Conical spring with round cross section

Determine mechanical properties of a conical helical spring with wire diameter d, radii r0 r1, shear modulus G and nw as number of active windings. Calculate also deflection v and tension σ as a function of force F.

The shear modulus can be calculated using Young's Modulus and Poisson's ratio of a material.

\( G=\dfrac {E}{2(1+ν)}\)

F d r 1 r 0 n w = 6

\(k=\dfrac {\ Gd^{4}}{16n_{w}(r_{0}+r_{1})(r_{0}^{2}+r_{1}^{2})}\)

\( v_{(F)}=\dfrac {\ 16Fn_{w}(r_{0}+r_{1})(r_{0}^{2}+r_{1}^{2})} {Gd^{4}} \)

\( v_{max}=\dfrac {\ n_{w}(r_{0}+r_{1})(r_{0}^{2}+r_{1}^{2})πσ} {Gdr_{1}} \)

\(σ_{(F)}=\dfrac {\ 16Fr_{1}} {πd^{3}} \)