Bicycle and motorcycle dynamics
Encyclopedia
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
of the motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...
of bicycle
Bicycle
A bicycle, also known as a bike, pushbike or cycle, is a human-powered, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A person who rides a bicycle is called a cyclist, or bicyclist....
s and motorcycle
Motorcycle
A motorcycle is a single-track, two-wheeled motor vehicle. Motorcycles vary considerably depending on the task for which they are designed, such as long distance travel, navigating congested urban traffic, cruising, sport and racing, or off-road conditions.Motorcycles are one of the most...
s and their components, due to the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s acting on them. Dynamics is a branch of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, which in turn is a branch of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Bike motions of interest include balancing
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
, steering
Countersteering
Countersteering is the technique used by single-track vehicle operators, such as cyclists and motorcyclists, to initiate a turn toward a given direction by momentarily steering counter to the desired direction...
, braking
Brake
A brake is a mechanical device which inhibits motion. Its opposite component is a clutch. The rest of this article is dedicated to various types of vehicular brakes....
, accelerating
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
, suspension
Suspension (vehicle)
Suspension is the term given to the system of springs, shock absorbers and linkages that connects a vehicle to its wheels. Suspension systems serve a dual purpose — contributing to the car's roadholding/handling and braking for good active safety and driving pleasure, and keeping vehicle occupants...
activation, and vibration
Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
. The study of these motions began in the late 19th century and continues today.
Bicycles and motorcycles are both single-track vehicle
Single-track vehicle
A single-track vehicle is a vehicle that leaves a single ground track as it moves forward. Single-track vehicles usually have little or no lateral stability when stationary but develop it when moving forward or controlled...
s and so their motions have many fundamental attributes in common and are fundamentally different from and more difficult to study than other wheeled vehicles such as dicycles
Dicycle (vehicle)
A dicycle is a vehicle with two wheels side by side, unlike single-track vehicles such as motorcycles and bicycles, which have one wheel followed by another....
, tricycle
Tricycle
A tricycle is a three-wheeled vehicle. While tricycles are often associated with the small three-wheeled vehicles used by pre-school-age children, they are also used by adults for a variety of purposes. In the United States and Canada, adult-sized tricycles are used primarily by older persons for...
s, and quadracycle
Quadracycle
A quadracycle is a four-wheeled human-powered vehicle. It is also referred to as a quadricycle, quadcycle pedal car or four-wheeled bicycle, amongst other terms....
s. As with unicycle
Unicycle
A unicycle is a human-powered, single-track vehicle with one wheel. Unicycles resemble bicycles, but are less complex.-History:One theory of the advent of the unicycle stems from the popularity of the penny-farthing during the late 19th century...
s, bikes lack lateral stability when stationary, and under most circumstances can only remain upright when moving forward. Experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
ation and mathematical analysis
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
have shown that a bike stays upright when it is steered to keep its center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
over its wheels. This steering is usually supplied by a rider, or in certain circumstances, by the bike itself. Several factors, including geometry, mass distribution, and gyroscopic effect all contribute in varying degrees to this self-stability, but long-standing hypotheses and claims that any single effect, such as gyroscopic or trail, is solely responsible for the stabilizing force have been discredited.
While remaining upright may be the primary goal of beginning riders, a bike must lean in order to maintain balance in a turn: the higher the speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
or smaller the turn radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
, the more lean is required. This balances the roll torque about the wheel contact patches generated by centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...
due to the turn with that of the gravitational force. This lean is usually produced by a momentary steering in the opposite direction, called countersteering
Countersteering
Countersteering is the technique used by single-track vehicle operators, such as cyclists and motorcyclists, to initiate a turn toward a given direction by momentarily steering counter to the desired direction...
. Countersteering skill is usually acquired by motor learning
Motor learning
Motor learning is a “relatively permanent” change, resulting from practice or a novel experience, in the capability for responding...
and executed via procedural memory
Procedural memory
Procedural memory is memory for how to do things. Procedural memory guides the processes we perform and most frequently resides below the level of conscious awareness. When needed, procedural memories are automatically retrieved and utilized for the execution of the integrated procedures involved...
rather than by conscious thought. Unlike other wheeled vehicles, the primary control
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
input on bikes is steering torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
, not position.
