# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 4 Results and Discussion
These figures revealed that the fundamental dimensionless frequencies of the AFG CC-CC double nanobeams were larger compared to those of the AFG CS-CS and SS-SS double nanobeams. In Figure 10, the fundamental dimensionless frequencies of the CC-CC double nanobeams decreased by 38.7%, the CS-CS by 31.86%, whereas the SS-SS by 20.95% as \(\mu\) varied from 0 to 1. Figure 11 shows that \(\Omega_{1}\) of the CC-CC double nanobeams rose by 23.6%, the CS-CS by 42.9%, and the SS-SS by 78.7% as \(K\) grew over the range 0 to 500. From Figure 12, it is depicted that \(\Omega_{1}\) of the CC-CC double nanobeams grew by 11%, the CS-CS by 21.5%, and the SS-SS by 41% as the Winkler modulus parameter \(K_{W}\) varied over the range 0-400. Figure 8: The fundamental dimensionless transverse frequency of CS-CS double AFG nanobeam as function of the stiffness of the elastic layer and the nonlocal scale coefficient \(\left(c_{h_{1}}=\ c_{h_{2}}=c_{h_{2}}=c_{h_{1}}=0.2,\ K_{G}=10,\text{and}\ K_{W}=400\right)\). Figure 7: The first three dimensionless natural transverse frequencies as function of height taper ratio of nanobeam-2 \(c_{h_{2}}\) of a nonlocal AFG Euler–Bernoulli double nanobeam \(\left(c_{h_{1}}=\ c_{h_{2}}=c_{h_{1}}=0.8,\ \mu=0.1,\ K_{G}=10,K_{W}=400\ \text{, and}\ K=800\right)\). (**a**) CC-CC and (**b**) SS-SS. Figure 11: The fundamental dimensionless natural transverse frequency versus the stiffness of the coupling elastic layer \(K\) of a nonlocal AFG Euler–Bernoulli double nanobeam at different boundary conditions \(\left(c_{b_{1}}=\ c_{b_{2}}=\ c_{b_{2}}=c_{b_{1}}=0.1,\ \ K_{G}=10,K_{W}=400\,\ \text{and}\ \mu=0.2\right)\). Figure 10: The fundamental dimensionless natural transverse frequency as function of the nonlocal scale coefficient of AFG Euler–Bernoulli double nanobeam with different boundary conditions \(\left(c_{b_{1}}=\ c_{b_{2}}=\ c_{b_{2}}=c_{b_{1}}=0.1,\ \ K_{G}=10,K_{W}=400\,\ \text{and}\ \ K=500\right)\). Figure 9: The fundamental dimensionless natural transverse frequency of CC-CC AFG double beam as function of the breadth taper ratio \(c_{b_{1}}\) and the height taper ratio \(c_{h_{1}}\) of nanobeam-I (\(c_{h_{2}}=\ c_{b_{2}}=0.1,\ \ K_{G}=10,K_{W}=400,\ K=500\) and \(\mu=0.2\)). The mode shapes of AFG SS-SS double nanobeams are displayed in Figures 13a-d and 14a-d. These figures indicate that the mode shapes were sensitive to the change in the nonlocal scale coefficient \(\mu\). For instance, it is clearly seen in Figure 13b-d, that the mode shapes of nanobeam-1 (in blue) looked different at several values of \(\mu\). Furthermore, Figure 14b shows that the mode shapes of nanobeam-2 (in green) were sensitive to the change in the value of the stiffness parameter \(K\). Similar observations could be obtained from Figure 14c,d. Figure 12: The fundamental dimensionless natural transverse frequency versus the Winkler modulus parameter \(K_{\rm W}\) of a nonlocal AFG Euler–Bernoulli double nanobeam at different boundary conditions \(\left(c_{b_{1}}=\ c_{h_{2}}=\ c_{b_{2}}=c_{h_{1}}=0.2,\ K_{G}=10,K=400\,\ {\rm and}\ \mu=0.1\right)\). Figure 4:0 The first four mode shapes of a SS-SS AFG nonlocal Euler–Bernoulli double nanobeam (Blue: beam 1 and green: beam 2) at different values of the nonlocal parameter (solid: \(\mu=0\), dashed: \(\mu=0.2\), and dotted: \(\mu=0.8\)) \(\left(c_{b_{1}}=\ c_{b_{2}}=0.2,c_{h_{1}}=c_{h_{2}}=0.1,\ \ K_{W}=200,\ K_{G}=10,{\rm and}\ K=100\right)\). (**a**) First mode shape, (**b**) second mode shape, (**c**) third mode shape, and (**d**) fourth mode shape.