No 1 (2025)

A problem of motion of a dumbbell-shaped body on a horizontal rough plane is considered. It is assumed that the dumbbell is a weightless inextensible rod, with masses being concentrated at two points of it, and there is dry friction between these points and the plane. It is also assumed that a constant force acts perpendicular to the rod on some fixed point on it. The conditions under which the rod is at rest, as well as the conditions under which the rod uniformly rotates around one of its points of support, are determined. The relationship between the magnitude of the angular velocity of uniform rotation and the force providing such a rotation is revealed. Bifurcation diagrams are constructed and analyzed.

Regular quaternion differential equations of the perturbed orbital motion of a cosmic body (in particular, a spacecraft, an asteroid) in the Earth’s gravitational field are considered, which take into account zonal, tesseral and sectorial harmonics of the field. These equations, unlike classical equations, are regular (do not contain special points such as singularity (division by zero)) for perturbed orbital motion in the central gravitational field of the Earth. In these equations, the main variables are four-dimensional Kustaanheim–Stiefel variables (KS-variables) or four-dimensional variables proposed by the author of the article, in which the equations of orbital motion have a simpler and symmetric structure compared to equations in KS-variables. Additional variables in the equations are orbital energy and time. The new independent variable is related to time by a differential relation containing the distance from the cosmic body to the Earth’s center of mass (the Sundman differential time transformation is used). Regular equations of perturbed orbital motion in quaternion osculating (slowly changing) variables are proposed. The equations are convenient for using methods of nonlinear mechanics and high-precision numerical calculations, in particular, for forecasting and correcting the orbital motion of spacecraft. In the case of orbital motion in the Earth’s gravitational field, the description of which takes into account the central and zonal harmonics of the field, the first integrals of the equations of orbital motion of the eighth order are given, changes of variables and transformations of these equations are considered, which made it possible to obtain closed systems of differential equations of the sixth order for the study of orbital motion, as well as systems of differential equations of the fourth and third orders, including a system of differential equations of the third order with respect to the distance from the cosmic body to the center of mass of the Earth and the sine of geocentric latitude, as well as a system of two integro-differential equations of the first order with respect to these two variables.

A theoretical and experimental method is poroposed for identification of mechanical properties of the surface layers of highly elastic materials by the results of their dynamic indentation for small depths (nanoDMA). The method is based on an approximate solution of the contact problem for a rigid ball in contact with a deformable specimen, the contact being loaded by an oscillating normal force. The specimen is modeled by a linear viscoelastic half-space with the relaxation kernel presented as a sum of exponential terms. The method allows one to determine sets of parameters defining the relaxation and creep functions of a material in a time interval corresponding to the experimental range of frequencies, as well as to calculate the dynamic storage and loss moduli for each frequency. The application of the method is shown by an example of the analysis of the mechanical properties of surface layers for two types of frost-resistant rubber (butadiene-nitrile and isoprene) depending on the degree of wear of their surfaces. It is established that the wear of surfaces of the rubbers under investigation leads to an increase of the surface layers stiffness and to a decrease in their relaxation properties; these changes are more pronounced for rubber based on nitrile butadiene than for that based on isoprene.

The asymptotic method for studying the behavior of non-stationary waves in thin shells generally involves using the separation method of solutions in the phase plane into components with different indices of variability in coordinates and time. In the case of normal type of impact, one of these components is an elliptical boundary layer occurring in a small neighborhood of the surface Rayleigh wave front. Its equations are derived by the method of asymptotic integration from the three-dimensional equations of elasticity theory. And they are partial differential equations of elliptic type with boundary conditions specified by hyperbolic equations. The article presents a general asymptotic method for solving the equations of the boundary layer under consideration in the case of the arbitrary form shell of revolution as an example. It is based on a preliminary study of basic problems for shells of revolution of zero Gaussian curvature using integral Laplace and Fourier transforms. The equations of this boundary layer for different types of normal loading have a common characteristic property: the asymptotically principal components coincide with the corresponding equations for shells of revolution of zero Gaussian curvature. This property, together with the property of different variability of the components of the stress-strain state and geometric parameters, allows, when using the method of exponential representations in the Laplace transform space, to functionally relate the solutions in the case of the arbitrary form shell of revolution with the solutions for shells of revolution of zero Gaussian curvature. The developed general approach is applied in this article to solving the problem of an elliptical boundary layer in shells of revolution under normal type loading. A numerical calculation of the shear stress for the obtained asymptotic solution in the case of a spherical shell is given.

The mechanical properties of metamaterials with a cellular chiral internal structure were experimentally studied during normal penetration by a rigid spherical striker. The metamaterial samples were 3D printed from TPU 95A plastic (thermoplastic polyurethane). They had auxetic and non-auxetic chiral structures of cells in the form of concave and convex hexagons, respectively. The results of the experiments on sample penetration, conducted for two temperature and two speed modes, are presented. The relative loss of kinetic energy of the striker during penetration of auxetic samples was significantly higher than that of non-auxetic ones. It was found that for the studied types of flexible metamaterials, the resistance to striker penetration increases with increasing temperature in the considered temperature range. The dependence of the striker deviation on exit from the flexible sample on the type of chirality of the structure being penetrated was established.

We conduct the qualitative analysis of differential equations describing the rotation of a dynamically asymmetric rigid body around a fixed point. The body is enclosed in a spherical shell, to which one ball and one disk adjoin. The motion of the body by inertia and under the action of potential forces is considered. It is established that in the absence of external forces, the differential equations have the families of solutions corresponding to the equilibrium positions of the body, and in the case of potential forces there exist manifolds of pendulum motions. For a number of the solutions, the necessary and sufficient conditions of the Lyapunov stability are derived.

Single crystals of cobalt-base alloy Co8.4Al9.4W1.9T, at. % with axial macro-segregation of tungsten and aluminum (gradient castings) were directionally solidified with a flat solidification front. Mini-specimens of different chemical compositions were cut from the obtained single-crystals at different casting heights for compression and oxidation tests. The tests performed at 900 °C showed that tungsten increases the yield strength of the alloy, while aluminum improves its oxidation resistance. It is shown that the method of directional solidification with a flat front can be effectively applied to optimize the physical and mechanical characteristics of multicomponent alloys of metals.