WHEN I'M GOING APPROX. 40 MPH, THE CAR STARTS TO SHAKE? THE TRANSMISSION SEEMS OK. I WAS TOLD I NEEDED NEW BACK STRUTS, WOULD THAT BE THE PROBLEM?

balance the wheels????

Check for codes, may have a bad speed sensor either input(turbine) or output (vehicle) speed sensor. They are a common problem with the Chrysler transmission.

Struts

need to change your top strut mounts i just put new struts on and noticed the noise she was hearing was still there just change strut mounts you will be fine

Does it have an automatic transmission? If so it's common on these to have poor connections on the multipin plug going to the transmission which can cause these symptoms!

I have this same car with exactly the same problem. Begins shaking at 40mph and stops shaking as soon as you let off the gas, but then begins again as soon as you press on the gas and smooths out as you get over 50. And the shaking is not felt through the steering wheel. Solution: Replace the transmission mounts. There are two of them, one on each side mounted on the frame right next to the shock strut. There is also a front and rear engine mount, but I replaced the two as described and shaking is gone.

If your talking about the 97 Jeep, then no, it doesn't need a quick Learn procedure done on it because it isn't a "adaptive learning" transmission.

I would get the car aligned, and also check the motor mounts on the engine.

These cars have a lock up torque converter meaning 4 gear going above 35 to 40 MPH, the engine and transmission lock together and turn at the same rate. I know that older GM cars had a plug that needed to be plugged into the transmission for the lock function to work on the torque converter and if you unplugged it the RPM's would run high as you describe. I have never had a 1999 Silhouette and I'm not 100% sure that these cars have that type of plug but it sounds like that type of problem. I agree with you. You know your car and I would take it to a transmission mechanic and describe your problem to them.

Good luck!

This happened to my 2001 Regal... There is a factory recall on the Catalytic Converter going bad. I had mine replaced with the recall and haven't had a problem since