Spur Gears Help

Spur Gears is a web application that creates involute spur gears and provides the following main tasks:

  • calculate the geometric parameters of the gear,
  • draw an accurate image of the gear with a detail of the teeth,
  • adjust the tooth thickness by use of correction (optimization of toothing),
  • calculate the tooth root bending stress with the Lewis method,
  • return a DXF file containing the 2D profile of the gear to be used with 2D and 3D CAD software.

This help page gives a complete clarification of the meaning of the terms used in the software and additional information regarding spur gears as formulas, an useful bibliography, a numerical example and tips about gear tooth correction.

Figure #1 contains a visual representation of the terms related to spur gears and to the tooth root stress calculation.

#1 - Detail of a spur gear

Detail of a spur gear

In figure #2 there is the schema used to generate the involute profile of the tooth of the gear.

#2 - Involute schema to create the gear

Involute schema to create the gear

Figure #3 represents the rack cutter used to generate the gear, as defined in ISO 53:1998.

#3 - Rack tooth profile

Rack tooth profile

In the following table there are a numerical example and some basic formulas related to standard spur gears valid if R/m = 0 and x/m = 0.

Element Formula Example
number of teeth $$z$$ 30
module $$m$$ 5 mm
pressure angle $$ \alpha $$ 20°
rack shift coefficient $$ x/m $$ 0
coefficient of fillet radius of the rack cutter $$ R/m $$ 0
static nominal torque $$ C $$ 250 Nm
face width $$ b $$ 10 mm
$$ l_0 = \frac{d} {2} \cdot {sin}^2(\alpha)$$ 8.77 mm
$$ \frac{y} {d/2} = \frac{2.5} {z} - {sin}^2(\alpha)$$ -0.0336
$$ l = l_0 + y$$ 6.25 mm
pitch $$p = {m \cdot \pi}$$ 15.71 mm
reference diameter $$d = {m \cdot z}$$ 150 mm
base diameter $$d_b = d \cdot cos(\alpha)$$ 140.95 mm
involute tooth limit diameter $$d_{lim} = 2 \cdot \sqrt{ \left( r-l \right)^2+\left( \frac{l}{tan(\alpha)}\right)^2}$$ 141.72 mm
root diameter $$d_f = d - 2 \cdot l$$ 137.5 mm
addendum diameter $$d_t = d + 2 \cdot m $$ 160 mm
tooth addendum $$t_a = m $$ 5 mm
tooth dedendum $$t_f = 1.25 \cdot m $$ 6.25 mm
circular reference tooth thickness $$s = \frac{m \cdot \pi}{2} $$ 7.85 mm
$$z_{min} = 1.25 \cdot \frac {2} {{sin}^2(\alpha)} $$ 22
rack addendum $$h_a = 1.25 \cdot m $$ 6.25 mm
rack dedendum $$h_f = 1.25 \cdot m $$ 6.25 mm
nominal load, normal to the line of contact $$F_{bn} = \frac {C} {d/2 \cdot cos(\alpha)} $$ 3547.26 N
$$ \alpha_1 $$ 26.92°
nominal transverse load in plane of action $$F_{bt} = F_{bn} \cdot cos(\alpha_1) $$ 3162.85 N
tooth root chord at the critical section $$ s_{Fn} $$ 9.74 mm
bending moment arm relevant to load application at the tooth tip $$h_{Fe} $$ 9.4 mm
tooth form factor - Lewis method $$Y_{L} = \frac {{s_{Fn}}^2} {6 \cdot h_{Fe} \cdot m} $$ 0.3361
tooth root bending stress at point T $$\sigma_{f} = \frac {F_{bt}} {Y_L \cdot b \cdot m} $$ 188.21 N/mm2

Gear ratio

The gear ratio $\tau$ of a gear train is the ratio of the angular velocity of the input gear to the angular velocity of the output gear: $$\tau=\frac {\omega_1} {\omega_2}=\frac {d_2} {d_1}=\frac {z_2} {z_1}$$ where

$\omega_1$ is the angular velocity of the input gear e $\omega_2$ is the angular velocity of the output gear;
$d_1$ is the reference diameter of the input gear e $d_2$ is the reference diameter of the output gear;
$z_1$ is the number of teeth of the input gear e $z_2$ is the number of teeth of the output gear.

Center distance

For a pinion and a wheel without correction (x/m = 0) or in case of complementary correction (e.g. the pinion with a positive correction x/m = +0.5 and the wheel with a negative correction x/m = -0.5), the center distance $i$ is calculated with the formula: $$i = \frac {d_1} {2} + \frac {d_2} {2} = \frac {m \cdot (z_1 + z_2)} {2} $$ In case $x_1+x_2\neq0$, the center distance $i'$ is different from $i$ and may be calculated solving the following formulas: $$ inv(\alpha')= \frac {2 \cdot (x_1+x_2) \cdot tan(\alpha)} {m \cdot (z_1 + z_2)} + inv(\alpha)$$ $$i'=i\cdot\frac {cos(\alpha)} {cos(\alpha')}$$ where $ \alpha' $ is the working pressure angle, different from the pressure angle $ \alpha $ of the rack cutter.


The pinion-wheel clearance $c$ depends from the value of $(x_1+x_2)$ and may be calculated with the formula $$c=m\cdot\left[0.25-\frac {x_1+x_2} {m}+\frac {z_1+z_2} {2}\cdot \left( \frac {cos(\alpha)} {cos(\alpha')}-1\right)\right]$$ For gears with $x_1+x_2=0$, the clearance is equal to 0.25m (type A basic rack tooth profile - ISO 53:1998).


In the software, it is possible to set the resolution of the involute generating process of the gear.

Here are the meaning and the associated values of the Resolution parameter:

Pitch of the movement of the rack cutter to create the gear image Number of points of the involute profile of the flank of the tooth
Coarse 4 deg 5
Medium 2 deg 10
Fine 1 deg 20


[1] - Georges Henriot - Ingranaggi - Trattato teorico e pratico - Vol. I e II - Tecniche Nuove - Ed. 1977
[2] - Lodovico Soria - Tecnica degli ingranaggi : trattato teorico-pratico di calcolo, correzione, dentatura, misura, trattamento termico, finitura e controllo ingranaggi cilindrici, elicoidali, a catena, conici dritti e conici spiroidali - Editore Viglongo - Torino - Ed. 1971
[3] - prof. Paolo Righettini - Progettazione funzionale di sistemi meccanici - Ruote Dentate - Università degli Studi di Bergamo - Italy
[4] - ISO 6336-1:1996 - Calculation of load capacity of spur and helical gears - Part 1: Basic principles, introduction and general influence factors
[5] - ISO 6336-3:2006 - Calculation of load capacity of spur and helical gears - Part 3: Calculation of tooth bending strength
[6] - ISO 53:1998 - Cylindrical gears for general and heavy engineering - Standard basic rack tooth profile
[7] - Gear - en.wikipedia.org/wiki/Gear