1. Results reported in Figs. 8.4, 8.6 and Figs. 8.5, 8.7 with size-dependent behaviour and in the case of linear vibrations exhibit increase of the eigenfrequencies
and increase of the quality factor of the beam resonator.
2. The occurrence of nonlinearity in both tested boundary (simple-simple) and
(clamped-clamped) conditions implies a significant increase in the frequency
of the fundamental vibration mode as well as the quality factor of the resonator
(Figs. 8.8–8.11).
3. The analysis of the obtained results shows that in the case of the thermoelastic
damping, it is necessary to take into account the nonlinear behaviour as well as
size-dependent effects of the microbeams. Both mentioned features are crucial
for the quality factor of the beam resonators.
In Chap. 9, problems associated with multifunctional requirements with respect
to effective characteristics of composites consisting of two components as well as
composites with holes or technological inclusions have been studied. In the process
of investigation, a strong dependence of the optimal topology of the distribution of
materials in the microstructure of composites on the form of the target functions has
been detected. The study of transformations of the optimal topology of the composite microstructure with a change in the weight coefficient from x ¼ 0 (maximization of the heat transfer) up to x ¼ 1 (maximization of the mechanical moduli)
has been conducted. Moreover, a set of alternatives optimal in the Pareto sense has
been constructed. The considered examples clearly indicate the inability to achieve
the best/required properties simultaneously in both cases, which is caused by
conflicting criteria in the target function. In addition, nonlinear dynamics of the
size-dependent Euler-Bernoulli beams embedded into temperature field with
topologically optimized microstructure is studied. The following new results are
presented:
(i) We have developed a mathematical model of the size-dependent nonlinear
beam, taking into account the topological optimization under the criterion of
maximum stiffness. The mathematical model is based on the
Bernoulli-Euler, von Kármán and Duhamel-Newmann hypotheses. Also, an
algorithm and a computer program for numerical computations of the
optimized beam microstructure for the given boundary conditions, the form
of external load and the temperature have been developed (both static and
dynamic problems have been considered).
(ii) Reliability of the results was confirmed by investigating the convergence
along the spatial variable as well as by examining the solution to the Cauchy
problems, and investigating Lyapunov exponents and time evolution of the
frequency obtained using wavelet analysis.
(iii) The analysis of the reliability of the results obtained for different numbers of
spatial partitions was carried out based on the analysis of the power spectrum for chaotic system states. The reliability of chaos was validated by
computing LLEs using four different methods.
xiv
Introduction
and increase of the quality factor of the beam resonator.
2. The occurrence of nonlinearity in both tested boundary (simple-simple) and
(clamped-clamped) conditions implies a significant increase in the frequency
of the fundamental vibration mode as well as the quality factor of the resonator
(Figs. 8.8–8.11).
3. The analysis of the obtained results shows that in the case of the thermoelastic
damping, it is necessary to take into account the nonlinear behaviour as well as
size-dependent effects of the microbeams. Both mentioned features are crucial
for the quality factor of the beam resonators.
In Chap. 9, problems associated with multifunctional requirements with respect
to effective characteristics of composites consisting of two components as well as
composites with holes or technological inclusions have been studied. In the process
of investigation, a strong dependence of the optimal topology of the distribution of
materials in the microstructure of composites on the form of the target functions has
been detected. The study of transformations of the optimal topology of the composite microstructure with a change in the weight coefficient from x ¼ 0 (maximization of the heat transfer) up to x ¼ 1 (maximization of the mechanical moduli)
has been conducted. Moreover, a set of alternatives optimal in the Pareto sense has
been constructed. The considered examples clearly indicate the inability to achieve
the best/required properties simultaneously in both cases, which is caused by
conflicting criteria in the target function. In addition, nonlinear dynamics of the
size-dependent Euler-Bernoulli beams embedded into temperature field with
topologically optimized microstructure is studied. The following new results are
presented:
(i) We have developed a mathematical model of the size-dependent nonlinear
beam, taking into account the topological optimization under the criterion of
maximum stiffness. The mathematical model is based on the
Bernoulli-Euler, von Kármán and Duhamel-Newmann hypotheses. Also, an
algorithm and a computer program for numerical computations of the
optimized beam microstructure for the given boundary conditions, the form
of external load and the temperature have been developed (both static and
dynamic problems have been considered).
(ii) Reliability of the results was confirmed by investigating the convergence
along the spatial variable as well as by examining the solution to the Cauchy
problems, and investigating Lyapunov exponents and time evolution of the
frequency obtained using wavelet analysis.
(iii) The analysis of the reliability of the results obtained for different numbers of
spatial partitions was carried out based on the analysis of the power spectrum for chaotic system states. The reliability of chaos was validated by
computing LLEs using four different methods.
xiv
Introduction