Bicycles, Motorcycles, and Models - Spiral

Bicycles, Motorcycles, and Models - Spiral
Bicycles, Motorcycles,
and Models
he potential of human-powered transportation was
recognized over 300 years ago. Human-propelled
vehicles, in contrast with those that utilized wind
power, horse power, or steam power, could run on
that most readily available of all resources:
willpower. The first step beyond four-wheeled horse-drawn
vehicles was to make one axle cranked and to allow the
rider to drive the axle either directly or through a system of
cranks and levers. These vehicular contraptions [1], [2] were
» OCTOBER 2006
so cumbersome that the next generation of machines was
fundamentally different and based on only two wheels.
The first such development came in 1817 when the
German inventor Baron Karl von Drais, inspired by the
idea of skating without ice, invented the running machine,
or draisine [3]. On 12 January 1818, von Drais received his
first patent from the state of Baden; a French patent was
awarded a month later. The draisine shown in Figure 1
features a small stuffed rest, on which the rider’s arms are
laid, to maintain his or her balance. The front wheel was caught their legs when cornering. As a result, machines
steerable. To popularize his machine, von Drais traveled with centrally hinged frames and rear-steering were tested
to France in October 1818, where a local newspaper but with little success [4].
Speed soon became an obsespraised his skillful handling of the
sion, and the velocipede sufdraisine as well as the grace and
fered from its bulk, its harsh
speed with which it descended a
ride, and a poor gear ratio to the
hill. The reporter also noted that
driven wheel. In 1870, the first
the baron’s legs had “plenty to
do” when he tried to mount his
appeared. The “ordinary” or
vehicle on muddy ground. Despite
penny farthing had its pedals
a mixed reception, the draisine
attached directly to a large front
enjoyed a short period of Eurowheel,
pean popularity. In late 1818, the
improved gearing (see Figure 3).
draisine moved to England, where
Indeed, custom front wheels
Denis Johnson improved its
were available that were as
design and began manufacturing
large as one’s leg length would
the hobby horse. Despite the puballow. Solid rubber tires and the
lic’s enduring desire for rider-prolong spokes of the large front
pelled transportation, the draisine
wheel provided a smoother ride
was too flawed to survive as a
than its predecessors. This
viable contender; basic impedimachine, which was the first to
ments were the absence of drive
be called a “bicycle,” was the
and braking capabilities.
world’s first single-track vehicle
Although the history of the
to employ the center-steering
invention of the pedal-drive bicycle
is riven with controversy [2], tradi- FIGURE 1 The draisine, or running, machine. This head that is still in use today.
vehicle, which was first built in Germany in 1816, is
tional credit for introducing the early in a long line of inventions leading to the con- These bicycles enjoyed great
first pedal-driven two wheeler, in temporary bicycle. (Reproduced with permission of popularity among young men of
means during their hey-day in
approximately 1840, goes to the the Bicycle Museum of America, New Bremen, Ohio.)
the 1880s. Thanks to its
Scotsman Kirkpatrick Macmillan
adjustable crank and several
[1]. Another account has it that
other new mechanisms, the
pedals were introduced in 1861 by
penny farthing racked up
the French coach builder Pierre
record speeds of about 7 m/s.
Michaux when a customer brought
As is often said, pride comes
a draisine into his shop for repairs
before a fall. The high center of
and Michaux instructed his son
gravity and forward position of
Ernest to affix pedals to the broken
the rider made the penny fardraisine. In September 1894, a
thing difficult to mount and dismemorial was dedicated in honor
mount as well as dynamically
of the Michaux machine. Shown in
challenging to ride. In the event
Figure 2, this vehicle weighed an
unwieldy 60 pounds and was FIGURE 2 Velocipede by Pierre Michaux et Cie of that the front wheel hit a stone
Paris, France circa 1869. In the wake of the draisine,
known as the velocipede, or bone the next major development in bicycle design was the or rut in the road, the entire
shaker. This nickname derives velocipede, which was developed in France and machine rotated forward about
from the fact that the velocipede’s achieved its greatest popularity in the late 1860s. The its front axle, and the rider, with
construction, in combination with velocipede marks the beginning of a continuous line of his legs trapped under the hanthe cobblestone roads of the day, developments leading to the modern bicycle. Its most dlebars, was dropped unceresignificant improvement over the draisine was the
made for an extremely uncomfort- addition of cranks and pedals to the front wheel. Dif- moniously on his head. Thus
able ride. Although velocipedoma- ferent types of (not very effective) braking mecha- the term “taking a header”
nia only lasted about three years nisms were used, depending on the manufacturer. In came into being.
Another important invention
(1868–1870), the popularity of the the case of the velocipede shown, the small spoon
machine is evidenced by the large brake on the rear wheel is connected to the handlebar was the pneumatic tire introand is engaged by a simple twisting motion. The
number of surviving examples. A wheels are wooden wagon wheels with steel tires. duced by John Boyd Dunlop in
common complaint among veloci- (Reproduced with permission of the Canada Science 1899. The new tires substantially
improved the cushioning of the
pedists was that the front wheel and Technology Museum, Ottawa, Canada.)
ride and the achievable top speed. Dunlop sold the marketing rights to his pneumatic tire to the Irish financier Harvey
Du Cros, and together they launched the Pneumatic Tyre
Company, which supplied inflatable tires to the British
bicycle industry. To make their tires less puncture prone,
they introduced a stout canvas lining to the inner surface of
the tire carcass while thickening the inner tube [2].
A myriad of other inventions and developments have
made the bicycle what it is today. For bicycles using wheels
of equal size, key innovations include chain and sprocket
drive systems, lightweight stiff steel frames, caliper brakes,
sprung seats, front and rear suspension systems, free-running
drive hubs, and multispeed Derailleur gear trains [1], [5].
A comprehensive and scholarly account of the history
of the bicycle can be found in [2]. Archibald Sharp’s book
FIGURE 3 Penny farthing, or ordinary. This bicycle is believed to
have been manufactured by Thos Humber of Beeston, Nottinghamshire, England, circa 1882. The braking limitations of this vehicle’s layout are obvious! (Reproduced with permission of the Glynn
Stockdale Collection, Knutsford, England.)
FIGURE 4 Sylvester Roper steam motorcycle. This vehicle is powered by a two-cylinder steam engine that uses connecting rods fixed
directly to the rear wheel. (Reproduced with permission of the
Smithsonian Museum, Washington, D.C.)
[1] gives a detailed account of the early history of the bicycle and a thorough account of bicycle design as it was
understood in the 19th century. Archibald Sharp was an
instructor in engineering design at the Central Technical
College of South Kensington (now Imperial College).
Although Sharp’s dynamical analysis of the bicycle is only
at a high school physics course level, it is sure footed and
of real interest to the professional engineer who aspires to
a proper appreciation of bicycle dynamics and design.
If one considers a wooden frame with two wheels and a
steam engine a “motorcycle,” then the first one was probably
American. In 1867, Sylvester Howard Roper demonstrated a
motorcycle (Figure 4) at fairs and circuses in the eastern
United States. His machine was powered by a charcoal-fired,
two-cylinder engine, whose connecting rods drove a crank
on the rear wheel. The chassis of the Roper steam velocipede
was based on the bone-shaker bicycle.
Gottlieb Daimler is considered by many to be the inventor of the first true motorcycle, or motor bicycle, since his
machine was the first to employ an internal combustion
engine. After training as a gunsmith, Daimler became an
engineer and worked in Britain, France, and Belgium before
being appointed technical director of the gasoline engine
company founded by Nikolaus Otto. After a dispute with
Otto in 1882, Daimler and Wilhelm Maybach set up their
own company. Daimler and Maybach concentrated on producing the first lightweight, high-speed gasoline-fueled
engine. They eventually developed an engine with a surfacemounted carburetor that vaporized the petrol and mixed it
with air; this Otto-cycle engine produced a fraction of a kilowatt. In 1885 Daimler and Maybach combined a Daimler
engine with a bicycle, creating a machine with iron-banded,
wooden-spoked front and rear wheels as well as a pair of
smaller spring-loaded outrigger wheels (see Figure 5).
The first successful production motorcycle was the
Hildebrand and Wolfmueller, which was patented in
Munich in 1894 (see Figure 6). The engine of this vehicle
was a 1,428-cc water-cooled, four-stroke parallel twin,
which was mounted low on the frame with cylinders in a
fore-and-aft configuration; this machine produced less than
2 kW and had a top speed of approximately 10 m/s. As
with the Roper steamer, the engine’s connecting rods were
coupled directly to a crank on the rear axle. The Hildebrand
and Wolfmueller, which was manufactured in France under
the name Petrolette, remained in production until 1897.
Albert Marquis de Dion and his engineering partner
Georges Bouton began producing self-propelled steam
vehicles in 1882. A patent for a single-cylinder gasoline
engine was filed in 1890, and production began five years
later. The De Dion Bouton engine, which was a small,
lightweight, high-rpm four-stroke “single,” used batteryand-coil ignition, thereby doing away with the troublesome hot-tube ignition system. The engine had a bore of 50
mm and a stroke of 70 mm, giving rise to a swept volume
of 138 cc. De Dion Bouton also used this fractional kilowatt
engine, which was widely copied by others including the
Indian and Harley-Davidson companies in the United
States, in road-going tricycles. The De Dion Bouton engine
is arguably the forerunner of all motorcycle engines.
Testosterone being what it is, the first motorcycle race
probably occurred when two motorcyclists came across
each other while out for a spin. From that moment on, the
eternal question in motorcycling circles became: “How do I
make my machine faster?” As one would imagine, the
quest for speed has many dimensions, and it would take
us too far afield to try to analyze these issues in detail. In
the context of modeling and control, it is apparent that the
desire for increased speed as well as the quest to more
fully utilize machine capability, requires high-fidelity
models, control theory, and formal dynamic analysis. One
also needs to replace the fractional kilowatt Otto-cycle
engine used by Daimler with a much more powerful one.
Indeed, modern high-performance two- and four-stroke
motorcycle engines can rotate at almost 20,000 rpm and
produce over 150 kW. In combination with advanced
materials, modern tires, sophisticated suspension systems,
stiff and light frames, and the latest in brakes, fuels, and
lubricants, these powerful engines have led to Grand Prix
machines with straight-line speeds of approximately 100
m/s. Figure 7 shows Ducati’s Desmosedici GP5 racing
motorcycle currently raced by Loris Capirossi.
The parameters and geometric layout that characterize
the dynamic behavior of modern motorcycles can vary
widely. Ducati’s Desmosedici racing machine has a steep
steering axis and a short wheelbase. These features produce the fast steering and the agile maneuvering required
for racing. The chopper motorcycle, such as the one
shown in Figure 8, is at the other extreme, having a heavily raked steering axis and a long wheelbase. “Chopped”
machines are not just aesthetically different; they also
have distinctive handling properties that are typified by a
very stable feel at high straight-line speeds as compared
with more conventional machine geometries. However,
as with many other modifications, this stable feel is
accompanied by less attractive dynamic features such as
a heavy feel to the front end and poor responsiveness at
slow speeds and in corners.
Web sites and virtual museums dedicated to bicycles
and motorcycles are ubiquitous. See, for example, [6]–[10]
for bicycles and [11]–[15] for motorcycles.
FIGURE 5 Daimler petrol-powered motorcycle. Gottlieb Daimler, who
later teamed up with Karl Benz to form the Daimler-Benz Corporation, is credited with building the first motorcycle in 1885. (Reproduced with permission of DaimlerChrysler AG, Stuttgart, Germany.)
FIGURE 6 Hildebrand and Wolfmueller motorcycle. This machine,
patented in 1894, was the first successful production motorcycle.
(Reproduced with permission of the Deutsches Zweirad- und NSUMuseum, Neckarsulm, Germany.)
From a mathematical modeling perspective, single-track
vehicles are multibody systems; these vehicles include bicycles, motorcycles, and motor scooters, all of which have
broadly similar dynamic properties. One of the earliest
FIGURE 7 Loris Capirossi riding the Ducati Desmosedici GP5.
Ducati Corse’s MotoGP racing motorcycle is powered by a V-4 fourstroke 989-cc engine. The vehicle has a maximum output power of
approximately 161 kW at 16,000 rpm. The corresponding top speed
is in excess of 90 m/s. (Reproduced with permission of Ducati
Corse, Bologna, Italy.)
FIGURE 8 “Manhattan” designed and built by Vic Jefford of Destiny Cycles. Manhattan received the Best in Show award at the
2005 Bulldog Bash held at the Shakespeare County Raceway,
Warwickshire, England. Choppers, such as the one featured, are
motorcycles that have been radically customized to meet a particular taste. The name chopper came into being after the Second World War when returning GIs bought up war surplus
motorcycles and literally chopped off the components they did
not want. According to the taste and purse of the owner, high
handle bars, stretched and heavily raked front forks, aftermarket
exhaust pipes, and chrome components are added. Custom-built
choppers have extreme steering-geometric features that have a
significant impact on the machine’s handling properties. These
features include a low head angle, long forks, a long trail, and a
long wheelbase. The extreme steering geometry of Manhattan
includes a steering head angle of 56°! (Reproduced with the permission of Destiny Cycles, Kirkbymoorside, Yorkshire, England.)
attempts to analyze the dynamics of bicycles appeared in
1869 as a sequence of five short articles [16]. These papers
use arguments based on an heuristic inverted-pendulumtype model to study balancing, steering, and propulsion.
Although rear-wheel steering was also contemplated, it
was concluded that “A bicycle, then, with the steering
wheel behind, may possibly be balanced by a very skillful
rider as a feat of dexterity; but it is not suitable for ordinary
use in practice.” These papers are interesting from a historical perspective but are of little technical value today.
The first substantial contribution to the theoretical bicycle literature was Whipple’s seminal 1899 paper [17],
which is arguably as contributory as anything that followed it; see “Francis John Welsh Whipple.” This remarkable paper contains, for the first time, a set of nonlinear
differential equations that describe the general motion of a
bicycle and rider. The possibility of the rider applying a
steering torque input by using a torsional steering spring is
also considered. Since appropriate computing facilities
were not available at the time, Whipple’s general nonlinear
equations could not be solved and consequently were not
pursued beyond simply deriving and reporting them.
Instead, Whipple studied a set of linear differential equations that correspond to small motions about a straightrunning trim condition at a given constant speed.
Whipple’s model, which is essentially the model considered in the “Basic Bicycle Model” section, consists of
two frames—the rear frame and the front frame—which
are hinged together along an inclined steering-head assem-
bly. The front and rear wheels are attached to the front and
rear frames, respectively, and are free to rotate relative to
them. The rider is described as an inert mass that is rigidly
attached to the rear frame. The rear frame is free to roll and
translate in the ground plane. Each wheel is assumed to be
thin and thus touches the ground at a single ground-contact point. The wheels, which are also assumed to be nonslipping, are modeled by holonomic constraints in the
normal (vertical) direction and by nonholonomic constraints [18] in the longitudinal and lateral directions.
There is no aerodynamic drag representation, no frame
flexibility, and no suspension system; the rear frame is
assumed to move at a constant speed. Since Whipple’s linear straight-running model is fourth order, the corresponding characteristic polynomial is a quartic. The
stability implications associated with this equation are
deduced using the Routh criteria.
