ВЛАСНІ ЗГИННІ КОЛИВАННЯ БАЛКИ ЗІ СПЕЦІАЛЬНИМ ЗАКОНОМ ЗМІНИ ШИРИНИ

Abstract

Purpose. The aim of the work is to construct a closed analytical solution of the problem of natural vibrations of the beam, the width of which varies according to the law exp (αx2). Methodology. The approach is based on the provisions of symmetric analysis of differential equations with variable coefficients. This approach allows you to find a way to obtain an analytical solution of the corresponding differential equation and, ultimately, the boundary value problem. Findings. The main result is the construction of the algorithm and obtaining the solution of the differential equation of the IV order, which describes the transverse bending vibrations of the beam with a special law of change of width (the thickness of the beam is a constant value). Two examples of the analysis of oscillations of such beam in case of its bilateral rigid fastening and cantilever fastening are resulted. For these cases, the frequency equations are obtained, the natural frequencies and amplitude coefficients are found, which are necessary for the construction of natural forms of oscillations. Originality.The approach presented in this paper is based on the idea of symmetries of differential equations and is characterized by a sim- plified analysis of the solution of the problem of bending oscillations of the beam with a special law of width. The method proposed for solving the boundary value problem is convenient and simple, because the results are found without the use of numerical research methods. Practical value.Examples of the exact analysis of fluctuations which allow to assert about real possibility of expansion of an existing number of configurations of a beam, both on width, and on ways of fastening are resulted. Such beams can, for example, be used as prototype samples for resonant tests of materials for fatigue strength. Сonclusions. The given algorithm for constructing the solution of the problem on eigenvalues for a beam with the given special law of change of width is universal and can be extended to other constructions of beams. Since in this case the question arises only about the choice of the approximation function, which allows to use the method of symmetries and obtain an exact solution of the corresponding differential equation of the IV order, which in turn describes the transverse oscillations of the beams.