# OS-V: 0350 Simply-Supported Thin Square Plate Periodic Forced Vibration Response

Test 13P OptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stress 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-sin 3ωt) is induced in the z-direction. (F0 = 100 N/m2 over whole plate, ω = 2πf, f = 1.2 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)
Reference Solution 2.863 2.018
HOE:
Direct Solution 2.928 2.418
Normalized 0.977800546 0.834574028
Modal Solution 2.929 2.426
Normalized 0.977466712 0.831821929
LOE:
Direct Solution 2.825 1.956
Normalized 1.013451327 1.031697342
Modal Solution 2.826 1.961
Normalized 1.013092711 1.029066803

## Reference

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