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

Table 8

Frequency parameter Ω=𝜔𝑎²(𝜌ℎ₀/𝐷₀)^1/2 for elastic support moderately thick plates (𝑎/𝑏=1) with different gradient (𝑘_𝑥0=𝑘_𝑥𝑎=𝑘_𝑦0=𝑘_𝑦𝑏=Γ_𝑤=2,   𝐾_𝑥0=𝐾_𝑥𝑎=𝐾_𝑦0=𝐾_𝑦𝑏=Γ_𝑥=2).

𝛼, 𝛽	ℎ0/𝑎	Model sequence
		1	2	3	4	5	6
𝛼=𝛽=0.1	0.05	31.275	52.615	52.657	68.954	76.146	79.850
		(31.303^a)	(52.664)	(52.675)	(69.033)	(76.174)	(79.948)
	0.1	17.270	25.244	25.250	33.949	46.050	46.625
		(17.271^a)	(25.255)	(25.258)	(34.089)	(46.122)	(46.708)
	0.15	10.190	16.266	16.274	24.164	37.327	37.526
		(10.189^a)	(16.269)	(16.280)	(24.323)	(37.401)	(37.622)
	0.2	6.780	12.258	12.260	19.685	31.452	31.754
		(6.780^a)	(12.265)	(12.270)	(19.851)	(31.514)	(31.864)
𝛼=𝛽=0.2	0.05	32.530	52.947	53.141	68.954	77.034	80.589
		(32.577^a)	(53.014)	(53.203)	(69.172)	(77.088)	(80.773)
	0.1	16.882	24.979	24.979	34.154	48.057	48.575
		(16.883^a)	(24.983)	(25.001)	(34.433)	(48.134)	(48.667)
	0.15	9.804	16.284	16.309	24.683	38.844	39.085
		(9.803^a)	(16.283)	(16.322)	(24.980)	(38.909)	(39.191)
	0.2	6.499	12.290	12.295	20.133	32.354	32.770
		(6.498^a)	(12.296)	(12.307)	(20.420)	(32.387)	(32.895)
𝛼=𝛽=0.3	0.05	33.516	52.845	53.308	68.704	78.203	81.561
		(33.563^a)	(52.939)	(53.389)	(69.055)	(78.271)	(81.789)
	0.1	16.419	24.799	24.830	34.527	50.061	50.583
		(16.419^a)	(24.790)	(24.869)	(34.947)	(50.143)	(50.688)
	0.15	9.430	16.334	16.378	25.236	40.193	40.506
		(9.429^a)	(16.329)	(16.395)	(25.660)	(40.243)	(40.625)
	0.2	6.235	12.288	12.293	20.572	33.102	33.663
		(6.235^a)	(12.296)	(12.306)	(20.965)	(33.090)	(33.811)
𝛼=𝛽=0.4	0.05	34.215	52.463	53.264	68.430	79.654	82.865
		(34.252^a)	(52.575)	(53.346)	(68.921)	(79.732)	(83.113)
	0.1	15.921	24.695	24.793	35.027	52.011	52.571
		(15.921^a)	(24.674)	(24.842)	(35.583)	(52.098)	(52.688)
	0.15	9.074	16.388	16.444	25.797	41.379	41.789
		(9.074^a)	(16.381)	(16.464)	(26.336)	(41.404)	(41.928)
	0.2	5.990	12.236	12.238	21.002	33.735	34.461
		(5.989^a)	(12.249)	(12.250)	(21.487)	(33.653)	(34.641)
𝛼=𝛽=0.5	0.05	34.615	51.972	53.088	68.255	81.344	84.510
		(33.806^a)	(52.100)	(53.157)	(68.891)	(81.433)	(84.761)
	0.1	15.416	24.659	24.829	35.615	53.879	54.494
		(15.045^a)	(24.625)	(24.888)	(36.303)	(53.970)	(54.626)
	0.15	8.740	16.420	16.481	26.353	42.417	42.946
		(8.529^a)	(16.414)	(16.499)	(26.996)	(42.398)	(43.114)
	0.2	5.761	12.127	12.127	21.434	34.286	35.192
		(5.622^a)	(12.134)	(12.148)	(21.998)	(34.110)	(35.415)

aResults in parentheses are taken from FEM.