Research on the dynamics of beam structures

Authors

UDC

531.39

DOI:

https://doi.org/10.31429/vestnik-20-4-11-24

Abstract

The design of beam structures is an actual scientific task of the development of modern technology. It is often difficult to obtain exact solutions when studying natural and forced vibrations of beam structures within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom. Therefore, in the article, the model of a beam structure is endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of the finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is assumed that the elements of beam structures work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal vibrations (the frequency of longitudinal vibrations is two orders of magnitude higher than the frequency of bending vibrations). The resolving system of differential equations of vibrations of beam structures, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the 2nd kind. In the article using Green's functions, stiffness, mass, malleability matrices, etc. several problems of free oscillations of beam structures were solved: a cantilever beam with a distributed mass and a tower-type structure (a television tower). The numerical results obtained in the article, when compared with exact solutions implemented using direct and indirect of boundary element methods, as well as with other little-known numerical solutions, with an increase in the number of degrees of freedom (the number of concentrated masses modeling the distributed mass of the beam), showed rapid convergence to exact solutions.

Keywords:

oscillation of beams, oscillation frequencies, Green's function, stiffness matrix, mass matrix, malleability matrix

Funding information

The study did not have sponsorship.

Author info

  • Peter G. Velikanov

    канд. физ.-мат. наук, доцент кафедры теоретической механики Казанского (Приволжского) федерального университета, доцент кафедры реактивных двигателей и энергетических установок Казанского национального исследовательского технического университета им. А.Н. Туполева — КАИ

  • Yuri P. Artyukhin

    д-р физ.-мат. наук, профессор кафедры теоретической механики Казанского (Приволжского) федерального университета

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Issue

Vol. 20 No. 4 (2023)

Pages

11-24

Section

Mechanics

Dates

Submitted

January 24, 2023

Accepted

June 10, 2023

Published

December 31, 2023

How to Cite

[1]
Velikanov, P.G., Artyukhin, Y.P., Research on the dynamics of beam structures. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2023, т. 20, № 4, pp. 11–24. DOI: 10.31429/vestnik-20-4-11-24

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Copyright (c) 2024 Великанов П.Г., Артюхин Ю.П.

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