Astrom, K. J., R. E. Klein, et al. (2005). Bicycle Dynamics and Control:
Adapted bicycles for education and research. IEEE Control Systems Magazine.
2p) Lowell, J. and H. D. McKell (1982). "The Stability
of Bicycles." American Journal of Physics 50(12): 7.
Jones, D. E. H. (1970). The Stability of the Bicycle. Physics Today.
This is probably
the most famous piece written on bicycle stability. Almost every paper
on the subject references this article. Jones may have been the first
person to show the connection between front fork geometry, namely trail,
and the inherent stability of the bicycle in an article suited to the
layman. Whitt and Wilson [3b] give a great
summary of Jones' paper and add more information than is in the paper
Jones built several
experimental bicycles to prove and disprove some theories. His first
bicycle was configured so that the gyroscopic forces from the front
wheel were canceled. The bicycle was easily ridden hands-on, difficult
to ride hands-off and showed no signs of self stability when released
riderless. One experimental error that may have been introduced to the
first bicycle configuration was that the counter rotating wheel was
hung off the side of the fork, thus creating moment about the steering
axis. Better designed gyro cancellation bicycles can be found in [1p].
The second bicycle
configuration had increased trail due to the fork being rotated 180
degrees. This bicycle was awkward to ride but showed incredible self
stability when released riderless. This lead Jones to look more closely
at front fork geometry and how it affects stability. He theorized that
the when the bicycle rolls the steering angle adjusts to minimize the
potential energy of the fork/front wheel assembly. So he iteratively
calculated the height change of a point on the fork as a function of
steer angle and roll angle. From this he determined that the theory
wasn't correct because the steer angle never adjusts to the minimal
potential energy angle in normal riding conditions.
Jones then realized
that the change in height of the point on the fork with respect to the
steering angle was proportional to the torque about the steer axis for
small steering angles. This led to calculating the constant of proportionality
associated with this relationship. This constant was used as his stability
index and turned out to simply be the mechanical trail divided by two
times the front wheel diameter as shown in [5b]
on page 290. Jones used this stability index to make a plot that compared
various bicycles. All of the bicycles he found had a negative stability
index and most of the modern bicycles were grouped fairly close together.
Using this information
Jones made his next bicycle that had a frame geometry which gave a positive
stability index. This bicycle had negative trail. It was rideable, although
difficult, and fell instantly when released riderless. These three bicycles
experimentally verified Jones' self stability index.
This study essentially
gave a way to classify, in terms of self stability, bicycles based on
their frame geometry. The stability index can then be used as a design
tool to allow a new bicycle design to have the same stability characteristics
as a previously designed bicycle.
Vincenti, W. G. (1988). How Did It Become 'Obvious' That an Airplane Should
Be Inherently Stable? Invention & Technology: 50-56.
This is a great
piece that presents a timeline of how the aeronautical community came
to understand handling qualities and the balance between stability and
Schwab, A. L., J. P. Meijaard, et al. (2004). Benchmark Results on the
Linearized Equations of Motion of an Uncontrolled Bicycle. The Second
Asian Conference on Multibody Dynamics. Seoul, Korea, KSME.
Limebeer, D. J. N. and R. S. Sharp (2006). Bicycles, Motorcycles, and
Models. IEEE Control Systems Magazine. 26: 34-61.