Helical spring deflection ratio – effect of mean coil diameter\n\nTwo closely coiled helical springs A and B are made of the same material, have the same number of active turns, and the same wire diameter. Under the same axial load W, the mean coil diameter of spring A is twice that of spring B. What is the ratio of deflections δ_B / δ_A?

Introduction / Context:
Spring sizing depends strongly on geometry. For closely coiled helical springs, deflection varies with the cube of the mean coil diameter, making diameter a dominant factor in compliance.

Given Data / Assumptions:

• Springs A and B: same material (same shear modulus G), same number of active turns n, and same wire diameter (common symbol often d_w).
• Both carry the same axial load W.
• Mean coil diameters: D_A = 2 D_B.

Concept / Approach:
The deflection of a closely coiled helical spring under axial load is δ = 8 W D^3 n / (G d_w^4). With all factors except D identical, δ ∝ D^3.

Step-by-Step Solution:
δ_A / δ_B = (D_A^3) / (D_B^3).Given D_A = 2 D_B → δ_A / δ_B = (2^3) = 8.Therefore δ_B / δ_A = 1 / 8.

Verification / Alternative check:
Dimensional check: the formula shows cubic dependence on D, consistent with torsional deformation of coils.

Why Other Options Are Wrong:
1/4, 2, 4, 8 result from linear or squared assumptions about diameter influence; the correct relationship is cubic.

Common Pitfalls:
Confusing wire diameter with mean coil diameter; using outer instead of mean diameter; ignoring inactive end coils (not affecting the ratio here since n is the same).

Final Answer:
1/8