Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment
Abstract
:1. Introduction
2. Theoretical Formulations
2.1. Description of the Model
2.2. Governing Equations and Boundary Conditions
2.3. Meshfree TRPIM Shape Function
2.4. Discretization of Governing Equations and Boundary Conditions
2.5. Continuous Condition
3. Numerical Results and Discussions
3.1. Verification and Convergence Study
3.2. Numerical Examples
4. Conclusions
- (1)
- The meshless Chebyshev-PRIM technique is effective and has relatively high accuracy in the vibration solution of rotating structures. This method has the advantage of fast convergence, and relatively accurate results can be obtained with a smaller number of nodes.
- (2)
- The increase of the half-apex angle of the conical shell reduces the structural rigidity, so the structural frequency decreases. For the combined structure under the CC boundary, after the cone angle increases to a certain extent, the effect of the rotational speed will decrease, and the frequencies corresponding to different rotational speeds will gradually approach.
- (3)
- If the temperature is too high, thermal stress is accumulated inside the structure, the stiffness of the structure is reduced, and the frequency of the combined structure will also decrease. For the boundary conditions with weakened constraints, such as the CF boundary, thermal buckling also occurs with the increase of the temperature difference.
Author Contributions
Funding
Conflicts of Interest
References
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N | CC | SS | FC | CF | ||||
---|---|---|---|---|---|---|---|---|
n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 | |
5 | 0.2328 | 0.1812 | 0.2232 | 0.1754 | 0.1171 | 0.0803 | 0.0494 | 0.0339 |
6 | 0.2284 | 0.1790 | 0.2262 | 0.1761 | 0.1127 | 0.0825 | 0.0464 | 0.0317 |
7 | 0.2306 | 0.1805 | 0.2255 | 0.1754 | 0.1142 | 0.0833 | 0.0457 | 0.0317 |
8 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0464 | 0.0324 |
9 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
10 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
11 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
12 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
13 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
14 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
15 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
16 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
17 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
18 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
19 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
20 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
