determine the ratio between moment and loadany

Question

Determine the ratio between moment and load:

Any close coiled helical spring of circular section extends 1 mm while subjected to an axial load of W and then given an angular rotation of 1 radian while a moment M is independently applied around the axis. Determine the ratio M / W

Solution

For a close coiled spring :

Δ=64 W R³ n/ Gd ⁴ (Δ = 1 mm)

∴          W = Gd ⁴/64 R³ n                   ------------- (1)

φ= 128 M Rn  / Ed ⁴

∴          M =      E d ⁴/128 R n                        -------------- (2)

Eq. (2) divided by Eq. (1), we get

M/ W = (Ed ⁴/128 R n) × (64 R³n/ Gd ⁴)

=   E   R² / 2 G                      E = 2 G (1 + ν)

M / W  = R²  (1 + ν)