Bicycle and motorcycle dynamics

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A simplified mathematical model of bike and rider demonstrating the dynamic of a right turn

Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles, in entirety or in parts, due to the forces acting on them during balancing, steering, braking, and suspension. Experimentation and mathematical analysis have shown that a bike stays upright when it is steered to keep its center of mass over its wheels. This steering is usually supplied by a rider, or in certain circumstances, by the bike itself. Long-standing hypotheses and claims that gyroscopic effect is the main stabilizing force have been refuted.^[1]^[2] While remaining upright may be the primary goal of beginning riders, a bike must lean in order to turn. The higher the speed or smaller the turn radius, the more lean is required. This is necessary in order to balance centrifugal forces due to the turn with gravitational forces due to the lean. When braking, depending on the location of the combined center of mass of the bike and rider with respect to the point where the front wheel contacts the ground, bikes can either skid the front wheel or flip the bike and rider over the front wheel.

[edit] Forces

The forces that act on a bike and its components include gravity, ground reaction (usually at the front and rear wheel contact patches with both horizontal and vertical components), centrifugal (in the accelerating reference frame of the bike and rider; in a stationary, inertial reference frame this is simply inertia and not a force at all), gyroscopic (due to rotating parts such as wheels, engine, transmission, etc.), propulsive (usually at the rear wheel contact patch), torques applied between the steering mechanism (front fork, handlebars, front wheel, etc.) and rear frame, and between the rider and the rear frame, friction (between any parts that move against each other: at the tire contact patches, as mentioned above, in the drive train, between the steering mechanism and the rear frame, etc.), and aerodynamic (mostly in the form of drag, but also possibly from a crosswind).

[edit] Balance

A bike remains upright when it is steered so that the ground reaction forces exactly balance all the other forces it experiences such as gravitational, inertial or centrifugal if in a turn, and aerodynamic if in a crosswind.^[3] Steering may be supplied by a rider or, under certain circumstances, by the bike itself. This self-stability is generated by a combination of several effects that depend on the geometry, mass distribution, and forward speed of the bike. Tires, suspension, steering damping, and frame flex can also influence it, especially in motorcycles.

If the steering of a bike is locked, it becomes virtually impossible to ride. Instead, if just the gyroscopic effect of rotating bike wheels is cancelled by adding counter-rotating wheels, it can still be easily ridden.^[1]^[2]

[edit] Forward speed

Steering inputs counter lean to maintain balance. At high speeds, the input required to return the bike to upright only needs to be small; a much greater input is required to maintain balance at low speed. As such, it is easier to maintain balance at high speeds.^[4]

[edit] Center of mass location

The farther forward (closer to front wheel) the center of mass of the combined bike and rider, the less the front wheel has to move laterally in order to maintain balance. Conversely, the further back (closer to the rear wheel) the center of mass is located, the more front wheel lateral movement or bike forward motion will be required to regain balance. This can be noticeable on long-wheelbase recumbents and choppers.

A bike is also an example of an inverted pendulum. Thus, just as a broomstick is easier to balance than a pencil, tall bikes (with a high center of mass) can be easier to balance than short ones because their lean rate will be slower.^[5]

[edit] Trail

Bicycle head angle, rake, and trail

A factor that influences how easy or hard a bike will be to ride is trail, the distance by which the front wheel ground contact point trails behind the point where a line through the steering axis, the axis about which the steering mechanism (fork, handlebars, front wheel, etc.) pivots, intersects the ground. In traditional bike designs, with a steering axis tilted back from the vertical, trail causes the front wheel to steer into the direction of a lean, independent of forward speed.^[3] This can be seen by pushing a stationary bike to one side. The front wheel will usually also steer to that side. In a lean, gravity provides this force.

The more trail a bike has, the more stable it feels. Bikes with negative trail (where the contact patch is actually in front of where the steering axis intersects the ground), while still ridable, feel very unstable. Bikes with too much trail feel difficult to steer. Normally, road racing bicycles have more trail than mountain bikes or touring bikes. In the case of mountain bikes, less trail allows more accurate path selection off-road, and also allows the rider to recover from obstacles on the trail which might knock the front wheel off-course. Touring bikes are built with small trail to allow the rider to control a bike weighed down with luggage. As a consequence, an unloaded touring bike can feel rather unstable to ride. In bicycles, fork rake, often a curve in the fork blades forward of the steering axis, is used to diminish trail.^[6] In motorcycles, rake instead refers to the head angle, and offset created by the triple tree is used to diminish trail.^[7]

Trail is a function of head angle, fork offset or rake, and wheel size. Their relationship can be described by this formula:^[8]

$Trail = \frac{(R_w \cos(A_h) - O_f)}{\sin(A_h)}$

where $R w$ wheel radius, $A h$ is the head angle measured clock-wise from the horizontal and $O f$ is the fork offset or rake. Trail can be increased by increasing the wheel size, decreasing or slackening the head angle, or decreasing the fork rake.

A small survey by Whitt and Wilson^[3] found:

touring bicycles with head angles between 72° and 73° and trail between 43 mm and 60 mm
racing bicycles with head angles between 73° and 74° and trail between 28 mm and 45 mm
track bicycles with head angles of 75° and trail between 23.5 mm and 37 mm

At the same time, Lemond offers,^[9] both with forks that have 45 mm of offset or rake and the same size wheels:

a 2007 Filmore, designed for the track, with a head angle that varies from 72.5° to 74° depending on frame size
a 2006 Tete de Course, designed for road racing, with a head angle that varies from 71.25° to 74°, depending on frame size

[edit] Steering mechanism mass distribution

Another factor that can also contribute to the self-stability of traditional bike designs is the distribution of mass in the steering mechanism, which includes the front wheel, the fork and the handlebar. If the center of mass for the steering mechanism is forward of the steering axis, then the pull of gravity will also cause the front wheel to steer in the direction of a lean. This can be seen by leaning a stationary bike to one side. The front wheel will usually also steer to that side independent of any interaction with the ground.^[10]

More subtle effects, such as the fore-to-aft position of the center of mass and the elevation of the center of mass also contribute to the dynamic behavior of a bike.^[10]^[3]

[edit] Gyroscopic effects

The role of the gyroscopic effect in most bike designs is to help steer the front wheel into the direction of a lean. This phenomenon is called precession and the rate at which an object precesses is inversely proportional to its rate of spin. The slower a front wheel spins, the faster it will precess when the bike leans, and vice-versa.^[11] The rear wheel is prevented from precessing as the front wheel does by friction of the tires on the ground, and so continues to lean as though it were not spinning at all. Hence gyroscopic forces do not provide any resistance to tipping.

At low forward speeds, the precession of the front wheel is too quick, contributing to an uncontrolled bike’s tendency to oversteer, start to lean the other way and eventually oscillate and fall over. At high forward speeds, the precession is usually too slow, contributing to an uncontrolled bike’s tendency to understeer and eventually fall over without ever having reached the upright position.^[12] This instability is very slow, on the order of seconds, and is trivial to counteract for most riders. Thus a fast bike may feel stable even though it is actually not self-stable and would fall over if it were uncontrolled.

[edit] Self stability

Between these two extremes, there may be a range of forward speeds for a given bike design at which the effects described above steer an uncontrolled bike upright.^[13] However, even without self-stability a bike may be ridden by steering it to keep it over its wheels.^[2]

See a video of a riderless bicycle exhibiting this self-stability.

Note that the effects mentioned above that combine to produce self stability may be overwhelmed by additional factors such as headset friction and stiff control cables.^[3]

[edit] Instability

Bikes, as complex mechanisms, have a variety of unstable modes: ways that they can be unstable. Some of the names given to these instabilities include capsize, weave, and wobble. These different modes are differentiated by the speed at which they occur and the relative phases of leaning and steering.

In this context, "stability" is used to mean that an uncontrolled bike will continue rolling forward without eventually falling over. Conversely "instability" means that an uncontrolled bike will eventually fall over.

[edit] Modes

There are three main unstable modes that a bike can experience: capsize, weave, and wobble. A lesser known mode is rear wobble, and it is usually stable.

[edit] Capsize

Capsize is the word used to describe a bike falling over, without oscillation. An uncontrolled front wheel usually steers in the direction of lean during capsize until a very high lean angle is reached, at which point the steering may turn in the opposite direction. A capsize can happen very slowly if the bike is moving forward rapidly. Because the capsize instability is so slow, on the order of seconds, it is easy for the rider to control, and is actually used by the rider to initiate the lean necessary for a turn.^[14]

[edit] Weave

Weave is the word used to describe a slow (0-4 Hz) oscillation between leaning left and steering right and vice-versa. The entire bike is affected with significant changes in steering angle, lean angle (roll), and heading angle (yaw). The steering is 180° out of phase with the heading and 90° out of phase with the leaning.^[14]

For most bikes, depending on geometry and mass distribution, weave is unstable at low speeds, and becomes less pronounced as speed increases until it is no longer unstable. While the amplitude may decrease, the frequency actually increases with speed.

