170
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
are studied. It is found that the Bernoulli–Euler model is more suited for more length
beams, whereas the Timoshenko model is preferred for the more short beams. The
second work is devoted to the analysis of the natural frequencies of nonlinear free
transversal vibrations of the Bernoulli–Euler nanobeam.
Investigation on vibrations of nonlinear nanobeam based on the Bernoulli–Euler
model in the framework of the gradient theory is carried out by Rajabi and Ramezani
[29] and Kong et al. [62]. The governing PDEs are obtained with the help of the
first-order Galerkin method, and then they are reduced to ODEs. It is shown that
the geometric nonlinearity plays a key role in increase of the beam natural vibration
frequency for the case of relatively thick beams. However, when the beam thickness
becomes compatible with the length scale material parameter, then a key role is
played by the size-dependent behaviour.
Ke et al. [54] and Ke and Wang [63] reported results for the mathematical models
constructed on a basis of the modified couple stress theory of elasticity. They investigated dynamical stability of linear Timoshenko microbeam and nonlinear beam free
vibrations. The obtained results show that the size-dependent effect for the characteristics of the dynamic stability plays a crucial influence only if the beam thickness
is comparable with the scale parameter of the material length.
Surface theory of elasticity is employed to construct the mathematical models
of nanobeam vibrations in [64]. There were considered transversal and longitudinal vibrations of two nanobeams within framework of the Euler–Bernoulli model
in linear statement. The mentioned theory is adopted to the case of nanoporous
Timoshenko beam [65]. The obtained results indicate that the critical value of stability loss of nanoporous materials depends on the characteristic dimensions of the
microstructures.
Besides the already mentioned works, dynamical problems of nanobeams were
considered in [24, 54, 66–73].
The so far carried out overview of the state of the art points out that in spite of the
large number of publications devoted to study of beams and plates in the framework
of various models and different theories aimed at inclusion of the size-dependent
parameters, there is a gap in reason of focused on chaotic dynamics of nanobeams.
In this section, we consider numerous problems of nonlinear dynamics of homogeneous straight-linear nanobeams in the framework of the modified couple theory
of elasticity for the mathematical beam models of Bernoulli–Euler, Timoshenko and
Sheremetev–Pelekh with an account of geometric nonlinearity. The comparison of
the dynamic characteristics (Fourier and wavelet spectra, phase portraits, Poincaré
maps, LLEs) for the given physical-geometric parameters with and without the sizedependent behaviour is carried out [74]. The charts of the character of vibrations are
constructed and studied. Moreover, scenarios of transition from regular to chaotic
vibrations are illustrated and discussed.
Derivation of the governing equations and boundary conditions is reported in
Sect. 6.6. We begin with investigation aimed at the determination of convergence of
the numerical algorithms and reliability of the obtained results. For this purpose,
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
are studied. It is found that the Bernoulli–Euler model is more suited for more length
beams, whereas the Timoshenko model is preferred for the more short beams. The
second work is devoted to the analysis of the natural frequencies of nonlinear free
transversal vibrations of the Bernoulli–Euler nanobeam.
Investigation on vibrations of nonlinear nanobeam based on the Bernoulli–Euler
model in the framework of the gradient theory is carried out by Rajabi and Ramezani
[29] and Kong et al. [62]. The governing PDEs are obtained with the help of the
first-order Galerkin method, and then they are reduced to ODEs. It is shown that
the geometric nonlinearity plays a key role in increase of the beam natural vibration
frequency for the case of relatively thick beams. However, when the beam thickness
becomes compatible with the length scale material parameter, then a key role is
played by the size-dependent behaviour.
Ke et al. [54] and Ke and Wang [63] reported results for the mathematical models
constructed on a basis of the modified couple stress theory of elasticity. They investigated dynamical stability of linear Timoshenko microbeam and nonlinear beam free
vibrations. The obtained results show that the size-dependent effect for the characteristics of the dynamic stability plays a crucial influence only if the beam thickness
is comparable with the scale parameter of the material length.
Surface theory of elasticity is employed to construct the mathematical models
of nanobeam vibrations in [64]. There were considered transversal and longitudinal vibrations of two nanobeams within framework of the Euler–Bernoulli model
in linear statement. The mentioned theory is adopted to the case of nanoporous
Timoshenko beam [65]. The obtained results indicate that the critical value of stability loss of nanoporous materials depends on the characteristic dimensions of the
microstructures.
Besides the already mentioned works, dynamical problems of nanobeams were
considered in [24, 54, 66–73].
The so far carried out overview of the state of the art points out that in spite of the
large number of publications devoted to study of beams and plates in the framework
of various models and different theories aimed at inclusion of the size-dependent
parameters, there is a gap in reason of focused on chaotic dynamics of nanobeams.
In this section, we consider numerous problems of nonlinear dynamics of homogeneous straight-linear nanobeams in the framework of the modified couple theory
of elasticity for the mathematical beam models of Bernoulli–Euler, Timoshenko and
Sheremetev–Pelekh with an account of geometric nonlinearity. The comparison of
the dynamic characteristics (Fourier and wavelet spectra, phase portraits, Poincaré
maps, LLEs) for the given physical-geometric parameters with and without the sizedependent behaviour is carried out [74]. The charts of the character of vibrations are
constructed and studied. Moreover, scenarios of transition from regular to chaotic
vibrations are illustrated and discussed.
Derivation of the governing equations and boundary conditions is reported in
Sect. 6.6. We begin with investigation aimed at the determination of convergence of
the numerical algorithms and reliability of the obtained results. For this purpose,