Helical wound spring with round cross section

Determine mechanical properties of a conical helical spring with wire diameter d, radius r, Young's modulus E and nw as number of active windings. Calculate also deflection θ and tension σ as a function of force F.

F d θ D m n w = ± 3,3 r

\( k=\dfrac {\ Ed^{4}}{64n_{w}D_{m}}\)

\( θ_{(F)}=\dfrac {\ 64Fn_{w}D_{m}r} {Ed^{4}} \)

\( θ_{max}=\dfrac {\ 2D_{m}n_{w}πσ} {Ed} \)

\( σ_{(F)}=\dfrac {\ 32Fr} {πd^{3}} \)