Abstract

A formulation of a boundary value problem to find natural frequencies of an inhomogeneous beam in the framework of the linear theory of elasticity is represented. The main attention in the article is paid to the analysis of transverse vibrations of beams with a variable cross-section. A system of two connected ordinary differential equations with variable coefficients is obtained in the framework of the linear theory of elasticity, by using semidiscrete approximations and a projection approach. Various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are proposed. In the final part of the paper the advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed based on a numerical example.

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