# Vol 25, No 1 (2019)

**Year:**2019**Articles:**8**URL:**https://journals.ssau.ru/est/issue/view/374

## Full Issue

## Статьи

### THE CORRECTNESS OF A DIRICHLET TYPE PROBLEM FOR THE DEGENERATE MULTIDIMENSIONAL HYPERBOLIC-ELLIPTIC EQUATIONS

#### Abstract

Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that the oscillations of elastic membranes in space according to Hamilton’s prism can be modeled by multidimensional degenerate hyperbolic equations. Assuming that the membrane is in equilibrium in half the bend, from Hamilton’s principle we also obtain degenerate elliptic equations. Consequently, vibrations of elastic membranes in space can be modeled as multidimensional degenerate hyperbolic-elliptic equations. When studying these applications, it is necessary to obtain an explicit representation of the investigated boundary value problems. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where the unique solvability of this problem is shown, which essentially depends on the height of the cylindrical domain under consideration. However, the Dirichlet problem in a cylindrical domain for multidimensional degenerate hyperbolic-elliptic equations has not been studied previously. In this paper, the Dirichlet problem is studied for a class of degenerate multidimensional hyperbolic-elliptic equations. Moreover, the existence and uniqueness of the solution depends on the height of the considered cylindrical domain and on the degeneration of the equation. A uniqueness criterion for a regular solution is also obtained.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):7-20

### A PROBLEM WITH AN INTEGRAL CONDITION OF THE FIRST KIND FOR AN EQUATION OF THE FOURTH ORDER

#### Abstract

The article deals with a non-local problem with an integral condition for fourth-order pseudo-hyperbolic equation. The equation contains both a mixed derivative and a fourth order derivative in the spatial variable. The integral condition is a condition of the first kind, which leads to difficulties in the study of solvability of a problem. One of the successful methods of overcoming the difficulties of such a plan is the transition from the conditions of the first kind to the conditions of the second kind. The article proves the equivalence of the conditions of the first kind to the conditions of the second kind for this problem. The conditions on the coefficients of the equation and the input data are obtained and they guarantee the existence of a single problem solving. In the literature, such an equation is called the Rayleigh-Bishop equation.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):21-31

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 1. MOTION EQUATIONS AND CLASSIFICATION OF MALFUNCTIONS

#### Abstract

In the proposed cycle of work, we begin the study of the motion of an aircraft which is described bynonlinear ordinary differential equations. Based on these equations, the probable malfunctions in the motioncontrol system are classified, the concepts of reference malfunctions and their neighborhoods are introduced,the mathematical modeling of these malfunctions and their neighborhoods is carried out, the concept ofdiagnostic space is introduced, and the mathematical structure of this space is defined. Proposed work isthe first in the cycle, therefore, the classification of malfunctions is given. This activity is also a preparatorypart of the diagnostic problem, which can be represented in the form of two successively solved problems,i.e., control problem, that is the problem of determining the presence of a malfunction in the system, anddiagnostic problem, that is the recognition problem of malfunction specification. This activity is just anillustration of the proposed approach.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):32-43

### RAYLEIGH — RITZ METHOD AND METHOD OF INITIAL PARAMETERS IN THE PROBLEM OF CALCULATION OF DYNAMIC CHARACTERISTICS OF MULTIBEAM ELASTIC STRUCTURES

#### Abstract

The solution of the problem of dynamic synthesis based on the application of the Rayleigh — Ritz method is considered. A method is proposed for determining the dynamic characteristics of a composite beam structure, taking into account the calculation of the shapes of oscillations of partial substructures using the method of initial parameters. Two variants of the formation of coordinate functions, using static and dynamic condensation, are considered. When carrying out static condensation, in order to increase the accuracy of the result, the internal physical degrees of freedom of the element were added to the boundary degrees of freedom to the reduced model degrees of freedom. When conducting dynamic condensation, modal degrees of freedom were added to the physical degrees of freedom of the boundary nodes of the reduced model, which are, in fact, coefficients in the accepted decomposition of the field of partial forms calculated with fixed boundary degrees of freedom. In the framework of the proposed approach, test calculations were carried out for a beam with variable mass-stiffness characteristics along the length, which showed good convergence of the target parameters to exact values. The proposed approach can be used to carry out calculations of composite elastic structures based on the method of initial parameters in cases where the application of the finite element method is irrational or difficult. In addition, if necessary, on the basis of the approach considered in this paper, the combined use of these two methods can be organized.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):44-56

