Collection of formulae

A collection of useful formulas, which I have gathered over the years.

Gears with involute teeth

Spur gear teeth

General formulae for spur gears (Gears with \(\beta = 0^\circ\)). The list would need some adjustments to also work with helical gears, especially the module \(m\) would need to be split into \(m_n\) and \(m_t\).

Name Formula Unit
Gear index \(i \in \{1, 2\}\) \(1\)
Number of teeth \(z_i\) \(1\)
Helix angle at reference circle \(\beta\) \(1^\circ\)
Normal pressure angle \(\alpha_n = 20^\circ\) \(1^\circ\)
Reference pressure angle \(\alpha_t = \arctan \frac{\tan \alpha_n}{\cos \beta}\) \(1^\circ\)
Operating pressure angle \(\alpha_{wt} = \arccos\frac{d_{bi}}{d_{wi}}\) \(1^\circ\)
Operating pitch diameter \(d_{wi} = \frac{2 \cdot z_i}{z_1 + z_2} \cdot a\) \(mm\)
Reference center distance \(a_d = \frac{d_1 + d_2}{2} = m \cdot \frac{z_1 + z_2}{2}\) \(mm\)
Center distance \(a = a_d \cdot \frac{\cos\alpha_t}{\cos\alpha_{wt}}\) \(mm\)
Module \(m\) \(mm\)
Pitch on reference circle \(p = m \cdot \pi\) \(mm\)
Profile shift factor \(x_i\) \(1\)
Profile shift \(x_i \cdot m\) \(mm\)
Profile shift factor sum \(\sum x_i = x_1 + x_2\) \(1\)
Addendum \(h_{ap} = m\) \(mm\)
Dedendum \(h_{fp} = m + c\) \(mm\)
Tooth profile height \(h_p = 2 \cdot m + c = h_{ap} + h_{fp}\) \(mm\)
Addendum alteration \(k \cdot m = a - a_d - m \cdot \sum x_i\) \(mm\)
Bottom clearance \(c = 0.25 \cdot m\) \(mm\)
Reference/Pitch diameter \(d_{i} = z_i \cdot m\) \(mm\)
Base circle diameter \(d_{bi} = d_i \cdot \cos\alpha_t\) \(mm\)
Tip diameter \(d_{ai} = d_i + 2 \cdot ( x_i \cdot m + h_{ap} + k \cdot m)\) \(mm\)
Root diameter \(d_{fi} = d_i + 2 \cdot ( x_i \cdot m - h_{fp})\) \(mm\)

Determining parameters of a gear

An easy method to determine most of the parameters of a physical gear is shown here. All you need to do is to measure a pair of outer distances between a number of teeths, while that number is being denoted as \(n\) or \(o\) in the following formulas. The correct amount of teeths needed depends on the gear geometry, in practice you need to make sure that you are measuring tangentially to the flanks of the teeth. If you have those two distances, you can easily calculate the module \(m\) and the profile shift factor \(x_i\) of a given gear.

These formulas work with spur gears teeth, but i haven't checked them if they work with helical gear teeth.

Name Formula Unit
Base tangent length \(W_n = m \cdot \cos\alpha_t \cdot ((n-0.5)\cdot \pi + z_i \cdot \text{inv}\alpha_t) + 2 \cdot x_i \cdot m \cdot \sin\alpha_t\) \(mm\)
Module \(m = \frac{W_n - W_o}{\pi \cdot \cos\alpha_t \cdot (n - o)}\) \(mm\)
Profile shift factor \(x_i = \frac{W_n - m \cdot \cos\alpha_t \cdot ((n - 0.5) \cdot \pi + z_i \cdot \text{inv}\alpha_t)}{2 \cdot m \cdot \sin\alpha_t}\) \(1\)
Involut function \(\text{inv}\alpha = \tan\alpha - \alpha\) \(rad\)

\(\text{inv}\alpha_t \approx 0.0149\) with \(\alpha_t = 20^\circ\)