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

Table 13

Benchmark 9. Simply supported isotropic spherical shell panel. First three vibration modes in terms of no-dimensional circular frequency 𝜔=𝜔(𝑎²/ℎ)𝜌/𝐸 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
	(𝑚=0, 𝑛=1)
I mode	999.01	99.924	10.213	5.3986
II mode	2040.3	204.02	20.360	10.115
III mode	3785.7	378.55	37.677	18.556
	(𝑚=1,  𝑛=0)
I mode	999.01	99.924	10.213	5.3986
II mode	2040.3	204.02	20.360	10.116
III mode	3785.7	378.55	37.677	18.556
	(𝑚=1,  𝑛=1)
I mode	1046.3	104.76	11.564	6.9554
II mode	2885.4	288.53	28.793	14.302
III mode	5111.6	511.13	50.821	24.937
	(𝑚=1, 𝑛=2)
I mode	1076.3	108.52	16.906	12.400
II mode	4562.2	456.21	45.524	22.598
III mode	7857.9	785.72	77.933	37.847
	(𝑚=2, 𝑛=1)
I mode	1076.3	108.52	16.906	12.400
II mode	4562.2	456.21	45.524	22.598
III mode	7857.9	785.72	77.933	37.847
	(𝑚=2,  𝑛=2)
I mode	1084.1	110.77	23.273	17.513
II mode	5770.8	577.07	57.581	28.565
III mode	9869.7	986.87	97.645	46.869