Here is a video that shows weave: AVI movie

[edit] Wobble or shimmy

Wobble, shimmy, tank-slapper, speed wobble, and even death wobble are all words and phrases used to describe a quick (4 - 10 Hz) oscillation of primarily just the front end (front wheel, fork, and handlebars). The rest of the bike remains mostly unaffected. This instability occurs mostly at high speed and is similar to that experienced by shopping cart wheels, airplane landing gear, and automobile front wheels.^[12]^[14]

While wobble or shimmy can be easily remedied by adjusting speed, position, or grip on the handlebar, they can be fatal if left uncontrolled.^[15]

Wobble or shimmy begins when some otherwise minor irregularity accelerates the wheel to one side. The restoring force is applied in phase with the progress of the irregularity, and the wheel turns to the other side where the process is repeated. If there is insufficient damping in the steering the oscillation will increase until system failure. Speed changes, making the bike stiffer or lighter, or increasing the stiffness of the steering (of which the rider is the main component) can change the oscillation frequency, though only speed change is applicable in the situation.^[3]

Here is a video that shows wobble: AVI movie

[edit] Rear wobble

The term rear wobble is used to describe a mode of oscillation in which lean angle (roll) and heading angle (yaw) are almost in phase and both 180° out of phase with steer angle. The rate of this oscillation is moderate with a maximum of about 6.5 Hz. Rear wobble is heavily damped and falls off quickly as bike speed increases.^[14]

[edit] Design criteria

The design characteristics of a bike can affect these instabilities in the following ways:^[16]

[edit] Frame stiffness

Lateral and torsional stiffness of the rear frame and the wheel spindle affect the wobble-mode damping substantially with significant changes in the wobble frequency and slight reduction in the weave-mode damping at high speeds.

[edit] Geometry

Long wheelbase and trail and a flat steering-head angle have been found to increase weave-mode damping. Lateral distortion can be countered by locating the front fork torsional axis as low as possible.

[edit] Suspension

Degraded damping of the rear suspension amplify cornering weave tendencies.

[edit] Tire characteristics

Cornering and camber stiffnesses and relaxation length of the rear tire, make the largest contribution to weave damping, but not so much from the same parameters of the front tire. Inflation pressures are also important variables in the behavior of a motorcycle at high speeds.

[edit] Loading

Rear loading amplifies cornering weave tendencies. Rear load assemblies with appropriate stiffness and damping were successful in damping out weave and wobble oscillations.

[edit] Turning

The forces acting on a leaning bike in the rotating reference frame of a turn where

N

is the normal force,

F

is friction, and

m

is mass

In order to turn, that is change their direction of forward travel, bikes must lean to balance the relevant forces: gravitational, inertial, frictional, and ground support. The angle of lean, $θ$ , can easily be calculated using the laws of circular motion:

$\theta = \arctan \left (\frac{gr}{v^2} \right )$

where $v$ is the forward speed, $r$ is the radius of the turn and $g$ is the acceleration of gravity.^[11]

For example, a bike in a 10 m (33 ft) radius steady-state turn at 10 m/s (22 mph) must be at an angle of ca. 45°. A rider can lean with respect to the bike in order to keep either the torso or the bike more or less upright if desired. The angle that matters is the one between the horizontal plane and the plane between the tire contacts and the location of the center of mass of bike and rider.

