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An accurate beam theory and its first-order approximation in free vibration analysis

Xie, L., Wang, S., Ding, J., Banerjee, J. R. and Wang, J. (2020). An accurate beam theory and its first-order approximation in free vibration analysis. Journal of Sound and Vibration, 485, 115567. doi: 10.1016/j.jsv.2020.115567

Abstract

An infinite system of one-dimensional differential equations is derived from the two-dimensional theory of elasticity by expanding the displacement field in a series of trigonometrical functions together with a linear term. Since the trigonometrical functions are pure thickness-vibration modes of infinite plates or beams with the top and bottom surfaces being free, the differential equations and the corresponding boundary conditions serve as the basis of an accurate beam theory for vibration analysis, named Lee's beam theory (LBT). Naturally, a high-order set of the infinite system should be quite useful in the analysis of beams at high frequencies. With the objective of vibration analysis of beams, this paper focuses on the first-order approximation, which leads to a first-order shear deformation beam theory for flexural vibrations (LBT1st). The differential equations in LBT1st are equivalent to those in Timoshenko's beam theory (TBT). The most important difference between LBT1st and TBT is the different field displacements. For the assessment of the accuracy of LBT1st, the numerical results of frequencies of free vibrations, frequency spectra and mode shapes of beams with classical boundary conditions are obtained and compared with those by TBT and plane stress problem of elasticity. Considering the plane stress problem as a reference, LBT1st is slightly more accurate in describing the field shapes of beams than TBT. Therefore, LBT1st, as well as LBT, is an addition to the existing beam theories with improved accuracy for the vibration analysis of beams and their combinations.

Publication Type: Article
Additional Information: This article has been published in Journal of Sound and Vibration, Elsevier. DOI: https://doi.org/10.1016/j.jsv.2020.115567 . © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords: Vibration, Beam, Higher-order, Shear deformation, Frequency
Subjects: T Technology > TJ Mechanical engineering and machinery
Departments: School of Mathematics, Computer Science & Engineering > Engineering > Mechanical Engineering & Aeronautics
Date available in CRO: 16 Jul 2021 08:59
Date deposited: 16 July 2021
Date of acceptance: 3 July 2020
Date of first online publication: 11 July 2020
URI: https://openaccess.city.ac.uk/id/eprint/26446
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