Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates
Abstract
:1. Introduction
2. Contribution of Current Study
3. FGM Circular Plate with Porosities
4. Constitutive Relations and Governing Equations
4.1. Coupled Equations of Motion
4.2. Decoupled Equation of Motion
5. Solution of the Problem
- Clamped:
- Simply supported:
- Sliding supported:
- Free:
- Clamped (C):
- Simply supported (SS):
- Sliding supported (S):
- Free (F):
6. Parametric Study
6.1. Imperfect FGM Circular Plate
6.2. Perfect FGM Circular Plate
7. Discussion
7.1. Imperfect FGM Circular Plate
- the fundamental eigenfrequency of the axisymmetric and non-axisymmetric vibrations of the circular plate decreases with the increasing value of the power-law index for the two considered distributions of porosities and all considered values of the porosity volume fraction ;
- for the evenly distributed porosities, the fundamental eigenfrequency of the axisymmetric and non-axisymmetric vibrations of the plate increases with the increasing value of the porosity volume fraction for and decreases for ;
- for the unevenly distributed porosities, the fundamental eigenfrequency of the axisymmetric and non-axisymmetric vibrations of the plate increases with the increasing value of the porosity volume fraction for and decreases for ;
- the influence of values of the porosity volume fraction on the values of the fundamental eigenfrequency of the axisymmetric and non-axisymmetric vibrations of the plate is smaller for the unevenly distributed porosities than for the evenly distributed porosities;
- for the evenly distributed porosities, the fundamental eigenfrequency of the axisymmetric and non-axisymmetric vibrations of plate decreases faster for with the increasing values of the power-law index than for ;
- for the unevenly distributed porosities, the fundamental eigenfrequency of the axisymmetric and non-axisymmetric vibrations of the plate decreases slowly with the increasing values of the power-law index for all considered values of the porosity volume fraction .
7.2. Perfect FGM Circular Plate
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Clamped | Simply Supported | Sliding Supported | Free | ||||||
---|---|---|---|---|---|---|---|---|---|
0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | ||
Present | 10.215 | 21.260 | 4.935 | 13.898 | 14.682 | 3.082 | 9.003 | 20.474 | |
[35] | 10.216 | 21.260 | 4.935 | 13.898 | 14.682 | 3.082 | 9.003 | 20.475 | |
[36] | 10.215 | 21.260 | 4.935 | 13.898 | - | - | 9.003 | 20.474 | |
