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2020 | OriginalPaper | Chapter

On Different Representations of Equations for Beam with Variable Cross-Section

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Abstract

The main part of the paper is devoted to the formulations of the boundary value problem to finding natural frequencies of an inhomogeneous beam in the framework of the Euler-Bernoulli hypotheses. Three different formulations of the beam equation in displacements, momentums, and bending moments are discussed. Next, the possibilities to constructing various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are investigated. In the final part, based on a numerical model example, the advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed.

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Literature
1.
go back to reference Saurin, V.: Analysis of dynamic behavior of beams with variable cross-section. Lobachevskii J. Math. 40, 364–374 (2019)MathSciNetCrossRef Saurin, V.: Analysis of dynamic behavior of beams with variable cross-section. Lobachevskii J. Math. 40, 364–374 (2019)MathSciNetCrossRef
2.
go back to reference Kostin, G., Saurin, V.: Dynamics of Solid Structures. Method Using Integrodifferential Relations. De Gruyter, Berlin (2017)CrossRef Kostin, G., Saurin, V.: Dynamics of Solid Structures. Method Using Integrodifferential Relations. De Gruyter, Berlin (2017)CrossRef
Metadata
Title
On Different Representations of Equations for Beam with Variable Cross-Section
Authors
Vladimir Poliakov
Vasily Saurin
Copyright Year
2020
DOI
https://doi.org/10.1007/978-3-030-47883-4_29

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