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

Table 10

Benchmark 6. Simply supported orthotropic cylindrical shell panel. First three vibration modes in term 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	1248.9	124.89	12.489	6.2444	1096.5	109.67	11.281	6.0108
II mode	1721.6	172.16	17.230	8.6360	1248.9	124.89	12.489	6.2444
III mode	3858.5	385.85	38.590	19.293	6057.8	605.78	60.570	30.272
	(𝑚=1,𝑛=0),𝜃=0^∘				(𝑚=1,𝑛=0),𝜃=90^∘
I mode	8.4450	8.4392	7.6484	6.1872	2.3272	2.4079	2.3754	2.2863
II mode	2385.2	238.52	23.861	11.941	2385.2	238.52	23.861	11.943
III mode	12194	1219.4	121.48	51.358	3476.0	347.58	34.633	17.114
	(𝑚=1,𝑛=1),𝜃=0^∘				(𝑚=1,𝑛=1),𝜃=90^∘
I mode	230.37	24.568	8.0953	6.3709	352.48	35.490	5.3604	4.1227
II mode	2939.9	294.00	29.454	14.782	3588.6	358.85	35.788	17.735
III mode	12257	1225.7	122.10	52.106	6559.0	655.90	65.574	32.756
	(𝑚=1,𝑛=2),𝜃=0^∘				(𝑚=1,𝑛=2),𝜃=90^∘
I mode	641.70	64.887	10.877	6.8670	644.27	65.496	12.428	9.0851
II mode	4155.7	415.58	41.604	20.824	4164.3	416.43	41.609	20.709
III mode	12445	1244.4	123.94	54.240	12370	1237.0	123.62	54.652
	(𝑚=2,𝑛=1),𝜃=0^∘				(𝑚=2,𝑛=1),𝜃=90^∘
I mode	79.744	39.112	28.352	18.587	160.90	20.150	11.545	9.9019
II mode	5062.8	506.29	50.681	25.371	6476.9	647.66	64.461	31.673
III mode	23492	2349.1	205.91	56.149	7983.5	798.33	79.684	39.571
	(𝑚=2,𝑛=2),𝜃=0^∘				(𝑚=2,𝑛=2),𝜃=90^∘
I mode	243.62	46.084	29.034	19.077	364.43	40.611	16.527	13.080
II mode	5849.5	584.96	58.566	29.293	6951.1	695.08	69.132	33.884
III mode	23598	2359.6	208.24	58.169	13115	1311.5	131.01	62.003