[37] | 10.215 | 21.260 | 4.935 | 13.898 | - | - | 9.003 | 20.474 | |
[38] | 10.215 | 21.260 | 4.935 | 13.898 | - | - | 9.003 | 20.474 | |
Present | 39.771 | 60.828 | 29.720 | 48.478 | 49.218 | 28.398 | 38.443 | 59.812 | |
[35] | 39.771 | 60.829 | 29.720 | 48.478 | 49.218 | 28.399 | 38.443 | 59.812 | |
[36] | 39.771 | 60.828 | 29.720 | 48.479 | - | - | 38.443 | 59.811 | |
[37] | 39.771 | 60.828 | 29.720 | 48.478 | - | - | 38.443 | 59.811 | |
[38] | 39.771 | 60.828 | 29.719 | 48.478 | - | - | 38.443 | 59.812 | |
Present | 89.104 | 120.079 | 74.156 | 102.773 | 103.499 | 72.859 | 87.750 | 118.957 | |
[35] | 89.104 | 120.079 | 74.156 | 102.772 | 103.499 | 72.859 | 87.750 | 118.957 | |
[36] | 89.104 | 120.079 | 74.156 | 102.773 | - | - | 87.705 | 118.957 | |
[37] | 89.104 | 120.080 | 74.156 | 102.773 | - | - | 87.750 | 118.957 | |
[38] | 89.104 | 120.079 | 74.156 | 102.773 | - | - | 87.750 | 118.957 |
0 | 0.2 | 0.4 | 0.6 | 1 | 2 | 3 | 4 | 5 | |||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 10.215 | 9.481 | 8.896 | 8.436 | 7.797 | 7.090 | 6.867 | 6.777 | 6.724 | |
0.05 | 10.286 | 9.522 | 8.905 | 8.414 | 7.718 | 6.920 | 6.661 | 6.559 | 6.503 | ||
0.1 | 10.362 | 9.566 | 8.914 | 8.387 | 7.623 | 6.712 | 6.401 | 6.280 | 6.219 | ||
0.2 | 10.535 | 9.668 | 8.932 | 8.315 | 7.374 | 6.113 | 5.612 | 5.402 | 5.305 | ||
0.3 | 10.745 | 9.792 | 8.948 | 8.207 | 6.993 | 5.034 | 3.949 | 3.312 | 2.923 | ||
1 | 0 | 21.260 | 19.731 | 18.514 | 17.557 | 16.228 | 14.756 | 14.292 | 14.105 | 13.993 | |
0.05 | 21.406 | 19.816 | 18.533 | 17.510 | 16.062 | 14.402 | 13.863 | 13.650 | 13.533 | ||
0.1 | 21.564 | 19.909 | 18.552 | 17.454 | 15.866 | 13.968 | 13.222 | 13.069 | 12.942 | ||
0.2 | 21.925 | 20.121 | 18.590 | 17.304 | 15.346 | 12.723 | 11.680 | 11.242 | 11.041 | ||
0.3 | 22.362 | 20.380 | 18.622 | 17.081 | 14.554 | 10.478 | 8.220 | 6.894 | 6.084 | ||
0 | 0 | 10.215 | 9.481 | 8.896 | 8.436 | 7.797 | 7.090 | 6.867 | 6.777 | 6.724 | |
0.05 | 10.288 | 9.544 | 8.949 | 8.478 | 7.819 | 7.079 | 6.844 | 6.751 | 6.698 | ||
0.1 | 10.364 | 9.611 | 9.004 | 8.521 | 7.840 | 7.065 | 6.816 | 6.719 | 6.666 | ||
0.2 | 10.523 | 9.751 | 9.120 | 8.612 | 7.882 | 7.023 | 6.738 | 6.630 | 6.577 | ||
0.3 | 10.696 | 9.903 | 9.246 | 8.710 | 7.923 | 6.959 | 6.622 | 6.495 | 6.438 | ||
1 | 0 | 21.260 | 19.731 | 18.514 | 17.557 | 16.228 | 14.756 | 14.292 | 14.105 | 13.993 | |
0.05 | 21.411 | 19.864 | 18.624 | 17.644 | 16.272 | 14.733 | 14.244 | 14.050 | 13.940 | ||
0.1 | 21.568 | 20.001 | 18.738 | 17.734 | 16.316 | 14.703 | 14.182 | 13.983 | 13.874 | ||
