A parked bicycle falls over at once. Give it a push and it can roll a surprisingly long way on its own. The trick is steering under the fall: whenever the bike leans, the front wheel turns toward the lean and brings the wheels back under the weight. Riders do this without thinking, and the fork geometry and spinning wheels help the bike do it by itself.
A bike balanced on its tires is like a broom balanced on your palm: the slightest lean (tilt to one side) grows, and it falls. To catch a broom you move your hand under it. A rider does the same with the wheels: when the bike leans right, they steer (turn the handlebars) right. The bike then curves to the right, and the curve swings the wheels back under the rider. Faster bikes need smaller steering for the same correction.
How to read this: two identical bikes, seen from behind, start leaning 5° right. The green rider turns the handlebars into the lean; the faded red one doesn't. The graph traces both leans over 6 seconds: green settles back toward zero, red runs away as it falls.
Try this: drag Speed down and press Push and replay to see how the same steering copes when the bike is slow. Steer per lean is how far the rider turns for each degree of tilt; Steer per lean rate adds extra turning when the tilt is changing fast. Turn radius only changes the readout of how far a bike leans in a steady turn.
The bike and rider are one point mass at height h = above the ground. The wheels are w = apart. Lean φ and steer angle δ are in radians. The rider steers in proportion to the lean and to how fast the lean changes, with the bars limited to 40°.
Gravity tips the bike over. Steering into the lean makes the bike turn, and the turn pushes the wheels sideways under the mass. For small angles this settles down only if kp > g·w/v² and kd > 0. Halve the speed and the rider needs four times as much steering.
With no steering at all, the lean grows like cosh(t/τ) with τ = √(h/g) ≈ 0.30 s, the same as a falling stick.
In a steady turn of radius r, the tires push sideways with the force m·v²/r that bends the path into a circle. Gravity pulls down with m·g. The rider balances by leaning at angle θ so the sum of the two forces points straight through the tire contact. Too little lean and the bike stands up out of the turn. Too much and it falls inward.
The front fork turns about a tilted line called the steering axis. The head angle is how steep that line is, measured from the ground. Extend the axis down to the road. On most bikes it lands a little ahead of where the tire touches the ground. That gap is the trail. Because the tire touches down behind the axis, the front wheel trails along like a shopping-cart caster, and when the bike leans, the wheel swings toward the lean on its own.
How to read this: the gold dashed line is what the handlebars turn around. Where it hits the road (gold dot) is ahead of where the tire touches (green dot), and that gap is the trail. The small "from above" box shows which way the front wheel swings when the bike leans: green means toward the lean (helps), red means away.
Try this: click Negative trail and watch the gold gap turn red and the wheel in the box swing the wrong way.
R is the wheel radius, θ the head angle from horizontal, and the offset is how far the fork bends the hub forward of the axis. More offset means less trail. The torque line uses the front-wheel load N ≈ 309 N from the benchmark bike in Lab 3 and a small lean φ in radians.
When the bike leans, the road pushes up on the tire at a point behind the steering axis. That push twists the fork toward the low side, so the bike steers into its own fall, which is exactly what the rider does in Lab 1. The front assembly's own weight, if it sits ahead of the axis, adds to the same twist. A torque is a force times its lever arm, as in E03 · Levers, pulleys & gears, and here the lever arm is the normal trail.
In 2011 Kooijman, Meijaard, Papadopoulos, Ruina and Schwab (Science) built a bicycle with small negative trail and extra counter-spinning wheels that cancel the gyroscopic effect, and it still balanced itself above a certain speed. Their analysis indicates that neither trail nor gyroscopic action is strictly required. The way the bike's mass and steering geometry work together decides it.
A spinning wheel resists being tipped. Physicists call its "amount of spin" angular momentum: how heavy the rim is, times how fast it turns. Tip a spinning wheel and it swings sideways instead, which is called precession. On a bike that sideways swing turns the front wheel toward the lean. Below, a standard textbook bicycle model is checked at every speed: does a small wobble die out on its own, or grow?
How to read this: on the wheel, the gold arrow shows how much spin it has (angular momentum, L) and grows as you speed up; ω is how fast the wheel turns. The chart checks a riderless bike at every speed. Such a bike can move in a few basic ways: a side-to-side wobble, a slow tip-over, and a quick flop of the front wheel. Each dot is one of those motions at one speed. Red dots above the middle line mean that motion grows; green dots below mean it fades. Inside the gold band every dot is green, so the bike balances itself.
Try this: drag Speed across the gold band, then switch Wheel spin effect Off and see what happens to the band.
The front wheel has radius R = and spin inertia I = 0.28 kg·m². It rolls without slipping, so it spins at ω = v/R. When the bike leans over at rate φ′, the spin axis tilts, and the wheel answers with a torque about the steering axis (tilted λ = 18° from vertical).
This is the model F. J. W. Whipple wrote down in 1899: four rigid parts, knife-edge wheels that roll without slipping, no rider input. Meijaard, Papadopoulos, Ruina and Schwab (2007) published a checked set of parameters for it: in total, wheelbase, trail. The page builds M, C₁, K₀ and K₂ from those parameters and finds the four growth rates s with det(M·s² + v·C₁·s + g·K₀ + v²·K₂) = 0.
A positive growth rate means that motion grows. Below about the weave (a side-to-side wobble) grows. Above about the capsize mode grows very slowly: the bike leans over gradually, and a rider corrects that with almost no effort. In between, this model bike balances itself.
Switch the wheel spin off and this bike has no self-stable speed at all. Change the trail and the band shifts and narrows. For this particular bike both effects matter, though Lab 2 explains why that is not true of every possible bike. The same conservation of angular momentum makes inflowing air spin faster in W00 · Hurricanes.