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Proc. R. Soc. A (2007) 463, 1983–2003
doi:10.1098/rspa.2007.1849
Published online 5 June 2007
Hands-free circular motions of a
benchmark bicycle
B Y P RADIPTA B ASU -M ANDAL 1 , A NINDYA C HATTERJEE 1, *
2
AND J. M. P APADOPOULOS
1
Department of Mechanical Engineering, Indian Institute of Science,
Bangalore 560 012, India
2
2802 West Carrera Court, Green Bay, WI 54311, USA
We write nonlinear equations of motion for an idealized benchmark bicycle in two
different ways and verify their validity. We then present a complete description of handsfree circular motions. Three distinct families exist. (i) A handlebar-forward family,
starting from capsize bifurcation off straight-line motion and ending in unstable static
equilibrium, with the frame perfectly upright and the front wheel almost perpendicular.
(ii) A handlebar-reversed family, starting again from capsize bifurcation but ending with
the front wheel again steered straight, the bicycle spinning infinitely fast in small circles
while lying flat in the ground plane. (iii) Lastly, a family joining a similar flat spinning
motion (with handlebar forward), to a handlebar-reversed limit, circling in dynamic
balance at infinite speed, with the frame near upright and the front wheel almost
perpendicular; the transition between handlebar forward and reversed is through
moderate-speed circular pivoting, with the rear wheel not rotating and the bicycle
virtually upright. Small sections of two families are stable.
Keywords: bicycle dynamics; circular motions; stability
1. Introduction
The dynamics of idealized bicycles is of academic interest due to the complexities
involved in the behaviour of this seemingly simple machine, and is also useful as
a starting point for studies of more complex systems, such as motorcycles with
suspensions, flexibility and real tyres. In addition, reliable analyses of bicycles
can provide benchmarks for checking general multibody dynamics simulation
software. Meijaard et al. (2007) provide a detailed review of the scattered bicycle
dynamics literature from 1869 to the present, with over 80 references. They then
present multiply verified equations and analyses of near-straight hands-free
motions of two benchmark bicycles. Here we present a detailed study of the
circular hands-free motions of the first of these benchmark bicycles.
* Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.1849 or via
http://www.journals.royalsoc.ac.uk.
Received 29 November 2006
Accepted 30 March 2007
1983
This journal is q 2007 The Royal Society
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P. Basu-Mandal et al.
Of the many published analyses of a rigid-wheeled bicycle’s near-straight motions,
only a few are both general and correct (Meijaard et al. 2007). For circular motions,
the analytical literature is even smaller and less verifiably correct. The fully nonlinear
equations for bicycles are long, and their explicit form varies with the approach used
in deriving them. For example, the two independent (and numerically cross-verified)
sets of equations presented below differ greatly in length and defy full manual
comparison. We advocate the view, therefore, that numerical agreement with many
digits (say, 10 or more) between outputs (accelerations) from two sets of equations for
several sets of randomly generated inputs may be taken as a reliable demonstration of
equivalence, and we will use such checks below. In this light, we note that full
nonlinear equations and/or simulations are presented by, among others, Collins
(1963), Roland (1973), Psiaki (1979), Franke et al. (1990) and Lennartsson (1999),
but we are unable to comment on their correctness here beyond a qualitative
comparison between their results and ours. We also acknowledge the significant
amount of work done on motorcycles, using more complex and therefore less easily
verifiable models (e.g. Cossalter & Lot (2002) and Meijaard & Popov (2006), as well
as references therein).
The topic of hands-free circular motions might be initially misunderstood because
human riders can easily follow a wide range of circles at quite arbitrary speeds. But, in
general, this is possible only by imposing handlebar torques or upper-body
displacement from the symmetry plane. With zero handle torque (i.e. hands-free)
and a centred rider, only a few discrete lean angles are possible at each speed.
We mention for motivation that straight-line motions of most standard
bicycles, including the benchmark bicycle with its handlebar forward (HF) or
reversed (HR), have a finite range of stable speeds. A bifurcation known as
capsize occurs at the upper limit, where (by linear analysis) steady turns at all
large radii can be sustained with zero steering torque. These HF and HR
bifurcations are the origins of two distinct circular motion families.
Prior work on hands-free circular motions of bicycles and motorcycles is limited,
incomplete and occasionally arguable in terms of the conclusions presented.
Kane’s paper (Kane 1977) on steady turns of a motorcycle with front and rear point
masses, linearized in the steer angle but not in the lean, contains plots of steer angle
(to 128) versus steer torque (which we take as zero), parameterized by lean, steer and
turn radius. The zero-torque axis seems difficult to relate to our results below. Later,
Man & Kane (1979) gave a fully nonlinear treatment, also incorporating distributed
masses, tyre slip relations and the possibility of rider lean relative to the frame. The
results in figs. 5 and 6 of that paper (and associated text) seem to imply that any
desired turn radius can be achieved at a given speed, without altering the steer angle,
and indeed that turn radius at a given speed is independent of the steer angle. This
result, arguably lying outside normal bicycling experience, may be due to details of
tyre slip modelling. No evidence of bifurcation from straight motion or other handsfree turning is apparent.
Psiaki (1979) found one solution family (off point B in figure 5), calculated
eigenvalues and reported stability up to a lean of approximately 188. The
mechanical parameters of the bicycle were different, and speeds studied ranged
from the capsize speed to approximately 2% below it. Our corresponding
solutions below are unstable.
Franke et al. (1990) studied the nonlinear motions of a bicycle, modelled using
some point mass simplifications, without providing full details of mechanical
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Circular motions of a benchmark bicycle
1985
parameters, and allowing for lateral displacement of the rider. Their results (for
zero rider displacement) show only one solution family (corresponding to our
curve BA in figure 5), which like ours is unstable.
Cossalter et al. (1999) evaluated steady turns of a motorcycle with toroidal
wheels and various tyre parameters. In the upper left plot of their fig. 5 (rigid
tyres), a line of zero steer torque is plotted on axes of curvature and speed. It
seems equivalent to most of our curve BA, with a bifurcation of around 5 m sK1,
and with turn radius decreasing to approximately 5 m at a speed of 4 m sK1. No
other circular motion families are apparent.
Lennartsson (1999) presented a thoroughgoing analysis of circular motions of a
bicycle, with different parameters, where he finds all but one of the solution
families, which we will present later (our curve CDE in figure 5—even though he
separately determines the HR bifurcation that gives rise to it). For yet other
parameter values, Aström et al. (2005) again overlooked the same family.