Concurrent with Whipple’s work, and apparently independently of it, Carvallo [19] derived the equations of
motion for a free-steering bicycle linearized around a
straight-running equilibrium condition. Klein and
Sommerfeld [20] also derived equations of motion for a
straight-running bicycle. Their slightly simplified model (as
compared with that of Whipple) lumps all of the frontwheel assembly mass into the front wheel. The main purpose of their study was to determine the effect of the
gyroscopic moment due to the front wheel on the
machine’s free-steering stability. While this moment does
indeed stabilize the free-steering bicycle over a range of
speeds, this effect is of only minor importance because the
rider can easily replace the stabilizing influence of the front
wheel’s gyroscopic precession with low-bandwidth rider
control action [21].
An early attempt to introduce side-slipping and forcegenerating tires into the bicycle literature appears in [22].
Other classical contributions to the theory of bicycle
dynamics include [23] and [24]. The last of these references, in its original 1967 version, appears to contain the
first analysis of the stability of the straight-running bicycle
fitted with pneumatic tires; several different tire models
are considered. Reviews of the bicycle literature from a
dynamic modeling perspective can be found in [25] and
[26]. The bicycle literature is comprehensively reviewed
from a control theory perspective in [27], which also
describes interesting bicycle-related experiments.
Some important and complementary applied work
has been conducted in the context of bicycle dynamics.
An attempt to build an unridable bicycle (URB) is
described in [21]. One of the URBs described had the
gyroscopic moment of the front wheel canceled by
another that was counterrotating. The cancellation of
the front wheel’s gyroscopic moment made little difference to the machine’s apparent stability and handling
qualities. It was also found that this riderless bicycle
was unstable, an outcome that had been predicted
theoretically in [20]. Three other URBs described in [21]
include various modifications to their steering geometry.
These modifications include changes in the front-wheel
radius and the magnitude and sign of the fork offset.
Experimental investigations of bicycle dynamics have
also been conducted in the context of teaching [28].
Francis John Welsh Whipple
rancis John Welsh Whipple (see Figure A) was born on 17
March 1876. He was educated at the Merchant Taylors’
School and was subsequently admitted to Trinity College, Cambridge, in 1894. His university career was brilliant, and he
received his B.A. degree in mathematics in 1897 as second
wrangler. (Wrangler is a term that refers to Cambridge honors
graduates receiving a first-class degree in the mathematics tripos; the senior wrangler is the first on the list of such graduates.) In 1898, he graduated in the first class in Part II of the
mathematics tripos. Whipple received his
M.A. degree in 1901 and an Sc.D. in 1929.
In 1899, he returned to the Merchant Taylors’ School as mathematics master, a
post he held until 1914. He then moved to
the Meteorological Office as superintendent of instruments.
Upon his death in 1768, Robert Smith,
master of Trinity College, Cambridge and
previously Plumian professor of astronomy, left a bequest establishing two annual
prizes for proficiency in mathematics and
natural philosophy to be awarded to junior
bachelors of arts. The prizes have been
awarded every year since, except for 1917
when there were no candidates. Throughout its existence, the competition has
played a significant role by enabling graduates considering an academic career, and the majority of prize
winners have gone on to become professional mathematicians
or physicists. In 1883, the Smith Prizes ceased to be awarded
through examination and were given instead for the best two
essays on a subject in mathematics or natural philosophy.
On 13 June 1899, the results of the Smith Prize competition
were announced in the Cambridge University Reporter [84].
Whipple did not win the prize, but it was written: “The adjudicators are of the opinion that the essay by F.J.W. Whipple, B.A.,
of Trinity College, ‘On the stability of motion of a bicycle,’ is
worthy of honorable mention.”
The main results of this essay depend on the work of another Cambridge mathematician, Edward John Routh, who
received his B.A. degree in mathematics from Cambridge in
1854. He was senior wrangler in the mathematical tripos examinations, while James Clerk Maxwell placed second. In 1854,
Maxwell and Routh shared the Smith Prize; George Gabriel
Stokes set the examination paper for the prize, which included
the first statement of Stokes’ theorem.
Figure B, which was generated directly from a quartic equation given in Whipple’s paper, shows the dynamic properties of a
forward- and reverse-running bicycle as a function of speed.
Whipple found the parameters by experiment on a particular
machine. It is surely the case that Whipple would have loved to
have seen this figure—derived from the remarkable work of a
young man of 23, working almost 100 years before the widespread availability of MATLAB!
FIGURE A Francis John Welsh Whipple by
Elliot and Fry. Francis Whipple was assistant
director of the Meteorological Office and
Superintendent of the Kew Observatory from
1925–1939. He served as president of the
Royal Meteorological Society from 1936–
1938. Apart from his seminal work on bicycle
dynamics, he made many other contributions to knowledge, including identities for
generalized hypergeometric functions, several of which have subsequently become
known as Whipple’s identities and transformations. He devised his meteorological slide
rule in 1927. He introduced a theory of the
hair hygrometer and analyzed phenomena
related to the great Siberian meteor. (Picture
reproduced with the permission of the
National Portrait Gallery, London.)
... (1/s) xxx (rad/s)
Auto-Stable Region
Speed (m/s)
FIGURE B Stability properties of the Whipple bicycle. Real and
imaginary parts of the eigenvalues of the straight-running Whipple bicycle model as functions of speed. Plot generated using
equation (XXVIII) in [17].
Point-Mass Models
Bicycles and motorcycles are now established as nonlinear
systems that are worthy of study by control theorists and
vehicle dynamicists alike. In most cases, control-theoretic
work is conducted using simple models, which are special
cases of the model introduced by Whipple [17]. An early
example of such a model can be found in [29] (see equations (e) and (j) on pages 240 and 241, respectively, of [29]).
These equations describe the dynamics of a point-mass
bicycle model of the type shown in Figure 9; [29] presents
both linear and nonlinear models. Another early example
of a simple nonholonomic bicycle study in a control systems context can be found in [30], which gives a servorelated interpretation of the self-steer phenomenon. A
more contemporary nonholonomic bicycle, which is essentially the same as that presented in [29], was introduced in
[31] and [32]. This model is studied in [32] and [33] in the
context of trajectory tracking. A model of this type is also
examined in [27] in the context of the performance limitations associated with nonminimum phase zeros.
The coordinates of the rear-wheel ground-contact point
of the inverted pendulum bicycle model illustrated in Figure 9 are given in an inertial reference frame O-xyz. The
Society of Automotive Engineers (SAE) sign convention is
used: x-forward, y-right, and z-down for axis systems and
a right-hand-rule for angular displacements. The roll angle
ϕ is around the x-axis, while the yaw angle ψ is around the
z-axis. The steer angle δ is measured between the front
frame and the rear frame.
The vehicle’s entire mass m is concentrated at its mass
center, which is located at a distance h above the ground
and distance b in front of the rear-wheel ground-contact
point. The acceleration due to gravity is denoted g, and w
is the wheelbase. The motion of the bicycle is assumed to
be constrained so that there is no side slipping of the vehicle’s tires and thus the rolling is nonholonomic. The kinematics of the planar motion are described by
ẋ = v cos ψ,
ẏ = v sin ψ,
v tan δ
ψ̇ =
w cos ϕ
where v is the forward speed.
The roll dynamics of the bicycle correspond to those of
an inverted pendulum with an acceleration influence
applied at the vehicle’s base and are given by
hϕ̈ = g sin ϕ− (1 − hσ sin ϕ)σ v2
cos ϕ,
+ b ψ̈ + v̇ σ −
where the vehicle’s velocity and yaw rate are linked by the
curvature σ satisfying vσ = ψ̇ . Using (3) to replace ψ̈ in (4)
by the steer angle yields
hϕ̈ =g sin ϕ − tan δ
+ tan ϕ
ϕ̇ − 2 tan δ
w cos2 δ
Equation (5) represents a simple nonholonomic bicycle
with the control inputs δ and v. The equation can be linearized about a constant-speed, straight-running condition
to obtain the simple small-perturbation linear model
Hϕδ (s) = −
FIGURE 9 Inverted pendulum bicycle model. Schematic diagram of
an elementary nonholonomic bicycle with steer δ, roll ϕ, and yaw ψ
degrees of freedom. The machine’s mass is located at a single
point h above the ground and b in front of the rear-wheel groundcontact point. The wheelbase is denoted w. Both wheels are
assumed to be massless and to make point contact with the ground.
Both ground-contact points remain stationary during maneuvering
as seen from the rear frame. The path curvature is σ (t) = 1/R(t).
δ̇ .
In the constant-speed case, the only input is the steer angle.
Taking Laplace transforms yields the single-input, single-output transfer function
(x, y)
ϕ̈ =
bv s + v/b
wh s2 − g/ h
which has the speed-dependent gain (−bv)/(wh), a speed
dependent zero at −v/b, and fixed poles at ± g/ h; the
unstable pole g/ h corresponds to an inverted-pendulumtype capsize mode. The zero −v/b, which is in the left-half
plane under forward-running conditions, moves through
the origin into the right-half plane as the speed is reduced
and then reversed in sign. Under backward-running
model has three degrees of freedom—the roll angle ϕ of
the rear frame, the steering angle δ, and the angle of
conditions, the right-half plane zero, which for some
speeds comes into close proximity to the right-half plane
pole, is associated with the control difficulties found in
rear-steering bicycles [34].
Basic Bicycle Model
We use AUTOSIM [35] models, which are derivatives of
that given in [26], to illustrate the important dynamic properties of the bicycle. As with Whipple’s model, the models
we consider here consist of two frames and two wheels.
Figure 10 shows the axis systems and geometric layout
of the bicycle model studied here. The bicycle’s rear frame
assembly has a rigidly attached rider and a rear wheel that
is free to rotate relative to the rear frame. The front frame,
which comprises the front fork and handlebar assembly,
has a front wheel that is free to rotate relative to the front
frame. The front and rear frames are attached using a
hinge that defines the steering axis. In the reference configuration, all four bodies are symmetric relative to the bicycle midplane. As with Whipple’s model, the nonslipping
road wheels are modeled by holonomic constraints in the
normal (vertical) direction and by nonholonomic constraints in the longitudinal and lateral directions. There is
no aerodynamic drag, no frame flexibility, no propulsion,
and no rider control. Under these assumptions, the bicycle
FIGURE 10 Basic bicycle model with its degrees of freedom. The
model comprises two frames pinned together along an inclined
steering head. The rider is included as part of the rear frame. Each
wheel is assumed to contact the road at a single point.
TABLE 1 Parameters of the benchmark bicycle. These parameters are used to populate the AUTOSIM model described in [26]
and its derivatives. The inertia matrices are referred to body-fixed axis systems that have their origins at the body’s mass
center. These body-fixed axes are aligned with the inertial reference frame 0 − xyz when the machine is in its nominal state.
Wheel base
Head angle
Forward speed
1.02 m
0.08 m
9.81 N/kg
variable m/s
Rear wheel (rw)
Mass moments of inertia
(Axx ,Ayy ,Azz)
0.3 m
2 kg
(0.06,0.12,0.06) kg-m2
Rear frame (rf)
Position center of mass
(xrf ,yrf ,zrf )
(0.3,0.0,-0.9) m
85 kg
9.2 0 2.4
11 0
Bx x
Mass moments of inertia
Bx z
Front frame (ff)
Position center of mass
(xff ,yff ,zff )
Cx x
Mass moments of inertia
Front wheel (fw)
Mass moments of inertia
Cx z
Rf w
mf w
(Dxx ,D yy ,Dzz)
(0.9,0.0,-0.7) m
4 kg
0.35 m
3 kg
(0.14,0.28,0.14) kg-m2
rotation θr of the rear wheel relative to the rear frame. The
steering angle δ represents the rotation of the front frame
with respect to the rear frame about the steering axis.
The dimensions and mechanical properties of the
benchmark model are taken from [26] and presented in
Table 1. All inertia parameters use the relevant body-mass
centers as the origins for body-fixed axes. The axis directions are then chosen to align with the inertial O-xyz axes
when the bicycle is in its nominal state, as shown in Figure
10. Products of inertia Axz , Bxz and so on are defined as
m(x, z)xzdxdz.
As derived in [17] and explained in [26], the linearized
equations of motion of the constant-speed, straight-running
nonholonomic bicycle, expressed in terms of the generalized coordinates q = (ϕ, δ)T , have the form
Mq̈ + vCq̇ + (v2 K2 + K0 )q = mext ,
where M is the mass matrix, the damping matrix C is multiplied by the forward speed v, and the stiffness matrix has
a constant part K0 and a part K2 that is multiplied by the
square of the forward speed. The right-hand side mext contains the externally applied moments. The first component
of mext is the roll moment mϕ that is applied to the rear
frame. The second component is the action-reaction steering moment mδ that is applied between the front frame
and the rear frame. This torque could be applied by the
rider or by a steering damper. In the uncontrolled bicycle,
both external moments are zero. This model, together with
nonslipping thin tires and the parameter values of Table 1,
constitute the basic bicycle model.
... (1/s) xxx (rad/s)
10 12
Speed (m/s)
FIGURE 11 Basic bicycle straight-running stability properties. The
real and imaginary parts of the eigenvalues of the straight-running
basic bicycle model are plotted as functions of speed. The (blue)
dotted lines correspond to the real part of the eigenvalues, while the
(red) crosses show the imaginary parts for the weave mode. The
weave mode eigenvalue stabilizes at vw = 4.3 m/s, while the capsize mode becomes unstable at vc = 6.1 m/s giving the interval of
auto-stability vc ≥ v ≥ vw .
To study (8) in the frequency domain, we introduce the
matrix-valued polynomial
P(s, v) = s2 M + svC + (v2 K2 + K0 ) ,
which is quadratic in both the forward speed v and in the
Laplace variable s. The associated dynamic equation is
P11 (s)
P21 (s, v)
P12 (s, v)
P22 (s, v)
mϕ (s)
mδ (s)
where P11 is independent of v. When studying stability,
the roots of the speed-dependent quartic equation
det(P(s, v)) = 0
need to be analyzed using the Routh criteria or found by
numerical methods. Figure 11 shows the loci of the roots of
(11) as functions of the forward speed. The basic bicycle
model has two important modes—the weave and capsize
modes. The weave mode begins at zero speed with the two
real, positive eigenvalues marked A and B in Figure 11. The
eigenvector components corresponding to the A-mode
eigenvalue have a steer-to-roll ratio of −37; the negative
sign means that as the bicycle rolls to the left, for instance,
the steering rotates to the right. This behavior shows that
the motion associated with the A mode is dominated by the
front frame diverging toward full lock as the machine rolls
over under gravity. Because real tires make distributed contact with the ground, a real bicycle cannot be expected to
behave in exact accordance with this prediction. The eigenvector components corresponding to the B-mode eigenvalue have a steer-to-roll ratio of −0.57. The associated motion
involves the rear frame toppling over, or capsizing, like an
unconstrained inverted pendulum to the left, for instance,
while the steering assembly rotates relative to the rear
frame to the right with 0.57 of the roll angle.
Note that the term “capsize” is used in two different
contexts. The static and very-low-speed capsizing of the
bicycle is associated with the point B in Figure 11 and the
associated nearby locus. The locus marked capsize in Figure 11 is associated with the higher-speed unstable toppling over of the machine. This mode crosses the stability
boundary and becomes unstable when the matrix
v2 K2 + K0 in (8) is singular.
As the machine speed builds up from zero, the two
unstable real modes combine at approximately 0.6 m/s to
produce the oscillatory fish-tailing weave mode. The
basic bicycle model predicts that the weave mode frequency is approximately proportional to speed above 0.6
m/s. In contrast, the capsize mode is a nonoscillatory
motion, which when unstable corresponds to the riderless bicycle slowly toppling over at speeds above 6.057
m/s. From the perspective of bicycle riders and designers, this mode is unimportant because it is easy for the
rider to stabilize it using a low-bandwidth steering control torque. In practice, the capsize mode can also be stabilized using appropriately phased rider body motions,
as is evident from hands-free riding.
