# Mechanics

## Study of the mechanical properties of carbon molecular structures in the form of multilayer graphene with vertically oriented carbon nanotubes

In this work, we performed a theoretical study of the Young's modulus of carbon molecular structures in the form of multilayer graphene with vertically oriented carbon nanotubes (VO-CNTs). The carbon nanotubes that make up the molecular structures were of two types (zigzag and armchair). The studies were carried out by the molecular-mechanical method with the energy potential of AIREBO.

## On differential approximations of difference schemes

The concept of the first differential approximation was introduced in the 1950s for the analysis of difference schemes by A. I. Zhukov and then was used to study the quality of difference schemes approximating equations in partial derivatives. In the present work, the first differential approximation is considered as a universal construction that allows to use computer algebra methods for investigation difference schemes, bypassing the direct use of the methods of difference algebra.

## Waves in a viscoelastic cylindrical waveguide with a defect

In this paper, we consider a direct problem on waves in a viscoelastic inhomogeneous cylindrical waveguide with annular delamination and investigate an inverse problem on the identification of the delamination parameters on the basis of the additional data on the displacement field at the outer boundary of the waveguide. In order to account rheological properties within the framework of the complex modules concept, we use a model of a standard viscoelastic body.

## 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.