version 9 - Solving differential equations with constraints

The problem is $$ x''(t)=F(t) - kx'(t) \\ x'(0)=x(0)=0$$ with the additional constraints $$ 0 < x(t) WhenEvent[...].

Example:

First@First@NDSolve[{
  x''[t] == 10 Sin[0.1 t] - 0.1 x'[t], x[0] == 0, 
  x'[0] == 0, WhenEvent[x[t] > 400, x[t] -> 399], 
  WhenEvent[x'[t] > 40, x'[t] -> 39], 
  WhenEvent[x[t] < 0, x[t] -> 1], 

  WhenEvent[x'[t] < -40, x'[t] -> -39]}, 
 x, {t, 0, 200}, 
 MaxSteps -> 50000];
Plot[{x[t] /. %, 10 x'[t] /. %}, {t, 0, 200}, PlotRange -> All, 
 PlotLegends -> {"x[t]", "x'[t]"}, ImageSize -> 700]

What is the proper way to solve this type of problem?

Answer

I guess that can be done. I don't know if you'll love this solution though.

{s, a} = NDSolveValue[{
   x''[t] == alive[t] 10 Sin[0.1 t] - 0.1 x'[t],
   x[0] == 0, x'[0] == 0,
   alive[0] == 1,
   speed[0] == 0,
   WhenEvent[x[t] > 400, {
     speed[t] -> x'[t], x'[t] -> 0, alive[t] -> 0}],
   WhenEvent[Sin[.1 t] < 0, {
     x'[t] -> -speed[t], alive[t] -> 1}],
   WhenEvent[x[t] < 0, {

     speed[t] -> x'[t], x'[t] -> 0, alive[t] -> 0}],
   WhenEvent[Sin[.1 t] > 0, {
     x'[t] -> -speed[t], alive[t] -> 1}]},
  {x, alive},
  {t, 0, 100},
  DiscreteVariables -> {alive, speed}]

Roughly, I stored the first derivative and temporarily killed it together with force and then restored both at the appropriate moment.

Plot[{s[t], 300 a[t]},
 {t, 0, 100},

 PlotRange -> All,
 PlotStyle -> Thick]