In this context, precise numerical values of coordinates and velocities
corresponding to several circular motions of the benchmark bicycle will serve a
benchmarking purpose of their own. With this motivation, as also to present for the
first time a complete picture of the circular motions of at least one idealized bicycle,
we take up for study the primary benchmark bicycle of Meijaard et al. (2007).
We find, nominally, four different one-parameter families of circular motions of
the benchmark bicycle, most easily understood through the eight limiting points
they connect pairwise. Physically, two of these families merge into one, leaving
three in all. We also examine the stability of these families, and find only two
small intervals of stability (for the benchmark bicycle parameters).
Beyond the confirmed nonlinear equations, and some precise results for handsfree turns, the main contribution of this paper is to describe previously unknown
non-trivial limiting cases and the circular motion families that connect them.
2. Mechanical model, coordinates and notation
Our bicycle model is mechanically identical to that of Meijaard et al. (2007),
though our chosen coordinates and symbols differ on the surface, as enumerated
in the electronic supplementary material.
The bicycle model has four rigid bodies: a rear frame with rider rigidly
attached; a front frame (fork and handlebar assembly); and two wheels. These
are connected by frictionless hinges at the steering head and two wheel hubs. The
axisymmetric wheels make knife-edged dissipationless no-slip contact with the
horizontal ground. There is no propulsion. The parameter values used here also
have lateral symmetry, though our equations allow asymmetry.
We begin with nine generalized coordinates, not all independent. Three
coordinates x, y and z specify the position of the centre of the rear wheel in a
global reference frame XYZ (figure 1). The same point is attached to the rear
frame as well. Three angles q, j and f (in (3, 1, 3) Euler angle sequence as
described below) specify the rear frame configuration. The rear wheel rotation
relative to the rear frame is br. The front fork rotation (steering angle) relative
to the rear frame is jf. Finally, the front wheel rotation relative to the front
fork is bf.
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P. Basu-Mandal et al.
front fork and
handle-bar
front wheel
F
yf
X
rear wheel
m
bf
O
rear frame (3–1–3) Euler angles
(q−y−f)
Y
notation
br
F
t
wb
R
Figure 1. The bicycle in its reference configuration, flat on the ground. Unit vectors n^ P and n^ Q , at
points P and Q, point out of the page. S is a point on the fork axis. G and H are rear and front
frame centres of mass (not including wheels). Notation for vectors is shown boxed. When the
bicycle is in the reference configuration, the position vector from P to Q is denoted r Q=P;ref . At a
later arbitrary configuration, it is simply r Q=P .
There are six constraints on the bicycle. Of these, two are holonomic, requiring
the two wheels to touch the ground. Four are non-holonomic, expressing no-slip
at each wheel (two equations per wheel).
Note that we have introduced at least one unnecessary coordinate, namely z. It
is eliminated in Lagrange’s equations below using the holonomic constraint of
normal contact between the rear wheel and the ground, but retained in the
Newton–Euler equations. Another holonomic constraint, involving normal
contact between the front wheel and the ground, could in principle be used to
eliminate one more generalized coordinate (such as the third Euler angle f), but
the contact condition is analytically complicated and this constraint is retained
in Lagrange’s equations below along with the four non-holonomic constraints of
rolling without slip.
We now describe how an instantaneous configuration of the bicycle is obtained,
using the generalized coordinate values, starting from the reference configuration. In
the non-standard reference configuration chosen here (figure 1), the bicycle is flat on
the ground. Its lateral symmetry plane coincides with the XY plane.
The rear frame is first rotated by q about e3 (the Z-direction). This determines
the eventual heading direction through a yawing motion. (In discussing rear
frame rotations, it may help to imagine a point like P being held fixed, though
the rotation matrix does not depend on it.) Next, the rear frame is rotated by j
about the body-fixed e1 axis (what was X before the first rotation). This
determines the lean of the bicycle through a rolling motion. Finally, the rear
frame is rotated by f about the body-fixed e3 axis (what was Z at the start). This
rotation (pitching motion) is not arbitrary: it will eventually be determined by
front wheel-to-ground contact. Holding the rear frame fixed, the rear wheel is
rotated by br, the front fork by jf and the front wheel relative to the front fork by
bf. After these rotations, the bicycle is translated so that the rear wheel centre is
at (x, y, z). Now the bicycle is in the instantaneous configuration. A straightrunning bicycle will have jZp/2 (initially putting the bicycle below ground) and
fZp (bringing the bicycle back up to point along the negative x -axis).
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Circular motions of a benchmark bicycle
1987
Note that z and f must ensure contact between the wheels and the ground (j
determines z and j and jf together determine f as discussed later). The bicycle thus
has seven independent configuration variables, though we use nine for convenience.
Finally, we describe here our notation for rotation matrices, in the form of products
of matrices corresponding to rotations about known axes. Given vectors a and b, the
cross product a!b is written using matrix components as S(a)b, where
8 9
2
3
0 Kaz ay
b
>
< x>
=
6 a
7
0 Kax 5 and b Z by :
SðaÞ Z 4 z
>
: >
;
Kay ax
0
bz
Let a body rotate about unit vector n^ through angle q. A vector r fixed in the body
then gets rotated to r 0 ZR(n, q)r, where the rotation matrix
Rðn; qÞ Z ½cos qI C ð1Kcos qÞnn T C sin qSðnÞ:
In particular, r may be the position vector from any point fixed in the body to any
other point fixed in the body, with neither necessarily lying on the axis of rotation.
3. Lagrange’s equations of motion
Finding Lagrange’s equations for the bicycle is routine, if tedious and errorprone. Details of the calculations outlined below may be found in Basu-Mandal
(in preparation) and in the electronic supplementary material for this paper.
First, the kinetic and potential energies of the system at an arbitrary configuration
are found. This is straightforward and not presented here to save space.
For the contact constraint equations, we equate the vector velocities of the
ground contact points to zero, giving six scalar equations, including four no-slip
conditions and two vertical direction equations, which are differentiated
holonomic constraints. Of the latter two, the one for the rear wheel is easily
integrated and is used in our Lagrangian formulation to eliminate z in terms of
lean j; the one for the front wheel is retained in differentiated form due to
analytical difficulties.
Thus, for Lagrange’s equations, we retain five velocity constraints.
We now have eight degrees of freedom (z being eliminated). Very long
equations of motion are found in the usual way (the MATLAB m-file is 3.5 MB in
size; for the MAPLE file, see electronic supplementary material).
