Free Vibration Investigations of Rotating FG Beams Resting on Elastic Foundation with Initial Geometrical Imperfection in Thermal Environments

Table 1

The comparative frequencies () of the FGM resting on the two-parameter elastic foundation, , and S-S.

	a/h = 100
	n	Mode	Exact [44]	This work
				6 elements	8 elements	10 elements	12 elements	14 elements
Without elastic foundation and	0	1	5.483	5.483	5.483	5.483	5.483	5.483
		2	21.933	21.931	21.924	21.922	21.921	21.921
		3	49.350	49.389	49.318	49.296	49.288	49.284
	1	1	4.221	4.220	4.220	4.220	4.220	4.220
		2	16.884	16.884	16.879	16.877	16.876	16.876
		3	37.989	38.030	37.975	37.958	37.952	37.949
	2	1	3.852	3.851	3.851	3.851	3.851	3.851
		2	15.407	15.408	15.403	15.402	15.401	15.401
		3	34.666	34.707	34.657	34.641	34.635	34.633
With elastic foundation and	0	1	6.015	6.014	6.014	6.014	6.014	6.014
		2	22.445	22.442	22.436	22.434	22.433	22.433
		3	49.858	49.891	49.824	49.803	49.796	49.792
	1	1	5.008	5.007	5.007	5.007	5.007	5.007
		2	17.666	17.664	17.660	17.659	17.658	17.658
		3	38.77	38.801	38.753	38.738	38.732	38.729
	2	1	4.752	4.752	4.752	4.752	4.752	4.752
		2	16.316	16.315	16.311	16.310	16.310	16.309
		3	35.577	35.606	35.564	35.550	35.545	35.543