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Noob 4WD: Locked Center Diff but only one wheel spinning in snow – Is this normal?

I much prefer a manual in snow because I find it a lot easier to regulate power and obtain feedback about traction. But . . . vehicles with automatic transmissions can be very capable in snow and low traction. For starting from a stop in extremely slippery conditions, the technique I'd often use is to select a higher gear to start. This reduces the amount of torque transmitted to the wheels and reduces initial slipping. Instead of slipping a manual clutch, I'm slipping the torque converter or clutch bands. Do this enough - like when you're stuck and trying to rock out of a hole - and you can absolutely burn up a clutch band that is nearing end-of-life (done that). So just like slipping a manual clutch, it needs to be done sparingly.

I used this technique everyday for several years driving to work in a '68 Plymouth Fury III in the early morning winters in Alaska. The Plymouth was a big 18' long 4-door sedan with almost no weight on the rear axle and a 318 with 230 HP up front. Going to work, I'd pull into a left turn lane that was aimed uphill with about a 3% grade. In Anchorage, the roads would get covered with ice and snow in October and stay that way until May. Left turns lanes were almost always polished ice due to lots of sliding stops, wild spinning starts (dumb drivers) and the heat of idling vehicles melting the ice and snow (which would immediately re-freeze). The technique to get moving without spinning was to manually select 2nd gear on the automatic transmission (or sometimes even 3rd gear in extremely slippery conditions ), let off the brake and let idle or a tiny amount of throttle get the car moving. There was almost always enough traction to move slowly - like at a slow walk. Try this in Drive (which would select 1st gear to start from a stop) and the tires would immediately break free and spin in place. Less informed drivers could hold everyone up as they madly spun their wheels, going no where through two or three light cycles before finally getting through the intersection by dumb luck.
Many autos won’t allow you to select the gear, rather they’ll let you select the maximum gear. It’s super annoying. I’ve been in road conditions where I want to select 2nd and curse the engineers that took that choice away from me.
 
100% - if one wheel was spinning, I don’t believe the center differential was locked. As mentioned, you’ll know it’s locked because of the indicator on the telltale board. But interesting fact - when you go to unlock it, when the light goes away, it doesn’t mean you’re totally unlocked. I had a nasty surprise a couple months ago where my engine went into limp mode on the highway. Turns out, when I unlocked my center diff about a week before after crawling out of a snowbank, it was not absolutely in the unlocked high slot. Enough not to grind, enough where the light went off - but about a week later, maybe a bump in the road or something and it thought I was trying to lock the diff at speed and so went into limp mode. Took my dealers mechanic a bit to figure that one out. You might THINK you’ve got it fully in the locked or unlocked position, but you can’t be shy with it. You’ve really got to make sure you haul that thing into position - go full-on exorcist with it, like you’re dealing with one of the Nine Princes of Hell. Your buddies may wonder why you are yelling “the Power of Christ compels you!” as you wrestle with your center diff, but the good news is they will probably all agree to let you have the tent to yourself and plenty of space to stretch out at the campfire.
 
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(1) An open differential splits input torque equally between two outputs.
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I'd say that's almost correct. An open differential will transfer the same amount of torque as the side with the least amount of torque to the other side. So if the driveshaft provides 100 Nm but the left tire can only make 5 Nm of torque against some ice, only 5 Nm will be transmitted to the right tire.
 
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