## Thursday, October 27, 2011

### Liberty Science Center

Liberty Science Center
is an interactive science museum and learning center located in Liberty State Park in Jersey City, New Jersey.
The center, which first opened in 1993 as New Jersey's first major state science museum, has science exhibits, the largest IMAX Dome theater in the United States, numerous educational resources, and the original Hoberman sphere, a silver, computer-driven engineering artwork designed by Chuck Hoberman. The museum opened with another artistic exhibit that is related to the sciences, Jim Gary's Twentieth Century Dinosaurs sculpture exhibition, as the exhibit on the ground floor. [1]

## History

#### Center steering

Flevobike with 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, similar to articulated steering. An early implementation of the concept was the Phantom bicycle in the early 1870s promoted as a safer alternative to the penny-farthing.[37] This design allows for simple front-wheel drive and current implementations appear to be quite stable, even ridable no-hands, as many photographs illustrate.[38][39]
These designs, such as the Python Lowracer, 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.[40]

#### 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 cruiser bicycles, 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.[41]

#### Tires

Tires have a large influence over bike handling, especially on motorcycles.[9][25] Through a combination of cornering force and camber thrust, tires generate the lateral forces necessary for steering and balance. Tire inflation pressures have also been found to be important variables in the behavior of a motorcycle at high speeds.[42] 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.[9] Also, 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.
Bike tire contact patch during a right-hand turn
One torque generated by a tire, called the self aligning torque, is caused 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).[9] 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.[10]

#### High side

highsiderhighside, or high side is a type of bike motion which is caused by a rear wheel gaining traction when it is not facing in the direction of travel, usually after slipping sideways in a curve.[9] This can occur under heavy braking, acceleration, a varying road surface, or suspension activation, especially due to interaction with the drive train.[43] It can take the form of a single slip-then-flip or a series of violent oscillations.[25]

### Maneuverability and handling

Bike maneuverability and handling is difficult to quantify for several reasons. The geometry of a bike, especially the steering axis angle makes kinematic analysis complicated.[2] Under many conditions, bikes are inherently unstable and must always be under rider control. Finally, the rider's skill has a large influence on the bike's performance in any maneuver.[9] Bike designs tend to consist of a trade-off between maneuverability and stability.

#### Rider control inputs

Graphs showing the lean and steer angle response of an otherwise uncontrolled bike, traveling at a forward speed in its stable range (6 m/s), to a steer torque that begins as an impulse and then remains constant. Torque to right causes initial steer to right, lean to left, and eventually a steady-state steer, lean, and turn to left.
The primary control input that the rider can make is to apply a torque directly to the steering mechanism via the handlebars. Because of the bike's own dynamics, due to steering geometry and gyroscopic effects, direct position control over steering angle has been found to be problematic.[8]
A secondary control input that the rider can make is to lean the upper torso relative to the bike. As mentioned above, the effectiveness of rider lean varies inversely with the mass of the bike. On heavy bikes, such as motorcycles, rider lean mostly alters the ground clearance requirements in a turn, improves the view of the road, and improves the bike system dynamics in a very low-frequency passive manner.[8]

#### Differences from automobiles

The need to keep a bike upright to avoid injury to the rider and damage to the vehicle even limits the type of maneuverability testing that is commonly performed. For example, while automobile enthusiast publications often perform and quote skidpad results, motorcycle publications do not. The need to "set up" for a turn, lean the bike to the appropriate angle, means that the rider must see further ahead than is necessary for a typical car at the same speed, and this need increases more than in proportion to the speed.[8]

#### Rating schemes

Several schemes have been devised to rate the handling of bikes, particularly motorcycles.[9]
• The roll index is the ratio between steering torque and roll or lean angle.
• The steering ratio is the ratio between the theoretical turning radius based on ideal tire behavior and the actual turning radius.[9] Values less than one, where the front wheel side slip is greater than the rear wheel side slip, are described as under-steering; equal to one as neutral steering; and greater than one as over-steering. Values less than zero, in which the front wheel must be turned opposite the direction of the curve due to much greater rear wheel side slip than front wheel have been described as counter-steering. Riders tend to prefer neutral or slight over-steering.[9] Car drivers tend to prefer under-steering.
• The Koch index is the ratio between peak steering torque and the product of peak lean rate and forward speed. Large, touring motorcycles tend to have a high Koch index, sport motorcycles tend to have a medium Koch index, and scooters tend to have a low Koch index.[9] It is easier to maneuver light scooters than heavy motorcycles.

