We would like to think that this is the definitive review paper on the linearized equations of motion for a bicycle.

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**The bricycle: A bicycle in zero gravity can be balanced or steered but not
both**

O. Dong, C. Graham, A. Grewal, C. Parrucci and A. RuinaAccepted J
of Veh. Dyn. Aug 2014.

3 minute video

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.

We would like to think that this is the definitive review paper on the linearized equations of motion for a bicycle.

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.

Suresh Goyal, Cornell technical report TR 89-1043, October 1989.

(A possibly related paper, Jim thinks, is "Les mouvements de roulement en dynamique")

(A possibly related paper, Jim thinks, is "Les mouvements de roulement en dynamique")

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.

J.M. Papadopoulos, Dec 1987

This is Jim Papadopoulos' summary of the bicycle dynamics equations and what he knew about them.

J.M. Papadopoulos (1986?)

J.M. Papadopoulos, Dec 1987

J.M. Papadopoulos, A. Ruina, Feb 1987

J. Olsen, J.M. Papadopoulos, Bike Tech, vol. 7, issue 6, pages 13-15, Dec 1988

Supplementary Material (PDF)

Supplementary Material (PDF)

This article contains scaling laws, and a 100 year history of bicycle dynamics equations from Hand's thesis. (1987?)

J.M. Papadopoulos, May 1988

Papadopoulos (Incomplete, 1987?)

J.M. Papadopoulos, R.S. Hand, A. Ruina, Jul 1990

If you know the whereabouts of any of these authors or copyright holders, please let us know.

(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.

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) Some original paperwork from the prize [posted with permission from the French Academy] Link

More information about the Prix is available, ask us.

More information about the Prix is available, ask us.

"On the Motion of a Bicycle", The Messenger of Mathematics, Vol 28 (May 1898-April 1899), pp 151-169.

(Note G.R.R. Routh is the son of the famous E.J. Routh and grandson of Airy.).

(Note G.R.R. Routh is the son of the famous E.J. Routh and grandson of Airy.).

"Apercu sur la thèorie de la bicyclette", Journal de Mathèmatique Pures et appliquèes, Tome Cinquième, pp 117-135.

"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.

"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.

"The Stability of the Motion of a Bicycle", Quarterly of Applied Mathematics paper based on Whipple's 1899 Trinity College Cambridge Undergraduate Thesis.

"Theory des Kreisels" (Theory of Gyroscopes), Chapter IX, Section 8. pages 863-884. English translation arranged by Jim Papadopoulos.

German original available by search from University of Michigan or by direct link.

German original available by search from University of Michigan or by direct link.

"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.

Brunswick. "Stability of Single-Track Vehicles", 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.

PhD thesis, Univ of Wisconsin, "A Mathematical Analysis of the Stability of Two Wheeled Vehicles". From microfilm, [Posted without permission].

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].

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).

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].

"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].

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].

A correction of the equations in Niemark and Fufaev.

Last edits by Jen Cheu 6/16/09