7.4 Chaotic Dynamics of Size-Dependent Graded Timoshenko Beams
229
the wavelet spectrum and the LLE. On the contrary, the functionally graded beam
(Table 7.6) starts to chaotically vibrate in the beginning of the time interval.
Therefore, the functionally graded beam with the stiffer layer located on the upper
side can be employed for a given amplitude and frequency of the excitation load.
The analysis of the obtained results for the variants corresponding to a lack of
the size-dependent behaviour (γ 2 = 0) validates a conclusion that the beam with
the stiffer layer located on the upper side can be employed to carry the dynamic
load. It should be mentioned that all characteristics of the vibration process, i.e.
Fourier and wavelet spectra, Poincaré section, and the LLEs qualitatively coincide.
The carried-out analysis and comparison with the previous variants (γ 2 = 0.3) for
the homogeneous beam and the beam with the stiffer layer located on the bottom
side implies the essential difference in all characteristics of the vibrational process.
In other words, for the studied structures, the size-dependent behaviour plays an
essential role.
The analysis of the results associated with the homogeneous beams (Tables 7.4,
7.7, 7.8, 7.9) allows one to extend the conclusions formulated with respect to static
problems. Namely, application of the material of large stiffness has an essential
influence on the character of vibrations.
In the case of the beams with a single (initial) stiffness, chaotic vibrations occur
(Tables 7.4, 7.7), whereas in the case of beams with a doubled stiffness (Tables 7.8,
7.9), quasi-periodic vibrations are exhibited.
In order to validate the reliability of the LLEs computation using Wolf’s algorithm
[129], we have computed them using three other methods, i.e. Rosenstein’s [130],
Kantz’s [131] and neural network (NW) approaches [121]. The investigations have
been carried out for all mentioned variants. In what follows, we give exemplary results
regarding the case 1 (see Table 7.4). The numbers of time intervals correspond to the
following values: 1—t ∈ [300; 2100], 2—t ∈ [2105; 2160], 3—t ∈ [2165; 3900],
4—t ∈ [3901; 5000], 5—t ∈ [5001; 8000].
All four methods yield positive values of the LLEs (λ 1 ) on all time intervals, which
implies chaotic vibrations. However, there are some differences with respect to the
computed values. The qualitative changes of λ 1 are similar for three methods (Wolf’s,
Rosenstein’s, Kantz’s) on the intervals 1–3. Beginning from the third time interval,
all three characteristics (Rosenstein’s, Kantz’s, neural networks (NW)) shown in
Fig. 7.9 approach each other, and hence essentially differ from the values obtained
using Wolf’s method.
Figure 7.10 reports time evolutions of Lyapunov spectrum obtained with the neural
network approach. One may observe the qualitative similarity between the curves
corresponding to the first four Lyapunov exponents.
229
the wavelet spectrum and the LLE. On the contrary, the functionally graded beam
(Table 7.6) starts to chaotically vibrate in the beginning of the time interval.
Therefore, the functionally graded beam with the stiffer layer located on the upper
side can be employed for a given amplitude and frequency of the excitation load.
The analysis of the obtained results for the variants corresponding to a lack of
the size-dependent behaviour (γ 2 = 0) validates a conclusion that the beam with
the stiffer layer located on the upper side can be employed to carry the dynamic
load. It should be mentioned that all characteristics of the vibration process, i.e.
Fourier and wavelet spectra, Poincaré section, and the LLEs qualitatively coincide.
The carried-out analysis and comparison with the previous variants (γ 2 = 0.3) for
the homogeneous beam and the beam with the stiffer layer located on the bottom
side implies the essential difference in all characteristics of the vibrational process.
In other words, for the studied structures, the size-dependent behaviour plays an
essential role.
The analysis of the results associated with the homogeneous beams (Tables 7.4,
7.7, 7.8, 7.9) allows one to extend the conclusions formulated with respect to static
problems. Namely, application of the material of large stiffness has an essential
influence on the character of vibrations.
In the case of the beams with a single (initial) stiffness, chaotic vibrations occur
(Tables 7.4, 7.7), whereas in the case of beams with a doubled stiffness (Tables 7.8,
7.9), quasi-periodic vibrations are exhibited.
In order to validate the reliability of the LLEs computation using Wolf’s algorithm
[129], we have computed them using three other methods, i.e. Rosenstein’s [130],
Kantz’s [131] and neural network (NW) approaches [121]. The investigations have
been carried out for all mentioned variants. In what follows, we give exemplary results
regarding the case 1 (see Table 7.4). The numbers of time intervals correspond to the
following values: 1—t ∈ [300; 2100], 2—t ∈ [2105; 2160], 3—t ∈ [2165; 3900],
4—t ∈ [3901; 5000], 5—t ∈ [5001; 8000].
All four methods yield positive values of the LLEs (λ 1 ) on all time intervals, which
implies chaotic vibrations. However, there are some differences with respect to the
computed values. The qualitative changes of λ 1 are similar for three methods (Wolf’s,
Rosenstein’s, Kantz’s) on the intervals 1–3. Beginning from the third time interval,
all three characteristics (Rosenstein’s, Kantz’s, neural networks (NW)) shown in
Fig. 7.9 approach each other, and hence essentially differ from the values obtained
using Wolf’s method.
Figure 7.10 reports time evolutions of Lyapunov spectrum obtained with the neural
network approach. One may observe the qualitative similarity between the curves
corresponding to the first four Lyapunov exponents.