Screw trick

f[r_, t_] := r (t (4 Pi - t)/(4 Pi^2))^0.2;
screw[x_, dir_] := Show[
  ParametricPlot3D[
   x + {f[r, t] Sin[ 2 t], f[r, t] Cos[ 2 t], t}, {t, 0, 4 Pi}, {r, 0,
     1}, Axes -> None, Boxed -> False, Lighting -> "Neutral",
   Mesh -> {2, 2}, PlotRange -> All,
   PlotStyle -> RGBColor[0.7, 0.2, 0.2]],
  Graphics3D[{RGBColor[0.7, 0.7, 0.7],
    If[dir == 1,
     {Cylinder[{x, x + {0, 0, 6 Pi}}, 0.4], GrayLevel[0.2],
      Cylinder[{x + {0, 0, 6 Pi}, x + {0, 0, 6 Pi + 0.5}}, 1]
      }
     ,
     {Cylinder[{x - {0, 0, 2 Pi}, x + {0, 0, 4 Pi}}, 0.4],
      GrayLevel[0.2],
      Cylinder[{x - {0, 0, 2 Pi}, x - {0, 0, 2 Pi + 0.5}}, 1]}
     ]
    }]];
frame[t_] :=
 Show[screw[{0, 0, 0}, 1], screw[{1.3 Cos[t], 1.3 Sin[t], 0}, -1],
  ImageSize -> {300}, ViewPoint -> {4, 5 Cos[t/3], 5 Cos[t/3]}];
Manipulate[frame[t], {t, 0, 6 Pi}]