In a closely coiled helical spring of circular wire, if d = diameter of spring wire; D = Mean diameter of spring coil; n = no. of active coils; C = spring index (D/d); G = Modulus of rigidity of the spring wire material, then the stiffness of the spring is:
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To remember the formula, look at the units. Rigidity ($G$) and diameter ($d$) are on top because they make the spring stronger/stiffer. The index ($C$) and number of coils ($n$) are on the bottom because adding more "springiness" or "width" makes it easier to compress.
1. Basic Stiffness Formula: The standard formula for the stiffness of a helical spring is:
$$k = \frac{G d^4}{8 D^3 n}$$
2. Incorporating the Spring Index (C): The spring index $C$ is defined as the ratio of the mean diameter of the coil to the diameter of the wire:
$$C = \frac{D}{d} \implies D = C \cdot d$$
3. Derivation: Substituting $D = C \cdot d$ into the stiffness formula:
$$k = \frac{G d^4}{8 (C d)^3 n}$$
$$k = \frac{G d^4}{8 C^3 d^3 n}$$
Simplifying by dividing both the numerator and denominator by $d^3$:
$$k = \frac{G d}{8 C^3 n}$$