6.5.2.1 Explicit solution for DWCNT

Considering a DWCNT, the system of Eq. (6.48) is reduced to two equations:

$\begin{array}{c}({\mathit{B}}_1-\mathit{\rho }h{\mathit{\omega }}^2){\mathit{W}}_1+{\mathit{c}}_{12}{\mathit{W}}_2=0,\\ {\mathit{c}}_{21}{\mathit{W}}_1+({\mathit{B}}_2-\mathit{\rho }h{\mathit{\omega }}^2){\mathit{W}}_2=0,\end{array}$ (6.50)

(6.50)

and the natural frequencies of the DWCNT are obtained by solving

${(\mathit{\rho }h{\mathit{\omega }}^2)}^2-({\mathit{B}}_1+{\mathit{B}}_2)\mathit{\rho }h{\mathit{\omega }}^2+{\mathit{B}}_1{\mathit{B}}_2-{\mathit{c}}_{12}{\mathit{c}}_{21}=0$ (6.51)

(6.51)

Due to the fact that from the equilibrium distance, any increase (w_i−w_j≥0, i, j=1, 2) or decrease (w_i−w_j≤0) in the distance between the inner and outer tubes will cause an attractive or repulsive vdW force, respectively. Note that we assume the inward pressure to be positive. Thus, from Eq. (6.42) we have c₁₂≤0 and c₂₁≥0, and it is obvious that

${({\mathit{B}}_1-{\mathit{B}}_2)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}\ge 0$ (6.52)

(6.52)

and note that

$\begin{array}{ll}{\mathit{B}}_1{\mathit{B}}_2-{\mathit{c}}_{12}{\mathit{c}}_{21}=\hfill & \left\{\frac{\mathit{Eh}}{(1\mathit{-}{\mathit{\nu }}^2)}\frac{1}{{\mathit{R}}_1^2}+\mathit{D}{\left[{\left(\frac{m\mathit{\pi }}{\mathit{L}}\right)}^2+{\left(\frac{\mathit{n}}{{\mathit{R}}_1}\right)}^2\right]}^2\right\}\times \left\{\frac{\mathit{Eh}}{(1\mathit{-}{\mathit{\nu }}^2)}\frac{1}{{\mathit{R}}_2^2}+\mathit{D}{\left[{\left(\frac{m\mathit{\pi }}{\mathit{L}}\right)}^2+{\left(\frac{\mathit{n}}{{\mathit{R}}_2}\right)}^2\right]}^2\right\}\hfill \\ \hfill & -{\mathit{c}}_{21}\left\{\frac{\mathit{Eh}}{(1\mathit{-}{\mathit{\nu }}^2)}\frac{1}{{\mathit{R}}_1^2}+\mathit{D}{\left[{\left(\frac{m\mathit{\pi }}{\mathit{L}}\right)}^2+{\left(\frac{\mathit{n}}{{\mathit{R}}_1}\right)}^2\right]}^2\right\}\hfill \\ \hfill & -{\mathit{c}}_{12}\left\{\frac{\mathit{Eh}}{(1\mathit{-}{\mathit{\nu }}^2)}\frac{1}{{\mathit{R}}_2^2}+\mathit{D}{\left[{\left(\frac{m\mathit{\pi }}{\mathit{L}}\right)}^2+{\left(\frac{\mathit{n}}{{\mathit{R}}_2}\right)}^2\right]}^2\right\}\ge 0\hfill \end{array}$ (6.53)

(6.53)

Hence, one can easily arrive at

${({\mathit{B}}_1-{\mathit{B}}_2)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}={({\mathit{B}}_1\mathit{+}{\mathit{B}}_2)}^2-4({\mathit{B}}_1{\mathit{B}}_2-{\mathit{c}}_{12}{\mathit{c}}_{21})<{({\mathit{B}}_1\mathit{+}{\mathit{B}}_2)}^2$ (6.54)

(6.54)

Thus, the solution to Eq. (6.51) is given by

$\mathit{\rho }h{\mathit{\omega }}_1^2=\frac{1}{2}\left[{\mathit{B}}_1\mathit{+}{\mathit{B}}_2-\sqrt{{({\mathit{B}}_1-{\mathit{B}}_2)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}}\right]$ (6.55)

