376
9 Topologic Optimization of Vibrations of Size-Dependent Beams
The nonlinear forced vibrations of a microbeam have been investigated in [120],
employing the strain gradient elasticity theory. The geometrically nonlinear equation
of motion, the microbeam, taking into account the size effect, has been obtained
employing a variational approach. Results of the frequency response of the system
have been studied.
Arbind and Reddy [42] have considered functionally graded microstructuredependent beams made from a material non-homogeneously distributed along the
beam thickness, taking into account the von Kármán nonlinearity. The modified
couple stress theory has been used and the counterpart governing equations for the
Euler-Bernoulli and the Timoshenko beams have been derived. In particular, the
influence of the scale length parameter, the law of the beam properties change along
with the thickness, shear deformations, and geometric nonlinearity on the static beam
deflections have been studied. Arbind et al. [43] have derived nonlinear PDEs of the
size-dependent Sheremetev-Pelekh-Reddy beam made from NM by using Hamilton’s principle.
In the present work, we are aimed at carrying out a comparative analysis of the
static and dynamic behaviour of size-dependent beams [121, 122]. An investigation
of nonlinear dynamics of micro/nanobeams, plates, and shells has been carried out in
references [103, 123, 124]. The studies, which can be found in the available literature
devoted to the microstructure of beams, have been conducted without using the optimization methods. Furthermore, the overviewed references did not offer a rigorous
analysis of the reliability of results. Also, there is a lack of works devoted to studying the chaotic dynamics of size-dependent beams made from NM in the existing
literature, which stands for the motivation of the present study. In this work, a mathematical model of a functionally graded size-dependent nonlinear Euler-Bernoulli
beam is derived, the algorithms and programs for construction of a microstructure
beam (working in the given conditions), and then investigation of its static/dynamic
features, are developed.
At the first stage of our analysis, for the given conditions of the beam loading,
boundary conditions and the temperature field, the topological optimization was
introduced following the criterion of keeping the maximum stiffness. In all case
studies, the achieved optimal microstructure is original and not obtained by others.
As a result of the carried out optimization, the optimal microstructure beam exhibits
functional grading in two directions, i.e. along its length and thickness.
The second stage of our study was aimed at analyzing static and dynamic beam
behaviour. The mathematical model is constructed based on the Euler-Bernoulli
hypotheses, the modified couple stress theory and the von Kármán geometric nonlinearity. The influence of the temperature field on the model follows the classical
Duhamel-Neumann relations.
The carried out comparative analysis of the statics and nonlinear dynamics of
homogeneous and non-homogeneous (functionally and optimally graded) beams
yielded the main observation that the stress-strain states and magnitudes of the fundamental frequencies of vibrations of these structures that essentially depend on the
used temperature.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
The nonlinear forced vibrations of a microbeam have been investigated in [120],
employing the strain gradient elasticity theory. The geometrically nonlinear equation
of motion, the microbeam, taking into account the size effect, has been obtained
employing a variational approach. Results of the frequency response of the system
have been studied.
Arbind and Reddy [42] have considered functionally graded microstructuredependent beams made from a material non-homogeneously distributed along the
beam thickness, taking into account the von Kármán nonlinearity. The modified
couple stress theory has been used and the counterpart governing equations for the
Euler-Bernoulli and the Timoshenko beams have been derived. In particular, the
influence of the scale length parameter, the law of the beam properties change along
with the thickness, shear deformations, and geometric nonlinearity on the static beam
deflections have been studied. Arbind et al. [43] have derived nonlinear PDEs of the
size-dependent Sheremetev-Pelekh-Reddy beam made from NM by using Hamilton’s principle.
In the present work, we are aimed at carrying out a comparative analysis of the
static and dynamic behaviour of size-dependent beams [121, 122]. An investigation
of nonlinear dynamics of micro/nanobeams, plates, and shells has been carried out in
references [103, 123, 124]. The studies, which can be found in the available literature
devoted to the microstructure of beams, have been conducted without using the optimization methods. Furthermore, the overviewed references did not offer a rigorous
analysis of the reliability of results. Also, there is a lack of works devoted to studying the chaotic dynamics of size-dependent beams made from NM in the existing
literature, which stands for the motivation of the present study. In this work, a mathematical model of a functionally graded size-dependent nonlinear Euler-Bernoulli
beam is derived, the algorithms and programs for construction of a microstructure
beam (working in the given conditions), and then investigation of its static/dynamic
features, are developed.
At the first stage of our analysis, for the given conditions of the beam loading,
boundary conditions and the temperature field, the topological optimization was
introduced following the criterion of keeping the maximum stiffness. In all case
studies, the achieved optimal microstructure is original and not obtained by others.
As a result of the carried out optimization, the optimal microstructure beam exhibits
functional grading in two directions, i.e. along its length and thickness.
The second stage of our study was aimed at analyzing static and dynamic beam
behaviour. The mathematical model is constructed based on the Euler-Bernoulli
hypotheses, the modified couple stress theory and the von Kármán geometric nonlinearity. The influence of the temperature field on the model follows the classical
Duhamel-Neumann relations.
The carried out comparative analysis of the statics and nonlinear dynamics of
homogeneous and non-homogeneous (functionally and optimally graded) beams
yielded the main observation that the stress-strain states and magnitudes of the fundamental frequencies of vibrations of these structures that essentially depend on the
used temperature.