96
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
require convergence with respect to the signal/time history also in the case of
chaotic vibrations. It should be mentioned that in the previous investigations [7],
the signal convergence was required rather only in the case of periodic vibrations,
whereas for chaos, the integral convergence was accepted.
2. Cauchy problem is also solved numerically, and hence solutions essentially
depend on both the chosen method and the time step integration. Therefore, in
order to achieve reliable results, Cauchy problem is solved using Runge-Kutta
of the fourth (RK4) and the second (RK2) orders [8], Runge-Kutta-Fehlberg of
the fourth order (RKF4) [9, 10], the Cash-Karp of the fourth order (RKCK) [11],
Runge-Kutta-Prince-Dormand of the eighth order (RKPD8) [12] as well as the
implicit Runge-Kutta methods of the second (IRK2) and the fourth (IRK4) orders.
The explicit method is characterized by the triangular form of the matrix coefficients (including zeroth main diagonal). On the contrary, in the implicit method,
the governing matrix has an arbitrary shape. Moreover, control of integration
errors can be implemented in a relatively easy manner.
3. For each spatial coordinate partition as well as for the chosen method, the signals/time histories, 2D/3D phase portraits, Fourier power spectra, the Morlet
wavelets, snapshots of beam deflections, the Poincaré sections and the 2D wavelet
spectra are constructed while solving Cauchy problem. Haar [13], ShannonKotelnikov, Meyer [14] and Daubechies (from db2 to db16) [15] wavelets, coiflets
and simlets, Morlet wavelet as well as the complex arbitrary Gauss function of the
order higher than 8 (see also [16–18]) are implemented. The Haar and ShannonKotelnikov wavelets are rather not feasible for investigation of the beam constructions, i.e. the first one is badly localized in time. The carried out analysis of the
wavelet spectra obtained with the help of the Daubechies wavelets, coiflets and
simlets yielded a conclusion that an increase in the used wavelets order implies the
improvement of the distinction of the frequency localization. However, no matter how different wavelets and different filters are employed, the wavelet spectra
obtained by the Daubechies wavelets, simlets and coiflets are practically the same,
but still not enough accurate for investigation of the vibration characteristics of
the studied continuous mechanical systems. On the other hand, employment of
the arbitrary Gauss function shows that an increase in the derivative order implies
better resolution of the detected frequencies. It should be emphasized that the
spectra obtained on a basis of the Meyer wavelets [17, 18] (smoothened variant of the Shannon-Kotelnikov wavelets) are localized better in the case of the
low-frequency band in comparison to the Morlet wavelets. However, the higher
spectrum part is better identified using the Morlet wavelets. In what follows, we
report results obtained based on the Morlet wavelets. Observe that real wavelets
exhibit a lack of the scaling function ϕ, whereas the function ψ does not have a
compact carrier and is given explicitly. The complex Morlet and Gauss wavelets
exhibit better localization with respect to frequency than their real counterparts.
Therefore, in order to analyse complex vibrations of continuous systems composed of beams, one may employ either complex or real Morlet wavelets as well
as wavelets based on the derivatives of the Gauss function of the order higher
than 16.
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
require convergence with respect to the signal/time history also in the case of
chaotic vibrations. It should be mentioned that in the previous investigations [7],
the signal convergence was required rather only in the case of periodic vibrations,
whereas for chaos, the integral convergence was accepted.
2. Cauchy problem is also solved numerically, and hence solutions essentially
depend on both the chosen method and the time step integration. Therefore, in
order to achieve reliable results, Cauchy problem is solved using Runge-Kutta
of the fourth (RK4) and the second (RK2) orders [8], Runge-Kutta-Fehlberg of
the fourth order (RKF4) [9, 10], the Cash-Karp of the fourth order (RKCK) [11],
Runge-Kutta-Prince-Dormand of the eighth order (RKPD8) [12] as well as the
implicit Runge-Kutta methods of the second (IRK2) and the fourth (IRK4) orders.
The explicit method is characterized by the triangular form of the matrix coefficients (including zeroth main diagonal). On the contrary, in the implicit method,
the governing matrix has an arbitrary shape. Moreover, control of integration
errors can be implemented in a relatively easy manner.
3. For each spatial coordinate partition as well as for the chosen method, the signals/time histories, 2D/3D phase portraits, Fourier power spectra, the Morlet
wavelets, snapshots of beam deflections, the Poincaré sections and the 2D wavelet
spectra are constructed while solving Cauchy problem. Haar [13], ShannonKotelnikov, Meyer [14] and Daubechies (from db2 to db16) [15] wavelets, coiflets
and simlets, Morlet wavelet as well as the complex arbitrary Gauss function of the
order higher than 8 (see also [16–18]) are implemented. The Haar and ShannonKotelnikov wavelets are rather not feasible for investigation of the beam constructions, i.e. the first one is badly localized in time. The carried out analysis of the
wavelet spectra obtained with the help of the Daubechies wavelets, coiflets and
simlets yielded a conclusion that an increase in the used wavelets order implies the
improvement of the distinction of the frequency localization. However, no matter how different wavelets and different filters are employed, the wavelet spectra
obtained by the Daubechies wavelets, simlets and coiflets are practically the same,
but still not enough accurate for investigation of the vibration characteristics of
the studied continuous mechanical systems. On the other hand, employment of
the arbitrary Gauss function shows that an increase in the derivative order implies
better resolution of the detected frequencies. It should be emphasized that the
spectra obtained on a basis of the Meyer wavelets [17, 18] (smoothened variant of the Shannon-Kotelnikov wavelets) are localized better in the case of the
low-frequency band in comparison to the Morlet wavelets. However, the higher
spectrum part is better identified using the Morlet wavelets. In what follows, we
report results obtained based on the Morlet wavelets. Observe that real wavelets
exhibit a lack of the scaling function ϕ, whereas the function ψ does not have a
compact carrier and is given explicitly. The complex Morlet and Gauss wavelets
exhibit better localization with respect to frequency than their real counterparts.
Therefore, in order to analyse complex vibrations of continuous systems composed of beams, one may employ either complex or real Morlet wavelets as well
as wavelets based on the derivatives of the Gauss function of the order higher
than 16.