Bicycle Mechanics and Dynamics - Papers

Link to Overview and External Links


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. RuinaIn Press, J of Veh. Dyn. Sept 2014.
3 minute video

A bicycle can be self-stable without gyroscopic or caster effects ,
J. D. G. Kooijman1, J. P. Meijaard2, Jim M. Papadopoulos3, Andy Ruina4, and A. L. Schwab1,
Science Magazine, April 15, 2011

Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review - including appendix (PDF)

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.

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 (PDF)

Suresh Goyal, Cornell technical report TR 89-1043, October 1989.
(A possibly related paper, Jim thinks, is "Les mouvements de roulement en dynamique")

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 (PDF)

Papadopoulos (Incomplete, 1987?)

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 - from the University of Michigan, in French (PDF)

(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.

Bourlet, C. 1899 (PDF)

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.

G.R.R. Routh, 1899 (PDF)

"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.).

Valentine Joseph Boussinesq, 1899 (PDF)

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

Valentine Joseph Boussinesq, 1899 (PDF)

"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.

Valentine Joseph Boussinesq, 1899 (All PDF PP.766-781, PP.859-862, Art. 11, Art. 21)

"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.

F.J.W. Whipple, 1901 (PDF)

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

Klein, F. and Sommerfeld, A. (actually written by Fritz Noether), 1910, in English (PDF)

"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.

Herfkens, B. D., 1949, in Dutch and English (PDF)

"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.

E. Dohring (Döhring), 1955 (PDF)

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.

Robert Neil Collins 1963 (PDF)

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

Digvijai Singh 1964 (PDF)

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].

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).

David Jeffrey Eaton 1973 (PDF)

PhD thesis, Univ of Michigan, "Man-Machine Dynamics in the Stabilization of Single-Track Vehicles". Advised by Leonard Segel. [Posted with permission from].

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, +1 650-793-7012, +1-650-967-1163].

Mark L. Psiaki 1979 (10MB PDF)

Batchelor's thesis, Princeton, "Bicycle Stability: A Mathematical and Numerical Analysis". Advised by F.C. Shoemaker. [Posted with permission from Mark Psiake,].

Dikarev, Dikarev and Fufaev, 1981 (PDF)

A correction of the equations in Niemark and Fufaev.