doi: 10.3850/S2010428611000134

Analysis of Non-Linear Vibrations of Single-Layered Euler-Bernoulli Beams using Wavelets

Jan Awrejcewicz¹, Olga A. Saltykova^2,a, Maxim V. Zhigalov², Peter Hagedorn³ and
Vadim A. Krysko^{2,b 1}Department of Automation and Biomechanics, Technical University of Lodz, 1/15 Stefanowski Str,
90-924 Lodz, Poland. jan.awrejcewicz@p.lodz.pl ²Department of Mathematics and Modeling Saratov State Technical University, 410054 Saratov, Politehnicheskaya 77, Russian Federation. ^aaolga_a_saltykova@mail.ru ^btak@san.ru ³SzM, Dynamics & Vibrations Group, LOEWE-Zentrum AdRIA, TU Darmstadt Bartning Str. 53,
64289 Darmstadt, Germany. peter.hagedorn@dyn.tu-darmstadt.de

ABSTRACT

In this work the investigation of non-linear vibrations of an Euler-Bernoulli beam for two types of boundary conditions has been carried out using the Gauss wavelets 1-8, and the Morlet wavelets. It has been shown that the latter ones give the most complete information about vibrations of the Euler-Bernoulli beams. In particular, the scenarios of transitions from a regular to chaotic beam dynamics are studied. In contrary to the standard approach based on the FFT (Fast Fourier Transform), the wavelets analysis allows following time evolutions of the beam frequency spectrum, and hence one may trace either an appearance or disappearance of frequency components. In addition, as it has been illustrated, the wavelet-oriented analysis allows us to investigate the redistribution of the beam energy over the frequency vibrations spectrum. Namely, we have shown how the beam energy located in a vicinity of the excitation frequency is transmitted into the other frequencies, when finally the system transits into the multi-frequency regimes.

Keywords: Bernoulli beams, Wavelets, Gauss function, Chaos.

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