4.5 Numerical Experiment
105
beam and two “thick rings” for the second beam. This implies the occurrence of the
period doubling bifurcation.
For n = 80, the frequency power spectra of both beams contain the following
frequencies:
ω p
2
,
ω p
5
,
4ω p
5
. The 2D phase portrait of the first beam presents a ring,
whereas the 3D phase portrait exhibits a more complex structure, i.e. two rings (the
structure of the phase portrait is more unique for the second beam). The wavelet
spectrum for ω ∈ [0.2; 5.1] exhibits only the frequency ω p , whereas the wavelet
spectrum ω ∈ [0.2; 2] shows a set of frequencies, which are not visible in the power
spectrum, whereas the Poincaré pseudo-map presents a thick ring for both beams.
For n = 120 and n = 160, the mentioned characteristics fully coincide in the
frequency power spectrum for both the first and the second beams, and the following
frequencies are distinguished ω p , ω 1 , ω 2 , ω 3 , ω 4 . The 2D phase portrait of the first
beam presents a ring, whereas the second beam phase portrait is composed of five
rings. The wavelet spectrum ω ∈ [0.2; 5.1] exhibits only the frequency ω p , whereas
the wavelet spectrum for ω ∈ [0.2; 2] possesses frequencies which are not reported
by the power spectrum.
The Poincaré pseudo-map presents a thick ring for both the first and the second
beams.
Since the wavelet spectrum computed for ω ∈ [0.2; 5.1] and for t ∈ [0; 1000]
does not exhibit any other frequencies than ω p (frequency of excitation), it has not
been presented. An increase in the number of nodes, i.e. of beams partitions, yields
a decrease in the frequencies number to 5, i.e. the so-called regularization of beam
vibrations takes place. Finally, let us emphasize the occurrence of the chaotic phase
synchronization of vibrations of beams.
In Table 4.5, we report surfaces of the beams deflections for t ∈ [95; 100] and
for n = 40; 80; 120; 160. The horizontal axis denotes the beam length, whereas the
vertical axis stands for the beam deflection, and the remaining axis denotes time.
We have achieved convergence of the deflection surfaces of the middle lines of
the beams with respect to time for n = 120 and n = 160. Therefore, the convergence
of the numerical results is guaranteed for the whole time and space intervals.
Figure 4.2a, b presents the curves obtained based on an estimation of the
largest Lyapunov exponents by Wolf, Rosenstein and Kantz methods versus n. The
obtained values result from the computation using the eighth-order Runge-Kutta
method. It should be emphasized that different methods of computation of Lyapunov exponent are needed to obtain reliable/true value and, consequently, reliable estimation of chaos. When using the FDM, for the number of beams partitions
n = 40, 80, 120; 160 for any mentioned method, Lyapunov exponents coincide with
an accuracy up to the third decimal digit. Furthermore, all of the largest Lyapunov
exponents (LLEs) are positive.
In what follows, we define how the LLEs depend on the method employed for
the solution of Cauchy problem. For this purpose, we have computed Lyapunov
exponents using Wolf, Kantz and Rosenstein algorithms for n = 160. On the contrary
to the dynamic characteristics, we have not observed full coincidence of the results.
However, the difference between the minimum and the maximum of the exponents
105
beam and two “thick rings” for the second beam. This implies the occurrence of the
period doubling bifurcation.
For n = 80, the frequency power spectra of both beams contain the following
frequencies:
ω p
2
,
ω p
5
,
4ω p
5
. The 2D phase portrait of the first beam presents a ring,
whereas the 3D phase portrait exhibits a more complex structure, i.e. two rings (the
structure of the phase portrait is more unique for the second beam). The wavelet
spectrum for ω ∈ [0.2; 5.1] exhibits only the frequency ω p , whereas the wavelet
spectrum ω ∈ [0.2; 2] shows a set of frequencies, which are not visible in the power
spectrum, whereas the Poincaré pseudo-map presents a thick ring for both beams.
For n = 120 and n = 160, the mentioned characteristics fully coincide in the
frequency power spectrum for both the first and the second beams, and the following
frequencies are distinguished ω p , ω 1 , ω 2 , ω 3 , ω 4 . The 2D phase portrait of the first
beam presents a ring, whereas the second beam phase portrait is composed of five
rings. The wavelet spectrum ω ∈ [0.2; 5.1] exhibits only the frequency ω p , whereas
the wavelet spectrum for ω ∈ [0.2; 2] possesses frequencies which are not reported
by the power spectrum.
The Poincaré pseudo-map presents a thick ring for both the first and the second
beams.
Since the wavelet spectrum computed for ω ∈ [0.2; 5.1] and for t ∈ [0; 1000]
does not exhibit any other frequencies than ω p (frequency of excitation), it has not
been presented. An increase in the number of nodes, i.e. of beams partitions, yields
a decrease in the frequencies number to 5, i.e. the so-called regularization of beam
vibrations takes place. Finally, let us emphasize the occurrence of the chaotic phase
synchronization of vibrations of beams.
In Table 4.5, we report surfaces of the beams deflections for t ∈ [95; 100] and
for n = 40; 80; 120; 160. The horizontal axis denotes the beam length, whereas the
vertical axis stands for the beam deflection, and the remaining axis denotes time.
We have achieved convergence of the deflection surfaces of the middle lines of
the beams with respect to time for n = 120 and n = 160. Therefore, the convergence
of the numerical results is guaranteed for the whole time and space intervals.
Figure 4.2a, b presents the curves obtained based on an estimation of the
largest Lyapunov exponents by Wolf, Rosenstein and Kantz methods versus n. The
obtained values result from the computation using the eighth-order Runge-Kutta
method. It should be emphasized that different methods of computation of Lyapunov exponent are needed to obtain reliable/true value and, consequently, reliable estimation of chaos. When using the FDM, for the number of beams partitions
n = 40, 80, 120; 160 for any mentioned method, Lyapunov exponents coincide with
an accuracy up to the third decimal digit. Furthermore, all of the largest Lyapunov
exponents (LLEs) are positive.
In what follows, we define how the LLEs depend on the method employed for
the solution of Cauchy problem. For this purpose, we have computed Lyapunov
exponents using Wolf, Kantz and Rosenstein algorithms for n = 160. On the contrary
to the dynamic characteristics, we have not observed full coincidence of the results.
However, the difference between the minimum and the maximum of the exponents