9.7 Size-Dependent Euler-Bernoulli Beams with Topologically Optimized Microstructure 387
Therefore, the frequency-deflection ω(w) dependency exhibits the difference of
the results obtained for the optimal and homogeneous beam in all studied cases
(θ = −100, θ = 0, θ = 100).
9.7.3.3 Dynamic Problems
To investigate the influence of the topologically optimal beam microstructure on the
nonlinear dynamics of the beam, Fourier and wavelet spectra, Poincaré maps, phase
portraits and the largest Lyapunov exponents (LLEs) can be employed. In this study,
all results were obtained for the harmonic load q = q 0 sin(ω p t). The dissipation
coefficient was taken as ε = 1. In order to investigate the reliability of the computed
LLEs based on the Wolf method [130], the LLEs (λ 1 ) were additionally estimated
using the methods of Rosenstein [131]], Kantz [132] and neural networks NW [133].
We compared the results for different mother wavelets (Morlet, Daubechies, Gauss,
Haar, etc.), and eventually used the Morlet wavelet, which is the most suitable for
our purpose.
Tables 9.8, 9.9 and 9.10 contain the following results: (a) time series (signal)
w(0.5; t); (b) Fourier spectra S(ω) computed based on the FFT (Fast Fourier Transforms); (c) 2D wavelet spectra based on the Morlet wavelets; (d) phase portraits
˙
w [w(t)]; (e) Poincaré maps w t+T [w t ]; (f) values of LLEs estimated based on
employment of four qualitatively different approaches. Column designations (1, 1*,
2, 2*) of the Tables 9.8, 9.9 and 9.10 correspond to the types of tasks described in
Sect. 9.6.3.2.
We began with studying the beam dynamics for θ =0, focusing on estimation and tracing the changes in the frequency band. For the lack of the sizedependent behaviour (k 4 = 0), the following excitation parameters were fixed: q 0 =
18500, ω p = 8.5. For taking into account the size-dependent behaviour (k 4 = 0.3),
the excitation amplitude q 0 = 19000, and its frequency ω p = 8.7.
Case study 1 (k 4 = 0, homogeneous beam). The signal (1a) exhibits complexity,
the frequency spectrum band (9.71b) contains 20 fundamental frequencies separated
by the interval ω n − ω n−1 ≈ 0.42, and a set of frequencies of small powers. One can
observe two subharmonic frequencies ω 1 = ω p /2 and ω 2 = ω p /4 resulting from
the period doubling bifurcation. The wavelet spectrum (9.71c) demonstrates nonuniform character and some frequencies are switched on and/or switched off, i.e.
their births/deaths depend on the time evolution. The Poincaré section (9.71e) shows
a set of points in the form of a loop. The phase portrait indicates the occurrence of a
strange chaotic attractor (9.71d).
Case study 1* (k 4 = 0, optimal microstructure). Signal (9.71*a) is periodic.
The Fourier spectrum (9.71b) exhibits two frequencies, i.e. ω p and ω 1 = 0.6538 =
ω p /13 . The Morlet wavelet (9.71*c) reports, besides of ω 1 , also a few frequencies
of small powers in the low-frequency band. Phase portrait (9.71*d) is structured
and more compressed in comparison to the previous case. Poincaré map (9.71*e)
presents a point proving the existence of periodic vibrations. Therefore, for the given
Therefore, the frequency-deflection ω(w) dependency exhibits the difference of
the results obtained for the optimal and homogeneous beam in all studied cases
(θ = −100, θ = 0, θ = 100).
9.7.3.3 Dynamic Problems
To investigate the influence of the topologically optimal beam microstructure on the
nonlinear dynamics of the beam, Fourier and wavelet spectra, Poincaré maps, phase
portraits and the largest Lyapunov exponents (LLEs) can be employed. In this study,
all results were obtained for the harmonic load q = q 0 sin(ω p t). The dissipation
coefficient was taken as ε = 1. In order to investigate the reliability of the computed
LLEs based on the Wolf method [130], the LLEs (λ 1 ) were additionally estimated
using the methods of Rosenstein [131]], Kantz [132] and neural networks NW [133].
We compared the results for different mother wavelets (Morlet, Daubechies, Gauss,
Haar, etc.), and eventually used the Morlet wavelet, which is the most suitable for
our purpose.
Tables 9.8, 9.9 and 9.10 contain the following results: (a) time series (signal)
w(0.5; t); (b) Fourier spectra S(ω) computed based on the FFT (Fast Fourier Transforms); (c) 2D wavelet spectra based on the Morlet wavelets; (d) phase portraits
˙
w [w(t)]; (e) Poincaré maps w t+T [w t ]; (f) values of LLEs estimated based on
employment of four qualitatively different approaches. Column designations (1, 1*,
2, 2*) of the Tables 9.8, 9.9 and 9.10 correspond to the types of tasks described in
Sect. 9.6.3.2.
We began with studying the beam dynamics for θ =0, focusing on estimation and tracing the changes in the frequency band. For the lack of the sizedependent behaviour (k 4 = 0), the following excitation parameters were fixed: q 0 =
18500, ω p = 8.5. For taking into account the size-dependent behaviour (k 4 = 0.3),
the excitation amplitude q 0 = 19000, and its frequency ω p = 8.7.
Case study 1 (k 4 = 0, homogeneous beam). The signal (1a) exhibits complexity,
the frequency spectrum band (9.71b) contains 20 fundamental frequencies separated
by the interval ω n − ω n−1 ≈ 0.42, and a set of frequencies of small powers. One can
observe two subharmonic frequencies ω 1 = ω p /2 and ω 2 = ω p /4 resulting from
the period doubling bifurcation. The wavelet spectrum (9.71c) demonstrates nonuniform character and some frequencies are switched on and/or switched off, i.e.
their births/deaths depend on the time evolution. The Poincaré section (9.71e) shows
a set of points in the form of a loop. The phase portrait indicates the occurrence of a
strange chaotic attractor (9.71d).
Case study 1* (k 4 = 0, optimal microstructure). Signal (9.71*a) is periodic.
The Fourier spectrum (9.71b) exhibits two frequencies, i.e. ω p and ω 1 = 0.6538 =
ω p /13 . The Morlet wavelet (9.71*c) reports, besides of ω 1 , also a few frequencies
of small powers in the low-frequency band. Phase portrait (9.71*d) is structured
and more compressed in comparison to the previous case. Poincaré map (9.71*e)
presents a point proving the existence of periodic vibrations. Therefore, for the given