The differential gearbox

If you are an engineer then it is a must for you to know how a differential gearbox is working. It is a matter of self-respect.

If you still didn’t learn it, you my try it here. There is enough beauty in a differential gearbox and your effort shall be awarded.

A differential gearbox is a completely mechanical device that consist of an enclosure, some gears and shafts. It has three input/output shafts (I will call them x, y and z shaft). I said input/output because any of these three shafts can serve as either input or output shaft – but in any given moment at least one shaft must be output and at least one shaft must be input.

Its main purpose is to sum or differentiate rotation rates (N) applied on its shafts, while maintaining constant torque (T) ratio between shafts at any given moment. In general we can say:

   - Nx, Ny and Nz are rotation rates [rotations per second] for axes x, y and z
   - Tx, Ty and Tz are torques [Newton-meter] on axes x, y and z
   - A, B, and C are constants – they depend on gear ratios inside the differential gearbox (note that only two constants are actually needed, the third one is redundant)

If two shafts are used as input shafts then the gearbox will add their rotation rates (usually weighted by some constants) and will deliver the result to third output shaft. On the other hand, if only single shaft is used as an input shaft, the gearbox can split its rotation rate to two other shafts (also usually weighted by some constants) – but note that the percentage of this “rotation rate splitting” doesn’t have to have 50:50 or any other fixed ratio. Instead, this ratio is leveled to maintain constant torque ratio between shafts (this works especially nice if applied loads have such characteristic that increased rotation rate increases the torque).

Okay, more about torques.... The torque ratio between any two shafts is always maintained constant (a normal, two-shafted gearbox behaves the same). The torque on all differential gearbox shafts can be increased or decreased only simultaneously and proportionally. For example, if your car is driving along a road and then one wheel steps into a mud, the increased drag will be simultaneously felt (in form of increased torque) also by both, the other wheel and the engine. But if the mud is so slippery that your wheel loses its grip, then reduced torque will also be felt by the other wheel and by the engine. One wheel may spin in the mud, but the reduced torque on the other wheel will not be enough to pull you out.

You see that if you leave one shaft disconnected from load (zero torque applied) then you will not have any torque even on other two shafts.

Enough talking! How do they construct a differential gearbox? This is the charming part. It is amazing how clever it uses gears inside it.

It is not easy to draw a differential gearbox in two-dimensional page and I was thinking hard on how to present it. Finally I came up with the following picture.

There are more useful designs around than the one depicted above, but I think this one is the easiest to draw. Three output shafts, together with belonging gears, are colored blue, yellow and green. The simplest is the blue (x) shaft . It only has one gear directly attached to it.

Somewhat more complex is the yellow (y) shaft. The yellow gear attached to it drives (or is driven by) another yellow double-gear (piggyback). The piggyback gear rotates around the green shaft (it is not fixed to the green shaft). Otherwise there is no difference between blue gear and yellow piggyback gear – they both work symmetrically inside the differential gearbox.

The most complex one is the green (z) shaft and its two green gears. These two gears are mounted to short stubs that go vertically from the shaft. As the shaft rotates both gears rotate all around, carried by the stubs. In addition, every of these two gears can also rotate around the stub it is mounted on.

There is no much sense to discuss it any further. If you still don’t get it, you should spend some time imagining it in your head. You can also try to find some animations on the web. As a final step, you can go visiting some workshop and check it for real.

Anything else?

Well there is one charming thing. The design shown below is invented to make things simpler. It is used sometimes in industry (for small gearboxes used in servo systems).

There are (only) two gears, the blue one and the green one. The green one is special - it is somewhat flexible. Streched by the yellow axle from inside, the flexible green gear touches (becomes meshed with) the blue gear. The yellow axle can rotate inside the green 'ring-like' gear causing it to touch the blue gear at different positions, 360 degrees around.

The thing is that the blue gear and the green gear don't have the same number of teeth. For example the green could have 100, while the blue could have 101 teeth. After the yellow axle is rotated for one revolution, the green and the blue will travel one tooth relative to each other.

(The above picture is only 2-D crossection of such differential mechanism. In reality, the green gear must be very much extruded 'deeply into your monitor'. It looks like a cup - the bottom end is non-deformable and its shaft is atteched there.)

Sure, there are many disadvantages of such differential gearbox mechanism. Teeth on blue and green gears must be relatively small (there must be many of them), and the difference between number of teeth must be small. Due to deformation of the green gear, teeth become non-idealy-shaped. Also, heat is disipated when material is deformed.

See also the train wheel article (works without a differential).
See also the gasoline vs diesel engine comparation article.
See also the power vs torque article that explains what really accelerates your ride.