History of bicycle steer and dynamics equations

Delft University of Technology

This partial survey, of analyses and simulations of rigid-body models of  bicycles and motorcycles, had its genesis in the M.Sc. thesis of Hand (1988). Included are several papers which do not contain comparable equations, but which are well known as related to bicycle dynamics. Note that, from distant past to present, much of the key literature has not been peer-reviewed.

Most motorcycle models include tire compliance, which permits precise comparison with rigid-wheel models only if the tire stiffness can be made infinite. So we have largely neglected the motorcycle literature.

Papers marked with a * have equations at least nearly as general as ours and also have equations which we either have verified as at least close to correct or which we believe are likely to be close to correct. Papers marked with a ** are at least as general as ours and we are confident that they agree with ours exactly (but for minor typographical errors).

Although surely there are gaps to be filled and corrections to be made, this survey is more complete than anything available in the open literature.

1869

Rankine (famed civil engineer and thermodynamics theorist) presents fundamental, semi-quantitative observations on leaning and steering of Velocipedes (an early name for bicycles). Seemingly the vehicles he considers have vertical steering axes, straight forks, and pedals directly driving the front wheel. His exposition contains the earliest known description of 'countersteering' -- briefly turning left in order to enter a rightward curve.

1896

Archibald Sharp's authoritative book on the design of bicycles and tricycles includes a calculation of the torque required to execute a steady turn on a bicycle. His model is much simplified and his calculation is slightly wrong. Most importantly for bicycle dynamics analysis, his assumption that steering must self-center in order to be self-stable is now known to be wrong.

1895

Bourlet's book on bicycles is published. It is criticized by Klein and Sommerfeld (1910). We did not check it.
Bourlet1895JPM.txt , Bourlet1899JMP.txt

1898

McCaw determines under what conditions a biycle can make a steady turn with a zero steering torque. He considers a road with superelevation, as in an in-door cycle track. One can argue that the paper considers a special case, nothing is said about stability, and the calculation has an error, and Whipple (1899) did not interpret the paper correctly.
McCaw1898JPM.txt

1898-1899

G. R. RRouth,  On the motion of a bicycle.
Routh1898JPM.txt

1899

Bourlet,  Étude théorique sur la bicyclette.
Bourlet1899JPM.txt

1899

Boussinesq, four summaries + two papers.
Boussinesq1899JPM.txt

1899**

Meteorologist and mathematician Francis Whipple, just after his graduation as second wrangler from Trinity College (Cambridge), writes an "honourable mention" thesis and then paper on bicycle dynamics. Whipple writes non-linear governing equations which are nearly correct. A cosine psi term is missing in the constraint equation for the pitch angle, and this error propagates into the non-linear equations of motion. Fortunately this has no influence on the linearized equations. Therefore, he is the first to derive the correct linearized equations for a bicycle.  But for typographical errors, his linearization agrees with our linearized equations. He also considered the stabilization of the bicycle by steering torques and body lean.
Whipple1899Hand88.pdf, Whipple1899JPM.txt

1900-1901*

Carvallo writes a 300 page prize-winning monograph on the dynamics of monocycles and bicycles (a monocycle is a single-wheeled vehicle where the rider sits inside the wheel). Carvallo's model is not quite as general as the model presented here (see Hand 1988). When our model is simplified to correspond to his, the equations agree exactly. Carvallo  and Whipple (1899) seem unaware of each other.
Carvallo1901Hand88.pdf, Carvallo1901JPM.txt, Carvallo1901JMPJPM2.txt, Carvallo1901MJC.txt, Carvallo1901_comp_coleman2.pdf

1910

Henri Bouasse uses a much simplified model of a bicycle to show how a bicycle can be balanced by steering. Therefore only the lean equation is considered, with the steering input appearing as a forcing term in the right-hand side. When so simplified our equations agree with these. The spirit, although perhaps not the rigour, is much in the line of more modern control-theory-oriented work such as that of Getz & Marsden (1995).
Bouasse1910JPM.txt