[edit] Countersteering

Main article: Countersteering

In order to initiate a turn, a bike must momentarily steer in the opposite direction. This is often referred to as countersteering. This brief turn moves the wheels out from directly underneath the center of mass, and thus causes a lean in the desired direction. Where there is no external influence such as an opportune side wind to create the force necessary to lean the bike, countersteering happens in every turn.^[11]

As the lean approaches the desired angle, the front wheel must be steered in the direction of the turn, depending on the forward speed, the turn radius, and the need to maintain the lean angle. Once in a turn, the radius can only be changed with an appropriate change in lean angle, and this can only be accomplished by additional countersteering out of the turn to increase lean and decrease radius, then into the turn to decrease lean and increase radius. To exit the turn, the bike must again countersteer and momentarily steer more into the turn to decrease the radius to increase inertial forces in order decrease the angle of lean.^[17]

[edit] No hands

While countersteering is usually initiated by applying torque directly to the handlebars, on lighter vehicles such as bicycles, it can also be accomplished by shifting the rider’s weight. If the rider leans to the right relative to the bike, the bike will lean to the left to conserve angular momentum, and the combined center of mass will remain in the same vertical plane. This leftward lean of the bike will cause it to steer to the left and initiate a right hand turn as if the rider had countersteered to the left by applying a torque directly to the handle bars.^[11]

Note that this technique may be complicated by additional factors such as headset friction and stiff control cables.

[edit] Two-wheel steering

Because of theoretical benefits, such as a tighter turning radius at low speed, attempts have been made to construct motorcycles with two-wheel steering. One working prototype by Ian Drysdale in Australia is reported "by all accounts it seems to work very well."^[18]^[19]

Issues in the design include whether to provide active control on the rear wheel or let it swing freely. In the case of active control, the control algorithm needs to decide between steering with or opposite of the front wheel, when, and how much.

The Sideways bike is another implementation of two-wheel steering which lets the rider control the steering of both wheels directly.

[edit] Rear-wheel steering

Because of the theoretical benefits, especially a simplified front-wheel drive mechanism, attempts have been made to construct a ridable rear-wheel steering bike. For example, the Bendix Company built a rear-wheel steering bicycle, and the U.S. Department of Transportation commissioned construction of a rear-wheel steering motorcycle: both proved to be unridable. One documented case of someone successfully riding a rear-wheel steering bicycle is of L. H. Laiterman at MIT on a specially designed recumbent.^[3]

Rainbow Trainers, Inc. in Alton, IL, offers "A cash prize of US$5,000 ... to the first living person, hereafter referred to as the challenger, who can successfully ride the rear-steered bicycle, Rear Steered Bicycle I."^[20]

The difficulty is due to the fact that turning left, accomplished by turning the rear wheel to the right, initially moves the center of mass to the right, and vice versa. This makes compensating for leans induced by the environment tricky.^[21] Examination of the eigenvalues shows that the configuration is inherently unstable.

[edit] Center steering

Between classical front-wheel steering, and strictly rear-wheel steering, is a class of bikes with a pivot point somewhere between these two extremes and referred to as center-steering. These design allow for simple front-wheel drive and appear to be quite stable, even ridable no-hands, as many photographs attest.^[22]^[23] These designs usually have very lax head angles (40° to 65°) and positive or even negative trail. The builder of a bike with negative trail states that steering the bike from straight ahead forces the seat (and thus the rider) to rise slightly and this offsets the destabilizing effect of the negative trail.^[24]

[edit] Tiller effect

Tiller effect is the expression used to describe how handlebars that extend far behind the steering axis (head tube) act like a tiller on a boat in that one moves the bars to the right in order to turn the front wheel to the left, and visa versa. This situation is commonly found on cruisers, some recumbents, and even some cruiser motorcycles. It can be problematic when it limits the ability to steer because of interference or the limits of arm reach.^[25]

[edit] Braking

Most of the braking force of standard upright bikes comes from the front wheel. If the brakes themselves are strong enough, the rear wheel is easy to skid, while the front wheel often can generate enough stopping force to flip the rider and bike over the front wheel. This is called a stoppie or an ‘endo’. However, long or low bikes, such as cruiser motorcycles and recumbent bicycles, can also skid the front tire, causing a loss of the ability to balance.

[edit] Front wheel braking

The limiting factors in front wheel braking are:

the maximum, limiting value of static friction between the tire and the ground,
the kinetic friction between the brake pads and the rim or disk,
pitching over the front wheel.

For an upright bicycle on dry asphalt with excellent brakes, pitching will apply. The center of mass on a typical diamond frame will be about 600 mm back from the front wheel contact patch and 1200 mm above, allowing a maximum deceleration of 0.5 g (4.9 m/s² or 16 ft/s²).^[3] However, if the rider modulates the brakes properly, pitching can be avoided. If the rider moves his weight back and down even larger decelerations are possible.

