I first noticed a surging while slowly turning while in reverse. It was as if one piston was pushing twice as hard as the rest. Now it's starting to surge while in drive and I'm afraid it is going to damage itself.

I checked the wheels, all on tight. Visually checked the 1500 mile-old flex discs and they look and feel tight, all the bolts and nuts are there, no rips that I can see or feel. The differential has some wetness, though. Haven't checked the level yet. Could the differential produce that sort of surge?

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1982 300CD Petrol/Black Leather

Differential is a logical place to start. You may want to drain it and take a look. If it's all good, you can refill and see if that helps.

Check your half shafts from the differential to the wheels. If one of them is binding up, it can cause a very strange "lurching" sensation.

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]

Before draining the Differential be sure you can get the Fill Plug out. They can be extremely difficult to remove.
If you drain the differential and cannot fill it because the plug won't come out what will you do?

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84 300D, 82 Volvo 244Gl Diesel

Maybe there is a brake caliper stuck or some other drive train issue. Check the driveshaft support.

What if it is a fuel supply to the fuel injection pump issue?

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84 300D, 82 Volvo 244Gl Diesel

Easy enough to check. Drive car forward or reverse, see if vibration is present. If it is, stop and hold brake, apply accelerator pedal to raise engine rpm. If no vibration, it's driveline, not engine/fuel.

__________________
Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]

That would be my second guess. How would I test the axles? They're old, but the boots aren't torn yet.

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1982 300CD Petrol/Black Leather

RPM's rise smoothly (for a 617) with feet on both pedals.

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1982 300CD Petrol/Black Leather

Rules out fuel or engine as your vibration source.

Jack up the rear of the car and support it on jack stands. Put the car in Neutral and rotate the wheels by hand. Make sure they rotate smoothly. Watch the axle shafts and make sure they don't have any sort of runout or binding as they rotate.

__________________
Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]

I'll spin the wheels before taking the lid off the differential. Assuming they would show signs of failure without load.
Is lifting the rear end by the differential okay?

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1982 300CD Petrol/Black Leather

I started lifting the car, and noticed a bulge in the right front tire. It was a NOS wheel I got a couple years ago from eBay.

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1982 300CD Petrol/Black Leather