In the recent measurement program [36], an instrumented bicycle was used to validate the basic bicycle model
described in [17] and [26]. The measurement data show
close agreement with the model in the 3–6 m/s speed
range; the weave mode frequency and damping agreement
is noteworthy. The transition of the weave mode from stable to unstable speed ranges is also accurately predicted by
the basic bicycle model. These measurements lend credibility to the idea that tire and frame compliance effects can be
neglected for benign maneuvering in the 0–6 m/s range.
Special Cases
Several special cases of the basic bicycle model are now used
to illustrate some of the key features of bicycle behavior.
These cases include the machine’s basic inverted-pendulumlike characteristics, as well as its complex steering and selfstabilizing features. Some of these features are the result of
carefully considered design compromises.
Point-Mass Model with Trail and Inclined Steering
Interesting connections can now be made between the
Timoshenko-Young-type point-mass model and the more
complex basic bicycle model. To forge these links, we set to
zero the masses of the wheels and the front frame, as well
as all the inertia terms in (10). The trail and steering inclination angle are left unaltered.
We first reconcile (7) and the first row of equation (10),
which is
ϕ(s) =
−P12 (s, v)
P11 (s, v)
when the roll torque is mϕ (s) = 0. As in [26], we denote the
trail by t and the steering inclination angle as measured
from the vertical by λ. Direct calculation gives
Hϕδ (s, v) =
(s, v)
cos(λ)(tbs2 + sv(b + t) + v2 − gtb/ h)
. (16)
wh(s2 − g/ h)
Equation (16) reduces to (7) when λ and t are set to zero. It
follows from (10) and mϕ = 0 that
Locked Steering Model
The dynamically simple locked steering case is considered
first. If the steering degree of freedom is removed, the
steering angle δ(s) must be set to zero in (10), and consequently the roll freedom is described by
Hϕmδ (s, v) =
(s, v) ,
which reduces to
mϕ (s) = P11 (s)ϕ(s)
= (s2 Txx + gmtzt)ϕ(s) .
The roots of P11 (s) are given by
p± = ±
where mt is the total mass of the bicycle and rider, zt is the
height of the combined mass center above the ground,
and Txx is the roll moment of inertia of the entire machine
around the wheelbase ground line. In the case of the basic
bicycle model, p± = ±3.1348 . For the point-mass,
Timoshenko-Young model, zt = h and Txx = mh2 and so
p± = ± g/ h.
Since the steering freedom is removed, the A mode (see
Figure 11) does not appear. The vehicle’s inability to steer
also means that the weave mode disappears. Instead, the
machine’s dynamics are fully determined by the speedindependent, whole-vehicle capsize (inverted pendulum)
mode seen at point B in Figure 11 and given by (13). Not
surprisingly, motorcycles have a tendency to capsize at
low speeds if the once-common friction pad steering
damper is tightened down far enough to lock the steering
system; see [37].
Hϕmδ (s, v) =
w(tbs2 + sv(t + b) + v2 − gtb/ h)
mtbg(s2 − g/ h)(hw sin(λ) − tb cos(λ))
under the present assumptions. In contrast to the analysis
given in [27], (18) shows that the poles of Hϕmδ (s, v) are
fixed at ± g/ h and that the steering inclination and trail
do not alone account for the self-stabilization phenomenon
in bicycles.
We now compute Hδmδ (s, v) as
Hδmδ (s, v) =
(s, v)
det P
mtbg cos(λ)(hw sin(λ) − tb cos(λ))
which is a constant. Equation (19) shows that in a pointmass specialization of the Whipple model, the steer angle δ
and the steering torque mδ are related by a virtual spring
whose stiffness depends on the trail and steering axis inclination. Physically, this static dependence means that the
steer angle of the point-mass bicycle responds instantaneously to steering torque inputs. It also follows from (19)
that this response is unbounded in the case of a zero-trail
(t = 0) machine [29] because in this case the connecting
spring has a stiffness of zero.
No Trail or Steering Inclination
... (1/s) xxx (rad/s)
Speed (m/s)
FIGURE 12 Bicycle straight-running stability properties. This plot
shows the real and imaginary parts of the eigenvalues of the
straight-running basic bicycle model with the gyroscopic moment
associated with the front road wheel removed by setting D yy = 0.
The (blue) dotted lines correspond to the real parts of the eigenvalues, while the (red) pluses show the imaginary parts for the
weave mode.
No Trail, Steering Inclination, or Front-Frame Mass Offset
We now remove the basic bicycle’s trail (by setting
t = 0), the inclination of the steering system (by setting
λ = 0), and the front-frame mass offset by setting
xff = w. This case is helpful in identifying some of the
key dynamical features of the steering process. The first
row in (10) relates the roll angle to the steer angle when
mϕ = 0, and shows how the inverted pendulum system
is forced by the steer angle together with δ̇ and δ̈. The
second row of (10) is
(s2 Cxz − sfw D yy)ϕ(s) + {s2 (Czz + Dzz )
+s(Czz + Dzz )v/w)}δ(s) = mδ (s),
where fw (s) is the angular velocity of the front wheel.
The ϕ(s) term in (20), which is the self-steering term,
shows how the roll angle influences the steer angle.
The first component of the self-steering expression is a
product of inertia, which generates a steering moment
from the roll acceleration. The second self-steering
term represents a gyroscopic steering moment generated by the roll rate. The expression for P22 in (20) relates
the steering torque to the steering angle through the
steered system inertia and a physically obscure speedproportionate damper, apparently coming from the
rear-wheel ground-contact model.
We now modify the previous special case by including
front-frame mass offset effects (xff = w). As before, the first
row of (10), which relates the roll angle to the steering
angle when mϕ = 0, represents steer angle forcing of the
inverted pendulum dynamics. The second row of (10) in
this case is shown in (21), found at the bottom of the page.
The quadratic self-steering term in (21) contains a new
term involving xff − w that comes from the fact that the
front-frame mass is no longer on the steering axis, implying an increase in the effective xz-plane product of inertia
of the front frame. The constant self-steering term in (21)
represents a mass-offset-related gravitational moment,
which is proportional to the roll angle. The steering mass
offset also increases the moment of inertia of the steering
system, enhances the steering damping, and introduces a
new speed-dependent stiffness term.
By comparing (20) and (21), it is suggested that the
bicycle equations become too complicated to express in
terms of the original data set when trail and steering inclination influences are included. Indeed, when these elaborations are introduced, it is necessary to resort to the use of
intermediate variables and numerical analysis procedures
[26]. In the case of state-of-the-art motorcycle models, the
equations of motion are so complex that they can only be
realistically derived and checked using computer-assisted
multibody modeling tools.
Gyroscopic Effects
Gyroscopic precession is a favorite topic of conversation in
bar-room discussions among motorcyclists. While it is not
surprising that lay people have difficulty understanding
these effects, inconsistencies also appear in the technical literature on single-track vehicle behavior. The experimental
evidence is a good place to begin the process of understanding gyroscopic influences. Experimental bicycles
whose gyroscopic influences are canceled through the
inclusion of counterrotating wheels have been designed
and built [21]. Other machines have had their gyroscopic
influences exaggerated through the use of a high-momentof-inertia front wheel [27]. In these cases, the bicycles were
found to be easily ridable. As with the stabilization of the
capsize mode by the rider, the precession-canceled bicycle
appears to represent little more than a simple low-bandwidth challenge to the rider. As noted in [21], in connection
with his precession-canceled bicycle, “. . . Its ‘feel’ was a bit
strange, a fact I attributed to the increased moment of inertia about the front forks, but it did not tax my (average) riding skill even at low speed . . . ”. It is also noted in [21] that
s2 (Cxz + mff zff (w − xff )) − sfw D yy + gmff (w − xff ) ϕ(s)+
s2 (mff (w − xff )2 + Czz + Dzz ) + sv(Czz + Dzz − mff xff (w − xff ))/w + v2 mff (w − xff )/w δ(s) = mδ (s).
the precession-canceled bicycle has no autostable speed range,
thereby verifying by experiment the findings reported in [20].
When trying to ride this particular bicycle without hands,
however, the rider could only just keep it upright because
the vehicle seemed to lack balance and responsiveness.
In their theoretical work, Klein and Sommerfeld [20]
studied a Whipple-like quartic characteristic equation
using the Routh criteria. While the basic bicycle model has
a stable range of speeds, which Klein and Sommerfeld
called the interval of autostability, this model with the
spin inertia of the front wheel set to zero is unstable up to
a speed of 16.4 m/s. This degraded stability can be seen in
Figure 12, where the capsize mode remains stable with the
damping increasing with speed; due to its stability, the
capsize nomenclature may seem inappropriate in this
case. In contrast, the weave mode is unstable for speeds
below 16.4 m/s, and the imaginary part is never greater
than 1.8 rad/s. Klein and Sommerfeld attribute the stabilizing effect of front-wheel precession to a self steering
effect; as soon as a bicycle with spinning wheels begins to
roll, the resulting gyroscopic moment due to the sfw D yy
term in (20) causes the bicycle to steer in the direction of
the fall. The front contact point, consequently, rolls
towards a position below the mass center.
The Klein and Sommerfeld finding might leave the
impression that gyroscopic effects are essential to auto-stabilization. However, it is shown in [38] that bicycles without
trail or gyroscopic effects can autostabilize at modest speeds
by adopting extreme mass distributions, but the design
choices necessary do not make for a practical machine.
tmb cos λ(v2 cos λ − gw sin λ)
k2 (v) = 2
v cos λ − wg sin λ
k1 (v) =
Basic Bicycle as a Feedback System
Although this stiffness-only model represents the lowfrequency behavior of the steering system, the approximation obscures some of the basic bicycle model’s structure.
The poles and zeros of Hδmδ (s, v), as a function of speed,
are shown in Figure 14. Except for the pair of speed-independent zeros, this diagram contains the same information
To study the control issues associated with bicycles, we
use the second row of (10) to solve for δ(s), which yields
In [27], (22) is simplified to
δ(s) = k1 (v)mδ (s) + k2 (v)ϕ(s) ,
Equations (22) and (14) are shown diagrammatically in the
feedback configuration given in Figure 13. Eliminating ϕ(s)
yields the closed-loop transfer function
Hδmδ (s, v) =
(s, v) .
Imaginary Part (rad/s)
−P21 (s, v)
ϕ(s) +
mδ (s).
P22 (s, v)
P22 (s, v)
FIGURE 13 Block diagram of the basic bicycle model described in
[26]. The steer torque applied to the handlebars is mδ (s), ϕ(s) is the
roll angle, and δ(s) is the steer angle.
A Feedback System Perspective
δ(s) =
in which the mass and damping terms are neglected. If the
wheel and front frame masses, as well as all of the inertia
terms, are set to zero, these velocity-dependent gains are
given by
Real Part (1/s)
FIGURE 14 Poles and zeros of Hδmδ (s, v) as functions of speed.
The speed v is varied between ±10 m/s. The poles are shown as
blue dots for forward speeds and red crosses for reverse speeds.
There are two speed-independent zeros shown as black squares
at ± 3.135 1/s.
as that given in Figure 11. As the speed of the bicycle
increases, the unstable poles associated with the static capsize modes coalesce to form the complex pole pair associated with the weave mode. The weave mode is stable for
speeds above 4.3 m/s [26]. As the machine’s speed increases further, it becomes unstable due to the dynamic capsize
mode at 6.06 m/s.
The zeros of Hδmδ (s, v), which derive from the roots of
P11 (s) as shown in (13) [see (23)], are associated with the
speed-independent whole-vehicle capsize mode. The backward-running vehicle is seen to be unstable throughout
the speed range, but this vehicle is designed for forward
motion and, when running backwards, it has negative trail
and a divergent caster action. See “Caster Shimmy” and
note that the cubic terms of (38) and (39) are negative for
negative speeds, indicating instability in this case.
A control theoretic explanation for the stabilization difficulties associated with backward-running bicycles centers on the
positive zero fixed at + gmtzt/Txx , which is in close proximity to a right-half plane pole in certain speed ranges [34].
perturbation yaw rate response for the model described in
[26] can be calculated using
ψ̇ =
v cos λδ
w + t/ cos λ
which corresponds to (3) for the Timoshenko-Young bicycle with small perturbation restrictions. In the case of small
perturbations from straight running, (2) becomes
ẏ = vψ.
It now follows that the transfer function linking the lateral
displacement to the steer angle is
Hyδ (s, v) =
v2 cos λ
s2 (w + t/ cos λ)
and that the transfer function linking the lateral displacement to the steering torque is given by Hyδ (s, v)Hδmδ (s, v),
with Hδmδ (s, v) given in (23).
This transfer function is used
F (s)
in the computation of
responses to step steering
r (s)
torque inputs.
To study the basic bicycle
model’s steering response at
different speeds, including
those outside the autostable
speed range, it is necessary to
introduce stabilizing rider
control. The rider can be
FIGURE 15 Steering torque prefilter F(s) described in (29). This filter is an open-loop realization of the
emulated using the roll-angle
roll-angle-plus-roll-rate feedback law described in (28). As readers familiar with control systems are
plus roll-rate feedback law
aware, open- and closed-loop systems can be represented in equivalent ways if there are no disturbances and no modeling uncertainties.
mδ (s) = r(s) + (kϕ + skϕ̇ )ϕ(s) ,
An appreciation of the subtle nature of bicycle steering
goes back over 100 years. Archibald Sharp records [1, p.
222] “. . . to avoid an object it is often necessary to steer for
a small fraction of a second towards it, then steer away
from it; this is probably the most difficult operation the
beginner has to master. . . ” While perceptive, such historical accounts make no distinction between steering torque
control and steering angle control. They do not highlight
the role played by the machine speed, and timing estimates are based on subjective impressions rather than
experimental measurement.
As Whipple [17] surmised, the rider’s main control
input is the steering torque. While in principle one can
steer through leaning (by applying a roll moment to the
rear frame), the resulting response is too sluggish to be
practical in an emergency situation. The steer-torque-tosteer-angle response of the bicycle can be deduced from
(23). Once the steer angle response is known, the small
in which r(s) is a reference torque input and kϕ and kϕ̇ are
the roll and roll-rate feedback gains, respectively. This
feedback law can be combined with (17) to obtain the
open-loop stabilizing steer-torque prefilter
F(s) =
det(P(s, v))
det(P(s, v)) + (kϕ + skϕ̇ )P12 (s, v)k(s)
which maps the reference input r(s) into the steering
torque mδ (s) as shown in Figure 15. In the autostable
speed range, the stabilizing prefilter is not needed and
F(s) is set to unity in this case. The bicycle’s steering
behavior can now be studied at speeds below, within, and
above the autostable speed range. Prior to maneuvering,
the machine is in a constant-speed straight-running trim
condition. For an example of each of the three cases, the
filtered steering torque and the corresponding roll-angle
in the steer angle and lateral displacement responses is
attributable to the right-half plane zero in Hδmδ (s, v) given
by the roots of P11 (s) = 0 and corresponding to the lockedsteering whole-machine capsize mode as illustrated in
(13). Toward the end of the simulation shown, the steer
angle settles into an equilibrium condition, in which the
bicycle turns left in a circle with a fixed negative roll
angle. In relation to the nonminimum phase response in
the lateral displacement behavior, the reader is reminded
of the control difficulty that arises if one rides near to a
curb or a drop [39]; to escape, one has to go initially closer
to the edge. Body lean control is unusually useful in such
At speeds below the autostable range, a stabilizing steering-torque prefilter must be utilized to prevent the machine
from toppling over. In the low-speed (3.7 m/s) case, the steer
torque illustrated in Figure 16 is the unit-step response of the
prefilter, which is the steer torque required to establish a
steady turn. The output of the prefilter is unidirectional apart
from the superimposed weave-frequency oscillation required
to stabilize the bicycle’s unstable weave mode. In the case
considered here, the steady-state steer torque is more than
twice the autostable unit-valued reference torque required to
bring the machine to a steady-state roll angle of approximately −0.65 rad. To damp the weave oscillations in the roll and
Steer Angle (rad)
10 15
Time (s)
Lateral Displacement (m)
Roll Angle (rad)
Steer Torque (N-m)
responses are shown in Figure 16, while the steer angle
and lateral displacement responses are shown in Figure
17. In each case, the filter gains are chosen to be stabilizing
and to achieve approximately the same steady-state roll
angle; numerical gain values appear in the figure captions.
