Plenary Lecture

Regular vs. Chaotic Dynamics of Beams and Plates

Professor Jan Awrejcewicz
Department of Automation and Biomechanics
Lodz University of Technology (TUL)
E-mail: awrejcew@p.lodz.pl

Abstract: Bifurcation and chaos exhibited by the Euler-Bernoulli beams and Mindlin-Kirchhoff plates subjected either to transversal local external or shear load in the frame of the classical non-linear theories are monitored and studied. A transition from PDEs to ODEs is carried out using the Finite Difference Method (FDM) and the Finite Element Method (FEM). Reliability and validity of the obtained results are verified and discussed. Stability, bifurcation and chaos of the mentioned objects and in particular, an influence of harmonic external load parameters, system parameters and boundary conditions are studied. This research part presents both novel approach to analyze bifurcation and chaos exhibited by vibrated beams and plates as well novel results associated with stability, bifurcation and spatial-timing chaos of the analyzed structural members. It is shown that an application of the classical and widely used Fourier analysis does not allow to obtain real picture of the frequency vibration characteristics in each time instant. On the other hand, it is illustrated that application of the wavelets approach allows to follow frequency time evolutions. Presented numerical results indicate that vibrations in different plate points occur with the same frequencies set although their power is different. Hence, the vibration characteristics can be represented by one arbitrary taken plate point. Furthermore, using wavelets scenarios of transitions from regular to chaotic dynamics are illustrated and discussed including two novel scenarios not reported so far in the existing literature. In addition, analysis 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, scenarios of transitions from a regular to chaotic beam dynamics are revisited. 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, the wavelet-oriented analysis yields the redistribution of the beam energy over the frequency vibrations spectrum. Namely, it is 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 chaotic regimes.

Brief Biography of the Speaker: J. Awrejcewicz has been graduated from the Technical University of Lodz in 1977 (Mechanics) and from the University of Lodz in 1978 (Philosophy). He obtained PhD (Habilitation) in 1981 (1990), and he became a Full Professor in 1997. Now he is a chair person of Department of Automation and Biomechanics, a head of a 4-year PhD Study on Mechanics, and a head of the Mechatronics Study at the Technical University of Lodz. His research is focused on Nonlinear Mechanics. J. Awrejcewicz authored and/or co-authored: monographs-42; textbooks-2; edited books-3; editor conference proceedings-12; journal papers-260; conference papers-300; chapters in books-31. He served as an editor of 5 books, and as an Guest-Editor of 13 journal special issues. He supervised 18 PhD theses. He served in Editorial Boards of 33 journals, gave 60 seminars at international universities, delivered 26 plenary/keynote lectures, actively participated at 230 international and 65 national conferences, as well as he was a member of scientific committees of 100 conferences. He spent 10 years abroad carrying out research at University of California, Berkeley, USA (2001); University of Illinois, Urbana Champaign, USA (1999/2000); Tokyo University, Japan (1990-1992); University of Carolo Wilhelmina in Braunschweig, Germany (1987-1990, 1993); ENTPE, Lyon, France (1995, 2005); Central European University, Budapest, Hungary (2003/2004); Waikato University, Hamilton, New Zealand (1996/1997). J. Awrejcewicz obtained the Humboldt Research Award, Germany, 2011; MASTER Grant Award, Foundation for Polish Science, 2010-2012; Golden Lamp Award (PGNiG) in Technical Sciences, Poland, 2006; awards of the Ministry of Science and Education for monographs in 1996, 2004, 2006, 2008.