For the close coil helical spring of the maximum deflection is A. $$\frac{{{\text{W}}{{\text{D}}^3}{\text{n}}}}{{{{\text{d}}^4}{\text{N}}}}$$ B. $$\frac{{2{\text{W}}{{\text{D}}^3}{\text{n}}}}{{{{\text{d}}^4}{\text{N}}}}$$ C. $$\frac{{4{{\text{W}}^2}{{\text{D}}^3}{\text{n}}}}{{{{\text{d}}^4}{\text{n}}}}$$ D. $$\frac{{8{\text{W}}{{\text{D}}^3}{\text{n}}}}{{{{\text{d}}^4}{\text{n}}}}$$

The correct answer is $\frac{{2{\text{W}}{{\text{D}}^3}{\text{n}}}}{{{{\text{d}}^4}{\text{N}}}}$.

The maximum deflection of a close coil helical spring is given by the following equation:

$$\delta = \frac{{2{\text{W}}{{\text{D}}^3}{\text{n}}}}{{{{\text{d}}^4}{\text{N}}}}$$

where:

• $\delta$ is the maximum deflection,
• $W$ is the load applied to the spring,
• $D$ is the mean coil diameter,
• $n$ is the number of active coils,
• $d$ is the wire diameter, and
• $N$ is the spring rate.

The spring rate is a measure of how much the spring will deflect under a given load. It is calculated by the following equation:

$$N = \frac{{F}}{{\delta}}$$

where:

• $F$ is the force applied to the spring, and
• $\delta$ is the deflection of the spring.

The maximum deflection of a close coil helical spring is proportional to the load applied to the spring, the cube of the mean coil diameter, the number of active coils, and the inverse of the fourth power of the wire diameter.