Although longitudinally stable when stationary, bikes often have a high enough center of mass and a short enough wheelbase to lift a wheel off the ground under sufficient acceleration or deceleration. 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. A similar situation is possible while accelerating, but with respect to the rear wheel.
History
William John Macquorn Rankine
William John Macquorn Rankine was a Scottish civil engineer, physicist and mathematician. He was a founding contributor, with Rudolf Clausius and William Thomson , to the science of thermodynamics....
, Appell
Paul Émile Appell
Paul Appell , also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris...
, and Whipple
Francis John Welsh Whipple
Francis John Welsh Whipple was a British mathematician and meteorologist. From 1925 to 1939 he was superintendent of the Kew Observatory.-Biography:...
. In the early 19th century Karl von Drais
Karl Drais
Karl Drais was a German inventor and invented the Laufmaschine , also later called the velocipede, draisine or "draisienne" , also nicknamed the dandy horse. This incorporated the two-wheeler principle that is basic to the bicycle and motorcycle and was the beginning of mechanized personal...
, credited with inventing the two-wheeled vehicle variously called the laufmaschine, velocipede
Velocipede
Velocipede is an umbrella term for any human-powered land vehicle with one or more wheels. The most common type of velocipede today is the bicycle....
, draisine
Draisine
A draisine primarily refers to a light auxiliary rail vehicle, driven by service personnel, equipped to transport crew and material necessary for the maintenance of railway infrastructure....
, and dandy horse
Dandy horse
The dandy-horse, also known as hobby-horse, is a human-powered vehicle that, being the first means of transport to make use of the two-wheeler principle, is regarded as the forerunner of the bicycle. The dandy horse was invented by Baron Karl Drais in Mannheim, Germany, and patented in January 1818...
, showed that a rider could balance his device by steering the front wheel. By the end of the 19th century, Emmanuel Carvallo and Francis Whipple showed with rigid-body dynamics that some safety bicycle
Safety bicycle
A safety bicycle is a type of bicycle that became very popular beginning in the late 1880s as an alternative to the penny-farthing or ordinary and is now the most common type of bicycle. Early bicycles of this style were known as safety bicycles because they were noted for, and marketed as, being...
s 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 E. H. Jones
David E. H. Jones
David E. H. Jones is best known as Daedalus, the fictional inventor for DREADCO. Jones' columns as Daedalus were published weekly in the New Scientist starting in the mid-sixties. He then moved on to the journal Nature, and continued to publish for many years. He published two books with columns...
published an article in Physics Today
Physics Today
Physics Today, created in 1948, is the membership journal of the American Institute of Physics. It is provided to 130,000 members of twelve physics societies, including the American Physical Society...
showing that gyroscopic effects are not necessary to balance a bicycle. Since 1971, when he identified and named the wobble, weave and capsize modes, Robin Sharp has written regularly about the behavior of motorcycles and bicycles. While at Imperial College, London, he worked with David Limebeer and Simos Evangelou. In 2007, Meijaard, et al., published the canonical linearized equations of motion
Equation of motion
Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...
, in the Proceedings of the Royal Society A, along with verification by two different methods. These equations assumed the tires to roll without slip, that is to say, to go where they point, and the rider to be rigidly attached to the rear frame of the bicycle.
In 2011, Kooijman, et al., published an article in Science
Science (journal)
Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
showing that neither gyroscopic effects nor so-called caster effects due to trail are necessary for a bike to balance itself. They designed a two-mass-skate bicycle
Two-mass-skate bicycle
A two-mass-skate bicycle is a theoretical model created by a team of researchers at Cornell University, University of Wisconsin-Stout, and Delft University of Technology to show that it is neither sufficient nor necessary for a bike to have gyroscopic effects or positive trail to be self-stable...
that the equations of motion predict is self-stable even with negative trail, the front wheel contacts the ground in front of the steering axis, and with counter-rotating wheels to cancel any gyroscopic effects. Then they constructed a physical model to validate that prediction. This may require some of the details provided below about steering geometry or stability to be re-evaluated.
