182
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
In the interval [1300; 1600], the frequency spectrum does not undergo changes in
time and LLE is decreased. The signal possesses non-periodic structure on contrary
to the previous interval, whereas the phase portrait has washed out shape since the
frequency ω 4 undergoes periodic change.
Interval [1600; 1700] is associated with the occurrence of the frequency ω 3 in
the wavelet spectrum which achieves in the beginning its maximum value (see the
wavelet spectrum). It should be emphasized that the remaining frequencies of the
spectrum perhaps without ω 2 loos their power S(ω). In this time interval, the second
hum exhibited by LLE is observed. In the interval [1700; 5300], there are no changes
in the frequency spectrum, and the LLE smoothly decreases though remain positive.
The signal possesses complex character but regular. The phase portrait exhibits a tori
with the surrounding rings.
Taking into account the same parameters, size coefficient l/ h = 0.3, the sinusoidal load parameters q 0 = 4000, ω p = 9, the vibration characteristics (signals,
Fourier and wavelet spectra, phase portrait, Poincaré sections, time histories of the
Lyapunov exponents), they differ with each other depending on the chosen model.
Increase of complexity of the mathematical model, i.e. an account of the normal
rotation in the Timoshenko model and normal curving in the Sheremetev–Pelekh
model, implies an increase of complexity of the vibration character.
Each of the models has its own, and different numbers of frequencies exhibited
by the Fourier spectrum: Bernoulli–Euler—6, Timoshenko—3 and the Sheremetev–
Pelekh—7 (including the excitation frequency). Each of the studied 2D wavelets of
beam models (Tables 6.17c, 6.4c, 6.7c) shows that the frequency spectrum undergoes
changes in time which indicates changes of energy in time.
Both phase portraits and Poincaré maps (Table 6.7d, e) of the Timoshenko model
differ from the counterpart characteristics of the Bernoulli–Euler (Table 6.1d, e) and
Sheremetev–Pelekh (7.13d, e) models. Pictures (Table 6.7d, e) exhibit essential role
of three frequencies on the system dynamics.
In spite of the difference between all models with regard to frequency spectra,
time histories of LLE are qualitatively similar; in a given time instant, there observed
a sudden jump into positive, and then an LLE smoothly decreases though all the times
keeping positive value.
6.6.1.6 The Sheremetev–Pelekh Model l/h = 0
In Table 6.17, dynamic characteristics of the Sheremetev–Pelekh model in time
window t ∈ [300; 5300] are presented.
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
In the interval [1300; 1600], the frequency spectrum does not undergo changes in
time and LLE is decreased. The signal possesses non-periodic structure on contrary
to the previous interval, whereas the phase portrait has washed out shape since the
frequency ω 4 undergoes periodic change.
Interval [1600; 1700] is associated with the occurrence of the frequency ω 3 in
the wavelet spectrum which achieves in the beginning its maximum value (see the
wavelet spectrum). It should be emphasized that the remaining frequencies of the
spectrum perhaps without ω 2 loos their power S(ω). In this time interval, the second
hum exhibited by LLE is observed. In the interval [1700; 5300], there are no changes
in the frequency spectrum, and the LLE smoothly decreases though remain positive.
The signal possesses complex character but regular. The phase portrait exhibits a tori
with the surrounding rings.
Taking into account the same parameters, size coefficient l/ h = 0.3, the sinusoidal load parameters q 0 = 4000, ω p = 9, the vibration characteristics (signals,
Fourier and wavelet spectra, phase portrait, Poincaré sections, time histories of the
Lyapunov exponents), they differ with each other depending on the chosen model.
Increase of complexity of the mathematical model, i.e. an account of the normal
rotation in the Timoshenko model and normal curving in the Sheremetev–Pelekh
model, implies an increase of complexity of the vibration character.
Each of the models has its own, and different numbers of frequencies exhibited
by the Fourier spectrum: Bernoulli–Euler—6, Timoshenko—3 and the Sheremetev–
Pelekh—7 (including the excitation frequency). Each of the studied 2D wavelets of
beam models (Tables 6.17c, 6.4c, 6.7c) shows that the frequency spectrum undergoes
changes in time which indicates changes of energy in time.
Both phase portraits and Poincaré maps (Table 6.7d, e) of the Timoshenko model
differ from the counterpart characteristics of the Bernoulli–Euler (Table 6.1d, e) and
Sheremetev–Pelekh (7.13d, e) models. Pictures (Table 6.7d, e) exhibit essential role
of three frequencies on the system dynamics.
In spite of the difference between all models with regard to frequency spectra,
time histories of LLE are qualitatively similar; in a given time instant, there observed
a sudden jump into positive, and then an LLE smoothly decreases though all the times
keeping positive value.
6.6.1.6 The Sheremetev–Pelekh Model l/h = 0
In Table 6.17, dynamic characteristics of the Sheremetev–Pelekh model in time
window t ∈ [300; 5300] are presented.