21 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |
m | n = 0 | n = 1 | n = 2 | |||
---|---|---|---|---|---|---|
FEM | Present | FEM | Present | FEM | Present | |
1 | 0.50375 | 0.50305 | 0.29287 | 0.29279 | 0.10203 | 0.09996 |
2 | 0.60986 | 0.60985 | 0.63581 | 0.63506 | 0.50290 | 0.50217 |
3 | 0.93092 | 0.93082 | 0.81123 | 0.81141 | 0.69148 | 0.69116 |
4 | 0.95632 | 0.95612 | 0.93088 | 0.93137 | 0.85890 | 0.85888 |
5 | 0.97160 | 0.97134 | 0.94850 | 0.95183 | 0.91607 | 0.91544 |
6 | 1.01188 | 1.01142 | 0.99145 | 0.99156 | 0.96048 | 0.96007 |
n = 3 | n = 4 | n = 5 | ||||
FEM | Present | FEM | Present | FEM | Present | |
1 | 0.09377 | 0.08750 | 0.14460 | 0.14441 | 0.20390 | 0.19930 |
2 | 0.39220 | 0.39115 | 0.33034 | 0.32996 | 0.29633 | 0.29579 |
3 | 0.51518 | 0.51434 | 0.39562 | 0.39537 | 0.37623 | 0.37013 |
4 | 0.75359 | 0.75289 | 0.64458 | 0.64594 | 0.58167 | 0.57874 |
5 | 0.79698 | 0.79629 | 0.69114 | 0.69248 | 0.61422 | 0.61285 |
6 | 0.91939 | 0.91893 | 0.87194 | 0.87098 | 0.81980 | 0.81642 |
Ω* | n | FEM | Present | ||
---|---|---|---|---|---|
w*b | w*f | w*b | w*f | ||
0.01 rad/s | 1 | 0.5264 | 0.5264 | 0.5267 | 0.5267 |
2 | 0.3769 | 0.3769 | 0.3774 | 0.3774 | |
3 | 0.2873 | 0.2873 | 0.2869 | 0.2869 | |
4 | 0.236 | 0.236 | 0.2363 | 0.2363 | |
5 | 0.2231 | 0.2231 | 0.2246 | 0.2246 | |
6 | 0.2474 | 0.2474 | 0.2469 | 0.2469 | |
100 rad/s | 1 | 0.5430 | 0.5097 | 0.5432 | 0.5103 |
2 | 0.3906 | 0.3648 | 0.3904 | 0.3645 | |
3 | 0.3005 | 0.2816 | 0.3010 | 0.2822 | |
4 | 0.2527 | 0.2383 | 0.2528 | 0.2387 | |
5 | 0.2455 | 0.234 | 0.2469 | 0.2352 | |
6 | 0.2747 | 0.2647 | 0.2740 | 0.2645 | |
500 rad/s | 1 | 0.6085 | 0.4422 | 0.6090 | 0.4421 |
2 | 0.4605 | 0.3308 | 0.4609 | 0.3304 | |
3 | 0.4174 | 0.322 | 0.4174 | 0.3222 | |
4 | 0.4484 | 0.3756 | 0.4480 | 0.3762 | |
5 | 0.5212 | 0.4629 | 0.5220 | 0.4633 | |
6 | 0.6157 | 0.5612 | 0.6161 | 0.5608 |
φ | Mode | CC | CS | FC | ||||||
---|---|---|---|---|---|---|---|---|---|---|
FEM | Present | Diff,% | FEM | Present | Diff,% | FEM | Present | Diff,% | ||
π/6 | 1 | 234.96 | 237.01 | 0.872 | 227.4 | 228.08 | 0.299 | 120.2 | 119.54 | −0.549 |
2 | 250.73 | 251.66 | 0.371 | 227.84 | 228.85 | 0.443 | 133.66 | 133.66 | 0 | |
3 | 252.19 | 254.21 | 0.801 | 244.34 | 243.77 | −0.233 | 234.47 | 234.16 | −0.132 | |
4 | 265.13 | 264.69 | −0.166 | 247.19 | 247.93 | 0.299 | 240.92 | 241.13 | 0.087 | |
5 | 272.86 | 274.39 | 0.561 | 265.13 | 264.69 | −0.166 | 270.66 | 270.72 | 0.022 | |
6 | 285.89 | 286.71 | 0.287 | 281.3 | 282.09 | 0.281 | 272.89 | 274.28 | 0.509 | |
π/4 | 1 | 270.23 | 272.39 | 0.799 | 239.64 | 239.73 | 0.038 | 145.93 | 145.81 | −0.082 |
