Table 2 Variation of the first three non-dimensional natural frequencies, with respect to the rotational speed parameter, γ, and the hub radius parameter, δ (τb = 0, τh = 0.5)
From: Flapwise bending vibration analysis of functionally graded rotating double-tapered beams
δ | γ | First frequency | Second frequency | Third frequency | |||
|---|---|---|---|---|---|---|---|
Present approach | Ref. (Ozgumus and Kaya 2006) | Present approach | Ref. (Ozgumus and Kaya 2006) | Present approach | Ref. (Ozgumus and Kaya 2006) | ||
0.0 | 1 | 4.017 | 3.986 | 18.501 | 18.474 | 47.448 | 47.417 |
2 | 4.547 | 4.436 | 19.045 | 18.936 | 47.995 | 47.871 | |
3 | 5.308 | 5.092 | 19.917 | 19.683 | 48.891 | 48.619 | |
4 | 6.211 | 5.878 | 21.077 | 20.685 | 50.117 | 49.645 | |
2.0 | 1 | 4.482 | 4.386 | 18.974 | 18.879 | 47.939 | 47.830 |
2 | 6.033 | 5.742 | 20.819 | 20.473 | 49.902 | 49.486 | |
3 | 7.955 | 7.452 | 23.562 | 22.879 | 52.995 | 52.120 | |
4 | 10.027 | 9.310 | 26.918 | 25.866 | 57.013 | 55.581 | |
0.5 | 1 | 4.138 | 4.090 | 18.621 | 18.576 | 47.571 | 47.521 |
2 | 4.961 | 4.797 | 19.504 | 19.332 | 48.480 | 48.281 | |
3 | 6.082 | 5.777 | 20.892 | 20.531 | 49.953 | 49.520 | |
4 | 7.359 | 6.905 | 22.689 | 22.099 | 51.940 | 51.201 | |
1.0 | 1 | 4.256 | 4.191 | 18.739 | 18.677 | 47.694 | 47.624 |
2 | 5.343 | 5.132 | 19.953 | 19.720 | 48.959 | 48.686 | |
3 | 6.766 | 6.386 | 21.821 | 21.343 | 50.990 | 50.403 | |
4 | 8.346 | 7.793 | 24.187 | 23.425 | 53.692 | 52.706 | |