9.7 Size-Dependent Euler-Bernoulli Beams with Topologically Optimized Microstructure 391
frequency and amplitude of the harmonic load, the topologically optimal beam structure exhibits a transition from chaotic vibrations to two-frequency periodic vibrations.
Case study 2 (k 4 = 0.3, homogeneous beam). The presented signal (9.72a),
Fourier spectrum (9.72b), and in particular Morlet wavelet spectrum (9.72c) indicate
chaotic vibrations of the beam. The power spectrum comprises numerous frequencies. The order in the frequencies distribution is remarkable. Pairs of the frequencies
satisfy the following resonance relationship: ω 3 − ω 2 = ω 4 − ω 3 = ω 6 − ω 5 =
ω 8 − ω 7 = ω 9 − ω 8 = ω p − ω 10 = 0.766, whereas the remaining frequency pairs
are governed by formulas ω 2 − ω 1 = ω 10 − ω 9 = 0.862 and ω 5 − ω 4 = ω 7 −
ω 6 = 0.804. Therefore, the frequencies of the spectrum depend on the excitation
frequency ω p . The wavelet spectrum (9.72c) exhibits non-uniform character, since
certain frequencies are switched on and switched off in time. Phase portrait (9.72d)
exhibits similarity to the phase portrait of the first case, but its form is more compressed with respect to velocity w t . The Poincaré map consists of the gathered points
in the form of a spiral.
Case study 2* (k 4 = 0.3, optimal microstructure). Signal (9.72*a) is periodic,
but the amplitude is twice lower than in the previous case. Fourier spectrum
(9.72*b), besides ω p , exhibits the period doubling frequency ω 1 = 4.367 ≈ ω p /2 .
The wavelet spectrum (besides ω 1 ) contains a series of frequencies located in the lowfrequency zone. The phase portrait (9.72*d) qualitatively repeats the phase portrait
(9.71*d). The Poincaré map (9.72*e) consists of one point. Therefore, in the case of
neglecting of the size-dependent behaviour, the optimized beam material essentially
influences the vibration characteristics of the beams.
In what follows, we consider the dynamic characteristics of the homogeneous and
optimized beam in the temperature field. All further presented results were obtained
for q 0 = 50000, ω p = 8.5. Table 9.9 reports the vibration characteristics for the temperature (θ = 100C
◦ ).
Case study 1 (k 4 = 0, homogeneous beam). Signal (9.71a) exhibits a chaotic
structure. The Fourier spectrum is typical for chaos spanned on 3 frequencies, i.e.
ω p = 8.5 and two dependent frequencies ω 1 = 0.998, ω 2 = 3.832. The following
resonance relationship between them holds ω p = 3 · (ω 2 − ω 1 ). The wavelet spectrum (9.71c), besides the frequencies ω 2 , ω 1 , presents noisy-type frequencies localized in the low-frequency band. The phase portrait (9.71d) and the Poincaré map also
confirm the chaotic character of vibrations.
Case study 1* (k 4 = 0, optimal microstructure). Signal (9.71*a) has the form
of a sinusoid embedded in a sinusoid-like envelope. The Fourier spectrum (9.71b),
besides ω p , contains the following dependent frequencies: ω 1 , ω 2 , ω 3 , ω 4 , ω 5 . The
following resonance relations hold: ω p − ω 5 = ω 5 − ω 4 , ω 1 + ω 2 + ω 3 = ω p /4.
The wavelet spectrum (9.71*c) gives additional information on the results obtained
by the FFT. Namely, the frequencies ω 1 , ω 2 , ω 4 , ω 5 are born at time instants
t ≈ 2380, 1550, 2350, 1570, respectively. Therefore, up to time t ≈ 1500, the beam
vibrations exhibit only two frequencies ω p and ω 3 . The phase portrait (9.71*d) and
frequency and amplitude of the harmonic load, the topologically optimal beam structure exhibits a transition from chaotic vibrations to two-frequency periodic vibrations.
Case study 2 (k 4 = 0.3, homogeneous beam). The presented signal (9.72a),
Fourier spectrum (9.72b), and in particular Morlet wavelet spectrum (9.72c) indicate
chaotic vibrations of the beam. The power spectrum comprises numerous frequencies. The order in the frequencies distribution is remarkable. Pairs of the frequencies
satisfy the following resonance relationship: ω 3 − ω 2 = ω 4 − ω 3 = ω 6 − ω 5 =
ω 8 − ω 7 = ω 9 − ω 8 = ω p − ω 10 = 0.766, whereas the remaining frequency pairs
are governed by formulas ω 2 − ω 1 = ω 10 − ω 9 = 0.862 and ω 5 − ω 4 = ω 7 −
ω 6 = 0.804. Therefore, the frequencies of the spectrum depend on the excitation
frequency ω p . The wavelet spectrum (9.72c) exhibits non-uniform character, since
certain frequencies are switched on and switched off in time. Phase portrait (9.72d)
exhibits similarity to the phase portrait of the first case, but its form is more compressed with respect to velocity w t . The Poincaré map consists of the gathered points
in the form of a spiral.
Case study 2* (k 4 = 0.3, optimal microstructure). Signal (9.72*a) is periodic,
but the amplitude is twice lower than in the previous case. Fourier spectrum
(9.72*b), besides ω p , exhibits the period doubling frequency ω 1 = 4.367 ≈ ω p /2 .
The wavelet spectrum (besides ω 1 ) contains a series of frequencies located in the lowfrequency zone. The phase portrait (9.72*d) qualitatively repeats the phase portrait
(9.71*d). The Poincaré map (9.72*e) consists of one point. Therefore, in the case of
neglecting of the size-dependent behaviour, the optimized beam material essentially
influences the vibration characteristics of the beams.
In what follows, we consider the dynamic characteristics of the homogeneous and
optimized beam in the temperature field. All further presented results were obtained
for q 0 = 50000, ω p = 8.5. Table 9.9 reports the vibration characteristics for the temperature (θ = 100C
◦ ).
Case study 1 (k 4 = 0, homogeneous beam). Signal (9.71a) exhibits a chaotic
structure. The Fourier spectrum is typical for chaos spanned on 3 frequencies, i.e.
ω p = 8.5 and two dependent frequencies ω 1 = 0.998, ω 2 = 3.832. The following
resonance relationship between them holds ω p = 3 · (ω 2 − ω 1 ). The wavelet spectrum (9.71c), besides the frequencies ω 2 , ω 1 , presents noisy-type frequencies localized in the low-frequency band. The phase portrait (9.71d) and the Poincaré map also
confirm the chaotic character of vibrations.
Case study 1* (k 4 = 0, optimal microstructure). Signal (9.71*a) has the form
of a sinusoid embedded in a sinusoid-like envelope. The Fourier spectrum (9.71b),
besides ω p , contains the following dependent frequencies: ω 1 , ω 2 , ω 3 , ω 4 , ω 5 . The
following resonance relations hold: ω p − ω 5 = ω 5 − ω 4 , ω 1 + ω 2 + ω 3 = ω p /4.
The wavelet spectrum (9.71*c) gives additional information on the results obtained
by the FFT. Namely, the frequencies ω 1 , ω 2 , ω 4 , ω 5 are born at time instants
t ≈ 2380, 1550, 2350, 1570, respectively. Therefore, up to time t ≈ 1500, the beam
vibrations exhibit only two frequencies ω p and ω 3 . The phase portrait (9.71*d) and