Random but violent front end vibration

I had a similar probem on my black 73 Dart 340 Sport. It would start randomly it seemed, rattling the steering wheel back and forth, and I would have to slow way down to get it to quit. Even though it only had 30K plus miles, I replaced ball joints, tie rods, idler arm, pitman arm, control arm bushings, etc. Nothing changed. I had factory Rallye wheels on front with reproduction Goodyear Poly-Glas E70 tires. Turned out it was the tires. I had them installed and bubble balanced.

After going through all that, I had the tires checked for trueness and then speed balanced, the problem was gone.