Why is my Mazda shaking from Side to Side?

My 1991 Mazda RX-7 Turbo Convertible has started shaking from side to side. And it's very bad. But it only does it in spurts, like random occurrences. It seems to do it around speeds of 45-60 mph the most, though it has done it at lower speeds like 20. It does it occasionally when I slow down, and then when I'm just driving mid gear as well. Doesn't seem to do it as much when accelerating. The shakes are bad too, they make it feel like the car is going to fall apart. I have brand new tires, (probably less than 450 miles on them) as well as they should be balanced too. I do have speed bumps in my apartments and a friend suggested that I might have knocked some weights of hitting them, scrapping the bottom of my car. Which it's almost impossible to get over them, my car is so low to the ground. Anyway, it's getting scary to drive at times and I figured someone somewhere might know whats going on. Thanks. =D