(6.55)

and

$\mathit{\rho }h{\mathit{\omega }}_2^2=\frac{1}{2}\left[{\mathit{B}}_1\mathit{+}{\mathit{B}}_2+\sqrt{{({\mathit{B}}_1-{\mathit{B}}_2)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}}\right]$ (6.56)

(6.56)

It is worth mentioning that when the innermost radius is very large, ${\mathit{c}}_{12}\approx {\mathit{c}}_{21}$ and ${\mathit{R}}_1\approx {\mathit{R}}_2$, and thus ${\mathit{B}}_1\approx {\mathit{B}}_2$. Hence, we have

$\mathit{\rho }h{\mathit{\omega }}_1^2\approx \frac{1}{2}\left\{\frac{\mathit{Eh}}{(1\mathit{-}{\mathit{\nu }}^2)}\frac{1}{{\mathit{R}}_1^2}+\mathit{D}{\left[{\left(\frac{m\mathit{\pi }}{\mathit{L}}\right)}^2+{\left(\frac{\mathit{n}}{{\mathit{R}}_1}\right)}^2\right]}^2+\frac{\mathit{Eh}}{(1\mathit{-}{\mathit{\nu }}^2)}\frac{1}{{\mathit{R}}_2^2}+\mathit{D}{\left[{\left(\frac{m\mathit{\pi }}{\mathit{L}}\right)}^2+{\left(\frac{\mathit{n}}{{\mathit{R}}_2}\right)}^2\right]}^2\right\}$ (6.57)

(6.57)

Eq. (6.57) indicates that the effect of the vdW interaction on the lower frequency is very small and then can be neglected when the innermost radius is very large.

The ratios of the amplitudes of the vibration modes associated with ${\mathit{\omega }}_1$ and ${\mathit{\omega }}_2$ can be obtained by substituting Eqs. (6.55) and (6.56) into one of Eq. (6.50) and are expressed, respectively, as

${\mathit{R}}_{2,1}={\frac{{\mathit{W}}_2}{{\mathit{W}}_1}|}_{\mathit{\omega }={\mathit{\omega }}_1}=\frac{1}{2{\mathit{c}}_{12}}\left[{\mathit{B}}_2-{\mathit{B}}_1-\sqrt{{({\mathit{B}}_2-{\mathit{B}}_1)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}}\right]$ (6.58)

(6.58)

${\mathit{R}}_{2,2}={\frac{{\mathit{W}}_2}{{\mathit{W}}_1}|}_{\mathit{\omega }={\mathit{\omega }}_2}=\frac{1}{2{\mathit{c}}_{12}}\left[{\mathit{B}}_2-{\mathit{B}}_1+\sqrt{{({\mathit{B}}_2-{\mathit{B}}_1)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}}\right]$ (6.59)

(6.59)

for the second mode. Note that $\sqrt{{({\mathit{B}}_2-{\mathit{B}}_1)}^2+4{\mathit{c}}_{12}{\mathit{c}}_{21}}>{\mathit{B}}_2-{\mathit{B}}_1$ and ${\mathit{c}}_{12}<0$, that means the amplitude ratio associated with ${\mathit{\omega }}_1$ is ${\mathit{R}}_{2,1}={\mathit{W}}_2/{\mathit{W}}_1>0$ and the amplitude ratio associated with ${\mathit{\omega }}_2$ is ${\mathit{R}}_{2,2}={\mathit{W}}_2/{\mathit{W}}_1<0$. Thus, for the vibration associated with ${\mathit{\omega }}_1$, both layers are always vibrating in the same direction. In contrast, for the vibration associated with ${\mathit{\omega }}_2$, both layers are vibrating in the opposite direction always.

6.5.2.2 Explicit solution for TWCNT

Considering a TWCNT, the system of Eq. (6.48) is reduced to the following three equations:

$\begin{array}{c}({\mathit{B}}_1-\mathit{\rho }h{\mathit{\omega }}^2){\mathit{W}}_1+{\mathit{c}}_{12}{\mathit{W}}_2+{\mathit{c}}_{13}{\mathit{W}}_3=0,\\ {\mathit{c}}_{21}{\mathit{W}}_1+({\mathit{B}}_2-\mathit{\rho }h{\mathit{\omega }}^2){\mathit{W}}_2+{\mathit{c}}_{23}{\mathit{W}}_3=0,\\ {\mathit{c}}_{31}{\mathit{W}}_1+{\mathit{c}}_{32}{\mathit{W}}_2+({\mathit{B}}_3-\mathit{\rho }h{\mathit{\omega }}^2){\mathit{W}}_3=0,\end{array}$ (6.60)

