Bicycle mechanics and dynamics -
overview,
papers and links
Bicycle mechanics and dynamics -
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Overview. Cornell bicycle research began in 1985 when Jim Papadopoulos came to Cornell to work with Andy Ruina, or vice versa. Undergraduate projects have included the application of constraints to pedals, making a geared unicycle, designing a new suspension, measuring the efficiency of a bicycle transmission, designing a constrained pedal, tests of stability, automatic wheel truing, tests of what people can perceive, measurement of the effect of inertia on pedaling efficiency, etc.. Bike research had a lull from about 1988-2002. Starting in 2002-03, with the visit of Arend Schwab from Delft, collaborating with graduate student Andrew Dressel, the stability research has progressed.
VIDEO: Bicycle Stability Demonstration (MP4) (3 MB)
The clearly-observable asymptotic stability (quickly decaying oscillations) of the bicycle is shown. This is a random bicycle. The aymptotic stability of a bicycle is reflected in the equations of motion of an ideal conservative bicycle (with disk wheels and point contact).
A program for calculating stability eigenvalues (JBike6).
Papers
Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review, , J.P. Meijaard, J.M. Papadopoulos, A. Ruina, and A.L. Schwab. Proc. Roy. Soc. A., Volume 463, Number 2084 / August 08, 2007, pages 1955-1982. DOI 10.1098/rspa.2007.1857. Please contact an author if you would like to see a pre-print.)
We would like to think that this is the definitive review paper on the linearized equations of motion for a bicycle.
Comparisons and stability analysis of linearized equations of motion for a basic bicycle model (pdf), R.S. Hand, ScM. Thesis, May 1988
This thesis on the stability of an uncontrolled bicycle is thorough and has the only relatively complete survey of papers on bicycle stability. The advisors were Jim Papadopoulos and Andy Ruina.
Second order kinematic
constraint between two bodies rolling, twisting and slipping against each
other while maintaining point contact, Suresh
Goyal, Cornell technical report TR 89-1043, October 1989. (pdf)
(A possibly related paper, Jim thinks, is “Les mouvements de roulement
en dynamique”
http://gallica.bnf.fr/ark:/12148/bpt6k82007h/f1.item.)
Additional "Cornell'' Articles
Here are other articles on bicycle mechanics, mostly by Jim Papadopoulos, mostly incomplete, highly redundant. Some have contributions from Scott Hand, John Olson, and Andy Ruina.
Bicycle Steering Dynamics and Self-Stability: A Summary Report on Work in Progress (Preliminary Draft) (PDF), J.M. Papadopoulos, Dec 1987
This is Jim Papadopoulos' summary of the bicycle dynamics equations and what he knew about them.
Forces in
Bicycle Pedaling (PDF)
J.M. Papadopoulos (1986?)
Bicycle Handling Experiments You Can Do
(Early Draft) (PDF)
J.M. Papadopoulos, Dec 1987
Discussion of Le Henaff's Paper
(PDF)
J.M. Papadopoulos, A. Ruina, Feb 1987
Bicycle Dynamics - The
Meaning Behind the Math (PDF)
J. Olsen, J.M. Papadopoulos, Bike Tech, vol. 7, issue 6, pages 13-15, Dec 1988
Supplementary Material (PDF)
Scaling Rules, and Chronological Comparison of the Linearized Equations of Motion for the Bicycle Model (PDF) This article contains scaling laws, and a 100 year history of bicycle dynamics equations from Hand's thesis. (1987?)
A Single-Step Method of
Aligning a Bicycle Frame so No Handlebar Torque is Required for Straight-Line
Riding (Draft) (PDF)
J.M. Papadopoulos
May 1988
Governing Equations by Papadopoulos (Incomplete, 1987?) (PDF)
Bicycle and Motorcycle
Balance and Steer Dynamics (Draft)
(PDF)
J.M. Papadopoulos,
R.S. Hand, A. Ruina, Jul 1990
Some other good bicycle papers that are hard to find
If you know the whereabouts of any of these authors or copyright holders, please let us know.
Carvallo, E. 1899 (awarded Prix Fourneyron 1898, submitted 1897).
Also published as
Theorie du mouvement du Monocycle et de la Bicyclette. Journal de L'Ecole Polytechnique,
Series 2,
Part 1, Volume 5, ``Cerceau et Monocyle'', 1900, pp.~119--188, 1900. Part 2,
Volume 6, ``Theorie de la Bicyclette'',
pp.~1--118, 1901. (pdf from
the University of Michigan, in French)
Bourlet, C. 1899: Etude theorique sur la bicyclette,Bulletin
de la Societe Mathematique de France, 27,
part I pp. 47--67, Part II p. 76--96. (Originally submitted in 1897 for the Prix
Fourneyron, awarded first place 1898.)
