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[Phys-l] Bicycle question



Bob is not confused; Leigh is. Thanks for pointing that out. I edited that piece through so many iterations that my critical faculties apparently fried. I really don't want to try to repair it now, but I can identify the source of the confusion if my fallible memory serves me at all.

The problem arises because the line of contact with the pavement changes as the front wheel is steered. The torque exerted by the contact force in its newly steered position does exist with respect to the contact point line before the wheel was steered because there is a lever arm about the old line after the wheel is steered. If that is unclear to you, we are on the same wavelength; it is unclear. (See followup under topic "Teaching torque".)

Let me recommend an article I have not yet perused: "Bicycle Dynamics and Control" by Karl J. Åström, Richard E. Klein, and Anders Lennardtsson, IEEE Control Systems Magazine, August 2005, pp. 26-47. Skimming I have found much to reinforce my prejudices, which is why I recommend the article. It seems that the authors have not optimized the head tube angle (or "head angle" as they term it) from first principles, but they have done a lot that is new to me. I will share a delightful anecdote that appears as a sidebar in the article:

The NHSA Rear-Steered Motorcycle

The U.S. National Highway Safety Administration (NHSA) funded a project aimed at developing a safe motorcycle in the late 1970s. The key ideas included having a low center of mass, a long wheel base, and separation of steering and braking. The last requirement leads naturally to a design with rear-wheel steering, since the front wheel provides the major contribution to the braking force. Rear-wheel steering combined with a long wheel base also makes it possible to have a low center of mass. A contract to analyze a motorcycle with rear-wheel steering and to build a test vehicle was given to South Coast Technology in Santa Barbara, California, with Robert Schwarz as principal investigator.

A theoretical study was performed by taking the mathematical model developed by Sharp [50] and simply reversing the sign of the wheel- angular velocity. A derivation based on first principles showed that reversing the sign does indeed give the correct model. The model was linearized, and the eigenvalues were investigated. Two complex pole pairs and a real pole representing weave, wobble, and capsize dominate the dynamics. A range of geometrical configurations was investigated. The eigenvalues were plotted as functions of velocity for each configuration. The real part of the unstable poles typically covered 4–12 rad/s for speeds of 3–50 m/s. It was concluded that the instability was too fast to be stabilized by a human rider.

The result was reported to NHSA with a recommendation that it was pointless to build a prototype because the motorcycle could apparently not be ridden. NHSA was of a different opinion based on an in-house study, and they insisted that a prototype should be built and tested. The tests showed conclusively that the motorcycle was unridable, even by the most skilled riders.

The outriggers were essential; in fact, the only way to keep the machine upright for any measurable period of time was to start out down on one outrigger, apply a steer input to generate enough yaw velocity to pick up the outrigger, and then attempt to catch it as the machine approached vertical. Analysis of film data indicated that the longest stretch on two wheels was about 2.5 s. (from R. Schwarz, "Accident avoidance characteristics of unconventional motorcycle configurations," Warrendale, PA, SAE Paper 790258, 1979.)

This demonstrates the great respect in which theoreticians are held by inventors with fixed ideas spending other people's money.

Leigh


On 26 Aug 2006, at 09:00 Bob Sciamanda wrote:

Leigh, I'm confused. The road forces occur at the tire/road contact points - how can they then produce a torque about an axis through those points (isn't the "lever arm" zero?) ?

[deletia]

----- Original Message ----- From: "Leigh Palmer" <palmer@sfu.ca>
To: <phys-l@carnot.physics.buffalo.edu>
Sent: Thursday, August 24, 2006 7:31 PM
Subject: Re: [Phys-l] Bicycle question

. . .
The reaction force of the road in this new position exerts a
torque about the line containing the contact points of both tires
with the road . . .