# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 3 Method of Solution
### Discrete Singular-Convolution Differential Quadrature Method (DSCQQM)
\begin{array}{ll}(-1)^{i-j}\left(\frac{4\pi^{2}}{h_{x} ^{4}(i-j)^{2}}+\frac{4\pi^{2}}{h_{x}^{3}2\sigma^{2}}-\frac{24}{h_{x}^{4}(i-j)^ {4}}-\frac{12}{h_{x}^{2}(i-j)^{2}\sigma^{2}}-\frac{4h_{x}x^{2}(i-j)^{2}}{\sigma ^{6}}\right)}e^{-h_{x}^{2}\left(\frac{(i-j)^{2}}{2\sigma^{2}}\right)},&i\
eq j \\ \frac{3}{\sigma^{4}}+\frac{2\pi^{2}}{h_{x}^{2}\sigma^{2}}+\frac{\pi^{4}}{5h_{ x}},&i=j\end{array} \right. \tag{46}\]  
By appropriately substituting equations involving weighting coefficients (46) into Equations (24)-(27), the issue can be simplified into the subsequent nonlinear eigenvalue problem. \[k_{s}\bar{A}_{44}\left[\sum_{j=1}^{N}C_{ij}^{(2)}W_{j}+\eta\sum_{ j=1}^{N}C_{ij}^{(1)}\Theta_{j}\right]-k_{s}\left(\bar{E}_{15}\sum_{j=1}^{N}C_{ij}^{ (2)}\Phi_{j}+\bar{Q}_{15}\sum_{j=1}^{N}C_{ij}^{(2)}\Psi_{j}\right)+\] \[(\bar{N}_{E}+\bar{N}_{T}+\bar{N}_{M}+\bar{N}_{P})\left(\sum_{j=1}^ {N}C_{ij}^{(2)}W_{j}-\mu^{2}\sum_{j=1}^{N}C_{ij}^{(4)}W_{j}\right)-k_{1}\left( \sum_{j=1}^{N}\delta_{ij}W_{j}-\mu^{2}\sum_{j=1}^{N}C_{ij}^{(2)}W_{j}\right)+\] \[k_{2}\left(\sum_{j=1}^{N}C_{ij}^{(2)}W_{j}-\mu^{2}\sum_{j=1}^{N}C _{ij}^{(4)}W_{j}\right)+k_{3}\sum_{j=1}^{N}\delta_{ij}W_{j}^{3}=-\bar{I}_{0} \omega^{2}\left[\sum_{j=1}^{N}\delta_{ij}W_{j}-\mu^{2}\sum_{j=1}^{N}C_{ij}^{(2 )}W_{j}\right], \tag{47}\]  
\[\bar{D}_{11}\sum_{j=1}^{N}C_{ij}^{(2)}\Theta_{j}-k_{s}\bar{A}_{44 }\eta\left(\sum_{j=1}^{N}C_{ij}^{(1)}W_{j}+\eta\sum_{j=1}^{N}\delta_{ij} \Theta_{j}\right)+(\bar{Q}_{31}+k_{s}\bar{Q}_{15})\eta\] \[\sum_{j=1}^{N}C_{ij}^{(1)}\Psi_{j}+(\bar{E}_{31}+k_{s}\bar{E}_{15 })\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Phi_{j}=-\bar{I}_{2}\omega^{2}\left[\sum_{j=1 }^{N}\delta_{ij}\Theta_{j}-\mu^{2}\sum_{j=1}^{N}C_{ij}^{(2)}\Theta_{j}\right], \tag{48}\]  
\[\bar{E}_{31}\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Theta_{j}+\bar{E}_{15 }\left[\sum_{j=1}^{N}C_{ij}^{(2)}W_{j}+\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Theta_{j }\right]+\bar{X}_{11}\sum_{j=1}^{N}C_{ij}^{(2)}\Phi_{j}-\bar{X}_{33}\eta^{2} \sum_{j=1}^{N}\delta_{ij}\Phi_{j}+\] \[\bar{Y}_{11}\sum_{j=1}^{N}C_{ij}^{(2)}\Psi_{j}-\bar{Y}_{33}\eta^{ 2}\sum_{j=1}^{N}\delta_{ij}\Psi_{j}=0 \tag{49}\]  
\[\bar{Q}_{31}\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Theta_{j}+\bar{Q}_{15 }\left[\sum_{j=1}^{N}C_{ij}^{(2)}W_{j}+\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Theta_{j }\right]+\bar{Y}_{11}\sum_{j=1}^{N}C_{ij}^{(2)}\Phi_{j}-\bar{Y}_{33}\eta^{2} \sum_{j=1}^{N}\delta_{ij}\Phi_{j}+\] \[\bar{\overline{T}}_{11}\sum_{j=1}^{N}C_{ij}^{(2)}\Psi_{j}-\bar{ \overline{T}}_{33}\eta^{2}\sum_{j=1}^{N}\delta_{ij}\Psi_{j}=0 \tag{50}\]  
\[\bar{Q}_{31}\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Theta_{j}+\bar{Q}_{15} \left[\sum_{j=1}^{N}C_{ij}^{(2)}W_{k+1,j}+\eta\sum_{j=1}^{N}C_{ij}^{(1)}\Theta_{j }\right]+\bar{Y}_{11}\sum_{j=1}^{N}C_{ij}^{(2)}\Phi_{j}-\] \[(\bar{N}_{E}+\bar{N}_{T}+\bar{N}_{M}+\bar{N}_{P})\left(\sum_{j=1}^ {N}C_{ij}^{(2)}W_{j}-\mu^{2}\sum_{j=1}^{N}C_{ij}^{(4)}W_{j}\right)-k_{1}\left( \sum_{j=1}^{N}\delta_{ij}W_{j}-\mu^{2}\sum_{j=1}^{N}C_{ij}^{(2)}W_{j}\right)+\]0  
Next, an iterative method is used to solve the nonlinear system until convergence is achieved.