(6.60)

Using the condition for the nonzero solutions of W₁–W₃, one obtains the following determinant equation for the natural frequencies of a TWCNT:

${(\mathit{\rho }h{\mathit{\omega }}^2)}^3+{\mathit{a}}_1{(\mathit{\rho }h{\mathit{\omega }}^2)}^2+{\mathit{a}}_2(\mathit{\rho }h{\mathit{\omega }}^2)+{\mathit{a}}_3=0,$ (6.61)

(6.61)

where

$\begin{array}{l}{\mathit{a}}_1=-{\mathit{B}}_1-{\mathit{B}}_2-{\mathit{B}}_3,\hfill \\ {\mathit{a}}_2={\mathit{B}}_1{\mathit{B}}_2+{\mathit{B}}_1{\mathit{B}}_3+{\mathit{B}}_2{\mathit{B}}_3-{\mathit{c}}_{31}{\mathit{c}}_{13}-{\mathit{c}}_{21}{\mathit{c}}_{12}-{\mathit{c}}_{23}{\mathit{c}}_{32},\hfill \\ {\mathit{a}}_3=-{\mathit{B}}_1{\mathit{B}}_2{\mathit{B}}_3-{\mathit{c}}_{21}{\mathit{c}}_{32}{\mathit{c}}_{13}-{\mathit{c}}_{31}{\mathit{c}}_{12}{\mathit{c}}_{23}+{\mathit{c}}_{13}{\mathit{c}}_{31}{\mathit{B}}_2+{\mathit{c}}_{21}{\mathit{c}}_{12}{\mathit{B}}_3+{\mathit{c}}_{23}{\mathit{c}}_{32}{\mathit{B}}_1\hfill \end{array}$ (6.62)

(6.62)

The solutions to Eq. (6.61) are

$\begin{array}{l}\mathit{\rho }h{\mathit{\omega }}_1^2=-\frac{2}{3}\sqrt{{\mathit{a}}_1^2-3{\mathit{a}}_2}\cos \frac{\mathit{\alpha }}{3}-\frac{{\mathit{a}}_1}{3},\hfill \\ \mathit{\rho }h{\mathit{\omega }}_2^2=-\frac{2}{3}\sqrt{{\mathit{a}}_1^2-3{\mathit{a}}_2}\cos \frac{\mathit{\alpha }+4\mathit{\pi }}{3}-\frac{{\mathit{a}}_1}{3},\hfill \\ \mathit{\rho }h{\mathit{\omega }}_3^2=-\frac{2}{3}\sqrt{{\mathit{a}}_1^2-3{\mathit{a}}_2}\cos \frac{\mathit{\alpha }+2\mathit{\pi }}{3}-\frac{{\mathit{a}}_1}{3}\hfill \end{array}$ (6.63)

(6.63)

where

$\mathit{\alpha }={\cos }^{-1}\frac{27{\mathit{a}}_3+2{\mathit{a}}_1^3-9{\mathit{a}}_1{\mathit{a}}_2}{2\sqrt{{({\mathit{a}}_1^2-3{\mathit{a}}_2)}^3}}$ (6.64)

(6.64)

The associated amplitude ratios R_2,i and R_3,i of the TWCNT can be determined as

${\mathit{R}}_{2,\mathit{i}}={\frac{{\mathit{W}}_2}{{\mathit{W}}_1}|}_{\mathit{\omega }={\mathit{\omega }}_{\mathit{i}}}=\frac{{\mathit{c}}_{23}({\mathit{B}}_1\mathit{-}\mathit{\rho }h{\mathit{\omega }}_{\mathit{i}}^2)\mathit{-}{\mathit{c}}_{21}{\mathit{c}}_{13}}{{\mathit{c}}_{13}({\mathit{B}}_2\mathit{-}\mathit{\rho }h{\mathit{\omega }}_{\mathit{i}}^2)-{\mathit{c}}_{12}{\mathit{c}}_{23}}$ (6.65)

