Chapter 1
Nanostructural Members in Various
Fields: A Literature Review
1.1 Introduction
An overview of studies on the dynamics of nanoshells, nanoplates and nanobeams in
natural/thermal/electric/magnetic fields, obtained on the basis of some of the abovelisted theories of higher order, is given. The review of existing papers and monographs
shows that the problem of analysing the statics and dynamics of MEMS/NEMS
devices is only at the initial stage of development. Namely, most of the research
is limited to the analysis of Duffing-type equations, which are obtained using the
Bubnov-Galerkin variational method in the first approximation, thus searching for
solutions of strongly reduced order models with one degree of freedom. At present,
when studying the problems of statics and dynamics of plates and shells, kinematic
models of the first and second approximations are mainly employed. Moreover,
the majority of the approaches are based on a linear formulation and deal with a
small number of degrees of freedom. Often, in the problems under consideration, the
solution is not entirely reliable. Furthermore, since it is obtained by a limited number
of methods, their convergence is not shown or proved. Thus, the nonlinear dynamics
of nanostructures has not been sufficiently investigated, and no reliable scenarios
of the transition from periodic to chaotic vibrations have been revealed. The use of
such a tool for studying nonlinear dynamics as wavelet analysis and the spectrum of
Lyapunov exponents is at the initial stage of its development. Furthermore, chaotic
vibrations, hyperchaotic vibrations, hyper-hyperchaotic and other types of vibrations
are not sufficiently studied. The critical overview of the latest nonlinear dynamics of
micro/nanostructures presented in this chapter demonstrates the need for innovative
theoretical and numerical tools to achieve reliable and proven results.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_1
1
Nanostructural Members in Various
Fields: A Literature Review
1.1 Introduction
An overview of studies on the dynamics of nanoshells, nanoplates and nanobeams in
natural/thermal/electric/magnetic fields, obtained on the basis of some of the abovelisted theories of higher order, is given. The review of existing papers and monographs
shows that the problem of analysing the statics and dynamics of MEMS/NEMS
devices is only at the initial stage of development. Namely, most of the research
is limited to the analysis of Duffing-type equations, which are obtained using the
Bubnov-Galerkin variational method in the first approximation, thus searching for
solutions of strongly reduced order models with one degree of freedom. At present,
when studying the problems of statics and dynamics of plates and shells, kinematic
models of the first and second approximations are mainly employed. Moreover,
the majority of the approaches are based on a linear formulation and deal with a
small number of degrees of freedom. Often, in the problems under consideration, the
solution is not entirely reliable. Furthermore, since it is obtained by a limited number
of methods, their convergence is not shown or proved. Thus, the nonlinear dynamics
of nanostructures has not been sufficiently investigated, and no reliable scenarios
of the transition from periodic to chaotic vibrations have been revealed. The use of
such a tool for studying nonlinear dynamics as wavelet analysis and the spectrum of
Lyapunov exponents is at the initial stage of its development. Furthermore, chaotic
vibrations, hyperchaotic vibrations, hyper-hyperchaotic and other types of vibrations
are not sufficiently studied. The critical overview of the latest nonlinear dynamics of
micro/nanostructures presented in this chapter demonstrates the need for innovative
theoretical and numerical tools to achieve reliable and proven results.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_1
1