4. Numerical solution of the nonlinear equations
We have eight second-order ordinary differential equations (ODEs; Lagrange’s
equations) and five velocity constraints which we can differentiate to get secondorder ODEs also.1 The original eight ODEs also have five Lagrange multipliers:
there are 13 unknowns and 13 equations.
1
More robust numerical simulation would use coordinate partitioning on velocities, where we use a
partial state vector, and a subset of velocities is used, in each time step, to find the other velocities
from the velocity constraint equations, ensuring sustained satisfaction thereof (electronic
supplementary material).
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P. Basu-Mandal et al.
k
Q
ground
F
Figure 2. The front wheel. The lowermost point on the wheel touches the ground. The unit vector
^ say l^f , points out of the page, and r Q=F is then along l^f ! n^ Q .
along n^ Q ! k,
Figure 3. Eight ways in which the wheels of a vertical bicycle may, mathematically, touch the
ground. Only the first two can occur in our formulation. However, only the first is of interest. Here
we consider wheel contact configurations (i.e. whether the top or bottom point of a given wheel is
touching the ground, and whether the rider is above or below ground). The direction of overall
bicycle motion is relevant here, although for consistency all sketches suggest the bicycle is moving
to the right.
Of the 16 initial conditions needed to specify the initial-value problem, five
velocities are obtained from the five wheel constraint equations. Of the remaining
11 initial conditions (eight coordinates and three velocities), we can arbitrarily
specify only 10 because j(0) and jf(0) determine f(0). We choose to solve
_
_
_
_
for xð0Þ;
yð0Þ;
qð0Þ;
fð0Þ
and b_ f ð0Þ from the velocity constraint equations in
terms of the remaining, arbitrarily specified, initial conditions.
We now consider calculation and uniqueness of f(0). Figure 2 shows a view in
which the front wheel and the ground plane both reduce to lines. We have
r F Z r P C Rrf r S=P;ref C Rff r Q=S;ref C r F=Q ;
ð4:1Þ
where P, S and Q are as shown in figure 1; R rf and R ff are the rear frame and
front frame rotation matrices; and (for notation) r S=P;ref is the position vector
from P to S when the bicycle is in the reference position.
^ Q Þ, where r2 is the radius of the front wheel and
Also, r F=Q ZKr2 ðlf ! n
^
^
lf Z n^ Q ! k=jn^ Q ! kj. The z component of r F , equated to zero, yields an implicit
relation in f, j and jf only, free of velocities. Thus, f is determined by j and jf.
There are eight possible configurations in which the wheels could touch the
ground, as sketched in figure 3, with only one of them being of actual interest.
For a vertical, straight-running bicycle, the appropriate initial condition is
f(0)Zp and jZp/2 (figure 3, leftmost). This is because, in the reference
configuration (figure 1), both wheels touch the x -axis and the bicycle lies in the
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Circular motions of a benchmark bicycle
(a)
Faxle,rw
(b)
(c)
Faxle,fw
M
M1
S
H
G
P
Wrf
Wrw
Q
Wfw
R
Frw
R
Frw
M2
G
Ff
P
Wrw
(d )
F
Q
Wfw
F Wff
Ffw
P
Wrw
Wrf
R
Frw
Ffw
Figure 4. Free body diagrams. (a) rear wheel, (b) rear wheel and frame, (c) front wheel and (d )
entire bicycle.
negative y half-plane. Choosing f(0)Z0 here corresponds to figure 3, second from
left. Only these two are recognized by our formulation (lowermost point of the
wheel touches the ground).
In general (non-vertical, non-straight) initial choices of coordinates, there
again seems to be two possible choices of f(0) (the first two cases of figure 3).
Choosing the one closer to p usually gives the solution of interest, which we
verify post facto from animation (electronic supplementary material). We ignore
the cases where no solutions for f(0) exist.
With the above determination of f(0), we can now numerically integrate the
equations of motion. A nonlinear simulation of the benchmark bicycle is provided by
Meijaard et al. (2007). On simulating the bicycle using identical initial conditions, we
obtain a visual match with their results (see our electronic supplementary material
and fig. 4 of their paper). We will make more accurate comparisons in §5.
5. Newton–Euler equations
We now obtain the equations of motion using the Newton–Euler approach, which
is easily programmed into, say, MATLAB for a fully numerical evaluation of
the second derivatives needed for numerical integration. Though straightforward, the implementation is less routine than Lagrange’s equations and is
presented here fully.
(a ) Momentum balance
We begin with free body diagrams of the rear wheel, the rear wheel and the
rear frame, the front wheel and the entire bicycle (figure 4a–d ).
Consider the rear wheel (figure 4a). There are three forces acting on it: its
weight through P; an axle force F axle;rw also at P; and a force F rw at the ground
contact R. There is also an unknown bearing moment M with no component
along the bearing axis n^ P . Angular momentum balance about P gives
r R=P !F rw C M Z I cm;rw arw C urw !I cm;rw urw ;
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P. Basu-Mandal et al.
which in matrix notation is
SðrR=P ÞFrw C M Z Icm;rw arw C Sðurw ÞIcm;rw urw ;
ð5:1Þ
where S is a skew symmetric matrix and a is the angular acceleration (a vector).
We eliminate M by taking the dot product with n^ P , obtaining a scalar equation
(rearranged so that unknowns are on the left-hand side)
KnPT Icm;rw arw C nPT SðrR=P ÞFrw Z nPT Sðurw ÞIcm;rw urw :
ð5:2Þ
In equation (5.2), we could write KSðFrw ÞrR=P in place of SðrR=P ÞFrw , but
keeping the unknown Frw on the right-hand side eases assembly of the matrix
equations.
We now consider angular momentum balance for the rear wheel and the rear
frame, together, about the point S on the fork axis (figure 4b). Eliminating M1 by
taking the dot product with n^ f , we have the scalar equation
nT
f ðSðrR=S ÞFrw K Icm;rf arf K Icm;rw arw KSðrG=S Þm rf a G KSðrP=S Þmrw aP Þ
Z nT
f ðSðrP=S Þm rw ge3 C SðrG=S Þm rf ge3 C Sðurf ÞIcm;rf urf C Sðurw ÞIcm;rw urw Þ;
ð5:3Þ
where a is the acceleration (a vector). For clarity, we mention the roles of various
terms in equation (5.3). The first term on the left-hand side and the first two on
the right-hand side involve the form r!F (moment of a force). Each rigid body
contributes the terms I cm $aC u !I cm $u, of which the first appears on the lefthand side and the second on the right-hand side. The last two terms on the lefthand side are r!ma terms, one each for the rear frame and wheel.
