Automotive differential problem

[hexhamallstar]

Hi there,

I'm currently doing some research into differentials and have been unable to find any indepth information on them. Every article I read covers the basics that one axle turns one direction relative to the diff casing and the other in the opposite direction so that the average of the 2 is always the casing speed.

However, I'm looking for some detail on the bevel gears inside the differential, and the effect that their gear ratio (between the planet bevel gears and sun bevel gears) has on the operation of the diff. This is with regards to a simple open diff, with 4 bevel gears on the interior of the differential.

I would also like to know the advantages of using 4 planet gears instead of just 2 (I have taken apart 2 differentials and one of them used 2 planet gears and one used 4).

I am not, I repeat, am not talking about the pinion and crown wheel, I know what effect that has!

Thanks for reading,

Phil

 

[jack action]

The differential is an epicyclic gear arrangement. On this site, you have all the math behind the concept.

On the previously linked site, the differential would be represented by example #2:

http://www.roymech.co.uk/images9/gear_epi3.gif

The casing would rotate the planetary arm (L), one wheel is connected to the sun gear (S) and the other wheel is connected to the ring gear (A). In a car differential, N_A would be equal to N_S.

In the same example, you have this table:

                      Rotation (CW= +ve) from action......Nx = number of teeth
Action                          S       L	     P	             A
Fix L and rotate Sun 1 rev CW	1	0	- NS / NP	- NS / NA

Which, if N_A = N_S, gives the case where the casing is held fixed and one wheel rotates one way and the other rotates at the same speed in the reverse direction, or:

                      Rotation (CW= +ve) from action......Nx = number of teeth
Action                          S       L	     P	             A
Fix L and rotate Sun 1 rev CW	1	0	- NS / NP	   - 1

Now, in real life, the casing is also rotating, so you have to add that rotation to the whole package as well. Say the casing rotates with a speed of 10 instead of 0, then you have to add this speed to the other gears as well:

                      Rotation (CW= +ve) from action......Nx = number of teeth
Action                          S       L	     P	             A
Add rotation L          	1+10	0+10	- NS / NP	   - 1+10

or:

                      Rotation (CW= +ve) from action......Nx = number of teeth
Action                          S       L	     P	             A
Add rotation L          	11	10	- NS / NP	     9

So S is faster than A and the speed of L is the average of the S and A.

When the differential is «inactive» (straight line), there is no rotation of P, so all gears rotate at the same speed:

                      Rotation (CW= +ve) from action......Nx = number of teeth
Action                          S       L	     P	             A
No speed difference       	10	10	     0	             10

For planetary gears, only one is necessary for the differential to work, as shown in this picture:

The only gear ratio important is the one between the sun and ring gears (or the left and right wheels, in our case) which is always equal to 1 if you want the differential working properly. The planetary gears are only idlers and their speed has no effect on the output. I suspect that they're made with the smallest number of teeth as possible for them to work properly and thus, giving the smallest package possible.

Why use more than one planetary gear? To spread the load on many teeth. Remember that in a straight line, the gears don't rotate with respect to one another. The idler gear becomes some sort a «beam» locking both axle together and transmitting the torque through the gear teeth (which do not turn w.r.t. one another). The more idlers you have, the more «beams» you have, the stronger the differential.

 

[hexhamallstar]

Thanks for the excellent reply jack, this is exactly what I was after! I did think that the bevel gear ratio is irrelevant but it seems strange to me that such an important piece of kit has an arbitrary form.

Can anybody direct me to a textbook/website that explains how to find the smallest gear that can be used while still meshing properly?

Also, I was wondering about Torque in this case. Because the planet gears are attached to the casing they aren't meshing with the crown wheel so surely torque isn't transferred in the normal way (stepped down with size)? Does all of the torque in the crown wheel simply transfer to the sun gears via the mesh with the planet gears? Or does the ratio come into play here?

Thanks,

Phil

 

[jack action]

For gear design:

http://www.roymech.co.uk/Useful_Tables/Drive/Gears.html
http://www.roymech.co.uk/Useful_Tables/Drive/Bevel_Gears.html
http://viewmold.com/Products/Technical%20Reference%20Sheet/Gears/SPUR%20GEARS-BACKLASH%20UNDERCUT%20FORMULAS.pdf
http://www.akgears.com/pdf/direct_gear_design.pdf

For the torque, it is the same everywhere, so the force acting on the planetary gear(s) is proportional to the force acting on the ring gear, considering the radii where the forces are applied. Just like in the case of a stepped gear arrangement:

 

[hexhamallstar]

Again excellent reply jack thanks!

My last question is concerning a specific bit of software - MITcalc. I'm currently trying to use this software to tell me whether a competitors gear set can handle the power of a given engine. I keep on getting very low values for safety factor, i.e 0.05 etc. I think this is because the power I'm putting through the gear set is simply 1/2 of the engine peak power (just trying to get a rough idea of size of gears). I've realized though that the only thing MITcalc will be able to deal with is when the planet gears are actually spinning.

Does anybody know how I can model the differential meshes properly in MITcalc? (I can take a screencap of the required inputs if need be)

Thanks,

Phil