Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
The first four normalized frequencies of square plate (h = a/10, a/200)
| Mode | a/h = 10, (SSSS) | ||
|---|---|---|---|
| [28] | Article | [29] | |
| 1 | 4.403 | 4.366 | 4.37 |
| 2 | 6.940 | 6.744 | 6.74 |
| 3 | 6.940 | 6.744 | 6.74 |
| 4 | 8.608 | 8.354 | 8.35 |
| Mode | a/h = 10, (CCCC) | ||
| [28] | Article | [29] | |
| 1 | 5.808 | 5.703 | 5.71 |
| 2 | 8.226 | 7.876 | 7.88 |
| 3 | 8.226 | 7.876 | 7.88 |
| 4 | 9.731 | 9.325 | 9.33 |
| Mode | a/h = 200, (SSSS) | ||
| [28] | Article | [29] | |
| 1 | 4.481 | 4.443 | 4.443 |
| 2 | 7.252 | 7.025 | 7.025 |
| 3 | 7.252 | 7.025 | 7.025 |
| 4 | 9.200 | 8.885 | 8.886 |
| Mode | a/h = 200, (CCCC) | ||
| [28] | Article | [29] | |
| 1 | 6.124 | 5.998 | 5.999 |
| 2 | 9.060 | 8.567 | 8.568 |
| 3 | 9.060 | 8.567 | 8.568 |
| 4 | 11.019 | 10.403 | 10.407 |
The first frequency of FGM plates resting on the elastic foundation with n = 1 and h = a/20
| (k1, k2) | Method | p=q | Meshing | ||
|---|---|---|---|---|---|
| 4 × 4 | 6 × 6 | 8 × 8 | |||
| (100,100) | Article | 1 | 0.0662 | 0.0510 | 0.0474 |
| 2 | 0.0398 | 0.0397 | 0.0396 | ||
| 3 | 0.0396 | 0.0396 | 0.0396 | ||
| 4 | 0.0396 | 0.0396 | 0.0396 | ||
| [30] | 0.0388 | ||||
| [31] | 0.0386 | ||||
The dimensionless frequencies of the FGS-HC plate resting on the elastic foundation_
| a/h | n | (k1, k2) | (2-1-2) | (1-1-1) | (2-2-1) | |||
|---|---|---|---|---|---|---|---|---|
| Article | [32] | Article | [32] | Article | [32] | |||
| 10 | 0 | (0,0) | 1.29918 | 1.29692 | 1.29918 | 1.29692 | 1.29918 | 1.29692 |
| (10,10) | 1.61828 | 1.61603 | 1.61828 | 1.61603 | 1.61828 | 1.61603 | ||
| (100,100) | 3.32217 | 3.31161 | 3.32217 | 3.31161 | 3.32217 | 3.31161 | ||
| 10 | (0,0) | 0.93921 | 0.93742 | 0.95419 | 0.95372 | 0.98414 | 0.98239 | |
| (10,10) | 1.37299 | 1.37067 | 1.38093 | 1.37733 | 1.40154 | 1.39522 | ||
| (100,100) | 3.30716 | 3.29462 | 3.30545 | 3.2805 | 3.30531 | 3.28023 | ||
| 100 | 0 | (0,0) | 1.34067 | 1.34038 | 1.34067 | 1.34038 | 1.34067 | 1.34038 |
| (10,10) | 1.65962 | 1.65899 | 1.65962 | 1.65899 | 1.65962 | 1.65899 | ||
| (100,100) | 3.36947 | 3.36942 | 3.36947 | 3.36942 | 3.36947 | 3.36942 | ||
| 10 | (0,0) | 0.96026 | 0.96023 | 0.97591 | 0.97582 | 1.00790 | 1.00620 | |
| (10,10) | 1.39691 | 1.39670 | 1.40294 | 1.40285 | 1.42311 | 1.42192 | ||
| (100,100) | 3.34844 | 3.34801 | 3.33422 | 3.33315 | 3.33346 | 3.33266 | ||