1910*

Physicists Felix Klein & Arnold Sommerfeld write a book about gyroscopes which includes a chapter on bicycles. This chapter appears in part 4 which has been edited and completed by Fritz Noether. The model in this book is not as general as the model here, for example the mass of the front frame is effectively limited to a line along the steering axis. After our model is simplified it agrees with theirs exactly (see Hand 1988).
KleinSommerfeld1903Hand88.pdf, KleinSommerfeld1910JPM.txt, KleinSommerfeld1910JPM2.txt

1915

G. S. Bower, apparently unaware of all of the work above, derives incorrect equations for a highly simplified bicycle model (see Hand 1988).
Bower1915Hand88.pdf

1920

Richard Grammel has a simplified bicycle model in his book on spinning tops (`Der Kreisel`) to illustrate the gyroscopic effect. He takes into account two important effects but forgets about the trail. We did not check.
Grammel1920JPM.txt

1922

R. H. Pearsall attempts to extend Bower's analysis (1915) in order to explain "speedmans wobble" (which may correspond to what we would now call shimmy). His model is highly simplified and his equations for that model are not correct (see Hand 1988).
Pearsall1922Hand88.pdf

1934

L. G. Loicjanskii & A. G. Lu\'re derive equations for a simplified bicycle. These are the basis for the model in Neimark & Fufaev (1972). We have not seen the work. The more general but incorrect equations in Neimark & Fufaev, when reduced to apply to this simpler model, give equations which agree with our equations when similarly simplified. Neimark & Fufaev presumably used this for checking their more general model, thus there is a good chance that the 1934 equations of the simplified model are correct.

1948

Timoshenko and Young's dynamics text book use the same model and assumptions as Bouasse (1910).
TimoshenkoYoung1948Hand88.pdf, TimoshenkoYoung1948JPM.txt

1948

J. P. Den Hartog's excellent textbook on dynamics includes a discussion (see article 61) of how a bicycle can be balance, with control, by steering the wheels to a position underneath the rider as he/she falls. Den Hartog also explains that the gyroscopic torque of the front wheel naturally accomplishes steer in the right direction even without rider input. This work is in the line of Grammel (1920). There are no detailed equations of any bicycle model.

1949

B. D. Herfkens writes a report (in Dutch) on the stability of a bicycle. This is one of 13 reports of the Dutch Bicycle Research Institute (in Dutch: Instituut voor Rijwielontwikkeling) written between 1948 and 1952. He derives linearized equations of motion for the Whipple model by a Lagrange method with nonholonomic constraints. He gives reference to Carvallo (1900) and Whipple (1899) but makes no comparison. He addresses the question of design changes and the effect on the weave speed. He makes a graph showing the effect of changes in the front wheel inertia on the stable forward speed range. This is an intelligent report and we suspect, by first impression, that his equations are correct.
Herfkens1949JPM.txt

1953-1955**

E. Döhring, University of Technology Braunschweig, Germany, writes a PhD thesis on the stability of a straight ahead running motorcycle. He builds on  the model by Klein & Sommerfeld (1901) and ends up with the same model as presented here. His 1955 paper is translated by Lotsof at Calspan into English: these equations agree with ours in detail and are correct. Döhring also did experiments on a motor-scooter and two different motorcycles to validate his results. Some of these results are reported in his 1954 paper.
Doehring1953JPM.txt

Doehring1955Hand88.pdf, Doehring1955JPM.txt

1963

R. N. Collins' University of Wisconsin PhD thesis, sponsored by Harley Davidson, considers a model of a motorcycle that is equivalent to the model here with the addition of drive and drag forces. He writes non-linear differential equations and then linearizes. There appears to be a small error but the form of the equations also seems generally correct. The equations were deemed too complicated in form to check in detail (see Hand 1988).
Collins1963Hand88.pdf

1964

D. V. Singh, at Wisconsin with Collins (1963) adds a tire slip to Collins model. However the Collins and Singh equations seem to be mutually incompatible. And neither of them compare their equations with previous equations (see Hand 1988).
Singh1964Hand88.pdf