(6.65)

and

${\mathit{R}}_{3,\mathit{i}}={\frac{{\mathit{W}}_3}{{\mathit{W}}_1}|}_{\mathit{\omega }={\mathit{\omega }}_{\mathit{i}}}=-\frac{({\mathit{B}}_1\mathit{-}\mathit{\rho }h{\mathit{\omega }}_{\mathit{i}}^2)({\mathit{B}}_2\mathit{-}\mathit{\rho }h{\mathit{\omega }}_{\mathit{i}}^2)\mathit{-}{\mathit{c}}_{12}{\mathit{c}}_{21}}{{\mathit{c}}_{13}({\mathit{B}}_2\mathit{-}\mathit{\rho }h{\mathit{\omega }}_{\mathit{i}}^2)-{\mathit{c}}_{12}{\mathit{c}}_{23}}$ (6.66)

(6.66)

In the examples MWCNTs with an innermost radius R_I and an outermost radius R_O are analyzed. Each tube has the same length L and is modeled as an individual cylindrical shell. No internal or external lateral pressures are considered for the vibration analysis of the MWCNTs. Following most published papers on CNT, the initial interlayer separation between two adjacent layers is taken as 0.34 nm. The parameters used to calculate the vdW interaction coefficient c_ij (Ref. [22]) are taken as ε=2.968 meV and σ=3.407 Å [41]. For all examples, the bending and axial stiffnesses of MWCNTs are D=0.85 eV and Eh=360 J/m² [3], respectively, and the length to the outermost radius ratio is L/R_O=10.

Let us first consider a 10-walled MWCNT with innermost radius R_I=5 nm. Based on the proposed vdW interaction model, the natural frequencies ${\mathit{f}}_{\mathit{i}}={\mathit{\omega }}_{\mathit{i}}/2\mathit{\pi }$ which are associated with the radial mode for a 10-walled MWCNT have been calculated by using Eq. (6.48) and are presented in Table 6.3 for various combinations of the wave numbers (m, n). It can be seen from Eq. (6.48) that there are 10 equations that give 10 different natural frequencies f₁≤f₂≤… f_i …≤f₉≤f₁₀ for every combination of m and n. Table 6.3 shows that the natural frequencies of the 10-layered MWCNT are very high, in the order of terahertz. As the mode order increases, the lower natural frequencies such as f₁, f₂, and f₃ increase significantly. However, all the other higher natural frequencies, from f₄ to f₁₀, are insensitive to the mode order, and thus the effect of the mode order on the higher natural frequencies can be neglected. It should be noted that the natural frequencies for the breathing mode with n=0 are almost the same for various m. This is because the first and third terms in Eq. (6.49) are around 10⁷ times larger than the second term which is related to the wave number m, and thus the influence of the axial half wave number on the natural frequencies is negligible.

In Table 6.4 a comparison is made between the results for the radial modes of a five-walled CNT with three inner radii using the present multilayer interaction of vdW forces and the results which are obtained using the single neighboring layer on each side, as was used in Ref. [32]. It can be seen that the additional coupling that exists in the present model between remote layers will lead to higher frequencies for all the CNTs in this table. The differences are in the range of 0–5% and one cannot predict for which case the influence will be larger, as there are several coupling effects involved. In general the second, third, and fourth radial frequencies have larger differences as the radii are reduced.

Table 6.4

Radial Natural Frequencies (THz) of a Five-Walled CNT With L/R_O=10: Present vdW Interaction Model Compared to the Model in Ref. [32] for m=1, n=0 [24]

	RI=0.5 nm			RI=1 nm			RI=5 nm
	Present	Ref. [32]	%	Present	Ref. [32]	%	Present	Ref. [32]	%
${\mathit{\omega }}_{R1}$	2.1850	2.1620	1.06	1.7510	1.7370	0.81	0.6183	0.6181	0.03
${\mathit{\omega }}_{R2}$	3.3690	3.2500	3.66	2.7210	2.6490	2.72	1.2960	1.2710	1.97
${\mathit{\omega }}_{R3}$	4.1580	3.9520	5.21	3.4040	3.2770	3.88	2.2710	2.1820	4.08
${\mathit{\omega }}_{R4}$	5.0370	4.8320	4.25	3.9190	3.7260	5.18	3.1010	2.9430	5.37
${\mathit{\omega }}_{R5}$	7.3600	7.2600	1.38	4.4320	4.2020	5.47	3.6410	3.4370	5.94

