Free vibrations of elastically embedded stocky single-walled carbon nanotubes acted upon by a longitudinally varying magnetic field

Abstract

The mechanical properties of nano-scaled composites reinforced by carbon nanotubes are enhanced by application of appropriate magnetic fields, however, little is known on the free dynamic response of the magnetically affected stocky single-walled carbon nanotubes (SWCNTs) with elastic supports. Using nonlocal Rayleigh, Timoshenko, and higher-order beam theory, the equations of free transverse vibration of elastically embedded SWCNTs subjected to a longitudinally varying magnetic field are obtained. Since finding an analytical solution to the equations of motion is a very difficult task, an efficient meshless method is proposed. The frequencies of the magnetically affected stocky SWCNTs are evaluated for different boundary conditions. The convergence checks of the proposed numerical models are carried out. In a special case, the obtained results are also compared with those of assumed mode method, and a reasonably good agreement is achieved. Subsequently, the roles of the slenderness ratio of the SWCNT, small-scale parameter, strength of the magnetic field, lateral and rotational interactions of the SWCNT with its surrounding medium on the fundamental frequency are addressed in some detail. The capabilities of the proposed models in capturing the frequencies of the magnetically affected nanostructure are also comprehensively investigated.

Appendices

Appendix A: Exact solution to the flexural frequencies of a simply supported SWCNT acted upon by a uniform longitudinal magnetic field

In such a case, for free vibration analysis of the nanostructure, the dimensionless deformation fields of the MASWCNT based on the NRBT, NTBT, and NHOBT can be considered as follows:

$$\begin{aligned} \overline{w}^R= & {} \sum _{n=1}^{\infty }\,\overline{w}_{n0}^R\, e^{{\rm i}\varpi _n^R\tau }\sin (n\pi \xi ),\end{aligned}$$

(35a)

$$\begin{aligned} \overline{w}^T= & {} \sum _{n=1}^{\infty }\,\overline{w}_{n0}^T\, e^{{\rm i}\varpi _n^T\tau }\sin (n\pi \xi ), \\ \overline{\theta }^T= & {} \sum _{n=1}^{\infty }\,\overline{\theta }_{n0}^T\, e^{{\rm i}\varpi _n^T\tau }\cos (n\pi \xi ),\end{aligned}$$

(35b)

$$\begin{aligned} \overline{w}^H= & {} \sum _{n=1}^{\infty }\,\overline{w}_{n0}^H\, e^{{\rm i}\varpi _n^H\tau }\sin (n\pi \xi ), \\ \overline{\psi }^H= & {} \sum _{n=1}^{\infty }\,\overline{\psi }_{n0}^H\, e^{{\rm i}\varpi _n^H\tau }\cos (n\pi \xi ), \end{aligned}$$

(35c)

where $\overline{w}_{n0}^{[.]}$ and $\varpi _n^{[.]}$ are the dimensionless deflection amplitude and natural frequency associated with the $n$th vibration mode, $\overline{\theta }_{n0}^T$ and $\overline{\psi }_{n0}^H$ represent the angle of deformation associated with the $n$th vibration mode of the NTBT and NHOBT, respectively. By substituting Eqs. (35a), (35b), and (35c) into Eqs. (9), (19), and (29), respectively, and by setting the determinant of the resulted matrices of the coefficients equal to zero, the dimensionless natural frequencies can be calculated for the proposed beam models as will be displayed in the following parts.

1.1 A.1 Exact frequency analysis of MASWCNTs based on the NRBT

The $n$th flexural frequency, $\varpi _{n}^R$, is calculated as:

$$\begin{aligned} \varpi _n^R=\sqrt{\frac{(n\pi )^2\left( \left( \overline{H}_x^R\right) ^2 +\overline{K}_r^R\right) +\overline{K}_t^R}{1+\left( \frac{n\pi }{\lambda }\right) ^2} +\frac{(n\pi )^4}{\left( 1+(n\pi \mu )^2\right) \left( 1+\left( \frac{n\pi }{\lambda }\right) ^2\right) }}. \end{aligned}$$

(36)

1.2 A.2 Exact frequency analysis of MASWCNTs based on the NTBT

The $n$th flexural frequency, $\varpi _{n}^T$, is obtained as:

$$\begin{aligned} \varpi _n^T=\sqrt{\frac{\alpha _{2n}^2+\left( \frac{\alpha _{3n}}{\lambda }\right) ^2 -\sqrt{\left( \alpha _{2n}^2-\left( \frac{\alpha _{3n}}{\lambda }\right) ^2\right) ^2+ 4\left( \frac{n\pi }{\lambda }\right) ^2}}{2\alpha _{1n}^2}}, \end{aligned}$$

(37)

where $\alpha _{in}^2;\,i=1,2,3$ are as:

$$\begin{aligned} \alpha _{1n}^2=\lambda ^{-2}\left( 1+(n\pi \mu )^2\right) , \\ \alpha _{2n}^2=1+\eta (n\pi )^2+\overline{K}_r^T\left( 1+(n\pi \mu )^2\right) , \\ \alpha _{3n}^2=(n\pi )^2+\left( 1+(n\pi \mu )^2\right) \left( \overline{K}_t^T+(n\pi )^2\left( \overline{H}_x^T\right) ^2\right) . \end{aligned}$$