Angular momentum balance for the front wheel about Q (figure 4c) gives
T
T
T
KnQ
Icm;fw afw C nQ
SðrF=Q ÞFfw Z nQ
Sðufw ÞIcm;fw ufw :
ð5:4Þ
Now consider the entire bicycle (figure 4d ). Linear momentum balance gives
Frw C Ffw K m rw aP K m rf aG K m ff aH K m fw aQ Z m tot ge3 ;
ð5:5Þ
where m tot is the total mass. Angular momentum balance about G gives
SðrR=G ÞFrw C SðrF=G ÞFfw KSðrP=G Þm rw aP K Icm;rw arw K Icm;rf arf
KSðrH=G Þm ff aH K Icm;ff aff KSðrQ=G Þm fw a Q K Icm;fw afw
Z SðrP=G Þm rw ge3 C SðrH=G Þm ff ge3 C SðrQ=G Þm fw ge3 C Sðurw ÞIcm;rw urw
C Sðurf ÞIcm;rf urf C Sðuff ÞIcm;ff uff C Sðufw ÞIcm;fw ufw :
ð5:6Þ
The two momentum balance equations for the entire bicycle have three scalar
components each. Thus, so far we have nine scalar equations.
(b ) Constraint equations
The constraints are the same as before, but now we will not eliminate z. That is, we
will retain six constraint equations. Similar to differentiation of five velocity
constraint equations above, we now directly differentiate two vector equations.
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Circular motions of a benchmark bicycle
To begin, we write (in figure 2, read ‘rear’ in place of ‘front’, P in place of Q
and R in place of F)
r P=R Z r1 ðlr ! n^ P Þ;
ð5:7Þ
where
lr Z
n^ P ! k^
;
^
jn^ P ! kj
KSðe3 ÞnP
ffi:
or lr Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nPT S T ðe3 ÞSðe3 ÞnP
The velocity of P can be found in two ways
v P h x_ ^i C y_ ^j C z_ k^ h vR C urw !r P=R ;
where vR is zero (no-slip contact). Differentiating and rearranging, we get
aP C SðrP=R Þarw Z r1 Sðurw ÞðSðl_ r ÞnP C Sðlr Þn_ P Þ;
ð5:8Þ
in matrix notation, where n_ P Z Sðurw ÞnP and l_ r is
KSðe3 Þn_ P
Sðe Þn n_ T S T ðe3 ÞSðe3 ÞnP
ffiC 3 P P
l_ r Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3=2 :
nPT S T ðe3 ÞSðe3 ÞnP
nPT S T ðe3 ÞSðe3 ÞnP
ð5:9Þ
Similarly, the front wheel constraint equation in matrix form is
aQ C SðrQ=F Þafw Z r2 Sðufw ÞðSðl_ f ÞnQ C Sðlf Þn_ Q Þ;
ð5:10Þ
with similar expressions for n_ Q , lf and l_ f . We do not reproduce them here. We
thus have six more scalar equations (equations (5.8) and (5.10)).
So far, we have the following unknown vectors: a P , a G , a H , a Q , arw , arf , aff ,
afw , F rw and F fw . Thus, we have 30 scalar unknowns and 15 equations so far.
More equations come from kinematic relations among the unknowns.
(c ) Further kinematic relations
We begin with
v G Z v P C urf !r G=P :
Differentiating, we get (in matrix notation)
a G K aP C SðrG=P Þarf Z Sðurf ÞSðurf ÞrG=P :
ð5:11Þ
Similarly, we can write relations between aH and aG and between aQ and aH as
aH K aG C SðrS=G Þarf C SðrH=S Þaff
Z Sðurf ÞSðurf ÞrS=G C Sðuff ÞSðuff ÞrH=S ;
ð5:12Þ
aQ K aH C SðrQ=H Þaff Z Sðuff ÞSðuff ÞrQ=H :
ð5:13Þ
With nine more scalar equations (equations (5.11)–(5.13)), we still need six.
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P. Basu-Mandal et al.
The angular accelerations of the four rigid bodies are interrelated. We have
urw Z urf C b_ r n^ P :
Differentiating with respect to time, we have (in matrix notation)
arw K arf Kb€r nP Z b_ r Sðurf ÞnP :
ð5:14Þ
Similarly, we can write
€ f n f Z j_ f Sðurf Þn f ;
aff K arf Kj
ð5:15Þ
afw K aff Kb€f n Q Z b_ f Sðuff ÞnQ :
ð5:16Þ
€ f and b€f , giving a total
Thus, we have introduced three more unknowns, b€r , j
of 33 unknowns. But we have added equations (5.14)–(5.16), all vector
equations. So we now have 33 independent simultaneous linear algebraic
equations as well.
A key step remains. We have arf , the angular acceleration of the rear frame.
€ j
€ and f.
€ For the rear frame, it can be shown that
But we still need q,
8 9
q_ >
>
>
< >
=
urf Z ½ e3 R1 e1 R2 R1 e3 j_ ;
>
>
>
: >
;
f_
_ 2 R1 e3 fC
_ 1 e1 j:
€ SðR1 e1 jC
_ Then arf Z AQC
_ e3 qÞR
_ Sðe3 qÞR
_
of the form uZ AQ.
€
The above can be used to solve for Q, once arf is known from Newton–Euler
equations.
We now have a numerical procedure for obtaining the second derivatives of
the system coordinates. The choice of initial conditions is subject to the same
constraints as for the Lagrange’s equations case, except that initial conditions for
z and z_ must also be given and must satisfy the system constraints.
(d ) Verification
For several arbitrary (random) choices of coordinates and their first
derivatives, the second derivatives obtained from the Lagrange and Newton–
Euler sets of equations matched to machine precision (table 1). We conclude that
our two sets of equations are equivalent. We will verify in §6 that our MAPLE
equations, linearized about straight motion and with numerical benchmark
parameters inserted, match the numbers of Meijaard et al. (2007).
6. Stability of straight motion
To study the stability of straight motion, we use our Lagrangian equations. Since
the equations are very long, we substituted the benchmark system parameters
before the linearization.
