shaking when Topless

[32valvesftw]

This is a puzzler for me. We just got back from taking the Vette for a drive in the country. Along the way we stopped to remove the Targa top. It was then stored in its place. When I went to pull back not he road, the car was shaking badly, noticeable through the wheel. The shaking was pretty bad so much so that I did go over 50, cause I thought I had a wheel falling off. I did stop and check every lug nut was tight. I did visual of the drive shafts as best I could. I had the wife put the car in gear with he hood up, the engine did not jump, so no broken motor mounts. Then we proceeded on our trip and shaking resumed, I drove slowly for about 10 more miles. Then I stopped to put the roof panel back on, so I could hear better in the car. With the roof in place and bolted in, no more shake???WTH I am puzzled, How could the roof panel have that much to do with it?