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Current Development in Theory and Applications of Wavelets

Current Development in Theory and Applications of Wavelets
Volume 4, Issue 2, Pages 109 - 129 (August 2010)

APPROXIMATED SOLUTION OF A LINEAR DIFFERENTIAL EQUATION BASED ON THE CONTINUOUS WAVELET TRANSFORM
F. Castellanos, M. Ordaz and A. Contreras-Cristán

Abstract:
The Fourier transform solution of a linear differential equation implies the definition of a transfer function in the frequency domain, which depends only on the dynamical properties of the system. A similar solution is approximated through the continuous wavelet transform (CWT). The linear differential equation to be solved corresponds to the equation of motion for a linear damped single-degree-of-freedom (SDOF) oscillator subjected to motion at its base. The wavelet basis must accomplish some restrictions in order for the transfer function to depend only on the dynamical properties of the system. To fulfill these restrictions, a new wavelet must be defined, which consists of a sinusoid in the central part of its compact support, with linear exponential decay toward its extremes. The definition of the time-frequency transfer function demands the fulfillment of conditions that yield an inconsistent system of equations, which is solved by a least square approach. Using this transfer function, it is possible to estimate the response of a linear damped SDOF oscillator from the CWT of the corresponding excitation. When resonance takes place, an underestimated response is obtained; this diminishes as the wavelet basis tends to a Fourier basis.

Keywords and phrases: time-frequency transfer function, linear differential equation, continuous wavelet transform, single-degree-of-freedom oscillator, resonance, convolution.

Communicated by Ruqiang Yan

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