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 language | English |
|---|---|
| Pages (from-to) | 11-29 |
| Number of pages | 19 |
| Journal | Nonlinear Dynamics |
| Volume | 79 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 1 Jan 2015 |
| Externally published | Yes |
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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 journal › Article
}
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
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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 -