(38)

1.3 A.3 Exact frequency analysis of MASWCNTs based on the NHOBT

The $n$th flexural frequency, $\varpi _{n}^H$, is derived as follows:

$$\begin{aligned} \varpi _n^H=\sqrt{\frac{B_n^H-\sqrt{\left( B_n^H\right) ^2-4A_n^H C_n^H}}{2A_n^H}}, \end{aligned}$$

(39)

where

$$\begin{aligned} A_n^H= & {} \left( 1+(n\pi \mu )^2\right) ^2\left( 1+\left( \gamma _2^2+\gamma _1^2\gamma _6^2\right) (n\pi )^2\right) , \\ B_n^H= & {} \left( 1+(n\pi \mu )^2\right) \left( \begin{array}{l} (n\pi \gamma _3)^2+(n\pi )^4+\left( \overline{K}_t^H+\left( n\pi \overline{H}_x^H\right) ^2\right) \left( 1+(n\pi \mu )^2\right) \\ +\left( 1+\left( n\pi \gamma _2\right) ^2\right) \left( \gamma _7^2+\left( n\pi \gamma _8\right) ^2+\overline{K}_r^H\left( 1+\left( n\pi \mu \right) ^2\right) \right) \\ +(n\pi )^2\left( \gamma _6^2\left( \gamma _3^2-\left( n\pi \gamma _4\right) ^2\right) +\gamma _1^2\left( \gamma _7^2-\left( n\pi \gamma _9\right) ^2\right) \right) \end{array}\right) , \\ C_n^H= & {} \left( \gamma _7^2+\left( n\pi \gamma _8\right) ^2+\overline{K}_r^H\left( 1+\left( n\pi \mu \right) ^2\right) \right) \\&\times \left( \left( n\pi \gamma _3\right) ^2+\left( n\pi \right) ^4+\left( \overline{K}_t^H+\left( n\pi \overline{H}_x^H\right) ^2\right) \left( 1+(n\pi \mu )^2\right) \right) \\&-(n\pi )^2\left( \gamma _7^2-\left( n\pi \gamma _9\right) ^2\right) \left( \gamma _3^2-\left( n\pi \gamma _4\right) ^2\right) . \end{aligned}$$

(40)

Appendix B: Application of AMM for frequency analysis of simply supported SWCNTs subjected to a sinusoidal magnetic field

1.1 B.1 Frequency analysis of MASWCNTs modeled based on the NRBT using AMM

In order to analyze the problem via AMM, the dimensionless transverse displacement of the MASWCNT is stated in terms of appropriate mode shape functions pertinent to the considered boundary conditions:

$$\begin{aligned} \overline{w}^R(\xi ,\tau )=\sum _{k=1}^{NM} \phi _k^{w}(\xi ) \overline{w}_{k}^R(\tau ), \end{aligned}$$

(41)

where $\phi _k^{w}$ denotes the $k$th mode shape function of SWCNT’s deflection, and $NM$ is the number of vibration modes. For a simply supported MASWCNT, such modes are considered as: $\phi _k^{w}(\xi )=\sqrt{2}\sin (k\pi \xi )$. By substituting Eq. (41) into Eqs. (14b) and (14c),

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^R\right] _{kl}= & {} \left( 1+\left( \frac{\pi k}{\lambda }\right) ^2\right) \left( 1+\left( \mu \pi k\right) ^2\right) \delta _{kl};\,k,l=1,2,\ldots ,NM,\end{aligned}$$

(42a)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^R\right] _{kl}= & {} \left( k^2 l^2 \pi ^4+\left( \frac{\pi ^2}{2}\left( k^2+1\right) \left( \overline{H}_{x0}^R\right) ^2+\,\overline{K}_t^R+\overline{K}_r^R\pi ^2 k l \right) \left( 1+(\mu \pi k)^2\right) \right) \,\delta _{kl}, \end{aligned}$$

(42b)

where $\delta _{kl}$ is the Kronecker delta. By following the procedure mentioned in Sect. 3.2, the natural frequencies of the magnetically affected nanostructure could be determined.

1.2 B.2 Frequency analysis of MASWCNTs modeled based on the NTBT using AMM

For a simply supported MASWCNT in which modeled based on the NTBT, the dimensionless deformation fields are expressed by:

$$\begin{aligned} \overline{w}^T(\xi ,\tau )=\sum _{k=1}^{NM} \phi _k^{w}(\xi ) \overline{w}_{k}^T(\tau ), \overline{\theta }^T(\xi ,\tau )=\sum _{k=1}^{NM} \phi _k^{\theta }(\xi ) \overline{\theta }_{k}^T(\tau ), \end{aligned}$$

