NationMaster - Encyclopedia: Bicycle and motorcycle dynamics

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links Portal. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Bicycle (disambiguation). ... For other uses, see Motorcycle (disambiguation). ... For other uses, see Force (disambiguation). ... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... Countersteering is the name given to the counter-intuitive technique used by cyclists and motorcyclists to turn corners. ... This article is about the vehicle component. ... The front suspension components of a Ford Model T. Suspension is the term given to the system of springs, shock absorbers and linkages that connects a vehicle to its wheels. ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to retain or falsify a hypothesis or research concerning phenomena. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... A gyroscope is a device which demonstrates the principle of conservation of angular momentum, in physics. ... This article does not cite any references or sources. ... This article is about an authentication, authorization, and accounting protocol. ... Centrifugal force (from Latin centrum centre and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... This article covers the physics of gravitation. ...

History

The history of the study of bike dynamics is nearly as old as the bicycle itself. It includes contributions from famous scientists such as Rankine, Appell, and Whipple. In the early 1800s Karl von Drais himself showed that a rider could balance his device by steering the front wheel. By the end of the 1800s, Emmanuel Carvallo and Francis Whipple showed with rigid-body dynamics that some safety bicycles could actually balance themselves if moving at the right speed. It is not clear to whom should go the credit for tilting the steering axis from the vertical which helps make this possible. In 1970, David Jones published an article in Physics Today that showed that gyroscopic effects are not necessary to balance a bicycle. In 2007, Meijaard, Papadopoulos, Ruina, and Schwab published the canonical linearized equations of motion, in the Proceedings of the Royal Society A, along with verification by two different methods.^[3] William John Macquorn Rankine (July 2, 1820 - December 24, 1872) was a Scottish engineer and physicist. ... Paul Appell (September 27, 1855 in Strasbourg â€“ October 24, 1930 in Paris), also known as Paul Appel, was a French mathematician and Rector of the University of Paris. ... Francis John Welsh Whipple (1876-1943) was a British mathematician and meteorologist. ... Karl Drais ca 1820, then still a baron Karl Drais (April 29, 1785 â€“ December 10, 1851) was a German inventor and invented the Laufmaschine (running machine), also later called the velocipede, draisine (English) or draisienne (French), or nick-named, dandy horse. ... It has been suggested that SUVAT equations be merged into this article or section. ... For other uses, see Royal Society (disambiguation). ...

Forces

If the bike and rider are considered to be a single system, the forces that act on that system and its components can be roughly divided into two groups: internal and external. The external forces are due to gravity, inertia, contact with the ground, and contact with the atmosphere. The internal forces are due to the rider and interaction between components.

External forces

As with all masses, gravity pulls the rider and all the bike components toward the earth. There is also a gravitational attraction between each component, but this is minuscule and can be neglected compared to all the other forces involved. Gravity redirects here. ...

At each tire contact patch there are ground reaction forces with both horizontal and vertical components. The vertical components mostly counteract the force of gravity, but also vary with braking and accelerating. For details, see the section on longitudinal stability below. The horizontal components, due to friction between the wheels and the ground, including rolling resistance, are in response to propulsive forces, braking forces, and turning forces. Contact patch is the name applied to the area of a vehicles tire that is in contact with the road surface. ... In classical mechanics, Newtons third law states that forces occur in pairs, one called the Action and the other the Reaction (actio et reactio in Latin). ... For other uses, see Friction (disambiguation). ... Rolling resistance, sometimes called rolling friction, is the resistance that occurs when an object (e. ... Vehicle propulsion refers to the act of moving an artificial carrier of people or goods over a distance. ...

The turning forces are generated during maneuvers for balancing in addition to just changing direction of travel. These may be interpreted as centrifugal forces in the accelerating reference frame of the bike and rider or simply as inertia in a stationary, inertial reference frame and not a force at all. Centrifugal force (from Latin centrum centre and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... This article or section is in need of attention from an expert on the subject. ... This article is about inertia as it applies to local motion. ... An inertial frame of reference, or inertial reference frame, is one in which Newtons first and second laws of motion are valid. ...

Gyroscopic forces acting on rotating parts such as wheels, engine, transmission, etc., are also due to the inertia of those rotating parts. They are discussed further in the section on gyroscopic effects below. A gyroscope For other uses, see Gyroscope (disambiguation). ...

