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Research Papers

Analysis of the Nonlinear Dynamics of the Timoshenko Flexible Beams Using Wavelets

[+] Author and Article Information
J. Awrejcewicz

 Technical University of Lodz, Department of Automation and Biomechanics, 1/15 Stefanowski St., 90-924 Lodz, Polandawrejcew@p.lodz.pl

A. V. Krysko

 Saratov State Technical University, Department of Mathematics and Modeling, 77 Polytechnical, Saratov, 410054, Russiatan@san.ru

V. Soldatov

 Saratov State Technical University, Department of Mathematics and Modeling, 77 Polytechnical, Saratov, 410054, Russia

V. A. Krysko

 Saratov State Technical University, Department of Mathematics and Modeling, 77 Polytechnical, Saratov, 410054, Russiatak@san.ru

J. Comput. Nonlinear Dynam 7(1), 011005 (Aug 09, 2011) (14 pages) doi:10.1115/1.4004376 History: Received November 28, 2008; Revised May 16, 2011; Published August 09, 2011; Online August 09, 2011

Regular and chaotic dynamics of the flexible Timoshenko-type beams is studied using both the standard Fourier (FFT) and the continuous wavelet transform methods. The governing equations of motion for geometrically nonlinear Timoshenko-type beams are reduced to a system of ODEs using both finite element method (FEM) and finite difference method (FDM) to ensure the reliability of numerical results. Scenarios of transition from regular to chaotic vibrations and beam dynamical stability loss are analyzed. Advantages and disadvantages of various wavelet functions are discussed. Application of continuous wavelet transform to the investigation of transitional and chaotic phenomena in nonlinear dynamics is illustrated and discussed.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

The analyzed one layer beam

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Figure 2

The amplitude of beam vibrations versus q0 obtained via FDM and FEM and the vibration “scales” (FDM—above; FEM—below)

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Figure 3

Gauss-1 [1,5], Gauss-2 [2,6], Gauss-8 [3,7], and Morlet [4,8] 3D and 2D wavelet spectra for ω=4.0,q0=10,000,t∈[0,1300]

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Figure 4

Wavelet history [1], time history [2], phase curve [3], Fourier spectrum [4], and the Lyapunov exponent computations corresponding to the methods W [5] and K [6] for ω=4.0, q0=10,000,t∈[0,1300]

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Figure 5

Wavelet history [1], phase curve [2] and Fourier spectrum [3] for ω=4.0, q0=10,600, t∈[0,1300]

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Figure 6

Wavelet history [1], time history [2], phase curve [3], and Fourier spectrum [4] for ω=4.0, q0=10,600, t∈[200,400]

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Figure 7

Gauss-1 [1], Gauss-2 [2], Gauss-8 [3], and Morlet [4] wavelet time histories for ω=4.0, q0=10,600, t∈[1100,1300]

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Figure 8

Wavelet time history [1], time history [2], phase curve [3], and Fourier spectrum [4] for ω=4.0, q0=10,600, t∈[1100,1300]

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Figure 9

Wavelet history [1], phase curve [2] and Fourier spectrum [3] for ω=4.0, q0=12,200, t∈[0,1300]

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Figure 10

Gauss-1 [1,5], Gauss-2 [2,6], Gauss-8 [3,7], and Morlet [4,8] wavelet 2D and 3D spectra for ω=4.0, q0=12,200, t∈[0,1300]

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Figure 11

Wavelet history [1], time history [2], phase curve [3], Fourier spectrum [4], and the Lyapunov exponents (5W,6K) for ω=4.0, q0=12,200, t∈[200;400]

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Figure 12

Wavelet history [1], time history [2], phase curve [3], and Fourier spectrum [4] for ω=4.0, q0=12,200, t∈[600,800]

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Figure 13

Wavelet history [1], phase curve [2], and Fourier spectrum [3] for ω=4.0, q0=14,200, t∈[0,450]

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Figure 14

Gauss-1 [1,5], Gauss-2 [2,6], Gauss-8 [3,7], and Morlet [4,8] 2D and 3D wavelet spectra for ω=4.0, q0=14,200, t∈[0,450]

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Figure 15

Wavelet history [1], time history [2], phase curve [3], Fourier spectrum [4], and the Lyapunov exponents (5W, 6K) for ω=4.0, q0=14,200, t∈[100,150]

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Figure 16

Wavelet history [1], time history [2], phase curve [3], Fourier spectrum [4], and the Lyapunov exponents (5W,6K) for ω=4.0, q0=14,200, t∈[300,450]

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Figure 17

Transition to chaos via intermittency (Paumeau-Manneville scenario) with different length of the quasi-periodic windows (wavelet spectra versus Fourier spectra for ω=7.0)

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Figure 18

Gauss-1 [1,5], Gauss-2 [2,6], Gauss-8 [3,7], and Morlet [4,8] wavelet spectra for ω=7.0, q0=29,400, t∈[200,220]

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Figure 19

Wavelet history [1], time history [2], phase curve [3], and Fourier spectrum [4] for ω=7.0, q0=29,400, t∈[200,220]

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Figure 20

Wavelet history [1], time history [2], phase curve [3], and Fourier spectrum [4] for ω=7.0, q0=29,000, t∈[110,130]

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Figure 21

Wavelet history [1], time history [2], phase curve [3], and Fourier spectrum [4] for ω=7.0, q0=29,200, t∈[145,160]

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