This post describes and explain the mathematical model of a car differential. We will consider the following device:

## Framework

First, let’s define the framework, axis and positive direction of rotation:

Let’s name each gear. The gear number is labelled with name and is composed of teeth:

## Right angle transmission

Let’s have a look at the right angle transmission:

This is a classical right angle gear, the transmission is given by:

and

We will no longer consider this transmission in the following.

## Carrier’s frame

We will now work in a new referential attached to the carrier and focus on gears , , and . The term is the angular velocity of gear expressed in the carrier’s referential.

According to the previous direction of rotation, the transmissions between gears , , and are given by:

Note that the gear is redundant with the gear . The relation between gear and is given by:

As , the relation between angular velocities is :

## Global frame

Let’s now come back in the global referential. The carrier angular velocity is given by . In the carrier’s referential, the angular velocity of gear gear is equal to . The angular velocity of gear in the global referential is thus given by:

With the same reasoning, we can also express the angular velocity of gear in the global referential:

Gathering the previous equations gives the following system:

Solving the system provides the final equation of the model:

## Torque

The relation between gears , and is given by:

As :

And finally :

## Power

The input power must be equal to the output power:

As , the output torque can be rewrite as:

As , the previous equation can be simplified:

As , the previous equation can be rewriten:

Finally:

## Conclusion

The relation on velocities is given by:

The relation on torques is given by: