A crush sleeve is used in the assembly of a differential between the front bearing and the rear bearing in which the pinion gear rides on.
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It's called a "crush" sleeve because it's designed to collapse slightly. Allowing you to set the bearing pre-loads for the front and rear pinion bearing without having to add or remove shims. You tighten the pinion nut until the bearing preload provides the proper required rotation torque the pinion gear should require to spin freely. Basically you tighten the pinion nut until the pinion gear won't turn unless your physically turning it with your hand.
The difference between a crush sleeve and a solid pinion spacer is
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The solid spacer is well, solid it doesn't allow any give or flex the pre-load of the bearings are set with shims If just installing the spacer and not re-gearing you can measure the factory crush sleeve and start at that spacing, then back down on the shims from there.
The shim in the first image is the shim that places the pinion gear biting position in the center of the ring gear meshing position. (so you can see during assembly if re-gearing your going to go through a lot of, take apart, put together, check, take apart, put together, check, etc.) If using the factory gears and just installing a spacer you won't have to worry to much about the gear meshing position it should wind up where it was before it was disassembled.
The back lash is set by bearing caps shown with a red arrow here:
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These are located on both sides of the differential and adjust where the ring gear sits in relation to the pinion gear (setting backlash). These caps also adjust the bearing pre load of the carrier bearings, So once you get the backlash set you also have to tighten these rings to the proper torque (without changing the backlash) ensuring the bearings do not have any end play as well as are not tight enough to cause damage. (there is a rotational torque allowance here as well to tight the diff wont spin too loose it's going to wine like crazy and your talking 1000th's of inches)
By installing a solid pinion spacer you can prevent the pinion gear from deflecting:
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away from the ring gear. As you can see if the pinion gear moves away from the ring gear the less surface material the teeth have to mesh against thus causing the ring gear teeth to break away typically breaking through several teeth before lining up again.
hope this helps understand whats going on in there a little more. A crush sleeve works fine for daily driving and common torque loads. It doesn't do much good for the fluent off-roader though.