Authors:I.G. Brykina; G.A. Tirskiy Abstract: Publication date: Available online 31 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): I.G. Brykina, G.A. Tirskiy The interaction of a large meteoroid with the Earth's atmosphere is considered in the case when it moves as a single body and when it moves as a cloud of fragments with a common shock wave. Using the literature data, an expression has been obtained for the radiative heat transfer coefficient per unit area of the midsection of a meteoroid modelled by an oblate spheroid as a function of its velocity, size, the atmospheric density, and its oblateness coefficient. The regions of predominant influence of the convective and radiative fluxes are estimated. An expression is obtained for the drag coefficient of a spheroid. Assuming that the mass of the meteoroid decreases faster than its velocity, analytical solutions of equations of the physical theory of meteors have been obtained for the mass loss of a meteoroid of spheroidal shape, the profile of the light curve, and the altitude at which the maximum of this curve is reached. The fragmentation model, which considers the meteoroid as a cloud of fragments with the spaces between them filled by a vapour, expanding in the lateral directions and flattening in the flight direction with a rate that depends on the degree of expansion is proposed. Using this model and the analytical solutions obtained, the interaction of the Chelyabinsk meteoroid with the atmosphere is considered. Good agreement between the found solution for the profile of the light curve and the observed data–light curves based on different video recordings–is demonstrated down to an altitude of 27km.

Authors:G. Ye. Yakunina Abstract: Publication date: Available online 9 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): G. Ye. Yakunina The possibilities of increasing the margin of stability of motion of optimum bodies having minimum drag or maximum penetration depth during high-velocity motion in a dense medium are investigated. It is assumed that the stresses generated by the medium acting on a surface element of the body are described within the framework of the local interaction model by binomial formulae quadratic in the velocity. A study has been performed for the case when the body shape is taken to be prescribed and when it is possible to vary it without departing from the class of optimum bodies. It is shown that for a fixed shape the simplest ways to increase the margin of stability of motion of the body are to decrease its mass or to move the centre of mass of the body closer to its vertex. It is possible to increase the margin of stability of motion of the body without decreasing its mass and without breaching the homogeneity of the body if it is equipped with fins. A method has been developed for constructing homogeneous optimum bodies with fins, whose bow (nose or leading part) is an optimum cone (OC) and whose stern (aft part) is constructed from segments of an OC and planes tangent to an OC of shorter length. It is shown that for prescribed mass, length, and base area of the body it is always possible to construct a homogeneous optimum body with positive margin of stability of motion. A test of the analytical results was carried out, based on a numerical solution of the Cauchy problem for the system of equations of motion of the body, constructed without simplifying restrictions on the shape of the body or the nature of its motion.

Authors:A.M. Gaifullin; V.V. Zhvik Abstract: Publication date: Available online 8 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): A.M. Gaifullin, V.V. Zhvik The flow of an inviscid incompressible fluid about a wing of low aspect ratio with a parabolic planform (a Nikolsky wing) is investigated; a protrusion, whose height grows according to a parabolic law, is mounted on the leeward side of the wing in the symmetry plane. It is shown that for symmetric boundary conditions, along with a symmetric solution, an asymmetric solution also exists. The dependence of the asymmetric solution on the wing geometry is examined. It is shown that critical values of the wing curvature and the height of the protrusion exist, for which the asymmetric solution continuously transitions to the symmetric one. In addition, it is shown that a limit asymmetric solution exists which corresponds to an infinitely large protrusion. The stability of the solutions found is discussed.

Authors:L.A. Klimina; B. Ya. Lokshin; V.A. Samsonov Abstract: Publication date: Available online 8 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): L.A. Klimina, B. Ya. Lokshin, V.A. Samsonov An autonomous dynamical system with one degree of freedom is considered which possesses properties such that an asymptotically stable equilibrium becomes unstable after a certain parameter passes through zero and two new symmetrically arranged equilibria are created alongside it. It is known that, for sufficiently small values of the above mentioned parameter, bifurcation can be accompanied by the occurrence of periodic trajectories (cycles). To describe them, a bifurcation diagram of the relation between the amplitude of the cycles and the parameter, which characterizes the dissipation and takes finite values, is constructed. The results obtained are illustrated using the example of an investigation of the self-induced oscillatory modes in a model of an aerodynamic pendulum that takes account of the displacement of the pressure centre when the angle of attack is changed.