The aerodynamic forces due to the atmosphere are mostly in the form of drag, but can also be from crosswinds. At normal bicycling speeds on level ground, aerodynamic drag is the largest force resisting forward motion.^[4] Full aerodynamic force is the resultant of all aerodynamic forces, acting on a body in a stream of air. ... An object moving through a gas or liquid experiences a force in direction opposite to its motion. ... A crosswind is any wind that is blowing perpendicular to a line of travel, or perpendicular to a direction. ...

Internal forces

Internal forces are mostly due to the rider or to friction. The rider can apply torques between the steering mechanism (front fork, handlebars, front wheel, etc.) and rear frame, and between the rider and the rear frame. Friction exists between any parts that move against each other: in the drive train, between the steering mechanism and the rear frame, etc. For other senses of this word, see torque (disambiguation). ... For other uses, see Friction (disambiguation). ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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.^[4] 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 balance while riding. Instead, if just the gyroscopic effect of rotating bike wheels is cancelled by adding counter-rotating wheels, it is still easy to balance while riding.^[1]^[2]

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.^[5]

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. Tandem recumbent bicycle manufactured by BikeE A recumbent bicycle is a variety of bicycle which places the rider in a seated or supine position (rarely, in a prone position). ... Chopper refers to a motorcycle that was radically customized, especially the Harley-Davidsons as seen in the 1969 movie Easy Rider. ...

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.^[6] This is opposite of the impression a rider can have of a stationary bike. A top-heavy bike can require more effort to keep upright, when stopped in traffic for example, than a bike with a lower center of mass. This is why touring cyclists are advised to carry loads low on a bike.^[7] A schematic drawing of the inverted pendulum on a cart. ...

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 steering axis ground contact point. The steering axis is the axis about which the entire steering mechanism (fork, handlebars, front wheel, etc.) pivots. 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.^[4] 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. Image File history File links This is a lossless scalable vector image. ... Î¸ is the caster angle, red line is the pivot line, grey area is the tire Caster (or castor) angle is the angular displacement from the vertical axis of the suspension of a steered wheel in a car or other vehicle, measured in the longitudinal direction. ... This article or section needs to be wikified. ... Bike wheelbase, head angle, fork offset, and trail Bicycle and motorcycle geometry is the collection of key measurements (lengths and angles) that define a particular bike configuration. ...

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.^[8] In motorcycles, rake instead refers to the head angle, and offset created by the triple tree is used to diminish trail.^[9] bicycle fork A bicycle fork is the portion of a bicycle that holds the front wheel and allows one to steer. ... A triple tree is a two-part motorcycle component that attaches the fork tubes to the frame to comprise the fork and make steering possible. ...

Trail is a function of head angle, fork offset or rake, and wheel size. Their relationship can be described by this formula:^[10] bicycle fork A bicycle fork is the portion of a bicycle that holds the front wheel and allows one to steer. ...

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.

At the same time, Lemond offers,^[11] both with forks that have 45 mm of offset or rake and the same size wheels: A touring bicycle is a road bicycle designed for long-distance travel, and especially bicycle touring. ... An aluminum racing bicycle made by Raleigh and built using Shimano components. ... A track bicycle A track bicycle is a type of fixed-gear bicycle specially designed for track cycling in a velodrome. ...

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.^[12] Additional parameters, 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.^[12]^[4]

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.^[13] 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. Precession redirects here. ...

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.^[14] 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.

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.^[15] However, even without self-stability a bike may be ridden by steering it to keep it over its wheels.^[2] Note that the effects mentioned above that would combine to produce self stability may be overwhelmed by additional factors such as headset friction and stiff control cables.^[4] This video shows a riderless bicycle exhibiting self-stability. Parts of a threadless headset before installation. ... Invented by Frank Bowden, a bowden cable is a type of flexible cable used to transmit mechanical force or energy by the movement of an inner cable (most commonly of steel or stainless steel) relative to a hollow outer cable housing. ...

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.SdgSgfdbdfbdfbdfbf

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.

Modes

There are three main unstable modes that a bike can experience: capsize, weave, and wobble.^[16] A lesser known mode is rear wobble, and it is usually stable.^[17] For other types of mode, see mode. ...

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.^[17]

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.^[17] This AVI movie shows weave.

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.

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.^[14]^[17] While wobble or shimmy can be easily remedied by adjusting speed, position, or grip on the handlebar, they can be fatal if left uncontrolled.^[18] This AVI movie shows wobble. Speed wobble or shimmy is the spontaneous oscillation of the front wheel(s), or wobbling of a vehicle, at a set speed or speed range. ...

