On the efficacy of the wavelet decomposition for high frequency vibration analyses

This paper reports the extraordinary ability of the wavelet decomposition for vibration analyses under the framework of Rayleigh–Ritz method. Using a beam as an example, Daubechies wavelet scale functions are used as admissible functions for decomposing the flexural displacement of the structure, along with the artificial springs at the boundary, to predict vibration of an Euler–Bernoulli beam in an extremely large frequency range. It is shown that the use of wavelet basis allows reaching very high frequencies, typically covering more than 1000 modes using conventional computational facility within the available numerical dynamics of the computers with no particular care needed for round-off errors. As a side benefit, the use of spring boundary also allows handling any elastic boundary conditions through a dynamic contribution in the Hamiltonian of the beam. The wavelet decomposed approach combines the flexibility of the global methods and the accuracy of local methods by inheriting the versatility of the Rayleigh–Ritz approach and the superior fitting ability of the wavelets. Numerical results on both free and forced vibrations are given, in excellent agreement with predictions of classical methods.