Vol 54, No 1 (2019)

A new theory and a new algorithm for numerical solution of the problem of optimal spacecraft orbit reorientation by means of a pulsed (large) thrust, orthogonal to the osculating orbit plane, for a non-fixed number of thrust impulses are set out in a strict non-linear formulation. As a control, a vector of reactive acceleration from engine thrust is used. The combined functional is minimized, equal to the weighted sum of the reorientation time and the reactive acceleration impulse (characteristic speed) during the time of the spacecraft orbit reorientation (a special case of this functional is the case of minimization of the characteristic speed). To construct the theory, a solution for the problem of optimal spacecraft orbit reorientation in continuous formulation (using limited (low) thrust) is described in the first part of this article.

It is shown that the problem of optimal impulse spacecraft orbit reorientation in the case when optimal control consists of two reactive acceleration impulses, applied to the spacecraft at initial and final moments of time of motion, is solved analytically. Examples of numerical solution of the problem of optimal impulse spacecraft orbit reorientation are given, illustrating the capabilities of the proposed method.

The analysis of the constraint equations for strain deviator and V. V. Novozhilov stress deviator that contain uncertainty in the form of an additional characteristic, which is called the similarity phase of deviators is performed. This additional characteristic is a limitation of direct use of these equations for processing the experimental data. The transformation of the equations has allowed us to give the material functions a physical meaning of the shift characteristics in areas with main tangential stresses and to find relations for the generalized shear modulus and phase, which remove the indicated limitation and allow reflecting the effect of form change on bulk deformation and their interdependence.

The transformation of the initial equations, which include the constraint equation for average strain and stress tensor invariants, is reduced to the form that is characteristic for the equations of an anisotropic medium. Transformed equations make it possible to find another form of the equation for the average (bulk) deformation. This mathematical apparatus has allowed us to develop a method for determining all the characteristics of elasticity, including moduli and compliances that are capable to reflect the loosening effect due to the breaking of bonds and changing of the interaction between the elements of structure of composite materials, as well as the changing of elasticity.

Using the initial values of the bulk elasticity moduli, called “apparent” ones, since they characterize not the stiffness of the medium, but its change, three parameters of varying elasticity are determined. They are the ratio of the “classical” modulus assumed to be constant, when the average strain changes, to the “apparent” moduli. They make it possible to estimate the resulting deformation anisotropy and elasticity, which is preserved and acquired as a result of a change in the state of the bonds.

To verify the proposed method, the comparison of the calculations for the elastic and additional (dilatancy) parts of the bulk deformation according to the initial equation and equation found after the transformation that includes the compliance of the bulk elasticity and the loosening parameter with the experimental data has been performed. It has been shown that the obtained theoretical results are in a good agreement with experimental ones. Moreover, It has been stressed out that the loosening effect is the main and only reason for the change in elasticity, multimodularity, and anisotropy.

A quasi-static coupled contact problem of thermoelasticity that deals with a sliding frictional contact with taking into account the frictional heating is considered. Exact solutions of the problem are constructed in the form of Laplace convolutions, after calculating which the solution has been written in form of infinite series over eigenvalues of problem. The study of these eigenvalues in relation to three dimensionless parameters of the problem is carried out. Based on the analysis of the solutions obtained, it is possible to distinguish the domains of stable and unstable solutions in the space of dimensionless parameters. The properties of the obtained solutions are studied in relation to the dimensional and dimensionless parameters of the problem. Within the framework of the main research problem, partial problems of monitoring the sliding parameters as well as problems of controlling contact parameters in order to avoid thermoelastic instability are formulated and solved.

A qualitative analysis of the equations of motion of the Kovalevskaya top in a double constant field of forces is carried out. In the framework of this study, it was established that in the case of parallel force fields, the equations of motion of the body have families of permanent rotations, in the case of orthogonal ones they have pendulum-type movements, with special orientation of force fields they have families of equilibrium positions. It is shown that the solutions belong either to one of the two invariant manifolds of codimension 2, or to their intersection. For the solutions found, necessary and sufficient conditions of Lyapunov stability are obtained.

The forced transverse oscillations of an elastic hinge-supported rod under the action of a normal concentrated time-periodic force are investigated. The problem is solved by the method proposed in [1] using combined conditions, including the dynamic impact on the rod and the rotational motion relative to the bending wave front. In the framework of the linear theory of thin rectilinear inextensible rods the equation of motion and the system of equations of transverse vibrations of an elastic rod are obtained using the Hamilton-Ostrogradsky principle. The solution of the problem is built in the form of a number of own forms of vibrations. Two types of forced transverse oscillations and new resonant frequencies were obtained. Numerical results of calculations are given in the form of tables, graphs; the analysis of the results is provided.

There was a mistake in the affiliations. The right second affiliation is “Volgograd State Socio-Pedagogical University, pr. im. V.I. Lenina 27, Volgograd, 400066 Russia.”