# Mechanics

## Free vibration frequencies of a circular thin plate with nonlinearly perturbed parameters

Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant ones and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem.

## Analytical algorithm of energy and time quasioptimal turn of a spacecraft under arbitrary boundary conditions

The optimal attitude maneuver control problem without control constraints is studied in the quaternion statement for an axially symmetric spacecraft as a rigid body under arbitrary boundary conditions on angular position and angular velocity of a spacecraft. The performance criterion is given by a functional combining the time and energy used for the attitude maneuver. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution.

## Artificial libration points in the task of towing space debris by an ion beam

The work is devoted to the problem of towing space debris from a geostationary orbit to a graveyard orbit by a non-contact method using an ion beam created by the engine of an active spacecraft. For the planar case using the modified Hill task, the points of relative equilibrium (libration points) of the active spacecraft relative to the object of removal are determined and their stability is estimated. It is shown that, depending on the values of radial acceleration, there are up to 6 libration points, but only one point is suitable for towing a space debris object.

## Approximation of the orientation equations of the orbital coordinate system by the weighted residuals method

In the quaternion formulation, the problem of mathematical modeling of the spacecraft movement in an elliptical orbit was considered. Control is an acceleration vector from jet thrust. Control modulus is constant. The control is directed orthogonally to the plane of the spacecraft orbit. The quaternion differential equation of an orbital coordinate system orientation was used to describe spacecraft movement.

## Constructing the dependence between the Young’s modulus value and the Hounsfield units of spongy tissue of human femoral heads

Patient-specific biomechanical modeling requires not only the geometric model of the studied object of a particular patient, but also the mechanical properties of its tissues. Quantitative computed tomography provides the initial data for geometric modeling, as well as data on X-ray density (Hounsfield units) of the object. It is known that Hounsfield units correlate with mineral density of the scanned objects, as well as with their strength properties.

## Bounded finite-time stabilization of the prey – predator model via Korobov’s controllability function

The problem of finite-time stabilization for a Leslie-Gower prey – predator system through a bounded control input is solved. We use Korobov’s controllability function. The trajectory of the resulting motion is ensured for fulfilling a physical restriction that prey and predator cannot achieve negative values. For this purpose, a certain ellipse depending on given data and the equilibrium point of the considered system is constructed. Simulation results show the effectiveness of the proposed control methodology.

## Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness.

## Features of complex vibrations of flexible micropolar mesh panels

In this paper, a mathematical model of complex oscillations of a flexible micropolar cylindrical mesh structure is constructed. Equations are written in displacements. Geometric nonlinearity is taken into account according to the Theodore von Karman model. A non-classical continual model of a panel based on a Cosserat medium with constrained particle rotation (pseudocontinuum) is considered. It is assumed that the fields of displacements and rotations are not independent.

## On the identification problem of the thermomechanical characteristics of the finite functionally graded cylinder

The problem of axisymmetric vibrations of a functionally graded finite hollow cylinder is considered. The ends of the cylinder are thermally insulated and are in a sliding fit. Zero temperature is maintained on the inner surface of the cylinder, free from stress, and a combined thermal and power load acts on the outer surface. The direct problem after applying the Laplace transform is solved based on the method of separation of variables.

## The flexural strength of anisotropic composite plates with free edges

Modern technology shows increased demands on the strength properties of machines, their parts, as well as various structures, reducing their weight, volume and size, which leads to the need to use anisotropic composite materials. Finding criteria to determine the ultimate strength characteristics of structural elements, engineering structures is one of the urgent problems of solid mechanics. Strength problems in structures are often reduced to finding out the nature of the local stress state at the vertices of the joints of the constituent parts.