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. The oscillation frequency can be changed by changing the forward speed, making the bike stiffer or lighter, or increasing the stiffness of the steering, of which the rider is a main component.^[4] A Steering damper (or Sprint damper) is a damping device designed to inhibit an undesirable, uncontrolled movement or oscillation of a steering mechanism - a phenomenon known in the motorcycling community as wobble, or in extreme cases, a tank-slapper. Modern bikes are unlikely to exhibit this behaviour in daily use...

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.^[17]

Design criteria

The effect that the design characteristics of a bike has on these instabilities can investigate by examining the eigenvalues of the linearized equations of motion.^[19] For more details on the equations of motion and eigenvalues, see the section on theory below. Some general conclusions that have been drawn are described here.

The lateral and torsional stiffness of the rear frame and the wheel spindle affects wobble-mode damping substantially. 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. For other uses, see Motorcycle (disambiguation). ... The trail of a bicycle or motorcycle is the distance between the point where the steering axis meets the ground, and the point where the wheel meets the ground. ... The trail of a bicycle or motorcycle is the distance between the point where the steering axis meets the ground, and the point where the wheel meets the ground. ... The trail of a bicycle or motorcycle is the distance between the point where the steering axis meets the ground, and the point where the wheel meets the ground. ... 1968 BMW R60US with conventional telescopic fork Yamahas inverted telescopic fork The worlds first oil-damped telescopic fork, on a 1939 BMW R12 Trailing link fork on a 1928 BMW R57 Unusual trailing bottom link on a Honda Rune Earles front forks on three BMWs BMW Telelever fork...

Degraded damping of the rear suspension amplifies cornering weave tendencies. 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. A motorcycles suspension is similar to the suspension in an automobile in its purpose: But a motorcycle suspension is usually simpler, since it does not have to contend with lateral forces such as body roll. ... Firestone tire This article is about pneumatic tires. ...

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

Turning

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: In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ...

where

v

is the forward speed,

r

is the radius of the turn and

g

is the acceleration of gravity.^[13] Gravity is a force of attraction that acts between bodies that have mass. ...

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.

As a bike leans, the tires' contact patches move farther to the side causing wear. The portions at either edge of a motorcycle tire that remain unworn by leaning into turns is sometimes referred to as chicken strips.

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.^[13] Countersteering is the name given to the counter-intuitive technique used by cyclists and motorcyclists to turn corners. ...

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.^[20]

Steady-state turning

Once a turn is established, the torque that must be applied to the steering mechanism in order to maintain a constant radius at a constant forward speed depends on the forward speed and the geometry and mass distribution of the bike.^[14] At speeds below the capsize speed, described below in the section on Eigenvalues and also called the inversion speed, the self-stability of the bike will cause it to tend to steer into the turn, thus righting itself and exiting the turn, unless a torque is applied opposite the direction of the turn. At speeds above the capsize speed, the capsize instability will cause it to tend to steer out of the turn, thus increasing the lean, unless a torque is applied in the direction of the turn. At the capsize speed no input steering torque is necessary to maintain the steady-state turn. A computer-generated, simplified model of bike and rider demonstrating an uncontrolled 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. ...

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.^[13] Note that this technique may be complicated by additional factors such as headset friction and stiff control cables. This gyroscope remains upright while spinning due to its angular momentum. ...

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."^[21]^[22]

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. One implementation of two-wheel steering lets the rider control the steering of both wheels directly: Sideways bike.

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.^[4] Front-wheel drive is the most common form of engine/transmission layout used in modern passenger cars, where the engine drives the front wheels. ...

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."^[23] 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.^[24] Examination of the eigenvalues shows that the configuration is inherently unstable. A computer-generated, simplified model of bike and rider demonstrating an uncontrolled 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. ...

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.^[25]^[26] 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.^[27]

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 vice 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.^[28] A tiller or till is a lever attached to a rudder post (American terminology) or rudder stock (English terminology) of a boat in order to provide the leverage for the helmsman to turn the rudder. ... A cruiser bicycle is a bicycle designed for riding on roads and paths in comfort and style over performance. ... For other uses, see Motorcycle (disambiguation). ...

