Thank you. We do know how they work, and that it's nothing to do with gyroscopes and everything to do with (what parent said, but simplifying a bit) front fork geometry.
Pretty much anyone can see this for themselves. Walk a bike with your hand holding the back of the seat, not the handlebars. Steer the front wheel by leaning the bike. Lean father to correct faster. If it's a cheap bike, let go and watch this happen on its own till it wobbles too far to counteract.
This is one of those idiotic tropes like bumblebees not being able to fly.
If you read the actual paper referenced, we know some of how it works, but not all.
Two of the proposed theories (that it has to do with gyroscopic effects, and that it has to do with trail), have been disproven by creating a self-stable bike with no gyroscopic effects and (slightly) negative trail. The paper introduces one additional factor, the difference in center of mass between the steering assembly and the rigid body of the bike; the steering assembly having a lower center of mass causes it to fall faster, providing the necessary corrective steering to achieve self-stability.
So, there are several factors we know about, which can increase stability. We know how to locally optimize stability for certain designs. But we don't yet have a full set of necessary and sufficient conditions for a bike to be self-stable. We haven't even proven, analytically, the intuitive notion that a bicycle must lean toward a fall, though given our intuition it is believed to be true.
Here are the two necessary conditions that the paper provides:
> To hold a self-stable bicycle in a right steady turn requires a left torque on the handlebars. Equivalently, if the hands are suddenly released from holding a self-stable bicycle in a steady turn to the right, the immediate first motion of the handlebars will be a turn further to the right. This is a rigorous version of the more general, as-yet-unproved claim that a stable bicycle must turn toward a fall.
> Another simple necessary condition for self-stability is that at least one factor coupling lean to steer must be present [at least one of Mδϕ, Cδϕ, or Kδϕ must be nonzero (SOM text S3)]. These coupling terms arise from combinations of trail, spin momentum, steer axis tilt, and center of mass locations and products of inertia of the front and rear assemblies.
That's what is meant when people say "we don't know how bicycles work." We do know some of how they work; we know that the designs that we create steer into a fall, and do so in such a way that damps the wobbles and eventually goes straight again. And we do know some necessary conditions for a self-stable bicycle, like a requirement that something that couples lean and steering must be present; but we don't know if the steering into the fall is absolutely necessary, or if you could build a bike that managed to achieve self-stability without it.
So, I would say that a more accurate summary is "we know how current bicycle designs work, but we don't know exactly what aspects of them are necessary, or how to completely characterize the sets of designs that work or don't work." But that's a bit more of a mouthful than "we don't know how bicycles work", so that's what gets repeated.
So we do know how the bikes that actually are 'bikes' work at least to a very large degree (engineering vs math), but we don't know all the possible physics which can enable an arbitrary two wheeled construct to be selfstablize when perturbed while in forward motion.
I completely understand the point your making. However, at when making a technical argument and then generalizing the end results, we can end up in a situation where the truth of the technical argument no longer strictly implies the truth of the generalized/summarized result. I feel the statement 'we do not know how bikes work' has crossed that line.
I suppose it depends on how you look at it, or perhaps on whether you're interested in how a bike works versus why a bike works. How is relatively easily answered; as the bike tilts it steers into the tilt, moving its base back under its center of gravity, in a way which damps itself thus getting back to upright without oscillating repeatedly or falling over.
Why it works is the open question. We know that it's some combination of gyroscopic effects, rake, trail, and the different centers of gravity of the frame and fork, but we don't know the precise relationship between them that allows it to work.
You could build a bike with no gyroscopic effects, slightly negative rake AND the steering column center of mass at the same height as the rest of the bike. So you could cancel out and test the influence of the last effect.
If that failed to be stable, all that would prove is that particular design was unstable, not that there is no design with those features that is stable. This is a problem of finding a general rule, not just a particular design that is stable or unstable.
The issue is that no one has found a way to characterize all possible designs (within certain constraints, such as two wheels each attached to a rigid frame, the frames joined by a hinge) which are self-stable. They know some conditions that are necessary, such as at least one factor linking lean to steering and the design needing a steering force applied to turn in a steady turn. They have not yet characterized what conditions are sufficient for a stable design.
What you want, to say that you fully undertsand how a bicycle works, is a set of conditions which are both necessary and sufficient for a bicycle to be self-stable. If you build a bicycle which meets those conditions, then it will be self-stable (at some speed; certain designs may be self-stable over a wider range of speeds while some may only be self-stable at a narrow range of speeds); if you build a bicycle which does not meet those conditions, it will not be self-stable at any speed.
We've gotten closer over the last century; initially it was believed that gyroscopic force was necessary, but that was disproved. Later it was believed that trail was necessary (or either gyroscopic force or trail was necessary; I haven't read the older paper), but that has now been disproved. We now know a couple of necessary conditions (listed above), but they are somewhat weak necessary conditions, and we don't yet have (as far as I know) a set of sufficient conditions (conditions which, if they hold true, will guarantee that the bicycle will be stable, regardless of other changes to the design), beyond a few known designs which are demonstrably stable.
In fact, if you follow from the paper in Science to the "Supplementary Online Materials" (which is actually the full-length paper; what's published in Science is really an extended abstract), you will see that they prove that "no combination of positive gyroscopic action, positive trail, or positive steer axis tilt are either necessary or sufficient for self-stability over at least a small range of speeds." They construct models of bicycles that lack each of these things but are stable, and have all of these things but are unstable.
Pretty much anyone can see this for themselves. Walk a bike with your hand holding the back of the seat, not the handlebars. Steer the front wheel by leaning the bike. Lean father to correct faster. If it's a cheap bike, let go and watch this happen on its own till it wobbles too far to counteract.
This is one of those idiotic tropes like bumblebees not being able to fly.