Free Vibration Analysis of Functionally Graded Spherical Torus Structure with Uniform Variable Thickness along Axial Direction

Table 7

Results of FG spherical shell with uniform variable thickness under different h₀/R ratio (θ₀ = π/4, θ_ζ = π/2, n = 2, a = 1/b = 0.5/c = 1/p = 1).

Type	Edge conditions	h0/R	m
			1	2	3	4	5
FGMI	F–C	0.03	0.4432	1.0481	1.3822	2.0679	2.4996
		0.05	0.4738	1.1650	1.8301	2.4691	2.9391
		0.07	0.5088	1.3057	2.2010	2.5746	2.9715
		0.09	0.5477	1.4510	2.3658	2.7963	3.0043
		0.11	0.5889	1.5890	2.4201	2.9234	3.1775
	C–C	0.03	1.2589	1.3556	2.1041	3.0770	3.3356
		0.05	1.3466	1.7804	2.9556	3.3752	4.0456
		0.07	1.4468	2.1796	3.2748	3.7918	4.1087
		0.09	1.5525	2.5145	3.3055	4.1287	4.3264
		0.11	1.6557	2.7807	3.3195	4.1461	4.7406
	C–SD	0.03	0.9675	1.2376	1.8193	2.0536	2.8547
		0.05	1.0130	1.5538	2.0093	2.6982	3.5972
		0.07	1.0726	1.8177	2.1034	3.3445	3.6618
		0.09	1.1398	1.9186	2.3257	3.5585	4.0219
		0.11	1.2093	1.9482	2.5766	3.5856	4.4516
	C–E1	0.03	1.0025	1.3522	1.9276	2.1570	3.1102
		0.05	1.0767	1.7613	2.0304	2.9668	3.6263
		0.07	1.1679	1.9748	2.2419	3.4964	3.8330
		0.09	1.2610	2.0140	2.5679	3.5702	4.3233
		0.11	1.3466	2.0409	2.8447	3.5890	4.7302
FGMII	F–C	0.03	0.4424	1.0457	1.3782	2.0609	2.4954
		0.05	0.4722	1.1600	1.8209	2.4612	2.9311
		0.07	0.5062	1.2973	2.1878	2.5618	2.9619
		0.09	0.5438	1.4387	2.3527	2.7753	2.9904
		0.11	0.5837	1.5724	2.4057	2.9053	3.1498
	C–C	0.03	1.2562	1.3515	2.0971	3.0666	3.3306
		0.05	1.3414	1.7711	2.9406	3.3662	4.0339
		0.07	1.4386	2.1640	3.2621	3.7676	4.0941
		0.09	1.5409	2.4924	3.2907	4.1107	4.2912
		0.11	1.6405	2.7525	3.3018	4.1242	4.6963
	C–SD	0.03	0.9658	1.2344	1.8146	2.0492	2.8447
		0.05	1.0103	1.5469	2.0032	2.6846	3.5870
		0.07	1.0687	1.8088	2.0913	3.3260	3.6433
		0.09	1.1345	1.9096	2.3052	3.5433	3.9864
		0.11	1.2024	1.9376	2.5488	3.5665	4.4067
	C–E1	0.03	1.0001	1.3483	1.9236	2.1507	3.0996
		0.05	1.0721	1.7532	2.0238	2.9514	3.6172
		0.07	1.1607	1.9669	2.2262	3.4814	3.8102
		0.09	1.2509	2.0039	2.5452	3.5550	4.2877
		0.11	1.3335	2.0281	2.8156	3.5706	4.6856