334
9 Topologic Optimization of Vibrations of Size-Dependent Beams
(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.
(iv) The Cauchy problem was solved by numerous methods and the fourth-order
Runge-Kutta method was chosen as the most efficient. The optimal step was
chosen using the Runge principle.
(v) Based on the tests carried out for numerous wavelets (Daubechies, Gauss,
Haar and Morlet), the Morlet wavelets were chosen as the most feasible for
our problem.
(vi) The static analysis was carried out for three values of temperature and two
values of size-dependent parameter. The investigated ‘frequency-deflection’
dependency exhibits different results for homogeneous and non-homogeneous
(optimized) beams for all values of the length-dependent parameter.
(vii) The use of beams with the optimized microstructure allows for an increase
in the range of working loads regimes compared to homogeneous beams for
which the vibration regimes are either periodic or quasi-periodic.
(viii) The analysis of the scenarios of transition from periodic to chaotic vibrations
was carried out. In all cases, the transition into chaotic vibrations followed a
scenario similar to the classical Pomeau-Manneville scenario (a few exceptions were observed for some values of temperature and the length-dependent
parameter).
9.2 Literature Review
Functionally graded materials (FGM) are fabricated based on the multi-phase composites with volume part phases and with mechanical/thermomechanical properties
being changed along the chosen directions in order to achieve engineering required
characteristics. Those new fabricated materials possess the efficient strength characteristics without harmful concentration of stresses of some of their parts. Owing
to the mentioned properties, the beams made from FGM found wide application in
numerous engineering constructions including wings of gas and wind turbines, rotor
blades of helicopters, propeller blades as well as stator blades, feathering paddles,
wicket gates and others (see for instance [1]).
A change of the material property along a given direction can be incorporated by
either stepwise or exponential rules, which are widely used for the description of the
variable FGM properties, towards the material thickness, its longitudinal direction
or towards both directions simultaneously.
The classical (standard) way to change the elastic properties of a material element,
i.e. its microstructure, is introduced by a design engineer (constructor). The majority
of investigations devoted to FGM beams’ design are focused on tailoring the beams’
dynamical properties along their thickness.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
(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.
(iv) The Cauchy problem was solved by numerous methods and the fourth-order
Runge-Kutta method was chosen as the most efficient. The optimal step was
chosen using the Runge principle.
(v) Based on the tests carried out for numerous wavelets (Daubechies, Gauss,
Haar and Morlet), the Morlet wavelets were chosen as the most feasible for
our problem.
(vi) The static analysis was carried out for three values of temperature and two
values of size-dependent parameter. The investigated ‘frequency-deflection’
dependency exhibits different results for homogeneous and non-homogeneous
(optimized) beams for all values of the length-dependent parameter.
(vii) The use of beams with the optimized microstructure allows for an increase
in the range of working loads regimes compared to homogeneous beams for
which the vibration regimes are either periodic or quasi-periodic.
(viii) The analysis of the scenarios of transition from periodic to chaotic vibrations
was carried out. In all cases, the transition into chaotic vibrations followed a
scenario similar to the classical Pomeau-Manneville scenario (a few exceptions were observed for some values of temperature and the length-dependent
parameter).
9.2 Literature Review
Functionally graded materials (FGM) are fabricated based on the multi-phase composites with volume part phases and with mechanical/thermomechanical properties
being changed along the chosen directions in order to achieve engineering required
characteristics. Those new fabricated materials possess the efficient strength characteristics without harmful concentration of stresses of some of their parts. Owing
to the mentioned properties, the beams made from FGM found wide application in
numerous engineering constructions including wings of gas and wind turbines, rotor
blades of helicopters, propeller blades as well as stator blades, feathering paddles,
wicket gates and others (see for instance [1]).
A change of the material property along a given direction can be incorporated by
either stepwise or exponential rules, which are widely used for the description of the
variable FGM properties, towards the material thickness, its longitudinal direction
or towards both directions simultaneously.
The classical (standard) way to change the elastic properties of a material element,
i.e. its microstructure, is introduced by a design engineer (constructor). The majority
of investigations devoted to FGM beams’ design are focused on tailoring the beams’
dynamical properties along their thickness.