### Abstract

In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler-Bernoulli beams is carried out. The Karman-type geometric non-linearity is taken into account. The algorithms reducing the studied objects associated with the boundary value problems are to the Cauchy problem using the finite difference method (FDM) with an approximation of and the finite element method (FEM). The constructed Cauchy problem is solved using the fourth and six 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: The amplitude and the frequency 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 beam thicknesses. 2O(c).

Original language | English |
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Title of host publication | Proceedings of the 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013 |

Publisher | Civil-Comp Press |

Volume | 102 |

ISBN (Print) | 9781905088577 |

Publication status | Published - 1 Jan 2013 |

Externally published | Yes |

Event | 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013 - Cagliari, Sardinia, Italy Duration: 3 Sep 2013 → 6 Sep 2013 |

### Conference

Conference | 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013 |
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Country | Italy |

City | Cagliari, Sardinia |

Period | 3.9.13 → 6.9.13 |

### Fingerprint

### Keywords

- Attractors
- Bifurcations
- Chaotic vibrations
- Phase portraits
- Temporal-space chaos
- The euler-bernoulli beams

### ASJC Scopus subject areas

- Environmental Engineering
- Civil and Structural Engineering
- Computational Theory and Mathematics
- Artificial Intelligence

### Cite this

*Proceedings of the 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013*(Vol. 102). Civil-Comp Press.

**On the lyapunov exponents computation of coupled non-linear euler-bernoulli beams.** / Awrejcewicz, J.; Krysko, A. V.; Dobriyan, V.; Papkova, I. V.; Krysko, V. A.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013.*vol. 102, Civil-Comp Press, 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013, Cagliari, Sardinia, Italy, 3.9.13.

}

TY - GEN

T1 - On the lyapunov exponents computation of coupled non-linear euler-bernoulli beams

AU - Awrejcewicz, J.

AU - Krysko, A. V.

AU - Dobriyan, V.

AU - Papkova, I. V.

AU - Krysko, V. A.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler-Bernoulli beams is carried out. The Karman-type geometric non-linearity is taken into account. The algorithms reducing the studied objects associated with the boundary value problems are to the Cauchy problem using the finite difference method (FDM) with an approximation of and the finite element method (FEM). The constructed Cauchy problem is solved using the fourth and six 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: The amplitude and the frequency 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 beam thicknesses. 2O(c).

AB - In this work the mathematical modeling and analysis of the chaotic dynamics of flexible Euler-Bernoulli beams is carried out. The Karman-type geometric non-linearity is taken into account. The algorithms reducing the studied objects associated with the boundary value problems are to the Cauchy problem using the finite difference method (FDM) with an approximation of and the finite element method (FEM). The constructed Cauchy problem is solved using the fourth and six 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: The amplitude and the frequency 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 beam thicknesses. 2O(c).

KW - Attractors

KW - Bifurcations

KW - Chaotic vibrations

KW - Phase portraits

KW - Temporal-space chaos

KW - The euler-bernoulli beams

UR - http://www.scopus.com/inward/record.url?scp=84893924279&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893924279&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781905088577

VL - 102

BT - Proceedings of the 14th International Conference on Civil, Structural and Environmental Engineering Computing, CC 2013

PB - Civil-Comp Press

ER -