The autostable case is considered first, because no stabilizing torque demand filtering is required. In this case, the
clockwise (when viewed from above) unit-step steer
torque demand is applied directly to the bicycle’s steering
system (see Figure 16). The machine initially steers to the
right and the rear wheel ground-contact point starts moving to the right also (see Figure 17). Following the steer
torque input, the bicycle immediately rolls to the left (see
Figure 16) in preparation for a left-hand turn. After
approximately 0.6 s, the steer angle sign reverses, while
the rear-wheel ground-contact point begins moving to the
left after approximately 1.2 s. The oscillations in the roll
angle and steer angle responses have a frequency of about
0.64 Hz and are associated with the weave mode of the
bicycle (see Figure 11). Therefore, to turn to the left, one
must steer to the right so as the make the machine roll to
the left. This property of the machine to apparently roll in
the wrong direction is sometimes referred to as countersteering [39], [27], but an alternative interpretation is also
possible, as seen below. The nonminimum phase behavior
10 15
Time (s)
FIGURE 16 Step responses of the prefilter and the roll angle of the
basic bicycle model. The steering torque and roll angle response at
the autostable speed of 4.6 m/s are shown in blue; the prefilter gains
are kϕ = 0 and kϕ̇ = 0. The low-speed 3.7 m/s case, which is below
the autostable speed range, is shown in red; the stabilizing preflter
gains are kϕ = −2 and kϕ̇ = 3. The high-speed 8.0-m/s case, which
is above the autostable speed range, is shown in green; the stabilizing prefilter gains are kϕ = 2.4 and kϕ̇ = 0.02. In each case, a clockwise steering moment (viewed from above) causes the machine to
roll to the left. This tendency of the machine to apparently roll “in the
wrong direction” is sometimes referred to as countersteering. In the
high-speed case (green curves), the steering torque is positive initially
and then negative. This need to steer in one direction to initiate the
turning roll response, and then to later apply an opposite steering
torque that stabilizes the roll angle, is a high-speed phenomenon.
0.5 1 1.5
Time (s)
0.5 1 1.5
Time (s)
FIGURE 17 Response of the simple bicycle model to a steering
moment command. The steer angle and the rear-wheel ground-contact point displacement responses at the autostable speed of 4.6
m/s are shown in blue; the prefilter gains are kϕ = 0 and kϕ̇ = 0.
The responses at a speed of 3.7 m/s, which is below the autostable
speed range, are shown in red; the stabilizing prefilter gains are
kϕ = −2 and kϕ̇ = 3. The responses at a speed of 8.0 m/s, which is
above the auto-stable speed range, are shown in green; the stabilizing prefilter gains are kϕ = 2.4 and kϕ̇ = 0.02. The steer angle and
lateral displacement responses show the influence of the right-halfplane zero of P11 (s). This zero is associated with the unstable
whole-vehicle capsize mode. See point A in Figure 11 and (13).
steer angle responses, the torque demand filter, which mimics the rider, introduces weave-frequency fluctuations into
the steering torque. The steer angle and lateral displacement
responses are similar to those obtained in the autostable case.
If the trim speed is increased to the upper limit of the
autostable range (in this case 6.1 m/s; see Figure 11), then
the steady-state steering torque required to maintain an
equilibrium steady-state turn falls to zero; this response is
due to the singularity of the stiffness matrix v2 K2 + K0 at
this speed. At speeds above the autostable range, stabilizing rider intervention is again required. As before, in
response to a positive steer torque input, the steer angle
and lateral displacement initially follow the steer torque
(see Figure 17). At the same time the machine rolls to the
left (see Figure 16). Moments later, one observes the nonminimum phase response in the steer angle and the lateral
displacement responses. The interesting variation in this
case is in the steering torque behavior. This torque is initially positive and results in the machine rolling to the left.
However, if this roll behavior were left unchecked, the
bicycle would topple over, and so to avoid the problem the
steer torque immediately reduces and then changes sign
after approximately 4 s. The steer torque then approaches a
steady-state value of −0.6 N-m to stabilize the roll angle
and maintain the counterclockwise turn. This need to steer
in one direction to initiate the turning roll angle response,
and then to later apply an opposite steering torque that
stabilizes the roll angle is a high-speed phenomenon, providing the alternative interpretation of countersteering
mentioned earlier. Countersteering in the first sense is
always present, while in the second sense it is a high-speed
phenomenon only. It is interesting to observe that the prefilter enforces this type of countersteering for all stabilizing
values of kϕ and kϕ̇ . First note that the direct feedthrough
(infinite frequency) gain of F(s) is unity. Since kϕ and kϕ̇
are stabilizing, all of the denominator coefficients of F(s)
have the same sign as do all of the numerator terms in the
autostable speed range. As the speed passes from the
autostable range, det(v2 K2 + K0 ) changes sign, as does the
constant coefficient in the numerator of F(s). Therefore, at
speeds above the autostable range, F(s) has a negative
steady-state gain, thereby enforcing the sign reversal in the
steering torque as observed in Figure 16.
We conclude this section by associating the basic bicycle model’s nonminimum phase response (in the steer
angle) with its self-steering characteristics. To do this, consider removing the basic bicycle’s ability to self-steer by
setting α = π/2, t = 0, Cxz = 0, D yy = 0, and xff = w. With
these changes in place, it is easy to see from (20) that
P21 (s, v) = 0. This identity means that
P22 (s, v)
s(Czz + Dzz )(s + v/w)
which is clearly minimum phase and represents the
response one would expect when applying a torque to a
pure inertia with a damper to ground.
Pneumatic Tires, Flexible Frames, and Wobble
A modified version of the basic bicycle model is now considered in which a flexible frame and side-slipping tires
are included. The flexibility of the frame is modeled by
including a single rotational degree of freedom located
between the steering head and the rear frame. In the model
studied here, the twist axis associated with the frame flexibility freedom is in the plane of symmetry and perpendicular to the steering axis, and the associated motion is
restrained by a parallel spring-damper combination. In
this modified model, the nonholonomic lateral ground
contact constraints are replaced by (31) and (32); see “Tire
Modeling.” These equations represent tires that produce
lateral forces in response to sideslip and camber, with time
lags dictated by the speed and the tires’ relaxation lengths.
The tire and frame flexibility data used in this study are
given in Table 2; two representative values for the frame
stiffness KP and frame damping CP are included. The higher values of KP and CP are associated with a stiff frame,
while the lower values correspond to a flexible frame.
First, we examine the influence of frame compliance
alone on the system eigenvalues, which can be seen in Figure 18. The dotted curve corresponds to the rigid frame that
was studied in Figure 11 and is included here for reference
purposes. The cross- and circle-symbol loci correspond to
the stiff and flexible frames, respectively. The first important
observation is that the model predicts wobble when frame
compliance is included; see “Wobble.” In the case of the
flexible frame, the damping of the wobble mode reaches a
minimum at about 10 m/s and the mode has a resonant frequency of approximately 6 Hz. In the case of the stiff frame,
the wobble mode’s resonant frequency increases, while its
damping factor decreases, with increasing speed. Figure 18
also illustrates the impact of frame flexibility on the damping of the weave mode. At low speeds, frame flexibility has
no impact on the characteristics of the weave mode. At
intermediate and high speeds, the weave-mode damping is
TABLE 2 Bicycle tire and frame flexibility parameters. Tire
parameters include relaxation lengths and cornering- and
camber-stiffness coefficients. The frame flexibility is
described in terms of stiffness and damping coefficients. All
of the parameter values are given in SI units.
σ f , σr
0.1, 0.1
C f s , Cr s
14.325, 14.325
C f c , Cr c
1.0, 1.0
KPlow ,
CPlow ,
2,000, 10,000
20, 50
Tire Modeling
lassical bicycle models, such as those developed by Whipple
[17], and Timoshenko and Young [29], describe the wheelroad contact as a constraint. The wheel descriptions involve rotational coordinates to specify the wheels’ orientation and
translational coordinates that describe the location of the road
contact points. The rolling constraints connect these coordinates
so that translational changes are linked to rotational ones. In the
case of general motions, the rotational and translational coordinates cannot be linked algebraically, since this linkage is path
dependent; thus the nomenclature “nonholonomic,” or incomplete
constraint [18, p. 14]. Instead, it is the rotational and translational
velocities that are linked, and the rolling constraint renders the
wheels’ ground-contact points, or lines, absolutely stationary [24],
[51], [52]. During motion, the wheel-ground contact points change
with time, with each point on the wheel periphery coming into contact with the ground once per wheel revolution. In the case of the
bicycle, it is illustrated that (nonholonomic) tire constraint modeling
limits the fidelity of the vehicle model to low speeds only.
By 1950, the understanding of tire behavior had improved substantially, and it had become commonplace, although not universal,
to regard the rolling wheel as a force producer rather than as a constraint on the vehicle’s motion. With real tire behavior, the tread
material at the ground contact “slips” relative to the road and so has
a nonzero absolute velocity and the linkage between the wheels’
rotational and translational velocities is lost. To model this behavior,
it is necessary to introduce a slip-dependent tire force-generation
To understand the underlying physical mechanisms underpinning tire behavior, it is necessary to analyze the interface between
the elastic tire tread base and the ground. This distributed contact
involves the tire carcass and the rubber tread material, which can
be thought of as a set of bristles that join the carcass to the ground.
Under dynamic conditions, these bristles move, as a continuous
stream, into and out of the ground-contact region. Under freerolling conditions, in common with the nonholonomic rolling model,
the tread-base material is stationary; consequently, the bristles
remain undeformed in bending as they pass through the contact
region. When rolling resistance is neglected, no shear forces are
developed. Free-rolling corresponds to zero slip, and, if a slip is
developed, it has in general both longitudinal and lateral components [50], [51]. In contrast to the physical situation, tire models
usually rely on the notion of a ground-contact point.
To assemble these ideas in a mathematical framework, let vf
denote the velocity of the tread base material at the ground contact
point. In the case of no longitudinal slipping, vf is perpendicular to
the line of intersection between the wheel plane and the ground
plane; the unit vector i lies along this line of intersection and the
unit vector j is perpendicular to it. The velocity of a tread base point
with respect to the wheel axle is given by vf r = ωf Rf i, where ωf is
the wheel’s spin velocity and Rf is the wheel radius. If we now
associate with this ground contact point an “unspun” point, its
velocity is vfus = vf + vf r . The slip (for the front wheel) is defined as
ssf =
< vfus , i >
where < ·, · > denotes the inner product. The slip is in the j direction in
the case of no longitudinal slipping, as is assumed here. If the bristle
bending stiffness is constant and the frictional coupling between the
bristle tips and the ground is sufficient to prevent sliding, the lateral
force developed is proportional to ssf and acts to oppose the slip.
When the rolling wheel is leaned over, then even with no slip,
the tread base material becomes distorted from its unstressed state.
This distortion leads to the development of a lateral force that is
approximately equal to the normal tire load multiplied by the camber
angle [51], [52]. If the tire is not working hard, the force due to camber simply superimposes on the force due to slip. The elemental lateral forces due to camber are distributed elliptically over the contact
length, while those due to sustained slip increase with the longitudinal distance of the tire element from the point of first contact. As the
sideslip increases, the no-sliding condition is increasingly challenged as the rear of the contact patch is approached. Thus, as the
tire works harder in slip, sliding at the rear of the contact patch
becomes more pronounced. Force saturation is reached once all
the tire elements (bristles) in contact with the road begin to slide.
When the tire operates under transient conditions, following for
example a step change in steering angle, the distortion of the tire
tread material described above does not develop instantly. Instead,
the distortion builds up in a manner that is linked to the distance covered from the time of application of the transient. For vehicle modeling purposes, a simple approximation of this behavior is to treat the
dynamic force development process as a speed-dependent firstorder lag. The characterizing parameter, called the relaxation length
σ f , is similar to a time constant except that it has units of length
rather than time. The relaxation length is a tire characteristic that can
be determined experimentally. The lateral force response of the tire
due to steering, and therefore side slipping, is a dynamic response to
the slip and camber angles of the tire, which is modeled as
Ẏf + Yf = Zf (Cfs ssf + Cf c ϕf ) ,
|vfus |
where Zf is the normal load on the front tire and ϕf is the front wheel’s
camber angle relative to the road (pavement). The force Yf acts in the j
direction and opposes the slip. The product Zf Cf s is the tire’s cornering
stiffness, while Zf Cf c is its camber stiffness. The sideforce associated
with the rear tire is given analogously by
Ẏr + Yr = Zr (Cr s ssr + Cr c ϕr ) ,
r |
where each term has an interpretation that parallels that of the front
wheel. Equations (31) and (32) are suitable only for small perturbation
Contemporary large perturbation tire models are based on
magic formulas [51] and [53]–[55], which can mimic accurately measured tire force and moment data over a wide range of operating
phenomenon known variously as speedman’s wobble,
speed wobble, or death wobble is well known among
cyclists [85] and [86]. As the name suggests, wobble is a steering oscillation belonging to a more general class—wheel shimmy. The oscillations are similar to those that occur with
supermarket trolley wheels, aircraft nose wheels, and automobile steering systems. Documentation of this phenomenon in
bicycles is sparse, but a survey [86] suggests that wobble at
speeds between 4.5–9 m/s is unpleasant, while wobble at
speeds between 9–14 m/s is dangerous. The survey [86] also
suggests a wide spread of frequencies for the oscillations with
the most common being between 3–6 Hz, somewhat less than
for motorcycles. The rotation frequency of the front wheel is
often close to the wobble frequency, so that forcing from wheel
or tire nonuniformity may be an added influence. Although
rough surfaces are reported as being likely to break the regularity of the wobble and thereby eliminate it, an initial event is normally needed to trigger the problem. Attempting to damp the
vibrations by holding on tightly to the handlebars is ineffective, a
result reproduced theoretically for a motorcycle [87]. The survey [86] recommends “pressing one or both legs against the
frame, while applying the rear brake” as a helpful practical procedure, if a wobble should commence. The possibility of accelerating out of a wobble is mentioned, suggesting a worst-speed
condition. The influences of loading are discussed with special
emphasis on the loading of steering-frame-mounted panniers.
