Friday, February 17, 2006

Counterintuitive

The bicycle was invented in 1790, and the motorcycle around a century later. Thus it is interesting to find that the physics of riding both were a mystery until quite recently. For nearly two centuries it was assumed that gyroscopic forces stabilized a bicycle and made it ridable. In 1970, physicist David Jones decided to test the received wisdom. He mounted another set of wheels on the bicycle. The second set was engineered to rotate in the opposite direction to the normal wheels at the same speed. If the gyroscope theory was true, the counter-rotating wheels should have canceled out the effect and the bicycle should have been difficult to ride. He found no difference in ridability, which then raised the question “why in fact is a bike stable?”

It turns out that the critical factor is something called “trail” – the position where the tire touches the ground compared to where the steering axis hits the ground. If the wheel touches behind the imaginary extension of the steering axis, then the bicycle is stable. The basic reason why this makes a motorcycle is stable is that the friction of the tire on the ground pulls the tire in line behind the steering axis. This acts to straighten up the wheel.

Now riding a motorcycle or bicycle involves more than just keeping it stable. If you are going to go anywhere except straight ahead, then you need to steer. Steering a motorcycle or bicycle is counterintuitive; to turn right, you must steer left initially, and vice versa. Most bicycle riders don't notice that they are doing this, because the bicycle is light, and they can easily 'muscle' the bike onto the path they want. Not so with a motorcycle, which can easily weigh 500 pounds. The only way to get any sized motorcycle turning is through an initially counter-directed turn by turning the handlebars explicitly (called countersteering) or by throwing your hips to the side. Gyroscopic forces play only a limited role in balancing and steering; and there is no way you are going to muscle this bad boy into doing your bidding!

Centrifugal forces will throw your bike over on its side if you steer the handlebars in the direction of a desired turn without first leaning the bike into the turn (not a healthy turn of events). Indeed, bike crashes are often caused by road obstacles like railroad tracks or sewer grates turning the front wheel and handlebars abruptly. Leaning the bike into the turn allows gravitational forces to balance the centrifugal forces, leading to a controlled and stable turn. Thus steering a bike involves a complicated interaction between centrifugal and gravitational forces, and torques applied to the handlebars, all mediated by the bike geometry.

Countersteering is employed by both motorcyclists and bicyclists, though most bicyclists countersteer unconsciously. You may have noticed, however, that while on a bicycle, it is surprisingly difficult to ride clear of a nearby high curb or sharp drop. This is because you must steer towards the edge to get away from the edge. It is easy to directly demonstrate counter-steering on a bicycle. While riding at a brisk pace (possibly downhill to avoid the complications of peddling), let go with your left hand while pushing the right handlebar with the open palm of your right hand. Since your hand is open, you can only turn the handlebar left, but the bike will turn right. Bicycles are also designed with large (for their weight) handlebars, that provide substantial leverage over tire motion. Motorcycles (except for dirt bikes) tend to have short clip-on bars, which are more or less useless for steering; they are used to nudge the bike into countersteer.

The process of making a countersteered right turn (right and left are from the perspective of the rider) can be broken into five steps:

1. You initiate the turn by applying a torque to the handlebars, steering the front wheel to the left.

2. The wheel steers to the left. The rate at which the steer-ing angle increases is set primarily by the moment of inertia I, of the wheel, fork, and handlebars around the steering axis, and by the “trail”

3. As the bike is now turning to the left, a centrifugal torque leans both you and the bike frame to the right. Gyroscopic action also leans the bike to the right, but, as I will show later, its effect is negligible.

4. Transmitted by the fork, the increasing lean attempts to lean the front wheel over as well. For the first time, gyroscopic action becomes important, as the wheel responds to this “leaning” torque by attempting to steer to the right, thus counteracting the steering torque. The steering angle stops increasing.

5. The leaning torque overcomes the steering torque and the wheel steering angle decreases. Note that the lean continues to increase because the bike is still turning left.

6. As the bike has now acquired substantial leaning velocity, the lean increase cannot end instantly. Driven by the still increasing lean, the wheel steering angle passes smoothly through zero and then points right. The centrifugal torques reverse direction, eventually halting the lean increase and balancing the gravitational torques. As no more leaning torque is applied to the wheel, the steering angle stabilizes, and the bike executes the desired right turn.

Alternately, the required lean can be generated by throwing your hips in the direction counter to the turn. Throwing your hips is how a bike is steered no-hands. The sign of the effect is subtle, but a half-hour session in an empty parking lot should convince you that while riding no-handed, you steer the bike by leaning your shoulders in the direction of the desired turn. Since angular momentum is conserved by a sudden shift of your shoulders, your hips move the opposite way, thereby leaning the bike the opposite way as well. With the bike now leaning, the bike’s “trail” becomes important. As the steering axis is not vertical, the point of contact of the wheel with the road “trails” the intersection of the steering axis with the road. The trail makes the bike self-steer: when the bike leans to the left, the front wheel steers left; when the bike leans to the right, the front wheel steers right. This effect is easily demonstrated by standing beside a bicycle and leaning it from side to side.

The trail is the single most important geometric parameter which enters into the handling of a bike. Countersteer is the single most counterintuitive thing a biker needs to learn.

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