Understanding Helical Spring Deflection

When a helical coil spring is subjected to an axial load, it undergoes deflection or extension. The amount of deflection ($\delta$) depends on various factors, including the applied load, the material properties, and the geometry of the spring.

Formula for Helical Spring Deflection

The standard formula for the deflection ($\delta$) of a close-coiled helical spring under an axial load (P) is given by:

$\delta = \frac{64 P n D^3}{G d^4}$

Where:

• P is the axial load applied to the spring.
• n is the number of active turns in the spring.
• D is the mean coil diameter.
• d is the wire diameter.
• G is the shear modulus of the spring material.

Condition for Independent Extension

The question states that the extension (deflection $\delta$) of the spring must be independent of both the mean coil diameter (D) and the wire diameter (d). Looking at the deflection formula:

$\delta = \frac{64 P n}{G} \times \frac{D^3}{d^4}$

In this formula, P, n, and G are typically constant for a given spring and load scenario. The geometric properties that influence deflection are D and d, specifically through the term $\frac{D^3}{d^4}$.

For the deflection $\delta$ to be independent of D and d, the term $\frac{D^3}{d^4}$ must effectively behave as a constant that does not change when D or d are varied (as long as the specified ratio is maintained). This implies that the ratio $\frac{D^3}{d^4}$ itself must be a constant value.

If the ratio $\frac{D^3}{d^4} = \text{Constant}$, then the deflection formula becomes:

$\delta = \left(\frac{64 P n}{G}\right) \times \text{Constant}$

Since P, n, and G are constant, and we are maintaining the ratio $\frac{D^3}{d^4}$ as a constant, the deflection $\delta$ will then be independent of the individual values of D and d. Instead, it will only depend on the constant ratio maintained between them, the applied load, the number of coils, and the material properties.

Therefore, for the extension of the spring to be independent of D and d while maintaining a constant ratio between D and d, that constant ratio must be $\frac{D^3}{d^4}$.

Conclusion

The ratio that must be maintained constant for the extension of the helical spring to be independent of the mean coil diameter (D) and the wire diameter (d) is $\frac{D^3}{d^4}$.