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Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

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journal contribution
posted on 15.12.2021, 22:57 authored by J Ranjan Banerjee, Stanislav O Papkov, Thuc VoThuc Vo, Isaac Elishakoff
Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

History

Publication Date

12/10/2021

Journal

Journal of Vibration and Control

Pagination

(p. 107754632110482-107754632110482)

Publisher

SAGE Publications

ISSN

1077-5463

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The Author reserves all moral rights over the deposited text and must be credited if any re-use occurs. Documents deposited in OPAL are the Open Access versions of outputs published elsewhere. Changes resulting from the publishing process may therefore not be reflected in this document. The final published version may be obtained via the publisher’s DOI. Please note that additional copyright and access restrictions may apply to the published version.

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