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# 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.

$$G=\dfrac {E}{2(1+ν)}$$

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

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

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

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