Evidently, these influences are closely connected with the first
term in each of (20) and (21), representing roll-acceleration-tosteer-torque feedback. Sloppy wheel or steering-head bearings
and flexible wheels are described as contributory. Increasing
the mechanical trail is considered stabilizing with respect to
compromised by the flexible frame, although this mode
remains well damped. As is now demonstrated, the more
realistic tire model has a strong impact on the predicted
properties of both wobble and weave.
Figure 19 shows the influence of frame flexibility in combination with relaxed side-slipping tires. Again, the crossand circle-symbol loci correspond to the stiff and flexible
frames, respectively, while the dotted loci belong to the
rigid-framed machine. As can be seen from this dotted locus,
the introduction of side-slipping tires also produces a wobble mode, which is not a property of the basic bicycle. The
predicted resonant frequency of the wobble mode varies
from approximately 12.7–4.8 Hz, depending on the frame
stiffness. Lower stiffnesses correspond to lower natural frequencies. With a rigid frame, the wobble damping is least at
low and high speeds. With a compliant frame, the damping
is least at an intermediate speed. The frame flexibility can be
such that the resonant frequency aligns with the practical
evidence. Frame flexibility modeling can also be used to
align the wobble-mode damping with experimental mea-
wobble, raising both the frequency and the worst-case speed,
but is not necessarily advantageous overall.
In an important paper from a practical and experiential viewpoint, [37] implies that wobble was a common motorcycling phenomenon in the 1950s. Machines of the period were usually fitted
with a rider-adjustable friction-pad steering damper. The idea was
that the rider should make the damper effective for high-speed
running and ineffective for lower speeds; see also [88]. Reference [37] offers the view that steering dampers should not be
necessary for speeds under 45 m/s, indicating that, historically,
wobble of motorcycles has been a high-speed problem. Reference [37] also points to the dangers of returning from high speed
to low speed while forgetting to lower the preload on the steering
damper. A friction lock on the steering system obliges a rider to
use fixed (steering position) control, which we have earlier
demonstrated to be difficult. The current status of motorcycle
wobble analysis is covered in the “Motorcycle Modeling” section.
Wheel shimmy in general is discussed in detail in [51], where
a whole chapter is devoted to the topic. Ensuring the stability of
wheel shimmy modes in aircraft landing gear, automotive steering systems, and single-track vehicles is vital due to the potential
violence of the oscillations in these contexts. An idea of how
instability arises can be obtained by examining simple cases
(see “Caster Shimmy”), but systems of practical importance are
sufficiently complex to demand analysis by automated multibody
modeling tools and numerical methods.
A simple system quite commonly employed to demonstrate
wheel shimmy, both experimentally and theoretically [51], [89],
is shown in Figure C. If the tire-to-ground contact is assumed
to involve nonholonomic rolling, the characteristic equation of
the system of Figure C is third order, and symbolic results for
surement. Comparison with Figure 18 shows that the introduction of the side-slipping tire model causes a marked
reduction in the wobble-mode frequency for the stiff-framed
machine, while the impact of the side-slipping tires on the
wobble mode of the flexible-framed machine is less marked.
As with the flexible frame, side-slipping tires have little
impact on the weave mode at very low speeds. However, as
the speed increases, the relaxed side-slipping tires cause a
significant reduction in the intermediate and high-speed
weave-mode damping. By extension from measured motorcycle behavior, there is every reason to suspect that the accurate reproduction of bicycle weave- and wobble-mode
behavior requires a model that includes both relaxed sideslipping tires and flexible frame representations.
Several factors differentiate bicycles from motorcycles. A
large motorcycle can weigh at least ten times as much as a
the conditions for stability are
obtained in “Caster Shimmy.” For
higher levels of complexity, the
system order is increased and
m, Jz
analytical stability conditions
become significantly more comδ
plex. In [51], a base set of parameter values is chosen, and
stability boundaries are found
numerically for systematic variae
tions in speed and mechanical
trail . The resulting stability
boundaries are plotted in the (v ,
FIGURE C Plan view of a simple system capable of shimmy. This example is adapted from [51]
e) parameter space for several
and [89, pp. 333, 334, ex. 215 p. 414]. The wheel is axisymmetric and free to spin relative to
values of the lateral stiffness k of
the forks that support it; the wheel is deemed to have no spin inertia. The wheel has mechanical trail e and mass offset f with respect to the vertical king-pin bearing. The king-pin is free to
the king-pin mounting. The least
translate laterally with displacement y from static equilibrium, while the whole assembly
oscillatory system is that having
moves forward with constant speed v. The king-pin mounting has stiffness k, while the moving
the highest stiffness, with the
assembly has mass m. The steer angle is δ. The king-pin is assumed massless so that analyking-pin compliance contributing
sis deals with only one body; see “Caster Shimmy.” The tire-ground contact can be treated on
to the system behavior in much
one of three different levels. First, pure (nonholonomic) rolling, implying no sideslip, can be
assumed. Second, the tire may be allowed to sideslip thereby producing a proportionate and
the same way tire lateral compliinstantaneous side force. Third, the side force may be lagged relative to the sideslip by a firstance contributes.
order lag determined by the tire relaxation length; see “Tire Modeling.”
Significant from the point of
view of single-track vehicles, and
aircraft nose-wheels, is the lateral compliance at the king-pin. If
[51] to create a second area of instability in the (v , e) space at
this compliance allows the assembly to rotate in roll about an
higher speeds, which have a substantially different mode
axis well above the ground, as with a typical bicycle or motorcyshape. The gyroscopic mode involves a higher ratio of lateral
cle frame or aircraft fuselage, lateral motions of the wheel contact point velocity to steer velocity than occurs in situations
assembly are accompanied by camber changes. If, in addition,
in which a roll freedom is absent. This new phenomenon is
the wheel has spin inertia, gyroscopic effects have an important
called gyroscopic shimmy, and it is this shimmy variant that is
influence on the shimmy behavior. These effects are shown in
particularly relevant to the single-track vehicle [40], [41], [47].
bicycle, and, consequently, in the case of a motorcycle, the
rider’s mass is a much smaller fraction of the overall ridermachine mass. A modern sports motorcycle can achieve
top speeds of the order 100 m/s, while a modern sports
bicycle might achieve a top speed of approximately 20
m/s. As a result of these large differences in speed, our
understanding of the primary modes of bicycles must be
extended to speeds that are usually irrelevant to bicycle
behavioral studies. At high speeds, aerodynamic forces are
important and need to be accounted for.
In his study of bicycles, Whipple [17] introduced a
nondimensional approach to bicycle dynamic analysis,
which is helpful when seeking to deduce the behavior of
motorcycles from that of bicycles. The dimensionless
model was obtained by representing each mass by
m = αw, where α is dimensionless and w has the units of
mass (kilograms, for example) and each length quantity by
l = β b, where β is dimensionless and b has the units of
length (meters, for example). As a result, the moments and
products of inertia are expressed as J = γ wb2 , where γ is
also dimensionless. These changes of variable allowed
Whipple to establish that the roots of the quartic characteristic equation, which represents the small perturbation
behavior around a straight-running trim state, are independent of the mass units used. Therefore, for the nonholonomic bicycle model, increasing the masses and inertias
of every body by the same factor makes no difference to
the roots of the characteristic equation. In this restricted
sense, a grown man riding a motorcycle is dynamically
equivalent to a child riding a bicycle.
Whipple then showed that the characteristic equation
p(λ, v) = 0 can be replaced with p̃(ξ, ) = 0 using a change
of variables. In the first case, the speed v has units such as
m/s, the characteristic equation has roots λi having the
units of circular frequency (rad/s for example). The new
variables: ξ = λb/v and = gb/v 2 , where g is the gravitational constant, are dimensionless as are the polynomial’s
coefficients. Therefore, all of the nondimensional singletrack vehicles corresponding to p̃(ξ, ) = 0, where is a
constant, have the same dynamical properties in terms of
Caster Shimmy
aster wheel shimmy can occur in everyday equipment such
as grocery trolleys, gurneys, and wheelchairs. These selfexcited oscillations, which are energetically supported by the
vehicle prime mover, are an important consideration in the
design of aircraft landing gear and road vehicle suspension
and steering systems. In the context of bicycles and motorcycles, this quantitative analysis is conducted by including the
appropriate frame flexibility freedom and dynamic tire descriptions in the vehicle model. The details are covered in the
“Pneumatic Tires, Flexible Frames, and Wobble” section.
By its nature, a caster involves a spinning wheel, a king-pin
bearing, and a mechanical trail sufficient to provide a self-centering steering action. Our purpose here is to demonstrate how
oscillatory instability can be predicted for the simple system of
Figure C. In the case of small perturbations, the tire sideslip is
eδ̇ − ẏ
It follows from (31) that the resulting tire side force F is given by
Ẏ + Y = Cs,
in which C is the tire’s cornering stiffness and σ is the relaxation
length. The equations of motion for the swivel wheel assembly in
Figure D are
m(ÿ − f δ̈) + ky − Y = 0
Jz δ̈ + (e − f )Y + kyf = 0,
where Jz is the yaw-axis moment of inertia of the swiveled wheel
assembly around the mass center. The characteristic polynomial
associated with small motions in the system in Figure D is derived
directly from (33)–(36). The resulting quintic polynomial is
ms2 + k
−f ms2
s2 Jz
−C(1 + (es)/v ) 1 + σ s/v
Two interesting special cases can be deduced from the general problem by making further simplifying assumptions. In the
case of the nonholonomic wheel, the cornering stiffness becomes
arbitrarily large for all values of σ, thereby preventing tire sideslip
det[·] (m(e − f )2 + Jz )s3
e2 s
m(e − f )s2
+ e,
ξ . The modal frequencies and decay/growth rates scale
according to eλi t translating to e(ξi v/bt ) , where t is dimensionless. This analysis provides a method for predicting
the properties of a family of machines from those of a sin-
where det[·] comes from (37). It follows from (38) and the Routh
criterion that shimmy occurs if m f (e − f ) ≤ Jz , and in the case
that m f (e − f ) = Jz the frequency of oscillation is
ω = (ke)/(m(e − f )). These results show the role played by the
steering system geometry, and the mass and inertia properties of
the moving assembly in determining the stability, or otherwise, of
the system. The king-pin stiffness influences the frequency of oscillation. The case of m f (e − f ) = Jz corresponds to a mass distribution in which the rolling contact is at the center of percussion
relative to the kingpin. In this situation the rolling constraint has no
influence on the sping force.
In the case of a rigid assembly
k →∞
det[·] σ (f 2 m + Jz )s3
e2 s
(f 2 m + Jz )s2
+ e.
It follows from (39) that shimmy occurs if e ≤ σ , and in the case
that e = σ the frequency of oscillation is ω = Ce/(f 2 m + Jz ).
The tire properties dictate both conditions for the onset of shimmy
and its frequency when it occurs. Interestingly, the tire relaxation
length alone determines the onset, or otherwise, of shimmy, while
the frequency of oscillation is dictated by the tire’s cornering stiffness alone. The Pirelli company reports [90] on a tire tester that
relies on this precise result. The test tire is mounted in a fork trailing
a rigidly mounted king-pin bearing and runs against a spinning
drum to represent movement along a road. Following an initial
steer displacement of the wheel assembly, the exponentially
decaying steering vibrations are recorded, and the decrement
yields the tire relaxation length, while the frequency yields the cornering stiffness. Unlike the bicycle case, the shimmy frequency is
independent of speed.
As discussed in the “Basic Bicycle Model” section, in connection with the zero-speed behavior, the simple caster does not in
reality oscillate at vanishingly small speeds due to the distributed
contact between the tire and the ground. The energy needed to
increase the amplitude of unstable shimmy motions comes from
the longitudinal force that sustains the forward speed of the kingpin. This longitudinal force is given by
F = m(δ ÿ + f δ̇ 2 ) + δky
for the small perturbation problem described in (33)–(36). In the
case of a pure-rolling (nonholonomic) tire, δ̇ should be eliminated
from the above equation using the zero-sideslip constraint
δ̇ = (ẏ − v δ)/e.
gle nondimensional vehicle. For example, if b is halved so
as to represent a child’s bicycle in this alternative lengthscaling sense,√then a simultaneous reduction of the speed
by a factor of 2 leaves the roots of p̃(ξ, ) unchanged. The
Straight-Running Motorcycle Models
An influential contribution to the theoretical analysis of the
straight-running motorcycle is given in [47]. The model
developed in [47] is intended to provide the minimum level
of complexity required for predicting the capsize, weave,
and wobble modes. This research is reminiscent of Whipple’s analysis in terms of the assumptions concerning the
rider and frame degrees of freedom. In contrast to Whipple,
[47] treats the tires as force generators, which respond to
both sideslip and camber; tire relaxation is included (see
“Tire Modeling”), while aerodynamic effects are not.
A linearized model is used for the stability analysis
through the eigenvalues of the dynamics matrix, which is
a function of the vehicle’s (constant) forward speed. Two
cases are considered: one with the steering degree of freedom present, giving rise to the free-control analysis, and
the other with the steering degree of freedom removed,
giving rise to the fixed-control analysis. The free-control
model predicts the existence of capsize, weave, and wobble modes. As with the bicycle, the capsize mode is a
slowly divergent instability of the whole vehicle, which
corresponds to the machine toppling over onto its side.
This mode is relatively unimportant because it is easily
(and subconsciously) controlled by the rider. As with the
bicycle, weave is a low-frequency (2–3 Hz) oscillation of
the whole vehicle involving roll, yaw, and steer motions,
and is well damped at moderate speeds but becomes
Imaginary Part (rad/s)
Real Part (1/s)
FIGURE 18 Root loci of the basic bicycle model with a flexible frame.
The speed is varied from 0–20 m/s; the zero-speed end is represented by a square and the high-speed end by a diamond. The
(blue) dotted loci correspond to the rigid frame, the (black) crosses
to the high frame stiffness and damping values, and the (red) circles
to the low stiffness and damping values.
Imaginary Part (rad/s)
associated variation in
√ the time domain response comes
from λi translating to 2λi .
Whipple’s scaling rules, in combination with observations, lead one to conclude that a viable motorcycle model
1) must be consistent with bicycle-like behavior at low
speed, 2) must reproduce the autostability properties predicted by Whipple [17], 3) must reproduce the motorcycle’s
inclination to wobble at intermediate and high speeds, and
4) must reproduce the observed high-speed weave characteristics of modern high-performance motorcycles.
High-powered machines with stiff frames have a tendency to wobble at high speeds [40]–[42]; see “Tommy
Smith’s Wobble.” A primary motivation for studying wobble and weave derives from the central role they play in
performance and handling qualities. These modes are also
associated with a technically challenging class of stabilityrelated road accidents. Several high-profile accidents of this
type are reviewed and explained in the recent literature
[43]. Central to understanding the relevant phenomena is
the ability to analyze the dynamics of motorcycles under
cornering, where the in-plane and out-of-plane motions,
which are decoupled in the straight-running situation,
become interactive. Consequently, cornering models tend
to be substantially more complex than their straight-running counterparts. This added complexity brings computerassisted multibody modeling to the fore [42], [44], [45].
In the remainder of this article, we study several contributions, both theoretical and experimental, that have
played key roles in bringing the motorcycle modeling art
to its current state of maturity. Readers who are interested
in the early literature are referred to the survey paper [46],
which reviews theoretical and experimental progress up to
the mid 1980s. That material focuses almost entirely on the
straight-running case, which is now considered.
