2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Calculation hypothesis: The deformation of the skate is overlooked.
Load 1 — Normal pressure \({P}_{n}:\mathrm{300N}/{\mathrm{mm}}^{2}\)
We check:
The good transmission of normal forces at the level of the contact zone: the normal pressure at the level of the contact zone is equal to the pressure applied
That the vertical movement of the skate at contact zone \(\mathrm{AB}\) is equal to the game.
Load 2 — Normal pressure \({P}_{n}:\mathrm{300N}/{\mathrm{mm}}^{2}\) and tangential pressure \({P}_{t}=\mathrm{178.2N}/{\mathrm{mm}}^{2}\)
It is a situation of adherence. We check that the skate nodes located in the sliding zone (\(\mathrm{AB}\)) do not move tangentially:
Load 3 — Normal pressure \({P}_{n}:\mathrm{300N}/{\mathrm{mm}}^{2}\) and tangential pressure \({P}_{t}=\mathrm{181.8N}/{\mathrm{mm}}^{2}\)
It’s a slippery situation. We check that the skate nodes located in the sliding zone (\(\mathrm{AB}\)) move from \(\mathrm{9mm}\) to the following \(X\):
Determination of spring stiffness \(k\): we want to determine the spring stiffness as a function of the desired displacement. At the moment of sliding, the force in the spring is:
\({F}_{r}={F}_{t}–\mu\) \({F}_{n}=\mathrm{0,01}\mu {F}_{n}\) with (\({F}_{t}=181.8\times 20\), \({F}_{n}=300\times 40\))
\({F}_{r}=K{U}_{t}\) |
: |
spring force |
\({F}_{t}={P}_{t}\times {S}_{\mathrm{AD}}\) |
: |
tangential force |
\({F}_{n}={P}_{n}\times {S}_{\mathrm{CD}}\) |
: |
normal force |
\({U}_{t}\) |
: |
tangential displacement |
\({S}_{\mathrm{AD}}\) |
: |
surface |
\({S}_{\mathrm{DC}}\) |
: |
surface |
For a displacement of \(9.\mathrm{mm}\) the spring stiffness \(k\) must be \(0.01\mu {F}_{n}/9=4N/\mathrm{mm}\)
2.2. Benchmark results#
Load 1 (Normal Pressure \({P}_{n}\) ) :
Loading 2 ( \({P}_{n}:\mathrm{300N}/{\mathrm{mm}}^{2}\) and \({P}_{t}=\mathrm{178.2N}/{\mathrm{mm}}^{2}\) ) : we check that there is at least one node on the contact surface that does not slip. It is tested that at least one of the nodes located on the face opposite to the lateral loading application does not slide.
Loading 3 ( \({P}_{n}:\mathrm{300N}/{\mathrm{mm}}^{2}\) and \({P}_{t}=181.8/{\mathrm{mm}}^{2}\) ) : we check that all the nodes on the contact surface are sliding. We test that all the nodes located on the side opposite the side loading application slide.
2.3. Uncertainties about the solution#
Less than 0.1%
2.4. Bibliographical references#
Not applicable