Well, my specialty is more along the lines of automotive electrics and engine work, but I'll take a stab at it...
A differential is an assembly that allows two wheels on the ends of what would normally be a solid axle to turn at different speeds (like in a turn.) This is accomplished through the use of "spider" gears (between the carrier and the axle shafts) and "side" gears (into which the axle shafts are splined) coupling the ring & pinion assembly to the shaft. I don't know a good way to explain it - so experiment next time you have the cover off and the wheels jacked up so they'll spin freely. Block the front wheels, and put the transmission in neutral so you may turn all three elements by hand (left wheel, right wheel, and input/pinion gear) and watch what happens to the differential gears.
A "open" differential will allow a wheel that breaks traction to have ALL the spin - that's why you get stuck.
A "limited slip" (also "Posi" or "LSD") is a differential with clutch packs to engage the side gears. As I understand it, when a wheel "breaks free" and begins to spin faster, the clutches disengage on the "free" side, and engage on the "tight" side, and that allows the wheel with traction to spin.
An "automatic locking" differential is similar to an LSD, but uses a mechanical locking device (usually a pawl clutch of some variety) to accomplish the coupling.
A "selective locking" differential is either an "open" or LSD, with the user-controlled option (typically powered by compressed air, but now also electric) to "lock" the two axle shaft together. This forces them to turn at the same rate, and overrides the differential entirely.
Lastly, a "spool" or "Lincoln locker" (welded spider gears) locks the two axle shafts together at all times - by definition, it ain't a differential, since the two shafts can't spin at different speeds.
Some companies use model numbers for nomenclature (like Dana,) some use the number of cover bolts (popularised by GM,) and some use the ring gear outside diameter (ChryCo and Ford come to mind.) Some companies - particularly heavy-duty builders - use the weight capacity of the axle (like Rockwell and Eaton.)
I don't have the figures handy, but as the Dana model number gets larger (30/35/41/44/50/60/70/80,) the axle gets heavier, and the ring gear gets larger. Typically, a larger ring gear (outside diameter - the differential case has to fit inside) will make for a stronger differential - since the contact band will be larger and it allows for a larger case (which is typically cast.)
ChryCo axles - the XJ got the 8.25" stock, but some put a 9.25" under there, and there is also a 10.25" for heavy-duty pickups - put the ring gear size right out where it can be see. Ditto Ford - 8.8" and 9" come readily to mind.
You'll also hear talk of GM 10-bolt, 12-bolt, and 14-bolt axles. This refers to the numbrer of bolts that hold the cover down - but I've heard tell that it (coincidentally) matches the number of bolts holding the ring gear to the differential case. I have not had an opportunity to check this out - but again, more is better (because it's a larger housing.) The 14-bolt is typically found under 3/4-ton or larger pickups, the 10-bolt and 12-bolt may also be found under RWD passenger cars, 1/2-ton trucks, and S-10s.
Then, there are the "big boys" - if the rating isn't given, it can usually be inferred from context. For example, when someone here talks about "Rockwells," they're usually referring to a Rockwell 2-1/2-ton axle, like that found under the M35A2 6x6 2-1/2-ton truck. Rockwell also makes a 5-ton, and some real monsters for use under prime movers (you know them as semi-trucks.) This refers to drive axles - the ones under the trailer don't drive anything.
For splines, there is typically a given "spline count" for a shaft diameter - the shaft diameter determines (largely) the smallest useful spline that may be put on it. "Spline count" is simple - mark one spline, and count around the axle shaft.
A higher spline count for a given model of axle will usually indicate a larger shaft - which means that a greater torsional load is required to fail it (sudden traction on large tyres, for instance.) Typically, you will note this with ChryCo 8.25/9.25 axles, Ford 9", and Dana 44-up (there are typically two sizes - which also translates to two possible side gears, and two possible wheel bearings. Don't go by lug pattern - that usually means nothing.) Since splines are 'always engaged,' unlike gear teeth, more splines usually means a stronger coupling (unlike gear teeth - where you want fatter, and preferably helically-cut - teeth.)
This concludes the nickel tour on axles (and is probably worth what you paid for it.) Any questions? Like I said, this is out of specialty for me, so I honestly posted this in the hopes of being corrected somewhat - and learning something myself...
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