# 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).

## 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 harmonic forced
vibration F = F_{0} sin ωt is induced in the z-direction. (F_{0} =
10^{6} N/m^{2} 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 α_{1} = 5.772 and α_{2} =
6.926×10^{-5} 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 (mm) | Peak Stress (N/mm^{2}) |
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 |

## Reference

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