1967*

Neimark & Fufaev authoritative monograph on non-holonomic dynamics (translated in 1972) has a chapter on the equations for a bicycle. The model is based on Loicjanskii & Lu're (1937). The formulation by Neimark & Fufaev seems correct, and is imitated in Hand's thesis (1988). However there are various errors (missing terms, calculation errors, typographical errors) so that the final equations are incorrect.  The main error is the lacking of second order contribution of steer angle to pitch of rear frame, leading to the error in energy, which of course makes the Lagrangian incorrect. Although they mention that their equations have the same general form as those of Döhring (1955) they did not check for detailed agreement (see Hand 1988).
NeimarkFufaev1967Hand88.pdf

1970

Chemist D. E. H. Jones writes a popular-style, casual and widely remembered article about his attempt to build an unrideable (no-hands) bicycle. Jones gives up on dynamics equations and attempts an empirical understanding. He discovers that positive trail and angular momentum of the front wheel are both critical for riderless stability; while trail has a greater effect on making no-hands riding impossible.

1970

Commissioned by the National Commission on Product Safety, R. S. Rice and R. D. Roland write a report on the safe handling of a bicycle. The study involves experiments and an analytical model. They (Roland) derive equations of motion for a bicycle which includes radial and lateral tire stiffness and a lateral leaning rider. They show no simulation results, apparently due to lack of time and funding. We did not check the equations.

1971**

D. V. Singh & V. K. Goel use Döhring's (1955) correct model to analyze the stability of a motor-scooter and thus most likely have correct equations (see Hand 1988).
SinghGoel1971Hand88.pdf

1971*

R. S. Sharp considers a model that is slightly less general (mass moment of inertia of front assembly parallel to steering axis) than the model here but which has tire compliance. He is the first to label the two major rigid eigenmodes as weave and capsize mode. His nonlinear model is incorrect, he treats rear-frame pitch as zero, with a constant force acting upward on the front wheel. When he linearizes his non-linear model and takes the limit of infinite tire stiffness he introduces several algebraic and typographical errors. After correcting for these errors his model agrees with ours (see Hand 1988).
Sharp1971Hand88.pdf

1971-1972

In the only known study financed by a bicycle company (Schwinn), Roland & Massing (1971) and Roland & Lynch (1972) derive non-linear equations for a bicycle with radial and lateral tire deformation. The equations of motion contains several algebraic and typographical errors. The side-slip angle of the front wheel does not contain the steering rate angle and leads to discrepancies. Therefore, after linearizing and taking the non-slip limit we were unable to make agreement with our equations (see Hand 1988). The same bicycle model is presented in Roland (1973).
Roland1971Hand88.pdf, RolandMassing1971JPM.txt, Roland1973JPM.txt

1972**

D. H. Weir's UCLA motorcycle PhD thesis makes bicycle dynamics history by being the first researcher to explicitly compare his equations, in detail, with previous research. Weir's equations, when slightly simplified to apply to Sharp's (1971) simpler model, agrees with Sharp's equations. When, instead, Weir's model is left more general with regard to inertial properties, but side-slip is eliminated, his equations agree with our equations (see Hand 1988).
Weir1972Hand88.pdf, Weir1972JPM.txt

1973

D. J. Eaton's Michigan PhD thesis concerns vehicle-rider interactions. We have not checked his governing differential equations.

1975

Singh & Goel derive equations for a more general model than they used in 1971. They do not mention checking if their new equations reduce to their previous (Döhring's) equations and we did not attempt this check either (see Hand 1988).

1975

P. J. Van Zytveld's PhD thesis at Stanford includes equations for a bicycle with a rider as a fifth rigid body. If the rider is removed, his equations agree with ours. Van Zytveld was advised by John Breakwell who (private communication - Andreas von Flotow) is said to have had an independent derivation (see Breakwell 1982) that agreed with Van Zytveld.
VanZytveld1975JPM.txt

1975*

R. S. Sharp & C. J. Jones extended Sharp's 1971 model to include a different tire model. When we removed the tire deformation the equations agree with a simplified version of our model (see Hand 1988).
SharpJones1975Hand88.pdf

