# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 2 Formulation of the Problem
The following dimensionless parameters are used to normalize the field variables:  
\[\zeta=\frac{\mathrm{x}}{\mathrm{L}},\mathrm{w}=\frac{\mathrm{W}}{\mathrm{L}}, \eta=\frac{\mathrm{L}}{\mathrm{h}},\mu=\frac{\mathrm{e}_{0}\mathrm{a}}{\mathrm{ L}},\theta_{\mathrm{x}}=\Theta,\varphi=\frac{\phi_{\mathrm{E}}}{\phi_{0}}, \phi_{0}=\sqrt{\frac{\mathrm{A}_{11}}{\mathrm{X}_{33}}},\psi\psi_{0}=\sqrt{ \frac{\mathrm{A}_{11}}{\mathrm{T}_{33}}},\overline{\mathrm{A}_{44}}=\frac{ \mathrm{A}_{44}}{\mathrm{A}_{11}},\]  
\[\overline{\mathrm{D}}_{11}=\frac{\mathrm{D}_{11}}{\mathrm{A}_{11}\;\mathrm{h} ^{2}},\overline{\mathrm{I}}_{0}=\frac{\mathrm{I}_{0}}{\mathrm{I}_{0}},\overline {\mathrm{I}}_{2}=\frac{\mathrm{I}_{2}}{\mathrm{I}_{0}\;\mathrm{h}^{2}}, \bar{\mathrm{X}}_{11}=\frac{\mathrm{X}_{11}\phi_{0}^{2}}{\mathrm{A}_{11}\; \mathrm{h}^{2}},\bar{\mathrm{X}}_{33}=\frac{\mathrm{X}_{33}\phi_{0}^{2}}{ \mathrm{A}_{11}},\quad\overline{\mathrm{E}}_{15}=\frac{\mathrm{E}_{15}\phi_{0} }{\mathrm{A}_{11}\;\mathrm{h}},\]  
\[\overline{\mathrm{E}}_{31}=\frac{\mathrm{E}_{31}\phi_{0}}{\mathrm{A}_{11}\; \mathrm{h}}\overline{\mathrm{Q}}_{15}=\frac{\mathrm{Q}_{15}\psi_{0}}{\mathrm{A}_{ 11}\;\mathrm{h}},\overline{\mathrm{Q}}_{31}=\frac{\mathrm{Q}_{31}\psi_{0}}{ \mathrm{A}_{11}\;\mathrm{h}},\overline{\mathrm{N}}_{\mathrm{T}}=-\frac{\lambda_{1 }\;\mathrm{h}\;\mathrm{h}\;\mathrm{\Lambda}\mathrm{T}}{\mathrm{A}_{11}},\quad \bar{\mathrm{N}}_{\mathrm{E}}=\frac{2\mathrm{e}_{31}\;\mathrm{V}_{0}}{\mathrm{A}_ {11}\;\mathrm{h}},\overline{\mathrm{N}}_{\mathrm{P}}=\frac{\mathrm{P}_{0}}{ \mathrm{A}_{11}},\]  
\[\overline{\mathrm{N}}_{\mathrm{M}}=\frac{2\mathrm{q}_{31}\;\mathrm{A}_{0}}{ \mathrm{A}_{11}},\tau=\frac{\mathrm{t}}{\mathrm{L}}\sqrt{\frac{\mathrm{I}_{0}}{ \mathrm{A}_{11}}},\quad\mathrm{k}_{1}=\frac{\mathrm{K}_{1}\;\mathrm{L}^{4}}{ \piThe substitution of Equations (22) and (23) into Equations (11)-(14) transforms the time-dependent problem into a static eigenvalue problem.  
