On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams

J. Awrejcewicz, A. V. Krysko, N. A. Zagniboroda, V. V. Dobriyan, V. A. Krysko

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We present chaotic dynamics of flexible curvilinear shallow Euler–Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second-order accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth- and sixth-order Runge–Kutta methods with respect to time. This preserves reliability of the obtained results. Nonlinear dynamics is investigated with the help of a qualitative theory of differential equations. Frequency power spectra using fast Fourier transform, phase and modal portraits, autocorrelation functions, spatiotemporal dynamics of the beam, 2D and 3D Morlet wavelets, and Poincaré sections are constructed. Four first Lyapunov exponents are estimated using the Wolf algorithm. Transitions from regular to chaotic dynamics are detected, illustrated and discussed. Depending on signs of four Lyapunov exponents the chaotic, hyper chaotic, hyper-hyper chaotic, and deep chaotic dynamics is reported. Curvilinear beams are treated as systems with an infinite number of degrees of freedom. Charts of vibration character, elastic–plastic deformations, and stability loss zone versus control parameters of the studied beams are reported.

Original languageEnglish
Pages (from-to)11-29
Number of pages19
JournalNonlinear Dynamics
Volume79
Issue number1
DOIs
Publication statusPublished - 1 Jan 2015
Externally publishedYes

Fingerprint

Euler-Bernoulli Beam
Chaotic Dynamics
Lyapunov Exponent
Cauchy Problem
Second-order Accuracy
Frequency Spectrum
Fast Fourier transform
Autocorrelation Function
Runge-Kutta Methods
Power Spectrum
Chart
Control Parameter
Nonlinear Dynamics
Difference Method
Finite Difference
Wavelets
Vibration
Degree of freedom
Finite Element Method
Differential equation

Keywords

  • Chaos
  • Euler–Bernoulli beams
  • Fourier transform
  • Lyapunov exponents
  • Wavelets

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams. / Awrejcewicz, J.; Krysko, A. V.; Zagniboroda, N. A.; Dobriyan, V. V.; Krysko, V. A.

In: Nonlinear Dynamics, Vol. 79, No. 1, 01.01.2015, p. 11-29.

Research output: Contribution to journalArticle

Awrejcewicz, J. ; Krysko, A. V. ; Zagniboroda, N. A. ; Dobriyan, V. V. ; Krysko, V. A. / On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams. In: Nonlinear Dynamics. 2015 ; Vol. 79, No. 1. pp. 11-29.
@article{9450c3fa673c40749a0509020209fd64,
title = "On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams",
abstract = "We present chaotic dynamics of flexible curvilinear shallow Euler–Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second-order accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth- and sixth-order Runge–Kutta methods with respect to time. This preserves reliability of the obtained results. Nonlinear dynamics is investigated with the help of a qualitative theory of differential equations. Frequency power spectra using fast Fourier transform, phase and modal portraits, autocorrelation functions, spatiotemporal dynamics of the beam, 2D and 3D Morlet wavelets, and Poincar{\'e} sections are constructed. Four first Lyapunov exponents are estimated using the Wolf algorithm. Transitions from regular to chaotic dynamics are detected, illustrated and discussed. Depending on signs of four Lyapunov exponents the chaotic, hyper chaotic, hyper-hyper chaotic, and deep chaotic dynamics is reported. Curvilinear beams are treated as systems with an infinite number of degrees of freedom. Charts of vibration character, elastic–plastic deformations, and stability loss zone versus control parameters of the studied beams are reported.",
keywords = "Chaos, Euler–Bernoulli beams, Fourier transform, Lyapunov exponents, Wavelets",
author = "J. Awrejcewicz and Krysko, {A. V.} and Zagniboroda, {N. A.} and Dobriyan, {V. V.} and Krysko, {V. A.}",
year = "2015",
month = "1",
day = "1",
doi = "10.1007/s11071-014-1641-5",
language = "English",
volume = "79",
pages = "11--29",
journal = "Nonlinear Dynamics",
issn = "0924-090X",
publisher = "Springer Netherlands",
number = "1",

}

TY - JOUR

T1 - On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams

AU - Awrejcewicz, J.

AU - Krysko, A. V.

AU - Zagniboroda, N. A.

AU - Dobriyan, V. V.

AU - Krysko, V. A.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We present chaotic dynamics of flexible curvilinear shallow Euler–Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second-order accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth- and sixth-order Runge–Kutta methods with respect to time. This preserves reliability of the obtained results. Nonlinear dynamics is investigated with the help of a qualitative theory of differential equations. Frequency power spectra using fast Fourier transform, phase and modal portraits, autocorrelation functions, spatiotemporal dynamics of the beam, 2D and 3D Morlet wavelets, and Poincaré sections are constructed. Four first Lyapunov exponents are estimated using the Wolf algorithm. Transitions from regular to chaotic dynamics are detected, illustrated and discussed. Depending on signs of four Lyapunov exponents the chaotic, hyper chaotic, hyper-hyper chaotic, and deep chaotic dynamics is reported. Curvilinear beams are treated as systems with an infinite number of degrees of freedom. Charts of vibration character, elastic–plastic deformations, and stability loss zone versus control parameters of the studied beams are reported.

AB - We present chaotic dynamics of flexible curvilinear shallow Euler–Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second-order accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth- and sixth-order Runge–Kutta methods with respect to time. This preserves reliability of the obtained results. Nonlinear dynamics is investigated with the help of a qualitative theory of differential equations. Frequency power spectra using fast Fourier transform, phase and modal portraits, autocorrelation functions, spatiotemporal dynamics of the beam, 2D and 3D Morlet wavelets, and Poincaré sections are constructed. Four first Lyapunov exponents are estimated using the Wolf algorithm. Transitions from regular to chaotic dynamics are detected, illustrated and discussed. Depending on signs of four Lyapunov exponents the chaotic, hyper chaotic, hyper-hyper chaotic, and deep chaotic dynamics is reported. Curvilinear beams are treated as systems with an infinite number of degrees of freedom. Charts of vibration character, elastic–plastic deformations, and stability loss zone versus control parameters of the studied beams are reported.

KW - Chaos

KW - Euler–Bernoulli beams

KW - Fourier transform

KW - Lyapunov exponents

KW - Wavelets

UR - http://www.scopus.com/inward/record.url?scp=85027932073&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027932073&partnerID=8YFLogxK

U2 - 10.1007/s11071-014-1641-5

DO - 10.1007/s11071-014-1641-5

M3 - Article

VL - 79

SP - 11

EP - 29

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 1

ER -