# Mechanics

## Equilibrium Analysis of the Tethered Tug Debris System with Fuel Residuals

The problem of tethered transportation of space debris is considered. The system consists of orbit tug, tether, and passive spacecraft with fuel residuals. The planar motion on circular orbit is studied in the orbital frame. Nonlinear motion equations are obtained by Lagrangian formalism. They consider action of the space tug-thrust and gravitational moments. Two variants of stable positions of relative equilibrium are deﬁned. They depend on main parameters of the tethered system: aspect ratio and mass ratio.

## The Motion of Propping Agent in an Opening Crack in Hydraulic Fracturing Plast

## Application of Generalized Differential Quadrature Method to Two-dimensional Problems of Mechanics

## Bending of a Sandwich Beam by Local Loads in the Temperature Field [Изгиб трехслойной балки локальными нагрузками в температурном поле]

Deformation of sandwich beam in a temperature field under the action of uniformly distributed and sinusoidal local loads is considered. An analytical view of the loads was set by using functions of Heaviside. To describe kinematic properties of an asymmetric through thickness of sandwich beam we have accepted the hypotheses of a broken line as follows: Bernoulli’s hypothesis is true in the thin bearing layers; Timoshenko’s hypothesis is true in the compressible through thickness filler with a linear approximation of displacements through the layer thickness.

## Identification of Properties of Inhomogeneous Plate in the Framework of the Timoshenko Model

We consider an inverse problem on identification of properties of an inhomogeneous circular plate for the Timoshenko model. The identification procedure is based on the analysis of

## Calculating of the Fastest Spacecraft Flights between Circular Orbits

The problem of optimal reorientation of spacecraft orbit is considered in quaternion formulation. Control (jet thrust vector orthogonal to the plane of the orbit) is limited in magnitude. It is necessary to minimize the duration of the process of reorientation of the spacecraft orbit. To describe the motion of the spacecraft center of mass quaternion differential equations of the orientation of the orbital coordinate system was used.

## An Asymptotic Model for the Far-Field of Rayleigh Wave in Multilayered Plate

An asymptotic model is proposed, which allows to calculate farfield of Rayleigh wave in an infinite multilayered plate subjected to non-stationary surface load. The model is derived by using of the standard asymptotic techniques. As a result, a system of two onedimensional integro-differential equations (head system) is obtained, which describes the propagation of Rayleigh waves along the plate surfaces. For the decaying wave fields in layers the boundary problems for elliptic equations are obtained.

## Propagation and Reflection of Harmonic Waves in a Plane Acoustic Layer with Non-Homogeneous Flexible Walls

A plane acoustic layer bounded by elastic membranes, one of which has an insert with different material properties, is considered. The propagation and reflection of harmonic waves in such a layer is studied. The source of vibrations is an incident mode, coming from infinity. The solution in three regions (before the insert, under the insert, after the insert) is sought as modal expansion. The numerical results for the reflected power coefficient are presented.

## Scheme Models Development of Integro-Differential Equations Numeral Calculation of Processes Dynamics in Electric Circuits

Scheme models of numeral calculation of integral-differential equations, describing transients in electric circuits are developed. It is shown that offered modeling has the best fast-acting concerning to the known calculations.

## Cross-Coupled Type-III Thermoelastic Waves of a Given Azimuthal Number in a Waveguide under Sidewall Heat Interchanging

The paper is devoted to a study of cross-coupled type-III generalized thermoelastic waves of a given azimuthal order propagating via a long cylindrical waveguide with circular cross-section. Sidewall of the waveguide is assumed free from tractions and permeable to heat. The study is carried out in the framework of coupled generalized theory of type-III thermoelasticity (GNIII) consistent with the fundamental principles of continuum thermomechanics. The type-III theory combines the both possible mechanisms of heat transfer: thermodiffusion and wave.