\[\mathrm{k}_{\mathrm{s}}\,\overline{\mathrm{A}}_{44}\left[\frac{ \partial^{2}\,\mathrm{W}}{\partial\zeta^{2}}+\eta\frac{\partial\Theta}{\partial \zeta}\right]-\mathrm{k}_{\mathrm{s}}\left(\overline{\mathrm{E}}_{15}\frac{ \partial^{2}\Phi}{\partial\zeta^{2}}+\overline{\mathrm{Q}}_{15}\frac{ \partial^{2}\Psi}{\partial\zeta^{2}}\right)+\left(\overline{\mathrm{N}}_{ \mathrm{E}}+\overline{\mathrm{N}}_{\mathrm{T}}+\overline{\mathrm{N}}_{\mathrm{P }}+\overline{\mathrm{N}}_{\mathrm{m}}\right)\] \[\left(\frac{\partial^{2}\,\mathrm{W}}{\partial\zeta^{2}}-\mu^{2} \frac{\partial^{4}\,\mathrm{W}}{\partial\zeta^{4}}\right)-\mathrm{k}_{ \mathrm{I}}\bigg{(}\,\mathrm{W}-\mu^{2}\frac{\partial^{2}\,\mathrm{W}}{ \partial\zeta^{2}}\bigg{)}+\mathrm{k}_{\mathrm{Z}}\bigg{(}\frac{\partial^{2} \,\mathrm{W}}{\partial\zeta^{2}}-\mu^{2}\frac{\partial^{4}\,\mathrm{W}}{ \partial\zeta^{4}}\bigg{)}+ \tag{24}\] \[\mathrm{k}_{\mathrm{s}}\,\mathrm{W}^{3}=-\omega^{2}\overline{ \mathrm{I}}_{0}\bigg{[}\,\mathrm{W}-\mu^{2}\frac{\partial^{2}\,\mathrm{W}}{ \partial\zeta^{2}}\bigg{]}\] \[\overline{\mathrm{D}}_{11}\frac{\partial^{2}\Theta}{\partial \zeta^{2}}-\mathrm{k}_{\mathrm{s}}\,\overline{\mathrm{A}}_{44}\eta\bigg{(} \frac{\partial\,\mathrm{W}}{\partial\zeta}+\eta\Theta\bigg{)}+(\overline{ \mathrm{E}}_{31}+\mathrm{k}_{\mathrm{s}}\overline{\mathrm{E}}_{15})\eta\frac{ \partial\Phi}{\partial\zeta}+(\overline{\mathrm{Q}}_{31}+\mathrm{k}_{\mathrm{s }}\overline{\mathrm{Q}}_{15})\eta\frac{\partial\Psi}{\partial\zeta}=\] \[-\omega^{2}\overline{\mathrm{I}}_{2}\left[\Theta-\mu^{2}\frac{ \partial^{2}\Theta}{\partial\zeta^{2}}\right]\] (25) \[\overline{\mathrm{E}}_{31}\eta\frac{\partial\Theta}{\partial \zeta}+\overline{\mathrm{E}}_{15}\bigg{[}\frac{\partial^{2}\,\mathrm{W}}{ \partial\zeta^{2}}+\eta\frac{\Theta\Theta}{\partial\zeta}\bigg{]}+\overline{ \mathrm{X}}_{11}\frac{\partial^{2}\Phi}{\partial\zeta^{2}}-\overline{\mathrm{ X}}_{33}\eta^{2}\Phi+\overline{\mathrm{Y}}_{11}\frac{\partial^{2}\Psi}{ \partial\zeta^{2}}-\overline{\mathrm{Y}}_{33}\eta^{2}\Psi=0\] (26) \[\overline{\mathrm{Q}}_{31}\eta\frac{\partial\Theta}{\partial \zeta}+\overline{\mathrm{Q}}_{15}\bigg{[}\frac{\partial^{2}\,\mathrm{W}}{ \partial\zeta^{2}}+\eta\frac{\Theta\Theta}{\partial\zeta}\bigg{]}+\overline{ \mathrm{Y}}_{11}\frac{\partial^{2}\Phi}{\partial\zeta^{2}}-\overline{\mathrm{ Y}}_{33}\eta^{2}\Phi+\overline{\mathrm{T}}_{11}\frac{\partial^{2}\Psi}{ \partial\zeta^{2}}-\overline{\mathrm{T}}_{33}\eta^{2}\Psi=0 \tag{27}\]  
The substitution of the harmonic solutions (Equations (22) and (23)) into the boundary conditions (Equations (15)-(18)) results in the following boundary conditions.