### Lateral motion 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.[2] 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: under the right circumstances, 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.[28] Finally, in the language of control theory, a bike exhibits non-minimum phase behavior.[44] It turns in the direction opposite of how it is initially steered, as described above in the section on countersteering

#### Degrees of freedom

Graphs of bike steer angle and lean angle vs turn radius.
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):[2]
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,[2] the lean equation
$M_{\theta\theta}\ddot{\theta_r} + K_{\theta\theta}\theta_r + M_{\theta\psi}\ddot{\psi} + C_{\theta\psi}\dot{\psi} + K_{\theta\psi}\psi = M_{\theta}$
and the steer equation
$M_{\psi\psi}\ddot{\psi} + C_{\psi\psi}\dot{\psi} + K_{\psi\psi}\psi + M_{\psi\theta}\ddot{\theta_r} + C_{\psi\theta}\dot{\theta_r} + K_{\psi\theta}\theta_r = M_{\psi}\mbox{,}$
where
• θr is the lean angle of the rear assembly,
• ψ is the steer angle of the front assembly relative to the rear assembly and
• Mθ and Mψ are the moments (torques) applied at the rear assembly and the steering axis, respectively. For the analysis of an uncontrolled bike, both are taken to be zero.
These can be represented in matrix form as
$M\mathbf{\ddot{q}}+C\mathbf{\dot{q}}+K\mathbf q=\mathbf f$
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 v2 and is symmetric in g and asymmetric in v2,
• $\mathbf q$ 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.[2]
The equations show that the bicycle is like an inverted pendulum with the lateral position of its support controlled by terms representing roll acceleration, roll velocity and roll displacement to steering torque feedback. The roll acceleration term is normally of the wrong sign for self-stabilization and can be expected to be important mainly in respect of wobble oscillations. The roll velocity feedback is of the correct sign, is gyroscopic in nature, being proportional to speed, and is dominated by the front wheel contribution. The roll displacement term is the most important one and is mainly controlled by trail, steering rake and the offset of the front frame mass center from the steering axis. All the terms involve complex combinations of bicycle design parameters and sometimes the speed. The limitations of the benchmark bicycle are considered and extensions to the treatments of tires, frames and riders, and their implications, are included. Optimal rider controls for stabilization and path-following control are also discussed.[45]

#### Eigenvalues

Eigenvalues plotted against forward speed for a typicalutility bicycle simplified to have knife-edge wheels that roll without slip.
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 of one particular bicycle are calculated for forward speeds of 0–10 m/s (22 mph). When the real parts of all eigenvalues (shown in dark blue) are negative, the bike is self-stable. When the imaginary parts of any eigenvalues (shown in cyan) are non-zero, the bike exhibits oscillation. The eigenvalues are point symmetric about the origin and so any bike design with a self-stable region in forward speeds will not be self-stable going backwards at the same speed.[2]
There are three forward speeds that can be identified in the plot to the right at which the motion of the bike changes qualitatively:[2]
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, where the weave mode eigenvalues switch from positive to negative in a Hopf bifurcation 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, where the capsize mode eigenvalues switch from negative to positive in a pitchfork bifurcation at about 8 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.[9]
Experimentation with real bikes has so far confirmed the weave mode predicted by the eigenvalues. It was found that tire slip and frame flex are not important for the lateral dynamics of the bicycle in the speed range up to 6 m/s.[46] The idealized bike model used to calculate the eigenvalues shown here does not incorporate any of the torques that real tires can generate, and so tire interaction with the pavement cannot prevent the capsize mode from becoming unstable at high speeds, as Wilson and Cossalter suggest happens in the real world.

