History of bicycle steer and dynamics equations 


Delft University of Technology 
This partial survey, of analyses and simulations of rigidbody 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 peerreviewed.
Most motorcycle models include tire compliance, which permits precise comparison with rigidwheel 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, semiquantitative 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 selfcenter in order to be selfstable 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 indoor 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
 18981899
 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 nonlinear 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 nonlinear
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  19001901*
 Carvallo writes a 300 page prizewinning
monograph on the dynamics of monocycles and bicycles (a monocycle is
a singlewheeled 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 righthand side. When so simplified our equations agree with these. The
spirit, although perhaps not the rigour, is much in the
line of more modern
controltheoryoriented 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  19531955**
 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 motorscooter 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 nonlinear 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 nonholonomic
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 popularstyle, casual and widely remembered article about his attempt to build an unrideable (nohands) 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 nohands 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 motorscooter
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
rearframe pitch as zero, with a constant force acting upward on the front
wheel. When
he linearizes his nonlinear 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  19711972
 In the only known study financed by a bicycle company
(Schwinn), Roland & Massing (1971)
and Roland & Lynch (1972) derive nonlinear
equations for a bicycle with radial and lateral tire deformation. The equations
of motion contains several algebraic and typographical errors. The sideslip
angle of the front wheel does not contain the steering rate angle and leads
to discrepancies. Therefore, after linearizing and taking the
nonslip 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 sideslip is eliminated, his equations agree with
our equations (see Hand 1988).
Weir1972Hand88.pdf, Weir1972JPM.txt  1973
 D. J. Eaton's Michigan PhD thesis concerns vehiclerider 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 "BoltzmannHamel"
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 nonlinear 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 peerreviewed 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 nonlinear
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 nonlinear 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 passivedynamics, 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
DifferentialAlgebraic 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 rigidbody 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 selfstable.
References

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

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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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Updated Thursday, September 10, 2009 11:13 by Arend L. Schwab.