0.2 | 21.901 | 20.293 | 18.980 | 17.923 | 16.404 | 14.617 | 14.023 | 13.798 | 13.688 | ||
0.3 | 22.260 | 20.610 | 19.243 | 18.127 | 16.490 | 14.483 | 13.782 | 13.517 | 13.399 |
0 | 0.2 | 0.4 | 0.6 | 1 | 2 | 3 | 4 | 5 | |||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 4.935 | 4.580 | 4.297 | 4.075 | 3.767 | 3.425 | 3.317 | 3.274 | 3.248 | |
0.05 | 4.969 | 4.600 | 4.302 | 4.064 | 3.728 | 3.343 | 3.218 | 3.168 | 3.141 | ||
0.1 | 5.005 | 4.621 | 4.306 | 4.051 | 3.683 | 3.242 | 3.092 | 3.033 | 3.004 | ||
0.2 | 5.089 | 4.670 | 4.315 | 4.017 | 3.562 | 2.953 | 2.711 | 2.609 | 2.563 | ||
0.3 | 5.190 | 4.730 | 4.322 | 3.965 | 3.378 | 2.432 | 1.908 | 1.600 | 1.412 | ||
1 | 0 | 13.898 | 12.898 | 12.103 | 11.477 | 10.608 | 9.646 | 9.343 | 9.220 | 9.147 | |
0.05 | 13.993 | 12.954 | 12.115 | 11.446 | 10.500 | 9.415 | 9.062 | 8.923 | 8.847 | ||
0.1 | 14.097 | 13.015 | 12.127 | 11.410 | 10.372 | 9.131 | 8.708 | 8.543 | 8.460 | ||
0.2 | 14.333 | 13.153 | 12.152 | 11.312 | 10.032 | 8.317 | 7.635 | 7.349 | 7.218 | ||
0.3 | 14.618 | 13.322 | 12.173 | 11.166 | 9.514 | 6.849 | 5.373 | 4.506 | 3.977 | ||
0 | 0 | 4.935 | 4.580 | 4.297 | 4.075 | 3.767 | 3.425 | 3.317 | 3.274 | 3.248 | |
0.05 | 4.970 | 4.611 | 4.323 | 4.095 | 3.777 | 3.420 | 3.306 | 3.261 | 3.236 | ||
0.1 | 5.006 | 4.643 | 4.349 | 4.116 | 3.787 | 3.413 | 3.292 | 3.246 | 3.220 | ||
0.2 | 5.083 | 4.710 | 4.406 | 4.160 | 3.808 | 3.393 | 3.255 | 3.203 | 3.177 | ||
0.3 | 5.167 | 4.784 | 4.467 | 4.207 | 3.828 | 3.362 | 3.199 | 3.137 | 3.110 | ||
1 | 0 | 13.898 | 12.898 | 12.103 | 11.477 | 10.608 | 9.646 | 9.343 | 9.220 | 9.147 | |
0.05 | 13.997 | 12.985 | 12.174 | 11.534 | 10.637 | 9.631 | 9.311 | 9.185 | 9.113 | ||
0.1 | 14.099 | 13.075 | 12.249 | 11.592 | 10.666 | 9.611 | 9.273 | 9.141 | 9.069 | ||
0.2 | 14.317 | 13.266 | 12.408 | 11.716 | 10.724 | 9.555 | 9.167 | 9.020 | 8.948 | ||
0.3 | 14.551 | 13.473 | 12.580 | 11.850 | 10.780 | 9.468 | 9.009 | 8.836 | 8.759 |
0 | 0.2 | 0.4 | 0.6 | 1 | 2 | 3 | 4 | 5 | |||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 14.682 | 13.626 | 12.785 | 12.124 | 11.206 | 10.190 | 9.870 | 9.740 | 9.663 | |
0.05 | 14.782 | 13.685 | 12.798 | 12.092 | 11.092 | 9.946 | 9.573 | 9.426 | 9.346 | ||
0.1 | 14.892 | 13.749 | 12.811 | 12.053 | 10.956 | 9.646 | 9.199 | 9.025 | 8.938 | ||
0.2 | 15.141 | 13.895 | 12.837 | 11.950 | 10.597 | 8.786 | 8.066 | 7.764 | 7.625 | ||
0.3 | 15.442 | 14.074 | 12.860 | 11.796 | 10.051 | 7.236 | 5.676 | 4.761 | 4.201 | ||
