My car has suddenly taken to shuddering when I accelerate, but only between 60 and 75. I’m pretty sure it has nothing to do with the transmission, because there isn’t any hard shifting, but it feels like the bottom of my car is going to drop out from under me. It’s an '87, but short of rebuilding the engine, I’m not going to give it up.
I would begin my investigation with your CV axles.
Also check the condition of your tires. Do you remember the last time they were inspected, balanced, or rotated?
On a RWD car like a 1987 BMW 325?
I’d check out a drive line issue. Perhaps the u-joint(s), and/or the joints in the independant rear suspension. Sometimes the drive shaft itself can fail. It’s an old car so a mount may have failed knocking the drive shaft out of alignment.
The shutter at specific speeds, between 60 and 75, can indicate a tire tread or belt separation. These can be “internal” and you might see what looks like an OK tire that is really out of round and on its way to flying apart. If you have friend drive behind you at 70mph and the tire is causing the shutter sometimes you can see the offending tire bouncing or vibrating from the observer car.