2. Benchmark solution

2.1. Calculation method used for the reference solution

During the impact phase, the system is a solution to the differential equation:

\(m\mathrm{.}\ddot{u}+\mathit{k.u}+{K}_{c}<u{>}^{\text{+}}\mathrm{=}0\) with \({u}_{0}=0\) and \(\dot{{u}_{0}}=\dot{{U}_{0}}\).

\(<x{>}^{\text{+}}\) refers to the positive value of \(x\).

The analytical solution to this problem is:

\(u=\frac{\dot{{U}_{0}}}{{\omega }_{c}}\mathrm{sin}({\omega }_{c}t)\) where \({\omega }_{c}=\sqrt{\frac{k+{K}_{c}}{m}}\).

The speed is cancelled for \({t}_{\dot{u}=0}=\frac{\pi }{2{\omega }_{c}}\).

The shock force is then maximum and is worth \({F}_{\text{max}}={K}_{c}u({t}_{\dot{u}=0})={K}_{c}\frac{\dot{{U}_{0}}}{{\omega }_{c}}\).

By construction, the duration of the shock is worth \({T}_{\mathrm{choc}}=2{t}_{\dot{u}=0}\).

The system returns to position \(u=0\) with speed \(-\dot{{U}_{0}}\).

In field \(u<0\) the system has the equation \(\mathrm{m.}\ddot{u}+\mathrm{k.u}=0\) with the initial conditions \({u}_{1}=0\) and \(\dot{{u}_{1}}=-\dot{{U}_{0}}\) as initial conditions.

His solution is \(u=-\frac{\dot{{U}_{0}}}{{\omega }_{0}}\mathrm{sin}({\omega }_{0}t\text{'})\) or \({\omega }_{0}=\sqrt{\frac{k}{m}}\).

Speed cancels to: \(t{\text{'}}_{\dot{u}=0}=\frac{\pi }{2{\omega }_{0}}\).

By construction, free flight time is worth: \({T}_{\mathrm{vol}}=2t{\text{'}}_{\dot{u}=0}\).

The system is therefore periodic with alternately a shock time phase of duration \({T}_{\mathrm{choc}}\) where the system describes a sine arch in the \(u>0\) domain and a free flight phase of duration \({T}_{\mathrm{vol}}\) where the system describes a sine arch in the \(u<0\) domain.

The impetus for each impact is worth: \(I=\underset{0}{\overset{{T}_{\mathrm{choc}}}{\int }}{K}_{c}u(t)\mathrm{dt}=2{K}_{c}\frac{\dot{{U}_{0}}}{{\omega }_{c}^{2}}=\frac{2m\dot{{U}_{0}}}{1+\frac{k}{{K}_{c}}}\).

2.2. Benchmark results

The results taken as reference are the values of the maximum force moments, the maximum force value, the maximum force value, the duration of the shock time, the value of the impulse and the impact speed as well as the elementary impact number for the first two oscillations of the system.

2.3. Uncertainty about the solution

Analytical solution.

2.4. Bibliographical references

• 7. JACQUART: Post-treatment of core and internal stones REP under seismic stress - HP-61/95/074/A.