(43)

where $\phi _k^{w}(\xi )=\sqrt{2}\sin (k\pi \xi )$ and $\phi _k^{\theta }(\xi )=\sqrt{2}\cos (k\pi \xi )$ in order are the $k$th mode shapes pertinent to the deflection and rotational fields of the MASWCNT modeled based on the NTBT. By substituting Eq. (43) into Eqs. (24c)–(24h), the nonzero mass and stiffness matrices are calculated as:

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^T\right] _{kl}^{w w}= & {} \left( 1+kl\left( \mu \pi \right) ^2\right) \delta _{kl}, \end{aligned}$$

(44a)

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^T\right] _{kl}^{\theta \theta }=\, & {} \lambda ^{-2} \left( 1+kl\left( \mu \pi \right) ^2\right) \delta _{kl}, \end{aligned}$$

(44b)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^T\right] _{kl}^{w w}= & {} \left( kl\pi ^2+\left( 1+(\mu \pi k)^2\right) \left( \frac{\pi ^2}{2}\left( k^2+1\right) \left( \overline{H}_{x0}^T\right) ^2+\overline{K}_t^T \right) \right) \,\delta _{kl}, \end{aligned}$$

(44c)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^T\right] _{kl}^{w \theta }= & {} -k\pi \delta _{kl}, \end{aligned}$$

(44d)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^T\right] _{kl}^{\theta w}= & {} -l\pi \delta _{kl}, \end{aligned}$$

(44e)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^T\right] _{kl}^{\theta \theta }= & {} \left( 1+ \eta kl\pi ^2+kl\left( 1+(\mu \pi )^2\right) \overline{K}_r^T\right) \delta _{kl}. \end{aligned}$$

(44f)

Finally, by introducing Eqs. (44a)–(44f) to Eq. (23) and following the procedure mentioned in Sect. 3.2, the dimensionless frequencies of the SWCNT subjected to the sinusoidal magnetic field could be readily evaluated.

1.3 B.3 Frequency analysis of MASWCNTs modeled based on the NHOBT using AMM

For a simply supported SWCNT modeled according to the NHOBT, the dimensionless deformation fields can be stated as follows:

$$\begin{aligned} \overline{w}^H(\xi ,\tau )=\sum _{k=1}^{NM} \phi _k^{w}(\xi ) \overline{w}_{k}^H(\tau ), \overline{\psi }^H(\xi ,\tau )=\sum _{k=1}^{NM} \phi _k^{\psi }(\xi ) \overline{\psi }_{k}^H(\tau ), \end{aligned}$$

(45)

where $\phi _k^{w}(\xi )=\sqrt{2}\sin (k\pi \xi )$ and $\phi _k^{\psi }(\xi )=\sqrt{2}\cos (k\pi \xi )$. By introducing Eq. (45) to Eqs. (34c)–(34j), the nonzero mass and stiffness matrices are obtained as:

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^H\right] _{kl}^{\psi \psi }= & {} \left( 1+kl(\mu \pi )^2\right) \delta _{kl}, \end{aligned}$$

(46a)

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^H\right] _{kl}^{\psi w}= & {} -\gamma _6^2\, k\pi \left( 1+(\mu \pi l)^2\right) \delta _{kl}, \end{aligned}$$

(46b)

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^H\right] _{kl}^{w w}= & {} \left( 1+(\mu \pi kl)^2\right) \left( 1+\left( \pi k l \gamma _2\right) ^2\right) \delta _{kl}, \end{aligned}$$

(46c)

$$\begin{aligned} \left[ {\overline{{\mathbf {M}}}}_b^H\right] _{kl}^{w \psi }= & {} -\gamma _1^2\, k\pi \left( 1+(\mu \pi )^2 kl \right) \delta _{kl}, \end{aligned}$$

(46d)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^H\right] _{kl}^{w w}= & {} \left( \gamma _3^2 \pi ^2 kl + \pi ^4 k^2l^2+\left( 1+(\mu \pi k)^2\right) \left( \frac{\pi ^2}{2}\left( k^2+1\right) \left( \overline{H}_{x0}^H\right) ^2+\overline{K}_t^H\right) \right) \delta _{kl}, \end{aligned}$$

(46e)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^H\right] _{kl}^{w\psi }=\, & {} k\pi \left( \gamma _3^2-kl\pi ^2\gamma _4^2\right) \delta _{kl}, \end{aligned}$$

(46f)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^H\right] _{kl}^{\psi w}=\, & {} l\pi \left( \gamma _7^2-kl\pi ^2\gamma _9^2\right) \delta _{kl}, \end{aligned}$$

(46g)

$$\begin{aligned} \left[ {\overline{{\mathbf {K}}}}_b^H\right] _{kl}^{\psi \psi }= & {} \left( \gamma _7^2+ kl\pi ^2\gamma _8^2+\overline{K}_r^H\left( 1+kl(\mu \pi )^2\right) \right) \delta _{kl}. \end{aligned}$$

(46h)

By substituting Eqs. (46a)–(46h) into Eq. (33) and solving the resulting set of eigenvalue equations based on the procedure explained in Sect. 3.2, the dimensionless natural frequencies of the SWCNT acted upon by the sinusoidal magnetic field are calculated.