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 if the rear wheel is lifted but the bicycle does not flip or an ‘endo’ (abbreviated form of 'end-over-end') if the bicycle flips. 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. Linear-pull brake on rear wheel of a mountain bike Bicycle brake systems are used to slow down, or brake a bicycle. ... Stoppie 180, by Duke (French champion) during the Stunt Bike Show, in Carole Racetrack The stoppie, also known as the endo, is a motorcycle and bicycle trick in which the back wheel is lifted and the bike is ridden on the front wheel. ... A scooter (left) and a cruiser Motorcycles have been produced in myriad different models for innumerable purposes. ... Tandem recumbent bicycle manufactured by BikeE A recumbent bicycle is a variety of bicycle which places the rider in a seated or supine position (rarely, in a prone position). ...

Longitudinal stability

Mechanical analysis of the forces generated by a bike with a wheelbase

L

and a center of mass at height

h

and halfway between the wheels, with both wheels locked, reveals that the normal (vertical) forces at the wheels are:^[29] Bike wheelbase, head angle, fork offset, and trail Bicycle and motorcycle geometry is the collection of key measurements (lengths and angles) that define a particular bike configuration. ...

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 The coefficient of friction is a dimensionless quantity used to calculate the force of friction (static or kinetic). ... For other uses, see Mass (disambiguation). ...

then the normal force of the rear wheel will be zero (at which point the equation no longer applies) and the bike will begin to flip forward over the front wheel.

The coefficient of friction of rubber on dry asphalt is between 0.5 and 0.8.^[30] Using the lower value of 0.5, 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 forward over the front wheel. This does not cite any references or sources. ... The term asphalt is often used as an abbreviation for asphalt concrete. ...

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 traditional tandem bicycle. ...

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. A riding technique that takes advantage of how braking increases the downward force on the front wheel is known as trail braking. A motorcycles suspension is similar to the suspension in an automobile in its purpose: But a motorcycle suspension is usually simpler, since it does not have to contend with lateral forces such as body roll. ... The fork tube holds a motorcycles front wheel. ... A motorcycles suspension is similar to the suspension in an automobile in its purpose: But a motorcycle suspension is usually simpler, since it does not have to contend with lateral forces such as body roll. ... Trail braking is an advanced riding technique of motorcycle riders which requires professional training, and it is also used in car racing, meaning to continue to break into a turn. ...

Front wheel braking

For an upright bicycle on dry asphalt with excellent brakes, pitching will probably be the limiting fact. The combined center of mass of a typical upright bicycle and rider will be about 60 cm back from the front wheel contact patch and 120 cm above, allowing a maximum deceleration of 0.5 g (4.9 m/s² or 16 ft/s²).^[4] 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. For other uses, see Friction (disambiguation). ... For other uses, see Friction (disambiguation). ...

Front brakes on many inexpensive bikes are not strong enough so, on the road, they are the limiting factor. Cheap cantilever brakes, especially with "power modulators", and Raleigh-style side-pull brakes severely restrict the stopping force. 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 force is exceeded the wheel slides.

Rear wheel braking

The rear brake of a upright bicycle can only produce about 0.1 g deceleration at best.^[4] This is due to the decrease in normal force at the rear wheel as described above. All bikes with only rear braking are subject to this limitation: for example bikes with only a coaster brake and fixed-gear bikes with no other braking mechanism. Linear-pull brake on rear wheel of a mountain bike Bicycle brake systems are used to slow down, or brake a bicycle. ... A fixed gear bicycle A fixed-gear bicycle or fixed wheel bicycle, is any bicycle without a freewheel and usually only one gear ratio. ...

Theory

Although its equations of motion can be linearized, a bike is a nonlinear system. The variable(s) to be solved for cannot be written as a linear sum of independent components, i.e. its behavior is not expressible as a sum of the behaviors of its descriptors.^[16] Generally, nonlinear systems are difficult to solve and are much less understandable than linear system. In mathematics, a nonlinear system is a system which is not linear i. ...

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. This article is about the law of conservation of energy in physics. ... A computer-generated, simplified model of bike and rider demonstrating an uncontrolled 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. ...

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.^[13] In physics and mathematics, a nonholonomic system is a system in which a return to the original internal configuration does not guarantee return to the original system position. ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I â†’ X. The initial point of the path is f(0) and the terminal point is f(1). ...

Finally, in the language of control theory, a bike exhibits non-minimum phase behavior.^[31] It turns in the direction opposite of how it is initially steered, as described above in the section on countersteering For control theory in psychology and sociology, see control theory (sociology). ... In control theory and signal processing, a linear, time-invariant system is minimum-phase if the system and its inverse are causal and stable. ... A computer-generated, simplified model of bike and rider demonstrating an uncontrolled 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. ...

