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ARTICLE

Exact Solutions for the Free Vibration of Extensional Curved Non-uniform Timoshenko Beams

by Sen Yung Lee¹, Jyh Shyang Wu²

1 Corresponding author. Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China.Tel: +886-6-2757575 ext.62150; E-mail: sylee@mail.ncku.edu.tw
2 Department of Mechanical Engineering, National Cheng Kung University and Department of Mechanical Engineering, Far East University, Tainan, Taiwan, Republic of China

Computer Modeling in Engineering & Sciences 2009, 40(2), 133-154. https://doi.org/10.3970/cmes.2009.040.133

Abstract

The three coupled governing differential equations for the in-plane vibrations of curved non-uniform Timoshenko beams are derived via the Hamilton's principle. Three physical parameters are introduced to simplify the analysis. By eliminating all the terms with the axial displacement parameter, then reducing the order of differential operator acting on the flexural displacement parameter, one uncouples the three governing characteristic differential equations with variable coefficients and reduces them into a sixth-order ordinary differential equation with variable coefficients in term of the angle of the rotation due to bending for the first time. The explicit relations between the axial and the flexural displacements and the angle of the rotation due to bending are also revealed. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the in-plane vibrations of the beam can be obtained. Several limiting studies are illustrated. Finally, limiting cases are studied and the influence of the taper ratio and the arc length on the first two natural frequencies of the beams is explored.

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