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Circular motions of a benchmark bicycle
Table 1. Comparison of the results obtained using Lagrange and Newton–Euler. (The initial
conditions are consistent with the constraints. The first and the second columns represent the
inputs, while the third and the fourth columns represent the outputs. These were verified
independently by Arend Schwab using SPACAR.)
input
output
q
q_
Lagrange (€
q)
Newton–Euler (€
q)
xZ0
yZ0
zZ0.2440472102925
qZ0
jZ0.9501292851472
fZ3.1257073014894
jfZ0.2311385135743
brZ0
bfZ0
K2.8069345714545
K0.1480982396001
0.1058778746261
0.7830033527065
0.6068425835418
K0.0119185528069
0.4859824687093
8.9129896614890
8.0133620584155
K0.5041626315047
K0.3449706619454
—
0.8353281706379
K7.8555281128244
0.1205543897884
K4.6198904039403
1.8472554144217
2.4548072904550
K0.5041626315047
K0.3449706619454
K1.4604528332980
0.8353281706379
K7.8555281128244
0.1205543897884
K4.6198904039403
1.8472554144217
2.4548072904550
Let e be infinitesimal, v be the nominal forward velocity of the bicycle and t be
time. We make the following substitutions into the equations of motion:
~
fq Z eq;
~
j Z p=2 C ej;
br Z vt=r1 C eb~r ;
~
f Z p C ef;
~f ;
jf Z e j
x ZKvt C e~
x;
y Z e~
y;
bf Z vt=r2 C eb~f g:
Setting eZ0 gives the solution corresponding to exactly straight-line motion.
The constraint forces (or ls) still need to be found, from a set of linear
equations. Barring l5Z309.30353., all are zero. Accordingly, we substitute
fl1 Z el~1 ;
l2 Z el~2 ;
l3 Z el~3 ;
l4 Z el~4 ;
l5 Z 309:30353 /Cel~5 g:
We now expand the equations of motion about eZ0 and drop terms of O(e2). The
O(e) terms give the linearized equations of motion (omitted for brevity).
Note that we do not differentiate the five velocity constraint equations. Instead, we
~_ q~_ and b~_ r , and substitute into Lagrange’s eight equations of
solve them for x~_ , y~_ , f,
motion, differentiating as needed (e.g. differentiating x~_ where x€~ is needed). We solve
€~ and j
€~ . Of these, we
~ b€~f , j
the resulting eight linear equations for the five ls,
f
€~ and j
€~ . Dropping tildes, they are
~f h 0. There remain two equations giving j
find b€
f
(we retained more decimal places than shown here)
€ ZK0:10552v jK0:33052v
_
j
j_ f C 9:48977jKð0:57152 C 0:89120v 2 Þjf ;
ð6:1Þ
€ f Z 3:67681v jK3:08487v
_
j
j_ f C 11:71948j C ð30:90875K1:97172v 2 Þjf :
ð6:2Þ
These equations match with those of Meijaard et al. (2007) completely and the
eigenvalues obtained from the two systems match to 14 decimal places. In particular,
straight motion of the HF bicycle is stable at speeds between 4.2924 and 6.0243 m sK1.
Further straight-motion stability results are therefore not presented here.
Proc. R. Soc. A (2007)
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1994
P. Basu-Mandal et al.
(b)
F
E
lean (positive means left)
upright, 0
JD
K
A
0
p /2
steer (positive means left)
(a)
flat,p /2
G,H
K
A
J
D
E,F
0
BC
front wheel speed
8
8
B C
GH
front wheel speed
Figure 5. Circular motion families (schematic). (a) lean and (b) steer.
7. Circular motions
(a ) Finding hands-free circular motions
In circular motions, x and y vary sinusoidally and z is constant. The rear wheel centre
traverses a circle of radius (say) R. The first Euler angle q (heading) grows linearly
with time. The second Euler angle j (roll), the third Euler angle f (pitch), the
steering rotation angle jf and the wheel spin rates b_ f and b_ r are all constant.
We seek a triple ðj; jf ; b_ r Þ subject to some conditions, dependent on a free
parameter R, as follows.
(i) Given j and jf, f is found as discussed earlier. (ii) j determines z (used in
our Newton–Euler equations) and z_ Z 0. (iii) The initial values of q, x, y, br and bf
_ j_ f Z 0, and with b_ r given (or
are arbitrarily taken as zero. (iv) Setting jZ
_ f_ and b_ f . (We find that
_ y,
_ q,
chosen), the velocity constraint equations give x,
_ 0). (v) Having all initial conditions required for the Newton–Euler equations,
fZ
we find the second derivatives of the coordinates. (vi) Finally, we define a vector
function with R as a parameter
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T
T
2
2
_
_
€
€
_
_
fj; jf ; br g 1
x C y KRq; j; j f :
(vii) We numerically find an R-parameterized family of zeros of the above map,
where steer and lean acceleration vanish.
As a preamble to what follows, note that the existence of one circular handsfree motion implies the existence of three others, by symmetry. First, one may
create a mirror image of the configuration, for example, leaning and steering
rightwards instead of leftwards. Second, the rotational velocities of both wheels
may be reversed, without affecting inertial forces and moments.
(b ) Plotting hands-free circular motions
The number of families of circular motion solutions, not counting symmetries, is
three or four depending on how we count. Consider, initially, the schematic in
figure 5a,b, depicting lean and steer, respectively, as a function of front wheel speed.
Proc. R. Soc. A (2007)
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Circular motions of a benchmark bicycle
1995
In figure 5a, lean is defined as jKp/2, positive when the bicycle leans left, and
shown here from 0 (upright) to p/2 (flat on the ground). In figure 5b, steer angle
is plotted from 0 (straight) to p/2 (perpendicular leftwards). The front wheel
speed (really, speed of the front contact point) is b_ f r2 , and is plotted from 0 to N
in a non-uniform scaling (b_ f is obtained from b_ r using the velocity constraint
equations).
For the two thick curves (one solid and another dashed), b_ f r2 ! 0 and Kb_ f r2 is
plotted instead. For these thick curves, b_ f r2 ! 0 corresponds to the bicycle
moving forward with a reversed handlebar (i.e. turned beyond p/2). In figure 5b,
for the thick curves corresponding to reversed handlebars, p has been added to
the steer jf.
For visualization, separation between nearly coincident portions of light and
thick curves is exaggerated. Points E and F coincide in reality, as do G and H.