1 | 0 | 3.082 | 2.860 | 2.684 | 2.545 | 2.352 | 2.139 | 2.072 | 2.045 | 2.029 | |
0.05 | 3.103 | 2.873 | 2.687 | 2.538 | 2.328 | 2.088 | 2.010 | 1.980 | 1.962 | ||
0.1 | 3.126 | 2.886 | 2.690 | 2.530 | 2.300 | 2.025 | 1.931 | 1.894 | 1.876 | ||
0.2 | 3.178 | 2.917 | 2.695 | 2.509 | 2.225 | 1.844 | 1.693 | 1.630 | 1.600 | ||
0.3 | 3.242 | 2.954 | 2.700 | 2.476 | 2.110 | 1.519 | 1.191 | 0.999 | 0.882 | ||
0 | 0 | 14.682 | 13.626 | 12.785 | 12.124 | 11.206 | 10.190 | 9.870 | 9.740 | 9.663 | |
0.05 | 14.786 | 13.717 | 12.861 | 12.184 | 11.237 | 10.174 | 9.836 | 9.703 | 9.627 | ||
0.1 | 14.895 | 13.812 | 12.940 | 12.246 | 11.268 | 10.154 | 9.795 | 9.656 | 9.581 | ||
0.2 | 15.124 | 14.014 | 13.107 | 12.377 | 11.328 | 10.094 | 9.684 | 9.529 | 9.453 | ||
0.3 | 15.372 | 14.233 | 13.289 | 12.518 | 11.388 | 10.002 | 9.518 | 9.334 | 9.253 | ||
1 | 0 | 3.082 | 2.860 | 2.684 | 2.545 | 2.352 | 2.139 | 2.072 | 2.045 | 2.029 | |
0.05 | 3.104 | 2.880 | 2.700 | 2.558 | 2.359 | 2.136 | 2.065 | 2.037 | 2.021 | ||
0.1 | 3.127 | 2.890 | 2.716 | 2.571 | 2.365 | 2.131 | 2.056 | 2.027 | 2.011 | ||
0.2 | 3.175 | 2.942 | 2.752 | 2.598 | 2.378 | 2.119 | 2.033 | 2.000 | 1.984 | ||
0.3 | 3.227 | 2.988 | 2.790 | 2.628 | 2.391 | 2.100 | 1.998 | 1.960 | 1.942 |
0 | 0.2 | 0.4 | 0.6 | 1 | 2 | 3 | 4 | 5 | |||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 9.003 | 8.355 | 7.840 | 7.435 | 6.872 | 6.248 | 6.052 | 5.973 | 5.926 | |
0.05 | 9.064 | 8.391 | 7.848 | 7.415 | 6.802 | 6.099 | 5.870 | 5.780 | 5.731 | ||
0.1 | 9.132 | 8.431 | 7.856 | 7.391 | 6.718 | 5.915 | 5.641 | 5.534 | 5.480 | ||
0.2 | 9.284 | 8.521 | 7.872 | 7.328 | 6.498 | 5.388 | 4.946 | 4.761 | 4.675 | ||
0.3 | 9.469 | 8.630 | 7.886 | 7.233 | 6.163 | 4.437 | 3.481 | 2.919 | 2.576 | ||
1 | 0 | 20.474 | 19.002 | 17.830 | 16.908 | 15.628 | 14.211 | 13.764 | 13.584 | 13.476 | |
0.05 | 20.615 | 19.084 | 17.848 | 16.863 | 15.468 | 13.870 | 13.350 | 13.145 | 13.033 | ||
0.1 | 20.767 | 19.173 | 17.866 | 16.809 | 15.280 | 13.452 | 12.829 | 12.586 | 12.464 | ||
0.2 | 21.115 | 19.378 | 17.902 | 16.665 | 14.779 | 12.253 | 11.248 | 10.827 | 10.633 | ||
0.3 | 21.535 | 19.628 | 17.934 | 16.450 | 14.016 | 10.091 | 7.916 | 6.639 | 5.859 | ||
0 | 0 | 9.003 | 8.355 | 7.840 | 7.435 | 6.872 | 6.248 | 6.052 | 5.973 | 5.926 | |
0.05 | 9.067 | 8.411 | 7.886 | 7.471 | 6.890 | 6.239 | 6.032 | 5.950 | 5.903 | ||
0.1 | 9.133 | 8.470 | 7.935 | 7.509 | 6.909 | 6.226 | 6.007 | 5.921 | 5.875 | ||
0.2 | 9.274 | 8.593 | 8.037 | 7.590 | 6.947 | 6.190 | 5.938 | 5.843 | 5.796 | ||
