# 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

## 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/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 α_{1} = 5.772 and
α_{2} = 6.929×10^{-5} are given.

**Property****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} |
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