Real Part (1/s)
FIGURE 19 Root loci of the basic bicycle model with flexible frame
and relaxed sideslipping tires. The speed is varied from 0–20 m/s;
the zero-speed end is represented by a square and the high-speed
end by a diamond. The (blue) dotted loci correspond to the rigid
frame, the (black) crosses to the high frame stiffness and damping
values, and the (red) circles to the low stiffness and damping values. The properties illustrated here for the limited speed range of
the bicycle are remarkably similar to those of the motorcycle, with
its extended speed capabilities.
Tommy Smith’s Wobble
ommy Smith was born in 1933. He started riding motorized bicycles at the age of 13 and was racing motorcycles professionally
by the age of 17. In 1952, Tommy had the opportunity to ride a
modified 650 cc Triumph Thunderbird at the Bonneville Salt Flats in
Utah, United States. At that time, fuel (as opposed to gasoline)
motorcycles used about 70% methanol and 30% nitro methane;
crankcase explosions occurred when higher nitro percentages were
tried. To further increase the motorcycle’s engine power, the cylinder head was reversed, so that the intake ports were pointing forward to achieve a ram air effect. This engine configuration made it
impossible to sit on the machine in a conventional manner. For this
reason, the motorcycle was fitted with a plywood board for the rider
to lie prone on. Leathers were heavy and uncomfortable and so
Tommy rode the bike wearing a fiberglass helmet, goggles, tennis
shoes (with socks), and a Speedo bathing suit (see Figure D).
On the first high-speed run, the machine produced an eerie
“floating” sensation that was probably the result of a veneer of
loose salt on the running track combined with a lightly loaded
front wheel, resulting from the high speed and unusual riding
position. Engine revolution and speed measurements taken at
the time suggested that there was approximately 4.5 m/s of longitudinal tire-slip velocity. An accompanying lateral drifting phenomenon had to be corrected with small handlebar inputs that
were required every 5–10 s. The need for continuous steering
corrections may have also been associated with an unstable
capsize mode with an unusually large growth rate and the lack
of constraint between the rider and machine, both related to the
riding position. Detailed calculations relevant to the situation
described have not been carried out, so far as the authors are
aware. The official one-way speed achieved was 147.78 mi/h,
which was not to be exceeded by a 650-cc-motorcycle rider for
another ten years.
increasingly less damped and possibly unstable at higher
speeds. Wobble is a higher frequency (typically 7–9 Hz)
motion that involves primarily the steering system. In
contrast to the bicycle study presented in this article, [47]
predicts that the wobble mode is well damped at low
speeds, becoming lightly damped at high speeds.
In particular, the study shows that tire relaxation is an
important contributor to the prediction of wobble and the
quantitative characteristics of high-speed weave. The influences of parameter variations on the vehicle’s dynamic
behavior are also studied, and the results obtained are for
the most part aligned with the behavior of vehicles of the
time. Of particular importance is the predicted influence of
the steering damper on the wobble-mode damping and the
destabilizing effect that the damper has on the weave mode.
The positive effect of moving the rear frame mass center forward, the critical impact on stability of the steering-head
angle, the mechanical trail, and the front frame mass center
On 25 August 1952, Tommy made his third high-speed run.
Everything started normally—the floating sensation was the same
as it had been on previous tests. Suddenly, the motorcycle went
into a high-speed wobble and Tommy held tightly onto the handlebars to prevent himself from falling off. After a period of 3–5 s, the
wobble was so violent that Tommy “hit the salt” and slid through
the first 1/10 mi speed trap at an official speed of 139 mi/h. The
speed of the motorcycle was not recorded! Although the motorcycle was only slightly damaged, Tommy’s abrasion injuries were
severe enough to keep him out of the Korean War. At the time, it
was suggested that Tommy’s light weight contributed to the
motorcycle’s instability, because heavier riders did not experience
wobble at similar speeds. This suggestion that light riders might
be prone to instability has been investigated by computer simulation studies [43]. The mobility of the rider relative to the motorcycle, as well as his rearward positioning, which led to a reduction in
the front wheel load, are likely to have been important influences
on the machine problem treated above.
FIGURE D 24 Modified Triumph Thunderbird. Tommy Smith riding
a modified 650-cc Triumph Thunderbird at the Bonneville Salt
Flats in Utah. Note the forward-facing air intake ports.
offset from the steering axis are also demonstrated. A recurring theme is the need to find compromises under variations in these critical parameters.
Leaving briefly the constant-forward-speed case, [48]
represents the first attempt to study the effects of acceleration and deceleration on the stability of motorcycles. A
rather simple approach, in which the longitudinal equation
of motion is decoupled from the lateral equations, gives the
longitudinal acceleration as a parameter of the lateral
motion. The acceleration parameter contributes to longitudinal inertia forces, which are included in standard stability
computations. Such computations lead to some tentative
conclusions, which depend on knowledge of the influence
of loading on tire force and moment properties. More recent
results [49], which are based on a higher fidelity model, are
not supportive of the conclusions given in [48]. In [49], it is
found that braking and acceleration have little influence on
the frequency and damping of the weave mode. It is also
concluded in [49] that descending a hill or braking have a
substantial destabilizing effect on the wobble mode. Conversely, the wobble-mode damping increases substantially
under acceleration or ascending an incline, for small perturbations from straight running. An open issue is the influence of acceleration or braking on a cornering machine.
Tire Modeling
Modeling the generation of shear forces and moments by
pneumatic tires has been approached in various ways, which
recognize the physics of the situation in more or less detail. At
one extreme, physical models [50]–[52] contain detailed
descriptions of the tire structure and the tread-ground interactions, while, at the other, empirical formulas [50]–[53] come
from fitting curves to measured data. In the middle ground,
simple physical models provide good representations of the
basic geometry and the distributed tire-ground rolling contact.
The detailed models are effective in terms of accuracy and
range of behavior covered but are computationally demanding to use. Contemporary high-fidelity models, which can be
used over a wide variety of operating conditions, are almost
exclusively of the empirical variety. An overview of many of
these ideas in the context of car tires is given in [51].
The basis for contemporary tire models are magic formulas [51], [53]–[55], which are empirical models favored for
their ability to accurately match tire force and moment data
covering a full range of operating conditions. The original
development was for car tires [56], in which context magic
formula models are now dominant. These models describe
the steady-state longitudinal forces, side forces, aligning
moments, and overturning moments as functions of the longitudinal slip, sideslip, camber angle, and normal load. The
extension of magic formula ideas to motorcycle tires is relatively recent, with substantial changes needed to accommodate the changed roles of sideslip and cambering in the
force and moment generation process. When finding the
parameters that populate the magic formulas, constraints
must be placed on the parameter set to ensure that the tire
behavior is reasonable under all operating conditions, some
of which may be beyond those used in the parameter identification process. Although limited tire-parameter information can be found in the literature, models can be
augmented with available experimental force and moment
data. A full set of parameters for modern front and rear
high-performance motorcycle tires can be found in [42].
Additional data are available in [51], [53]–[55], and [57]–[60].
Aerodynamic Forces
The importance of aerodynamic forces on the performance
and stability of high-powered motorcycles at high speeds
was demonstrated in [61]. Wind tunnel data were obtained
for the steady-state aerodynamic forces acting on a wide
range of motorcycle-rider configurations. It appears from
the results in [61] that the effects of aerodynamic side forces,
yawing moments, and rolling moments on the lateral stabil-
ity of production motorcycles are minor. However, the drag,
lift, and pitching moments contribute significantly to
changes in the posture of the machine on its suspension and
also to the tire loads. Aiming to explain the high-speed
weave stability problem, [61] introduces these aerodynamic
effects into the model of [47] using aerodynamic parameters
corresponding to a streamlined machine. These results yield
the conclusions that aerodynamic effects lead to only minor
changes in the wobble mode and that high-speed weave difficulties cannot be attributed entirely to steady-state aerodynamic loading. As a result, it is postulated in [61] that the
problem may involve nonsteady aerodynamic influences. To
fully appreciate aerodynamic effects, it is necessary to
employ a state-of-the-art model that includes the suspension
system as well as tire models that recognize the influences
of load changes. In such models [44], the aerodynamic drag
and lift forces and the pitching moments are represented as
being proportional to the square of the speed.
Structural Flexibility
Motivated by the known deterioration in the steering behavior
resulting from torsional compliance between the wheels, [62]
extends the model of [47] by allowing the rear wheel to camber relative to the rear frame. This freedom is constrained by a
parallel spring-damper arrangement. It was found that
swingarm flexibility had very little influence on the capsize
and wobble modes, but it reduced the weave mode damping
at medium and high speeds. The removal of the damping
associated with the swingarm flexibility made no material difference to these findings. The results indicate that a swingarm
stiffness of 12,000 N-m/rad for a high-performance machine
would bring behavior approaching closely that for a rigid
frame. Product development over the last 30 years has clearly
involved substantial stiffening of the swingarm structure,
such that most contemporary designs are probably deep into
diminishing returns for additional stiffness.
Experimental work [63]–[66] shows that the theory
existing at the time overpredicted the wobble-mode damping at moderate speeds, at which the damping is often
quite small. In particular, [65] associates the low mediumspeed-wobble damping with front fork compliance and
shows improved behavior from stiffer forks. It is also
shown in [65] that stiffening the rear frame with additional
structures increased the damping of the weave mode.
The discrepancy between theory and experiment, mainly
with respect to the damping of the wobble mode and its
variation with speed, is substantially removed by the results
of [40] and [41], where mathematical models were extended
to include front frame compliances. In particular, [40]
employs three model variants A, B, and C. The A model
allows the front wheel to move laterally along the wheel
spindle. The B model allows torsional compliance in the
front frame about an axis parallel to the steer axis, while the
C model allows twisting of the front frame relative to the
rear frame about an axis perpendicular to the steering axis.
In each case, the new compliance involves a parallel springdamper arrangement. The parameters from four different
large production motorcycles are used. The following conclusions are drawn. 1) The torsional freedom parallel to the
steering axis makes very little difference to the results
obtained from the stiff-framed model. 2) The front-wheel
lateral compliance results in a decrease in the wobble-mode
damping, but the associated speed dependence is not supported by experiment. This flexibility also results in
improved weave-mode damping at moderate speeds but
worsens it for high speeds, which is where it matters. It is
suggested that the lateral stiffness should be made large but
that such stiffening brings diminishing returns beyond an
intermediate stiffness level. 3) The C-model freedom leads
to the prediction of the observed intermediate-speed low
damping of the wobble mode, with higher damping at high
speeds deriving from the frame compliance. Thus, the compliance may to some extent contribute to good behavior. In
an independent study, [41] confirms the findings described
above. Apart from varying the torsional stiffness, the effect
of changing the height of the lateral fork bending joint was
also examined. The analysis concluded that the lateral bending of the front fork should be reduced by stiffening and
that the bending axis should be located as close to the pavement as possible. It also concluded that the “best” frontwheel suspension system should be designed to have high
lateral stiffness without being excessively heavy.
Measured static torsional stiffness data for motorcycle
frames are given in [58] and [67]–[69], while [68] and [69]
also include the results of dynamic testing. Stiffnesses for
large motorcycles from the past apparently lie in the range
of 25,000–150,000 N-m/rad, where the influence on stability properties is marked. Predicting the wobble mode properly and understanding the need for a steering damper
depend on accounting for frame torsional compliance in
the steering-head region and lateral fork bending.
This frame flexibility work is consolidated in [70], where
a motorcycle model is developed for straight-running studies with design parameters and tire properties obtained
from laboratory experiments. The model constituents are, in
addition to those given in the earlier model [47], lateral and
frame twist flexibilities at the steering head, a flexibility of
the rear wheel assembly about an inclined hinge, a roll freedom associated with the rider’s upper body, in-plane aerodynamic effects, and more elaborate tire modeling.
Hands-on and hands-off cases are presented, and the results
are in agreement with empirical observations and experimental findings of [71]. The results show the advantage that
can be derived in respect of the weave mode damping from
a long wheelbase and a large steering-head angle. The
model of [70] was subsequently rebuilt using a modern
multibody simulation package [72], confirming the original.
In the context of contemporary high-performance machines,
the only frame flexibility deemed to be important is that
associated with the steering head and front forks [42], [44].
Rider Modeling
In early motorcycle and bicycle models, the rider is considered to be no more than an inert mass rigidly attached to the
rear frame [17], [47]. In [57] and [58], the rider’s lower body is
represented as an inert mass attached to the rear frame, while
the upper body is represented as an inverted pendulum that
has a single roll freedom constrained by a parallel springdamper arrangement. The parameter values come from simple laboratory experiments, which show that values can vary
significantly from rider to rider [73]. This single-degree-offreedom inverted pendulum rider model is also used in [70].
The straight-running stability of a combined motorcycle rider model, which focuses on the frame flexibilities
and the rider’s dynamic characteristics, is studied in [73].
This 12-degree-of-freedom model includes two rider freedoms. The first is associated with the rolling motion of the
rider’s upper body, while the second allows the rider’s
lower body to translate laterally relative to the motorcycle’s main frame. Both bodies associated with the rider are
restored to their nominal positions by linear springs and
dampers. The system parameters are found experimentally, and the rider data, in particular, is measured by means
of forced vibration experiments, whereby the frequency
responses from vehicle roll to rider body variables are
obtained. The frequency and damping ratios of the wobble and weave modes are calculated at various speeds and
compared with results obtained from experiments conducted with four motorcycles covering a range of sizes. A
model without rider freedoms (a reduction of two degrees
of freedom) is used for comparison. In general terms,
there is very good agreement between the experimental
results for each of the four machines and the detailed
model, with a tendency for the measured damping factors
to be a little greater than those predicted.
The effect of individual rider parameters on the ridermotorcycle system stability is also investigated analytically. It is found that the rider’s vibration characteristics
influence both wobble and weave. The parameters of the
rider’s upper body motion are most influential on weave,
while those concerned with the rider’s lower body primarily influence the wobble mode.
The role of the rider as an active controller is studied in
some detail in [74], where it is recognized that inadvertent
rider motions can have a significant influence on the vehicle’s behavior. The focus of [74] is to treat the rider as a feedback compensator that maps the vehicle’s roll angle errors
into a steering torque, where the controller’s characteristics
are chosen to mimic those of the rider’s neuromuscular system. The rider is modeled (roughly) as being able to control
his upper body roll angle as well as the steering torque; the
steering torque influence is found to be dominant.
A motorcycle rider model similar to that studied in [73] is
investigated in [75] to find those aspects of the rider’s control
action that are most important in the description of singlelane-change maneuvering behavior. In this case, the rider
are constant, while the suspension posture of the machine,
the tire force system, and the aerodynamic forces are all
functions of the lean angle. Essential components of highfidelity cornering models include [44]: 1) a rigid rear frame,
which has six degrees of freedom; 2) a front frame joined to
the rear frame using an inclined steering system with a compliance included between the steering head and the rear
frame; 3) spinning road wheels, which include thick profiled
tire descriptions, where the dynamic migration of the roadtire ground contact point under cornering is modeled; 4) an
elaborate tire force and moment representation informed by
extensive measurements; 5) lag mechanisms by which tire
forces are delayed with respect to the slip phenomena that
produce them; 6) aerodynamic effects, which allow the tire
loads and machine posture to be properly represented under
speed variations; 7) a realistic suspension model; and 8) the
freedom for the rider’s upper body to roll relative to the rear
frame of the vehicle. The accuracy of predicted behavior
depends not only on effective conceptual modeling and
multibody analysis but also on good parameter values.
