Chaotic dynamics of flexible Euler-Bernoulli beams

Jan Awrejcewicz , V.A. Krysko , I.E. Kutepov , N.A. Zagniboroda , V. Dobriyan , A.V. Krysko

Abstract

Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c2) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q0 and frequency ωp of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.
Author Jan Awrejcewicz (FACME / IV)
Jan Awrejcewicz ,,
- Institute of Vehicles
, V.A. Krysko
V.A. Krysko,,
-
, I.E. Kutepov
I.E. Kutepov,,
-
, N.A. Zagniboroda
N.A. Zagniboroda,,
-
, V. Dobriyan
V. Dobriyan,,
-
, A.V. Krysko
A.V. Krysko,,
-
Journal seriesChaos, ISSN 1054-1500
Issue year2013
Vol23
Pages043130
URL http://scitation.aip.org/content/aip/journal/chaos/23/4/10.1063/1.4838955
Languageen angielski
Score (nominal)45
ScoreMinisterial score = 45.0, 29-11-2017, ArticleFromJournal
Ministerial score (2013-2016) = 45.0, 29-11-2017, ArticleFromJournal
Publication indicators WoS Impact Factor: 2013 = 1.761 (2) - 2013=2.06 (5)
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