# Differentials

The basic shape of a differential is shown in the picture above. It consists of a central gear, the so-called sun wheel, several gears that we call planet gears, an internal gear and a planet carrier. The in- and outgoing shafts are on the same axis line. Usually there are three planet gears used to achieve a load stability to the planetary system. Planetary systems require an accurate production. A planetary system has a large gear ratio combined with a compact design.

## Relationship between the gears in a planetary system

To determine the relation of the number of teeth in a planetary system both of the solar wheel A (za), the planet gears B (zb), the internal gears C (zc) and the number of planet gears N, the following conditions must be met. This is necessary to make the assembly possible.

### 1st condition:

zc = za + 2•zb

This condition is the result from the centre distance that these gears should have. Deviating number of teeth are possible by corrections, but the centre distance of the Sun wheel A to the planet gears B (ax1) and the center distance of the planet gears B and the internal gears C (ax2) should always be the same.

ax1 = ax2

### 2nd condition:

(za + zc)/N = integer

This condition divides the planet gears evenly over the circumference of the Sun wheel A. If an odd number of planet gears is required the following formula should be submitted:

(za + zc)•θ/180= integer

where: θ is the half angle between two consecutive planet gears

### 3rd condition:

zb + 2 < (za + zb)•sin(180/N)

This condition ensures that two adjacent planet gears can work without further contact. This condition is valid for an equal division of the planet gears. For other versions the following formula should be submitted:

dab < 2•ax•sin θ

where:
dab = outer diameter of the planet gear
ax = centre distance between the Sun wheel and the planet gears.

Outside these three conditions still there might be problems with the interference between the internal gear and the planet wheels.

Rotational speeds of planetary systems

In planetary systems the rotational speeds and rotational directions are linked with each other and depend on which of the three different components is getting locked.

## The planetary type

With this type of planetary system the internal gear is stationary. The Sun wheel A acts as the input and the planet carrier D acts as output.

The table below shows the different speeds for each item in this mechanism:

 N° Description Sun wheel A za Planet gear B zb Crown C zc Bar D 1 Sun wheel A turns 1x Bar D stands still +1 – – 0 2 The unit turns + + + + + 3 Sum of 1 & 2 1+ – 0 (stationary) +

Ratio = (za/zc)/(1+za/zc) = 1/(zc/za+1)

The rotational direction of the incoming and outgoing axis are equal.

## The sun wheel type

With this type the sun wheel is stationary. The internal gear C acts as the input and the planet carrier D acts as output.

The table below shows the different speeds for each item in this mechanism:

 N° Description Sunwheel A za Planet gear B zb Crown C zc Bar D 1 Sun wheel A turns 1x Bar D stands still +1 – – 0 2 The unit turns + -1 -1 -1 -1 3 Sum of 1 & 2 0 (vast) –-1 –-1 -1

Ratio= -1/(-za/zc-1) = 1/(za/zc+1)

The rotational direction of the incoming and outgoing axis are equal.

## The pen type

With this type the planet carrier (pen) D is kept still. The planet gears runs just around a fixed pen D. This system loses its function as differential system and works like a normal gear system. The Sun wheel A works as input  and the internal gear C acts as output.

Ratio = -za/zc

The planet gears are intermediate gears. The ingoing and outgoing shaft rotate in opposite sense.