Editing Speed wobble (section)

== Theory ==
Sustained oscillation has two necessary components: an [[underdamped]] [[order (differential equation)|second- or higher-order]] system and a [[positive feedback]] mechanism. An example of an underdamped second order system is a [[spring–mass system]], where the mass can bob up and down (oscillate) when hanging from a spring.

If shimmy cannot be designed out of the system, a device known as a [[steering damper]] may be used, which is essentially a [[notch filter]] designed to damp the shimmy at its known natural frequency.<ref>{{cite magazine
| title = Control of Motorcycle Steering Instabilities
| author = Simos Evangelou, David J.N. Limebeer, Robin S. Sharp, and Malcolm C. Smith
| date = October 2006
| magazine = IEEE Control Systems Magazine
| quote =  For machines with a stiff front frame, a steering damper is required to stabilize the wobble mode at high speeds, while older, more flexible machines may require a steering damper at intermediate speeds.
| citeseerx = 10.1.1.227.9657
}}</ref>

Shimmy is usually associated with the deformation of (rubber) tires. However, it can also be observed in nondeformable (e.g., steel) wheels. The phenomenon can be explained by introducing multicomponent dry friction forces,<ref>
{{cite journal
| last1 = Zhuravlev
| first1 = V.Ph.
| last2 = Klimov
| first2 = D.M.
| title = Theory of the shimmy phenomenon
| journal = Mechanics of Solids
| year = 2010
| volume =4
| issue = 3
| pages =324–330
| doi =  10.3103/S0025654410030039
| bibcode = 2010MeSol..45..324Z
| s2cid = 122904947
}}
</ref> apart from the usual forces considered in the literature.

Another explanation is that speed wobble is a [[Hopf bifurcation]], a mathematical phenomenon in which a stable fixed point of a dynamical system loses stability as a parameter changes, giving rise to a [[limit cycle]] (a path that a system follows repeatedly, looping back on itself in a predictable pattern, which other nearby behaviors tend to mimic over time). In the case of a speed wobble, the system changes from one state (a stable ride) to a second state, (constant amplitude oscillation), when one parameter (forward speed, or air speed) progresses through a critical point.
{{r |ST |p=1 |q=An alternative theory (or maybe these theories aren&apos;t mutually exclusive) is that speed wobble is an example of a "Hopf Bifurcation" (no, Hopf is not a typo). Simply put, a system that is stable, operating normally, and as it should, can cease operating normally when one parameter changes, the parameter in the case of speed wobble being velocity. At that point the system becomes unstable, and a "periodic orbit" comes into existence, such orbit, as I understand it, expressed as the oscillation in your bike during a speed wobble. The aeroelastic flutter described above is a kind of Hopf Bifurcation.}}
{{r |VN_LZ |p=1 |q=A linear analysis leading to resonance is appropriate for any system where there is an oscillator that is being forced at a special frequency — the resonance frequency — and when this happens, the amplitude can simply build to infinity. This is not what happens in bicycle instability for two reasons: first, there is no periodic forcing that causes the high-speed wobble (in fact, it can happen on a smooth road); and second, there is not a phenomenon that shows a characteristic building of amplitude. Instead, the high-speed wobble is a critical phenomena, which is typical of bifurcations and bifurcation theory in general. Below the critical parameter value, you see one thing, in this case a stable equilibrium characteristic of a smooth ride, and slightly above the critical parameter, the smooth ride is no longer stable (but it still exists as an equilibrium, but an unstable equilibrium, just as standing a stick upright is an equilibrium but unstable because if it tips even slightly away from the exact equilibrium, it quickly drifts away), but the now unstable equilibrium gives way to a stable periodic orbit, which is the wobble. And as the parameter increases, the amplitude of the wobble can increase to some larger but fixed amplitude. —Erik M. Bollt, W. Jon Harrington Professor of Mathematics, Clarkson University}}