2 | 281.53 | 283.4 | 0.664 | 250.99 | 250.54 | −0.179 | 152.71 | 152.7 | −0.007 | |
3 | 293.78 | 293.32 | −0.157 | 271.81 | 272.52 | 0.261 | 256.96 | 256.67 | −0.113 | |
4 | 294.13 | 296.48 | 0.799 | 293.78 | 293.32 | −0.157 | 268.27 | 270.41 | 0.798 | |
5 | 319.31 | 321.55 | 0.702 | 302.2 | 300.76 | −0.477 | 281.31 | 283.08 | 0.629 | |
6 | 328.18 | 329.1 | 0.28 | 312.49 | 314.21 | 0.55 | 305.66 | 306.68 | 0.334 |
L2/L1 | Ω, rad/s | n | Forward | Backward | ||||||
---|---|---|---|---|---|---|---|---|---|---|
CC | SS | CF | FC | CC | SS | CF | FC | |||
0.5 | 50 | 1 | 0.4723 | 0.4561 | 0.1127 | 0.2549 | 0.4826 | 0.4664 | 0.1230 | 0.2667 |
2 | 0.3809 | 0.3478 | 0.0847 | 0.1157 | 0.3883 | 0.3559 | 0.0928 | 0.1245 | ||
3 | 0.3544 | 0.3153 | 0.1341 | 0.1709 | 0.3595 | 0.3212 | 0.1400 | 0.1776 | ||
100 | 1 | 0.4671 | 0.4509 | 0.1083 | 0.2490 | 0.4877 | 0.4715 | 0.1282 | 0.2726 | |
2 | 0.3772 | 0.3448 | 0.0818 | 0.1120 | 0.3927 | 0.3595 | 0.0980 | 0.1297 | ||
3 | 0.3522 | 0.3139 | 0.1326 | 0.1702 | 0.3640 | 0.3249 | 0.1459 | 0.1820 | ||
1 | 50 | 1 | 0.3735 | 0.3618 | 0.0759 | 0.1835 | 0.3846 | 0.3728 | 0.0862 | 0.1945 |
2 | 0.3448 | 0.3161 | 0.0361 | 0.0921 | 0.3536 | 0.3249 | 0.0449 | 0.1009 | ||
3 | 0.3404 | 0.3028 | 0.0663 | 0.1687 | 0.3470 | 0.3087 | 0.0729 | 0.1754 | ||
100 | 1 | 0.3676 | 0.3559 | 0.0707 | 0.1776 | 0.3905 | 0.3780 | 0.0914 | 0.2004 | |
2 | 0.3411 | 0.3124 | 0.0332 | 0.0884 | 0.3581 | 0.3293 | 0.0508 | 0.1061 | ||
3 | 0.3382 | 0.3006 | 0.0670 | 0.1680 | 0.3507 | 0.3131 | 0.0803 | 0.1805 | ||
1.5 | 50 | 1 | 0.2896 | 0.2829 | 0.0553 | 0.1392 | 0.3006 | 0.2947 | 0.0663 | 0.1510 |
2 | 0.2505 | 0.2358 | 0.0258 | 0.0781 | 0.2593 | 0.2446 | 0.0346 | 0.0869 | ||
3 | 0.2218 | 0.2019 | 0.0597 | 0.1665 | 0.2291 | 0.2092 | 0.0670 | 0.1724 | ||
100 | 1 | 0.2837 | 0.2778 | 0.0501 | 0.1334 | 0.3065 | 0.2999 | 0.0715 | 0.1562 | |
2 | 0.2461 | 0.2313 | 0.0236 | 0.0744 | 0.2645 | 0.2498 | 0.0420 | 0.0928 | ||
3 | 0.2203 | 0.2004 | 0.0612 | 0.1650 | 0.2336 | 0.2137 | 0.0752 | 0.1776 | ||
2 | 50 | 1 | 0.2328 | 0.2284 | 0.0420 | 0.1105 | 0.2439 | 0.2402 | 0.0530 | 0.1223 |
2 | 0.1864 | 0.1783 | 0.0214 | 0.0700 | 0.1952 | 0.1871 | 0.0302 | 0.0788 | ||
3 | 0.1606 | 0.1496 | 0.0575 | 0.1540 | 0.1672 | 0.1569 | 0.0641 | 0.1606 | ||
100 | 1 | 0.2269 | 0.2225 | 0.0368 | 0.1046 | 0.2498 | 0.2453 | 0.0589 | 0.1282 | |
2 | 0.1820 | 0.1739 | 0.0192 | 0.0670 | 0.2004 | 0.1923 | 0.0376 | 0.0847 | ||
3 | 0.1584 | 0.1481 | 0.0589 | 0.1525 | 0.1724 | 0.1621 | 0.0729 | 0.1658 |
φ | Ω, rad/s | n | Forward | Backward | ||||||
---|---|---|---|---|---|---|---|---|---|---|
CC | SS | CF | FC | CC | SS | CF | FC | |||
π/6 | 50 | 1 | 0.2763 | 0.2645 | 0.0729 | 0.1260 | 0.2881 | 0.2763 | 0.0840 | 0.1378 |
