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

Table 4

Frequency parameters Ω=𝜔𝑏²(𝜌ℎ/𝐷)^1/2/𝜋² for moderately thick plates (𝑎/𝑏=1) with uniform thickness and all edge elastic restraints (𝐾_𝑥0=𝐾_𝑥𝑎=𝐾_𝑦0=𝐾_𝑦𝑏=Γ_𝑥 and 𝑘_𝑥0=𝑘_𝑥𝑎=𝑘_𝑦0=𝑘_𝑦𝑏=Γ_𝑤).

Γ𝑥^a	Γ𝑤^a	ℎ0/𝑎	Model sequence
			1	2	3	4	5	6
0.5	0	0.05	0.986	1.967	3.893	3.893	4.850	4.850
			(0.988^b)	(1.971)	(3.897)	(3.897)	(4.852)	(4.852)
		0.1	0.916	1.835	3.515	3.522	4.375	4.375
			(0.917^b)	(1.837)	(3.519)	(3.528)	(4.341)	(4.341)
		0.15	0.789	1.635	2.983	3.027	3.771	3.771
			(0.792^b)	(1.638)	(2.988)	(3.029)	(3.775)	(3.775)
		0.2	0.646	1.444	2.497	2.601	3.249	3.249
			(0.652^b)	(1.448)	(2.508)	(2.621)	(3.254)	(3.254)
1	0.5	0.05	3.417	3.417	4.431	5.294	5.461	6.448
			(3.419^b)	(3.419)	(4.435)	(5.303)	(5.478)	(6.456)
		0.1	1.593	1.593	2.401	3.849	3.861	4.729
			(1.611^b)	(1.611)	(2.408)	(3.852)	(3.866)	(4.767)
		0.15	1.122	1.122	1.913	3.272	3.290	4.038
			(1.127^b)	(1.227)	(1.924)	(3.284)	(3.314)	(4.069)
		0.2	0.890	0.890	1.640	2.781	2.829	3.476
			(0.899^b)	(0.899)	(1.654)	(2.787)	(2.834)	(3.497)
1.5	1	0.05	4.325	4.325	5.619	6.273	6.571	7.619
			(4.334^b)	(4.334)	(5.681)	(6.299)	(6.642)	(7.642)
		0.1	2.013	2.013	2.812	4.088	4.116	4.993
			(2.024^b)	(2.024)	(2.827)	(4.122)	(4.187)	(4.994)
		0.15	1.338	1.338	2.108	3.430	3.443	4.196
			(1.342^b)	(1.342)	(2.112)	(3.467)	(3.482)	(4.200)
		0.2	1.038	1.038	1.772	2.936	2.965	3.613
			(1.042^b)	(1.042)	(1.786)	(2.937)	(2.977)	(3.624)
2	2	0.05	5.231	5.231	6.919	7.630	8.002	9.220
			(5.238^b)	(5.238)	(6.931)	(7.637)	(8.011)	(9.241)
		0.1	2.594	2.594	3.439	4.470	4.544	5.428
			(2.597^b)	(2.597)	(3.442)	(4.495)	(4.557)	(5.438)
		0.15	1.653	1.653	2.403	3.613	3.632	4.392
			(1.662^b)	(1.662)	(2.414)	(3.627)	(3.646)	(4.404)
		0.2	1.238	1.238	1.949	3.077	3.099	3.751
			(1.244^b)	(1.244)	(1.964)	(3.094)	(3.116)	(3.788)

^aΓ_𝑥=𝐾_𝑖/𝐾₀,(𝐾₀=1×10⁹Nm/rad,  𝑖=𝑥0,𝑥𝑎,𝑦0,𝑦𝑏),  Γ_𝑤=𝑘_𝑖/𝑘₀,(𝑘₀=1×10⁹N/m,  𝑖=𝑥0,𝑥𝑎,𝑦0,𝑦𝑏).
bResults in parentheses are taken from FEM.