The first four frequencies of square FGS plates (1-4–1) with a/h = 45, k1 = 50, k2 = 15
| BCs | Plate | n | Normalized frequencies | |||
|---|---|---|---|---|---|---|
| SSSS | FGS-HC | 0 | 2.0351 | 4.8527 | 4.8527 | 7.6586 |
| 0.5 | 1.8611 | 4.3877 | 4.3877 | 6.9055 | ||
| 1 | 1.7707 | 4.1436 | 4.1436 | 6.5090 | ||
| 2 | 1.6811 | 3.8982 | 3.8982 | 6.1097 | ||
| 5 | 1.5964 | 3.6633 | 3.6633 | 5.7257 | ||
| 10 | 1.5643 | 3.5740 | 3.5740 | 5.5797 | ||
| FGS-SC | 0 | 1.3185 | 2.7594 | 2.7594 | 4.1886 | |
| 0.5 | 1.6503 | 3.7252 | 3.7252 | 5.7897 | ||
| 1 | 1.7741 | 4.0725 | 4.0725 | 6.3581 | ||
| 2 | 1.8764 | 4.3540 | 4.3540 | 6.8179 | ||
| 5 | 1.9566 | 4.5755 | 4.5755 | 7.1788 | ||
| 10 | 1.9831 | 4.6487 | 4.6487 | 7.2979 | ||
| SSCC | FGS-HC | 0 | 2.7052 | 5.9047 | 5.9266 | 8.9704 |
| 0.5 | 2.4566 | 5.3292 | 5.3484 | 8.0838 | ||
| 1 | 2.3269 | 5.0266 | 5.0448 | 7.6161 | ||
| 2 | 2.1970 | 4.7216 | 4.7382 | 7.1440 | ||
| 5 | 2.0732 | 4.4288 | 4.4437 | 6.6894 | ||
| 10 | 2.0261 | 4.3177 | 4.3323 | 6.5166 | ||
| FGS-SC | 0 | 1.6225 | 3.2659 | 3.2746 | 4.8327 | |
| 0.5 | 2.1214 | 4.4804 | 4.4956 | 6.7338 | ||
| 1 | 2.3039 | 4.9129 | 4.9301 | 7.4055 | ||
| 2 | 2.4527 | 5.2635 | 5.2827 | 7.9484 | ||
| 5 | 2.5692 | 5.5387 | 5.5590 | 8.3744 | ||
| 10 | 2.6083 | 5.6297 | 5.6504 | 8.5155 | ||
| SCSC | FGS-HC | 0 | 2.8756 | 5.3557 | 6.7353 | 9.1380 |
| 0.5 | 2.6076 | 4.8374 | 6.0736 | 8.2344 | ||
| 1 | 2.4671 | 4.5654 | 5.7249 | 7.7575 | ||
| 2 | 2.3264 | 4.2913 | 5.3733 | 7.2769 | ||
| 5 | 2.1921 | 4.0284 | 5.0346 | 6.8132 | ||
| 10 | 2.1416 | 3.9287 | 4.9057 | 6.6368 | ||
| FGS-SC | 0 | 1.6935 | 2.9985 | 3.6657 | 4.9129 | |
| 0.5 | 2.2376 | 4.0841 | 5.0755 | 6.8517 | ||
| 1 | 2.4353 | 4.4723 | 5.5750 | 7.5362 | ||
| 2 | 2.5967 | 4.7870 | 5.9794 | 8.0896 | ||
| 5 | 2.7231 | 5.0344 | 6.2970 | 8.5240 | ||
| 10 | 2.7652 | 5.1161 | 6.4020 | 8.6677 | ||
| CCCC | FGS-HC | 0 | 3.5315 | 7.1179 | 7.1179 | 10.4331 |
| 0.5 | 3.1928 | 6.4172 | 6.4172 | 9.3997 | ||
| 1 | 3.0152 | 6.0476 | 6.0476 | 8.8533 | ||
| 2 | 2.8367 | 5.6747 | 5.6747 | 8.3016 | ||
| 5 | 2.6652 | 5.3157 | 5.3157 | 7.7691 | ||
| 10 | 2.6004 | 5.1788 | 5.1788 | 7.5661 | ||
| FGS-SC | 0 | 2.0033 | 3.8544 | 3.8544 | 5.5549 | |
| 0.5 | 2.7066 | 5.3520 | 5.3520 | 7.7846 | ||
| 1 | 2.9595 | 5.8820 | 5.8820 | 8.5694 | ||
| 2 | 3.1644 | 6.3109 | 6.3109 | 9.2038 | ||
| 5 | 3.3258 | 6.6474 | 6.6474 | 9.7021 | ||
| 10 | 3.3789 | 6.7588 | 6.7588 | 9.8669 | ||
