You are fantastic! Thank you so much. Everything is working fine now.

Thanks. That code works. I really appreciate you helping out.

Now that we know that the series evaluation works in the radian domain, it should be easy to extend it to the time domain. I tried modifying the code to this, but no luck. I think I am close, but missing something simple.

[Chi] = 1; n =.; t =.; f = 1/[Chi]; [Omega] = 2 [Pi] f; r = 1.; L = 5 r; [Theta] = [Omega] t;

Clear[fourier, n, nmax];

[Theta]2 = [Theta] + 2 [Pi]/3.; [Theta]3 = [Theta]2 + 2 [Pi]/3;

Clear[P1, P2, P3, coeff]; P1[[Theta]_] := -r Sin[[Theta]] - (r^2 Sin[[Theta]] Cos[[Theta]])/ Sqrt[L^2 - r^2 (Sin[[Theta]])^2]; P2[[Theta]_] := P1[[Theta] + 2 Pi/3];

P3[[Theta]_] := P1[[Theta] + 4 Pi/3];

fsum[t] := Max[P1[[Theta]], 0] + Max[P2[[Theta]], 0] + Max[P3[[Theta]], 0];

coeff[n_Integer] := coeff[n] = (1/[Chi]) NIntegrate[ fsum[s] Exp[-I ([Pi] n s)/[Chi]], {s, 0, [Chi]/6, [Chi]/3, [Chi]/2, 2 [Chi]/3, 5 [Chi]/6, [Chi]}];

fourier[nmaxInteger , t_] := Evaluate[Re@Sum[coeff[n] Exp[I ([Pi] n t)/[Chi]], {n, 0, nmax}]];

Manipulate[ Plot[Evaluate[{fsum, fourier[n, t]}], {t, 0, [Chi]}, PlotRange -> {0, 1.3}], {n, 10, 20, 1}]

I cut and paste your code in a new notebook and the computation hangs. What can I be doing wrong?

Yes, I did. Attached is the new file.

Attachments:

Thanks again for the suggestions and help.

I modified my file to explicitly compute the Fourier coefficients like this:

a[n_Integer] : = (1/[Chi]) NIntegrate [ fsum[t] Cos[( [Pi] n t)/[Chi]], {t, 0, [Chi]}] b[n_Integer] : = (1/[Chi]) NIntegrate [ fsum[t] Sin[( [Pi] n t)/[Chi]], {t, 0, [Chi]}] Print[{a0, a[1], b[1], a[2], b[2]}]

and the computation still hangs and provides errors on convergence.

The command NIntegrate only works for fully numerical functions. Symbolic parameters are not accepted. You can try like this:

a[0, \[Chi]_] := 
  a[0, \[Chi]] = (1/\[Chi])  NIntegrate [fsum[t], {t, 0, \[Chi]} ];

a[n_Integer, \[Chi]_] := 
  a[n] = (1/\[Chi])  NIntegrate [
     fsum[t] Cos[( \[Pi] n t)/\[Chi]], {t, 0, \[Chi]}];

b[n_Integer, \[Chi]_] := 
  b[n] = (1/\[Chi])  NIntegrate [
     fsum[t] Sin[( \[Pi] n t)/\[Chi]], {t, 0, \[Chi]}];
a[0, 1]
b[2, 3]

I would use radiants instead of degrees. Also, the If constructs do not play well with Integrate. I would write

A = 1;   f = 5;    r = 1;    L = 5 r;
\[Theta] =.; \[Theta]2 = \[Theta] + 
  120 Degree; \[Theta]3 = \[Theta]2 + 120 Degree;
\[Delta] = -30 Degree;

P1 = -r Sin[\[Theta]] - (r^2 Sin[\[Theta]] Cos[\[Theta]])/Sqrt[
   L^2 - r^2 (Sin[\[Theta]])^2];
P2 = -r Sin[\[Theta]2] - (r^2 Sin[\[Theta]2] Cos[\[Theta]2])/Sqrt[
   L^2 - r^2 (Sin[\[Theta]2])^2];
P3 = -r Sin[\[Theta]3] - (r^2 Sin[\[Theta]3] Cos[\[Theta]3])/Sqrt[
   L^2 - r^2 (Sin[\[Theta]3])^2];
tot[\[Theta]_] = Max[P1, 0] + Max[P2, 0] + Max[P3, 0]

Even with this, FourierSeries seems too slow. You can calculate the Fourier coefficients directly fast enough with NIntegrate:

NIntegrate[
  tot[\[Theta]] E^(-3 I \[Theta])/(2 Pi), {\[Theta], 0, 2 Pi/3, 
   4 Pi/3, 2 Pi}]