9.7 Size-Dependent Euler-Bernoulli Beams with Topologically Optimized Microstructure 393
Case study 2* (k 4 = 0.3, optimal microstructure). The signal (9.72*a) is composed of a sinusoid modulated regularly and having the amplitude three times lower
than in the case of the homogeneous beam (9.72). The Fourier spectrum (9.72*b)
contains, besides of ω p , the dependent frequencies ω 1 , ω 2 , ω 3 , ω 4 , ω 5 . The following resonance phenomena hold: ω p − ω 5 = ω 5 − ω 4 , ω 1 + ω 2 + ω 3 = ω p /4.
The wavelet spectrum (9.72*c), similar to the classical Fourier spectrum, exhibits
temporal behaviour of the frequencies ω 2 , ω 5 , beginning from the time instants
t ≈ 2050, 2150, respectively. In other words, before reaching t ≈ 2050, the vibrations are practically harmonic with the fundamental frequency ω p , since the power of
the frequencies ω 1 , ω 3 , ω 4 is extremely small. The Poincaré map consists of densely
distributed points, i.e. the trajectories are not divergent and the beam vibrations are
regular.
9.7.3.4 Scenarios of Transition into Chaos
Detection, construction and analysis of a scenario of transition from regular to
chaotic vibrations, and vice versa, are important and fundamental tasks when investigating nonlinear dynamics of continuous mechanical systems. These transitions
can be based on the classical Feigenbaum, Pomeau-Manneville, and Ruelle-TakensNewhouse scenarios. There are papers reporting scenarios of the transition for structural members, including beams, plates, and shells, by using the Bernoulli-Euler,
Timoshenko and Sheremetev-Pelekh-Reddy models without [134–136] and taking
[121, 137] into account of the size-dependent behaviour. Furthermore, scenarios of
transition from regular to chaotic dynamics of nonlinear non-homogeneous beams
have been detected and studied with respect to their thickness for different values of
the size-dependent parameter [138].
We begin the investigations with a construction and analysis of scenarios for
two temperature values (θ =100, θ = −100), for two cases, i.e. taking into account
the size-dependent behaviour γ 2 = 0.3 or not γ 2 = 0, for a non-homogeneous
(optimized) beam. The following parameters were fixed: ω p = 8.5, ε = 1, γ 1 =
40, ν = 0.3. All detected scenarios have a similar qualitative picture, i.e. a transition into chaotic vibrations which follows the Pomeau-Manneville scenario [139],
where load plays a role of the control parameter.
Chaotic vibrations are achieved after the third cycle of the process “periodic
vibrations-chaotic vibrations”. Each cycle is realized through different scenarios.
Let us consider the scenarios for the non-homogeneous (optimized) beam for the
temperature θ = −100, γ 2 = 0. The following dynamic characteristics are reported
in Tables 9.11, 9.12 and 9.13: (a) signal w(0.5; t); (b) Fourier spectrum S(ω) based
on the FFT; (c) 2D wavelet spectrum based on the Morlet mother wavelet; (d) phase
portrait ˙
w [w(t)]; (e) Poincaré section w t+T [w t ]; (f) LLEs computed by four methods.
The first cycle consist of three frequencies exhibiting the following resonance
behaviour ω p − ω 3 = ω 3 − ω 2 = ω 2 − ω 1 . An increase in the amplitude of load
Case study 2* (k 4 = 0.3, optimal microstructure). The signal (9.72*a) is composed of a sinusoid modulated regularly and having the amplitude three times lower
than in the case of the homogeneous beam (9.72). The Fourier spectrum (9.72*b)
contains, besides of ω p , the dependent frequencies ω 1 , ω 2 , ω 3 , ω 4 , ω 5 . The following resonance phenomena hold: ω p − ω 5 = ω 5 − ω 4 , ω 1 + ω 2 + ω 3 = ω p /4.
The wavelet spectrum (9.72*c), similar to the classical Fourier spectrum, exhibits
temporal behaviour of the frequencies ω 2 , ω 5 , beginning from the time instants
t ≈ 2050, 2150, respectively. In other words, before reaching t ≈ 2050, the vibrations are practically harmonic with the fundamental frequency ω p , since the power of
the frequencies ω 1 , ω 3 , ω 4 is extremely small. The Poincaré map consists of densely
distributed points, i.e. the trajectories are not divergent and the beam vibrations are
regular.
9.7.3.4 Scenarios of Transition into Chaos
Detection, construction and analysis of a scenario of transition from regular to
chaotic vibrations, and vice versa, are important and fundamental tasks when investigating nonlinear dynamics of continuous mechanical systems. These transitions
can be based on the classical Feigenbaum, Pomeau-Manneville, and Ruelle-TakensNewhouse scenarios. There are papers reporting scenarios of the transition for structural members, including beams, plates, and shells, by using the Bernoulli-Euler,
Timoshenko and Sheremetev-Pelekh-Reddy models without [134–136] and taking
[121, 137] into account of the size-dependent behaviour. Furthermore, scenarios of
transition from regular to chaotic dynamics of nonlinear non-homogeneous beams
have been detected and studied with respect to their thickness for different values of
the size-dependent parameter [138].
We begin the investigations with a construction and analysis of scenarios for
two temperature values (θ =100, θ = −100), for two cases, i.e. taking into account
the size-dependent behaviour γ 2 = 0.3 or not γ 2 = 0, for a non-homogeneous
(optimized) beam. The following parameters were fixed: ω p = 8.5, ε = 1, γ 1 =
40, ν = 0.3. All detected scenarios have a similar qualitative picture, i.e. a transition into chaotic vibrations which follows the Pomeau-Manneville scenario [139],
where load plays a role of the control parameter.
Chaotic vibrations are achieved after the third cycle of the process “periodic
vibrations-chaotic vibrations”. Each cycle is realized through different scenarios.
Let us consider the scenarios for the non-homogeneous (optimized) beam for the
temperature θ = −100, γ 2 = 0. The following dynamic characteristics are reported
in Tables 9.11, 9.12 and 9.13: (a) signal w(0.5; t); (b) Fourier spectrum S(ω) based
on the FFT; (c) 2D wavelet spectrum based on the Morlet mother wavelet; (d) phase
portrait ˙
w [w(t)]; (e) Poincaré section w t+T [w t ]; (f) LLEs computed by four methods.
The first cycle consist of three frequencies exhibiting the following resonance
behaviour ω p − ω 3 = ω 3 − ω 2 = ω 2 − ω 1 . An increase in the amplitude of load