TEST CASE 2.6 NATURAL FREQUENCIES OF THE ROUND PLATE CLAMPED ALONG THE CONTOUR
Reference:
V.N. Chelomey, Vibration in technics, Handbook in six volumes; V.V. Bolotin, Volume 1, The vibrations of linear systems, Moscow, Mechanical Engineering, 1978, p. 207.
Problem description:
Determine the natural frequencies of the round plate clamped along the contour.
Problem sketch:
Type of created problem:
Flat plate or grillage (Z, UX, UY).
Geometric characteristics:
r = 0.5 m; t = 0.01 m.
Material properties:
Elastic modulus: E = 2.06·10^{11} Pa;
Poisson’s ratio: μ = 0.3;
Density: γ = 7850 kg/m^{3}.
Boundary conditions:
On the perimeter: Z = UX = UY = 0.
Loads:
Dead load.
Model description:
The system is modeled using 288 finite elements of thin plate with additional nodes (FE type is 19), splitting in the radial direction with a step of 0.0625 m and in the tangential with a step of 10°. To solve the problem, a modal analysis was performed.
Analytical solution:
J_{n}, I_{n} – Bessel functions.
Calculation results:
Form № 
Frequency, rad/sec 
Deviation, % 

Analytical solution 
LIRA 10 

1 
633.5 
636.8 
0.52 
2, 3 
1318.3 
1324.9 
0.50 
4 
2162.7 
2173.1 
0.48 
5 
2162.7 
2173.2 
0.49 
6 
2466.1 
2477.8 
0.47 
7, 8 
3164.3 
3178.8 
0.46 
9, 10 
3771.9 
3788.1 
0.43 
11 
4319.8 
4338.6 
0.44 
12 
4319.8 
4338.7 
0.44 
13, 14 
5244.8 
5266.5 
0.41 
15 
5525.2 
5546.5 
0.39 