180
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
6.6.1.3 The Timoshenko Model for l/h = 0.3
In Table 6.7, there are given the dynamic characteristics of the Timoshenko beam
model for l/ h = 0.3 and for the time window t ∈ [300; 5300], whereas in Tables
6.8, 6.9 for the windows t ∈ [300; 1600] and t ∈ [1600; 5300].
The Fourier spectrum exhibits three frequencies ω p = 9, ω 1 = 1.805, ω 2 = 5.41.
They are coupled by the linear relation 2 · ω 1 + ω 2 = ω p . However, on contrary to
the Bernoulli–Euler model, the mentioned three frequencies have one more important features. Power S(ω) of the excitation frequency ω p is equal to the arithmetic
average of powers of frequencies ω 1 , ω 2 . The frequencies ω 1 , ω 2 are noisy. The phase
portrait presents three-rotational stable cycle. Noisy frequencies imply divergence of
the trajectories in the phase portrait. However, the Poincare sections do not exhibit
divergence effects.
The signal (a) for t ≈ 1700 exhibits a sudden amplitude increase. It corresponds
to occurrence of the frequencies in the wavelet spectrum. It implies also a jump of
LLE from zero to positive values. Further change of the spectrum is not observed,
and the magnitude of LLE decreases but remains positive. It should be mentioned
that the birth of new frequencies implies an increase in the power of frequency.
Comparison of dynamic characteristics in time windows (Tables 6.8 and 6.9)
shows that the birth of new frequencies qualitatively changes the signal character
though it remains periodic in contrary to the results reported in Tables 6.1 and 6.3.
The phase portrait shown in Table 6.7 stands for the combination of the phase portraits
from Tables 6.8 and 6.9. The latter conclusion holds also for the remaining cases
considered in this section.
6.6.1.4 The Timoshenko Model for l/h = 0
Table 6.10 presents results of numerical analysis of nonlinear dynamics of the Timoshenko model for the time interval t ∈ [300; 5300]. In Tables 6.14 and 6.15, there
are also given results for the time windows t ∈ [300; 1200] and t ∈ [1200; 5300].
The signal exhibits the amplitude jumps for t ≈ 1170, which is a consequence
of the birth of frequencies ω 3 , ω 4 . At the same time instants, the power of excitation
frequency ω p suddenly increases that is visible in 2D wavelet (c), the Fourier spectrum (b) shows occurrence of the frequencies ω 1 = 1.7696, ω 2 = 5.4634, which
obey the following dependence 2 · ω 1 + ω 2 = ω p . The time history of LLE suffers
for a sudden jump in time instant where both frequencies appear. The frequencies in
the Fourier spectrum are noisy which is confirmed by positive Lyapunov exponent.
The phase portrait possesses a complex structure in the form of the embedded in
each other cylinder and forces. Observe that the frequency spectrum contains three
frequencies, as it was observed for the Timoshenko beam with an account of the
size-dependent behaviour. The qualitative similarity concerns also the LLE but the
remaining characteristics (phase portrait and Poincaré map) are different.
The analysed results shown in Tables 6.11 and 6.12 allow to conclude that the
change of the frequency spectrum, i.e. the birth of the frequencies ω 3 , ω 4 and increase
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
6.6.1.3 The Timoshenko Model for l/h = 0.3
In Table 6.7, there are given the dynamic characteristics of the Timoshenko beam
model for l/ h = 0.3 and for the time window t ∈ [300; 5300], whereas in Tables
6.8, 6.9 for the windows t ∈ [300; 1600] and t ∈ [1600; 5300].
The Fourier spectrum exhibits three frequencies ω p = 9, ω 1 = 1.805, ω 2 = 5.41.
They are coupled by the linear relation 2 · ω 1 + ω 2 = ω p . However, on contrary to
the Bernoulli–Euler model, the mentioned three frequencies have one more important features. Power S(ω) of the excitation frequency ω p is equal to the arithmetic
average of powers of frequencies ω 1 , ω 2 . The frequencies ω 1 , ω 2 are noisy. The phase
portrait presents three-rotational stable cycle. Noisy frequencies imply divergence of
the trajectories in the phase portrait. However, the Poincare sections do not exhibit
divergence effects.
The signal (a) for t ≈ 1700 exhibits a sudden amplitude increase. It corresponds
to occurrence of the frequencies in the wavelet spectrum. It implies also a jump of
LLE from zero to positive values. Further change of the spectrum is not observed,
and the magnitude of LLE decreases but remains positive. It should be mentioned
that the birth of new frequencies implies an increase in the power of frequency.
Comparison of dynamic characteristics in time windows (Tables 6.8 and 6.9)
shows that the birth of new frequencies qualitatively changes the signal character
though it remains periodic in contrary to the results reported in Tables 6.1 and 6.3.
The phase portrait shown in Table 6.7 stands for the combination of the phase portraits
from Tables 6.8 and 6.9. The latter conclusion holds also for the remaining cases
considered in this section.
6.6.1.4 The Timoshenko Model for l/h = 0
Table 6.10 presents results of numerical analysis of nonlinear dynamics of the Timoshenko model for the time interval t ∈ [300; 5300]. In Tables 6.14 and 6.15, there
are also given results for the time windows t ∈ [300; 1200] and t ∈ [1200; 5300].
The signal exhibits the amplitude jumps for t ≈ 1170, which is a consequence
of the birth of frequencies ω 3 , ω 4 . At the same time instants, the power of excitation
frequency ω p suddenly increases that is visible in 2D wavelet (c), the Fourier spectrum (b) shows occurrence of the frequencies ω 1 = 1.7696, ω 2 = 5.4634, which
obey the following dependence 2 · ω 1 + ω 2 = ω p . The time history of LLE suffers
for a sudden jump in time instant where both frequencies appear. The frequencies in
the Fourier spectrum are noisy which is confirmed by positive Lyapunov exponent.
The phase portrait possesses a complex structure in the form of the embedded in
each other cylinder and forces. Observe that the frequency spectrum contains three
frequencies, as it was observed for the Timoshenko beam with an account of the
size-dependent behaviour. The qualitative similarity concerns also the LLE but the
remaining characteristics (phase portrait and Poincaré map) are different.
The analysed results shown in Tables 6.11 and 6.12 allow to conclude that the
change of the frequency spectrum, i.e. the birth of the frequencies ω 3 , ω 4 and increase