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Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions

Chellappan, Vinita and Gopalakrishnan, S and Mani, V (2014) Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions. In: CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES, 97 (2). pp. 131-174.

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Official URL: http://dx.doi.org/ 10.3970/cmes.2014.097.131

Abstract

The paper discusses the frequency domain based solution for a certain class of wave equations such as: a second order partial differential equation in one variable with constant and varying coefficients (Cantilever beam) and a coupled second order partial differential equation in two variables with constant and varying coefficients (Timoshenko beam). The exact solution of the Cantilever beam with uniform and varying cross-section and the Timoshenko beam with uniform cross-section is available. However, the exact solution for Timoshenko beam with varying cross-section is not available. Laplace spectral methods are used to solve these problems exactly in frequency domain. The numerical solution in frequency domain is done by discretisation in space by approximating the unknown function using spectral functions like Chebyshev polynomials, Legendre polynomials and also Normal polynomials. Different numerical methods such as Galerkin Method, Petrov- Galerkin method, Method of moments and Collocation method or the Pseudo-spectral method in frequency domain are studied and compared with the available exact solution. An approximate solution is also obtained for the Timoshenko beam with varying cross-section using Laplace Spectral Element Method (LSEM). The group speeds are computed exactly for the Cantilever beam and Timoshenko beam with uniform cross-section and is compared with the group speeds obtained numerically. The shear mode and the bending modes of the Timoshenko beam with uniform cross-section are separated numerically by applying a modulated pulse as the shear force and the corresponding group speeds for varying taper parameter in are obtained numerically by varying the frequency of the input pulse. An approximate expression for calculating group speeds corresponding to the shear mode and the bending mode, and also the cut-off frequency is obtained. Finally, we show that the cut-off frequency disappears for large in, for epsilon > 0 and increases for large in, for epsilon < 0.

Item Type: Journal Article
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Additional Information: Copy right for this article belongs to the TECH SCIENCE PRESS, 6825 JIMMY CARTER BLVD, STE 1850, NORCROSS, GA 30071 USA
Keywords: Partial differential equation; Spectral finite element method; Frequency domain method; Timoshenko beam; group speed; cut-off frequency
Department/Centre: Division of Mechanical Sciences > Aerospace Engineering (Formerly, Aeronautical Engineering)
Date Deposited: 19 Aug 2014 09:19
Last Modified: 19 Aug 2014 09:19
URI: http://eprints.iisc.ernet.in/id/eprint/49570

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