Problems with WhenEvent

Question

Any help/explanations will be highly appreciated:

---

This works:

sol = NDSolve[{x''[t] + x[t] == Sin[t], x[0] == 0, x'[0] == 1,
     WhenEvent[Abs[t x[t]] > 1, {x'[t] -> -x'[t]}]}, 
    x, {t, 0, 8}][[1]];

xt = x /. sol;

Plot[xt[t], {t, 0, 8}];

---

This does not:

TT = t;

sol = NDSolve[{x''[t] + x[t] == Sin[t], x[0] == 0, x'[0] == 1,
     WhenEvent[Abs[TT x[t]] > 1, {x'[t] -> -x'[t]}]}, 
    x, {t, 0, 8}][[1]];

xt = x /. sol;

Plot[xt[t], {t, 0, 8}]

---

This crashes the Kernel:

sol = NDSolve[{x''[t] + x[t] == Sin[t], x[0] == 0, x'[0] == 1,
     WhenEvent[(Abs[x[t]] > 1) && (x[t] > 5), {x'[t] -> -x'[t]}]}, 
    x, {t, 0, 8}][[1]];

xt = x /. sol;

Plot[xt[t], {t, 0, 8}]

Answer 1

Your second example doesn't work because the t in TT = t; is another t, outside the scope of the DSolve[].

Your third example doesn't kill the kernel here :

sol = NDSolve[{x''[t] + x[t] == Sin[t], x[0] == 0, x'[0] == 1, 
     WhenEvent[(Abs[x[t]] > 1) && (x[t] > 5), {x'[t] -> -x'[t]}]}, 
    x, {t, 0, 8}][[1]];

xt = x /. sol;

Plot[xt[t], {t, 0, 8}]

but the WhenEvent[] clause isn't working. I guess that what is happening is that the first part of the ÀND clause rises an "exception" each time it occurs and then the second part is checked, but it's false.
So, simply reversing the AND clauses works:

sol = NDSolve[{x''[t] + x[t] == Sin[t], x[0] == 0, x'[0] == 1, 
     WhenEvent[(x[t] > 1.5) && (Abs[x[t]] > 1), {x'[t] -> -x'[t]}]}, 
    x, {t, 0, 8}][[1]];

xt = x /. sol;

Plot[xt[t], {t, 0, 8}]

Please note that I've changed the x[t] value where the event is met only to make the effect more visible.

Answer 2

Indeed, it was a scope issue. I solved the second example, this works:

TT[t_] := t;
sol = NDSolve[{x''[t] + x[t] == Sin[t], x[0] == 0, x'[0] == 1,
   WhenEvent[Abs[TT[t] x[t]] > 1, {x'[t] -> -x'[t]}]}, 
x, {t, 0, 8}][[1]];
xt = x /. sol;
Plot[xt[t], {t, 0, 8}];