6.6 Chaotic Dynamics of the Size-Dependent Flexible Beams
183
Table 6.17 Numerical results for the Sheremetev–Pelekh model (l/ h = 0, t ∈ [300; 5300])
[reprinted with permission from International Journal of Non-Linear Mechanics publishers]
Time history/signal is homogeneous. The Fourier spectrum exhibits two frequencies: ω 1 = 1.8086, ω 2 = 5.3812 and the following frequency relation 2 · ω 5 + ω 6 =
ω p . The wavelet spectrum also points out the spectrum homogeneity versus time.
The phase portrait has the shape of an ellipse. The result difference occurs when the
size-dependent behaviour is taken into account, in all characteristics of the dynamical process. The LLE is negative in the whole investigated time interval. Therefore,
all dynamical characteristics point out the quasi-periodic vibrations. The wavelet
spectrum shows that the frequency characteristics do not depend on time.
The general conclusions are as follows:
(i) dynamic regimes are different for different models, and the frequency spectra
of the Bernoulli–Euler and Timoshenko models depend on time;
(ii) phase portraits for the whole time interval (Tables 6.10e and 6.13e) contain
information given in time windows (Tables 6.11b, 6.12b, 6.14b, 6.15b);
(iii) time history of the LLE is in agreement with the time history and wavelet
spectrum;
(iv) time histories of the LLE of the Euler–Bernoulli, Timoshenko and Sheremetev–
Pelekh models are qualitatively different, on contrary to the case of l/ h = 0.3.
183
Table 6.17 Numerical results for the Sheremetev–Pelekh model (l/ h = 0, t ∈ [300; 5300])
[reprinted with permission from International Journal of Non-Linear Mechanics publishers]
Time history/signal is homogeneous. The Fourier spectrum exhibits two frequencies: ω 1 = 1.8086, ω 2 = 5.3812 and the following frequency relation 2 · ω 5 + ω 6 =
ω p . The wavelet spectrum also points out the spectrum homogeneity versus time.
The phase portrait has the shape of an ellipse. The result difference occurs when the
size-dependent behaviour is taken into account, in all characteristics of the dynamical process. The LLE is negative in the whole investigated time interval. Therefore,
all dynamical characteristics point out the quasi-periodic vibrations. The wavelet
spectrum shows that the frequency characteristics do not depend on time.
The general conclusions are as follows:
(i) dynamic regimes are different for different models, and the frequency spectra
of the Bernoulli–Euler and Timoshenko models depend on time;
(ii) phase portraits for the whole time interval (Tables 6.10e and 6.13e) contain
information given in time windows (Tables 6.11b, 6.12b, 6.14b, 6.15b);
(iii) time history of the LLE is in agreement with the time history and wavelet
spectrum;
(iv) time histories of the LLE of the Euler–Bernoulli, Timoshenko and Sheremetev–
Pelekh models are qualitatively different, on contrary to the case of l/ h = 0.3.