On Free Bending Vibrations Of Concrete Beams With Variable Cross Section

Abstract

The paper consists of an introduction and three sections. The introduction discusses the relevance of issues related to the study of inhomogeneous beams vibrations. The analysis of publications and the results obtained in this area is performed. The first section is devoted to the formulation of the boundary-value problem of finding natural-vibration frequencies for an inhomogeneous beam under the Euler-Bernoulli hypotheses. By introducing new variables, the problem originally formulated in displacements reduces to an identical one, but formulated in terms of bending moment. The next section describes the method of integro-differential relations, which is an alternative to the classical variational approaches. Further, the possibilities of constructing various bilateral energy estimates of the quality of approximate solutions arising from the method of integro-differential relations are investigated. In the third section, using the example of free vibrations of a supported concrete beam, numerical aspects of constructing an approximate solution for boundary problems described by an ordinary differential equation with variable coefficients are studied. The proposed bilateral quality criteria of the approximate solution make it possible to obtain high-precision solutions for mathematical models of small dimension.