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, gravityGravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
pulls the rider and all the bike components toward the earth. At each tire contact patch
Contact patch
Contact patch is the portion of a vehicle's tire that is in actual contact with the road surface. It is most commonly used in the discussion of pneumatic tires, , where the term is strictly used to describe the portion of the tire’s tread that touches the road surface...
there are ground reaction
Reaction (physics)
The third of Newton's laws of motion of classical mechanics states that forces always occur in pairs. Every action is accompanied by a reaction of equal magnitude but opposite direction. This principle is commonly known in the Latin language as actio et reactio. The attribution of which of the two...
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
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
between the wheels and the ground, including rolling resistance
Rolling resistance
Rolling resistance, sometimes called rolling friction or rolling drag, is the resistance that occurs when a round object such as a ball or tire rolls on a flat surface, in steady velocity straight line motion. It is caused mainly by the deformation of the object, the deformation of the surface, or...
, are in response to propulsive
Vehicle propulsion
Vehicle propulsion refers to the act of moving an artificial carrier of people or goods over any distance. The power plant used to drive the vehicles can vary widely. Originally, humans or animals would have provided the propulsion system, later being supplemented by wind power...
forces, braking forces, and turning forces. Aerodynamic
Full aerodynamic force
Aerodynamic force is exerted on a body by the air in which the body is immersed, and is due to the relative motion between the body and the gas...
forces due to the atmosphere are mostly in the form of drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
, but can also be from crosswind
Crosswind
A crosswind is any wind that has a perpendicular component to the line or direction of travel. In aviation, a crosswind is the component of wind that is blowing across the runway making landings and take-offs more difficult than if the wind were blowing straight down the runway...
s. At normal bicycling speeds on level ground, aerodynamic drag is the largest force resisting forward motion. At faster speed, aerodynamic drag becomes overwhelmingly the largest force resisting forward motion.
Turning forces are generated during maneuvers for balancing in addition to just changing direction of travel. These may be interpreted as centrifugal
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...
forces in the accelerating reference frame
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
of the bike and rider; or simply as inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
in a stationary, inertial reference frame
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...
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.
Internal forces
Internal forces are mostly caused by the rider or by friction. The rider can apply torqueTorque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
s between the steering mechanism (front fork, handlebars, front wheel, etc.) and rear frame, and between the rider and the rear frame. Friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
exists between any parts that move against each other: in the drive train
Powertrain
In a motor vehicle, the term powertrain or powerplant refers to the group of components that generate power and deliver it to the road surface, water, or air. This includes the engine, transmission, drive shafts, differentials, and the final drive...
, between the steering mechanism and the rear frame, etc. Many bikes have front and rear suspensions
Bicycle suspension
A bicycle suspension is the system or systems used to suspend the rider and all or part of the bicycle in order to protect them from the roughness of the terrain over which they travel...
, and some motorcycles have a steering damper
Steering damper
A steering damper, steering stabiliser or sprint damper is a damping device designed to inhibit an undesirable, uncontrolled movement or oscillation of a vehicle steering mechanism, a phenomenon known in motorcycling as wobble, or in extreme cases, a tank-slapper...
to dissipate undesirable kinetic energy. On bikes with rear suspensions, feedback between the drive train and the suspension is an issue designers attempt to handle with various linkage
Linkage (mechanical)
A mechanical linkage is an assembly of bodies connected together to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for...
configurations and dampers
Dashpot
A dashpot is a mechanical device, a damper which resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly used in conjunction with a spring...
.
Motions
Motions of a bike can be roughly grouped into those out of the central plane of symmetry: lateral; and those in the central plane of symmetry: longitudinal or vertical. Lateral motions include balancing, leaning, steering, and turning. Motions in the central plane of symmetry include rolling forward, of course, but also stoppieStoppie
The stoppie , is a motorcycle and bicycle trick in which the back wheel is lifted and the bike is ridden on the front wheel by carefully applying brake pressure. It is also sometimes called a front wheelie.-General description:...
s, wheelie
Wheelie
In vehicle acrobatics, a wheelie is a vehicle maneuver in which the front wheel or wheels come off the ground due to extreme torque being applied to the rear wheel or wheels. Wheelies are usually associated with bicycles and motorcycles, but can be done with other vehicles such as cars, especially...
s, brake diving, and most suspension activation. Motions in these two groups are linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
ly decoupled, that is they do not interact with each other to the first order
Orders of approximation
In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...
