Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Mikhasev G. I., Le N. D. On the influence of surface stresses and inertia on the natural low-frequency vibrations of an elastic ultrathin strip-beam. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024, vol. 24, iss. 1, pp. 86-96. DOI: 10.18500/1816-9791-2024-24-1-86-96, EDN: RKHCCU

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2024
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Language: 
Russian
Heading: 
Mechanics
Article type: 
Article
UDC: 
534/539
DOI: 
10.18500/1816-9791-2024-24-1-86-96
EDN: 
RKHCCU

On the influence of surface stresses and inertia on the natural low-frequency vibrations of an elastic ultrathin strip-beam

Autors: 
Mikhasev Gennadi Ivanovich, Harbin Institute of Technology
Le Nguyen D., Belarusian State University
Abstract: 

A differential equation is derived that describes free long-wave vibrations of a low-dimensional elastic isotropic strip-beam, taking into account  effects on free surfaces.  Boundary conditions on external surfaces are formulated within the framework of the Gurtin – Murdoch surface theory of elasticity, which takes into account surface inertia and shear stresses, including residual ones. Additional geometric dimensions are introduced, associated with the face surfaces, which are assumed to be small compared to the main geometric dimension — the wavelength. The ratio of the thickness of the ultrathin strip to the wavelength of bending vibrations is considered as the main small parameter. Using the method of asymptotic integration of two-dimensional equations of the theory of elasticity over the thickness of the strip-beam, relations for displacements and stresses in the volume of the strip were obtained in explicit form. The main result of the paper is a differential equation for low-frequency vibrations of a beam, which takes into account surface effects and generalizes the well-known equations of beam theory. It is shown that the presence of surface stresses leads to an increase in natural frequencies from the lower spectrum, while taking into account surface inertia, as well as transverse shears in volume, leads to a decrease in frequencies.

Key words: 
ultrathin strip-beam
surface elasticity
long-wave asymptotics
natural frequencies
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Received: 
06.12.2023
Accepted: 
28.12.2023
Published: 
01.03.2024
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