# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 4 Numerical Results
The fundamental frequency of the nanobeam was inversely related to the temperature change, electric voltage, non-local parameter, and length-to-thickness ratio (Figures 2-6) but directly related to the axial force and magnetic potential (Figures 7-11). Figures 5, 6, 10, and 11 show the first three modes (transverse displacement and electric potential), revealing that increasing linear (and nonlinear) elastic foundation parameters amplified both displacement and electric potential amplitudes. The nanobeam's response was more strongly influenced by the axial force and electric and magnetic fields than by temperature variations. Comparisons between BiTiO\({}_{3}\)-COFe\({}_{2}\)O\({}_{4}\) and PTZ-5H-COFe\({}_{2}\)O\({}_{4}\) show that the former consistently exhibited a higher fundamental frequency, amplitude, and electric potential. The numerical results, accurate and convergent with other methods, offer valuable design insights for creating customized nanoelectronic and biotechnological smart nanostructures. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{6}{c}{**Normalized Frequencies**} \\ \hline
**B.C** & \(\mu\) & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{3}\) & \(\omega_{4}\) & \(\omega_{5}\) \\ \hline CH & 0 & 5.8127 & 16.5508 & 30.5288 & 46.4423 & 63.1618 \\  & 0.05 & 5.7455 & 15.8011 & 27.6732 & 39.4747 & 49.9575 \\  & 0.1 & 5.5596 & 14.0742 & 22.4378 & 29.383 & 34.673 \\  & 0.15 & 5.2935 & 12.2016 & 18.0756 & 22.5677 & 25.8567 \\  & 0.2 & 4.9905 & 10.5937 & 14.9913 & 18.2813 & 20.6293 \\  & 0.3 & 4.3971 & 8.3366 & 11.3117 & 13.4537 & 14.6772 \\ \hline CC & 0 & 7.8228 & 18.9123 & 32.7410 & 48.2897 & 64.5063 \\  & 0.05 & 7.7286 & 18.0251 & 29.6487 & 41.0333 & 51.045 \\  & 0.1 & 7.4687 & 15.9906 & 24.0051 & 30.5441 & 35.4677 \\  & 0.15 & 7.0987 & 13.8005 & 19.3378 & 23.4804 & 26.4795 \\  & 0.2 & 6.6802 & 11.9355 & 16.0648 & 19.0330 & 21.0858 \\  & 0.3 & 5.8704 & 9.3446 & 12.2038 & 13.922 & 14.8882 \\ \hline HH & 0 & 4.0711 & 14.1202 & 28.1803 & 44.4842 & 61.727 \\  & 0.05 & 4.0296 & 13.5050 & 25.2715 & 37.814 & 48.785 \\  & 0.1 & 3.9141 & 12.0759 & 20.7633 & 28.1311 & 33.799 \\  & 0.15 & 3.7468 & 10.506 & 16.7268 & 21.5699 & 25.1501 \\  & 0.2 & 3.5532 & 9.1397 & 13.8537 & 17.4392 & 20.0658 \\ \hline \end{tabular}
\end{table} 
Table 10: Conditions of boundaries and nonlocal parameter vs. normalized frequency (METE, nanobeam). (\(L=80\) nm; \(h=10\) nm; \(\Delta T=0\); \(V_{0}=0\); \(P_{0}=0\); \(A_{0}=0.02\); \(\mu=0\); \(L/\rm h=8\); \(k_{1}=10\); \(k_{2}=0.25\); \(k_{3}=0.15\)). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multicolumn{6}{c}{**Normalized Frequencies**} \\ \hline
**B.C** & \(L/h\) & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{3}\) & \(\omega_{4}\) & \(\omega_{5}\) \\ \hline CH & 6 & 65.3075 & 132.3290 & 193.2852 & 240.0726 & 274.7364 \\  & 8 & 41.9400 & 87.6475 & 130.6966 & 166.538 & 193.0549 \\  & 12 & 22.8307 & 48.2839 & 73.7153 & 96.2304 & 114.8452 \\  & 16 & 15.396 & 32.2589 & 49.1744 & 65.2192 & 78.5352 \\  & 20 & 11.8894 & 24.2939 & 37.0331 & 48.9753 & 59.8226 \\  & 30 & 8.1887 & 16.0752 & 24.0178 & 31.6047 & 38.7174 \\ \hline \end{tabular}
\end{table} 
Table 11: Comparing the L/t ratio, boundary conditions, and normalized frequencies for METE, nanobeams. (\(\Delta T=0\); \(V_{0}=0\); \(P_{0}=0\); \(A_{0}=0.02\); \(h=2\) nm; \(\mu=0\); \(k_{1}=10\); \(k_{2}=0.25\); \(k_{3}=0.15\)).