I have a 1997 Jeep Wrangler TJ that when you give it a lot of gas to accelerate off the line there is a noticable vibration and knocking noise(definite associated with vibration and not engine). Only off the line.Not noticable on the highway…
That is usually a sign of bad driveshaft joints - whether CV joints up front or universals between front & rear.
The other thing to check is the motor & transmission mounts.
Is the TJ lifted?
As cigroller stated, this is very symptomatic of bad u-joints on the rear driveshaft. The rear driveshaft is short and has a single u-joint out of the transfer case and a single u-joint at the differential. If the Jeep is lifted even a couple inches, you can get a bad angle on the u-joints causing just what you are experiencing. Fortunately, they are easy to replace on this vehicle.
The engine mounts are also not too difficult to replace (one on each side of the engine, and one at the rear of the transmission attached to the cross-member/skid plate).
I would like to agree with the U-Joint call but the no vibration at highway speed makes me scratch my head. If it is a u-joint why no highway speed vibration?
Is this a manual trans,my mind wanders to a clutch chatter problem.
To me it makes sense because the joints take the most stress under torque. At cruising speed there is a lot less than at accel. If this is a joint problem, I would expect it to start showing up at all speeds as it gets worse.
The guess about the joints, of course, could be wrong - and the OP should also explore other avenues.
It does have a lift. I would guess no more than 3 inches. It does not seem to have any play in the u-joints. Could they still be bad if they arent loose?
You might want to look at the TJ Technical Forum on jeepforum.com. There is a lot of information on lift issues.
Go over to one of the Jeep forums on the internet. But, yes a 3" lift without a new CV-joint type driveshaft can cause premature failure of the u-joints. TJs usually require a transfer case drop when lifted unless a better driveshaft is installed due to the extremely short rear driveshaft.
If it is unclear if the u-joints are bad because the driveshaft is installed remove the driveshaft and you can get a better feel if they are bad. Don’t get stuck on the u-joint possibility do something to either call them good or bad.