods of rectangular cross-section with the smallest thickness

The article presents the calculation of stress fields that arise in cylindrical rods of circular cross section in the case of Pochhammer–Chree waves. Examples of calculating the fields for the four lowest wave modes in a steel rod for two cases of phase velocities are considered. The structures of the isostat of the principal stresses in the longitudinal section of the rod are constructed and considered. The fields of the first principal stress and von Mises stress are constructed.

The fast decompositions method in analytical form was used to solve the elastic problem with mixed boundary conditions for stresses in a sharp wedge-shaped cutting tool of finite dimensions in the form of a truncated sector. A load is applied to the faces of the wedge and its nose, which rapidly decreases with distance from the nose. At a larger radius, the wedge is rigidly fixed. The resulting solution is valid for any angle of the wedge spread. If we take a small enough inner radius and the angle of the spread, then the shape of the proposed cutter will be close enough to real. The influence of the angle of the spread and the inner radius of the wedge on the magnitude and location of the highest stress σ̃ was investigated. The critical dimensions of the wedge were found, but for any of its sizes the stresses are finite everywhere.

A numerical analytical refined model of a short term forecast of the Earth’s pole motion is proposed. The model allows to increase the accuracy of predicting the coordinates of the pole with observed irregular effects in its motion. A numerical simulation of the oscillatory motion of the Earth’s pole is carried out in comparison with the data of observations and measurements of the International Earth Rotation Service, and the accuracy characteristics of the model are investigated.

The buckling and post-buckling behavior of initially imperfect, sinusoidally curved hinged-hinged beams undergoing very large deflections subjected to lateral concentrated loads acting at the midpoints is investigated via the geometrically nonlinear analysis of the problem. The transverse shear deformation is neglected. The amplitudes of the imperfection are considered to be relatively small. Therefore, the concerning curved beams (or arches) are shallow. The values of the extensional rigidity/length of the beams are assumed to be large enough not to permit considerable amount of change in the lengths of the beams during the deformation. The force-deflection curves corresponding to various amplitudes of the imperfection and the diagrams of the deflections and internal forces corresponding to various stages of the deformation, including those after the buckling, are presented. Not being able to solve the highly nonlinear problem analytically, numerical methods are used.