# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 4 Results and Discussion
Moreover, the results indicate that as the nonlocal scale parameter rises, the dimensionless transverse frequencies decrease and the reduction is more significant for the higher modes. In Figure 2a, \(\Omega_{1}\) of CC-CC double beam decreased by 55.2%, \(\Omega_{2}\) by 56.48%, and \(\Omega_{3}\) by 75.3% as the nonlocal scale parameter \(\mu\) varied from the range 0 to 1. In Figure 2b, \(\Omega_{1}\) of SS-SS double nanobeam decreased by 35.49%, \(\Omega_{2}\) by 48.09%, and \(\Omega_{3}\) by 67.38%. It is observed that the veering phenomenon took place as the scale parameter grew. The veering occurred when two consecutive frequencies came near each other and then veered apart as a parameter was changed (the nonlocal scale parameter in Figure 2a,b). For example, at \(\mu\) = 0.32, veering occurred between \(\Omega_{2}\) and \(\Omega_{3}\) for the CC-CC and SS-SS AFG double nanobeams. Figure 3a,b displays the impact of the shear modulus parameter \(K_{G}\) on \(\Omega_{1}\), \(\Omega_{2}\), and \(\Omega_{3}\) of AFG CC-CC and SS-SS double nanobeams with \(c_{b_{1}}=c_{b_{2}}=0.1,\;\;c_{b_{1}}=c_{b_{2}}=0.2,\;\;\mu=0.1,\;K_{W}=100\), and \(K=100\). These figures show that the dimensionless natural transverse frequencies grow as \(K_{G}\) increased, and the dimensionless frequencies for the CC-CC double beam were higher than that for the SS-SS double beam for the same value of \(K_{G}\). However, the rate at which these frequencies increase with \(K_{G}\) was higher for the SS-SS double nanobeam. For example, \(\Omega_{1}\) of a SS-SS double nanobeam increased by 7.5%, \(\Omega_{2}\) by 10.56%, and \(\Omega_{3}\) by 1.83% as the parameter \(K_{G}\) increased from 0 to 10. On the other hand, \(\Omega_{1}\) of a CC-CC double nanobeam increased by 2.17%, \(\Omega_{2}\) by 8.48%, and \(\Omega_{3}\) by 0.97% or the same range of \(K_{G}\). Figure 4a,b displays the effect of the Winkler modulus stiffness \(K_{W}\) on the first three dimensionless natural transverse frequencies of AFG CC-CC and SS-SS double nanobeams with \(c_{b_{1}}=c_{b_{2}}=0.1,\;c_{b_{1}}=c_{b_{2}}=0.2,\;\mu=0.2,\;K_{G}=10\), and \(K=100\). As predicted, it was revealed that _rising_\(K_{W}\) had a hardening effect on the frequencies. For the CC-CC and SS-SS double nanobeams, \(\Omega_{2}\) was more affected by increasing the Winkler modulus parameter than \(\Omega_{1}\) and \(\Omega_{3}\). For example, \(\Omega_{2}\) of a CC-CC double nanobeam increased by 23.11% as \(K_{W}\) rose from 0 to 400. The first and third dimensionless frequencies increased by 2.71% and 0.19%, respectively. It is noticed from Figure 4b that \(\Omega_{2}\) of a SS-SS double nanobeam increased nonlinearly with the parameter \(K_{W}\), unlike the second and third dimensionless frequencies that increased linearly. Figure 5 shows the influence of the stiffness of the coupling elastic springs \(K\) on the dimensionless natural frequencies of AFG CC-CC and SS-SS double nanobeams with \(c_{b_{1}}=c_{b_{2}}=\;\;c_{b_{1}}=c_{b_{2}}=0.1,\;\;\mu=0.2,\;K_{G}=10\), and \(K_{W}=200\), and the parameter \(K\) was taken in the range of 0-1000. The figures revealed that the first dimensionless frequencies increased nonlinearly with the parameter \(K\) for the CC-CC and SS-SS doubled nanobeams. For the CC-CC double nanobeam, the second  
dimensionless frequency increased linearly by a rate of 54.55% as the parameter \(K\) grew from 0 to 652, whereas it increased by 1.74% in the range of 625 and 1000. From this figure, it is noted that the rate at which the frequencies rose depended on the range of the parameter \(K\). Furthermore, the veering phenomena takes place between \(\Omega_{2}\) and \(\Omega_{3}\) at \(K=625\) and \(K=348\) for the CC-CC and SS-SS double nanobeams, respectively. The first three dimensionless natural transverse frequencies as function of the breadth taper ratio of nanobeam-1 \(c_{b_{1}}\) for CC-CC and SS-SS double nanobeams with \(c_{b_{2}}=\ c_{h_{1}}=c_{h_{2}}=0.8,\ \ \mu=0.1,\ K_{G}=10\), \(K_{W}=400\), and \(K=800\) are presented in Figure 6a,b.