Free Vibration Analysis of Moderately Thick Rectangular Plates with Variable Thickness and Arbitrary Boundary Conditions

Table 10

Frequency parameter Ω=𝜔𝑎²(𝜌ℎ₀/𝐷₀)^1/2 for S-S-S-S moderately thick plates (𝑎/𝑏=1) with linearly varying thickness and elastic rotation support (ℎ(𝑥,𝑦)=ℎ₀(1+0.5𝑥²)(1+0.5𝑦²), 𝐾_𝑥0=𝐾_𝑥𝑎=𝐾_𝑦0=𝐾_𝑦𝑏=Γ_𝑥).

Γ𝑥^a	ℎ0/𝑎	Model sequence
		1	2	3	4	5	6
0	0.05	23.462	54.999	55.573	82.950	100.900	101.187
		(23.741^b)	(55.114)	(55.983)	(83.146)	(101.004)	(101.347)
	0.1	18.606	42.725	42.971	60.040	71.167	71.761
		(18.936^b)	(43.014)	(43.148)	(60.247)	(71.642)	(71.993)
	0.15	53.055	65.251	85.404	106.700	114.880	143.531
		(53.467^b)	(65.472)	(85.984)	(107.211)	(115.687)	(144.007)
	0.2	17.439	21.531	32.170	44.987	59.438	64.179
		(17.874^b)	(21.748)	(32.651)	(45.264)	(60.188)	(64.879)
0.5	0.05	6.904	13.374	13.514	19.400	22.904	23.056
		(9.909^b)	(13.548)	(13.651)	(19.564)	(23.088)	(23.379)
	0.1	5.316	10.237	10.314	14.499	17.155	17.264
		(5.374^b)	(10.448)	(10.651)	(14.764)	(17.188)	(17.879)
	0.15	4.136	8.201	8.220	11.474	13.578	13.671
		(4.174^b)	(8.348)	(8.651)	(11.764)	(13.688)	(13.879)
	0.2	3.449	6.915	6.924	9.549	11.225	11.327
		(3.474^b)	(6.948)	(6.951)	(9.764)	(11.388)	(11.579)
1	0.05	7.040	13.617	13.745	19.726	23.233	23.402
		(7.074^b)	(13.748)	(13.851)	(19.964)	(23.488)	(23.879)
	0.1	5.712	10.696	10.793	14.991	17.594	17.744
		(5.774^b)	(10.748)	(10.851)	(15.054)	(17.788)	(17.879)
	0.15	4.489	8.507	8.540	11.759	13.803	13.922
		(4.514^b)	(8.548)	(8.611)	(11.864)	(13.988)	(14.119)
	0.2	3.677	7.084	7.090	9.692	11.330	11.445
		(3.684^b)	(7.148)	(7.191)	(9.764)	(11.588)	(11.879)
1.5	0.05	7.089	13.708	13.828	19.847	23.355	23.530
		(7.124^b)	(13.748)	(13.951)	(19.967)	(23.588)	(23.879)
	0.1	5.895	10.923	11.024	15.239	17.813	17.988
		(5.874^b)	(10.748)	(11.651)	(15.264)	(17.818)	(17.979)
	0.15	4.697	8.694	8.737	11.938	13.790	14.083
		(4.774^b)	(8.748)	(8.951)	(11.864)	(13.688)	(14.879)
	0.2	3.836	7.201	7.210	9.792	11.403	11.530
		(3.874^b)	(7.748)	(7.651)	(9.864)	(11.588)	(11.879)
2	0.05	7.114	13.754	13.871	19.911	23.418	23.597
		(7.174^b)	(12.848)	(13.851)	(19.864)	(23.588)	(23.879)
	0.1	6.001	11.060	11.159	15.391	17.944	18.136
		(6.004^b)	(11.148)	(11.151)	(15.464)	(17.988)	(18.379)
	0.15	4.836	8.821	8.872	12.061	14.037	14.189
		(4.874^b)	(8.848)	(8.951)	(12.164)	(14.188)	(14.879)
	0.2	3.954	7.289	7.301	9.871	11.457	11.594
		(3.974^b)	(7.348)	(7.651)	(9.964)	(11.488)	(11.679)

^aΓ_𝑥=𝐾_𝑖/𝐾₀,(𝐾₀=1×10⁹Nm/rad,  𝑖=𝑥0,𝑥𝑎,𝑦0,𝑦𝑏).
^bResults in parentheses are taken from FEM.