Mustang Vibration

I have a 1999 GT Mustang Convertible, automatic with 66,000 miles. I have recently developed a vibration that appears to be in the rear of the car. I had the tires balanced and the suspentioned checked out. Everything appears to be fine. The car starts to vibrate at about 50 MPH and 1500 rpm. When I back off the gas the vibration increases. There are no clunks or change in the transmission automatically shifting....also the battery light appears when I first start the car and stays on for about five minutes or so and goes out. It does not come back the rest of the day. Next morning it is on again. I have had the battery and alternator checked and everything checks out fine the aternator is putting out what it is suppose to....I need hellp these things are driving me crazy....Any suggestions......Paul

Welcome to AF!

For the vibration the u joints could be bad or the driveshaft is out of balance. The alt. light could be defective. Hook a voltmeter to the battery. It should be 12.8 volts at rest, 13.8 -14.3 volts running. Do this first thing when you start it up and your light comes on. Could be nothing wrong except---the light. If start up voltage is low check the alternator accessory belt for looseness or glazing.