0.3 | 9.426 | 8.728 | 8.149 | 7.676 | 6.983 | 6.133 | 5.836 | 5.724 | 5.674 | ||
1 | 0 | 20.474 | 19.002 | 17.830 | 16.908 | 15.628 | 14.211 | 13.764 | 13.584 | 13.476 | |
0.05 | 20.620 | 19.129 | 17.935 | 16.992 | 15.670 | 14.188 | 13.718 | 13.531 | 13.425 | ||
0.1 | 20.771 | 19.262 | 18.045 | 17.078 | 15.713 | 14.160 | 13.660 | 13.466 | 13.361 | ||
0.2 | 21.091 | 19.543 | 18.279 | 17.261 | 15.798 | 14.077 | 13.505 | 13.288 | 13.182 | ||
0.3 | 21.437 | 19.848 | 18.532 | 17.457 | 15.881 | 13.948 | 13.273 | 13.017 | 12.903 |
1 | 2 | 3 | 4 | 5 | ||||
---|---|---|---|---|---|---|---|---|
0 | Present | 7.797 | 7.090 | 6.867 | 6.777 | 6.724 | 5.199 | |
[27] | 8.498 | 8.123 | 7.911 | 7.733 | 7.573 | - | ||
Present | 30.357 | 27.604 | 26.737 | 26.386 | 26.177 | 20.243 | ||
[27] | 33.086 | 31.625 | 30.798 | 30.107 | 29.485 | - | ||
Present | 68.012 | 61.845 | 59.902 | 59.116 | 58.649 | 45.352 | ||
[27] | 74.127 | 70.855 | 69.002 | 67.453 | 66.059 | - | ||
1 | 16.228 | 14.756 | 14.292 | 14.105 | 13.993 | 10.821 | ||
Present | 46.430 | 42.219 | 40.893 | 40.357 | 40.038 | 30.961 | ||
91.655 | 83.344 | 80.725 | 79.667 | 79.037 | 61.118 |
1 | 2 | 3 | 4 | 5 | ||||
---|---|---|---|---|---|---|---|---|
0 | Present | 3.767 | 3.425 | 3.317 | 3.274 | 3.248 | 2.512 | |
[27] | 4.105 | 3.924 | 3.821 | 3.736 | 3.658 | - | ||
Present | 22.685 | 20.628 | 19.980 | 19.717 | 19.562 | 15.127 | ||
[27] | 24.724 | 23.633 | 23.015 | 22.498 | 22.033 | - | ||
Present | 56.602 | 51.470 | 49.853 | 49.199 | 48.810 | 37.744 | ||
[27] | 61.692 | 58.968 | 57.426 | 56.137 | 54.977 | - | ||
1 | 10.608 | 9.646 | 9.343 | 9.220 | 9.147 | 7.074 | ||
Present | 37.003 | 33.648 | 32.591 | 32.163 | 31.909 | 24.675 | ||
78.446 | 71.332 | 69.091 | 68.185 | 67.646 | 52.310 |
1 | 2 | 3 | 4 | 5 | |||
---|---|---|---|---|---|---|---|
0 | 6.872 | 6.248 | 6.052 | 5.973 | 5.926 | 4.582 | |
29.343 | 26.682 | 25.844 | 25.505 | 25.303 | 19.567 | ||
66.979 | 60.905 | 58.992 | 58.218 | 57.757 | 44.663 | ||
1 | 15.628 | 14.211 | 13.764 | 13.584 | 13.476 | 10.421 | |
45.653 | 41.513 | 40.209 | 39.682 | 39.368 | 30.443 | ||
90.799 | 82.565 | 79.971 | 78.922 | 78.298 | 60.547 |
1 | 2 | 3 | 4 | 5 | |||
---|---|---|---|---|---|---|---|
0 | 11.206 | 10.190 | 9.870 | 9.740 | 9.663 | 7.473 | |
37.568 | 34.161 | 33.088 | 32.654 | 32.396 | 25.051 | ||
79.000 | 71.836 | 69.579 | 68.667 | 68.124 | 52.680 | ||
1 | 2.352 | 2.139 | 2.072 | 2.045 | 2.029 | 1.568 | |
21.676 | 19.711 | 19.091 | 18.841 | 18.692 | 14.454 | ||
55.612 | 50.569 | 48.981 | 48.338 | 47.956 | 37.084 |