2 | 0.2269 | 0.2085 | 0.0766 | 0.1444 | 0.2365 | 0.2173 | 0.0855 | 0.1532 | ||
3 | 0.2807 | 0.2652 | 0.2033 | 0.2763 | 0.2881 | 0.2726 | 0.2107 | 0.2829 | ||
100 | 1 | 0.2704 | 0.2593 | 0.0670 | 0.1201 | 0.2940 | 0.2822 | 0.0899 | 0.1437 | |
2 | 0.2225 | 0.2041 | 0.0729 | 0.1400 | 0.2417 | 0.2225 | 0.0914 | 0.1584 | ||
3 | 0.2785 | 0.2630 | 0.2019 | 0.2741 | 0.2925 | 0.2770 | 0.2159 | 0.2873 | ||
π/4 | 50 | 1 | 0.2859 | 0.2756 | 0.0589 | 0.1297 | 0.2969 | 0.2866 | 0.0700 | 0.1415 |
2 | 0.2306 | 0.2115 | 0.0597 | 0.1606 | 0.2402 | 0.2210 | 0.0685 | 0.1695 | ||
3 | 0.2549 | 0.2365 | 0.1599 | 0.2520 | 0.2616 | 0.2431 | 0.1665 | 0.2586 | ||
100 | 1 | 0.2800 | 0.2697 | 0.0538 | 0.1238 | 0.3028 | 0.2925 | 0.0752 | 0.1474 | |
2 | 0.2269 | 0.2078 | 0.0560 | 0.1562 | 0.2453 | 0.2255 | 0.0744 | 0.1746 | ||
3 | 0.2527 | 0.2343 | 0.1577 | 0.2498 | 0.2667 | 0.2476 | 0.1717 | 0.2638 | ||
π/3 | 50 | 1 | 0.2918 | 0.2822 | 0.0494 | 0.1319 | 0.3028 | 0.2940 | 0.0597 | 0.1437 |
2 | 0.2343 | 0.2144 | 0.0501 | 0.1857 | 0.2431 | 0.2232 | 0.0597 | 0.1945 | ||
3 | 0.2424 | 0.2218 | 0.1356 | 0.2394 | 0.2498 | 0.2291 | 0.1422 | 0.2468 | ||
100 | 1 | 0.2859 | 0.2770 | 0.0442 | 0.1260 | 0.3087 | 0.2999 | 0.0648 | 0.1496 | |
2 | 0.2299 | 0.2100 | 0.0472 | 0.1812 | 0.2483 | 0.2284 | 0.0656 | 0.1989 | ||
3 | 0.2402 | 0.2196 | 0.1341 | 0.2372 | 0.2542 | 0.2336 | 0.1481 | 0.2512 | ||
π/2 | 50 | 1 | 0.2962 | 0.2873 | 0.0427 | 0.1334 | 0.3080 | 0.2991 | 0.0516 | 0.1451 |
2 | 0.2380 | 0.2181 | 0.0457 | 0.2092 | 0.2476 | 0.2269 | 0.0545 | 0.2181 | ||
3 | 0.2372 | 0.2144 | 0.1194 | 0.2321 | 0.2439 | 0.2218 | 0.1267 | 0.2387 | ||
100 | 1 | 0.2903 | 0.2814 | 0.0383 | 0.1275 | 0.3139 | 0.3043 | 0.0560 | 0.1510 | |
2 | 0.2343 | 0.2137 | 0.0427 | 0.2048 | 0.2520 | 0.2321 | 0.0604 | 0.2232 | ||
3 | 0.2350 | 0.2129 | 0.1186 | 0.2299 | 0.2483 | 0.2262 | 0.1326 | 0.2439 |
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Li, Z.; Hu, S.; Zhong, R.; Qin, B.; Zhao, X. Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment. Materials 2022, 15, 6177. https://doi.org/10.3390/ma15176177
Li Z, Hu S, Zhong R, Qin B, Zhao X. Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment. Materials. 2022; 15(17):6177. https://doi.org/10.3390/ma15176177
Chicago/Turabian StyleLi, Zhen, Shuangwei Hu, Rui Zhong, Bin Qin, and Xing Zhao. 2022. "Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment" Materials 15, no. 17: 6177. https://doi.org/10.3390/ma15176177
APA StyleLi, Z., Hu, S., Zhong, R., Qin, B., & Zhao, X. (2022). Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment. Materials, 15(17), 6177. https://doi.org/10.3390/ma15176177