# Mechanics

## About a problem of spacecraft's orbit optimal reorientation

The problem of optimal reorientation of the spacecraft's orbit is solved with the help of the Pontryagin maximum principle and quaternion equations. Control (thrust vector, orthogonal to the orbital plane) is limited in magnitude. Functional, which determines a quality of control process is weighted sum of time and module (or square) of control. We have formulated a differential boundary problems of reorientation of spacecraft's orbit.

## Моделирование трещинообразования в полосе переменной толщины

Проведено математическое описание модели зарождения трещины в полосе переменной толщины. Определение неизвестных параметров, характеризующих зародышевую трещину, сводится к решению системы сингулярных интегральных уравнений. Получено условие, определяющее критическое значение внешней нагрузки, при которой происходит трещинообразование.

## Determination of the Wall Temperature Change for a Cavity in a Solid as a Result of the Temperature Change of the Gas Flow in a Cavity

The wall temperature change for a cylindrical cavity in a solid was found as a response to the temperature change of the gas flowing in a cavity. Three important special cases of the gas temperature dependence on time are considered: temperature is constant; temperature changes according to the linear law; temperature changes according to the harmonic law. The plots of five «µ-functions» used to denote solutions are submitted.

## Biomechanical Assessment of the Bone Ingrowth Effect During Cementless Endoprosthesis Osteointegration

Finite elementmodel of porous titaniuminserts for cementless endoprosthesis was reconstructed usingX-ray tomography. The stress distribution is calculated for a model with open-cell foam and composite bone / titanium. The results explain the mechanism of the porous structure destruction and positive influence of the osteointegration effect on the strength properties. Numerical calculations are confirmed by experimental data of the porous samples during compression testing.

## Dual Matrix and Biquaternion Methods of Solving Direct and Inverse Kinematics Problems of Manipulators, for Example Stanford Robot Arm. I

The methology of solving the direct kinematics problem of manipulators by using screw mechanics methods (dual direction cosine matrices, Clifford biquaternions) is shown on the example of Stanford robot arm. Kinematic equations of motion of the manipulator are found. These equations will be used for solving the inverce kinematics problem with the help of biquaternion theory of kinematic control.

## Configuration Space in Second Boundary Value Problem of Non-classical Plate Theory

The article contains investigation of second boundary value problem for equilibrium equation «in mixed formulation» describing nonclassical mathematical model for hinged isotropic and uniform plate under generalized Timoshenko hypothesis taking into account initial irregularities. For this problem for the first time were proved the existance of generalized solution and weak compactness of the set of approximate solutions obtained with Bubnov–Galerkin method using V. Z. Vlasov scheme.

## Dynamical Simple Edge Effect in the Cylindrical Shell with the Edge of Arbitrary Form

The purpose of the article is to generalize the results derived in the cases of a circular shell and of a shell with a cut edge. Non-stationary wave process in a cylindrical shell with an arbitrary edge is considered. Half-geodesic frame is introduced on the middle surface of the shell and dynamical simple edge effect is studied. To find the solution Laplace transform is used while the inverse transform is realized via saddle-point method.

## Technique of Definition Areas Requiring a Quantum Description Within of the Hybrid Method (Quantum Mechanics / Molecular Mechanics)

The new model, which determines the active area (the region for which high-precision quantum methods must be used) of the structure,was developed within the of the hybrid method (QM/MM). Problem of determining atoms with the critical tension values is the basis of this model. The potential energy of these atoms and its nearest neighbours was calculated by quantum-chemical method. The potential energy of the rest structure was calculated by molecular mechanical method.

## The One-dimensional Problem of Unsteady-related Elastic Diffusion Layer

The problem of determining the stress strain state of an elastic medium, taking into account the structural changes caused by the presence of diffusion fluxes. The influence of diffusion processes on the stress-strain state of the environment is taken into account by using the locally equilibrium model of thermoelastic diffusion, which includes the coupled system of equations of motion of an elastic body and the equations of heat and mass transfer.

## Upper and low bounds of azimuthal numbers related to elementary wave functions of an elliptic cylinder

Numerical and analytical aspects of generating 2π-periodic solutions of the angular Mathieu equation obtained for the circumferential harmonics of an elliptic cylinder and localization problem for the Mathieu eigenvalues and corresponding azimuthal numbers are considred. Those are required in usual procedure of constructing the elliptic cylinder elementary wave functions playing a very important role in mathematical physics.