Andrew R Stewart 

The understanding of speed shimmy that I have come to believe in is it's being a harmonic vibration/oscillation and that by changing the frequency the parts can vibrate at can disrupt/change the nature of the shimmy. Stiffer frame "tubes" are one way to do that, IMO.

Rarely mentioned when frame stiffness and shimmy are discussed is the contribution that the fork has, more specifically the steerer as this is the one tube on the whole bike that is only solidly connected at one end, so the aspect of triangulation (which the rear end of the bike is an example of) adding to stiffness doesn't exist.

I have made 4 different touring frame/forks for myself over the years and as I became unhappy with the current version's shimmy I would tweak the design and make another. Initially I changed the steering geometry (and tried a second fork even) for #2. #3 saw heavier walled tubes but still traditional diameters. It was #4 that saw the shimmy essentially end. It had a 1.125" steerer and a 1.125 top tube but of a medium wall thickness. It is from this that I came to feel that the fork is more involved than most will talk about. Especially these days when a fork is just an off the shelf component that the frame builder really doesn't control much with. Andy (whose #4 touring bike doesn't have anywhere nearly as much "inch worming" when riding over bumps, the fork doesn't flex fore and aft as much)