Front brakes on most cheap bikes are not strong enough so, on the road, they are the limiting factor. Cheap cantilever brakes, especially with "power modulators", severely restrict the stopping forc. So do Raleigh-style side-pull brakes. In wet conditions they are even less effective.

Front wheel slides are more common off-road. Mud, water and loose stones reduce the friction between the tire and road. Knobby tires can mitigate this effect by grabbing the surface irregularities.

Front wheel slides are also common on corners, whether on road or off. This is because the centripetal acceleration adds to the forces on the tire/ground contact. When the friction is exceeded the wheel slides.

[edit] Rear wheel braking

The rear brake of a diamond frame bicycle can only give about 0.1 g deceleration at best.^[3] This is because of the braking force is not in line with the center of mass. This creates a pitching moment that reduces the rear-wheel contact force, and so limits the friction force.

On motorcycles, light rear-wheel braking, sometimes referred to as dragging, can be a useful aid to stability when turning at low speed.^{[citation needed]}

[edit] Longitudinal stability

Mechanical analysis of the forces generated by a bike with a wheelbase $L$ and a center of mass halfway between the wheels and at height $h$ , with both wheels locked, reveals that the normal (vertical) forces at the wheels are:^[26]

$N_r = mg\left(\frac{1}{2} - \mu \frac{h}{L}\right)$

for the rear wheel and

$N_f = mg\left(\frac{1}{2} + \mu \frac{h}{L}\right)$

for the front wheel, while the frictional (horizontal) forces are simply $F r = μ N r$ for the rear wheel and $F f = μ N f$ for the front wheel, where $μ$ is the coefficient of friction, $m$ is the mass and $g$ is the acceleration of gravity. Thus if

$\mu \frac{h}{L} > \frac{1}{2}$

then the normal force of the rear wheel will be negative and the bike will flip over.

The coefficient of friction of rubber on dry asphalt is between 0.5 and 0.8.^[27] Using the lower value, and if the center of mass height is greater than or equal to the wheel base, the front wheel can generate sufficient stopping force to flip the bike and rider over.

On the other hand, if the center of mass height is less than half the wheelbase and at least halfway towards the rear wheel, for example a tandem or a long-wheel-base recumbent, then even if the coefficient of friction is 1.0, it is impossible for the front wheel to generate enough braking force to flip the bike. It will skid unless it hits some fixed obstacle such as a curb.

A riding technique that takes advantage of how braking increases the downward force on the front wheel is known as trail braking.

In the case of a front suspension, especially telescoping fork tubes, this increase in downward force on the front end may cause the suspension to compress and the front end to lower. This is known as brake diving.

[edit] Theory

Although its equations of motion can be linearized, a bike is a nonlinear system: its behavior is not expressible as a sum of the behaviors of its descriptors.^[28]

In the idealized case, in which friction and any flexing is ignored, a bike is a conservative system. However, and perhaps surprisingly, it can still demonstrate damping: side-to-side oscillations will decrease with time. Energy added with a sideways jolt to a bike running straight and upright (demonstrating self stability) is converted into increased forward speed, not lost, as the oscillations die out.

A bike is a nonholonomic system because its outcome is path-dependent. In order to know its exact configuration, especially location, it is necessary to know not only the configuration of its parts, but also their histories: how they moved over time. This complicates mathematical analysis.^[11]

Finally, in the language of control theory, a bike exhibits non-minimum phase behavior.^[29] It turns in the direction opposite of how it is initially steered, as described above in the section on countersteering

[edit] Equations of motion

The equations of motion of an idealized bike, consisting of

a rigid frame,
a rigid fork,
two knife-edged, rigid wheels,
all connected with frictionless bearings and rolling without friction or slip on a smooth horizontal surface and
operating at or near the upright and straight ahead unstable equilibrium

can be represented by two linearized second-order ordinary differential equations,^[10] the lean equation

$M_{\theta\theta}\ddot{\theta_r} + K_{\theta\theta}\theta_r + M_{\theta\psi}\ddot{\psi} + C_{\theta\psi}\dot{\psi} + K_{\theta\psi}\psi = M_{\theta}$

and the steer equation

$M_{\psi\psi}\ddot{\psi} + C_{\psi\psi}\dot{\psi} + K_{\psi\psi}\psi + M_{\psi\theta}\ddot{\theta_r} + C_{\psi\theta}\dot{\theta_r} + K_{\psi\theta}\theta_r = M_{\psi}\mbox{,}$

where

$θ r$ is the lean angle of the rear assembly,
$ψ$ is the steer angle of the front assembly relative to the rear assembly and
$M θ$ and $M ψ$ are the moments (torques) applied at the rear assembly and the steering axis, respectively. For the analysis of an uncontrolled bike, both are taken to be zero.