Handlebar asymmetry plays a role in the solutions obtained. Turning the
handle by p (i.e. reversing the handlebar) effectively gives a slightly different
bicycle. If the front wheel centre was exactly on the fork axis, the front fork plus
handlebar centre of mass was exactly on the fork axis, and an eigenvector of the
front fork assembly’s central moment of inertia matrix coincided with the front
fork axis, then handlebar reversal would give exactly the same bicycle. The
reader might wish to consider the thick lines in figure 5 not as alternatively
plotted curves at all, but regularly plotted curves for a different bicycle whose
front assembly is a reversed version of the benchmark’s. Here, we avoid this
expedient in favour of consistency.
(c ) Limiting motions
We now discuss various special motions and limiting configurations which help
to understand the hands-free circular motions of the benchmark bicycle.
(i) HF and HR bifurcations to large-radius turns
By equations (6.1) and (6.2), four eigenvalues govern stability of straight
motion. For hands-free circular motions with very large radii, all sufficiently
small leans must give steady solutions, implying a zero eigenvalue. There is only
one such point with handlebar forward (HF): the ‘capsize’ bifurcation at the
upper limit of the stable speed range for straight motion (noted by Whipple
1899). With the handlebar reversed (HR), there is a similar stable speed range
with its own capsize point. These points are labelled B and C in figure 5. Near
these points, lean and steer (off 0 or Kp as appropriate) are both small and the
radius is large.
(ii) HF and HR flat spinning
There are limiting circular motions where the lean approaches p/2 (i.e. lying
flat on the ground), steer approaches zero, turn radius approaches a small finite
limit and velocity approaches infinity. (The contact points are substantially
displaced around each wheel.) Such solutions exist for both HF and HR
configurations, i.e. with steer approaching 0 and Kp. These points are labelled F
and E, respectively, in figure 5. These limits require infinite friction.
Proc. R. Soc. A (2007)
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1996
P. Basu-Mandal et al.
(iii) HF upright static equilibrium
With the handlebar turned almost p/2, there is a static equilibrium with a
lean angle of exactly zero, labelled A in figure 5. With a symmetric handlebar, it
is obvious that an upright equilibrium exists with exactly p/2 steer. But with the
left–right asymmetry due to finite steer of an asymmetric handlebar, it seems
surprising that the lean remains exactly zero, because equilibrium implies two
conditions (handlebar torque and net bicycle-tipping torque both zero) on the
one remaining variable (steer). To understand this, imagine locking the
handlebar at a variety of near-p/2 steer angles, at each of which the equilibrium
lean angle is determined. Select the locked steer angle that gives tipping
equilibrium with zero lean. Equilibrium ensures zero net moment on the bicycle
about the line (say L) of intersection between the rear frame symmetry plane and
the ground. Of all forces on the bicycle, the two not in the symmetry plane are
the front fork assembly weight and the front wheel ground contact force: they are
therefore in moment equilibrium about L. But these two forces are then in
moment equilibrium about the handle axis as well, and locking is not needed,
explaining the zero lean at A.
(iv) Pivoting about a fixed rear contact point
It seems possible, for each steer angle, to find an angle of lean such that the
normal to the front wheel rolling direction passes through the rear contact; in
such a configuration, the bicycle rotates about the rear contact, which remains
stationary. One imagines that by properly choosing both the steer angle and the
front wheel speed, we might simultaneously achieve roll and steer balance with
the rear contact at rest. Such a motion does exist: the steer is close to p/2
(handle turned left); the rear frame is nearly upright; the front wheel follows a
circle at a definite speed; and the bicycle pivots about a vertical axis through the
rear wheel contact. Such motions were found for both HF and HR configurations
(steer: Kp/2). Nearby points, defined for plotting convenience at exactly zero
lean, are labelled G and H, respectively, in figure 5. Noting that all motions of the
bicycle are time-reversible, a pivoting motion can be reversed to give another
motion where the front wheel speed and the handlebar are both reversed. Thus,
the HF and HR pivoting solutions actually coincide, as do G and H, although
they are sketched distinct for visualization.
(v) High-speed dynamic equilibrium
Envisioning that terminal points occur in pairs, an expected eighth is found as
K in figure 5. This configuration involves small lean, near-p/2 steer and speed
approaching infinity. It may be viewed as a perfectly dynamic counterpart to the
static solution at A. It seems that the normal ground force at one wheel must
become negative beyond some high speed for such a motion, but our analysis
assumes sustained contact and ignores this question.
(d ) Description of the circular motion families
We can now connect appropriate pairs of endpoints to describe four circular
motion families found for the benchmark bicycle.
Proc. R. Soc. A (2007)
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Circular motions of a benchmark bicycle
1997
One HF family connects points B and A. The bicycle first bifurcates from
HF straight motion, with steer and lean increasing while speed decreases,
until a maximum lean angle is reached. Thereafter, steer continues to increase
and velocity continues to decrease, while lean decreases towards upright. The
final perfectly upright state is approached via extremely slow motion,
superficially like the pivoting points GZH but with the rear contact not
quite fixed.
An HR family starts at C, bifurcating from HR straight motion. First, the
previous pattern is followed (attaining a maximum lean with continuously
decreasing speed), but then at a near-cusp point labelled D, a qualitatively
different curve is followed. The steer then decreases towards HR straightness,
while lean and speed increase, as the bicycle approaches the flat and fast limit
point E.
A third circular motion family, for continuity in the discussion, may be
thought of as starting from the HF flat and fast limit F (the path radii of the rear
wheel centre differ at F and E). Velocity and lean angle decrease, while steer
increasingly deviates until the rear frame is upright at pivoting motion G.
A fourth circular motion family starts with the identical pivoting motion at H
(now considered HR), with lean increasing/steer decreasing up to a near-cusp at
point J, and then reversing that trend to achieve a near upright lean and a nearly
perpendicular steer, as the speed goes to infinity at K. But since G is essentially
the same as H (except for an inconsequential speed reversal), the third and fourth
families are actually one (GZH could be removed from the list of terminal
points). This combined family—FGHJK—joins HF and HR configurations.
By this count, we have three circular motion families in all.
With this background, we consider qualitatively why thick curves ED and JH lie so
close to FG (figure 5). In figure 5a,b, we actually see the broken curve FGHJCDE
folded at GH. In essence, this says that starting in an upright condition with the steer
essentially p/2, an added leftward or rightward amount of steering leads to a bicycle
with HR and HF configuration, respectively; these two configurations may be viewed
as almost identical bicycles (due to ‘small’ handlebar asymmetry), and hence
dynamic equilibria obtained are almost identical as well. Without handlebar
asymmetry, the coincidence would be perfect.
