Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams

Gupta, R. K. ; Babu, Gunda Jagadish ; Ranga Janardhan, G. ; Venkateswara Rao, G. (2009) Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams Finite Elements in Analysis and Design, 45 (10). pp. 624-631. ISSN 0168-874X

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.finel.2009.04.001

Abstract

Large amplitude free vibration analysis of uniform, slender and isotropic beams is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). All possible boundary conditions where the von-Karman type nonlinearity is applicable, where the ends are axially immovable are considered. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method and is general for the type of the nonlinearity mentioned earlier. The nonlinear stiffness matrix derived in the present finite element formulation leads to symmetric stiffness matrix as compared to other recent formulations. Empirical formulas for the nonlinear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged-hinged, clamped-clamped and clamped-hinged beams. For the beams with nonclassical boundary conditions such as the hinged-guided and clamped-guided, the numerical results obtained, apparently for the first time and are in line with the physics of the problem.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Free Vibration; Large Amplitude; Beams; FEM; Iterative Solution
ID Code:53714
Deposited On:09 Aug 2011 11:22
Last Modified:09 Aug 2011 11:22

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