These can be represented in matrix form as $M\ddot{q}+C\dot{q}+Kq=f$ where

$M$ is the symmetrical mass matrix which contains terms that include only the mass and geometry of the bike,
$C$ is the so-called damping matrix, even though an idealized bike has no dissipation, which contains terms that include the forward speed $V$ and is asymmetric,
$K$ is the so-called stiffness matrix which contains terms that include the gravitational constant $g$ and $V 2$ and is symmetric in $g$ and asymmetric in $V 2$ ,
$q$ is a vector of lean angle and steer angle, and
$f$ is a vector of external forces, the moments mentioned above.

In this idealized and linearized model, there are many geometric parameters (wheelbase, head angle, mass of each body, wheel radius, etc.), but only four significant variables: lean angle, lean rate, steer angle, and steer rate. These equations have been verified by comparison with multiple numeric models derived completely independently.^[13]

[edit] Eigenvalues

Eigenvalues plotted against forward speed for a typical utility bicycle

It is possible to calculate eigenvalues, one for each of the four significant variables, from the linearized equations to analyze the self-stability of a particular bike design. In the plot to the right, eigenvalues are calculated for forward speeds of 0–10 m/s (22 mph). When the real parts of all eigenvalues (shown in dark blue) are negative, the bike is self-stable. When the imaginary parts of any eigenvalues (shown in cyan) are non-zero, the bike exhibits oscillation.

The forward speed at which oscillations do not increase, eventually causing the uncontrolled bike to fall over, is called the weave speed. The forward speed at which non-oscillatory leaning does not increase, eventually causing the uncontrolled bike to fall over, is called the capsize speed.^[28] Between these two speeds, if they both exist, is a range of forward speeds at which the particular bike design is self-stable. In the case of the bike whose eigenvalues are shown here, the self-stable range is 5.3–8.0 m/s (12-18 mph).

Note that this idealized model does not exhibit the wobble or shimmy and rear wobble instabilities described above. They are seen in models that incorporate tire interaction with the ground.^[14]

[edit] Experimentation

A variety of experiments have been performed in order to verify or disprove various hypotheses about bike dynamics.

David Jones built several bikes in a search for an unridable configuration.^[2]
Richard Klein built several bikes to confirm Jones's findings.^[1]
Richard Klein also built a "Torque Wrench Bike" and a "Rocket Bike" to investigate steering torques and their effects.^[1]
Keith Code built a motorcycle with fixed handlebars to investigate the effects of rider motion and position on steering.^[30]
Schwab and Kooijman have performed measurements with an instrumented bike.^[31]

[edit] Other hypotheses

Although bicycles and motorcycles can appear to be simple mechanisms with only four major moving parts (frame, fork, and two wheels), these parts are arranged in a way that makes them quite complicated to analyze.^[3]

While it is an observable fact that bikes can be ridden even when the gyroscopic effects of their wheels are canceled out,^[2]^[1] the hypothesis that the gyroscopic effects of the wheels are what keep a bike upright is common in print and online.^[1]^[11]

Examples in print:

"Angular momentum and motorcycle counter-steering: A discussion and demonstration", A. J. Cox, Am. J. Phys. 66, 1018–1021 ~1998
"The motorcycle as a gyroscope", J. Higbie, Am. J. Phys. 42, 701–702
The Physics of Everyday Phenomena, W. T. Griffith, McGraw–Hill, New York, 1998, pp. 149–150.
The Way Things Work., Macaulay, Houghton-Mifflin, New York, NY, 1989

And online:

