# 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

## 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/m^{2})^{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.

**Material Properties****Value**- Young’s Modulus
- 200 × 10
^{9}N/m^{2} - Poisson’s Ratio
- 0.3
- Density
- 8000 kg/m
^{3}

## Frequency Response Summary

`f*`

- Closed form solution

Peak Displacement PSD (mm^{2}/Hz) |
Peak Stress PSD ((N/mm^{2})^{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