Definitions

# Bicycle and motorcycle dynamics

Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles and their components, due to the forces acting on them. Dynamics is a branch of classical mechanics, which in turn is a branch of physics. Bicycles and motorcycles are both single-track vehicles and so their motions have many fundamental attributes in common.

Bike motions of interest include balancing, steering, braking, suspension activation, and vibration. 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 discredited.

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 the centrifugal forces of the turn with the gravitational forces of 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.

## 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, credited with inventing the two-wheeled vehicle variously called the laufmaschine, velocipede, draisine, and dandy horse, 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 showing 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.

## 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 caused by the rider and by 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 compared to all the other forces involved and can be ignored.

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.

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 forces at all.

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.

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.

### Internal forces

Internal forces are mostly caused by the rider or by 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. Many bikes have front and rear suspensions and some motorcycles have a steering damper to dissipate undesirable kinetic energy.

## 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. 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.

### Forward speed

The rider applies torque to the handlebars in order to turn the front wheel and so to control lean and maintain balance. At high speeds, small steering angles quickly move the ground contact points laterally; at low speeds, larger steering angles are required to achieve the same results in the same amount of time. Because of this, it is usually easier to maintain balance at high speeds.

### 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. It can also be an issue for touring bikes with a heavy load of gear over or even behind the rear wheel.

A bike is also an example of an inverted pendulum. Just as a broomstick is easier to balance than a pencil, a tall bike (with a high center of mass) can be easier to balance when ridden than a short one because its lean rate will be slower.

A rider can have the opposite impression of a bike when it is stationary. A top-heavy bike can require more effort to keep upright, when stopped in traffic for example, than a bike which is just as tall but with a lower center of mass. This is an example of a vertical second-class lever. A small force at the end of the lever, the seat or handlebars at the top of the bike, more easily moves a large mass if the mass is closer to the fulcrum, where the tires touch the ground. This is why touring cyclists are advised to carry loads low on a bike, and panniers hang down on either side of front and rear racks.

### Trail

A factor that influences how easy or difficult a bike will be to ride is trail, the distance that 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. 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 baggage. As a consequence, an unloaded touring bike can feel unstable. In bicycles, fork rake, often a curve in the fork blades forward of the steering axis, is used to diminish trail. In motorcycles, rake refers to the head angle instead, and offset created by the triple tree is used to diminish trail.

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

$Trail = frac\left\{\left(R_w cos\left(A_h\right) - O_f\right)\right\}\left\{sin\left(A_h\right)\right\}$
where $R_w$ is 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 found:

• touring bicycles with head angles between 72° and 73° and trail between 43.0 mm and 60.0 mm
• racing bicycles with head angles between 73° and 74° and trail between 28.0 mm and 45.0 mm
• track bicycles with head angles of 75° and trail between 23.5 mm and 37.0 mm.

At the same time, LeMond Racing Cycles offers, 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 72.50° to 74.00° depending on frame size
• a 2006 Tete de Course, designed for road racing, with a head angle that varies from 71.25° to 74.00°, depending on frame size.

### 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 in front 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. 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.

### 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. 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. This instability is very slow, on the order of seconds, and is easy for most riders to counteract. Thus a fast bike may feel stable even though it is actually not self-stable and would fall over if it were uncontrolled.

Another contribution of gyroscopic effects is a roll moment generated by the front wheel during countersteering. For example, steering left causes a moment to the right. The moment is small compared to the moment generated by the out-tracking front wheel, but begins as soon as the rider applies torque to the handlebars and so can be helpful in motorcycle racing. For more detail, see the countersteering article.

### Self-stability

Between the two unstable modes mentioned in the previous section, there may be a range of forward speeds for a given bike design at which the effects described above steer an uncontrolled bike upright. However, even without self-stability a bike may be ridden by steering it to keep it over its wheels. 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. This video shows a riderless bicycle exhibiting self-stability.

## Instability

Bikes, as complex mechanisms, have a variety of unstable modes: ways that they can be unstable. In this context, "stable" means that an uncontrolled bike will continue rolling forward without falling over as long as forward speed is maintained. Conversely, "unstable" means that an uncontrolled bike will eventually fall over, even if forward speed is maintained. The unstable modes are differentiated by the speed at which they occur and the relative phases of leaning and steering.

