Benchmark solution
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Calculation method used for the reference solution
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The simple oscillator verifies the following equation:

:math:`m\ddot{u}+c\dot{u}+ku={F}_{0}\mathrm{sin}(\omega t)`

with :math:`u(0)=0` and :math:`\dot{u}(0)=0`

:math:`\omega`: natural pulsation of the :math:`\omega \mathrm{=}\sqrt{\frac{k}{m}}` oscillator

Critical damping is :math:`{c}_{\mathit{critique}}\mathrm{=}\mathrm{2m}\omega`.

The solution for :math:`c={c}_{\mathrm{critique}}` is:

:math:`u(t)\mathrm{=}\frac{{F}_{0}}{\mathrm{2k}}\left[{e}^{\mathrm{-}\omega t}(1+\omega t)\mathrm{-}\mathrm{cos}(\omega t)\right]`

The solution for subcritical damping such as :math:`\frac{c}{{c}_{\mathit{critique}}}\mathrm{=}\xi` is:

:math:`u(t)\mathrm{=}{e}^{\mathrm{-}\xi \omega t}(\frac{{F}_{0}}{2\xi k}\mathrm{cos}({\omega }_{D}t)+\frac{{F}_{0}\omega }{2k{\omega }_{D}}\mathrm{sin}({\omega }_{D}t))\mathrm{-}\frac{{F}_{0}}{2\xi k}\mathrm{cos}(\omega t)`

with :math:`{\omega }_{D}\mathrm{=}\omega \sqrt{1\mathrm{-}{\xi }^{2}}`


Benchmark results
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Movement and speed of point :math:`B`.


Uncertainty about the solution
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Exact analytical solution.