# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 6 Parametric Investigation
Based on a comparative evaluation of the frequencies, we get the following sequencies of frequency for each selected boundary condition and different distribution patterns of the reinforcement phase:  
Figure 3: Effect of R/h on first dimensionless frequency of system in varied volume of CNTs and different boundary conditions: (**a**) S-S/S-S, (**b**) C-C/C-C, (**c**) C-F/C-F, (**d**) C-S/C-S.  
For a C-C/C-C system: FG-X > UD > FG-\(\Lambda\) > FG-V > FG-O,  
For a C-S/C-S system: FG-X > UD \(\approx\) FG-V > FG-O > FG-\(\Lambda\),  
For a S-S/S-S system: FG-V > UD > FG-X > FG-O > FG-\(\Lambda\),  
For a C-F/C-F system: FG-X > UD > FG-\(\Lambda\) > FG-V\(>\) FG-O. Furthermore, for V and O distribution patterns of CNTs, the boundary conditions do not affect the first frequency of the system. This statement is true for C-C/C-C, S-S/S-S, and C-S/C-S boundary conditions. Table 6 represent the frequency values for different CNTs volume fractions in the system. In this example, the CNT volume fraction at the top beam remains constant at 0.12, whereas \(V_{CNT}^{*}\) at the bottom beam increases from 0.12 to 0.28. CNTs are uniformly  
Figure 4: Effect of the interfacial stiffness on the first six dimensionless frequencies of the system with different boundary conditions: (**a**) S-S/S-S, (**b**) C-C/C-C, (**c**) C-F/C-F, (**d**) C-S/C-S.  
Figure 5: First dimensionless frequency of the system for different patterns and boundary conditions, for a fixed volume fraction of CNTs equal to 0.12.  
distributed along the beams and the stiffness factor is equal to 100. The geometry of the beams is the same as in the previous example. Based on Table 6, the frequencies of the system slightly increase by increasing the CNT volume in the bottom beam. It also seems that there are no differences in the system frequencies for an increased number of CNTs at the top beam and bottom beam. In other words, it is not important which beam contains a higher volume of CNTs to increase the frequency response of the system. Table 7 shows the frequency response of the system when only the bottom beam is reinforced by CNTs, and the top beam is made totally by PMMA. The volume fraction in the bottom beam varies from zero to 0.28 with an X-type functional graduation, where the condition \(V_{CNT}^{*}=0\) corresponds to a pure matrix material. The geometrical properties of the beams are \(\theta=45^{\circ}\), R\({}_{1}\) = R\({}_{2}\) = 10, R/h = 10, and \(L_{1}=L_{2}=2\ R\ \sin\left(\frac{\theta}{2}\right)\), with a stiffness factor equal to 100. For an increased volume fraction of CNTs from zero to 0.12 in the bottom beam, the frequencies of the system increase for all boundary conditions, while keeping constant for volume fractions from 0.12 to 0.28. This means that the volume of CNTs as reinforcement at the bottom beams can affect the frequencies of the system when the top beam remains unreinforced. Based on the same table, there is no difference in the frequency system when the top or bottom beam is reinforced with CNTs. Figure 6 shows the effect of curvature on the first dimensionless frequencies of the system. The CNTs volume fraction is 0.17 and distributed uniformly along the beams,