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

Table 6

Benchmark 2. Simply supported orthotropic square plate. 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=1 fictitious layers and 𝑁=18 order of expansion for the exponential matrix.

𝑎/ℎ	100	50	10	5	100	50	10	5
	(𝑚=0,  𝑛=1),𝜃=0^∘				(𝑚=0,𝑛=1),  𝜃=90^∘
I mode	2.8553	2.8537	2.8058	2.6731	10.007	9.9712	9.0075	7.2059
II mode	227.77	113.88	22.777	11.388	227.77	113.88	22.777	11.388
III mode	314.89	157.44	31.457	15.679	1104.7	552.35	110.41	46.861
	(𝑚=1,  𝑛=0),𝜃=0^∘				(𝑚=1,𝑛=0),𝜃=90^∘
I mode	10.007	9.9712	9.0075	7.2059	2.8553	2.8537	2.8058	2.6731
II mode	227.77	113.88	22.777	11.388	227.77	113.88	22.777	11.388
III mode	1104.7	552.35	110.41	46.861	314.89	157.44	31.457	15.679
	(𝑚=1,𝑛=1),𝜃=0^∘				(𝑚=1,𝑛=1),𝜃=90^∘
I mode	11.367	11.325	10.226	8.2183	11.367	11.325	10.226	8.2183
II mode	382.03	191.01	38.169	19.032	382.03	191.01	38.169	19.032
III mode	1130.2	565.09	112.94	49.636	1130.2	565.09	112.94	49.636
	(𝑚=1,𝑛=2),𝜃=0^∘				(𝑚=1,𝑛=2),𝜃=90^∘
I mode	17.719	17.648	15.817	12.656	41.011	40.436	29.614	19.220
II mode	652.40	326.17	65.004	32.100	549.29	274.64	54.882	27.380
III mode	1204.5	602.22	120.31	56.667	2222.2	1111.0	190.33	53.344
	(𝑚=2,𝑛=1),𝜃=0^∘				(𝑚=2,𝑛=1),𝜃=90^∘
I mode	41.011	40.436	29.614	19.220	17.719	17.648	15.817	12.656
II mode	549.29	274.64	54.882	27.380	652.40	326.17	65.004	32.100
III mode	2222.2	1111.0	190.33	53.344	1204.5	602.22	120.31	56.667
	(𝑚=2,𝑛=2),𝜃=0^∘				(𝑚=2,𝑛=2),𝜃=90^∘
I mode	45.302	44.664	32.873	21.768	45.302	44.664	32.873	21.768
II mode	764.05	381.98	76.128	37.625	764.05	381.98	76.128	37.625
III mode	2260.4	1130.1	198.54	59.836	2260.4	1130.1	198.54	59.836