I have a 2002 Grand Marquis. Every once in a while, it will have a sudden vibration when I accelerate, even a little bit. This vibration is much faster than a wheel, and it shakes the whole car. I think it may be coming from underneath. I once had another car (88 fifth avenue) with a similar symptom, and it needed a universal joint.
It doesn’t happen every day, every once in a while, it seems to happen during an entire one hour trip to and from work. One day it does it, the next day it won’t. I can go for weeks at a time where it doesn’t do this, then one day it starts doing it again.
Update: I took it to a mechanic, and he thinks it might be torque converter chatter.
bsaucer probably has it. I would call it torque converter “shudder”. Does it fell like you are running over rumble strips at the side of the interstate? That is about what TC shudder feels like. How many miles on this car? Has the transmission been serviced on schedule? I think that partway through 2002 the torque converter drain plug was deleted from this transmission as supplied new so the best way to do the service is with a machine that exchanges the fluid because you only get half of it out by draining the trans alone.
Take it somewhere that is willing to do a pan drop and change the filter and then do the exchange. The gasket is reusable if not damaged. Insist on Mercon V fluid, NOT Mercon, Dextron or anything universal. You should need about 15 quarts. Ideally you should go somewhere that will know signs of trouble in the transmission pan if there are any. It is very likely that new fluid will fix the problem.
P.S. This can sometimes be very difficult to differentiate from a mild missfire condition. If the trans is due for service, do it and see what happens. If it does not fix it, have the car evaluated by a driveablility expert.