Z[t_, a_, b_, c_, d_, ai_, bi_, ci_, di_] := 
 Module[{aim, bim, cim, dim},
  If[ai, aim = I, aim = 1];
  If[bi, bim = I, bim = 1];
  If[ci, cim = I, cim = 1];
  If[di, dim = I, dim = 1];
  aim*Exp[a I t] + bim*Exp[b I t]/2 + cim* Exp[c*I*t]/3 + dim* Exp[d*I*t]/10]
Manipulate[
 ListLinePlot[
  Table[{Re[Z[t, a, b, c, d, ai, bi, ci, di]], Im[Z[t, a, b, c, d, ai, bi, ci, di]]}, {t, 0, 2 Pi, 0.01}], 
  PlotStyle -> {RGBColor[.6, .84, .75], Thick}, Axes -> False, 
  ImageSize -> {500, 350}, AxesLabel -> {"real", "complex"}, 
  AspectRatio -> 1, PlotLabel -> TraditionalForm[
    z[t] \[LongEqual] Z[t, a, b, c, d, ai, bi, ci, di]]],
 {{a, -4, "Frequency of Rotation of Wheel 1"}, -30, 30, 1, Appearance -> "Labeled"},
 {{b, 9, "Frequency of Rotation of Wheel 2"}, -30, 30, 1, Appearance -> "Labeled"},
 {{c, 20, "Frequency of Rotation of Wheel 3"}, -30, 30, 1, Appearance -> "Labeled"},
 {{d, 10, "Frequency of Rotation of Wheel 4"}, -30, 30, 1, Appearance -> "Labeled"},
 Delimiter,
 "Multiply by \[ImaginaryI] (changing starting position):", 
 Row[{Control@{{ai, True, "First Term"}, {True, False}}, 
   Control@{{bi, True, "Second Term"}, {True, False}}, 
   Control@{{ci, True, "Third Term"}, {True, False}}, 
   Control@{{di, True, "Fourth Term"}, {True, False}}}, Spacer[7]],
 SaveDefinitions -> True]