A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

Table 1

Comparison and convergence of the first eight natural frequencies (Hz) for three open shells with F-F-F-F boundary condition.

Model		Model sequence
		1	2	3	4	5	6	7	8
Cylindrical shell	2	35.77	38.351	43.957	48.779	62.233	63.943	66.038	76.359
	4	16.32	25.170	28.765	31.603	35.307	56.076	60.732	61.657
	6	8.071	22.883	26.809	30.312	33.721	54.988	58.716	61.556
	8	7.467	22.336	26.337	30.125	33.328	54.699	58.374	61.541
	10	7.379	22.183	26.207	30.082	33.215	54.609	58.288	61.538
	12	7.354	22.133	26.165	30.068	33.177	54.578	58.261	61.537
	14	7.345	22.113	26.148	30.063	33.162	54.565	58.250	61.536
	16	7.341	22.105	26.141	30.061	33.156	54.560	58.245	61.535
	Ref. [37]	7.225	22.096	26.179	30.011	33.105	54.507	58.182	61.535
	ANSYS [37]	7.312	22.094	26.096	30.014	33.064	54.400	58.106	61.517
Conical shell	2	38.4	54.85	84.08	103.0	159.23	229.3	353.4	381.1
	4	18.7	25.30	32.04	36.89	49.87	72.59	88.88	96.85
	6	15.9	15.98	21.86	35.56	46.43	70.10	79.02	83.43
	8	10.2	14.45	20.23	35.32	45.74	69.56	76.99	78.38
	10	8.37	13.60	19.43	35.25	45.54	69.32	76.40	76.91
	12	7.75	13.16	19.04	35.21	45.47	69.18	76.18	76.41
	14	7.48	12.92	18.83	35.17	45.44	69.05	76.06	76.19
	16	7.48	12.92	18.83	35.17	45.44	69.05	76.06	76.19
	Ref. [37]	7.29	12.59	18.54	35.19	45.38	69.06	75.94	76.05
	Ref. [33]	7.21	12.32	18.21	34.40	44.32	67.78	75.43	76.05
	Experiment [33]	7.50	12.70	18.20	35.60	46.00	59.50	70.40	73.10
	Ref. [35]	7.33	12.85	18.80	35.96	46.18	70.46	76.52	76.69
Spherical shell	2	93.009	134.18	136.42	149.49	176.026	207.42	244.63	250.30
	4	36.887	53.459	95.559	97.201	126.99	182.01	219.35	225.24
	6	34.687	50.676	92.738	94.323	125.45	178.61	217.53	223.42
	8	34.462	50.256	92.318	93.893	125.19	178.05	217.20	223.08
	10	34.413	50.156	92.221	93.792	125.13	177.92	217.17	222.99
	12	34.398	50.125	92.191	93.762	125.11	177.88	217.09	222.96
	14	34.393	50.113	92.179	93.750	125.09	177.86	217.08	222.95
	16	34.390	50.108	92.175	93.745	125.09	177.85	217.07	222.95
	Ref. [37]	33.955	49.944	91.743	92.975	124.21	177.05	216.22	221.60
	Ref. [36]	33.460	48.906	91.297	—	121.80	173.11	211.57	—