# OS-V: 0396 Simply-Supported Thin Square Plate Harmonic Random Vibration Response

Test 13R OptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stress at undamped Natural Frequency (at the center of the plate).

## Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

## Benchmark Model

The 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05 m. The z-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x = 0 and x = 10 and y-rotation is fixed along the edge y = 0 and y = 10. A steady-state random forcing with uniform power spectral density (of force) PSD = (100 N/m2)2/Hz is induced in the z-direction. For modal analysis solution, a damping ratio of 0.02 is applied in all 16 modes and for direct solution, Rayleigh damping factor α1 = 0.299 and α2 = 1.339×10-3 are given.

The material properties are:
Material Properties
Value
Young’s Modulus
200 × 109 N/m2
Poisson’s Ratio
0.3
Density
8000 kg/m3

## Frequency Response Summary

The frequency of each targeted mode is normalized with the closed form solution.
f*
Closed form solution
Peak Displacement PSD (mm2/Hz) Peak Stress PSD ((N/mm2)2/Hz) Frequency (Hz)
Reference Solution 2063.20 1025.44 2.377
HOE
Direct Solutions 2232.98 1411.14 2.322
Normalized 0.923967075 0.726674887 1.023686477
Modal Solution 2241.33 1416.89 2.324
Normalized 0.920524867 0.723725907 1.022805508
LOE
Direct Solutions 2045.23 951.00 2.349
Normalized 1.008786298 1.078275499 1.011919966
Modal Solution 2065.73 960.22 2.345
Normalized 0.998775251 1.067921935 1.013646055

## Reference

NAFEMS R0016 - Selected Benchmarks for Forced Vibration, J Maguire, D J, Dawswell, L Gould 1989