The in-plane modes present under straight-running conditions are shown in Figure 20; similar plots can be found
in [44]. The in-plane modes that are associated with the suspension and tire flexibilities are referred to as the front
suspension pitch mode, the rear suspension pitch mode,
the front wheel-hop mode, and the rear wheel-hop mode.
These modes are insensitive to speed variations and are
decoupled from the out-of-plane modes described above.
The cornering situation is considerably more complex
than the straight-running case, since the in-plane and out-ofplane motions are coupled and these interactions tend to
increase with roll angle. As a consequence, several straightrunning modes merge together to form combined cornering
modes with the particular characteristics shown in the right-
model comprises upper and lower body masses that are both
free to roll relative to the motorcycle’s main frame. The rider is
assumed to generate three control torques that are applied to
the steering system from the rider upper body, the upper
body from the lower body, and the lower body from the rear
frame. The rider representation, which plays the role of a feedback controller tasked with tracking a heading, is as a proportional controller. Simulations for a single-lane-change
maneuver are compared with measurements generated by 12
different riders. The results show that, for a running speed of
17 m/s, a good match can be obtained between the simulation
model with suitably chosen controller parameters and the
measured responses of the different riders. The results also
suggest that the most important control input is the steering
torque. While it is possible to control the motorcycle with
lower body lean movement, much larger torques are required
in this case. Normally, lower body control is utilized to assist
steering torque control, while the upper body is controlled
only to keep the rider in the comfortable upright position.
A complex rider model that comprises 12 rigid bodies
representing the upper and lower body, the upper and
lower arms, and the upper and lower legs, with appropriate mass and inertia properties is introduced in [76]. The
various rider model masses are restrained by linear
springs and dampers so that rider motions such as steering, rolling, pitching, weight shifting and knee gripping
are possible. Rider control actions associated with these
degrees of freedom are also modeled using proportional
control elements. Steady-state cornering and lane-changing maneuvers are studied.
Suspension and Cornering Models
Under steady-state cornering it is clear that a motorcycle’s
forward speed, yaw rate, lateral acceleration, and lean angle
Imaginary (rad/s)
Front Wheel Hop
−18 −16 −14 −12 −10 −8 −6
Real (1/s)
Front Suspension Pitch
−18 −16 −14 −12 −10 −8 −6
Real (1/s)
FIGURE 20 Motorcycle root locus plots: (a) straight-running and (b) 30° of roll angle with speed the varied parameter. The speed is increased
from (a) 5 m/s (), (b) 6 m/s () to 60 m/s ().
hand root locus plot in Figure 20. Cornering weave is similar in frequency to straight-running weave at high speeds,
but for the machine studied here, the weave-mode damping
decreases as the lean angle increases. The suspension system contributes significantly to the machine characteristics,
as observed experimentally. The influence of suspension
damping on the weave mode is demonstrated both analytically and experimentally in [66] and [77]. Under cornering,
the wobble mode involves suspension movement, and the
previously speed-independent suspension-pitch and wheelhop modes now vary markedly with speed. An interaction
between the front wheel-hop and wobble modes occurs
when the two modes are close enough in terms of natural
frequency. This interaction is possibly linked to wheel patter,
which is known anecdotally [78]. The coupling of the inplane and out-of-plane motions also suggests the possibility
of road excitation signals being transmitted into the lateral
motions of the vehicle, causing steering oscillations [43].
The early literature [77] discusses the existence of a
modified weave mode that occurs under cornering conditions, where the suspension system plays an important role
in its initiation and maintenance. To investigate the effect of
suspension damping on cornering weave, [77] benchmarks
several front and rear suspension dampers in laboratory
experiments and riding tests. Motorcycle stability is found
to be sensitive to suspension damping characteristics, while
cornering weave instability is to some extent controllable
through rear suspension damper settings. As stated in [77],
“. . . slight road surface undulations exacerbate the problem,
which is generally confined to speeds above 60 mph and
roll angles in excess of 25 deg from pavement-perpendicular . . . ”. It is also found that, as the speed is increased, cornering weave is produced at smaller roll angles. A separate
study [79] demonstrates, using a simple analysis, the possibility of interaction between pitch and weave modes at
high speeds, where the lightly damped weave-mode natural frequency approaches that of the pitch mode. Although
for straight running the coupling of in-plane and out-of
plane motions is weak, for steady-state cornering the coupling between the two modes increases with increased roll
angle, indicating that the inclusion of pitch and bounce
freedoms in motorcycle models is essential for handling
studies involving cornering.
Cornering experiments described in [66] quantify the
influences of various motorcycle design parameters and
operating conditions on wobble and weave. Tests with a
range of motorcycles and riders are carried out for both
straight running and steady-state cornering. The wobble
mode, which is excited by a steering torque pulse input
from the rider, is seen to be self-sustained during handsoff straight running at a moderate speed of 18 m/s; the
measured wobble frequency is 5.4 Hz, which is lower than
the theoretical prediction. More importantly, under steadystate cornering, measurements of cornering weave
responses at 27 m/s, involving oscillations in the suspen-
sion system, indicate a frequency of 2.2 Hz, while at 36
m/s the frequency is 2.6 Hz. It is also found that the weave
oscillations die out once the rider reduces the roll angle.
Further, [66] also demonstrates that reduced rear suspension damping, increased rear loading, and increased speed
increases the tendency for the motorcycle to weave. As
predicted by theory, the frequency of wobble varies little
with speed, while that of weave increases with speed.
Significant steps in the theoretical analysis of motorcycle
behavior are documented in [57] and [58]. The model developed considers small perturbations about straight-running
conditions but also for the first time about steady-cornering
conditions. The model in [57] is used to calculate the eigenvalues of the small-perturbation linearized motorcycle,
where the results for straight running are consistent with
the conventional wisdom. The way the weave- and wobblemode characteristics are predicted as varying with speed is
conventional, with new front- and rear-suspension pitch
and wheel-hop modes almost independent of speed appearing. Under cornering conditions, the interaction of these
otherwise uncoupled modes produces more complicated
modal motions. The cornering weave and combined wheelhop/wobble modes are illustrated, and root loci are plotted
to observe the sensitivity of the results to parameter variations. Surprisingly, it is predicted that removing the suspension dampers hardly affects the stability of the cornering
weave mode, contrary to the experiences of [66] and [77].
One of the original aims of [44] is to investigate the
apparent conflict between the results of [80] on the negligible influence of suspension damping on the stability of
cornering weave and the anecdotal and experimental evidence of [66] and [77]. Cornering root loci with the rear
suspension damping varied are reproduced and the damping is found to have a significant influence, indicating a
probable error in the calculations in [80]. The model presented in [44] is enhanced in [42] to include magic formula
tire models with the additional features included in [81].
The influence of the front suspension system on the ride
qualities of a motorcycle is studied in [82]. A typical suspension unit is modeled on the basis of its inner structure and
functionality, which give rise to the spring forces, viscous
damping forces, friction forces, and oil lock forces. Sine-wave
excitation experiments show that the model represents the unit
accurately. Further experiments are conducted, this time to
check the validity of the fork unit model combined with a simplified motorcycle model that comprises the vertical and longitudinal dynamics. The results obtained for riding over bumps
and under braking agree with measurements. The influence of
the suspension characteristics on riding qualities of the vehicle
are found by simulation; experiments verify the findings.
Experimental cornering results obtained from an instrumented motorcycle are presented in [83]. The motorcycle is
fitted with steering torque and angle transducers. Fiberoptic gyros are used to measure the roll rate and yaw rate,
and strain gauges provide tire force and moment data.
This paper provides experimental data that are used for
model qualification.
A study of the effects of road profiling on motorcycle
steering responses is presented in [43]. The results show that
under cornering conditions, regular low-amplitude road
undulations that would not trouble four-wheeled vehicles
can be a source of considerable difficulty to motorcycle riders. At low machine speeds, the wobble and front suspension pitch modes are likely to respond vigorously to
resonant vertical-displacement road forcing, while, at higher
speeds, the weave and front wheel-hop modes may be similarly affected. Connections between resonant responses and
a class of single-vehicle loss-of-rider-control accidents are
postulated. This work has several practical consequences.
First, these results appear to explain the key features of many
stability-related road traffic accidents reported in the popular literature and help to show why motorcycles that behave
perfectly well for long periods can suddenly suffer serious
and dangerous oscillation problems. Such oscillations are
likely to be difficult to reproduce and study experimentally.
Second, road builders and maintainers, as well as motorcycle
manufacturers, should be aware of the possibility of strong
resonant responses to small but regular undulations under
certain critical running conditions. These conditions are characterized by the machine speed, the lean angle, the rider’s
mass and posture, and the road profile wavelength.
Research and scholarship relating to single-track vehicles
involves, to a large extent, two separate communities that
can benefit from a higher level of interaction. One group
favors the use of simple bicycle models, while the other is
concerned with high-performance motorcycles and the
development of models with a high level of quantitative
predictive capability over a wide operating envelope.
The simple models can be regarded as derivatives or
simplifications of Whipple’s model. In these models, the
lateral motion constraints at the road contact are nonholonomic and thus special techniques may be needed to form
correct equations of motion. When the tire is regarded as
constraining the motion of the vehicle, the model validity
is restricted to low speeds (< 10 m/s), low frequencies
(< 1.0 Hz), and low tire-force utilization associated with
benign maneuvering (< 20% of capacity).
The model of Timoshenko and Young [29] represents the
lowest level of complexity of any potential usefulness; their
model has no rake, no trail, no inertias, no front frame mass,
and a point-mass representation of the rear frame. The Timoshenko-Young model leads one to conclude that the steer
angle and speed completely determine the lateral motion of
the base point of an inverted pendulum that represents the
vehicle’s roll dynamics. In terms of understanding singletrack vehicle steering, this level of modeling complexity is
too low, since unrealistic steer angle control must be accommodated. The self-steering influences, which are vital to the
operation of a real single-track vehicle, are completely
absent. Nevertheless, steer-displacement control inputs that
allow a prescribed path to be followed while the rolling
motion is properly stabilized have been optimized on simple Timoshenko-Young-type models and applied to sophisticated machine models with some success.
Whipple’s model, when linearized for constant-speed
straight running, yields two second-order equations of
motion in rolling and steering. The Whipple model, while
simple enough for control system optimization studies,
contains a sufficient level of physical realism to make it
credible. Physical influences deriving from the vehicle’s
design can be seen to combine in complex ways to give an
effective steering inertia, steering damping, and steering
stiffness. The Whipple model also provides an appreciation
of the complex interactions between the roll angle, roll
velocity, and roll acceleration, and the steering torque. Linear versions of Whipple-type models are useful for explaining nonminimum phase responses, the benefits of feedback,
and achievable robustness margins in single-track vehicles.
In models of the Timoshenko-Young type, the roll dynamics are driven kinematically by the steer angle, the steer velocity, the steer acceleration, and the vehicle acceleration inputs.
The advantage of these models is that considerable insight
into the stability and steering control of single-track vehicles,
within the model applicability boundaries, can be gained
from their separable design parameter influences. Although
such insights cannot easily be developed by referring to the
numerical results derived from more complex models, comprehensive models surely have their place in enabling effective virtual product design and testing across the full
operating envelope. The essential features of modern motorcycle design models include: 1) multiple rigid bodies and a
complex set of allowed motion freedoms; 2) detailed tire force
and moment models, incorporating static behavior up to and
possibly beyond the tire saturation limits, as well as transient
behavior; 3) case-dependent frame and rider compliances; 4)
suspension systems; 5) aerodynamic forces and moments;
and 6) detailed geometric models for accurately describing all
of the external forces. When motorcycles are ridden “on the
limit,” the stability and performance of the machine are
restricted by the properties of the tires, the suspension setup,
the weight distribution, the frame stiffness properties, and the
steering damping. A practical virtual design and testing facility must be able to accurately predict every feature of this
limit behavior.
In relation to trajectory tracking on the boundaries of
the vehicle’s capability—essentially the racing problem—it
is clear that performance is limited by tire force saturation
and transient dynamics, among other things. It appears to
be a considerable act of faith to regard the ultimate performance as calculable on the basis of nonholonomic rolling
constraints! In the future, we hope to see more elaborate
models, of the motorcycle-fraternity type, applied to minimum lap time and optimal-trajectory tracking problems.
Complexity-related difficulties, implicit in comprehensive
motorcycle modeling activities, are an exciting opportunity,
rather than a threat to be feared and avoided. Indeed, the
thoughtful use of powerful multibody modeling tools makes
routine the study of problems that would have been deemed
intractable only a decade ago. The challenges facing modelers
include a systematic approach to removing redundancy in
nonlinear models and the retention of key insights, which tend
to be obscured or even destroyed, in model reduction exercises. The challenges facing control theorists include: 1) the development of general theories for reducing complex nonlinear
models that guarantee the reduced-order model’s dynamic
fidelity; 2) removing assumptions that currently make general
control theories inapplicable to nonlinear mechanics problems,
and 3) the parallel development of computational platforms
that support complex controller synthesis applications.
The authors would like to thank Prof. Karl Åström, Prof. Malcolm Smith, and the reviewers for their helpful suggestions.
David J.N. Limebeer ([email protected]) received the
B.Sc. degree in electrical engineering from the University of
Witwatersrand, Johannesburg, in 1974, the M.Sc. and Ph.D.
degrees in electrical engineering from the University of Natal,
Durban, South Africa, in 1977 and 1980, respectively, and the
D.Sc. degree from the University of London in 1992. He has
been with Imperial College London since 1984, where he is
currently the head of the Department of Electrical and Electronic Engineering. He has published over 100 papers and a
textbook on robust control theory. Three of his papers have
been awarded prizes, including the 1983 O. Hugo Schuck
Award. He is a past editor of Automatica and a past associate
editor of Systems and Control Letters and the International Journal of Robust and Nonlinear Control. He is a Fellow of the IEEE,
the IEE, the Royal Academy of Engineering, and the City and
Guilds Institute. His research interests include control system
design, frequency response methods, H-infinity optimization
and mechanical systems. He is qualified as an IAM senior
motorcycle instructor and received a RoSPA certificate for
advanced motorcycling. He can be contacted at Imperial College London, Department of Electrical and Electronic Engineering, Exhibition Road, London SW7 2AZ U.K.
Robin S. Sharp is a professorial research fellow in the
Department of Electrical and Electronic Engineering at Imperial
College London. He is a member of the Dynamical Systems and
Mechatronics Working Group of the International Union of
Theoretical and Applied Mechanics, the editorial board of Vehicle System Dynamics, and the editorial advisory board of Multibody System Dynamics. He was first vice-president and secretary
general of the International Association for Vehicle System
Dynamics, editorial panel member and book review editor for
The Proceedings of the Institution of Mechanical Engineers, Journal of
Mechanical Engineering Science, Part C, and editorial panel mem-
ber of The Proceedings of the Institution of Mechanical Engineers,
Part D, Journal of Automobile Engineering. From 1990–2002, he
was professor of automotive product engineering at Cranfield
University. He was a visiting associate research scientist at the
University of Michigan Transportation Research Institute, Ann
Arbor. His research covers topics in automotive dynamics and
control and in control and stability of single-track vehicles,
unmanned air vehicles, and the application of optimal preview
and learning control to road vehicle driving/riding.
[1] A. Sharp, Bicycles and Tricycles: An Elementary Treatise on Their Design and
Construction. White Plains, NY: Longman, 1896. (Reprinted as: Bicycles and
Tricycles: A Classic Treatise on Their Design and Construction. Mineola, NY:
Dover, 1977.)
[2] D.V. Herlihy, Bicycle: The History. New Haven, CT: Yale Univ. Press, 2004.
