Why doesn't this function definition work?

Question

I am trying to define a following function (which is a convolution integral)

force[τ_] := 
  Piecewise[{
    {F0 τ/(t1/2), 0 <= τ < t1/2}, 
    {2 F0 (1 - τ/t1), t1/2 <= τ < t1}, 
    {0, t1 <= τ}}]
x1[t_] := 
  Integrate[
    1/m force[τ] 1/ω0 Sin[ω0 (t - τ)], {τ, 0, t}, 
    Assumptions -> {0 <= t, t <= t1/2}]

F0 = 31.6*1000 (*N*);
t1 = 0.0109 (*s*);
m = 4200(*kg*);
k = 40000;
ω0 = Sqrt[k/m];

xprvi = Plot[x1[t], {t, 0, t1/2}, PlotRange -> All]

Yet this doesn't work at all. Not even close. The output is empty. I thought this is a correct definition, yet it looks like I was wrong.

Answer 1

You have made a couple of simple mistakes.

1. Define x1 with Set, not SetDelayed. You don't want to do the integral over and over again when you plot it.

2. Plot x1[t], not x1. Otherwise, the t in x1 will be in a different scope than the t in the 2nd argument to Plot.

F0 = 31.6*1000 (*N*);
t1 = 0.0109 (*s*);
m = 4200(*kg*);
k = 40000;
ω0 = Sqrt[k/m];

force[τ_] := 
 Piecewise[
    {{F0 τ/(t1/2), 0 <= τ < t1/2}, 
     {2 F0 (1 - τ/t1), t1/2 <= τ < t1}, 
     {0, t1 <= τ}}]
x1[t_] = 
  Integrate[
    1/m force[τ] 1/ω0 Sin[ω0 (t - τ)], {τ, 0, t}, 
    Assumptions -> {0 <= t, t <= t1/2}]

Piecewise[{{7.24771 (20. t - 6.48074 Sin[3.08607 t]), 0. < t <= 0.00545}}, 0.]

Plot[x1[t], {t, 0, t1/2}]