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

Table 12

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	30.187	51.102	51.141	67.066	73.954	77.412
		(30.188^a)	(51.105)	(51.144)	(67.068)	(73.961)	(77.477)
	0.1	16.995	24.719	24.724	33.071	44.343	44.819
		(16.996^a)	(24.725)	(24.729)	(33.084)	(44.347)	(44.822)
	0.15	10.077	15.872	15.883	23.443	35.937	36.065
		(10.081^a)	(15.877)	(15.892)	(23.447)	(35.941)	(36.084)
	0.2	6.713	11.953	11.958	19.086	30.372	30.604
		(6.713^a)	(11.957)	(11.965)	(19.092)	(30.378)	(30.612)
𝛼=𝛽=0.2	0.05	31.206	51.436	51.614	67.058	74.398	77.595
		(31.208^a)	(51.438)	(51.619)	(67.064)	(74.402)	(77.596)
	0.1	16.786	24.503	24.508	33.144	45.583	45.923
		(16.790^a)	(24.510)	(24.513)	(33.147)	(45.587)	(45.932)
	0.15	9.821	15.865	15.906	23.766	36.897	36.990
		(9.824^a)	(15.870)	(15.907)	(23.768)	(36.894)	(36.997)
	0.2	6.523	11.978	11.992	19.356	30.912	31.162
		(6.524^a)	(11.981)	(11.997)	(19.362)	(30.916)	(31.174)
𝛼=𝛽=0.3	0.05	32.097	51.462	51.900	66.832	74.991	77.863
		(32.098^a)	(51.466)	(51.907)	(66.849)	(74.996)	(77.872)
	0.1	16.507	24.332	24.383	33.330	46.806	47.074
		(16.511^a)	(24.327)	(24.380)	(33.339)	(46.810)	(47.086)
	0.15	9.562	15.885	15.964	24.103	37.748	37.828
		(9.562^a)	(15.892)	(15.977)	(21.105)	(37.751)	(37.832)
	0.2	6.338	11.992	12.013	19.603	31.338	31.634
		(6.340^a)	(11.994)	(12.011)	(19.612)	(31.344)	(31.639)
𝛼=𝛽=0.4	0.05	32.851	51.239	52.046	66.547	75.735	78.303
		(33.455^a)	(50.852)	(50.097)	(66.321)	(76.627)	(78.971)
	0.1	16.182	24.209	24.343	33.598	47.990	48.230
		(33.455^a)	(50.852)	(50.097)	(66.321)	(76.627)	(78.971)
	0.15	9.307	15.919	16.034	24.436	38.491	38.579
		(33.455^a)	(50.852)	(50.097)	(66.321)	(76.627)	(78.971)
	0.2	6.161	11.986	12.008	19.828	31.666	32.040
		(33.455^a)	(50.852)	(50.097)	(66.321)	(76.627)	(78.971)
𝛼=𝛽=0.5	0.05	33.453	50.854	52.088	66.311	76.610	78.947
		(33.455^a)	(50.852)	(50.097)	(66.321)	(76.627)	(78.971)
	0.1	15.832	24.130	24.372	33.924	49.118	49.362
		(15.830^a)	(24.127)	(24.374)	(33.931)	(49.147)	(49.382)
	0.15	9.060	15.956	16.099	24.754	39.134	39.249
		(9.064^a)	(15.961)	(16.102)	(24.755)	(39.138)	(39.452)
	0.2	5.992	11.955	11.971	20.034	31.908	32.392
		(5.991^a)	(11.957)	(11.978)	(20.042)	(31.910)	(32.398)

aResults in parentheses are taken from FEM.