This equation is problematic to my eye. The first alarm is that it contains Sign, which is discontinuous. Next, both Google and NDSolve claim that the solution becomes a nonzero constant after a certain instant, but this makes the two sides of the differential equation not the same:
eq = r''[t] == -r[t] - Sign[r'[t]]/10;
solNum = NDSolveValue[{eq, r[0] == 0, r'[0] == 1}, r, {t, 0, 20}]
Plot[solNum[t], {t, 0, 20}]
Plot[Evaluate[List @@ eq /. r -> solNum], {t, 15, 20}]
This is out of my expertise.
