Chemistry Reference
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using accurate theoretical model to acquire natural frequencies and mode
shapes, the elastic modulus of CNTs can be calculated indirectly.
Timoshenko's beam model used by Wang et al. [60] for free vibrations
of MWCNTs study; it was shown that the frequencies are significantly
over predicted by the Euler's beam theory when the aspect ratios are small
and when considering high vibration modes. They indicated that the Ti-
moshenko's beam model should be used for a better prediction of the fre-
quencies especially when small aspect ratio and high vibration modes are
considered.
Hu et al. [55] presented a review of recent studies on continuum mod-
els and MD simulations of CNT's vibrations briefly [61]. Three construct-
ed model of SWCNT consisting of Timoshenko's beam, Euler-Bernoulli's
beam and MD simulations are investigated and results show that funda-
mental frequency decreases as the length of a SWCNT increases and also
the Timoshenko's beam model provides a better prediction of short CNT's
frequencies than that of Euler-Bernoulli beam's model. Comparing the
fundamental frequency results of transverse vibrations of cantilevered
SWCNTs it can be seen that both beam models are not able to predict the
fundamental frequency of cantilevered SWCNTs shorter than 3.5 nm.
An atomistic modeling technique and molecular structural mechanics
are used by Li and Chou [62] to calculated fundamental natural frequency
of SWCNTs. A free and forced vibrations of SWCNT have been assessed
by Arghavan and Singh [63] using space frame elements with extensional,
bending and torsional stiffness properties to modeling SWCNTs and com-
pared their results with reported results by of Sakhaee pour [64] and Li and
Chou [62]. Their results were in close agreement with two other results,
within three to five percent.
Furthermore, free vibrations of SWCNTs have been studied by Gup-
ta et al. [65] using the MM3 potential. They mentioned that calculations
based on modeling SWCNTs as a beam will overestimate fundamental fre-
quencies of the SWCNT. They derived the thickness of a SWCNT/shell to
compare the frequencies of a SWCNT obtained by MM simulations with
that of a shell model and provided an expression for the wall thickness in
terms of the tube radius and the bond length in the initial relaxed configu-
ration of a SWCNT. Aydogdu [66] showed that axial vibration frequencies
of SWCNT embedded in an elastic medium highly over estimated by the
classical rod model because of ignoring the effect of small length scale and