(part 1, part
2). Some original paperwork from the prize [posted with permission from the
French Academy] (pdf). More
information about the Prix is available, ask us.)
G.R.R. Routh, 1899, "On the Motion of a Bicycle", The Messenger of Mathematics, Vol 28 (May 1898-April 1899), pp 151-169. (pdf) (Note G.R.R. Routh is the son of the famous E.J. Routh and grandson of Airy.).
Valentine Joseph Boussinesq, 1899, "Apercu sur la thèorie de la bicyclette", Journal de Mathèmatique Pures et appliquèes, Tome Cinquième, pp 117-135. (pdf)
Valentine Joseph Boussinesq, 1899, ``Complèment à une ètude rècente concernant la thèorie de la bicyclette (1): influence, sur l'èquilibre, des mouvements latèraux spontanès du cavalier ", Journal de Mathèmatique Pures et appliquèes, Tome Cinquième, pp 217-232. (pdf)
Valentine Joseph Boussinesq, 1899, "De l'effet produit, sur le mouvement d'inclinaison d'une bicylette en marche, par les dèplacements latèraux que s'imprime le cavalier.", Journal de Mathèmatique Pures et appliquèes, Tome Cinquième, CR, vol 128, pp 766-781. (pdf), and another (859-862), one more (art 11) and still another (art 21).
F.J.W. Whipple, 1901, "The Stability of the Motion of a Bicycle", Quarterly of Applied Mathematics paper based on Whipple's 1899 Trinity College Cambridge Undergraduate Thesis. (pdf)
Klein, F. and Sommerfeld, A. (actually written by Fritz Noether), 1910, "Theory des Kreisels" (Theory of Gyroscopes), Chapter IX, Section 8. pages 863-884. Here is an English translation arranged by Jim Papadopoulos (pdf). German original available from the University of Michigan at http://www.hti.umich.edu/u/umhistmath/ (then search) or here.
Herfkens, B. D., 1949,"De stabiliteit van
het rijwiel" (``The stability of the bicycle''),
Report S-98-247-50-10-'49, Instituut voor rijwielontwikkeling (``Institute for
bicycledevelopment''), 's-Gravenhage. (pdf, Dutch and English).
E. Dohring (Döhring), 1955 Brunswick. "Stability
of Single-Track Vehicles",
(pdf).
Forschung Ing.-Wes., 21, no. 2, 50-62 (1955). Translated by J. Lotsof for Cornell
Aeronautical Laboratory, Inc, Buffalo, N.Y., March 1957. Posted here with the
assumed courtesy of CALSPAN.
Robert Neil Collins 1963, PhD thesis, Univ of Wisconsin, "A Mathematical Analysis of the Stability of Two Wheeled Vehicles". From microfilm, [Posted without permission] (pdf).
Digvijai Singh 1964, PhD thesis, Univ. of Wisconsin, "Advanced Concepts of the Stability of Two-Wheeled Vehicles -- Application of Mathematical Analysis to Actual Vehicles". Advised by Archi H. Easton. [Posted without permission] (pdf).
David Herbert Weir 1972, PhD thesis, UC Los Angeles, "Motorcycle Handling Dynamics and Rider Control and the Effect of Design Configuration on response and Performance". Advised by John Lyman. [Permission for posting not granted]. (Can be obtained through standard channels. Weir's present work is: http://www.dynres.com/)
David Jeffrey Eaton 1973, PhD thesis, Univ of Michigan, "Man-Machine Dynamics in the Stabilization of Single-Track Vehicles". Advised by Leonard Segel. [Posted with permission from David.Eaton@pb.com] (pdf).
Paul Jacob P.J. Van Zytveld 1975, "Engineer" thesis, Stanford, "A Method for the Stabilization of an Unmannd Bicycle". Advised by John V. Breakwell. [Posted with permission from vanzytveldp@earthlink.net, +1 650-793-7012, +1-650-967-1163] (pdf)
Mark L. Psiaki 1979, Batchelor's thesis, Princeton, "Bicycle Stability: A Mathematical and Numerical Analysis". Advised by F.C. Shoemaker. [Posted with permission from Mark Psiake, mlp4@cornell.edu] (10 meg pdf).
Dikarev, Dikarev and Fufaev, 1981, A correction of the equations in Niemark and Fufaev. pdf
External Links
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Richard Klein's bicycle science pages (then click on "Bicycle Science") |
Please send Web site comments to ruina@cornell.edu.