# Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions
## 6 Parametric Investigation
The geometrical properties of the beams are \(\theta=45^{\circ}\), \(\text{R}_{1}=\text{R}_{2}=10\), and \(L_{1}=L_{1}=2R\sin\left(\frac{\theta}{2}\right)\). To investigate the effect of the CNT content, three different values of volume fraction are employed for CNTs, namely, \(V_{CNT}^{\text{a}}=0.12\), 0.17, and 0.28. A uniform distribution of CNTs is employed throughout the beam for comparative purposes, under the same assumption of CNT volume fraction for each arm of the specimen. The frequency response is obtained for a system with S-S/S-S, C-C/C-C, C-S/C-S, and C-F/C-F boundary conditions. The results show that the frequency of the system decreases for an increased value of \(R/h\). This is true for all boundary conditions and all CNT volume fractions. Also, for this example, we investigate the effect of the CNT volume on the frequency response of the system. Based on the results in Figure 3, a higher value of CNT volume fraction yields higher frequencies. For example, for a system with S-S/S-S boundary conditions and a fixed value of \(R/h=10\), the first dimensionless frequency of the system increases from about six to nine by increasing \(V_{CNT}^{\text{a}}\) from 0.12 to 0.28. By comparing the rate of reduction of frequencies for an increased R/h, we can deduce that the CNTs volume fraction does not affect the reduction rate. In addition, the reduction rate for S-S/S-S, C-C/C-C, and C-S/C-S boundary conditions is nonlinear, whereas it becomes approximately linear for a C-F/C-F example. Such effect of BCs can be observed from a comparative evaluation of the plots in Figure 3. It can be also noticed that the frequency results associated with clamped boundary conditions are higher compared to other boundary conditions, in the following decreasing order: C-C/C-C > C-S/C-S > S-S/S-S > C-F/C-F. Figure 4 shows the effect of the interfacial stiffness on the first six frequencies of the system, here defined by means of the coefficient \(A_{BM}\) which is obtained from the matrix properties. Nine different values are here considered as the stiffness factor, from 0.01 up to  
\(10^{6}\), whereas the CNTs volume fraction is assumed equal to 0.17 for a uniform distribution of CNTs. The geometry of beams is kept the same as in previous example, while assuming R/h is equal to 10. As already commented in the previous example, the higher values of frequencies are related to the system with C-C/C-C boundary conditions. For all boundary conditions, except C-F/C-F, the stiffness of the interfacial springs does not affect the first dimensionless frequency of system. Such sensitivity becomes more evident for higher modes of frequencies. The frequencies of the system increased for an increased interfacial stiffness up to a certain value of \(k_{s}\), above which frequencies remain constant. From this figure, it seems that the effect of the interfacial stiffness on the frequency response has a direct relationship with boundary conditions and mode frequency. Figure 5 shows the effect of the CNTs pattern on the first mode frequency of the system, with the same geometry discussed before, and for an interfacial stiffness equal to 100. The volume of CNTs is 0.17 and CNTs are assumed to be distributed linearly in the same pattern in both beams (UD). Based on this figure, the effect of the distribution pattern is related to the boundary condition of the system. In all BCs, except for S-S/S-S, the highest value of the first dimensionless frequency is associated with the FG-X pattern. The lowest values of frequencies are associated with the \(\Lambda\) distribution (for S-S/S-S and C-S/C-S boundary conditions), and to circular CNTs distribution (for C-C/C-C or C-F/C-F boundary conditions).