. An uncontrolled bike is laterally unstable when stationary and can be laterally self-stable when moving under the right conditions or when controlled by a rider. Conversely, a bike is longitudinally stable when stationary and can be longitudinally unstable when undergoing sufficient acceleration or deceleration.
Lateral dynamics
Of the two, lateral dynamics has proven to be the more complicated, requiring three-dimensionalThree-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
, multibody dynamic analysis with at least two generalized coordinates
Generalized coordinates
In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....
to analyze. At a minimum, two coupled, second-order differential equations are required to capture the principal motions. Exact solution
Exact solution
In mathematics and especially physics, an exact solution is a solution to a problem that encapsulates the whole mathematics or physics of the problem without using an approximation.-References:*...
s are not possible, and numerical methods
Numerical ordinary differential equations
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
must be used instead. Competing theories of how bikes balance can still be found in print and online. On the other hand, as shown in later sections, much longitudinal dynamic analysis can be accomplished simply with planar kinetics
Kinetics (physics)
In physics and engineering, kinetics is a term for the branch of classical mechanics that is concerned with the relationship between the motion of bodies and its causes, namely forces and torques...
and just one coordinate.
Balance
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.
Even when staying relatively motionless, a rider can balance a bike by the same principle. While performing a track stand
Track stand
The track stand is a technique that bicycle riders can use to maintain balanced while their bicycle remains stationary or moves only minimal distances. The technique originated in track cycling and is now used by other types of cyclists wishing to stop for a short time without putting a foot on the...
, the rider can keep the line between the two contact patches under the combined center of mass by steering the front wheel to one side or the other and then moving forward and backward slightly to move the front contact patch from side to side as necessary. Forward motion can be generated simply by pedaling. Backwards motion can be generated the same way on a fixed-gear bicycle
Fixed-gear bicycle
A fixed-gear bicycle is a bicycle that has no freewheel, meaning it cannot coast, as the pedals are always in motion when the bicycle is moving....
. Otherwise, the rider can take advantage of an opportune slope of the pavement or lurch the upper body backwards while the brakes are momentarily engaged.
If the steering of a bike is locked, it becomes virtually impossible to balance while riding. On the other hand, if 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 handlebarHandlebar
A handlebar is part of the steering mechanism, in lieu of a steering wheel, for vehicles that are ridden on, instead of in, such as bicycles, motorcycles, tricycles, ATVs, snowmobiles and personal watercraft. It may also be simply the handles on a device such as a zip line trolley or a floor buffer...
s 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 recumbentsRecumbent bicycle
A recumbent bicycle is a bicycle that places the rider in a laid-back reclining position. Most recumbent riders choose this type of design for ergonomic reasons; the rider's weight is distributed comfortably over a larger area, supported by back and buttocks...
and choppers
Chopper bicycle
A chopper bicycle is a bicycle that is designed to get attention and provide a smooth comfortable ride.- About :While the term "chopper" is generally used to describe a motorcycle or bicycle that has had some of its original parts replaced with custom parts, today's definition has grown to include...
. It can also be an issue for touring bikes
Touring motorcycle
A touring motorcycle is a type of motorcycle designed for touring. Although any motorcycle can be used to tour with, manufacturers have brought specific models designed to address the particular needs of these riders...
with a heavy load of gear over or even behind the rear wheel. Mass over the rear wheel can be more easily controlled if it is lower than mass over the front 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 low one because its lean rate will be slower. However, 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
Bicycle touring
Bicycle touring is cycling over long distances – prioritizing pleasure and endurance over utility or speed. Touring can range from single day 'supported' rides — e.g., rides to benefit charities — where provisions are available to riders at stops along the route, to multi-day...
cyclists are advised to carry loads low on a bike, and panniers hang down on either side of front and rear racks
Luggage carrier
A luggage carrier, also commonly called a rack, is a device attached to a bicycle to which cargo or panniers can be attached. This is popular with utility bicycles and touring bicycles....
.
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, positive trail tends to steer the front wheel into the direction of a lean, independent of forward speed.This can be simulated 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 dynamics of a moving bike are more complicated, however, and other factors can contribute to or detract from this effect.
