Chrysler Repair: I recently purchased a 1994..., chrysler lebaron, alignment rack

Question
I recently purchased a 1994 Chrysler LeBaron convertible.  The car has developed a severe shake from 30 to 50 mph.  It drives fine at lower and higher speeds.  It also stops shaking if you let up on the accelerator.  We have had work done on the axles, the motor mounts, and have had the tires inspected, balanced and rotated.  My mechanic says he has no idea what to do next.  What should we do next?

Answer
Hi Rich,
My thoughts are as follows:
There is a phenomenon like you describe that occurs when the radial run out of a wheel exceeds 0.04". That would be checked with a dial indicator placed on the tread and rotating the tire. If it does, then the tire and rim can be match mounted to reduce that out of roundness by taking advantage of any out of roundness of the rim and the tire  and trading them off against one another.
There is also the question of whether the engine and transmission are positioned properly in the body. With car on a level ground and wheel pointed straight ahead (this may be easier to measure on an alignment rack or platform rack than sitting on the ground because of space limitations):
Use a tape measure to find the distance between the inner edge of the outboard boot to the inner edge of the inboard boot. The right side should be 17.8 to 18.1 inches, and the left side should be 7.4 to 7.7 inches. If it is beyond spec then the mounts need to be loosened and the powertrain moved laterally to meet the spec.
Might the engine be missing in that range of rpm? See if your engine controller has noticed anything wrong with the engine (on board diagnostics) you can do yourself:
The most useful thing to do would be to try to get the fault codes that may stored in the engine controller memory to readout. Try using the ignition key: turn it "on-off-on-off-on" and leave it "on" (doing this quickly, no longer than 5 seconds). By "on" I mean just the normal position when the engine is running, not the cranking position. The 'check engine'light will remain on when you leave the key in the "on" position with the engine still not running. But then watch the 'check engine' light to begin flashing, then pause, flashing, pause, etc. Count the number of flashes before each pause and keep track of the numbers. Repeat the readout and verify the counts are correct. Then group them in pairs in the order that they came out, thus forming two digit numbers. You may notice that the pause is shorter between the digits of a given number, and longer between the numbers themselves. Then send me a 'follow-up' question telling me the results of your readout. By the way, 55 will be the last number (two groups of 5 flashes each) and that is the code for "end of readout".
There is a possibility of something being wrong with the transmission (inhale deeply). These electronically controlled units also have an onboard diagnostic capability but they require a handheld readout unit to do it. There is a plug under the dash around the fuse box to which a Chrysler body plug can be mated (6-pin) and then the trans controller asked if it has seen anything wrong. I would want to know what codes numbers if any came out, what they mean, what would be the repair procedure, and the cost. Then let me know what codes if any came out for corroboration. Hopefully there will be none. You may have to have the readout done at a Chrysler dealer because the body plug is unique to the Chrysler corp vehicles, and also the mechanic should have a program cartridge that is relevant to the '94 model year. An independent tranny shop (but probably avoid a franchise type of shop) might be able to do the readout too, but I would stand by to make sure they did it thru the under dash plug with the correct cartridge. The engine can be read out similarly but its plug is under the hood. So that would be a way of seeing that they knew what they were doing.
The last possibility is other loose fittings on the front suspension, so you might go to an alignment shop (again an independent) and ask them to drive the car to see what they think and then maybe they will find loose sway bar bushings or ball joints worn or similar such wear that can be corrected.
I haven't listed these in order of likelihood though I would try the engine code readout (its free), then maybe the alignment shop for the second and third one, and finally the tire/wheel approach, followed by the trans code readout.
I would very much like to know what you determine to be the cause.
Roland