Equations of motion

The equations of motion of an idealized bike, consisting of It has been suggested that SUVAT equations be merged into this article or section. ...

can be represented by two linearized second-order ordinary differential equations,^[12] the lean equation Steel frame and carbon fiber fork of 2000 LeMond Zurich racing bicycle A bicycle frame is the main component of a bicycle, onto which wheels and other components are fitted. ... â€œWheelsetâ€ redirects here. ... Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

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.^[15] Bike wheelbase, head angle, fork offset, and trail Bicycle and motorcycle geometry is the collection of key measurements (lengths and angles) that define a particular bike configuration. ...

Eigenvalues

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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... a Dutch utility bicycle A utility bicycle is one which is designed for a practical purpose, as opposed to sport bicycles which are designed for recreation and competition, such as touring bicycles, racing bicycles and mountain bicycles. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Oscillation is the variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. ...

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.^[16] 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.^[17] A computer-generated, simplified model of bike and rider demonstrating an uncontrolled 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. ... A computer-generated, simplified model of bike and rider demonstrating an uncontrolled 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

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

From American Scientist Online [1]: David E. H. Jones is a guest staff member in the chemistry department at the University of Newcastle-upon-Tyne, UK. In 1962, he earned a Ph. ...

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.^[4] 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]^[13] A gyroscope is a device which demonstrates the principle of conservation of angular momentum, in physics. ...

See also

Bike wheelbase, head angle, fork offset, and trail Bicycle and motorcycle geometry is the collection of key measurements (lengths and angles) that define a particular bike configuration. ... bicycle fork A bicycle fork is the portion of a bicycle that holds the front wheel and allows one to steer. ... Î¸ is the caster angle, red line is the pivot line, grey area is the tire Caster (or castor) angle is the angular displacement from the vertical axis of the suspension of a steered wheel in a car or other vehicle, measured in the longitudinal direction. ... Countersteering is the name given to the counter-intuitive technique used by cyclists and motorcyclists to turn corners. ... The highsider or highside is a type of motorcycle accident usually occurring in a curve which may be caused by a locked wheel due to excessive braking or more commonly by applying too much throttle when exiting a corner causing the rear tire to lose traction. ... The lowsider is a type of motorcycle accident usually occurring in a curve and caused by a locked wheel due to excessive braking. ... 1968 BMW R60US with conventional telescopic fork Yamahas inverted telescopic fork The worlds first oil-damped telescopic fork, on a 1939 BMW R12 Trailing link fork on a 1928 BMW R57 Unusual trailing bottom link on a Honda Rune Earles front forks on three BMWs BMW Telelever fork... Speed wobble or shimmy is the spontaneous oscillation of the front wheel(s), or wobbling of a vehicle, at a set speed or speed range. ... Stoppie 180, by Duke (French champion) during the Stunt Bike Show, in Carole Racetrack The stoppie, also known as the endo, is a motorcycle and bicycle trick in which the back wheel is lifted and the bike is ridden on the front wheel. ... Trail braking is an advanced riding technique of motorcycle riders which requires professional training, and it is also used in car racing, meaning to continue to break into a turn. ... An R/C truck pops a wheelie after a jump. ...

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.
^ Meijaard, Papadopoulos, Ruina, and Schwab (2007). "Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review". Proc. R. Soc. A. 463 (2084): 1955-1982.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k 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.
^ REI. Cycle Expert Advice: Packing for a Tour. Retrieved on 2007-11-13.
^ Zinn, Lennard. "Technical Q&A with Lennard Zinn — Rake, trail, offset", Velo News, 2004-12-21. 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 ^c 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 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.
^ ^a ^b ^c ^d ^e ^f 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. Sheldon Brown. 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. Sheldon Brown. 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.
^ 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.

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Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 356th day of the year (357th in leap years) in the Gregorian calendar. ... Year 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ... is the 258th day of the year (259th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Sheldon Brown and Igor Sheldon Brown (born July 14, 1944) is an American bicycle mechanic and technical authority. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 220th day of the year (221st in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 348th day of the year (349th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 348th day of the year (349th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 218th day of the year (219th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 349th day of the year (350th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 349th day of the year (350th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 349th day of the year (350th in leap years) in the Gregorian calendar. ... Sheldon Brown and Igor Sheldon Brown (born July 14, 1944) is an American bicycle mechanic and technical authority. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 220th day of the year (221st in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 216th day of the year (217th in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 306th day of the year (307th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 219th day of the year (220th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 219th day of the year (220th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 219th day of the year (220th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 219th day of the year (220th in leap years) in the Gregorian calendar. ...

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