108
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
Table 4.7 contains dynamic characteristics of the geometrically linear beam for
n = 160 (RKPD8).
For the same parameters, we have obtained the results for geometrically linear
beams taking into account their contact interactions (Table 4.6). Comparison of
the signals, 3D and 2D phase portraits, wavelet spectra and Poincaré pseudo-maps
shows their coincidence for the first beam. In the case of the second beam, although
its configuration form is repeated, the values of the signal do not coincide and the
number of loops exhibited by the 2D and 3D phase portraits is different. In addition,
the frequency power spectra differ from each other in the linear and nonlinear case.
Namely, in the linear case, the frequencies ω 1 and ω 2 are not exhibited. This can be
caused by their small power. Lyapunov exponents for both linearly geometric beams
are positive.
Therefore, carrying out the complex analysis of solutions to the problem, one
can conclude that vibrations of the studied two-layer beam with a small clearance
between the layers are chaotic. The efficient beam partition while using the FDM is
n = 160. Since the solutions for Cauchy problem coincide for all employed modifications of the Runge-Kutta methods, the second-order Runge-Kutta method can be
used. The comparison of the geometrically linear and nonlinear problems implies
the occurrence of nonlinear terms in the solution, despite the amplitude of beam
vibrations coincides with the linear approximation.
4.6 Application of the Principal Component Analysis
(PCA)
The PCA will be employed to find the fundamental frequencies. As it can be seen from
the reported characteristics (Table 4.8), for the beam partition number n = 160 and
for the design nonlinear problem, the frequency spectra for both beams exhibit five
linearly dependent frequencies ω p , ω 1 , ω 2 , ω 3 , and ω 4 , whereas for both geometrically
and design nonlinear problem of the contacting beams, the noisy components of the
frequency power spectra are exhibited.
After the time series reconstruction by means of the PCA, the number of frequencies has been reduced for the beam 1 in the case of the geometrically linear problem
and the locations of four frequencies have been detected (they coincide with the frequency amplitudes of the geometrically nonlinear problem). In the case of the signal
reconstruction obtained for the design nonlinear problem, all frequencies are exhibited. The beam 2, with no account of the geometric nonlinearity, exhibits chaotic
vibrations.
After time series reconstruction using the principal component analysis, the frequencies localization takes place. The frequencies are linearly dependent regarding
the geometrically nonlinear problem. As a result, the computation time has been
reduced eight times due to the decrease in the number of nodes while employing the
FDM, and the step of Runge-Kutta method has been increased. However, this method
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
Table 4.7 contains dynamic characteristics of the geometrically linear beam for
n = 160 (RKPD8).
For the same parameters, we have obtained the results for geometrically linear
beams taking into account their contact interactions (Table 4.6). Comparison of
the signals, 3D and 2D phase portraits, wavelet spectra and Poincaré pseudo-maps
shows their coincidence for the first beam. In the case of the second beam, although
its configuration form is repeated, the values of the signal do not coincide and the
number of loops exhibited by the 2D and 3D phase portraits is different. In addition,
the frequency power spectra differ from each other in the linear and nonlinear case.
Namely, in the linear case, the frequencies ω 1 and ω 2 are not exhibited. This can be
caused by their small power. Lyapunov exponents for both linearly geometric beams
are positive.
Therefore, carrying out the complex analysis of solutions to the problem, one
can conclude that vibrations of the studied two-layer beam with a small clearance
between the layers are chaotic. The efficient beam partition while using the FDM is
n = 160. Since the solutions for Cauchy problem coincide for all employed modifications of the Runge-Kutta methods, the second-order Runge-Kutta method can be
used. The comparison of the geometrically linear and nonlinear problems implies
the occurrence of nonlinear terms in the solution, despite the amplitude of beam
vibrations coincides with the linear approximation.
4.6 Application of the Principal Component Analysis
(PCA)
The PCA will be employed to find the fundamental frequencies. As it can be seen from
the reported characteristics (Table 4.8), for the beam partition number n = 160 and
for the design nonlinear problem, the frequency spectra for both beams exhibit five
linearly dependent frequencies ω p , ω 1 , ω 2 , ω 3 , and ω 4 , whereas for both geometrically
and design nonlinear problem of the contacting beams, the noisy components of the
frequency power spectra are exhibited.
After the time series reconstruction by means of the PCA, the number of frequencies has been reduced for the beam 1 in the case of the geometrically linear problem
and the locations of four frequencies have been detected (they coincide with the frequency amplitudes of the geometrically nonlinear problem). In the case of the signal
reconstruction obtained for the design nonlinear problem, all frequencies are exhibited. The beam 2, with no account of the geometric nonlinearity, exhibits chaotic
vibrations.
After time series reconstruction using the principal component analysis, the frequencies localization takes place. The frequencies are linearly dependent regarding
the geometrically nonlinear problem. As a result, the computation time has been
reduced eight times due to the decrease in the number of nodes while employing the
FDM, and the step of Runge-Kutta method has been increased. However, this method