- STRUCTURAL MECHANICS
- Dynamic Behavior Of Beams Of Variable Cross-Section Based On Variational Approaches
- UDC 539.3
doi: 10.33622/0869-7019.2022.12.04-09
Boris V. GUSEV1, info-rae@mail.ru
Vasiliy V. SAURIN2, saurin@ipmnet.ru
1 Russian University of Transport, ul. Obraztsova, 9, str. 9, Moscow 127994, Russian Federation
2 Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, prospekt Vernadskogo, 101, korр. 1, Moscow 119526, Russian Federation
Abstract. The boundary value problem of finding the natural frequencies of an inhomogeneous beam in the framework of the Euler-Bernoulli hypotheses is presented. Issues related to various classical variational formulations of the spectral problem arising in the theory of beams 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. A numerical example shows the advantages of the variational method in problems of free vibrations of inhomogeneous beams. Optimal forms of concrete beams with the maximum first natural frequency are constructed. The proposed two-sided criteria for the quality of the approximate solution make it possible to obtain high-precision solutions for low-dimensional mathematical models.
Keywords: beam with variable cross-section, natural vibrations, method of integro-differential relations, structural inhomogeneity, linear theory of elasticity, boundary value problem - REFERENCES
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