[ImperialCrown]

After a long drive, the differentials can be hot to the touch.
What did the shop mean when they said that the connection from the driveshaft to the differential was too tight? Were they indicating the rear u-joint or pinion gear preload? Gear preload is set at the factory and usually gets better as the bearings wear in. The gears will turn with resistance. That's normal.
A vibration under the floor does sound driveshaft/u-joint related. Shaft imbalance would be a consistent 'shudder', it wouldn't come and go and it would likely happen at certain speeds.
I would ask the dealer for a 2nd diagnosis. Did they actually witness the vibration? If not, they would be guessing.
Go for a road test with the technician and point out the shake. You can ask the service advisor if they can waive the 2nd diagnostic fee as you are back with the same complaint. It may have to be OK'd by the service manager, but that's OK as it may get him involved in your concern and get you an answer.

 

[tazdevil]

Try putting in a bottle of friction modifier.

 

[ImperialCrown]

While I agree that the friction modifier should have been added at the differential fluid change, the friction modifier is only to allow the limited-slip differential side gear clutches to slip smoothly around corners without chattering.
It won't affect straight ahead highway operation. This shake or shudder while driving is being caused by something else.

 

[Bearhawke]

I'd have the driveshaft balanced. Even if it's OK; it's money well spent IMHO.

 

[hermv2000]

Thanks all for your replies. ImperialCrown, the independent shop turned the input shaft into the differential and noticed the wheels turned immediately, i.e. there was no slack. They said there should be some slack or play before the wheels begin turning.

So, other than the vibration, are you saying the hot differential is not a concern? Also, why would the driveshaft suddenly be unbalanced? It was fine before and I havent' hit or run over anything.

Herman

 

[TWX]

Bear in mind that if one of the needles in the needle bearings gets out of place when pulling apart the u-joints it can look fine but actually offset the differential ever so slightly. A friend of mine had a gremlin of a shake on his '72 Monaco wagon that initially seemed to be the gear vendors overdrive, but turned out to be a single needle bearing having fallen into the cap from the side on the u-joint up at the slip yoke. Took years of aggravation to find it.

 

[ImperialCrown]

There should be a small amount of freeplay/backlash when turning the pinion yoke back and forth. If there is no geartooth backlash or slack, then yes, the gears are adjusted too tight.
Driveshaft weights are spotwelded to the shaft tube. Sometimes they can be thrown. It should leave a small bare metal patch if this has happened recently.