Trail is a function of head angle, fork offset or rake, and wheel size. Their relationship can be described by this formula:
where
is wheel radius, 
is the head angle measured clock-wise from the horizontal and 
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.The more trail a traditional 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, are reported to feel very unstable. Bikes with too much trail can 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
Bicycle fork
A bicycle fork is the portion of a bicycle that holds the front wheel and allows the rider to steer and balance the bicycle. A fork consists of two fork ends which hold the front wheel axle, two blades which join at a fork crown, and a steerer or steering tube to which the handlebars attach ...
rake, often a curve in the fork blades forward of the steering axis, is used to diminish trail. Bikes with negative trail exist, such as the Python Lowracer, and are ridable, and an experimental bike with negative trail has been shown to be self-stable.
In motorcycles, rake refers to the head angle instead, and offset created by the triple tree is used to diminish trail.
A small survey by Whitt and Wilson found:
- touring bicycleTouring bicycleA touring bicycle is a bicycle designed or modified to handle bicycle touring. To make the bikes sufficiently robust, comfortable and capable of carrying heavy loads, special features may include a long wheelbase , frame materials that favor flexibility over rigidity , heavy duty wheels , and...
s with head angles between 72° and 73° and trail between 43 mm and 60 mm - racing bicycleRacing bicycleA racing bicycle, also known as a road bike, is a bicycle designed for competitive road cycling, a sport governed by according to the rules of the Union Cycliste Internationale...
s with head angles between 73° and 74° and trail between 28 mm and 45 mm - track bicycleTrack bicycleA track bicycle or track bike is a bicycle optimized for racing at a velodrome or outdoor track. Unlike road bicycles, the track bike is a fixed-gear bicycle and so has a single gear and neither freewheel nor brakes. Tires are narrow and inflated to high pressure to reduce rolling resistance...
s with head angles of 75° and trail between 23.5 mm and 37 mm.
However, these ranges are not hard and fast. For example, LeMond Racing Cycles
LeMond Racing Cycles
LeMond Racing Cycles is a bicycle manufacturer originally founded by Greg LeMond, the first American to win the Tour de France . LeMond offered a geometry based on the racing frames he used in competition, which had a longer top tube and wheelbase in an otherwise traditional lightweight steel frame...
offers
both with forks that have 45 mm of offset or rake and the same size wheels:
- a 2006 Tete de Course, designed for road racing, with a head angle that varies from 71¼° to 74°, depending on frame size, and thus trail that varies from 69 mm to 51.5 mm.
- a 2007 Filmore, designed for the track, with a head angle that varies from 72½° to 74°, depending on frame size, and thus trail that varies from 61 mm to 51.5 mm.
The amount of trail a particular bike has may vary with time for several reasons. On bikes with front suspension, especially telescopic forks, compressing the front suspension, due to heavy braking for example, can steepen the steering axis angle and reduce trail. Trail also varies with lean angle, and steering angle, usually decreasing from a maximum when the bike is straight upright and steered straight ahead. Finally, even the profile of the front tire can influence how trail varies as the bike is leaned and steered.
A measurement similar to trail, called either mechanical trail, normal trail, or true trail, is the perpendicular distance from the steering axis to the centroid of the front wheel contact patch.
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
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
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. A bicycle wheel with an internal flywheel for enhanced gyroscopic effect is under development as a commercial product, the Gyrobike
Gyrobike
A Gyrobike is a bicycle with a Gyrowheel for its front wheel. The Gyrowheel contains a motor-driven flywheel that spins independently, and uses the principle of gyroscopic precession to help stabilize the bike and the rider...
, for making it easier to learn to ride bicycles.
Another contribution of gyroscopic effects is a roll moment
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...
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
Motorcycle racing
Motorcycle sport is a broad field that encompasses all sporting aspects of motorcycling. The disciplines are not all "races" or timed-speed events, as several disciplines test a competitor's various riding skills.-Motorcycle racing:...
. For more detail, see the countersteering article.
