Three-Dimensional Exact Free Vibration Analysis of Spherical, Cylindrical, and Flat One-Layered Panels

Table 14

Benchmark 10. Simply supported orthotropic spherical shell panel. First three vibration modes in terms of no-dimensional circular frequency 𝜔=𝜔(𝑎²/ℎ)𝜌/𝐸₂ for several combinations of half-wave numbers 𝑚 and 𝑛. 𝑀=100 fictitious layers and 𝑁=3 order of expansion for the exponential matrix.

𝑅𝛼/ℎ	1000	100	10	5	1000	100	10	5
	(𝑚=0, 𝑛=1), 𝜃=0^∘				(𝑚=0,  𝑛=1), 𝜃=90^∘
I mode	2385.2	238.51	23.802	11.825	1037.9	104.08	12.585	7.7554
II mode	2902.0	290.20	29.026	14.513	2385.2	238.51	23.802	11.824
III mode	4371.6	437.15	43.559	21.518	12223	1222.3	121.65	51.888
	(𝑚=1, 𝑛=0), 𝜃=0^∘				(𝑚=1, 𝑛=0), 𝜃=90^∘
I mode	1037.9	104.08	12.583	7.7554	2385.2	238.51	23.802	11.825
II mode	2385.2	238.51	23.802	11.824	2902.0	290.20	29.026	14.513
III mode	12223	1222.3	121.65	51.888	4371.6	437.15	43.559	21.518
	(𝑚=1, 𝑛=1), 𝜃=0^∘				(𝑚=1, 𝑛=1), 𝜃=90^∘
I mode	1209.7	121.32	14.754	9.1755	1209.7	121.32	14.754	9.1755
II mode	4016.5	401.63	40.041	19.819	4016.5	401.63	40.041	19.819
III mode	12456	1245.5	123.88	54.506	12456	1245.5	123.88	54.506
	(𝑚=1,  𝑛=2), 𝜃=0^∘				(𝑚=1, 𝑛=2), 𝜃=90^∘
I mode	1682.9	169.01	21.986	14.305	1114.0	117.90	30.685	19.645
II mode	6838.7	683.84	68.035	33.373	5758.3	575.82	57.401	28.398
III mode	13147	1314.6	130.57	60.847	23599	2359.8	208.50	58.572
	(𝑚=2,  𝑛=1), 𝜃=0^∘				(𝑚=2, 𝑛=1), 𝜃=90^∘
I mode	1114.0	117.90	30.685	19.645	1682.9	169.01	21.986	14.305
II mode	5758.3	575.82	57.401	28.398	6838.7	683.84	68.035	33.373
III mode	23599	2359.8	208.50	58.572	13147	1314.6	130.57	60.847
	(𝑚=2, 𝑛=2), 𝜃=0^∘				(𝑚=2, 𝑛=2), 𝜃=90^∘
I mode	1260.9	133.19	34.372	22.411	1260.9	133.19	34.372	22.411
II mode	8009.4	800.89	79.625	39.004	8009.4	800.89	79.625	39.004
III mode	23985	2398.3	216.62	65.349	23985	2398.3	216.62	65.349