BCs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 10 | 30 | 60 | ||||
Clamped | 0 | 8.498 | 8.123 | 7.911 | 7.733 | 7.573 | 6.977 | 6.064 | 5.687 | 5.199 | |
7.797 | 7.090 | 6.867 | 6.777 | 6.724 | 6.512 | 5.960 | 5.654 | 5.199 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
1 | 17.687 | 16.906 | 16.464 | 16.094 | 15.762 | 14.520 | 12.621 | 11.835 | 10.821 | ||
16.228 | 14.756 | 14.292 | 14.105 | 13.993 | 13.552 | 12.404 | 11.767 | 10.821 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
Simply supported | 0 | 4.105 | 3.924 | 3.821 | 3.736 | 3.658 | 3.370 | 2.929 | 2.747 | 2.512 | |
3.767 | 3.425 | 3.317 | 3.274 | 3.248 | 3.145 | 2.879 | 2.731 | 2.512 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
1 | 11.562 | 11.051 | 10.762 | 10.521 | 10.303 | 9.492 | 8.250 | 7.737 | 7.074 | ||
10.608 | 9.646 | 9.343 | 9.220 | 9.147 | 8.859 | 8.108 | 7.692 | 7.074 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
Sliding supported | 0 | 12.214 | 11.675 | 11.369 | 11.114 | 10.885 | 10.027 | 8.716 | 8.173 | 7.473 | |
11.206 | 10.190 | 9.870 | 9.740 | 9.663 | 9.359 | 8.566 | 8.126 | 7.473 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
1 | 2.564 | 2.451 | 2.387 | 2.333 | 2.285 | 2.105 | 1.830 | 1.716 | 1.569 | ||
2.352 | 2.139 | 2.072 | 2.045 | 2.029 | 1.964 | 1.798 | 1.706 | 1.569 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
Free | 0 | 7.489 | 7.159 | 6.972 | 6.815 | 6.674 | 6.149 | 5.344 | 5.012 | 4.582 | |
6.872 | 6.248 | 6.052 | 5.973 | 5.926 | 5.739 | 5.252 | 4.983 | 4.582 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 | |||
1 | 17.033 | 16.281 | 15.855 | 15.499 | 15.179 | 13.984 | 12.155 | 11.398 | 10.421 | ||
15.628 | 14.211 | 13.764 | 13.584 | 13.476 | 13.051 | 11.945 | 11.332 | 10.421 | |||
8.2 | 12.7 | 13.2 | 12.3 | 11.2 | 6.6 | 1.7 | 0.5 | 0 |
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Żur, K.K.; Jankowski, P. Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates. Symmetry 2019, 11, 429. https://doi.org/10.3390/sym11030429
Żur KK, Jankowski P. Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates. Symmetry. 2019; 11(3):429. https://doi.org/10.3390/sym11030429
Chicago/Turabian StyleŻur, Krzysztof Kamil, and Piotr Jankowski. 2019. "Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates" Symmetry 11, no. 3: 429. https://doi.org/10.3390/sym11030429
APA StyleŻur, K. K., & Jankowski, P. (2019). Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates. Symmetry, 11(3), 429. https://doi.org/10.3390/sym11030429