TEST CASE 2.4 NATURAL FREQUENCIES OF A SQUARE PLATE CLAMPED AROUND THE PERIMETER
Reference:
I.A. Birger, Ya.G. Panovko, Strength, Stability, Vibrations, Handbook in three volumes, Volume 3, Moscow, Mechanical engineering, 1968, p. 377.
Problem description:
Determine the natural frequencies of a square plate clamped around the perimeter.
Problem sketch:
Type of created problem:
Flat plate or grillage (Z, UX, UY).
Geometric characteristics:
a = 1 m; t = 0.01 m.
Material properties:
Elastic modulus: E = 2.1·10^{11} Pa;
Poisson’s ratio: μ = 0.3;
Density: γ = 7800 kg/m^{3}.
Boundary conditions:
On the perimeter: Z = UX = UY = 0.
Loads:
Dead load.
Model description:
The system is modeled using 100 finite elements of thin plate (FE type is 19) with additional nodes. To solve the problem, a modal analysis was performed.
Analytical solution:
Calculation results:
Form № 
Frequency, rad/sec 
Deviation, % 

Analytical solution 
LIRA 10 

1 
560.1 
557.80 
0.41 
2 
1143.2 
1137.57 
0.49 
3 
1143.2 
1137.57 
0.49 
4 
1686.6 
1677.33 
0.55 
5 
2054.0 
2039.12 
0.72 
6 
2054.0 
2048.76 
0.26 