# 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).

## Benchmark Model

The 2nd order and 1st order quad elements are used to model the square plate of thickness
0.05m. 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}=100 N/m^{2}
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 α_{2}=1.339×10^{-3} are given.

**Material Properties****Value**- Young’s Modulus
- 200 × 109 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 | 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 |

## Model Files

Refer to Access the Model Files to download the required model file(s).

- Test13HHOED.fem
- Test13HHOEM.fem
- Test13HLOED.fem
- Test13HLOEM.fem

## Reference

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