2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The direct calculation defines an assembled vector for the spatial distribution of the force and applies the spectral force density \({G}_{\mathrm{FF}}(\omega )\) to this distribution (method 1).
The decomposed calculation defines the excitation as an interspectral matrix of dimension 3 (equal to the number of excited nodes) and applies, in force imposed on the nodes, the following interspectral matrix (method 2):
\(\left[\begin{array}{ccc}\frac{1}{4}& \frac{1}{2}& \frac{1}{4}\\ \frac{1}{2}& 1& \frac{1}{2}\\ \frac{1}{4}& \frac{1}{2}& \frac{1}{4}\end{array}\right]{G}_{\mathrm{FF}}(\omega )\)
Both results should be the same without any approximation.
2.2. Benchmark results#
Node \(\mathrm{P3}\) displacement power spectral density at frequencies: \(4.,6.,8.,10.\) and \(12\mathrm{Hz}\).
2.3. Bibliographical references#
DUVAL « Dynamic response under random excitation in Code_Aster: theoretical principles and examples of use » - Note HP-61/92.148