# OS-V: 0340 Simply-Supported Thin Square Plate Harmonic Forced Vibration Response

Test 13H 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 harmonic forced vibration F = F0 sin ωt is induced in the z-direction. (F0 = 100 N/m2 over whole plate, ω = 2ωf, f = 0 to 4.16 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 = 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 (mm) Peak Stress (N/mm2) Frequency (HZ)
Reference Solution 45.42 30.03 2.377
HOE:
Direct Solution 47.254 37.57 2.323
Normalized 0.961188471 0.799307958 1.023245803
Modal Solution 47.34 37.64 2.324
Normalized 0.959442332 0.797821467 1.022805508
LOE:
Direct Solution 45.22 30.84 2.349
Normalized 1.004422822 0.973735409 1.011919966
Modal Solution 45.45 30.98 2.345
Normalized 0.999339934 0.969335055 1.013646055

## Reference

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