In figure 5b, we have exaggerated the closeness of points G and H. In reality,
due to the handlebar asymmetry, they have a small vertical separation, with G
lying slightly above p/2 and H slightly below it. Actual numerical and graphical
results presented below have no such misrepresentations.
Note that the hands-free-motion plots can also provide qualitative information
about the sign of steer torque away from the plotted curves. For example,
consider the light (HF) curves in the steer plot of figure 5. Recalling that the
horizontal axis is also a line of zero steer torque, one can imagine increasing the
steer angle at a speed just below B (such as 5 m sK1 in figure 6). The torque will
become non-zero (negative, as it happens), attain a peak negative value, reduce
to zero as the BA curve is crossed, then increase to a peak positive value and then
drop again as the FG curve is approached. Thus, steer torque may vary
significantly in both sign and slope (i.e. ‘stiffness’) as one alters turn radius,
posing something of a control problem for the rider attempting to corner quickly
at lean angles up to p/4 and steer angles up to p/12.
Proc. R. Soc. A (2007)
1998
(b)
E
3.0
K
–2
0
2
5
C
2
20
A
D
J
2.0
1.5
–5
H
A
G
B
1.0
steer yf (positive = left)
–20
3
1
B
0
H
D
0
–1
0
1
arctangent (scaled front wheel speed, bf r2/4)
F
–1
K
–2
0.5
G
–3
E
J
C
–1
0
1
arctangent (scaled front wheel speed, bf r2/4)
Figure 6. Circular motion families (numerics). (a) lean and (b) steer. The infinite speed range is mapped to a finite range using an arctangent mapping.
(a) Actual speeds at some locations are indicated using vertical dotted lines and labels. Curves plotted do not reach points B, C, E, F and K, and
reflections thereof, because the numerical continuation calculation was stopped when the approach to these points was clear; in reality, they continue
all the way to the indicated points.
P. Basu-Mandal et al.
roll y (p/ 2 = upright)
2.5
F
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(a)
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Circular motions of a benchmark bicycle
1999
Table 2. Some initial conditions for circular motion. (These were verified independently by Arend
Schwab using SPACAR. These points are also plotted in the electronic supplementary material.
Stability-governing eigenvalues of these solutions are given in table 3.)
no. (family)
roll angle (j)
steer angle (jf)
rear wheel spin
rate ðb_ r ; s K1 Þ
1
2
3
4
5
6
7
1.9893886377
1.9178291654
1.7670024274
1.7183161276
2.1950752979
2.0419972895
2.3535106155
K3.0755121969
0.4049333918
0.7254537952
0.8549190153
0.4266815552
K2.6133787369
K2.8688460258
26.3580011755
10.3899258905
5.5494771350
4.2289953550
14.4337001146
10.9563251310
19.4180569764
(CDE)
(AB)
(AB)
(AB)
(GE)
(HJK)
(CDE)
radius traversed
by rear wheel
centre (R, m)
13.8724247186
2.2588798195
1.1408878065
0.8939154494
1.7525375246
1.4016100055
2.3503396652
(e ) Accurate plots, with four-way symmetry
As mentioned above, for each circular motion, another is obtained if all speeds
are reversed, and every left-leaning solution also implies a right-leaning one,
where (j, jf) are replaced by (p Kj, Kjf). The resulting four-way symmetry in
the solutions is represented (actual numerics) in figure 6, where the infinite
horizontal scale of b_ f r2 is mapped to a finite range using the arctangent of b_ f r2 =4 (4 is
an arbitrary scaling parameter chosen for better visualization).
All the curves represent numerically obtained solutions, while the labelled
thick dots indicate the correspondence with figure 5. Here, roll (j) has been
plotted from 0 to p, instead of lean from Kp/2 to p/2. Steer (jf) has been plotted
from Kp to p. As mentioned above, in figure 5, the thick lines actually show pCjf
against Kb_ f r2 ; in figure 6, we plot jf against b_ f r2 , obtaining a curve in the
third quadrant.
The steer curves provide another vantage on the earlier described nearsymmetry about p/2. The lean curves are harder to untangle, unless one reflects
points HJK through the origin and CDE through the vertical axis. Then a
reflected curve (say C 0 D 0 E 0 ) is visible in the first quadrant and (similarly primed)
K 0 J 0 H 0 GF appears in the first and fourth.
(f ) Some precise (benchmark) numerical values
The graphical results discussed above were presented, after some trial and
error, in terms of variables allowing simple post facto interpretation. Here, in
terms of our original variables, we report some precise numerical results for
benchmarking.
Table 2 lists some initial conditions for steady circular hands-free motions.
The radius R traversed by the rear wheel centre is also provided. These were
independently verified through simulations by Arend Schwab (using SPACAR).
In addition, we now list some special numerical values.
We have separately sought and found a static equilibrium of the bicycle at
jfZ1.3397399115 and jZp/2 (corresponding to point A in figures 5 and 6). The
corresponding rear wheel centre radius R is 0.2771720012 m.
Proc. R. Soc. A (2007)
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2000
P. Basu-Mandal et al.
Point B in figures 5 and 6 corresponds to a straight-ahead capsize speed of
6.0243 m sK1. Point C corresponds to a straight-ahead (HR) capsize speed of
7.9008 m sK1. In linearized analysis of straight riding, capsize occurs at that
speed where the handlebar ‘torsional stiffness’ vanishes, permitting any arbitrary
turn to be maintained with zero handlebar torque. This forward speed is the
unique solution of a linear equation in V 2.
Point E corresponds to an HR flat motion with Rz3.3049 m (from numerical
extrapolation). Point F corresponds to another flat motion (handlebar forward)
with Rz3.0087 m (numerical extrapolation). These configurations are hard to
evaluate precisely due to geometrical (contact) and mathematical (Euler angles)
singularities.
Point G (ZH) represents the solution jZp/2, jfZ1.6416430491, b_ r Z
0:2735815731 and RZK0.0415586589 (here R!0 because jfOp/2 and the bicycle
moves in a circle that curves right instead of left). Since this point is not a limiting
motion, its definition is somewhat arbitrary: rather than an upright frame, we might
instead specify minimum front wheel velocity or some other condition.
Point K can be found precisely by setting gravity to zero, choosing any nonzero speed, and seeking a unique circular motion. (This is asymptotically
equivalent to finite gravity and infinite speed.) We have found jZ1.6679684551,
jfZK1.6922153670 and RZ0.0666827859 m.
