COMMUNICATION

Mathematics Applied to
the Study of Bicycles
Jim Papadopoulos

A riderless bicycle has remarkable properties of self-sta-

bility. If it is knocked sideways when rolling forward on
flat pavement, it may briefly wiggle left and right and
then return itself to upright straight rolling. This is sur-
prising to most of us, but in fact has been rediscovered
countless times by children, and it has been analyzed
mathematically by scores of investigators. The behavior
is documented online with titles like “Riderless bicycle,”
“Ghost riding motorcycle,” etc.
Such self-stability is not universal, however. The bicycle
must be in good shape, it must be symmetric, its dimen-
sions and masses must lie within certain limits, and the
speed must be neither too low nor too high.
A question that has inspired me for decades is: When
and why is a bicycle self-stable? I originally came to this
problem through bicycle racing, having grown skeptical of
the bike shop wisdom that very slight variations in bicycle
shape could cause dramatic changes in the ease of balance
and control by a rider. In 1986 I went to Cornell to look
into the problem with Andy Ruina.
Care is needed when formulating the precise question
to be studied. For example, there is a feel of stability or of
resistance to turning; there is the ability to ride no-hands;
and there is the behavior of a riderless bicycle, either with
or without 100 kg extra mass to represent the person.
These may be only loosely related.
Anything involving human behavior is potentially very
complex. For example, when struck by a wind gust, rider
response will depend on skill, fatigue, and the properties
of the bicycle. Different riders will respond in different
ways. Therefore I have focused on the simple question
of whether a given riderless bike is or is not intrinsically
stable at a given speed. Jim Papadopoulos demonstrates that a bicycle has
Inherent bicycle stability can be investigated either by remarkable properties of self-stability.
mathematical analysis of an ideal dynamic system or by
For permission to reprint this article, please contact: reprint- Jim Papadopoulos is assistant teaching professor of mechanical
permission@ams.org. and industrial engineering at Northeastern University. His e-mail
DOI: http://dx.doi.org/10.1090/noti1556 address is J.Papadopoulos@northeastern.edu.

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Figure 1. Below the stabilizing speed, the bicycle wobbles more and more, while above the stabilizing speed,
the wobbling disappears. Computed evolution of lean and steer angles provided by Andrew Dressel using his
MATLAB program, JBike6.

In the work culminating in the 2007 Royal Society

publication of “Linearized dynamics equations for the
When and why is a balance and steer of a bicycle: a benchmark and review,”
bicycle self-stable? we focused entirely on determining the correct mathe-
matical equations to describe a specific system of frame,
steering fork, and wheels rolling freely and nearly upright
on level ground. (When I say “we‚” I am referring to prin-
experiments on actual hardware with all its messy and
cipally Andy Ruina and Arend Schwab, who kindly invited
inconsistent friction and vibration behavior. We chose the me to participate in extending and publishing research
former. While it has proven hard for many to get the math initiated at Cornell. Andy also personally supported my
right, once that is done the result is solid—for the given involvement financially.)
ideal system. Thus my focus has been on mathematical The result is two coupled second-order differential
analysis of the ideal riderless bicycle, though with the equations, comparable to those describing two pendulums
unhappy awareness (a) that the ideal model may have left connected with a spring and a damper (a kind of shock
out physical phenomena that are important to real bicycle absorber). There are differences in detail, however. While
behavior and (b) that self-stability is only one part of the the coupled pendulums come to rest due to energy dis-
riding experience. sipation in the damper, the energy of the bicycle’s lateral
Our reported research rests on two foundation stones: motion can either fall or rise (due to interchange between
• One is the definition of a minimal problem. We forward kinetic energy and the energy of tipping).
stripped the problem down to simple questions of the The equations govern the lean angle L of a riderless bike
existence of linearized stability (i.e., the disappearance and the steer angle S of its front wheel and handlebars.
of small disturbances) for ideal dynamic systems ap- (For clarity, I have defined VL as the velocity of the lean
proximating bicycles. We specifically did not look at angle, that is, the rate of change of angle, and AL as the
ease of no-hands control by a rider. We also eliminated acceleration of the lean angle, that is, the rate of change
many factors, for example, the wiggling of a rider and of the velocity. Those who have studied calculus will rec-
steer-resisting friction of the flattened tire. Finally, we ognize these as the first and second derivatives of L with
did not relate the findings to rider preference. respect to time. Likewise I will use VS and AS.) They take
• The other is the care in analysis. Scores of previous the following form:
investigators made errors in setting up their equations
or deducing consequences. We spent a lot of effort to MLLAL+MLSAS+CLLVL+CLSVS+KLLL+KLSS = 0;
compare our work to previous results and to derive
our results in multiple ways that provided confirming MSLAL+MSSAS+CSLVL+CSSVS+KSLL+KSSS = 0.
matches.