The dimensionless frequencies of square FGS plates with a/h = 15, k1 = 10, k2 = 10
| BCs | Plate | n | Ratio of thicknesses | |||
|---|---|---|---|---|---|---|
| (1-0-1) | (1-1-1) | (1-2-2) | (1-6-1) | |||
| SSSS | FGS-HC | 0 | 1.9543 | 1.9543 | 1.9543 | 1.9543 |
| 0.5 | 1.5924 | 1.6596 | 1.6907 | 1.8175 | ||
| 1 | 1.4114 | 1.5053 | 1.5556 | 1.7477 | ||
| 2 | 1.2580 | 1.3579 | 1.4278 | 1.6794 | ||
| 5 | 1.1721 | 1.2350 | 1.3213 | 1.6209 | ||
| 10 | 1.1662 | 1.1949 | 1.2819 | 1.6067 | ||
| FGS-SC | 0 | 1.1769 | 1.1769 | 1.1769 | 1.1769 | |
| 0.5 | 1.7605 | 1.6906 | 1.6537 | 1.4641 | ||
| 1 | 1.8992 | 1.8417 | 1.7978 | 1.5740 | ||
| 2 | 1.9714 | 1.9446 | 1.8989 | 1.6657 | ||
| 5 | 1.9852 | 2.0037 | 1.9611 | 1.7343 | ||
| 10 | 1.9733 | 2.0155 | 1.9776 | 1.7504 | ||
| SSCC | FGS-HC | 0 | 2.5957 | 2.5957 | 2.5957 | 2.5957 |
| 0.5 | 2.0944 | 2.1903 | 2.2334 | 2.4092 | ||
| 1 | 1.8388 | 1.9751 | 2.0455 | 2.3131 | ||
| 2 | 1.6161 | 1.7658 | 1.8655 | 2.2194 | ||
| 5 | 1.4846 | 1.5889 | 1.7125 | 2.1385 | ||
| 10 | 1.4699 | 1.5283 | 1.6548 | 2.1181 | ||
| FGS-SC | 0 | 1.4774 | 1.4774 | 1.4774 | 1.4774 | |
| 0.5 | 2.3060 | 2.2035 | 2.1516 | 1.8838 | ||
| 1 | 2.5018 | 2.4142 | 2.3537 | 2.0374 | ||
| 2 | 2.6067 | 2.5587 | 2.4952 | 2.1638 | ||
| 5 | 2.6317 | 2.6435 | 2.5834 | 2.2584 | ||
| 10 | 2.6188 | 2.6624 | 2.6087 | 2.2805 | ||
| SCSC | FGS-HC | 0 | 2.7507 | 2.7507 | 2.7507 | 2.7507 |
| 0.5 | 2.2165 | 2.3197 | 2.3655 | 2.5530 | ||
| 1 | 1.9420 | 2.0896 | 2.1641 | 2.4509 | ||
| 2 | 1.7015 | 1.8651 | 1.9717 | 2.3510 | ||
| 5 | 1.5573 | 1.6737 | 1.8071 | 2.2649 | ||
| 10 | 1.5398 | 1.6082 | 1.7447 | 2.2430 | ||
| FGS-SC | 0 | 1.5457 | 1.5457 | 1.5457 | 1.5457 | |
| 0.5 | 2.4340 | 2.3225 | 2.2671 | 1.9804 | ||
| 1 | 2.6444 | 2.5476 | 2.4825 | 2.1437 | ||
| 2 | 2.7581 | 2.7025 | 2.6345 | 2.2782 | ||
| 5 | 2.7871 | 2.7944 | 2.7302 | 2.3788 | ||
| 10 | 2.7747 | 2.8154 | 2.7575 | 2.4024 | ||
| CCCC | FGS-HC | 0 | 3.3663 | 3.3663 | 3.3663 | 3.3663 |
| 0.5 | 2.7033 | 2.8327 | 2.8898 | 3.1229 | ||
| 1 | 2.3590 | 2.5461 | 2.6399 | 2.9967 | ||
| 2 | 2.0540 | 2.2641 | 2.3985 | 2.8727 | ||
| 5 | 1.8669 | 2.0222 | 2.1906 | 2.7658 | ||
| 10 | 1.8407 | 1.9381 | 2.1113 | 2.7389 | ||
| FGS-SC | 0 | 1.8436 | 1.8436 | 1.8436 | 1.8436 | |
| 0.5 | 2.9587 | 2.8163 | 2.7473 | 2.3877 | ||
| 1 | 3.2223 | 3.0967 | 3.0158 | 2.5904 | ||
| 2 | 3.3669 | 3.2904 | 3.2055 | 2.7581 | ||
| 5 | 3.4070 | 3.4069 | 3.3268 | 2.8827 | ||
| 10 | 3.3937 | 3.4346 | 3.3620 | 2.9119 | ||