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Elastic waves in Timoshenko beams: the 'lost and found' of an eigenmode

Elastic waves in Timoshenko beams: the 'lost and found' of an eigenmode
Elastic waves in Timoshenko beams: the 'lost and found' of an eigenmode
This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler-Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful 'branch' of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution ?(1)*??EB*[1+O(?2k*2)] and (ii) a singular solution ?(2)*?(1/2?2)+O(k*)2; ?, ?* and k* are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ?. The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.
flexural elastic waves, singular perturbation, agragian mechanics, hamiltonian, timoshenko beam
1364-5021
239-255
Bhaskar, Atul
d4122e7c-5bf3-415f-9846-5b0fed645f3e
Bhaskar, Atul
d4122e7c-5bf3-415f-9846-5b0fed645f3e

Bhaskar, Atul (2009) Elastic waves in Timoshenko beams: the 'lost and found' of an eigenmode. Proceedings of the Royal Society A, 465 (2101), 239-255. (doi:10.1098/rspa.2008.0276).

Record type: Article

Abstract

This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler-Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful 'branch' of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution ?(1)*??EB*[1+O(?2k*2)] and (ii) a singular solution ?(2)*?(1/2?2)+O(k*)2; ?, ?* and k* are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ?. The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.

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Submitted date: 6 July 2008
Published date: January 2009
Keywords: flexural elastic waves, singular perturbation, agragian mechanics, hamiltonian, timoshenko beam
Organisations: Computational Engineering and Design

Identifiers

Local EPrints ID: 64468
URI: https://eprints.soton.ac.uk/id/eprint/64468
ISSN: 1364-5021
PURE UUID: a9a9a438-4caf-4bbb-b504-e8ea0a5ed9f3

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Date deposited: 24 Dec 2008
Last modified: 13 Mar 2019 20:21

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