Here's my attempt at deriving the bounce height of a wheel hitting a step in the track.
wheel-and-step.001.png
The step is of height s and the wheel of radius r. The wheel approaches with horizontal velocity v and I assume that it looses no speed in climbing the step. I also assume that there's no deformation, so when first touching the step the wheel moves parallel to its tangent at the point of contact with the step (the thin blue arrow in the diagram). This gives it an upward velocity u and u = v sin(theta) where theta is the angle between the tangent and the horizontal. We can work out theta from the geometry of the green triangle in the picture, giving the expression u = v sin(acos(r-s/r)).
If you throw something up at speed u and let it fall back under gravity, it goes up to a height h = 0.5 u**2 / g where g is the downward acceleration due to gravity (neglecting air and pigeon resistance). For a given wheel and step, this height h depends only on v, since u is determined by v and the geometry. If h is greater than the step height s, then the wheel gets briefly airborne. This is not ideal, but we have no scale people to suffer from it, so it's possibly OK. If h is greater than s by more than a flange depth, however, then we probably have a derailment. (I'm effectively assuming that both wheels on the axle hit steps of the same height. If the step is only in one rail, the effect is less.)
Bunging some numbers in, it now looks like the limiting speeds are about 510mm/s for a 0.1mm step, 375mm/s for 0.25mm, 320 mm/s for 0.5mm and 300 mm/s for 0.75mm. All distances in the table below are in mm and speeds in mm/s.
r s r-s/r theta sin(theta) v u h h -s
6 0.25 0.958333333 0.289686994 0.285652275 10 2.85652275 0.000415888 -0.249584112
6 0.25 0.958333333 0.289686994 0.285652275 50 14.28261375 0.0103972 -0.2396028
6 0.25 0.958333333 0.289686994 0.285652275 100 28.5652275 0.041588798 -0.208411202
6 0.25 0.958333333 0.289686994 0.285652275 150 42.84784125 0.093574796 -0.156425204
6 0.25 0.958333333 0.289686994 0.285652275 200 57.130455 0.166355193 -0.083644807
6 0.25 0.958333333 0.289686994 0.285652275 250 71.41306875 0.259929989 0.009929989
6 0.25 0.958333333 0.289686994 0.285652275 300 85.69568251 0.374299185 0.124299185
6 0.25 0.958333333 0.289686994 0.285652275 375 107.1196031 0.584842476 0.334842476
6 0.5 0.916666667 0.411137862 0.399652627 10 3.996526269 0.000814079 -0.499185921
6 0.5 0.916666667 0.411137862 0.399652627 50 19.98263135 0.020351965 -0.479648035
6 0.5 0.916666667 0.411137862 0.399652627 100 39.96526269 0.08140786 -0.41859214
6 0.5 0.916666667 0.411137862 0.399652627 150 59.94789404 0.183167686 -0.316832314
6 0.5 0.916666667 0.411137862 0.399652627 200 79.93052539 0.325631442 -0.174368558
6 0.5 0.916666667 0.411137862 0.399652627 250 99.91315674 0.508799128 0.008799128
6 0.5 0.916666667 0.411137862 0.399652627 300 119.8957881 0.732670744 0.232670744
6 0.5 0.916666667 0.411137862 0.399652627 320 127.8888406 0.833616491 0.333616491
6 0.1 0.983333333 0.182828717 0.181811869 10 1.818118686 0.000168479 -0.099831521
6 0.1 0.983333333 0.182828717 0.181811869 50 9.090593429 0.004211972 -0.095788028
6 0.1 0.983333333 0.182828717 0.181811869 100 18.18118686 0.016847888 -0.083152112
6 0.1 0.983333333 0.182828717 0.181811869 150 27.27178029 0.037907747 -0.062092253
6 0.1 0.983333333 0.182828717 0.181811869 200 36.36237372 0.067391551 -0.032608449
6 0.1 0.983333333 0.182828717 0.181811869 250 45.45296714 0.105299298 0.005299298
6 0.1 0.983333333 0.182828717 0.181811869 300 54.54356057 0.151630989 0.051630989
6 0.1 0.983333333 0.182828717 0.181811869 510 92.72405297 0.438213558 0.338213558
6 0.75 0.875 0.50536051 0.484122918 10 4.841229183 0.001194572 -0.748805428
6 0.75 0.875 0.50536051 0.484122918 50 24.20614591 0.029864297 -0.720135703
6 0.75 0.875 0.50536051 0.484122918 100 48.41229183 0.119457187 -0.630542813
6 0.75 0.875 0.50536051 0.484122918 150 72.61843774 0.26877867 -0.48122133
6 0.75 0.875 0.50536051 0.484122918 200 96.82458366 0.477828746 -0.272171254
6 0.75 0.875 0.50536051 0.484122918 250 121.0307296 0.746607416 -0.003392584
6 0.75 0.875 0.50536051 0.484122918 300 145.2368755 1.075114679 0.325114679
The results in my previous post about this were not quite right, firstly because because I'd messed up the trig and secondly because I'd calculated h rather than h - s. The new results are rather less worrying.
You do not have the required permissions to view the files attached to this post.