Self-stability
Between the two unstable regimes mentioned in the previous section, and influenced by all the factors described above that contribute to balance (trail, mass distribution, gyroscopic effects, etc.), there may be a range of forward speeds for a given bike design at which these effects steer an uncontrolled bike upright. It has been proven that neither gyroscopic effects nor positive trail are sufficient by themselves or necessary for self-stability, although they certainly can enhance hands-free control.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
Headset (bicycle part)
The headset is the set of components on a bicycle that provides a rotatable interface between the bicycle fork and the head tube of the bicycle frame itself. The short tube through which the steerer of the fork passes is called the head tube. A typical headset consists of two cups that are pressed...
friction and stiff control cables
Bowden cable
A Bowden cable is a type of flexible cable used to transmit mechanical force or energy by the movement of an inner cable relative to a hollow outer cable housing...
. This video shows a riderless bicycle exhibiting self-stability.
Turning
Cornering force
Cornering force or side force is the lateral force produced by a vehicle tire during cornering.Cornering force is generated by tire slip and is proportional to slip angle at low slip angles. The rate at which cornering force builds up is described by relaxation length...
and camber thrust
Camber thrust
Camber thrust and camber force are terms used to describe the force generated perpendicular to the direction of travel of a rolling tire due to its camber angle and finite contact patch...
. The radius of the turn of an upright (not leaning) bike can be roughly approximated, for small steering angles, by:
where
is the approximate radius, 
is the wheelbase, 
is the steer angle, and 
is the caster angle of the steering axis.Leaning
However, unlike other wheeled vehicles, bikes must also lean during a turn 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 motionCircular motion
In physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...
:
where v is the forward speed, r is the radius of the turn and g is the acceleration of gravity. This is in the idealized case. A slight increase in the lean angle may be required on motorcycles to compensate for the width of modern tires at the same forward speed and turn radius.
For example, a bike in a 10 m (33 ft) radius steady-state turn at 10 m/s (36 km/h, 22 mph) must be at an angle of 45.6°. 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 defined by the tire contacts and the location of the center of mass of bike and rider.
This lean of the bike decreases the actual radius of the turn proportionally to the cosine of the lean angle. The resulting radius can be roughly approximated (within 2% of exact value) by:
where r is the approximate radius, w is the wheelbase, θ is the lean angle, δ is the steer angle, and φ is the caster angle of the steering axis. 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 .
The finite width of the tires alters the actual lean angle of the rear frame from the ideal lean angle described above. The actual lean angle between the frame and the vertical must increase with tire width and decrease with center of mass height. Bikes with fat tires and low center of mass must lean more than bikes with skinnier tires or higher centers of mass to negotiate the same turn at the same speed.
The increase in lean angle due to a tire thickness of 2t can be calculated as
where φ is the ideal lean angle, and h is the height of the center of mass. For example, a motorcycle with a 12 inch wide rear tire will have t = 6 inches. If the combined bike and rider center of mass is at a height of 26 inches, then a 25o lean must be increased by 7.28°: a nearly 30% increase. If the tires are only 6 inches wide, then the lean angle increase is only 3.16°, just under half.
Countersteering
In order to initiate a turn and the necessary lean in the direction of that turn, a bike must momentarily steer in the opposite direction. This is often referred to as countersteering. With the front wheel now at a finite angle to the direction of motion, a lateral force is developed at the contact patch of the tire. This force creates a torque around the longitudinal (roll) axis of the bike. This torque causes the bike to roll in the opposite direction of the turn. Where there is no external influence, such as an opportune side wind to create the force necessary to lean the bike, countersteering is necessary to initiate a rapid turn.While the initial steer torque and steer angle are both opposite the desired turn direction, this may not be the case to maintain a steady-state turn. The sustained steer angle is usually in the same direction as the turn, but may remain opposite to the direction of the turn, especially at high speeds. The sustained steer torque required to maintain that steer angle is usually opposite the turn direction. The actual magnitude and orientation of both the sustained steer angle and sustained steer torque of a particular bike in a particular turn depend on forward speed, bike geometry, tire properties, and combined bike and rider mass distribution. Once in a turn, the radius can only be changed with an appropriate change in lean angle, and this can 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.
