# OS-V: 0370 Simply-Supported Thick Square Plate Harmonic Forced Vibration Response

Test 21H 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).

## Benchmark Model

The 2nd order and 1st order quad elements are used to model the square plate of thickness 1.0m. 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 harmonic forced vibration F=F0 sin ωt is induced in the z-direction. (F0=106 N/m2 over whole plate, ω=2πf, f=0 to 78.17 Hz). 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=6.926×10-5 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 (mm) Peak Stress (N/mm2) Frequency (Hz)
Reference Solution 58.33 800.8 45.90
HOE:
Direct Solution 62.633 943.67 45.21
Normalized 0.931298197 0.848601736 1.01526211
Modal Solution 62.67 944.57 45.23
Normalized 0.930748364 0.847793176 1.014813177
LOE:
Direct Solution 60 774.73 45.62
Normalized 0.972166667 1.033650433 1.006137659
Modal Solution 60.05 774.54 45.59
Normalized 0.971357202 1.033903995 1.006799737

## Model Files

The model files used in this problem include:
• Test21HHOED.fem
• Test21HHOEM.fem
• Test21HLOED.fem
• Test21HLOEM.fem

## Reference

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