### 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.

#### Capsize

Capsize is the word used to describe a bike falling over without oscillation. During capsize, an uncontrolled front wheel usually steers in the direction of lean during capsize, but never enough to stop the increasing lean, 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.

#### 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. 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 death wobble are all words and phrases used to describe a rapid (4–10 Hz) oscillation of primarily just the front end (front wheel, fork, and handlebars). The rest of the bike remains essentially 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. While wobble or shimmy can be easily remedied by adjusting speed, position, or grip on the handlebar, it can be fatal if left uncontrolled. This AVI movie shows wobble.

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 occurs. 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.

#### 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.

### Design criteria

The effect that the design characteristics of a bike have on these instabilities can be investigated by examining the eigenvalues of the linearized equations of motion. 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.

Cornering weave tendencies are amplified by degraded damping of the rear suspension. Cornering, camber stiffnesses and relaxation length of the rear tire make the largest contribution to weave damping. The same parameters of the front tire have a lesser effect. Rear loading also amplifies cornering weave tendencies. Rear load assemblies with appropriate stiffness and damping, however, were successful in damping out weave and wobble oscillations.

Finally, tire inflation pressures are important variables in the behavior of a motorcycle at high speeds.

## 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, $theta$, can easily be calculated using the laws of circular motion:

$theta = arctan left \left(frac\left\{v^2\right\}\left\{gr\right\}right \right)$
where $v$ is the forward speed, $r$ is the radius of the turn and $g$ is the acceleration of gravity.

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, causing 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.

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. 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, momentarily steering more into the turn in order to decrease the radius, thus increasing inertial forces, and thereby decreasing the angle of lean.

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. 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, righting itself and exiting the turn, unless a torque is applied in the opposite direction of the turn. At speeds above the capsize speed, the capsize instability will cause it to tend to steer out of the turn, 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.

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

### 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 to "work very well.

Issues in the design include whether to provide active control of the rear wheel or let it swing freely. In the case of active control, the control algorithm needs to decide between steering with or in the opposite direction of the front wheel, when, and how much. One implementation of two-wheel steering, the Sideways bike, lets the rider control the steering of both wheels directly.

### 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. The Bendix Company built a rear-wheel steering bicycle, and the U.S. Department of Transportation commissioned the construction of a rear-wheel steering motorcycle: both proved to be unridable. Rainbow Trainers, Inc. in Alton, IL, offers US\$5,000 to the first person "who can successfully ride the rear-steered bicycle, Rear Steered Bicycle I". One documented example of someone successfully riding a rear-wheel steering bicycle is that of L. H. Laiterman at MIT, on a specially designed recumbent bike. The difficulty is 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 complicates the task of compensating for leans induced by the environment. Examination of the eigenvalues shows that the rear-wheel steering configuration is inherently unstable.

### Center steering

Between the extremes of bicycles with classical front-wheel steering and those with strictly rear-wheel steering is a class of bikes with a pivot point somewhere between the two referred to as center-steering. This design allows for simple front-wheel drive and appears to be quite stable, even ridable no-hands, as many photographs illustrate. 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.

### 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 troublesome when it limits the ability to steer because of interference or the limits of arm reach.

### Tires

Because real tires have a finite contact patch with the road surface that can generate a scrub torque, and when in a turn, can experience some side slipping as they roll, they can generate torques about an axis normal to the plane of the contact patch.

One such torque is generated by asymmetries in the side-slip along the length of the contact patch. The resultant force of this side-slip occurs behind the geometric center of the contact patch, a distance described as the pneumatic trail, and so creates a torque on the tire. Since the direction of the side-slip is towards the outside of the turn, the force on the tire is towards the center of the turn. Therefore, this torque tends to turn the front wheel in the direction of the side-slip, away from the direction of the turn, and therefore tends to increase the radius of the turn.

Another torque is produced by the finite width of the contact patch and the lean of the tire in a turn. The portion of the contact patch towards the outside of the turn is actually moving rearward, with respect to the wheel's hub, faster than the rest of the contact patch, because of its greater radius from the hub. By the same reasoning, the inner portion is moving rearward more slowly. So the outer and inner portions of the contact patch slip on the pavement in opposite directions, generating a torque that tends to turn the front wheel in the direction of the turn, and therefore tends to decrease the turn radius.