[3] M. Hamer, “Brimstone and bicycles,” New Scientist, issue 2428, pp. 48–49,
Jan. 2005.
[4] N. Clayton, Early Bicycles. Princes Risborough, Bucks, UK: Shire Publications, 1986.
[5] F.J. Berto, The Dancing Chain: History and Development of the Derailleur
Bicycle. San Francisco, CA: Van der Plas Publications, 2005.
[6] “The Bicycle Museum of America” [Online]. Available: http://www.
[7] “Pedaling history bicycle museum” [Online]. Available: http://www.
[8] “Metz bicycle museum” [Online]. Available: http://www.
[9] “Canada science and technology museum” [Online]. Available:
[10] “Canberra bicycle museum” [Online]. Available: http://
[11] “National motorcycle museum” [Online]. Available: http://www.
[12] “London motorcycle museum” [Online]. Available:
[13] “Allen vintage motorcycle museum” [Online]. Available: http://www.
[14] “Sammy Miller museum” [Online]. Available:
[15] “Motorcycle hall of fame museum” [Online]. Available: http://
[16] W.J.M. Rankine, “On the dynamical principles of the motion of velocipedes,” The Engineer, pp. 2, 79, 129, 153, 175, 1869/1870.
[17] F.J.W. Whipple, “The stability of the motion of a bicycle,” Q. J. Pure
Appl. Math., vol. 30, pp. 312–348, 1899.
[18] H. Goldstein, Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.
[19] M.E. Carvallo, “Théorie du movement du monocycle, part 2: Théorie de
la bicyclette,” J. L’Ecole Polytechnique, vol. 6, pp. 1–118, 1901.
[20] F. Klein and A. Sommerfeld, Über die Theorie des Kreisels. Teubner,
Leipzig, 1910, Chap. IX, Sec. 8, “Stabilität des Fahrrads,” Leipzig, Germany:
B.G. Teubner, pp. 863–884.
[21] D.E.H. Jones, “The stability of the bicycle,” Phys. Today, vol. 23, no. 4, pp.
34–40, 1970.
[22] R.D. Roland, “Computer simulation of bicycle dynamics,” in Proc.
ASME Symp. Mechanics Sport, 1973, pp. 35–83.
[23] E. Döhring, “Stability of single-track vehicles,” Institut für Fahrzeugtechnik, Technische Hochschule Braunschweig, Forschung Ing.-Wes., vol. 21, no. 2,
pp. 50–62, 1955.
[24] J.I. Neĭmark and N.A. Fufaev, “Dynamics of nonholonomic systems,”
(Amer. Math. Soc. Translations Math. Monographs, vol. 33), 1972.
[25] R.S. Hand, “Comparisons and stability analysis of linearized equations
of motion for a basic bicycle model,” M.Sc. Thesis, Cornell Univ., 1988.
[26] A.L. Schwab, J.P. Meijaard, and J.M. Papadopoulos, “Benchmark results
on the linearized equations of motion of an uncontrolled bicycle,” in Proc.
2nd Asian Conf. Multibody Dynamics, Aug. 2004, pp. 1–9.
[27] K.J. Åström, R.E. Klein, and A. Lennartsson, “Bicycle dynamics and control,” IEEE Control Syst. Mag., vol. 25, no. 4, pp. 26–47, 2005.
[28] R.E. Klein, “Using bicycles to teach system dynamics,” IEEE Control
Syst. Mag., vol. 6, no. 4, pp. 4–9, 1989.
[29] S. Timoshenko and D.H. Young, Advanced Dynamics. New York:
McGraw-Hill, 1948.
[30] A.M. Letov, Stability in Nonlinear Control Systems. Princeton, NJ: Princeton Univ. Press, 1961.
[31] N.H. Getz, “Control of balance for a nonlinear nonholonomic nonminimum
phase model of a bicycle,” in Proc. American Control Conf., 1994, pp. 148–151.
[32] N.H. Getz and J.E. Marsden, “Control of an autonomous bicycle,” in
Proc. IEEE Conf. Robotics Automation, 1995, pp. 1397–1402.
[33] J. Hauser, A. Saccon, and R. Frezza, “Achievable motorcycle trajectories,”
in Proc. 43rd CDC, Paradise Island, Bahamas, 14–17 Dec. 2004, pp. 3944–3949.
[34] K.J. Åström, “Limitations on control system performance,” Euro. J. Control, vol. 6, no. 1, pp. 2–20, 1980.
[35] Anon., Autosim 2.5+ Reference Manual. Mech. Simulation Corp., Ann
Arbor MI, 1998 [Online]. Available:
[36] A.L. Schwab, J.P. Meijaard, and J.D.G. Kooijman, “Experimental validation of a model of an uncontrolled bicycle,” in Proc. III European Conf. Computational Mechanics: Solids, Structures Coupled Problems Engineering, Lisbon,
Portugal, 5-9 June 2006, paper 98.
[37] R.A. Wilson-Jones, “Steering and stability of single-track vehicles,” in
Proc. Auto. Div. Institution Mechanical Engineers, 1951, pp. 191–199.
[38] A.L. Schwab, J.P. Meijaard, and J. M. Papadopoulos, “A multibody
dynamics benchmark on the equations of motion of an uncontrolled bicycle,” in Proc. ENOC-2005, Eindhoven, The Netherlands Aug. 2005, pp. 1–11.
[39] J. Fajans, “Steering in bicycles and motorcycles,” Amer. J. Phys., vol. 68,
no. 7, pp. 654–659, 2000.
[40] R.S. Sharp and C.J. Alstead, “The influence of structural flexibilities on
the straight running stability of motorcycles,” Vehicle Syst. Dyn., vol. 9, no. 6,
pp. 327–357, 1980.
[41] P.T.J. Spierings, “The effects of lateral front fork flexibility on the vibrational modes of straight-running single-track vehicles,” Vehicle Syst. Dyn.,
vol. 10, no. 1, pp. 21–35, 1981.
[42] R.S. Sharp, S. Evangelou, and D.J.N. Limebeer, “Advances in the modelling
of motorcycle dynamics,” Multibody Syst. Dyn., vol. 12, no. 3, pp. 251–283, 2004.
[43] D.J.N. Limebeer, R.S. Sharp, and S. Evangelou, “Motorcycle steering oscillations due to road profiling,” J. Appl. Mech., vol. 69, no. 6, pp. 724–739, 2002.
[44] R. S. Sharp and D.J.N. Limebeer, “A motorcycle model for stability and
control analysis,” Multibody Syst. Dyn., vol. 6, no. 2, pp. 123–142, 2001.
[45] V. Cossalter and R. Lot, “A motorcycle multi-body model for real time
simulations based on the natural coordinates approach,” Vehicle Syst. Dyn.,
vol. 37, no. 6, pp. 423–447, 2002.
[46] R.S. Sharp, “The lateral dynamics of motorcycles and bicycles,” Vehicle
Syst. Dyn., vol. 14, no. 6, pp. 265–283, 1985.
[47] R.S. Sharp, “The stability and control of motorcycles,” J. Mech. Eng. Sci.,
vol. 13, no. 5, pp. 316–329, 1971.
[48] R.S. Sharp, “The stability of motorcycles in acceleration and
deceleration,” in Proc. Inst. Mech. Eng. Conf. Braking Road Vehicles, London:
MEP, 1976, pp. 45–50.
[49] D.J.N. Limebeer, R.S. Sharp, and S. Evangelou, “The stability of motorcycles
under acceleration and braking,” J. Mech. Eng. Sci., vol. 215, no. 9, pp. 1095–1109, 2001.
[50] H.B. Pacejka and R.S. Sharp, “Shear force development by pneumatic
tyres in steady state conditions: A review of modelling aspects,” Vehicle Syst.
Dyn., vol. 20, no. 3–4, pp. 121–176, 1991.
[51] H.B. Pacejka, Tyre and Vehicle Dynamics. Oxford, U.K.: Butterworth
Heinemann, 2002.
[52] S.K. Clark, Ed., Mechanics of Pneumatic Tires, 2nd ed. Washington DC:
NTIS, 1981.
[53] E.J.H. de Vries and H.B. Pacejka, “Motorcycle tyre measurements and
models,” Vehicle Syst. Dyn., vol. 29, pp. 280–298, 1998.
[54] E.J.H. de Vries and H.B. Pacejka, “The effect of tyre modeling on the stability analysis of a motorcycle,” in Proc. AVEC’98, Nagoya, Soc. Automotive
Engineers paper Japan, 1998, pp. 355–360.
[55] Y. Tezuka, H. Ishii, and S. Kiyota, “Application of the magic formula tire
model to motorcycle maneuverability analysis,” J. Soc. Auto. Eng. paper Rev.,
vol. 22, no. 3, pp. 305–310, 2001.
[56] E. Bakker, L. Nyborg, and H.B. Pacejka, “Tyre modelling for use in vehicle dynamics studies,” Soc. Auto. Eng., paper 870421, 1987.
[57] C. Koenen and H.B. Pacejka, “Vibrational modes of motorcycles in
curves,” in Proc. Int. Motorcycle Safety Conf., Washington, Motorcycle Safety
Foundation, 1980, vol. II, pp. 501–543.
[58] C. Koenen and H.B. Pacejka, “The influence of frame elasticity, simple
rider body dynamics, and tyre moments on free vibrations of motorcycles in
curves,” in Proc. 7th IAVSD Symp. Dynamics Vehicles Roads Railway Tracks,
Cambridge, 1981, pp. 53–65.
[59] V. Cossalter, A. Doria, R. Lot, N. Ruffo, and M. Salvador, “Dynamic
properties of motorcycle and scooter tires: Measurement and comparison,”
Vehicle Syst. Dyn., vol. 39, no. 5, pp. 329–352, 2003.
[60] H. Sakai, O. Kanaya, and H. Iijima, “Effect of main factors on dynamic
properties of motorcycle tires,” Soc. Auto. Eng., paper 790259, 1979.
[61] K.R. Cooper, “The effects of aerodynamics on the performance and stability of high speed motorcycles,” in Proc. 2nd AIAA Symp. Aerodynamics
Sport Competition Automobiles, Los Angeles, 1974.
[62] R.S. Sharp, “The influence of frame flexibility on the lateral stability of
motorcycles,” J. Mech. Eng. Sci., vol. 16, no. 2, pp. 117–120, 1974.
[63] D.J. Eaton, “Lateral dynamics of the uncontrolled motorcycle,” in Proc.
2nd Int. Congr. Automotive Safety, San Fransisco, 1973.
[64] M.K. Verma, R.A. Scott, and L. Segel, “Effect of frame compliance on the lateral dynamics of motorcycles,” Vehicle Syst. Dyn., vol. 9, no. 3, pp. 181–206, 1980.
[65] G.E. Roe and T.E. Thorpe, “A solution of the low-speed wheel flutter
instability in motorcycles,” J. Mech. Eng. Sci., vol. 18, no. 2, pp. 57–65, 1976.
[66] D.H. Weir and J.W. Zellner, “Experimental investigation of the transient
behaviour of motorcycles,” Soc. Auto. Eng., paper 790266, 1979.
[67] A. Clerx, “Stijfheid en sterkte van motorfietsframes,” Dept. Mech. Eng.,
Tech. Univ. Eindhoven, Tech. Rep., 1977.
[68] C.G. Giles and R.S. Sharp, “Static and dynamic stiffness and deflection
mode measurements on a motorcycle, with particular reference to steering
behaviour,” in Proc. Inst. Mech. Eng./MIRA Conf. Road Vehicle Handling, London, 1983, pp. 185–192.
[69] M.P.M. Bocciolone, F. Cheli, and R. Vigano, “Static and dynamic properties of a motorcycle frame: Experimental and numerical approach,” Dept.
Mech. Eng., Politecnico di Milano, Tech. Rep., 2005.
[70] R.S. Sharp, “Vibrational modes of motorcycles and their design parameter sensitivities,” in Proc. Int Conf. Vehicle NVH Refinement, Birmingham, 3–5
May 1994, pp. 107–121.
[71] B. Bayer, “Flattern und pendeln bei krafträdern,” Automobil Industrie,
vol. 2, pp. 193–197, 1988.
[72] S. Evangelou and D.J.N. Limebeer, “Lisp programming of the ‘sharp
1994’ motorcycle model,” 2000 [Online]. Available:
[73] T. Nishimi, A. Aoki, and T. Katayama, “Analysis of straight running stability of motorcycles,” in Proc. 10th Int. Technical Conf. Experimental Safety
Vehicles, Oxford, 1–5 July 1985, pp. 1080–1094.
[74] D.H. Weir and J.W. Zellner, “Lateral-directional motorcycle dynamics
and rider control,” Soc. Auto. Eng. paper 780304, pp. 7-31, 1978.
[75] T. Katayama, A. Aoki, and T. Nishimi, “Control behaviour of motorcycle riders,” Vehicle Syst. Dyn., vol. 17, no. 4, pp. 211–229, 1988.
[76] H. Imaizumi, T. Fujioka, and M. Omae, “Rider model by use of multibody
dynamics analysis,” Japanese Soc. Auto. Eng., vol. 17, no. 1, pp. 75–77, 1996.
[77] G. Jennings, “A study of motorcycle suspension damping
characteristics,” Soc. Auto. Eng., paper 740628, 1974.
[78] R.S. Sharp and C.G. Giles, “Motorcycle front wheel patter in heavy braking,” in Proc. 8th IAVSD Symp. Dynamics Vehicles Roads Railway Tracks,
Boston, 1983, pp. 578–590.
[79] R.S. Sharp, “The influence of the suspension system on motorcycle
weave-mode oscillations,” Vehicle Syst. Dyn., vol. 5, no. 3, pp. 147–154, 1976.
[80] C. Koenen, “The dynamic behaviour of motorcycles when running straight
ahead and when cornering,” Ph.D. dissertation, Delft Univ. Technol., 1983.
[81] R.S. Sharp, D.J.N. Limebeer, and M. Gani, “A motorcycle model for stability and control analysis,” in Proc. Euromech Colloquium 404, Advances Computational Multibody Dynamics, 1999, pp. 287–312.
[82] T. Kamioka, N. Yoshimura, and S. Sato, “Influence of the front fork on the
movement of a motorcycle,” in Proc. SETC’97, Yokohama, 1997, pp. 397–403.
[83] H. Ishii and Y. Tezuka, “Considerations of turning performance for
motorcycles,” in Proc. SETC’97, Yokohama, 1997, pp. 383–389.
[84] A. Hill, “Smith’s prizes: award,” Cambridge Univ. Reporter, p. 1027, 1899.
[85] D.G. Wilson, Bicycling Science. Cambridge, MA: MIT Press, 2004.
[86] C. Juden, “Shimmy,” Cycletouring, pp. 208–209, June/July 1988.
[87] R.S. Sharp and D.J.N. Limebeer, “On steering wobble oscillations of
motorcycles,” J. Mech. Eng. Sci., vol. 218, no. 12, pp. 1449–1456, 2004.
[88] T. Wakabayashi and K. Sakai, “Development of electronically controlled
hydraulic rotary steering damper for motorcycles,” in Proc. Int. Motorcycle
Safety Conf., Munich, 2004, pp. 1–22.
[89] J.P. Den Hartog, Mechanical Vibration. New York: Dover, 1985.
[90] P. Bandel and C. Di Bernardo, “A test for measuring transient characteristics of tires,” Tire Sci. Technol., vol. 17, no. 2, pp. 126–137, 1989.
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