Spiral Length

Overview

Engineering reference data for Spiral Length in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

General Archimedean Spiral Formula

For a perfect Archimedean spiral where the radius increases linearly from the center, the total arc length $L$ from the starting point (radius $r_0$) to the outer point (radius $r_n$) after $n$ turns is given by:

$$L = \frac{1}{2} \left[ \theta \sqrt{1 + \theta^2} + \ln\left( \theta + \sqrt{1 + \theta^2} \right) \right]_{\theta_0}^{\theta_n}$$

where $\theta = \frac{2\pi r}{a}$, and $a$ is the increase in radius per turn (spiral pitch). This integral form is necessary for precise geometric spirals, distinct from the practical approximation formula for coiled materials.

Application Note

The primary formula $L = \frac{\pi n (D + d)}{2}$ (using $\pi \approx 3.14$) provides a practical estimate for the length of materials with uniform thickness wound in a coil. This applies to belts, hoses, ropes, and similar items where the total length is approximated by the mean circumference multiplied by the number of turns.

References

• Original Source