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These equations are the physics consequences of This demonstrated that the widespread beliefs didn’t
Newton’s laws applied to the infinitesimal particles that hold water. Furthermore, we built a physical model of
make up the bicycle parts when in motion while subject this simplified bike and demonstrated the actual recovery
to gravity. Each involves the lean acceleration, steer from a bump, proving that the mathematics adequately
acceleration, lean rate, steer rate, lean angle, and steer described reality.
angle. The coefficients of A, V, and L or S (like CLS) are This work does not immediately revolutionize bicycle
complicated functions of the bicycle dimensions, mass design, however! For one thing, there is no room for a
distribution, and velocity. rider. The fact is that bicycles are generally quite good;
Given all the bicycle parameters, leaning and steering all we did is show that in exploring changes, one needn’t
motion can be determined numerically using Excel or strictly maintain the caster and gyroscopic components
MATLAB. At a specific instant, if the steer and lean rates as formerly believed. Furthermore, the work is far from
are known and the steer and lean angles are known, then complete. One important need is to develop a mathe-
the two equations can be cast as two easily solved linear matical model that works with a rider on the bike. For
equations for lean and steer acceleration in terms of all example, when a heavy person is on a bicycle, the tires
other quantities. Then the solution proceeds as follows: are flattened and the original equations don’t capture the
• The steer acceleration lets us compute the change in effects of their friction and flex. In addition, a rider adds
steer rate over a short time. tremendous complexity by being extremely flexible and
• In that same time the steer velocity lets us compute also by including unpredictable muscular efforts at the
the change in steer angle. neck, waist, and hips.
• The same steps are possible for the lean angle. The hope is eventually to understand something of
Thus, we have the means to what riders value in terms of the ease or difficulty of
transition from one instant to a balancing and steering. Then, it should be possible to
The slightly later time and will have
the information to repeat the com-
genuinely tune the design to suit a child, a windy-weather
commuter, or a racer seeking the most rapid leaning and
widespread putation as many times as de- turning response.
sired. Figure 1 shows how a slight
beliefs change in bicycle speed from 4.25 Acknowledgments
to 4.5 meters per second leads The early phase of this research at Cornell was supported by
didn’t hold from instability and growing os- Andy Ruina’s NSF PYI award.
cillation in the left plot to stability
water. and shrinking oscillation in the
Photo Credits
right plot.
Figure 1 courtesy of Andrew Dressel.
With correct equations, one
Images of the author courtesy of Jim Papadopoulos.
can define a given bicycle in a
given configuration and state of motion, then numerically
solve the equations to see whether it rights itself. This is
a kind of calculational experiment, perhaps of interest to
a specific situation, but not revealing any general results ABOUT THE AUTHOR
about bicycle design. Jim Papadopoulos used to make
Potentially much more revealing is a study of the sta- and race bicycles and remains ob-
bility of ALL solutions of the given equations. Some great sessed with all applications of math
theoretical results from the 1800s dictate that the system to bikes. Unexpected results relating
will be stable if products of the equation coefficients (such to spoke stresses, tire forces, unsta-
as MLL, KSL, etc.) satisfy a number of inequalities called the ble steer oscillation (shimmy), and
Routh-Hurwitz criteria. optimum muscle usage when pedal-
Using this approach, we were able to investigate a cou- ing may be seen in his co-authored Jim Papadopoulos
ple of general questions. Among bicycle makers and users, book Bicycling Science, and others
it was universally agreed that stability was only possible keep turning up.
if some specific design criteria were met. One was that
the wheels should exhibit adequate gyroscopic strength,
requiring extra torque to turn. In other words, it was be-
lieved that a bicycle on skates could never be self-stable.
Another presumed requirement was that the front wheel
should touch the ground behind the line of the steering
axis, a kind of caster distance.
We theoretically outlined a simplified bicycle shape,
and adjusted its dimensions to show that it could satisfy
the Routh-Hurwitz criteria (thus achieving stability in
forward motion) despite violating the above two criteria.

890 Notices of the AMS Volume 64, Number 8