1978

Weir & Zellner present in an appendix the same derivation as Weir (1972) but unfortunately introduce a number of typographical errors. Weir's dissertation (1972) is the more authoritative work (see Hand 1988).
WeirZellner1978Hand88.pdf

1978

L. G. Lobas (in the translated work his name gets misspelled into Gobas) presents equations for a model like ours, including the possibility of forward acceleration. He uses the "Boltzmann-Hamel" equations in the derivation but the final equations do not agree with ours and thus seem to be incorrect (see Hand 1988).
Lobas1978Hand88.pdf

1979

C. Adiele writes a Master of Engineering thesis in which he uses Kane's equations to derive equations of motion that do not agree with our equations (see Hand 1988).
Adiele1979Hand88.pdf

1979*

Mark Psiaki, now a Professor of Mechanical Engineering at Cornell, writes a Princeton Physics honor's thesis on the dynamics of a bicycle, writing full non-linear equations. We have compared the eigenvalues in a forward speed range for the example from his thesis with our model and the results agree within plotting accuracy. Recently Psiaki ( private communication) compared his results with those of Hand 1988 and also found agreement to within plotting accuracy. So Psiaki's equations show signs of being correct.
Psiaki1979JPM.txt

1982

Lowell & McKell write equations for a highly simplified model of a bicycle. When our equations are simplified to correspond to their model, the equations do not agree. Very specialist model, point masses and no front mass, vertical axis and it is incorrect.
LowellMcKell1982Hand88.pdf

1982

John V. Breakwell, the advisor of Van Zytveld (1975) gives a talk about bicycles. This talk, or a one presumably like it, is mentioned in the Breakwell memorial biography by Arthur Bryson. We have not seen any notes of any kind from Breakwell or his talks.

1983

C. Koenen, advised by Hans Pacejka, writes a Delft PhD thesis on motorcycle dynamics. He investigates the stability of motorcycles, running straight ahead and in steady state cornering. His model includes tires, rear and front wheel suspension and a passive hinged rider. He puts extra effort in visualizing the eigenmodes. We have not checked his equations.

1985

Robin Sharp writes a review paper on the lateral dynamics of single track vehicles.
SharpReviewsJPM.txt

1987

Arnold Schoonwinkel writes a PhD thesis at Stanford on the design and test of a computer stabilized unicycle.

1987

Jim Papadopoulos presents various results concerning the dynamics of bicycles. The present publication is the first to move some contents of his notes from the grey to the peer-reviewed literature. The compact derivation of linearized equations presented here is from Papadopoulos (1987).

1988

Scott Hand's Master thesis, advised by Papadopoulos and Ruina, presents equations of motion that are checked against the literature. The Hand derivation follows the approach of Neimark and Fufaev (1972) but corrects errors therein. Hand's equations agree with Papadopoulos (1987), Döhring (1955), Weir (1972) and, when simplified, with Whipple (1899). Hand was unaware of Van Zytveld (1975), and Breakwell (1982). Hand's FORTRAN program  for calculating stability eigenvalues has errors and the eigenvalue calculations in Hand's thesis should not be trusted.

1988**

B. C. Mears, R. E. Klein's PhD student at Urbana Champaign, confirms the correctness of the Papadopoulos (1987) and Hand (1988) equations.

1990*

In a substantial bicycle research program at Oldenburg, G. Franke, W. Suhr & F. Riess derive non-linear equations of a bicycle. We did not check the derivation in detail. The authors were unable to find agreement between integration of their differential equations for small angles and the integration of the Hand equations (private communication to JP from the authors). Recent comparison of the linearized stability on the benchmark bicycle showed agreement of the eigenvalues within plotting accuracy. This paper was the topic of an entertaining lead editorial in Nature by John Maddox (1990).
FrankeSuhrRiess1990JPM.txt, FrankeSuhrRiess1990JMP.txt

1995

The non-linear dynamics group at Caltech, including Jerry Marsden and colleagues, writes the first of several papers on bicycle control. Because the emphasis is on control, not passive-dynamics, highly simplified models are used. We have not checked any of these in detail.

