Benchmark solution
=====================


Calculation method used for the reference solution
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The reference solution is analytical [:ref:`bib1 <bib1>`]. The natural pulsation of the oscillator is :math:`\sqrt{\frac{k}{m}}`,

be :math:`{\omega }_{0}\mathrm{=}\sqrt{\frac{k}{m}}\mathrm{=}100\mathit{rad}\mathrm{/}s`, and :math:`{f}_{o}\mathrm{=}\mathrm{15,9155}\mathit{Hz}`.

In absolute motion, the DSP of the acceleration response noted :math:`{G}_{\ddot{R}\ddot{R}}(\omega )` is linked to the DSP of the excitation :math:`{G}_{\ddot{E}\ddot{E}}` in acceleration also by:

:math:`{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:

:math:`{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:

:math:`{G}_{\ddot{R}\ddot{R}}(\omega )\mathrm{=}{G}_{\ddot{E}\ddot{E}}(\omega )`


Benchmark results
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We test the DSP of the answer for :math:`\mathrm{0,}\mathrm{5,}\mathrm{10,}\mathrm{15,}20\mathit{Hz}` in the three cases of movement: absolute, relative, and differential.


Uncertainty about the solution
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Analytical solution.


Bibliographical references
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1. C. DUVAL "Dynamic response under random excitation in *Code_Aster*: theoretical principles and examples of use" - Note HP-61/92.148