#### Modes

Graphs that show (from left to right, top to bottom) weave instability, self-stability, marginal self-stability, and capsize instability in an idealized linearized model of an uncontrolled utility bicycle.
Bikes, as complex mechanisms, have a variety of modes: fundamental ways that they can move. These modes can be stable or unstable, depending on the bike parameters and its forward speed. 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 modes can be differentiated by the speed at which they switch stability and the relative phases of leaning and steering as the bike experiences that mode. Any bike motion consists of a combination of various amounts of the possible modes, and there are three main modes that a bike can experience: capsize, weave, and wobble.[2] A lesser known mode is rear wobble, and it is usually stable.[9]
##### 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, 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.[9]
For most bikes, depending on geometry and mass distribution, capsize is stable at low speeds, and becomes less stable as speed increases until it is no longer stable. However, on many bikes, tire interaction with the pavement is sufficient to prevent capsize from becoming unstable at high speeds.[9][10]
##### 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.[9] 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
Eigenvalues plotted against forward speed for amotorcycle modeled with frame flexibility and realistic tire dynamics. Additional modes can be seen, such as wobble, which becomes unstable at 43.7 m/s.
The same eigenvalues as in the figure above, but plotted on a root locus plot. Several additional oscillating modes are visible.
Wobbleshimmytank-slapperspeed 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.[9][10] While wobble or shimmy can be easily remedied by adjusting speed, position, or grip on the handlebar, it can be fatal if left uncontrolled.[47] This AVI movie shows wobble.
Wobble or shimmy begins when some otherwise minor irregularity, such as fork asymmetry,[48] 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.[14]
##### 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.[9]
##### Design criteria
The effect that the design parameters of a bike have on these modes can be investigated by examining the eigenvalues of the linearized equations of motion.[42] For more details on the equations of motion and eigenvalues, see the section on the equations of motion above. 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 flatsteering-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 tiremake 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.
One study has shown theoretically that, while a bike leaned in a turn, road undulations can excite the weave mode at high speed or the wobble mode at low speed if either of their frequencies match the vehicle speed and other parameters. Excitation of the wobble mode can be mitigated by an effective steering damper and excitation of the weave mode is worse for light riders than for heavy riders.[13]

### 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.[14] While it is an observable fact that bikes can be ridden even when the gyroscopic effects of their wheels are canceled out,[5][6] the hypothesis that the gyroscopic effects of the wheels are what keep a bike upright is common in print and online.[5][28]
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:

## Longitudinal dynamics

A bicyclist performing a wheelie.
Bikes may experience a variety of longitudinal forces and motions. On most bikes, when the front wheel is turned to one side or the other, the entire rear frame pitches forward slightly, depending on the steering axis angle and the amount of trail.[9][27] On bikes with suspensions, either front, rear, or both, trim is used to describe the geometric configuration of the bike, especially in response to forces of braking, accelerating, turning, drive train, and aerodynamic drag.[9]
The load borne by the two wheels varies not only with center of mass location, which in turn varies with the amount and location of passengers and luggage, but also with acceleration and deceleration. This phenomenon is known as load transfer[9] or weight transfer,[25][43] depending on the author, and provides challenges and opportunities to both riders and designers. For example, motorcycle racers can use it to increase the friction available to the front tire when cornering, and attempts to reduce front suspension compression during heavy braking has spawned several motorcycle fork designs.
The net aerodynamic drag forces may be considered to act at a single point, called the center of pressure.[25] At high speeds, this will create a net moment about the rear driving wheel and result in a net transfer of load from the front wheel to the rear wheel.[25] Also, depending on the shape of the bike and the shape of any fairing that might be installed, aerodynamic lift may be present that either increases or further reduces the load on the front wheel.[25]

### Stability

Though longitudinally stable when stationary, a bike may become longitudinally unstable under sufficient acceleration or deceleration, and Euler's second law can be used to analyze the ground reaction forces generated.[49] For example, the normal (vertical) ground reaction forces at the wheels for a bike with a wheelbase L and a center of mass at height h and at a distance b in front of the rear wheel hub, and for simplicity, with both wheels locked, can be expressed as:[9]
$N_r = mg\left(\frac{L-b}{L} - \mu \frac{h}{L}\right)$ for the rear wheel and $N_f = mg\left(\frac{b}{L} + \mu \frac{h}{L}\right)$ for the front wheel.
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 μ is the coefficient of frictionm is the total mass of the bike and rider, and g is the acceleration of gravity. Therefore, if
$\mu \ge \frac{L-b}{h},$
which occurs if the center of mass is anywhere above or in front of a line extending back from the front wheel contact patch and inclined at the angle
$\theta = \tan^{-1} \left( \frac{1}{\mu} \right) \,$
above the horizontal,[25] 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 or loop forward over the front wheel.
On the other hand, if the center of mass height is behind or below the line, as is true, for example on most tandem bicycles or long-wheel-base recumbent bicycles, 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.
Similarly, powerful motorcycles can generate enough torque at the rear wheel to lift the front wheel off the ground in a maneuver called a wheelie. A line similar to the one described above to analyze braking performance can be drawn from the rear wheel contact patch to predict if a wheelie is possible given the available friction, the center of mass location, and sufficient power.[25] This can also happen on bicycles, although there is much less power available, if the center of mass is back or up far enough or the rider lurches back when applying power to the pedals.[50]
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. The possibility of performing a wheelie increases when riding up hill,[50] and is a major factor in motorcycle hillclimbing competitions.