2001

Sharp and Limebeer derive the equations of motion for a motorcycle by means of the multibody dynamics software AutoSim. The model is based on Koenen (1983), but the results did not agree.
SharpLimebeer2001JPM.txt

2002

Cossalter and Lot derive the equations of motion for a motorcycle for real time simulations based on the natural coordinate approach. The equations are formulated as a large system of Differential-Algebraic Equations (DAEs). We have not checked it.
CossalterLot2002JPM.txt

2004

In a chapter of David Wilson's popular book Bicycling Science, Papadopoulos presents various issues related to bicycle steering and balance. This discussion includes an informal introduction to the equations here.

2005**

J. P. Meijaard in preparing for this publication, makes an independent derivation of the linearized equations of motion.

2005**

Schwab, Meijaard and Papadopoulos write a draft of the present paper and present it at a meeting. The paper here completely subsumes that conference paper.

2005

K. J. Åström, R. E. Klein & A. Lennartsson present an inspiring 3 part paper on bicycle dynamics. The experimental work of Klein and students, much in the spirit of Jones (1970), extends back to the mid 1980s and is interestingly documented well here for the first time. Lennartsson presents simulations from a general purpose rigid-body dynamics code and gets agreement with the equations here, although not with enough accuracy presented to be totally assured of correctness. The theoretical work and bibliography in the paper is largely based on Papadopoulos (1987), Hand (1988) and related documents. The equations used to describe the simplified model (in turn used to intuitively explain bicycle stability) do not agree with the equations later used in the paper nor with the equations here. And, opposite to what is written in the paper, that model is not self-stable.

References

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

Adiele, C. 1979 Two wheeled vehicle design. M.Eng. thesis, Sibley School of Mechanical and Aerospace Engineering, Cornell University.

Appell, P.E. 1896 Mecanique Rationnelle, Vol. II, pp.297-302, 6th edn. Paris: Gauthier-Villars.

Åström, K.J., Klein, R.E. & Lennartsson, A. 2005 Bicycle dynamics and control: Adapted bicycles for education and research. IEEE Control Systems Magazine 25(4), 26-47.

Besseling, J. F. 1964 The complete analogy between the matrix equations and the continuous field equations of structural analysis. In International symposium on analogue and digital techniques applied to aeronautics: Proceedings, Presses Académiques Européennes, Bruxelles, pp. 223-242.

Bouasse, H. 1910 Cours de Mechanique, part 2, pp. 620-623. Paris: Ch. Delagrave. (Bou1910pp620-623.pdf)

Bourlet, C. 1899 Étude théorique sur la bicyclette. Bulletin de la Société Mathématique de France, 27 (1899), p. 47-67 (Bourlet1899pp47-67.pdf)

Bourlet, C. 1899 Étude théorique sur la bicyclette. Bulletin de la Société Mathématique de France, 27 (1899), p. 76-96 (Bourlet1899pp76-96.pdf)

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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).

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Carvallo, M. E. 1900 Théorie 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, Part 2, Volume 6, "Théorie de la Bicyclette", 1901, pp. 1-118.(Carvallo1900.pdf)

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Cossalter, V. & Lot, R. 2002 A motorcycle multi-body model for real time simulations based on the natural coordinate approach. Vehicle System Dynamics 37(6), 423-447.

Davis, J. A. 1975  Bicycle tire testing – effects of inflation pressure & low coefficient surfaces, Cornell Aero. Lab. Report no. ZN-5431-V-3

Davis, J. A. & R. J. Cassidy 1974 The Effect of Frame Properties on Bicycling Efficiency, Cornell Aero. Lab. Report no. ZN-5431-V-2

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[Posted here with the assumed courtesy (without permission) of CALSPAN.]

Eaton, D. J. 1973 Man-machine dynamics in the stabilization of single-track vehicles. Ph.D. thesis, University of Michigan.[Posted without permission](EatonDavid1973.pdf)

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(availabe with other related reports at: http://ruina.tam.cornell.edu/research/).

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Roland, R. D. 1973 Computer simulation of bicycle dynamics. Mechanics and Sport 4, ASME, pp. 35-83.

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[Posted without permission from P. J. Van Zytveld. If you know his whereabouts please let us know.]

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