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Sep 2nd, 2014
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  1. params = {ν1 -> 1.0, ω1 -> 10.0, F -> 4.0};
  2. system =
  3. {D[x1[t], {t, 2}] == -ν1 D[x1[t], t] - ω1^2 x1[t] + F Cos[ω t], x1[0] == 1, x1'[0] == 0};
  4. soln = DSolve[system /. params, x1[t], t][[1]][[1]];
  5. Plot[x1[t] /. soln /. ω -> 8, {t, 1, 20}, Frame -> True, Axes -> False]
  6.  
  7. amp = Table[Max[Limit[x1[t] /. soln, t -> ∞]], {ω, 1, 20, 1}]
  8. ListPlot[amp]
  9.  
  10. params = {ν1 -> 40.0, ω1 -> 10.0, F -> 10.0};
  11. x1 = a Sin[ω t + ϕ];
  12. system = D[x1, {t, 2}] == -ν1 D[x1, t] - ω1^2 x1 + F Cos[ω t]
  13. amp = Solve[system /. params, a]
  14. phase = Solve[D[a /. amp, t] == 0, ϕ][[1]][[1]]
  15.  
  16. params = {ν1 -> 1, ω1 -> 10, F -> 4};
  17. system = {D[ x1[t], {t, 2}] == -ν1 D[x1[t], t] - ω1^2 x1[t] + F Cos[ω t], x1[0] == 1, x1'[0] == 0};
  18. soln = DSolve[system /. params, x1[t], t][[1, 1]];
  19.  
  20. (* and the steady state is*)
  21. lim = ((List @@ (Expand@soln[[2]])) /. x_ /; (Limit[x, t -> Infinity] == 0) :> 0)
  22. steadyState = Simplify@Together[Plus @@ lim]
  23. (* (-4 (-100 + ω^2) Cos[t ω] + 4 ω Sin[t ω])/(10000 - 199 ω^2 + ω^4) *)
  24.  
  25. Plot[NMaxValue[steadyState, t], {ω, 1, 20}, PlotRange -> All]
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