Authors:V.G. Bazhenov; V.L. Kotov Abstract: Publication date: Available online 8 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): V.G. Bazhenov, V.L. Kotov A method is presented of investigating the stability of rectilinear motion of a body of revolution in a compressible soil medium with nonlinear physical-mechanical properties of the soil and two-dimensional effects of flow taken into account. The parameters of the axisymmetric process are calculated numerically, whereas the perturbed motion – the radial displacement and rotation relative to the centre of mass – is determined analytically. In the particular case of a conical projectile and linear pressure distribution along the generatrix, an estimate is obtained of the critical position of the centre of mass as a function of the taper angle, the mass and velocity of the body, the coefficient of friction, and the hydrodynamic parameters of the soil medium. Unlike the usually implemented situation of constant pressure postulated by the local interaction models, a displacement of the critical position of the centre of mass by up to 20% of the length of the cone has been found, which leads to a substantial decrease in the margin of stability in a restricted sense. Here, the force parameters and the kinematic parameters of motion of the cone on the boundary of the stability region differ both qualitatively and quantitatively. The stability of motion of bodies in soil media with a nonlinear pressure distribution over the contact surface has not previously been investigated.

Authors:L.A. Tolokonnikov Abstract: Publication date: Available online 8 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): L.A. Tolokonnikov On the basis of analytical solutions of problems on the diffraction of a plane sound wave on a homogeneous elastic sphere with a discretely layered coating and with a continuously-inhomogeneous coating, the direction diagrams of the scattered field are calculated. It is shown that a radially inhomogeneous coating can be modelled by a system of homogeneous elastic layers. For a linear dependence of the inhomogeneity, the number of homogeneous layers in the discretely-layered coating needed to ensure a prescribed accuracy of matching of the direction diagrams of bodies with a discretely layered coating to a continuously-inhomogeneous coating has been determined.

Authors:S.M. Aizikovich; S.S. Volkov; B.E. Mitrin Abstract: Publication date: Available online 8 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): S.M. Aizikovich, S.S. Volkov, B.E. Mitrin Using the bilateral asymptotic method, a semi-analytical solution of a dual integral equation with its right-hand side in the form of a Fourier series is constructed. This equation arises in the solution of a number of contact problems of elasticity theory for bodies with inhomogeneous coatings. The efficiency of the method is illustrated in the example of the solution of the plane contact problem on bending of a beam lying on a functionally graded strip with arbitrary variation of the elastic moduli with depth. It is assumed that the strip is perfectly bonded to an elastic half-plane. Numerical results are presented for a strip whose Young's modulus varies harmonically with depth. In this case, Young's modulus of the substrate is 100 times greater than at the lower boundary of the coating.

Authors:I.A. Soldatenkov Abstract: Publication date: Available online 8 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): I.A. Soldatenkov An analytical solution of the problem on the wear of a punch sliding on a thin elastic strip under the condition that they are in complete contact is given for variable friction and wear coefficients. The particular case of a wavy punch (periodic problem) is considered. Some distinguishing features of the process of wear in the transient regime are revealed. It is established, for example, that the contact pressure distribution curve can have a narrow local maximum (peak), significantly exceeding the average level of the contact pressure in magnitude.

Authors:N.M. Grevtsov; S.A. Kumakshev; A.M. Shmatkov Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): N.M. Grevtsov, S.A. Kumakshev, A.M. Shmatkov Using Bellman's dynamic programming method, a fuel-consumption optimum flight trajectory of a typical mid-range aircraft is constructed for a prescribed range. The optimum trajectory was calculated for the full model of motion of the aircraft along a trajectory in a vertical plane. Optimization was realized simultaneously for the whole trajectory without decomposing the process into individual steps. The method made it possible to take into account all restrictions imposed both by technical peculiarities of the aircraft and also by other requirements – safety and comfort of the passengers.

