5. Conclusions

Figure 4.

Table 10.

Figure 5.

18

beams with different boundary conditions.

(b) second-order frequencies, and (c) third-order frequencies.

Finite segment model of the AFG beam.

Mechanics of Functionally Graded Materials and Structures

F Order C-F C-P S-S C-C

α ¼ 0:9 1 2.4641 2.4651 4.0787 4.0789 3.0888 3.0891 4.5585 4.5579

α ¼ 0:5 1 2.0774 2.0772 4.0055 4.0056 3.1344 3.1344 4.7098 4.7096

α ¼ 1:7 1 1.6098 1.6104 3.7738 3.7738 3.1279 3.1277 4.6896 4.6897

α ¼ 2:7 1 1.5370 1.5377 3.7214 3.7217 3.1188 3.1212 4.6629 4.6630

Comparisons between FEM and numerical calculation of linear dimensionless natural frequencies of AFG

Dimensionless natural frequencies of C-F beams vary with parameter F: (a) fundamental frequencies,

Present FEM Present FEM Present FEM Present FEM

2 5.2251 5.2265 7.1520 7.1516 6.2895 6.2893 7.6920 7.6908 3 8.2540 8.2560 10.2762 10.2778 9.4410 9.4420 10.8549 10.8560 4 11.3209 11.324 13.4075 13.4092 12.5854 12.5869 14.0136 14.0148

2 4.8497 4.8491 7.1104 7.1100 6.2859 6.2861 7.8364 7.8360 3 7.9496 7.9501 10.2396 10.2409 9.4278 9.4294 10.9827 10.9836 4 11.0455 11.0652 13.3748 13.3760 12.5668 12.5707 14.1256 14.1273

2 4.4786 4.4792 6.9816 6.9816 6.2887 6.2884 7.8189 7.8187 3 7.7326 7.7332 10.1477 10.1490 9.4315 9.4325 10.9679 10.9696 4 10.9067 10.9079 13.3045 13.3042 12.5726 12.5737 14.1139 14.1158

2 4.4142 4.4142 6.9470 6.9471 6.2907 6.2904 7.7947 7.7948 3 7.6920 7.6931 10.1207 10.1222 9.4350 9.4360 10.9482 10.9497 4 10.8759 108,772 13.2807 13.2823 12.5762 12.5774 14.0972 14.0988

FGMs are innovative materials and are very important in engineering and other applications. Despite the variety of methods and approaches for numerical and analytical investigation of nonuniform FG beams, no simple and fast analytical method applicable for such beams with different boundary conditions and varying cross-sectional area was proposed. In this topic, two analytical approaches, the asymptotic perturbation and the Meijer G-function method, were described to analyze the free vibration of the AFG beams.

Based on the Euler-Bernoulli beam theory, the governing differential equations and related boundary conditions are described, which is more complicated because of the partial differential equation with variable coefficients. For both the asymptotic perturbation and the Meijer G-function method, the variable flexural rigidity and mass density are divided into invariant parts and variable parts firstly. Different analytical processes are then carried out to deal with the variable parts applying perturbation theory and the Meijer G-function, respectively. Finally, the simple formulas are derived for solving the nature frequencies of the AFG beams with C-F boundary conditions followed with C-C, C-S, and C-P conditions, respectively. It is observed that natural frequency increases gradually with the increase of the gradient parameter.

Accuracy of the results is also examined using the available data in the published literature and the finite element method. In fact, it can be clearly found that result of the APM is more accurate in low-order mode, which is caused by the defect of the perturbation theory. However, the APM is simple and easily comprehensible, while the Meijer G-function method is more complex and unintelligible for engineers. In general, the results show that the proposed two analytical methods are efficient and can be used to analyze the free vibration of AFG beams.
