User:AndrewDressel/Sandbox

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 trial.^[1] 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.^[1]

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^[1] or weight transfer,^[2]^[3] 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.^[2] 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.^[2] 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.^[2]

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. 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:^[1]^[4]

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 $\mu \,$ is the coefficient of friction, $m\,$ is the total mass of the bike and rider, and $g\,$ is the acceleration of gravity. Therefore, if

\mu \geq {\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,^[2] 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 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.

Similarly, powerful motorcycles can generate enough torque at the rear wheel to lift the front wheel off the ground. This is called a wheelie. A similar line 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.^[2] This can also happen on bicycles, although there is much less power available, if the center of mass is back far enough or the rider lurches back when applying power to the pedals.^[5]

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, and is a major factor in motorcycle hillclimbing competitions.

Braking

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,^[6]
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 (4.9 m/s² or 16 ft/s²).^[7] 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 deceleration at best,^[7] 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^[8]

Slippery 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.
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.
Long mountain descents. Alternating between front and rear brakes can help reduce heat buildup which can cause a blowout.

References

^ ^a ^b ^c ^d Cite error: The named reference Cossalter was invoked but never defined (see the help page).
^ ^a ^b ^c ^d ^e ^f Cite error: The named reference Foale was invoked but never defined (see the help page).
^ Cocco, Gaetano (2005). Motorcycle Design and Technology. Motorbooks. pp. 40–46. ISBN 978-0-7603-1990-1.
^ Ruina, Andy (2002). Introduction to Statics and Dynamics (PDF). Oxford University Press. p. 350. Retrieved 2006-08-04. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Cassidy, Chris. "Bicycling Magazine: The Wheelie". Retrieved 2009-05-22.
^ Kurtus, Ron (2005-11-02). "Coefficient of Friction Values for Clean Surfaces". Retrieved 2006-08-07.
^ ^a ^b Cite error: The named reference whitt was invoked but never defined (see the help page).
^ Brown, Sheldon "Front Brake". "Braking and Turning". Retrieved 2009-05-22.

[Cossalter-1] Cite error: The named reference Cossalter was invoked but never defined (see the help page).

[Foale-2] ^ ^a ^b ^c ^d ^e ^f Cite error: The named reference Foale was invoked but never defined (see the help page).

[Cocco-3] Cocco, Gaetano (2005). Motorcycle Design and Technology. Motorbooks. pp. 40–46. ISBN 978-0-7603-1990-1.

[4] Ruina, Andy (2002). Introduction to Statics and Dynamics (PDF). Oxford University Press. p. 350. Retrieved 2006-08-04. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[5] Cassidy, Chris. "Bicycling Magazine: The Wheelie". Retrieved 2009-05-22.

[kurtus-6] Kurtus, Ron (2005-11-02). "Coefficient of Friction Values for Clean Surfaces". Retrieved 2006-08-07.

[whitt-7] Cite error: The named reference whitt was invoked but never defined (see the help page).

[8] Brown, Sheldon "Front Brake". "Braking and Turning". Retrieved 2009-05-22.

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