memo    Fig 8.1  p.176

eq = {x''[t] + 2 z wn x'[t] + wn^2  x[t] == 
          0, x'[0] == 5, 
        x[0] == 0}
para = {wn -> 2 Pi , z ->  0.1}
ans = NDSolve[eq /. para, 
          x[t], {t, 0, 5}][[1]]
Plot[{x[t] /. ans, 0.82  Exp[-z wn t] /. para}, {t, 0, 5}, 
      PlotRange ->  {-1, 1}]

example 8.1  Fig. 8.2  p.176

eq = {x''[t] + 2 z wn x'[t] + wn^2  x[t] == 
          0, x'[0] == v0, 
        x[0] == x0}
para = {wn -> 2 Pi , z ->  0.05, x0 ->  1, 
        v0 ->  0}
ans = NDSolve[eq /. para, 
          x[t], {t, 0, 8}][[1]]
Plot[x[t] /. ans, {t, 0, 8}, 
      PlotRange ->  {-1, 1}]

Print["T sec=", T = (6.25 - 1.23)/5, "  fn Hz=", 
      1/T, "  ζ=", Log[E, 0.734/0.536]/2/Pi]