On the Vibration Analysis of Rotating Piezoelectric Functionally Graded Beams Resting on Elastic Foundation with a Higher-Order Theory

Table 2

The nondimensional frequencies () of the SS FG beam placed on the two-parameter elastic medium (, , and ).

		Mode	Exact solution [60]	Present work
				4 elements	6 elements	8 elements	10 elements	12 elements
	0	1	5.483	5.505	5.491	5.484	5.482	5.480
		2	21.933	22.504	22.124	22.020	21.977	21.958
		3	49.350	53.548	50.621	49.919	49.669	49.533
	1	1	4.221	4.242	4.230	4.229	4.220	4.220
		2	16.884	17.335	17.044	16.966	16.931	16.919
		3	37.989	41.275	39.034	38.429	38.260	38.211
	2	1	3.852	3.874	3.860	3.860	3.852	3.854
		2	15.407	15.827	15.560	15.487	15.458	15.442
		3	34.666	37.665	35.637	35.105	34.924	34.867
	0	1	6.015	6.038	6.021	6.014	6.008	6.007
		2	22.445	23.024	22.637	22.533	22.491	22.471
		3	49.858	54.077	51.158	50.441	50.164	50.056
	1	1	5.008	5.030	5.014	5.015	5.001	5.000
		2	17.666	18.128	17.827	17.748	17.708	17.697
		3	38.77	42.043	39.791	39.247	39.037	38.930
	2	1	4.752	4.773	4.759	4.756	4.741	4.740
		2	16.316	16.748	16.472	16.395	16.360	16.353
		3	35.577	38.588	36.533	36.006	35.843	35.733