Research Papers

On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method

[+] Author and Article Information
Simon Jones

Assistant Professor
Department of Mechanical Engineering,
Rose-Hulman Institute of Technology,
Terre Haute, IN 47803
e-mail: jones5@rose-hulman.edu

Mathias Legrand

Assistant Professor
Department of Mechanical Engineering,
McGill University,
Montreal, QC H3C 1PA, Canada
e-mail: mathias.legrand@mcgill.ca

1Corresponding author.

Manuscript received August 29, 2012; final manuscript received January 19, 2013; accepted manuscript posted February 12, 2013; published online August 21, 2013. Editor: Yonggang Huang.

J. Appl. Mech 80(6), 061012 (Aug 21, 2013) (7 pages) Paper No: JAM-12-1428; doi: 10.1115/1.4023637 History: Received August 29, 2012; Revised January 19, 2013; Accepted February 12, 2013

The discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler–Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly.

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Fig. 1

Thickness variation h(x) of the beam

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Fig. 2

Convergence of the first four natural frequencies using the finite element method (FEM) and wavelet-Galerkin method (W-G)

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Fig. 3

First four mode shapes predicted by the finite element method (FEM) and wavelet-Galerkin method (W-G) using 128 degrees of freedom (J = 7)

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Fig. 4

Frequency response function at midspan (x = 1 m) under a 10Nm harmonic moment acting at x = 2 m




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