2. Benchmark solution

2.1. Calculation method used for the reference solution

The simple oscillator verifies the following equation:

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

with \(u(0)=0\) and \(\dot{u}(0)=0\)

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

Critical damping is \({c}_{\mathit{critique}}\mathrm{=}\mathrm{2m}\omega\).

The solution for \(c={c}_{\mathrm{critique}}\) is:

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

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

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

with \({\omega }_{D}\mathrm{=}\omega \sqrt{1\mathrm{-}{\xi }^{2}}\)

2.2. Benchmark results

Movement and speed of point \(B\).

2.3. Uncertainty about the solution

Exact analytical solution.