# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 4 Numerical Results
The kernel's bandwidth was adjusted between 3 and 9. As shown in Table 4, the numerical results obtained through the DSCQM-DLK method converged to the exact solutions [47] when the grid sizes and bandwidths were 3 or greater. Tables 4 and 5 indicate that the DSCQM-DLK method demonstrated greater computational efficiency compared to both the PDQM and the Sinc-DQ method. A uniform grid with three to nine nodes was used to develop the Discrete Singular Convolution Differential Quadrature Method utilizing the Regularized Shannon Kernel (DSCQM-RSK). The kernel bandwidth (3-7) and \(\sigma\) (\(h_{x}\) to 1.75 \(h_{x}\)), where \(h_{x}=1/(\)N-1\()\), were varied. Table 6 explains the convergence to the exact solutions [46; 47] for grid sizes, bandwidths, and regularization parameters that exceeded specific thresholds. Table 7 highlights the superior computational efficiency among the methods compared. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{6}{c}{**Fundamental Frequency**} \\ \hline
**Bandwidth** & **N** & **3** & **5** & **7** & **9** & **11** \\ \hline
3 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\
5 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\
7 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\
9 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\ \hline Computational time & 0.09 s & A uniform grid with more than 9 nodes & & & \\ \hline \end{tabular}
\end{table} 
Table 4: Fundamental frequency convergence: DSCQM-DLK ((C-C) METE nanobeam). (\(L=80\) nm; \(h=10\) nm; \(\Delta T=0\); \(V_{0}=0\); \(P_{0}=0\); \(A_{0}=0\); \(\mu=0\); \(k_{1}=k_{2}=k_{3}=0\)). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Grid Size, N** & \(\mathbf{\omega_{1}}\) & \(\mathbf{\omega_{2}}\) & \(\mathbf{\omega_{3}}\) & \(\mathbf{\omega_{4}}\) & \(\mathbf{\omega_{5}}\) \\ \hline
3 & 19.512 & 40.542 & 75.254 & — & — \\
5 & 16.3278 & 37.1149 & 68.3338 & 152.781 & 178.53 \\
7 & 9.5236 & 25.457 & 40.2145 & 60.214 & 78.254 \\
11 & 7.6469 & 18.6977 & 32.5948 & 48.1074 & 64.802 \\ \hline Exact results [46] & 7.6473 & 18.6692 & 32.4618 & — & — \\ PDQM [47] & 7.6267 & 18.6229 & 32.3964 & — & — \\ \hline Computational time & 0.12 s & A uniform grid with more than 9 nodes & & & \\ \hline \end{tabular}
\end{table} 
Table 3: Normalized frequencies: Sinc-DQ vs. exact and numerical solutions ((C-C) METE nanobeam). (\(L=80\) nm; \(h=10\) nm; \(\Delta T=0\); \(V_{0}=0\); \(P_{0}=0\); \(A_{0}=0\); \(\mu=0\); \(k_{1}=k_{2}=k_{3}=0\)). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Fundamental Frequency** & & & & & \\ \hline
**Bandwidth** & **N** & **3** & **5** & **7** & **9** & **11** \\ \hline
3 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\
5 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\
7 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\
9 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 & 7.6469 \\ \hline Computational time & 0.09 s & A uniform grid with more than 9 nodes & & & \\ \hline \end{tabular}
\end{table} 
Table 4: Fundamental frequency convergence: DSCQM-DLK ((C-C) METE nanobeam). (\(L=80\) nm; \(h=10\) nm; \(\Delta T=0\); \(V_{0}=0\); \(P_{0}=0\); \(A_{0}=0\); \(\mu=0\); \(k_{1}=k_{2}=k_{3}=0\)). This research investigated the influence of various factors on the vibrational behavior of a nanobeam. Using the DSCQM-RSK method (grid size, 3; bandwidth, 7; \(\sigma=1.75\)\(hx\)), a parametric study examined the effects of linear (and nonlinear) elastic foundation parameters, temperature, electric voltage, nonlocality, the aspect ratio, axial force, magnetic potential, and different boundary conditions (clamped-clamped, clamped-simply supported, simply supported-simply supported) on natural frequencies and mode shapes. The results (Tables 8-11) show that increased linear foundation parameters and magnetic potential raised the fundamental frequency, while the nonlinear foundation parameter had a negligible effect.