### Braking

A motorcyclist performing a stoppie.
Most of the braking force of standard upright bikes comes from the front wheel. As the analysis above shows, 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 bike does not flip, or an endo (abbreviated form of end-over-end) if the bike flips. On long or low bikes, however, such as cruiser motorcycles and recumbent bicycles, the front tire will skid instead, possibly causing a loss of balance.
In the case of a front suspension, especially telescoping fork tubes, the increase in downward force on the front wheel during braking 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, often between 0.5 and 0.8 for rubber on dry asphalt,[51]
• the kinetic friction between the brake pads and the rim or disk, and
• pitching or looping (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 60 cm (24 in) back from the front wheel contact patch and 120 cm (47 in) above, allowing a maximum deceleration of 0.5 g (5 m/s2 or 16 ft/s2).[14] 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.

#### Rear-wheel braking

The rear brake of an upright bicycle can only produce about 0.1 g (1 m/s2) deceleration at best,[14] 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. There are, however, situations that may warrant rear wheel braking[52]
• Slippery surfaces or bumpy surfaces. Under front wheel braking, the lower coefficient of friction may cause the front wheel to skid which often results in a loss of balance.[52]
• Front flat tire. Braking a wheel with a flat tire can cause the tire to come off the rim which greatly reduces friction and, in the case of a front wheel, result in a loss of balance.[52]
• Front brake failure.[52]

## Suspension

Mountain bike rear suspension
Bikes may have only front, only rear, full suspension or no suspension that operate primarily in the central plane of symmetry; though with some consideration given to lateral compliance.[25] The goals of a bike suspension are to reduce vibration experienced by the rider, maintain wheel contact with the ground, and maintain vehicle trim.[9]The primary suspension parameters are stiffnessdamping, sprung and unsprung mass, and tire characteristics.[25] Besides irregularities in the terrain, brake, acceleration, and drive-train forces can also activate the suspension as described above. Examples include bob and pedal feedback on bicycles, the shaft effect on motorcycles, and squat and brake dive on both.

## Vibration

The study of vibration in bikes includes its causes, such as engine balance,[53] wheel balance, ground surface, and aerodynamics; its transmission and absorption; and its effects on the bike, the rider, and safety.[54] 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.[55] A close match means mechanical resonance that can result in large amplitudes. A challenge in vibration damping is to create compliance in certain directions (vertically) without sacrificing frame rigidity needed for power transmission and handling (torsionally).[56] Another issue with vibration for the bike is the possibility of failure due to material fatigue[57] Effects of vibration on riders include discomfort, loss of efficiency, Hand-Arm Vibration Syndrome, a secondary form Raynaud's disease, and whole body vibration. Vibrating instruments may be inaccurate or difficult to read.[57]

### 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 todamp 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 forkseatpost, or handlebars; tube shapes, such as curved seat stays;[58] and special inserts, such as Zertz by Specialized,[59][60] 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 enginebalance shafts, rubber engine mounts,[61] and tire weights.[62] The problems that vibration causes have also spawned an industry of after-market parts and systems designed to reduce it. Add-ons include handlebarweights,[63] isolated foot pegs, and engine counterweights. At high speeds, motorcycles and their riders may also experience aerodynamic flutter or buffeting.[64] This can be abated by changing the air flow over key parts, such as the windshield.[65]

## 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.[6]
• Richard Klein built several bikes to confirm Jones's findings.[5]
• Richard Klein also built a "Torque Wrench Bike" and a "Rocket Bike" to investigate steering torques and their effects.[5]
• Keith Code built a motorcycle with fixed handlebars to investigate the effects of rider motion and position on steering.[66]
• Schwab and Kooijman have performed measurements with an instrumented bike.[67]

 Cycling portal

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