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. 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.Steering angle
Several effects influence the steering angle, the angle at which the front assembly is rotated about the steering axis, necessary to maintain a steady-state turn. Some of these are unique to single-track vehicles, while others are also experienced by automobiles. Some of these may be mentioned elsewhere in this article, and they are repeated here, though not necessarily in order of importance, so that they may be found in one place.First, the actual kinematic steering angle, the angle projected onto the road plane to which the front assembly is rotated is a function of the steering angle and the steering axis angle:
where
is the kinematic steering angle, 
is the steering angle, and 
is the caster angle of the steering axis.Second, the lean of the bike decreases the actual radius of the turn proportionally to the cosine of the lean angle. The resulting radius can be roughly approximated (within 2% of exact value) by:
where
is the approximate radius, 
is the wheelbase, 
is the lean angle, 
is the steering angle, and 
is the caster angle of the steering axis.Third, because the front and rear tires can have different slip angle
Slip angle
In vehicle dynamics, slip angle or sideslip angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing...
s due to weight distribution, tire properties, etc., bikes can experience understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...
or oversteer. When understeering, the steering angle must be greater, and when oversteering, the steering angle must be less than it would be if the slip angles were equal to maintain a given turn radius. Some authors even use the term counter-steering to refer to the need on some bikes under some conditions to steer in the opposite direction of the turn (negative steering angle) to maintain control in response to significant rear wheel slippage.
Fourth, camber thrust
Camber thrust
Camber thrust and camber force are terms used to describe the force generated perpendicular to the direction of travel of a rolling tire due to its camber angle and finite contact patch...
contributes to the centripetal force
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...
necessary to cause the bike to deviate from a straight path, along with cornering force
Cornering force
Cornering force or side force is the lateral force produced by a vehicle tire during cornering.Cornering force is generated by tire slip and is proportional to slip angle at low slip angles. The rate at which cornering force builds up is described by relaxation length...
due to the slip angle
Slip angle
In vehicle dynamics, slip angle or sideslip angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing...
, and can be the largest contributor. Camber thrust contributes to the ability of bikes to negotiate a turn with the same radius as automobiles but with a smaller steering angle. When a bike is steered and leaned in the same direction, the camber angle of the front tire is greater than that of the rear and so can generate more camber thrust, all else being equal.
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 momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
, and the combined center of mass will remain in the same vertical plane. This leftward lean of the bike, called counter lean by some authors, 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.
Gyroscopic effects
As mentioned above in the section on balance, one effect of turning the front wheel is a roll momentMoment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...
caused by gyroscopic precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
. The magnitude of this moment is proportional to the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
of the front wheel, its spin rate (forward motion), the rate that the rider turns the front wheel by applying a torque to the handlebars, and the cosine of the angle between the steering axis and the vertical.
For a sample motorcycle moving at 22 m/s (50 mph) that has a front wheel with a moment of inertia of 0.6 kg·m2, turning the front wheel one degree in half a second generates a roll moment of 3.5 N·m. In comparison, the lateral force on the front tire as it tracks out from under the motorcycle reaches a maximum of 50 N. This, acting on the 0.6 m (2 ft) height of the center of mass, generates a roll moment of 30 N·m.
While the moment from gyroscopic forces is only 12% of this, it can play a significant part because it begins to act as soon as the rider applies the torque, instead of building up more slowly as the wheel out-tracks. This can be especially helpful in motorcycle racing
Motorcycle racing
Motorcycle sport is a broad field that encompasses all sporting aspects of motorcycling. The disciplines are not all "races" or timed-speed events, as several disciplines test a competitor's various riding skills.-Motorcycle racing:...
.
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 bikeSideways bike
The sideways bike is an invention, patented in 2005 , by Michael Killian, a software engineer from Dublin. He was inspired by the way that snowboarding is preferred to skiing due to the greater artistic potential, and decided to design a snowboard equivalent for the conventional bicycle. The...
, lets the rider control the steering of both wheels directly.
Milton W. Raymond built a long low two-wheel steering bicycle, called "X-2", with various steering mechanisms to control the two wheels independently. Steering motions included "balance", in which both wheels move together to steer the tire contacts under the center of mass; and "true circle", in which the wheels steer equally in opposite directions and thus steering the bicycle without substantially changing the lateral position of the tire contacts relative to the center of mass. X-2 was also able to go "crabwise" with the wheels parallel but out of line with the frame, for instance with the front wheel near the roadway center line and rear wheel near the curb
Curb (road)
A curb, or kerb , is the edge where a raised pavement/sidewalk/footpath, road median, or road shoulder meets an unraised street or other roadway.-Function:...
