# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 2 Formulation of the Problem
The limitations of Equation (19) are significant because it assumes instantaneous thermal equilibrium, which is not accurate at the nanoscale, especially for transient thermal loading. The use of a generalized thermoelasticity theory would lead to a more accurate representation of the temperature field and its influence on the nanobeam's vibrations. This would involve modifying the governing Equations (11)-(14) to include the appropriate energy equation from the chosen generalized thermoelasticity model (LS or GL). The resulting system of equations would be more complex and likely require more computationally intensive numerical techniques to solve. So, we will acknowledge the limitations of the current thermal model and propose future work incorporating a generalized thermoelasticity theory. This could involve a detailed discussion of the implications of using a more sophisticated thermal model, including the expected changes in the natural frequencies and mode shapes of the nanobeam. While a complete re-analysis using a generalized thermoelasticity theory might be beyond the scope of the current paper, outlining a plan for such future work would strengthen the paper significantly and demonstrate a commitment to addressing the limitations identified by the reviewer. Furthermore, a comparison of the results obtained with the simplified model (Equation (19)) to those expected from a generalized thermoelasticity model would provide valuable insights into the accuracy of the current approach. Also,  
\[\left\{ \begin{array}{l}(\mathrm{A}_{11},\;\mathrm{A}_{44})=\left(\overline{C} _{11},\overline{C}_{44}\right)\mathrm{h}\\ \mathrm{D}_{11}=\overline{\mathrm{C}}_{11}\;\mathrm{h}^{3}/12\\ (\mathrm{E}_{15},\mathrm{Q}_{15})=2\frac{(\overline{\mathrm{e}_{15}}\overline{ \mathrm{q}_{15}})}{\beta}\sin\left(\frac{\beta\mathrm{h}}{2}\right),\\ (\mathrm{I}_{0},\mathrm{I}_{2})=\rho\Big{(}\mathrm{h},\;\mathrm{h}^{3}/12\big{)},\\ (\mathrm{E}_{31},\mathrm{Q}_{31})=\left(\overline{\mathrm{e}_{31}},\overline{ \mathrm{q}_{31}}\right)\Big{[}-\mathrm{h}\cos\left(\frac{\beta\;\mathrm{h}}{2 }\right)+\frac{2}{\beta}\sin\left(\frac{\beta\;\mathrm{h}}{2}\right)\Big{]},\\ (\mathrm{X}_{11},\mathrm{Y}_{11},\;\mathrm{T}_{11})=\frac{(\overline{\mathrm{ S}_{11}}\overline{\mathrm{x}_{11}},\overline{\mathrm{g}_{11}})}{2}\Big{[}\, \mathrm{h}+\frac{\sin(\beta\mathrm{h})}{\beta}\Big{]},\\ (\mathrm{X}_{33},\mathrm{Y}_{33},\;\mathrm{T}_{33})=\frac{(\overline{\mathrm{ s}_{33}},\overline{\mathrm{Q}_{33}},\overline{\mathrm{e}_{33}})\beta^{2}}{2}\Big{[}\, \mathrm{h}-\frac{\sin(\beta\mathrm{h})}{\beta}\Big{]},\\ \beta=\pi/\mathrm{h}\end{array} \right. \tag{20}\]  
And,  
\[\left\{ \begin{array}{l}(\overline{C}_{11},\overline{C}_{44})=\left(\mathrm{ C}_{11}-\frac{\mathrm{C}_{33}^{2}}{\mathrm{C}_{44}}\right),\\ (\overline{\mathrm{e}_{31}},\overline{\mathrm{e}_{15}})=\left(\mathrm{e}_{31}- \frac{\mathrm{C}_{33}\mathrm{e}_{33}}{\mathrm{C}_{33}},\mathrm{e}_{15}\right),\\ (\mathrm{q}_{31},\overline{\mathrm{q}_{15}})=\left(\mathrm{q}_{31}-\frac{ \mathrm{C}_{33}\mathrm{e}_{33}}{\mathrm{C}_{33}},\mathrm{q}_{15}\right),\\ (\overline{\mathrm{S}_{33}},\;\overline{\mathrm{S}_{11}})=\left(\mathrm{S}_{33}- \frac{\mathrm{e}_{31}^{2}}{\mathrm{C}_{33}},\;\mathrm{S}_{11}\right),\\ \left(\overline{\mathrm{q}_{33}},\;\overline{\mathrm{d}_{11}}\right)=\left( \mathrm{d}_{33}+\frac{\mathrm{q}_{15}\mathrm{e}_{33}}{\mathrm{C}_{33}},\; \mathrm{d}_{11}\right),\\ (\beta_{33},\beta_{11})=\left(\mu_{33}+\frac{\mathrm{q}_{12}^{2}}{\mathrm{C}_{ 33}},\mu_{11}\right),\\ (\tilde{\lambda}_{1},\overline{\mathrm{p}_{3}},\tilde{\beta}_{3})=\left( \lambda_{1}-\frac{\mathrm{C}_{31}\lambda_{3}}{\mathrm{C}_{33}},\mathrm{p}_{3}+ \frac{\lambda_{3}\mathrm{e}_{33}}{\mathrm{C}_{33}},\beta_{3}+\frac{\beta_{3} \mathrm{e}_{33}\mathrm{e}_{33}}{\mathrm{C}_{33}}\right)\end{array} \right.