2. Benchmark solution

2.1. Calculation method used for the reference solution

During a free flight phase, the equation of motion is written \(m\ddot{x}+{k}_{\mathrm{rappel}}x={F}_{\mathrm{ext}}(t)\). Given the sine form of \({F}_{\mathrm{ext}}(t)\), this equation has an analytical solution. We can calculate numerically, with as much precision as we want, the moment \({t}_{\text{in}}\) of entry into the contact, verifying \(x({t}_{\text{in}})={\mathrm{jeu}}_{\mathrm{choc}}\).

We are then in a contact phase, whose equation is \(m\ddot{x}+{c}_{\mathrm{choc}}\dot{x}+({k}_{\mathrm{rappel}}+{k}_{\mathrm{choc}})x={F}_{\mathrm{ext}}(t)\). There is also an analytical solution. With the boundary conditions resulting from the instant of contact, it is possible to numerically calculate the instant \({t}_{\text{out}}\) of exit from the contact.

By proceeding in this way iteratively, the complete solution of the problem is obtained.

Note: the analytical formulas are not given here. The files containing the formulas and allowing the complete solution to be calculated are attached to the command file.

2.2. Benchmark results

We test the energy balance, the adequacy between contact forces and kinematics, as well as kinematics.

For the energy balance, we calculate the kinetic \({E}_{i}^{\mathrm{cin}}=\frac{1}{2}m\dot{x}{}_{i}^{2}\), potential \({E}_{i}^{\mathit{pot}}\mathrm{=}\frac{1}{2}{k}_{\mathit{rappel}}{x}_{i}^{2}\), shock energies \({E}_{i}^{\mathit{choc}}\mathrm{=}\frac{1}{2}{k}_{\mathit{choc}}{p}_{i}^{2}\) (\(p\) is the penetration; this expression is only valid if there is no shock absorption), injected by the external force \({E}_{i}^{\mathrm{inj}}=\sum _{j=1}^{i}{f}_{j}^{\mathrm{ext}}\dot{x}{}_{j+\delta \frac{1}{2}}\Delta t\) (with \(\delta =1\) for the Euler diagram, \(\delta =0\) for the diagram of centered differences), are calculated. We get total energy \({E}_{i}^{\mathit{tot}}\mathrm{=}{E}_{i}^{\mathit{cin}}+{E}_{i}^{\mathit{pot}}+{E}_{i}^{\mathit{choc}}\). Finally, the global error on the energy balance is calculated by \({\mathit{erreur}}_{\mathit{globale}}^{\mathit{énergie}}\mathrm{=}\sqrt{\frac{\mathrm{\sum }_{i}{({E}_{i}^{\mathit{tot}}\mathrm{-}{E}_{i}^{\mathit{inj}})}^{2}}{\mathrm{\sum }_{i}{({E}_{i}^{\mathit{inj}})}^{2}}}\), which is ideally equal to 0.

For the adequacy between contact forces and kinematics, we calculate the quantity \({\mathit{erreur}}_{\mathit{globale}}^{\mathit{force}}\mathrm{=}\sqrt{\frac{\mathrm{\sum }_{i}{({F}_{i}^{\mathit{choc}}\mathrm{-}{k}_{\mathit{choc}}{p}_{i})}^{2}}{\mathrm{\sum }_{i}{({k}_{\mathit{choc}}{p}_{i})}^{2}}}\), which is ideally equal to 0.

For kinematics, the calculated contact entry and exit times are compared to the analytical times.

2.3. Uncertainty about the solution

The solution is analytic in pieces. The contact entry and exit times are determined numerically to the nearest \({10}^{-9}s\).