The two normal modes of a coupled pair at kc/k = 0.1

At its defaults it draws the two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else.

normal-modes is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else.

The growth rates of the small motions of a riderless bicycle against forward speed, from the four roots of the linearised equations for a benchmark machine, with the published eigenvalues reproduced before anything else was drawn. A positive rate means the motion grows. Below 4.29 m/s a pair of roots has positive real part: the weave, a swaying of the steering and lean together, grows. Above 6.02 m/s one real root turns barely positive: the bicycle slowly falls over into a spiral, the capsize. Between these speeds, in the shaded band, all four roots have negative real parts, and a shove dies away on its own with no hand on the bar. Nothing is stabilising the machine except the coupling of lean to steer.

The lean of a riderless benchmark bicycle after a shove of half a radian per second, integrated from the linear equations, at three speeds. At 3 m/s the sway grows with each swing, and the curve is stopped at half a radian, where a real machine is falling and the linear model no longer holds. At 5 m/s the sway dies in a couple of seconds and the bicycle runs on upright. At 8 m/s the sway dies at once, then the lean drifts away slowly on one side, at a rate of a small fraction of a radian a second, the capsize that a rider corrects without noticing. The middle curve is the one that surprises people, though a bicycle ridden off a hill with nobody on it does the same.

The lean and steering angle of the benchmark machine at 5 m/s after a shove to the right, integrated. The steering angle swings towards the side of the fall, turning the wheels under the falling centre of mass, and it overshoots, turning the bicycle back the other way; the two oscillate with the steer ahead of the lean, and both shrink. A rider feels this as the bar turning by itself when the hands come off. The whole stabilising mechanism is here: a lean makes the front turn, the turn moves the contact points beneath the mass, and the motion is passed from one coordinate to the other until the machine's dissipation-free coupling has sent the motion away into forward speed.

The range of speeds in which a riderless bicycle is stable, for the benchmark machine and for three variants of it, each computed from the same formulas with one thing changed. The benchmark is stable from 4.3 to 6.0 m/s. Take the spin out of the front wheel, and no speed up to 14 m/s is stable; take it from both wheels, and no speed up to 14 m/s is stable; take away the trail, and the band shrinks to 2.9–3.1 m/s. These are changes to one machine and they do not separate the causes cleanly, because each change also shifts other terms. In 2011 Kooijman and colleagues built a real machine with no net wheel spin and negative trail, by placing its masses carefully, and it was self-stable. Neither gyroscopes nor trail is necessary; no simple sufficient condition is known.

The edges of the stable band of the benchmark bicycle as its trail, the distance by which the front contact point falls behind the steering axis, is varied. Band edges are computed by scanning the speed at each trail. The band appears only above about −1 cm of trail on this machine and widens as the trail grows. The 8 cm of a benchmark bicycle sits in a broad and undemanding part of the range, and bicycle frames have converged on similar numbers without reference to any of this. Riders, however, are not the benchmark's rigid frame; a bicycle with a rider's mass moves its band, and a rider steers as a feedback controller anyway, so this figure describes the machine and not what riding one is like.

physicscheck asserts something about normal-modes that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.