Chaotic and synchronized dynamics of non-linear Euler-Bernoulli beams

J. Awrejcewicz, A. V. Krysko, V. Dobriyan, I. V. Papkova, V. A. Krysko

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Abstract In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler-Bernoulli beams is carried out. Algorithms reducing the studied objects associated with the boundary value problems are reduced to the Cauchy problem through both the Finite Difference Method (FDM) with approximation of O(c2) and the Finite Element Method (FEM). The constructed Cauchy problem is solved via the fourth- and sixth-order Runge-Kutta methods. The validity and reliability of the obtained results is rigorously discussed. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincarè and pseudo-Poincarè maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. In particular, we study a transition from symmetric to asymmetric vibrations and we explain this phenomenon. Vibration-type charts are reported regarding two control parameters: amplitude q0 and frequency ωp of the uniformly distributed periodic excitation. Furthermore, we have detected and illustrated chaotic vibrations of the Euler-Bernoulli beams for different boundary conditions and different beams thickness. In addition, we study chaotic dynamics and synchronization of multi-layer beams coupled only via boundary conditions. Computational examples of the theoretical investigations are given, where geometric, physical and design non-linearities are taken into account.

Original languageEnglish
Article number5359
Pages (from-to)85-96
Number of pages12
JournalComputers and Structures
Volume155
DOIs
Publication statusPublished - 3 Jun 2015
Externally publishedYes

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Euler-Bernoulli Beam
Vibration
Boundary conditions
Chaotic Dynamics
Runge Kutta methods
Cauchy Problem
Autocorrelation
Finite difference method
Boundary value problems
Flexible Beam
Chaotic Synchronization
Synchronization
Autocorrelation Function
Runge-Kutta Methods
Mathematical Analysis
Chart
Finite element method
Mathematical Modeling
Lyapunov Exponent
Control Parameter

Keywords

  • Attractors
  • Bifurcations
  • Chaotic vibrations
  • Phase portraits
  • Spatio-temporal chaos
  • The Euler-Bernoulli beams

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Modelling and Simulation
  • Materials Science(all)
  • Mechanical Engineering
  • Computer Science Applications

Cite this

Chaotic and synchronized dynamics of non-linear Euler-Bernoulli beams. / Awrejcewicz, J.; Krysko, A. V.; Dobriyan, V.; Papkova, I. V.; Krysko, V. A.

In: Computers and Structures, Vol. 155, 5359, 03.06.2015, p. 85-96.

Research output: Contribution to journalArticle

Awrejcewicz, J. ; Krysko, A. V. ; Dobriyan, V. ; Papkova, I. V. ; Krysko, V. A. / Chaotic and synchronized dynamics of non-linear Euler-Bernoulli beams. In: Computers and Structures. 2015 ; Vol. 155. pp. 85-96.
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AB - Abstract In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler-Bernoulli beams is carried out. Algorithms reducing the studied objects associated with the boundary value problems are reduced to the Cauchy problem through both the Finite Difference Method (FDM) with approximation of O(c2) and the Finite Element Method (FEM). The constructed Cauchy problem is solved via the fourth- and sixth-order Runge-Kutta methods. The validity and reliability of the obtained results is rigorously discussed. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincarè and pseudo-Poincarè maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. In particular, we study a transition from symmetric to asymmetric vibrations and we explain this phenomenon. Vibration-type charts are reported regarding two control parameters: amplitude q0 and frequency ωp of the uniformly distributed periodic excitation. Furthermore, we have detected and illustrated chaotic vibrations of the Euler-Bernoulli beams for different boundary conditions and different beams thickness. In addition, we study chaotic dynamics and synchronization of multi-layer beams coupled only via boundary conditions. Computational examples of the theoretical investigations are given, where geometric, physical and design non-linearities are taken into account.

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