Vol 26, No 4 (2022)

Boundary value problems are studied for a one-dimensional Sobolev type integro-differential equation with boundary conditions of the first and third kind with two fractional differentiation operators $\alpha$ and $\beta$ of different orders. Difference schemes of the order of approximation $O(h^2+{\tau }^2)$ for $\alpha =\beta$ and $O(h^2+{\tau }^{2-\max \{\alpha ,\beta \}})$ are constructed for α≠β. Using the method of energy inequalities, a priori estimates are obtained in the differential and difference interpretations, from which the existence, uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme follow. Numerical experiments were carried out to illustrate the results obtained in the paper.

In the paper, the initial problem for the equation of vibrations of a rectangular plate with boundary conditions of the “hinged attachment” type is studied. An energy inequality is established, from which the uniqueness of the solution of the stated initial-boundary problem follows. The corresponding existence and stability theorems for the solution of the problem in the classes of regular and generalized solutions are proved. The existence of a solution to the problem posed is carried out by the method of spectral analysis and it is constructed as the sum of an orthogonal series over a system of eigenfunctions corresponding to a two-dimensional spectral problem, which is constructed by the method of separation of variables. A complete substantiation of the convergence of the constructed three-dimensional series in the class of regular solutions of the considered equation is given. The generalized solution is defined as the uniform limit of the sequence of regular solutions of the initial boundary value problem.In the paper, the initial problem for the equation of vibrations of a rectangular plate with boundary conditions of the “hinged attachment” type is studied. An energy inequality is established, from which the uniqueness of the solution of the stated initial-boundary problem follows. The corresponding existence and stability theorems for the solution of the problem in the classes of regular and generalized solutions are proved. The existence of a solution to the problem posed is carried out by the method of spectral analysis and it is constructed as the sum of an orthogonal series over a system of eigenfunctions corresponding to a two-dimensional spectral problem, which is constructed by the method of separation of variables. A complete substantiation of the convergence of the constructed three-dimensional series in the class of regular solutions of the considered equation is given. The generalized solution is defined as the uniform limit of the sequence of regular solutions of the initial boundary value problem.

Aim. In the present paper, nonlinear vibrations of an elastic simply supported plate on a viscoelastic foundation under the action of a moving oscillating load are studied in the case of the internal resonance 1:1 accompanied by the external resonance. The properties of the viscoelastic foundation are given via the generalized Fuss–Winkler model with the damping term described by the standard linear solid model with the Riemann–Liouville fractional derivatives. The external load is presented by linear viscoelastic oscillator based on the Kelvin–Voigt model with a fractional derivative in the case when the viscosity of the oscillator is considered to be small value. The dynamic behavior of the plate is described by a set of nonlinear ordinary differential equations of the second order in time with respect to generalized displacements. Methods. To solve the resulting set of equations, the method of multiple time scales is used in combination with the method of expansion of the fractional derivative in a Taylor series. Results. Resolving equations for determining of the nonlinear amplitudes and phases of force driven vibrations of the plate are obtained. The governing set of equations allows one to control not only the damping properties of the environment and the foundation by changing the fractional parameters, but also to control the damping parameters of the external load. Conclusion. Numerical analysis has shown that in the system “a plate on a viscoelastic foundation + a moving oscillating load”, energy transfer between the interacting vibration modes is observed. A comparison of the results of numerical studies for various values of the external load is presented, and the dependence of the amplitudes of nonlinear vibrations on the values of the fractional parameters of the environment and the foundation is also shown.

An analysis of the accuracy of the orbital elements obtained according to the coordinates and components of the velocities, found using the coefficients of the Chebyshev polynomials of the DE405 planetary catalog, is carried out. We compared the elements of orbital elements in the time interval from 1600 to 2200 years found using the DE405 catalog and obtained by numerical integration of the equations of motion based on the interaction of moving material bodies with the surrounding space. On the example of the numerical integration of the Moon motion equations, the advantage of using the equations of motion based on the interaction of moving material bodies with the surrounding space is shown in comparison with relativistic equations. Based on a comparison of the elements of Mercury's orbits, found by coordinates obtained by solving equations based on the interaction of moving material bodies with the surrounding space, and obtained using the DE405 catalog, it is shown that the orbital elements practically coincide on a given interval time. The maximum discrepancy in the mean anomaly at the end of the integration interval is less than 1′′ (second). The discrepancies of the secular displacements of perihelions for Mercury, Venus, Earth + Moon and Mars were determined, the values of which for DE405 are respectively: 43.08′′, 8.4′′, 3.83′′ and 1.14′′. It is shown that the errors of the secular displacements of the perihelions of the planets Mercury, Venus, the barycenter of the Earth + Moon and Mars obtained using the DE405 catalog take the following values: 0′′, 6.06′′, 3.83′′ and 1.08′′. For the outer planets: Jupiter, Saturn, Uranus, Neptune and the dwarf planet Pluto, on the basis of the considered comparisons of various equations of motion, no discrepancies in the orbital elements were found. Based on the studies carried out, it is shown that the use of harmonic coordinates in relativistic equations when creating the DE405 catalog is justified only for Mercury and the outer planets: Jupiter, Saturn, Uranus, Neptune and the dwarf planet Pluto.

Based on the model of incomplete reversibility of creep deformation, constitutive equations for the non-uniaxial stress state of metals under complex loading paths are proposed. The tensors of the viscoelastic, viscoplastic, and viscous components of the creep deformation are assumed to develop independently. The deformation kinetics is associated with the initial and deformation anisotropy. The measure of creep intensity for initially orthotropic materials is the equivalent stress introduced by Hill. In this case, the similarity of the stress and strain deviators is not required. The nature of the anisotropy of the deformation is associated with the value of the viscoplastic component of the deformation in the direction of the principal axes of the stress tensor. A superposition of the initial and deformation anisotropy is assumed. Samples made of 3KhV4SF tool steel and EI437B heat resistant alloy were tested, which are initially isotropic materials. The rheological coefficients of 3KhV4SF steel and EI437B alloy were calculated from the results of the uniaxial tension test samples at various levels of initial stresses. A comparative analysis of the forecast under complex loading according to the proposed equations with the test results was carried out.