To show the effect of vdW interaction on natural frequencies of an MWCNT of small radius, natural frequencies that are associated with the radial mode for a TWCNT with innermost radius R_I=0.65 nm are calculated by using Eq. (6.63), and the comparison of the results by considering and ignoring the vdW interaction is presented in Table 6.5(A) for various combinations of the wave numbers (m, n). It is observed that the vdW interaction has a significant effect on the frequencies, especially on the lower frequencies when the innermost radius is very small. Also, it can be seen that the wave number n has a dominant influence on the frequencies. This is because all the three terms in Eq. (6.49) are in the same order when the innermost radius R_I is very small, e.g., R_I≤1 nm, and thus the effects of the vdW interaction and the mode order must be taken into consideration.

Table 6.5(B) shows the comparison of radial-mode associated frequencies of a TWCNT with large radius for various combinations of (m, n). In contrast to the TWCNT of small radius, the effect of vdW interaction on the lowest frequency is very small and is negligible due to the fact that the effect of the vdW interaction on the lowest frequency that is associated with the radial mode is very small when the innermost radius is large, as indicated in Eq. (6.57) for a DWCNT of big radius. However, the effect of vdW interaction is significant on the higher frequencies for various combinations of (m, n), which is obvious from Eq. (6.49) that when the innermost radius is big, the term related to the vdW interaction coefficient dominates.

Natural frequencies of a DWCNT, a TWCNT, and a five-walled MWCNT are calculated from Eq. (6.44) for various m and n as well as the innermost radii R_I=5, 1, and 0.5 nm, and shown in Table 6.6(A)–(C) for the DWCNT, Table 6.7(A)–(C) for the TWCNT, and Table 6.8(A)–(C) for the five-walled MWCNT. It should be noted that n=0 represents the breathing vibration of MWCNTs. In Table 6.6(A)–(C) the modes are also identified by the letters L, T, and R and their combinations to indicate the dominance of the longitudinal, torsional, and radial modes in the various modes. In most cases one can see that the modes are not purely of type L, T, or R, but rather a combination of these where the more dominant one is put first. As can be seen from these tables the frequencies increase as n is increased except for the lowest frequency which decreases first and then increases with the increase of n. Similarly, all the frequencies increase as m is increased. It is clear from the tables that there are changes of mode shapes for the same order of frequency when m or n are changed. For example, in Table 6.6(A), the mode shape of the DWCNT associated with ${\mathit{\omega }}_1$ is mainly torsional mode for the combination of m=1 and n=1, but the mode shape becomes mostly radial mode for the combination of m=1 and n=2. In addition, the frequencies increase significantly with the increase of the radius of MWCNTs, especially the frequencies associated with higher number of n. Also, it can be seen from Tables 6.6–6.8 that for MWCNTs with the same size but different numbers of layers, the highest frequency for the DWCNT, the TWCNT, and the five-walled MWCNT is almost the same, and thus the differences between the lowest and highest frequencies are almost the same for an MWCNT with different numbers of layers. This indicates that the more layers there are, the closer the natural frequencies will be.

Wang et al. [32] stated that the effect of vdW interaction is less pronounced for radial modes of small-radius MWCNTs due to the fact that the interlayer vdW interaction is characterized by a radius-independent vdW interaction coefficient. In contrast, our more refined model for the interlayer vdW interaction is characterized by a radius-dependent vdW interaction coefficient and the results for the natural frequencies did show that the effect of vdW interaction on the radial modes of small-radius MWCNTs is significant. Figs. 6.9 and 6.10 show the influence of vdW interaction on the natural frequencies of a DWCNT with small innermost radius R_I=0.65 nm. Indeed, the effect of the vdW interaction on the frequencies which are associated with the torsional and longitudinal vibrations is very small and can be neglected. However, for the lower mode-order vibration such as m=1 and n<5 or n=1 and m<10, frequencies associated with the radial vibration when considering the vdW interaction are significantly different from those when ignoring the vdW interaction, indicating again that vdW interaction does play an important role in the radial vibration of the small-radius DWCNT.

