open coiled helical springin an open coiled

Question

Open coiled helical spring:

In an open coiled helical spring along with α = 38^o, if the inclination of the coil is ignored, determine the percentage by which the axial extension is under estimated.

E = 200 GPa, G = 80 GPa.

Solution

Δ= (64 W R³ n / d ⁴ cos α)( (cos² α/G)+(2 sin ² α /E))

            = K/ cos α[cos² α/G     +2 sin ² α/E]

α= 38^o , Δ = K × (0.115/0.866) × 10^{- 4}  = K × 0.0147

If α is neglected,

α= 0, Δ₂  = K × 0.0125

% Error = ((Δ₁ - Δ₂ ) / Δ1 )× 100

= (0.0147 - 0.0125/0.0147) × 100 = 14.96%