differential equations - Harmonic oscillator with damping attrition

Considering the following physical situation:

and writing the following code:

m := 1.52
g := 9.81

us := 0.15
uk := 0.10
k := 2.12
xi := 4.00
vi := 0.00
tmax := 10

P := m g
Fs := us P
Fk := uk P

Fe[t_] := -k x[t]

sol = NDSolve[{
        Fe[t] - Sign[x'[t]] Fk == m x''[t],
        x[0] == xi,
        x'[0] == vi},
      x, {t, 0, tmax}];

Plot[Evaluate[{
        Sign[x'[t]] Fs, 

        Fe[t], 
        x[t]} /. sol],
    {t, 0, tmax},
    AxesLabel -> {"t", "fct[t]"},
    PlotLegends -> {"-Fs", "Fe", "x"}]

you get the following graph:

which shows that oscillations are over for $t \approx 8\,s$ causes kinematic friction.

On the other hand, putting us = 0.40 you get this other graph:

which shows that oscillations are over for $t \approx 3\,s$ causes static friction.

Question: is it possible to automate all this by making the x(t) graph plot until the motion stops?

Answer

This is a perfect use case for WhenEvent:

sol = NDSolve[
  {
    Fe[t] - Sign[x'[t]] Fk == m x''[t],
    x[0] == xi,
    x'[0] == vi,

    WhenEvent[x'[t] == 0 && Fs > Abs[k x[t]], tmax = t; "StopIntegration"]
  }
  , x, {t, 0, Infinity}]

Note that this automatically sets tmax, so you don't need to specify anything before. The only thing to note is that you can't replace k x[t] with Fe[t], since WithEvent doesn't see the x[t] in that case. You could write (note the Evaluate wrapped around the condition)

WhenEvent[Evaluate[x'[t] == 0 && Fs > Abs[Fe[t]]], tmax = t; "StopIntegration"]

if you really want to write Fe[t].

For us=0.15:

For us=0.4: