Abstract
A formulation of a boundary value problem to find natural frequencies of an inhomogeneous beam in the framework of the Euler–Bernoulli hypotheses are represented. Questions related to various classical variational formulations for a spectral problem arising in the beam theory are discussed. Particularities of the application of the Hamiltonian principles to boundary-value problems are considered. The method of integro-differential relations, which is an alternative to the classical variational approaches is discussed. 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 advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed based on a numerical example.
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