# OS-V: 0390 Deep Simply-Supported Beam Random Forced Vibration Response

Test 5R OptiStruct is used to investigate the Peak Displacement in y-direction and extreme fiber bending stress at undamped Natural Frequency (at the mid-span node).

## Benchmark Model

Timoshenko beam and Engineer’s beam elements are used to model the simply-supported beam which consists of 10 elements. The displacements in x, y, and z direction, as well as the rotation in x direction are fixed at the end A. In addition, the displacements in y and z direction are constrained at end B. A steady-state random forcing with uniform power spectral density (of force) PSD= (106 N/m)2/Hz is induced in the y-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=5.36 and α2=7.46×10-5 are given.

The material properties are:
Property
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) Frequency (Hz)
Reference Solution 180.90 58516 42.65
HOE:
Direct Solution 184.03 56999.67 43.14
Normalized 0.982991903 1.026602435 0.988641632
Modal Solution 183.93 56973.24 43.16
Normalized 0.983526342 1.027078678 0.988183503
LOE:
Direct Solution 150.9 56917.35 45.34
Normalized 1.198807157 1.028087218 0.94067049
Modal Solution 151.4 57105.24 45.34
Normalized 1.194848085 1.024704563 0.94067049

## Model Files

The model files used in this problem include:
• Test5RPBEAMLD.fem
• Test5RPBEAMLM.fem
• Test5RPBEAMD.fem
• Test5RPBEAMM.fem

## Reference

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