To show the effect of the innermost radius on the lower mode-order frequencies of a DWCNT, frequencies associated with the radial vibration are calculated from Eqs. (6.55) and (6.56) and presented in Fig. 6.11 for the breathing vibration (m=1, n=0). The frequencies associated with the torsional, longitudinal, and radial vibrations (m=1, n=1) are calculated from Eq. (6.46) and presented in Fig. 6.12. It is observed from Figs. 6.11 and 6.12 that the frequencies are the function of the tube radius. With a small radius, say, less than 5 nm, the frequencies drop rapidly, indicating that the radius effect plays an important role in the frequencies of a small-radius MWCNT. For a DWCNT with a large radius, say, greater than 10 nm, the frequencies vary very slowly, showing that the dependence of the frequencies on the radius can be neglected when the tube radius is large enough. Also, the higher frequency associated with the radial vibration is almost a constant and independent of the tube radius when the innermost radius is greater than 10 nm, indicating again that the vdW interaction plays a critical role on the higher radial vibration of MWCNTs. Also, it is clear from Figs. 6.3 and 6.4 that the interlayer vdW interaction plays a critical role on the higher frequency associated with radial vibration of an MWCNT with arbitrary radius, but the effect of vdW interaction on the lower frequency associated with the radial vibration and on the frequencies associated with the torsional and the longitudinal vibrations is very small and is negligible for MWCNTs with radii from very small to very large.

The effect of the innermost radius on the higher mode order (m=5, n=5) frequencies of a DWCNT is shown in Fig. 6.13 for various innermost radii. When the interlayer vdW interaction is ignored, the mode shapes associated with f₁–f₆ do not change for different tube radii, as shown in Fig. 6.13B. However, when the vdW interaction is considered, the mode shapes which are associated with some frequencies will change from radial, torsional, and longitudinal mode shapes to other shapes as the innermost radius is varied from small to large radii. For example, in Fig. 6.13A, as the innermost radius increases, the mode shape associated with the frequency f₆ changes from torsional mode (R_I<7 nm) to a mix of radial and torsional modes (7 nm≤R_I≤9 nm), and then to a radial mode (R_I≥9 nm), and the mode shape associated with frequency f₄ changes from longitudinal mode (R_I≤4 nm) to a mix of longitudinal and torsional modes (4 nm≤R_I≤8 nm), and then to a torsional mode (R_I>8 nm). Similarly, the mode shape associated with frequency f₂ changes from radial mode (R_I≤2 nm) to a mix of radial and longitudinal modes (2 nm≤R_I≤4 nm), and then to a longitudinal mode (R_I≤4 nm). In contrast, the mode shapes associated with the frequencies f₃ and f₁ do not change for different tube radii.

As the radius of a tube in an MWCNT is increased the stiffening effect of that particular tube on the entire MWCNT assembly is reduced, and therefore a different physical combination among all the factors is obtained. This is similar to the vibrations of beams on elastic foundations, where the mode number is increased as the stiffness of the elastic foundation is increased. In the current problem, on top of this, also the coupling in the displacement field makes it more complex, and one can see in Fig. 6.13 the continuous change in the modal composition when the vdW interaction is considered.

In summary, based on the more refined vdW interaction formulas, the free vibration of an MWCNT is investigated by using the Donnell shell model. In contrast to the work by Wang et al. [32] where they conclude that the effect of the vdW interaction is less significant for the radial vibration of small-radius MWCNTs, our numerical results reveal that the effect of the vdW interaction plays a significant role on the higher natural frequencies which are associated with radial vibration of small-radius MWCNTs. This is due to the fact that our continuum model is based on radius-dependent vdW interaction formulas, as compared to the radius-independent vdW interaction formulas that were used by Wang et al. [32] The effects of the tube radius and the interlayer vdW interaction are also examined on the mode shapes of MWCNTs. When ignoring the vdW interaction, the mode shapes of MWCNTs do not change as the radii of the MWCNTs are varied from very small to very large. However, when the vdW interaction is considered, it is observed that some of the mode shapes will change between radial, torsional, and longitudinal modes, due to the effect of the vdW interaction even for cases where the innermost radius is less than 2 nm, indicating again that the interlayer vdW interaction plays a significant role in the vibration of MWCNTs with small radius. In addition, the numerical simulations show that ultrahigh natural frequency can be achieved by using an MWCNT. For a given combination of m and n, the lowest natural frequencies are not influenced by the vdW interaction. However, other higher natural frequencies are significantly dependent on the vdW interaction. Especially, the vdW interaction plays a dominant role on the higher radial frequencies so that the effect of the mode order can be neglected.