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Some research into (uncontrolled) bicycle stability (e.g. Jason K. Moore, Human Control of a Bicycle, 2012) models the dynamic behavior of some bikes and determines "critical speeds" for "capsize" and "weave" modes (another source mentions "wobble").

What is the definition of these terms?

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The animated illustrations in "Vibration modes of motorcycles" by Vittore Cossalter, Roberto Lot (posted in the community wiki answer) do a great job of illustrating capsize, weave, and wobble.

Simplified verbal approximations might be:

Capsize - The bicycle moves as if to fall over on its side with no distinct oscillations.

Weave - The bicycle moves with an oscillating pattern as if arcing between cones on a straight path without inducing movement at the handlebars (perhaps subtly more complex because of the perpendicular twisting motions of yaw and roll axes of the entire bicycle object).

Wobble - The bicycle moves distinctly at the front due to twisting movements of the handlebar to turn the front wheel back and forth (isolating twisting motion largely at the fork steering tube axis). To further contrast with weave, the front wheel might arc back and forth between cones on a straight path while the handlebar flops side-to-side but the rear wheel remains largely in place striking each cone.

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https://moorepants.github.io/dissertation/eom.html?highlight=weave#linearized-equations-of-motion

Capsize is always real and stable at low speeds but becomes marginally unstable at a higher speed. It describes the roll of the rear frame. Weave is real at very low speeds and describes an inverted pendulum-like motion i.e. the bicycle falls over. As speed increases the eigenvalues coalesce into a complex conjugate pair at the weave bifurcation that describes an exponentially increasing sinusoidal motion of roll and steer, with steer lagging the roll. This mode becomes stable at a higher speed. The weave and capsize critical speeds bound the stable speed range.

Also

https://www.dynamotion.it/dinamoto/8_on-line_papers/wobble-weave/wobble-weave_eng.html

The wobble mode is essentially a steering oscillation of the front forks; this mode doesn't involve the rear frame in any significant way.

The weave mode is a side-to-side motion of the entire motorcycle characterised by a coupling of the yaw and roll oscillations with a significant lateral displacement of the steering axis.

Capsize
...
this motion is a non-oscillatory motion and consists in the capsize of the two-wheel vehicle.


Explanations and lots of references in https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics

Since 1971, when he identified and named the wobble, weave and capsize modes,[12] Robin Sharp has written regularly about the behavior of motorcycles and bicycles.[13]

and https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics#Modes

  • Capsize is the word used to describe a bike falling over without oscillation.
  • Weave is the word used to describe a slow (0–4 Hz) oscillation between leaning left and steering right, and vice versa
  • Wobble, shimmy, tank-slapper, speed wobble, and death wobble are all words and phrases used to describe a rapid (4–10 Hz) oscillation of primarily just the front end (front wheel, fork, and handlebars).

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