Mathematically show how torque is split 50:50 in a standard open axle differential

1. Jan 10, 2012

bugatti79

Folks,

I am trying to see how the torque is split 50:50 in a standard axle differential mathematically or via a free body diagram.

Does anyone know?

Thanks

2. Jan 10, 2012

Ranger Mike

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

power from the engine is input through the drive shaft yoke to the differential. It is split evenly by gears with the same number of teeth to each axle.

3. Jan 10, 2012

bugatti79

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

I am aware of that but I would like to see this on paper. I have a hard time visualising how its split and also mathematically....?

4. Jan 10, 2012

Ranger Mike

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

this may help

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• diff.jpg
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5. Jan 10, 2012

bugatti79

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

I have access to pictures..thanks. I will see can I muster up a free body diagram this evening or so and get back. Off the top of my head Im just puzzled how the 50/50 torque split is independant of the pinion/crown wheel ratio. I relaise the little bevels are just acting as levers/arms when running straight on tractive grounds etc but still......?

Later..

6. Jan 10, 2012

jimgram

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

The torque applied to the drive gear is split 50-50 to each wheel shaft. The torque applied to the input of the differential is (generally) increased by the 'differential ratio', that is the ratio of the pinion to the drive gear (e.g. 3.17;1). These components simply act as any other in-line gear reduce in the drive line.

7. Jan 10, 2012

mender

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

The final drive ratio comes before the power is transferred to the axles via the side gears, sometimes known as spider gears, which is why the final drive ratio doesn't affect the torque split.

As shown in that GM film in the other thread, the force path goes from the crown gear to the carrier then to the shaft that drives the pinion gears which are free to rotate like in the film. The pinion gears then act on the gears that are on the ends of the axles, and as long as both wheels have traction and the vehicle is moving in a straight line, the pinion gears don't rotate about the pinion shaft and the axles turn at the same speed.

The reason that the split is 50/50 is that the axle gears are identical and are driven by one set of pinion gears. To get something other than a 50/50 split, the axle gears would have to be different sizes or tooth counts.

For example if you wanted to have more torque going to the right side axle, that gear would have to be larger than the left side by the amount of the increased torque demand; say 60/40 so the right gear would be 50% bigger to get 50% more leverage. With even tooth spacing that'd be like having the right side with 15 teeth and the left side with 10 if you want to do the math. To make that work there'd have to be a means of driving the different sized axle gears or two sets of pinion gears on that pinion shaft.

In short, since the axle gears are identical in size and tooth count, they get exactly the same amount of torque applied to them by the pinion gears, making for a 50/50 split. For math, it's the ratio between the tooth counts of the axle gears.

Have a look at a torque splitting centre differential on an all wheel drive vehicle; they usually use planetary gears to get the desired torque split with usually more rear torque than front. If one of those was placed in a differential housing the same uneven torque split would apply (a tip for those racers out there who might benefit from a little torque biasing!).

48/52 and 35/65 front/rear split are numbers used by the Mercedes-Benz 4-matic equipped vehicles to get the balance they're after. The same math works for them.

Last edited: Jan 10, 2012
8. Dec 12, 2012

janell

Re: Mathematically show how torque is split 50:50 in a standard open axle differentia

This was very helpful, thank you @ mender.