8. Stability of circular motions
For stability analysis of circular motions, the generalized coordinates x and y are
replaced by new ones defined by xZKR sin c and yZR cos c. R and c_ remain
constant during origin-centred circular motions. Also, two new configurationdependent unit vectors are introduced. These are
e^R ZKsin c^i C cos c ^j;
and e^c ZKcos c^iKsin c ^j;
which are in radial and circumferential directions in the plane of the ground.
Finally, the in-ground-plane vector constraint equations of no-slip are not retained
in terms of x and y components, but instead retained in terms of components along
e^R and e^c . This makes the constraint forces (hence Lagrange multipliers) constant
during the circular motions of interest. Lagrange’s equations are then obtained in
the usual way, for the new set of generalized coordinates.
In these new equations, we seek circular motions by noting that R, j, jf and f
_ u is a constant; cZ
_ u as well (i.e. not an extra unknown); b_ r
are constants; qZ
_
and bf are constants; and all the five Lagrange multipliers are constants as well.
Thus, there are 12 constants to be determined. Meanwhile, we have eight
equations of motion, five velocity constraint equations (including one that is
actually a differentiated holonomic constraint) and a holonomic constraint
equation to enforce front wheel contact with the ground. That is, we have 14
equations and 12 unknowns. The following lines of thought help to clarify
the situation.
The Lagrange multiplier (say l2) corresponding to the no-slip constraint at the
rear wheel, in the e^c direction, turns out to be zero; this is expected because there
is no propulsive thrust, and one of the equations of motion reduces to exactly
Proc. R. Soc. A (2007)
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Circular motions of a benchmark bicycle
2001
Table 3. Non-trivial eigenvalues governing linearized stability of some circular motions reported
earlier in table 2. (In no. 5, the instability is oscillatory and nos. 6 and 7 are stable.)
no.
R (m)
eigenvalues of the linearized equations of motion
1
2
3
4
5
6
7
13.872424719
2.258879819
1.140887806
0.893915449
1.752537525
1.401610005
2.350339665
0.038127379
1.989869132
3.091516610
3.393903081
K2.000953101
K8.659556236
K13.209338580
K21.152660576
K4.886076369
K2.853827876
K2.608053659
K7.982680274
K0.795208976
K0.467653580
K2.265960434G7.986013290i
K2.744979704G5.459259375i
K2.485975489G5.783418042i
K2.342566567G5.945917170i
5.575539147G5.799303852i
K0.118995944G3.110982262i
K0.075503592G7.402547429i
l2Z0. We drop this equation, but retain l2 as an unknown and expect our
subsequent calculation to rediscover that l2Z0 (an automatic consistency
check). So we now have 13 equations and 12 unknowns.
We retain the holonomic (front wheel contact) constraint equation in our
calculations to ensure that a correct value for f is obtained. But this
automatically ensures that the velocity constraint equation in the normal
direction at the front wheel contact is identically satisfied, and so we drop that
equation. We then have 12 equations and 12 unknowns.
As may be anticipated, it turns out that the e^R - direction no-slip equation is
identically satisfied at the rear wheel, leaving 11 equations and 12 unknowns.
This suggests, in line with prior calculations, that there are one or more oneparameter solution families. As before, we choose R, and solve 11 equations and
11 unknowns (see electronic supplementary material for further discussion).
All quantities of interest (including the Lagrange multipliers) are now treated
as e-order time-varying perturbations of the nominal solutions corresponding to
circular motion; the equations of motion (including velocity constraint
equations) are linearized in terms of e. The O(e) equations obtained from the
velocity constraint equations are differentiated to get a full second-order system.
These are solved for the (perturbations in) Lagrange multipliers and second
derivatives of generalized coordinates; and a constant coefficient system is
obtained in terms of the eight degrees of freedom used in our formulation. We
then obtain a non-minimal set of 16 eigenvalues. Of these, 10 are zero (see
electronic supplementary material for discussion). Of the remaining six non-zero
eigenvalues, two are found to be exactly Giu (where u is already known for the
circular motion). These two eigenvalues merely represent the same circular
motion shifted to a nearby circle. There remain four non-trivial eigenvalues,
which are tabulated for the motions reported in table 3. These accurate
eigenvalues can serve a benchmarking purpose. They are consistent to three or
four decimal places with correspondingly (in)accurate eigenvalues found
numerically using finite differences from the Newton–Euler equations (electronic
supplementary material). The latter, being quicker, were used to check the
stability of the circular motions obtained above.
All of the circular motions of the benchmark bicycle with straight (forward)
handlebar turn out to be unstable. Of the reversed-handlebar motions, relatively
few are stable. These are shown in figure 7 (recognizable as the second-quadrant
Proc. R. Soc. A (2007)
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2002
P. Basu-Mandal et al.
3.0
roll y (p /2 = upright )
2.8
2.6
2.4
2.2
2.0
1.8
1.6
–1.5 –1.4 –1.3 –1.2 –1.1 –1 –0.9 –0.8 –0.7 –0.6 –0.5
arctangent (front wheel speed/4)
Figure 7. Stable hands-free circular motions of the benchmark bicycle.
representation of points D and J from figure 6), by means of individual thick dots
corresponding to our discrete sampling of the underlying continuous curves. We
avoid here the sign change on speed used in figure 5; so the two curves are
reflected versions of CDE and HJK of figure 5a.
9. Conclusions
In this paper we have, first, obtained two independent sets of fully nonlinear
equations of motion for a bicycle. Of these two, the first (Lagrange/MAPLE)
allows analytical linearization and is used to numerically cross-check with
Meijaard et al. (2007). The second set (Newton–Euler/MATLAB) is good for rapid
simulation.
We have studied circular motions of a benchmark bicycle, obtaining
mathematically four (physically, three) different one-parameter families of
such solutions. Barring Lennartsson (1999) and Aström et al. (2005), each
missing one solution family, no other study of circular motions has reliably
reported these multiple solution families. We have described the solution families
obtained in terms of their endpoints in the plotting plane. These endpoints have
been intuitively interpreted and described. Precise numerical values for some
motions have been provided for benchmark purposes. A stability analysis has
also been carried out of the circular motions, and precise eigenvalues reported for
some chosen points. Most of the circular motions obtained turn out to be
unstable for the benchmark bicycle, though this may not remain the case for
other reasonable designs.
Arend Schwab and Andy Ruina read drafts of the paper and provided useful technical and
editorial comments. Arend Schwab also verified several of our numerical results and helped locate
some errors.
Proc. R. Soc. A (2007)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 15, 2017
Circular motions of a benchmark bicycle
2003
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