# OS-V: 0395 Simply-Supported Thick Square Plate Random Forced Vibration Response

Test 21R 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 1.0 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 = (106 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 = 5.772 and α2 = 6.929×10-5 are given.

The material properties are:
Property
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 Frequency (Hz)
Reference Solution 3401.81 641200.00 45.90
HOE:
Direct Solution 3929.62 892303.43 45.24
Normalized 0.865684214 0.718589639 1.014588859
Modal Solution 3928.88 892421.36 45.27
Normalized 0.865847264 0.718494681 1.013916501
LOE:
Direct Solution 3607.25 600979.1 45.62
Normalized 0.943048028 1.066925622 1.006137659
Modal Solution 3606.23 600094.2 45.63
Normalized 0.943314764 1.068498912 1.00591716

## Reference

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