THE TOTAL BOUNDARY VALUE PROBLEMS FOR LONGITUDINAL OSCILLATIONS OF ROD

Abstract

Modeling of oscillatory processes is associated with second-order differential equations in partial derivatives (general boundary value problems). Longitudinal oscillations of rods are oscillatory processes for the study of which discrete-continuous mathematical models are used, the basis of which are general boundary value problems. Methods for solving nonstationary boundary value problems can be divided into direct, which are based on the method of separating variables, the method of sources (Green's function method), the method of integral transformations, approximate and numerical methods. In many cases, obtaining solutions to such problems in a closed form is very difficult. Difficulties can be avoided by reducing these problems to quasi-differential equations, the analytical solutions of which are relatively easier to obtain using matrix calculus. Substantiation of existence and construction of exact analytical solutions of quasi-differential equations, development of programs for approximate calculations of eigenvalues and eigenfunctions is an urgent task. The scheme of solution proposed in this paper belongs to the direct methods of solving boundary value problems. The paper considers general boundary value problems for longitudinal oscillations of a rod, which consists of two pieces of piece-stable cross-section and load in the right part. Five different cases of boundary conditions are considered. Solutions of such problems are found using the concept of quasi-derivatives, modern theory of systems of linear differential equations, the classical Fourier method and the reduction method. The concept of quasi-derivatives allows to bypass the problem of multiplication of generalized functions that occur in the right-hand side of the equation depending on the type of load. Using the reduction method, the solution of the problem is reduced to finding the solutions of two problems. One problem is a stationary inhomogeneous boundary value problem with initial boundary conditions. The second problem is a mixed problem with zero boundary conditions for a certain inhomogeneous equation. The integration interval is divided into segments. The problems are considered on each segment of the partition, and then the analytical expression of the solution is written with the help of matrix calculus. This approach allows you to apply software to the process of solving the problem, in particular to find eigenvalues and eigenfunctions.