Authors:K. Yu. Osipenko Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): K. Yu. Osipenko A mathematical model of the plane motion of a non-axisymmetric body in a resistive medium is constructed using the local interaction method. Criteria for the stability of rectilinear motion are obtained in a general form in the case of a frozen axial velocity. The stability of the motion of a regular triangular pyramid is investigated in detail when a constant friction and pressure, specified using an empirical Poncelet formula in the form of a sum of inertial and strength terms, acts on its lateral surface. The stabilities of a pyramid and a cone are compared. The effect of deceleration on the stability of the rectilinear motion is considered.

Authors:E.V. Teodorovich Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): E.V. Teodorovich Based on the concept of cascade mechanism of energy transfer over the wave-number spectrum and related to the locality property of intermodal interactions forming the energy spectrum, the unified formula for spectral energy density is obtained that is valid in a wide range of wave numbers including both inertial and dissipation ranges. The form of the spectrum is governed by three parameters, namely, a value of wave number chosen as the normalization point assigned to the edge of the domain of the stochasticity emergence due to the development of instability of large-scale flows and the turbulent energy production, the spectral energy flux through the boundary of energy production domain in the wave-number space (the energy pumping rate), and the fluid kinematic viscosity.

Authors:V.V. Bulatov; Yu.V. Vladimirov Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): V.V. Bulatov, Yu.V. Vladimirov The problem of an internal gravity wave field from a pulsating point source of perturbations in the flow of a stratified medium of finite depth with horizontal velocity of the source V less than the maximum value of the group velocity of the internal gravity waves c is considered, unlike to the previously considered case V > c. Constructed asymptotic solutions make it possible to describe the amplitude-phase characteristics of individual modes comprising the full field of internal gravity waves. The excited waves consist of waves of two types: ring-like waves and wedge-shaped waves. Features of the mode structure of the excited fields are considered as functions of the parameters of the stratified medium and the characteristics of the perturbation source.

Authors:A.V. Borisov; G.M. Rozenblat Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): A.V. Borisov, G.M. Rozenblat Two mathematical models of rods of variable length from which an exoskeleton can be created, providing comfortable movement of a human in it owing to duplication of the properties of a motion-support apparatus, are considered. Their structure is elucidated on the basis of an analysis of the differential equations of motion, allowing for representing them in vector-matrix form. General regularities of the construction of the matrix elements entering into the system of differential equations of motion are established and generalizing formulae for the matrix elements are obtained. A new matrix method of constructing the differential equations of motion is presented and illustrated by a specific example. This system of equations is solved numerically. The possibility of reinforcing the control actions for control of the exoskeleton motion with a human inside it is considered.

Authors:I.B. Bakholdin Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): I.B. Bakholdin Equations which describe the propagation of waves in tubes with elastic walls are investigated, methods for calculating them are developed, and solutions containing reversible discontinuity structures are analysed in the case of a fluid-filled tube. The model of a tube with elastic walls constructed on the basis of the complete model of a membrane and the non-linear theory of elasticity is generalized. The viscosity and compressibility of the material, the possibility of filling the tube with a gas, and the flexural rigidity of the tube walls are taken into account. The problem of the decay of an arbitrary discontinuity is solved numerically in the case of a fluid-filled tube. The results obtained correspond to the previously developed theory of reversible discontinuities. Simplified hyperbolic equations of long waves, as well as equations for small-amplitude waves which do not take into account longitudinal elastic waves and are similar to the Boussinesq equations, are derived for cases when a tube is filled with a liquid and a gas. The possibility of overturning of the waves is analysed. A procedure for correcting the numerical schemes by adding terms with high-order derivatives to the equations is developed, and the order of approximation of the numerical scheme remains unchanged, enabling the performance of calculations with low schematic dissipation.

Authors:N.I. Amelkin; V.V. Kholoshchak Abstract: Publication date: Available online 7 May 2018 Source:Journal of Applied Mathematics and Mechanics Author(s): N.I. Amelkin, V.V. Kholoshchak Lavrent’ev's model (a satellite is simulated by a rigid shell with a spherical damper) is used to study the effect of internal forces on the motion of a satellite in a central gravitational field assuming that both dissipative and elastic internal forces arise with relative displacements of the damper. All the steady rotations are determined within the framework of this model for a dynamically symmetric satellite in a circular orbit and their stability is investigated as a function of the values of the damping and stiffness coefficients.