Research Article
Free Vibration Analysis of Functionally Graded Spherical Torus Structure with Uniform Variable Thickness along Axial Direction
Table 6
Results of FG spherical shell with uniform variable thickness subject to classical-elastic edge conditions (θ0 = π/4, θζ = π/2, h0 = 0.05 m, R = 1 m, a = 1/b = 0.5/c = 1/p = 1).
| | Type | n | m | Edge conditions | | F–E1 | F–E2 | F–E3 | C–E1 | C–E2 | C–E3 | SD–E1 | SD–E2 | SD–E3 |
| | FGMI | 1 | 1 | 0.1745 | 0.4219 | 0.1731 | 0.9155 | 1.0662 | 0.6921 | 0.6555 | 0.9871 | 0.3861 | | 2 | 0.8606 | 1.0758 | 0.4417 | 1.7104 | 1.7134 | 1.4281 | 1.1713 | 1.4578 | 1.1026 | | 3 | 1.1189 | 1.4545 | 1.0907 | 2.0695 | 1.8432 | 1.7637 | 1.5714 | 1.5141 | 1.4661 | | 4 | 1.8136 | 1.7704 | 1.4641 | 2.7920 | 3.0021 | 2.2959 | 2.5230 | 2.5401 | 1.6211 | | 5 | 2.1632 | 2.3807 | 1.8299 | 3.1121 | 3.5931 | 3.0201 | 2.6604 | 2.8782 | 2.6603 | | 6 | 3.0061 | 2.9868 | 2.9843 | 4.4865 | 4.2580 | 3.8824 | 4.1531 | 3.7194 | 3.6927 | | 2 | 1 | 0.1933 | 0.3800 | 0.1874 | 1.0767 | 1.1380 | 0.7658 | 1.0242 | 1.0231 | 0.6860 | | 2 | 1.0412 | 1.1309 | 0.6800 | 1.7613 | 1.7723 | 1.7434 | 1.3829 | 1.5690 | 1.3580 | | 3 | 1.1660 | 1.8245 | 1.1514 | 2.0304 | 2.4699 | 2.0269 | 1.6101 | 2.0579 | 1.5706 | | 4 | 1.8698 | 1.9600 | 1.8267 | 2.9668 | 3.0760 | 2.4819 | 2.7003 | 2.6571 | 2.2832 | | 5 | 2.8811 | 2.7052 | 2.2975 | 3.6263 | 3.6515 | 3.0777 | 2.9070 | 3.2501 | 2.7146 | | 6 | 3.0606 | 3.1279 | 3.0580 | 4.5289 | 4.4547 | 4.4066 | 4.1992 | 3.9565 | 3.8826 | | 3 | 1 | 0.3178 | 0.3600 | 0.3176 | 1.1726 | 1.1774 | 0.9256 | 1.0892 | 1.0714 | 0.8863 | | 2 | 1.2268 | 1.2148 | 0.9903 | 1.8612 | 1.8500 | 1.8494 | 1.6712 | 1.6529 | 1.6088 | | 3 | 1.6334 | 1.9387 | 1.4638 | 2.2805 | 2.9585 | 2.1829 | 1.9613 | 2.6623 | 1.8107 | | 4 | 1.9615 | 2.3374 | 1.9387 | 3.0505 | 3.3536 | 2.9825 | 2.7730 | 2.7692 | 2.7441 | | 5 | 3.1516 | 3.1079 | 3.0559 | 4.2551 | 4.0116 | 3.4333 | 3.4013 | 3.9011 | 3.1271 | | 6 | 3.3389 | 3.6991 | 3.1786 | 4.6062 | 4.5850 | 4.5660 | 4.2698 | 4.2883 | 4.1731 |
| | FGMII | 1 | 1 | 0.1740 | 0.4204 | 0.1727 | 0.9116 | 1.0626 | 0.6890 | 0.6528 | 0.9840 | 0.3845 | | 2 | 0.8576 | 1.0713 | 0.4400 | 1.7025 | 1.7044 | 1.4237 | 1.1678 | 1.4534 | 1.0992 | | 3 | 1.1141 | 1.4503 | 1.0862 | 2.0630 | 1.8375 | 1.7553 | 1.5638 | 1.5068 | 1.4605 | | 4 | 1.8042 | 1.7622 | 1.4595 | 2.7821 | 2.9863 | 2.2892 | 2.5167 | 2.5292 | 1.6146 | | 5 | 2.1584 | 2.3732 | 1.8209 | 3.0990 | 3.5838 | 3.0043 | 2.6461 | 2.8676 | 2.6461 | | 6 | 2.9898 | 2.9728 | 2.9719 | 4.4628 | 4.2411 | 3.8726 | 4.1306 | 3.7097 | 3.6832 | | 2 | 1 | 0.1924 | 0.3784 | 0.1866 | 1.0721 | 1.1336 | 0.7622 | 1.0206 | 1.0205 | 0.6831 | | 2 | 1.0374 | 1.1261 | 0.6772 | 1.7532 | 1.7630 | 1.7353 | 1.3788 | 1.5614 | 1.3548 | | 3 | 1.1610 | 1.8151 | 1.1466 | 2.0238 | 2.4627 | 2.0202 | 1.6032 | 2.0527 | 1.5632 | | 4 | 1.8601 | 1.9550 | 1.8172 | 2.9514 | 3.0605 | 2.4748 | 2.6864 | 2.6447 | 2.2769 | | 5 | 2.8744 | 2.6961 | 2.2915 | 3.6172 | 3.6422 | 3.0621 | 2.8999 | 3.2396 | 2.7009 | | 6 | 3.0440 | 3.1135 | 3.0414 | 4.5051 | 4.4333 | 4.3916 | 4.1767 | 3.9465 | 3.8722 | | 3 | 1 | 0.3161 | 0.3581 | 0.3159 | 1.1675 | 1.1726 | 0.9212 | 1.0865 | 1.0692 | 0.8831 | | 2 | 1.2214 | 1.2096 | 0.9859 | 1.8514 | 1.8402 | 1.8395 | 1.6639 | 1.6456 | 1.6044 | | 3 | 1.6284 | 1.9287 | 1.4588 | 2.2741 | 2.9447 | 2.1770 | 1.9559 | 2.6526 | 1.8037 | | 4 | 1.9512 | 2.3310 | 1.9287 | 3.0340 | 3.3427 | 2.9677 | 2.7593 | 2.7590 | 2.7310 | | 5 | 3.1345 | 3.0914 | 3.0451 | 4.2446 | 4.0010 | 3.4229 | 3.3930 | 3.8867 | 3.1192 | | 6 | 3.3305 | 3.6897 | 3.1643 | 4.5823 | 4.5624 | 4.5424 | 4.2476 | 4.2748 | 4.1545 |
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