Modify PlotStyle in your code for model:
model = ParametricPlot3D[{Subscript[F, x][t, 2],
Subscript[F, y][t, 1], Subscript[F, z][t]}, {t, 0, 20},
PlotStyle -> Tube[.15], Axes -> False, Boxed -> False,
PlotRange -> {{-2.2, 2.2}, {-1.2, 1.2}, {0, 5.5}}]
And use Printout3D to put model in STL file: ("model.stl" attached)
Printout3D[model, "model.stl"]

Attachments: