Since you say it tends to happen at a specific speed, my first thought is tires.
I'd think if it were anything else you'd notice the vibration at 20MPH or 30MPH.. maybe not as intense, but you'd notice it.
I had a car once that just out of the blue would start rattlin' like it was going to fall apart right at 45MPH.. you could drive all day at 44MPH and it was smooth as silk.. but at 45MPH, it would rattle your teeth out of their sockets.
Turned out to simply be a weight had come off and put the passenger side tire out of balance.
If it was me, i'd take all 4 tires off and have them balanced.
