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

Table 6

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+0.5𝑥)(1+0.5𝑦), 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	28.809	70.080	70.891	108.967	134.884	135.295
		(28.897^b)	(70.334)	(71.129)	(109.112)	(135.254)	(135.772)
	0.1	27.261	62.398	62.926	92.423	111.540	111.735
		(26.606^b)	(61.784)	(62.303)	(91.193)	(111.323)	(111.648)
	0.15	25.231	54.192	54.505	77.118	91.228	91.312
		(25.204^b)	(55.297)	(55.595)	(77.503)	(91.871)	(91.941)
	0.2	23.103	47.010	47.191	65.018	75.832	75.866
		(23.947^b)	(47.071)	(47.208)	(65.404)	(76.531)	(76.589)
0.5	0.05	23.103	47.010	47.191	65.018	75.832	75.866
		(23.210^b)	(47.245)	(47.358)	(65.241)	(76.122)	(76.345)
	0.1	48.377	94.650	95.597	135.484	162.635	163.501
		(48.524^b)	(94.891)	(95.992)	(135.845)	(167.012)	(164.024)
	0.15	36.239	70.939	71.298	99.997	118.584	119.009
		(36.647^b)	(71.214)	(71.687)	(100.024)	(119.114)	(119.457)
	0.2	28.749	56.937	57.037	79.214	93.009	93.162
		(29.142^b)	(57.034)	(57.544)	(79.654)	(93.668)	(93.754)
1	0.05	49.651	96.759	97.619	138.089	165.368	166.304
		(49.755^b)	(97.245)	(97.964)	(138.544)	(165.864)	(166.878)
	0.1	38.951	73.735	74.251	102.748	121.079	121.709
		(39.021^b)	(74.145)	(74.664)	(102.874)	(121.613)	(121.817)
	0.15	30.630	58.406	58.533	80.432	93.984	94.255
		(30.638^b)	(58.684)	(58.644)	(80.874)	(94.121)	(94.478)
	0.2	25.548	48.636	48.674	66.165	76.708	76.824
		(25.687^b)	(48.788)	(48.992)	(66.544)	(76.824)	(76.944)
1.5	0.05	50.118	97.561	98.374	139.096	166.421	167.382
		(50.247^b)	(97.877)	(98.661)	(139.247)	(166.875)	(167.662)
	0.1	40.310	75.216	75.783	104.229	122.402	123.164
		(40.366^b)	(75.542)	(75.921)	(104.557)	(122.874)	(123.385)
	0.15	31.837	59.352	59.530	81.245	94.611	94.983
		(31.902^b)	(59.451)	(59.871)	(81.375)	(94.784)	(95.105)
	0.2	26.302	49.119	49.156	66.528	76.954	77.130
		(26.418^b)	(49.354)	(49.334)	(66.921)	(77.010)	(77.470)
2	0.05	50.360	97.983	98.769	139.631	166.979	167.954
		(50.472^b)	(98.005)	(99.114)	(140.154)	(167.188)	(167.986)
	0.1	41.130	76.146	76.724	105.165	123.229	124.082
		(41.421^b)	(76.642)	(78.114)	(105.661)	(123.278)	(124.974)
	0.15	32.687	60.025	60.245	81.832	95.052	95.508
		(32.778^b)	(60.244)	(60.549)	(82.146)	(95.114)	(95.842)
	0.2	26.893	49.492	49.539	66.815	77.138	77.370
		(26.922^b)	(49.874)	(49.924)	(66.987)	(77.367)	(77.588)

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