[edit] See also

[edit] References

^ ^a ^b ^c ^d ^e ^f Klein, Richard E.; et al. Bicycle Science. Retrieved on 2006-08-04.
^ ^a ^b ^c ^d ^e Jones, David E. H. (1970). "The stability of the bicycle" (PDF). Physics Today 23 (4): 34–40. Retrieved on 2006-08-04.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j Whitt, Frank R.; David G. Wilson (1982). Bicycling Science, Second edition, Massachusetts Institute of Technology, 198–233. ISBN 0-262-23111-5.
^ Fajans, Joel. Email Questions and Answers: Balancing at low speeds. Retrieved on 2006-08-23.
^ Fajans, Joel. Email Questions and Answers: Robot Bicycles. Retrieved on 2006-08-04.
^ Zinn, Lennard. "Technical Q&A with Lennard Zinn — Rake, trail, offset", Velo News, December 21, 2004. Retrieved on 2006-08-04.
^ Foale, Tony (1997). Balancing Act. Retrieved on 2006-08-04.
^ Putnam, Josh (2006). Steering Geometry: What is Trail?. Retrieved on 2006-08-08.
^ Lemond Racing Cycles (2006). Retrieved on 2006-08-08.
^ ^a ^b ^c Hand, Richard S. (1988). Comparisons and Stability Analysis of Linearized Equations of Motion for a Basic Bicycle Model (PDF). Retrieved on 2006-08-04.
^ ^a ^b ^c ^d ^e ^f Fajans, Joel (July 2000). "Steering in bicycles and motorcycles" (PDF). American Journal of Physics 68 (7): 654–659. Retrieved on 2006-08-04.
^ ^a ^b Wilson, David Gordon; Jim Papadopoulos (2004). Bicycling Science, Third Edition, The MIT Press, 263-390. ISBN 0-262-73154-1.
^ ^a ^b Schwab, Arend L.; Jaap P. Meijaard, Jim M. Papadopoulos (2005). "Benchmark Results on the Linearized Equations of Motion of an Uncontrolled Bicycle" (PDF). KSME International Journal of Mechanical Science and Technology 19 (1): 292–304. Retrieved on 2006-08-04.
^ ^a ^b ^c ^d ^e Cossalter, Vittore (2006). Motorcycle Dynamics, Second Edition, Lulu.com, 241-342. ISBN 978-1-4303-0861-4.
^ Kettler, Bill. "Crash kills cyclist", Mail Tribune, 2004-09-15. Retrieved on 2006-08-04.
^ Evangelou, Simos (2004). The Control and Stability Analysis of Two-wheeled Road Vehicles (PDF) 159. Imperial College London. Retrieved on 2006-08-04.
^ Brown, Sheldon (2006). Sheldon Brown's Bicycle Glossary. Retrieved on 2006-08-08.
^ Ian Drysdale. Retrieved on 2006-12-14.
^ Foale, Tony (1997). 2 Wheel Drive/Steering. Retrieved on 2006-12-14.
^ Klein, Richard E.; et al. (2005). Challenge. Retrieved on 2006-08-06.
^ Wannee, Erik (2005). Rear Wheel Steered Bike. Retrieved on 2006-08-04.
^ Wannee, Erik (2001). Variations on the theme 'FlevoBike'. Retrieved on 2006-12-15.
^ Mages, Jürgen (2006). Python Gallery. Retrieved on 2006-12-15.
^ Mages, Jürgen (2006). Python Frame Geometry. Retrieved on 2006-12-15.
^ Brown, Sheldon (2006). Sheldon Brown's Bicycle Glossary. Retrieved on 2006-08-08.
^ Ruina, Andy; Rudra Pratap (2002). Introduction to Statics and Dynamics (PDF), Oxford University Press, 350. Retrieved on 2006-08-04.
^ Kurtus, Ron (2005-11-02). Coefficient of Friction Values for Clean Surfaces. Retrieved on 2006-08-07.
^ ^a ^b Meijaard, J. P.; et al. (2006). Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review. Retrieved on 2006-12-22.
^ Klein, Richard E.; et al. (2005). Counter-Intuitive.. Retrieved on 2006-08-07.
^ Gromer, Cliff. "STEER GEAR So how do you actually turn a motorcycle?", Popular Mechanics, February 1, 2001. Retrieved on 2006-08-07.
^ Schwab, Arend; et al. (2006). Bicycle Dynamics. Retrieved on 2006-08-07.

[edit] Additional reading

‘An Introduction to Bicycle Geometry and Handling’, Karl Anderson, C.h.u.n.k. 666
‘Interesting Bike Science’ by Paul Pedriana
‘What keeps the bicycle upright?’ by Jobst Brandt
‘Report on Stability of the Dahon Bicycle’ by John Forester

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