# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 4 Results and Discussion
The breadth taper ratio was assumed in the range of 0-0.8. The figure revealed that as the value of \(c_{b_{1}}\) gets larger, the dimensionless natural frequencies increase. It was also noticed that \(\Omega_{2}\) and \(\Omega_{3}\) were less sensitive to the rise of \(c_{b_{1}}\) than \(\Omega_{1}\). As an illustration, \(\Omega_{2}\) and \(\Omega_{3}\) for a CC-CC double nanobeam increase by 21% and 18%, respectively, whereas \(\Omega_{1}\) increased by 21.11% as \(c_{b_{1}}\) was varied from 0 to 0.8. The influence of the height taper ratio of nanobeam-2 \(c_{h_{2}}\) on the first three dimensionless natural transverse frequencies of CC-CC and SS-SS double nanobeams with \(c_{b_{1}}=\ c_{b_{2}}=c_{h_{1}}=0.8,\ \ \mu=0.1,\ K_{G}=10,K_{W}=400\), and \(K=800\) are shown in Figure 7a,b. It is clear that the relation between these frequencies and \(c_{b_{2}}\) was non-monotonic. Figure 4: The first three dimensionless natural transverse frequencies as function of Winkler modulus stiffness \(K_{W}\) of a nonlocal AFG Euler–Bernoulli double nanobeam \(\left(c_{h_{1}}=c_{h_{2}}=0.1,\ c_{h_{1}}=c_{h_{2}}=0.2,\ \ \mu=0.2,\ K_{G}=10,\ \text{and}\ K=100\right)\). (**a**) CC-CC and (**b**) SS-SS. For example, as displayed in Figure 7a, \(\Omega_{1}\) of a CC-CC double AFG nano beam got smaller as \(c_{h_{2}}\) varied from 0 to 0.448. In contrary, \(\Omega_{1}\) of a SS-SS double AFG nano beam increased as \(c_{h_{2}}\) varied from 0 to 0.8 as presented in Figure 7b. In Figure 8, the fundamental dimensionless frequency of a double CS-CS nano beam with \(c_{b_{1}}=c_{h_{2}}=c_{b_{2}}=c_{h_{1}}=0.2,K_{G}=10\) and \(K_{W}=400\) was plotted versus the nonlocal scale parameter and the stiffness of the coupling elastic springs \(K\). The notation CS-CS denotes that both nanobeams were clamped at \(x=0\), and simply supported at \(x=1\). The figure revealed that as \(K\) grew, the percentage of the decrease in \(\Omega_{1}\) with the nonlocal scale parameter got smaller. For instance, at \(K=10\), the dimensionless natural frequency of the CS-CS AFG double nano-beam declined by 66.95% as \(\mu\) increased from 0 to 1. However, at \(K=100\), \(\Omega_{1}\) of the CS-CS double nanobeam decreased by 44.62% as \(\mu\) varied from 0 to 1. Additionally, the results show that at higher values of \(\mu\), the rate at which the fundamental dimensionless natural frequency rises with \(K\) got larger. For example, for \(\mu\) = 0.1 and 0.9, the dimensionless natural frequency rose by 24.59% and 80.86%, respectively, as \(K\) increased from 50 to 400. Figure 9 shows the influence of the breadth taper ratio \(c_{b_{1}}\) and the height  
Figure 5: The first three dimensionless natural transverse frequencies as function of the coupling elastic springs parameter \(K\) of a nonlocal AFG Euler–Bernoulli double nanobeam \(\left(c_{h_{1}}=c_{h_{2}}=c_{h_{1}}=c_{h_{2}}=0.1,\ \ \mu=0.2,\ K_{G}=10,\text{and}\ K_{W}=200\ \right)\). (**a**) CC-CC and (**b**) SS-SS. Figure 6: The effect of the breadth taper ratio of nanobeam-1 \(c_{b_{1}}\) on the first three dimensionless natural transverse frequencies of a nonlocal AFG Euler–Bernoulli double nanobeam \(\left(c_{h_{2}}=c_{h_{1}}=c_{h_{2}}=0.8,\ \ \mu=0.1,\ K_{G}=10,K_{W}=400\,\text{ and}\ K=800\right)\). (**a**) CC-CC and (**b**) SS-SS. taper ratio \(c_{h_{1}}\) of nanobeam-1 on the fundamental frequency of a CC-CC AFG double nano beam with \(c_{h_{2}}=c_{h_{2}}=0.1,\ K_{G}=10,K_{W}=400\), \(K=500\), and \(\mu=0.2\). It is noticed that \(\Omega_{1}\) grows as the parameter \(c_{h_{1}}\) rises. Moreover, as shown in the figure, the frequencies were reduced as \(c_{h_{1}}\) rose to a certain value, and then they tended to augment as \(c_{h_{1}}\) increased. Figures 10-12 present the influence of the boundary conditions on the fundamental dimensionless natural transverse frequencies of AFG double nanobeams with \(c_{h_{1}}=\ c_{h_{2}}=\ c_{h_{2}}=c_{h_{1}}=0.1,\ K_{G}=10\), \(K_{W}=400,\ \text{and}\ K=500\).