Here's a recounting from a Richard Jow article in my recently recovered trove of Bicycling Magazine, 1975-1990 (complete) :
Frame shimmy is correlated with "Trail", tire profile, and geometry. Imagine the wheels of a shopping cart. When you push the cart, the wheels spin into a direction so that they "trail" their spindles. On a shopping cart, draw an imaginary line from the pivot spindle perpendicular to the ground. The distance from this point (on the ground) to the contact point of the tire (on the ground) is the "trail", and is usually 2-4 inches.
The same thing is true for a bicycle. Draw an imaginary line through the head tube, along the fork (until it rakes) and downwards to the ground. The point where the line touches the ground will typically be 0.5" - 4" in front of where the tire actually touches the ground. This causes the tire to naturally swing around to follow the direction of travel, and enables no-handed riding. Imagine if the tire were in front of the imaginary line !! It would be so likely to spin that no-handed riding would be impossible.
The article went on to quote the "trail" from about 15 bikes of the 1970's, including cinelli and colnago, i believe. I remember distinctly that the colnago / cinelli trail was about 1.75" (I was looking for the secrets to the magic-carpet ride.) For loaded turing, it was 2.75 inches (don't remember which bike.)
Bikes with relaxed geometries, such as 72 degree angles, have larger trails, however, when making a turn they tend to "lay down". In other words, the wheel tilts sideways and you turn on the sidewall of the tire, rather than turning on the top of the tire. Imagine a frame with 0 degree angles (a super-chopper). Just a slight turn to the handlebars would cause a dramatic loss of height for the front axle & frame.
Too much "lay-down" causes another form of instability, so too much trail can be a bad thing.
Tire profile can also have an impact on stability. Perhaps the most perfect tire profile (from sidewall to sidewall) would be a true semicircle. Then as you learned into a turn the handlebars would have less tendency to "lay down". High profile ridged tires (e.g. early specialized turbo models) tend to "lay down" quicker into a turn, and they are more unstable. They turn quicker, and have lower rolling resistance, but you pay for it with instability.
I think that in some cases it may be possible to correct bicycle shimmy by (1) tightening the headset, or (2) changing to a different tire profile. Shimmy does not always come from "frame geometry" (which head tube angle and trail.)
- Don Gillies
San Diego, CA