The combination of these two opposite torques creates a resulting yaw torque on the front wheel, and its direction is a function of the side-slip angle of the tire, the angle between the actual path of the tire and the direction it is pointing, and the camber angle of the tire (the angle that the tire leans from the vertical). The result of this torque is often the suppression of the inversion speed predicted by rigid wheel models described above in the section on steady-state turning.

Because the front and rear tires can have different slip angles due to weight distribution, tire properties, etc., bikes can experience understeer or oversteer. Of the two, understeer, in which the front wheel slides more than the rear wheel, is more dangerous since front wheel steering is critical for maintaining balance.

## 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. Long or low bikes, however, such as cruiser motorcycles and recumbent bicycles, can also skid the front tire, causing a loss of balance.

### 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:
$N_r = mgleft\left(frac\left\{1\right\}\left\{2\right\} - mu frac\left\{h\right\}\left\{L\right\}right\right)$ for the rear wheel and $N_f = mgleft\left(frac\left\{1\right\}\left\{2\right\} + mu frac\left\{h\right\}\left\{L\right\}right\right)$ for the front wheel,
while the frictional (horizontal) forces are simply $F_r = mu N_r$ for the rear wheel and $F_f = mu N_f$ for the front wheel, where $mu$ is the coefficient of friction, $m$ is the mass, and $g$ is the acceleration of gravity. Therefore, if
$mu frac\left\{h\right\}\left\{L\right\} ge frac\left\{1\right\}\left\{2\right\}$
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. Using the lower value of 0.5, and assuming the center of mass height is greater than or equal to the wheelbase, the front wheel can generate enough stopping force to flip the bike and rider forward over the front wheel.

On the other hand, if the center of mass height is less than half the wheelbase and at least halfway towards the rear wheel, as is true, for example on 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.

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.

### Front wheel braking

The limiting factors on the maximum deceleration 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 (of bike and rider) over the front wheel.

For an upright bicycle on dry asphalt with excellent brakes, pitching will probably be the limiting factor. The combined center of mass of a typical upright bicycle and rider will be about back from the front wheel contact patch and above, allowing a maximum deceleration of 0.5 g (4.9 m/s² or 16 ft/s²). If the rider modulates the brakes properly, however, pitching can be avoided. If the rider moves his weight back and down, even larger decelerations are possible.

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 trail, although knobby tires can mitigate this effect by grabbing the surface irregularities. Front wheel slides are also common on corners, whether on road or off. Centripetal acceleration adds to the forces on the tire-ground contact, and when the friction force is exceeded the wheel slides.

Of course, the angle of the terrain can influence all of the calculations above. All else remaining equal, the risk of pitching over the front end is reduced when riding up hill and increased when riding down hill.

### Rear wheel braking

The rear brake of a upright bicycle can only produce about 0.1 g deceleration at best, because of 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.

## Vibration

The study of vibration in bikes includes its causes, such as engine balance, wheel balance, ground surface, and aerodynamics; its transmission and absorbtion; and its effects on the bike, the rider, and safety. An important factor in any vibration analysis is a comparison of the natural frequencies of the system with the possible driving frequencies of the vibration sources. A close match means mechanical resonance that can result in large amplitudes. A challange in vibration damping is to create compliance in certain directions (vertically) without sacrificing frame rigidity needed for power transmission and handling (torsionally).

Effects of vibration on riders include Hand-Arm Vibration Syndrome, a secondary form Raynaud's disease, and whole body vibration.

### In bicycles

The primary cause of vibrations in a properly functioning bicycle is the surface over which it rolls. In addition to pneumatic tires and traditional bicycle suspensions, a variety of techniques have been developed to damp vibrations before they reach the rider. These include materials, such as carbon fiber, either in the whole frame or just key components such as the front fork, seatpost, or handlebars; tube shapes, such as curved seat stays; and special inserts, such as Zertz by Specialized, and Buzzkills by Bontrager.

### In motorcycles

In addition to the road surface, vibrations in a motorcycle can be caused by the engine and wheels, if unbalanced. Manufacturers employ a variety of technologies to reduce or damp these vibrations, such as engine balance shafts, rubber engine mounts, and tire weights. The problems that vibration causes have also spawned an industry of after-market parts and systems designed to reduce it. Add-ons include handlebar weights, isolated foot pegs, and engine counterweights.

