# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 8 Results and Discussion
### Natural Frequency Analysis
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{\(\epsilon_{0}\widetilde{a}=1\) nm} \\ \hline  & & \(\omega_{\rm{1 mm}}\) & \(\omega_{\rm{2 mn}}\) & \(\omega_{\rm{3 mm}}\) & \(\omega_{\rm{4 mn}}\) \\ \hline  & n = 1 & 4.61953 \(\times\) 10\({}^{11}\) & 2.99085 \(\times\) 10\({}^{12}\) & 7.38604 \(\times\) 10\({}^{12}\) & 1.21507 \(\times\) 10\({}^{13}\) \\ m = 1 & n = 2 & 9.31595 \(\times\) 10\({}^{11}\) & 2.30851 \(\times\) 10\({}^{12}\) & 1.03814 \(\times\) 10\({}^{13}\) & 1.64871 \(\times\) 10\({}^{13}\) \\  & n = 3 & 1.46995 \(\times\) 10\({}^{12}\) & 2.15757 \(\times\) 10\({}^{12}\) & 1.15336 \(\times\) 10\({}^{13}\) & 1.82203 \(\times\) 10\({}^{13}\) \\ \hline  & n = 1 & 5.70087 \(\times\) 10\({}^{11}\) & 5.67846 \(\times\) 10\({}^{12}\) & 6.99782 \(\times\) 10\({}^{12}\) & 1.36309 \(\times\) 10\({}^{13}\) \\ m = 2 & n = 2 & 1.01025 \(\times\) 10\({}^{12}\) & 4.20984 \(\times\) 10\({}^{12}\) & 1.01672 \(\times\) 10\({}^{13}\) & 1.67344 \(\times\) 10\({}^{13}\) \\  & n = 3 & 1.52519 \(\times\) 10\({}^{12}\) & 3.45108 \(\times\) 10\({}^{12}\) & 1.13942 \(\times\) 10\({}^{13}\) & 1.82509 \(\times\) 10\({}^{13}\) \\ \hline  & n = 1 & 7.34647 \(\times\) 10\({}^{11}\) & 6.85678 \(\times\) 10\({}^{12}\) & 7.19172 \(\times\) 10\({}^{12}\) & 1.51839 \(\times\) 10\({}^{13}\) \\ m = 3 & n = 2 & 1.13034 \(\times\) 10\({}^{12}\) & 5.94912 \(\times\) 10\({}^{12}\) & 9.69245 \(\times\) 10\({}^{12}\) & 1.71264 \(\times\) 10\({}^{13}\) \\  & n = 3 & 1.61201 \(\times\) 10\({}^{12}\) & 4.78913 \(\times\) 10\({}^{12}\) & 1.11226 \(\times\) 10\({}^{13}\) & 1.83239 \(\times\) 10\({}^{13}\) \\ \hline  & & \multicolumn{4}{c}{\(\epsilon_{0}\widetilde{a}=2\) nm} \\ \hline  & & \(\omega_{\rm{1 mm}}\) & \(\omega_{\rm{2 mn}}\) & \(\omega_{\rm{3 mm}}\) & \(\omega_{\rm{4 mn}}\) \\ \hline  & n = 1 & 3.84471 \(\times\) 10\({}^{11}\) & 2.07360 \(\times\) 10\({}^{12}\) & 5.08588 \(\times\) 10\({}^{12}\) & 8.36769 \(\times\) 10\({}^{12}\) \\ m = 1 & n = 2 & 5.86577 \(\times\) 10\({}^{11}\) & 1.35492 \(\times\) 10\({}^{12}\) & 5.97785 \(\times\) 10\({}^{12}\) & 9.49374 \(\times\) 10\({}^{12}\) \\  & n = 3 & 8.24879 \(\times\) 10\({}^{11}\) & 1.19378 \(\times\) 10\({}^{12}\) & 6.20668 \(\times\) 10\({}^{12}\) & 9.80507 \(\times\) 10\({}^{12}\) \\ \hline  & n = 1 & 4.33619 \(\times\) 10\({}^{11}\) & 3.68099 \(\times\) 10\({}^{12}\) & 4.50109 \(\times\) 10\({}^{12}\) & 8.78749 \(\times\) 10\({}^{12}\) \\ m = 2 & n = 2 & 6.24265 \(\times\) 10\({}^{11}\) & 2.40523 \(\times\) 10\({}^{12}\) & 5.77111 \(\times\) 10\({}^{12}\) & 9.49987 \(\times\) 10\({}^{12}\) \\  & n = 3 & 8.54911 \(\times\) 10\({}^{11}\) & 1.86919 \(\times\) 10\({}^{12}\) & 6.10670 \(\times\) 10\({}^{12}\) & 9.78197 \(\times\) 10\({}^{12}\) \\ \hline  & n = 1 & 5.04771 \(\times\) 10\({}^{11}\) & 4.25083 \(\times\) 10\({}^{12}\) & 4.25083 \(\times\) 10\({}^{12}\) & 9.17758 \(\times\) 10\({}^{12}\) \\ m = 3 & n = 2 & 6.77858 \(\times\) 10\({}^{11}\) & 3.33059 \(\times\) 10\({}^{12}\) & 5.40006 \(\times\) 10\({}^{12}\) & 9.54656 \(\times\) 10\({}^{12}\) \\  & n = 3 & 8.95666 \(\times\) 10\({}^{11}\) & 2.56714 \(\times\) 10\({}^{12}\) & 5.92601 \(\times\) 10\({}^{12}\) & 9.76398 \(\times\) 10\({}^{12}\) \\ \hline  & & \multicolumn{4}{c}{\(\epsilon_{0}\widetilde{a}=3\) nm} \\ \hline  & & \(\omega_{\rm{1 mm}}\) & \(\omega_{\rm{2 mm}}\) & \(\omega_{\rm{3 mm}}\) & \(\omega_{\rm{4 mn}}\) \\ \hline  & n = 1 & 3.44521 \(\times\) 10\({}^{11}\) & 1.53112 \(\times\) 10\({}^{12}\) & 3.71222 \(\times\) 10\({}^{12}\) &In Table 2, the values of the natural frequencies are compared based on changes in the nonlocal parameter. It can be observed that the natural frequencies decrease as the values of the nonlocal parameter increase. The general mode shapes of the vibrations of the two coupled plates are presented in a paper by [PERSON] [42; 43].