Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach

Version 1 : Received: 22 January 2021 / Approved: 25 January 2021 / Online: 25 January 2021 (14:09:44 CET)
Version 2 : Received: 15 September 2021 / Approved: 15 September 2021 / Online: 15 September 2021 (15:33:01 CEST)

How to cite: Fogang, V. An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach . Preprints 2021, 2021010501 (doi: 10.20944/preprints202101.0501.v2). Fogang, V. An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach . Preprints 2021, 2021010501 (doi: 10.20944/preprints202101.0501.v2).

Abstract

This study presents an exact solution to the free vibration analysis of a uniform Timoshenko beam using an analytical approach, a harmonic vibration being assumed. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. In this paper, a moment-shear force-circular frequency-curvature relationship was presented. The complete study was based on this relationship and closed-form expressions of efforts and deformations were derived. The free vibration response of single-span systems, as well as that of spring-mass systems, was analyzed; closed-form formulations of matrices expressing the boundary conditions were presented and the natural frequencies were determined by solving the eigenvalue problem. Systems with intermediate mass, spring, or spring-mass system were also analyzed. Furthermore, first-order dynamic stiffness matrices in local coordinates were derived. Finally, second-order analysis of beams resting on an elastic Winkler foundation was conducted. The results obtained in this paper were in good agreement with those of other studies.

Keywords

Timoshenko beam; rotary inertia; bending shear curvature natural frequency relationship; spring mass system vibration; closed-form solutions; first-order dynamic stiffness matrix; second-order vibration analysis

Comments (1)

Comment 1
Received: 15 September 2021
Commenter: Valentin Fogang
Commenter's Conflict of Interests: Author
Comment: The abstract
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