### THEORETICALLY RECONSTRUCTED ISOCHROMATIC FRINGES IN THE VICINITY OF THE CRACK TIP

#### Abstract

Theoretically reconstructed isochromatic fringe in the vicinity of the crack tip was received. The problem for a plate with central horizontal crack under uniaxial tension was considered. Isochromatic fringe in the vicinity of the crack tip image generating programme using complete asymptotic expansion of M. Williams stress field with scaling factor for infinity plate was developed. It was demonstrated that using large term of sum in the asymptotic expansion is necessary. If target isochromatic fringe is located far from the vicinity of the crack tip, then using many terms of sum in the asymptotic expansion of M. Williams is required.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):57-62

### INFLUENCE OF THE HIGHER ORDER TERMS IN WILLIAMS’ SERIES EXPANSION OF THE STRESS FIELD ON THE STRESS-STRAIN STATE IN THE VICINITY OF THE CRACK TIP. PART I

#### Abstract

The paper is devoted to the multi-parameter description of the stress fields in the vicinity of two collinear crack of different length in an infinite isotropic elastic medium subjected to 1) Mode I loading; 2) Mode II loading; 3) mixed (Mode I + Mode II) mode loading. The multi-parameter asymptotic expansions of the stress field in the vicinity of the crack tip in isotropic linear elastic media under mixed mode loading are obtained. The amplitude coefficients of the multi-parameter series expansion are found in the closed form. Having obtained the coefficients of the Williams series expansion one can keep any preassigned number of terms in the asymptotic series. Asymptotic analysis of number of the terms in the Williams asymptotic series which is necessary to keep in the asymptotic series at different distances from the crack tip. It is shown that the more distance from the crack tip the more terms in the Williams asymptotic expansion need to be kept.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):63-79

### INFLUENCE OF THE HIGHER ORDER TERMS IN WILLIAMS’ SERIES EXPANSION OF THE STRESS FIELD ON THE STRESS-STRAIN STATE IN THE VICINITY OF THE CRACK TIP. PART II

#### Abstract

The description of mechanical fields at the vicinity of a bi-dimensional crack-tip can be performed using the classic Williams asymptotic series expansion. While the general structure is well known, complete expressions are rarely available for specific crack problems. The paper is devoted to the multi-parameter description of the stress field in the vicinity of two collinear crack of different length in an infinite isotropic elastic medium subjected to 1) Mode I loading; 2) Mode II loading; 3) mixed (Mode I + Mode II) mode loading. The multi-parameter asymptotic expansions of the stress field are obtained. The procedure used in the paper relates Williams series coefficients and the complex potentials of the plane elasticity. The amplitude coefficients of the multi-parameter series expansion are found in the closed form. Having obtained the coefficients of the Williams series expansion one can keep any preassigned number of terms in the asymptotic series. Asymptotic analysis of number of the terms in the Williams asymptotic series which is necessary to keep in the asymptotic series at different distances from the crack tip. It is shown that the more distance from the crack tip the more terms in the Williams asymptotic expansion need to be kept. Complete closed-form expressions can be used to derive, test and improve numerical and experimental approaches involving higher order terms in crack-tip expansions.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):80-96

### THE PREDICTOR-CORRECTOR METHOD FOR MODELLING OF NONLINEAR OSCILLATORS

#### Abstract

In the work physically reasonable algorithm of numerical modeling of nonlinear oscillatory and self-oscillatory systems are offered. The algorithm is based on discrete in time model of the linear oscillator. Nonlinearity is considered by the introduction to the oscillator of additional communications by the structural analysis of an initial system. For approximation of a temporary derivative in nonlinear communications it is offered to use the scheme of the prediction and correction. In spite of the fact that theoretically the algorithm has the second order of accuracy, within the numerical experiment with Van der Pol oscillator it shows better results, than a standard method of the second order — the Heun’s method.

**Vestnik of Samara University. Natural Science Series**. 2019;25(1):97-103