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Research Papers

Dynamic Stiffness Analysis of a Beam Based on Trigonometric Shear Deformation Theory

[+] Author and Article Information
Jun Li

Vibration, Shock & Noise Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People’s Republic of ChinaLJY60023@yahoo.com

Hongxing Hua, Rongying Shen

Vibration, Shock & Noise Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People’s Republic of China

J. Vib. Acoust 130(1), 011004 (Nov 12, 2007) (7 pages) doi:10.1115/1.2775513 History: Received March 13, 2006; Revised July 20, 2007; Published November 12, 2007

The dynamic stiffness matrix of a uniform isotropic beam element based on trigonometric shear deformation theory is developed in this paper. The theoretical expressions for the dynamic stiffness matrix elements are found directly, in an exact sense, by solving the governing differential equations of motion that describe the deformations of the beam element according to the trigonometric shear deformation theory, which include the sinusoidal variation of the axial displacement over the cross section of the beam. The application of the dynamic stiffness matrix to calculate the natural frequencies and normal mode shapes of two rectangular beams is discussed. The numerical results obtained are compared to the available solutions wherever possible and validate the accuracy and efficiency of the present approach.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Geometry and coordinate of a beam

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Figure 2

Sign convention for positive shear force Q(x), bending moment M(x) and generalized moment M̃(x)

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Figure 3

Boundary condition for displacements and forces of beam element

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Figure 4

First five normal mode shapes of clamped-free beam: (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4, and (e) mode 5

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