At high speeds, motorcycles and their riders may also experience aerodynamic flutter or buffeting. This can be abated by changing the air flow over key parts, such as the windshield.

## 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. Generally, nonlinear systems are difficult to solve and are much less understandable than linear systems.

In the idealized case, in which friction and any flexing is ignored, a bike is a conservative system. Damping, however, can still be demonstrated: 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 have moved over time. This complicates mathematical analysis.

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

### Degrees of freedom

The number of degrees of freedom of a bike depends on the particular model being used. The simplest model that captures the key dynamic features, four rigid bodies with knife edge wheels rolling on a flat smooth surface, has 7 degrees of freedom (configuration variables required to completely describe the location and orientation of all 4 bodies):

1. x coordinate of rear wheel contact point
2. y coordinate of rear wheel contact point
3. orientation angle of rear frame (yaw)
4. rotation angle of rear wheel
5. rotation angle of front wheel
6. lean angle of rear frame (roll)
7. steering angle between rear frame and front end

Adding complexity to the model, such as suspension, tire compliance, frame flex, or rider movement, adds degrees of freedom. While the rear frame does pitch with leaning and steering, the pitch angle is completely constrained by the requirement for both wheels to remain on the ground, and so can be calculated geometrically from the other seven variables. If the location of the bike and the rotation of the wheels are ignored, the first five degrees of freedom can also be ignored, and the bike can be described by just two variables: lean angle and steer angle.

### 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 a single fourth-order linearized ordinary differential equation or two coupled second-order differential equations, the lean equation

$M_\left\{thetatheta\right\}ddot\left\{theta_r\right\} +$
K_{thetatheta}theta_r + M_{thetapsi}ddot{psi} + C_{thetapsi}dot{psi} + K_{thetapsi}psi = M_{theta} and the steer equation
$M_\left\{psipsi\right\}ddot\left\{psi\right\} +$
C_{psipsi}dot{psi} + K_{psipsi}psi + M_{psitheta}ddot{theta_r} + C_{psitheta}dot{theta_r} + K_{psitheta}theta_r = M_{psi}mbox{,} where

• $theta_r$ is the lean angle of the rear assembly,
• $psi$ is the steer angle of the front assembly relative to the rear assembly and
• $M_\left\{theta\right\}$ and $M_\left\{psi\right\}$ 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 $Mmathbfddot q+Cmathbfdot q+Kmathbf q=mathbf f$ 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 $V^2$,
• $mathbf q$ is a vector of lean angle and steer angle, and
• $mathbf f$ 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.

### Eigenvalues

It is possible to calculate eigenvalues, one for each of the four state variables (lean angle, lean rate, steer angle, and steer rate), from the linearized equations in order to analyze the normal modes and 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.

There are three forward speeds that can be identified in the plot to the right at which the motion of the bike changes qualitatively:

1. The forward speed at which oscillations begin, at about 1 m/s (2.2 mph) in this example, sometimes called the double root speed due to there being a repeated root to the characteristic polynomial (two of the four eigenvalues have exactly the same value). Below this speed, the bike simply falls over as an inverted pendulum does.
2. The forward speed at which oscillations do not increase, at about 5.3 m/s (12 mph) in this example, is called the weave speed. Below this speed, oscillations increase until the uncontrolled bike falls over. Above this speed, oscillations eventually die out.
3. The forward speed at which non-oscillatory leaning increases, at about 8.0 m/s (18 mph) in this example, is called the capsize speed. Above this speed, this non-oscillating lean eventually causes the uncontrolled bike to fall over.

Between these last 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). The fourth eigenvalue, which is usually stable (very negative), represents the castoring behavior of the front wheel, as it tends to turn towards the direction in which the bike is traveling. 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 or other degrees of freedom.

## 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.
• Richard Klein built several bikes to confirm Jones's findings.
• Richard Klein also built a "Torque Wrench Bike" and a "Rocket Bike" to investigate steering torques and their effects.
• Keith Code built a motorcycle with fixed handlebars to investigate the effects of rider motion and position on steering.
• Schwab and Kooijman have performed measurements with an instrumented bike.

## 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 complicated to analyze. While it is an observable fact that bikes can be ridden even when the gyroscopic effects of their wheels are canceled out, the hypothesis that the gyroscopic effects of the wheels are what keep a bike upright is common in print and online.

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: