2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is analytical [bib1]. The natural pulsation of the oscillator is \(\sqrt{\frac{k}{m}}\),
be \({\omega }_{0}\mathrm{=}\sqrt{\frac{k}{m}}\mathrm{=}100\mathit{rad}\mathrm{/}s\), and \({f}_{o}\mathrm{=}\mathrm{15,9155}\mathit{Hz}\).
In absolute motion, the DSP of the acceleration response noted \({G}_{\ddot{R}\ddot{R}}(\omega )\) is linked to the DSP of the excitation \({G}_{\ddot{E}\ddot{E}}\) in acceleration also by:
\({G}_{\ddot{R}\ddot{R}}(\omega )\mathrm{=}\frac{{\omega }_{0}^{4}+4{\xi }_{0}^{2}{\omega }_{0}^{2}{\omega }^{2}}{{({\omega }_{0}^{2}\mathrm{-}{\omega }^{2})}^{2}+4{\xi }_{0}^{2}{\omega }_{0}^{2}{\omega }^{2}}{G}_{\ddot{E}\ddot{E}}(\omega )\)
In relative motion, we have:
\({G}_{\ddot{R}\ddot{R}}(\omega )\mathrm{=}{∣(\frac{{\omega }^{2}}{{\omega }_{0}^{2}\mathrm{-}{\omega }^{2}+2j{\xi }_{0}{\omega }_{0}\omega })∣}^{2}{G}_{\ddot{E}\ddot{E}}(\omega )\)
In differential motion, we have:
\({G}_{\ddot{R}\ddot{R}}(\omega )\mathrm{=}{G}_{\ddot{E}\ddot{E}}(\omega )\)
2.2. Benchmark results#
We test the DSP of the answer for \(\mathrm{0,}\mathrm{5,}\mathrm{10,}\mathrm{15,}20\mathit{Hz}\) in the three cases of movement: absolute, relative, and differential.
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
DUVAL « Dynamic response under random excitation in Code_Aster: theoretical principles and examples of use » - Note HP-61/92.148