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developed an elastic rod model based on local and nonlocal rod theories
to investigate the small scale effect on the axial vibrations of SWCNTs.
The vibration properties of two and three functioned of carbon nano-
tubes considering different boundary conditions and geometries were stud-
ied by Fakhrabadi et al. [67]. The results show that the tighter boundaries,
larger diameters and shorter lengths lead to higher natural frequencies.
Lee and Lee [68] fulfilled modal analysis of SWCNT's and nano cones
(SWCNCs) using finite element method with ANSYS commercial pack-
age. The vibration behaviors of SWCNT with fixed beam and cantilever
boundary conditions with different cross-section types consisting of circle
and ellipse were constructed using 3D elastic beams and point masses. The
nonlinear vibration of an embedded CNT's were studied by Fu et al.[69]
and the results reveal that the nonlinear free vibration of nano-tubes is ef-
fected significantly by surrounding elastic medium.
The same investigation has been accomplished by Ansari and Hem-
matnezhad [70] using the variational iteration method (VIM). Wang et al.
[71]have studied axis symmetric vibrations of SWCNT immerged in water
which in contrast to solid liquid system, a submerged SWCNT is coupled
with surrounding water via vdW interaction. The analysis of DWCNTs vi-
bration characteristics considering simply support boundary condition are
carried out by Natsuki et al. [72] based on Euler-Bernoulli's beam theory.
Subsequently, it was found that the vibration modes of DWCNTs are non-
coaxial intertube vibrations and deflection of inner and outer nanotube can
come about in same or opposite deflections.
More-over the vibration analysis of MWCNTs were implemented by
Aydogdu [73] using generalized shear deformation beam theory (GSD-
BT). Parabolic shear deformation theory (PSDT) was used in the specific
solutions and the results showed remarkable difference between PSDT
and Euler beam theory and also the importance of vdW force presence
for small inner radius. Lei et al. [74] have presented a theoretical vibra-
tion analysis of the radial breathing mode (RBM) of DWCNTs subjected
to pressure based on elastic continuum model. It was shown that the fre-
quency of RBM increases perspicuously as the pressure increases under
different conditions.
The influences of shear deformation, boundary conditions and vdW co-
efficient on the transverse vibration of MWCNT were studied by Ambro-
sini and Borbón [75, 76]. The results reveal that the noncoaxial intertube
frequencies are independent of the shear deformation and the boundary