. "Balance" steering allowed easy balancing despite long wheelbase and low center of mass, but no self-balancing ("no hands") configuration was discovered. True circle, as expected, was essentially impossible to balance, as steering does not correct for misalignment of the tire patch and center of mass. Crabwise cycling at angles tested up to about 45° did not show a tendency to fall over, even under braking. X-2 is mentioned in passing in Whitt and Wilson's Bicycling Science 2nd edition.
Rear-wheel steering
Because of the theoretical benefits, especially a simplified front-wheel driveFront-wheel drive
Front-wheel drive is a form of engine/transmission layout used in motor vehicles, where the engine drives the front wheels only. Most modern front-wheel drive vehicles feature a transverse engine, rather than the conventional longitudinal engine arrangement generally found in rear-wheel drive and...
mechanism, attempts have been made to construct a ridable rear-wheel steering bike. The Bendix Company
Bendix Corporation
The Bendix Corporation was an American manufacturing and engineering company which during various times in its 60 year existence made brake systems, aeronautical hydraulics, avionics, aircraft and automobile fuel control systems, radios, televisions and computers, and which licensed its name for...
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, Illinois, offered 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 Massachusetts Institute of Technology, 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 for bicycles with common geometries and mass distributions shows that the rear-wheel steering configuration is inherently unstable. However, designs have been published that do not suffer this problem.
Center steering
Penny-farthing
Penny-farthing, high wheel, high wheeler, and ordinary are all terms used to describe a type of bicycle with a large front wheel and a much smaller rear wheel that was popular after the boneshaker, until the development of the safety bicycle, in the 1880s...
. This design allows for simple front-wheel drive and current implementations appear to be quite stable, even ridable no-hands, as many photographs illustrate.
These designs, such as the Python Lowracer
Recumbent bicycle
A recumbent bicycle is a bicycle that places the rider in a laid-back reclining position. Most recumbent riders choose this type of design for ergonomic reasons; the rider's weight is distributed comfortably over a larger area, supported by back and buttocks...
, 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 tillerTiller
A tiller or till is a lever attached to a rudder post or rudder stock of a boat that provides leverage for the helmsman to turn the rudder...
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 bicycle
Cruiser bicycle
Cruiser bicycles, also known as beach cruisers, combine balloon tires, upright seating posture, single-speed drivetrains, and straightforward steel construction with expressive styling...
s, 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
TiresMotorcycle tyres
Motorcycle tyres provide the only contact with the ground, via the contact patch under normal conditions, and so have a very large influence over motorcycle handling characteristics. Motorcycle tyres have a round cross section to facilitate the leaning necessary when a motorcycle turns...
have a large influence over bike handling, especially on motorcycles. Through a combination of cornering force
Cornering force
Cornering force or side force is the lateral force produced by a vehicle tire during cornering.Cornering force is generated by tire slip and is proportional to slip angle at low slip angles. The rate at which cornering force builds up is described by relaxation length...
and camber thrust
Camber thrust
Camber thrust and camber force are terms used to describe the force generated perpendicular to the direction of travel of a rolling tire due to its camber angle and finite contact patch...
, 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. Because the front and rear tires can have different slip angle
Slip angle
In vehicle dynamics, slip angle or sideslip angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing...
s due to weight distribution, tire properties, etc., bikes can experience understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...
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.
Also, because real tires have a finite contact patch
Contact patch
Contact patch is the portion of a vehicle's tire that is in actual contact with the road surface. It is most commonly used in the discussion of pneumatic tires, , where the term is strictly used to describe the portion of the tire’s tread that touches the road surface...
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
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
to the plane of the contact patch.
Self aligning torque
Self aligning torque, also known as aligning torque, SAT, or Mz, is the torque that a tire creates as it rolls along, which tends to steer it, i.e. rotate it around its vertical axis...
, is caused by asymmetries in the side-slip along the length of the contact patch. The resultant force
Cornering force
Cornering force or side force is the lateral force produced by a vehicle tire during cornering.Cornering force is generated by tire slip and is proportional to slip angle at low slip angles. The rate at which cornering force builds up is described by relaxation length...
of this side-slip occurs behind the geometric center of the contact patch, a distance described as the pneumatic trail
Pneumatic trail
Pneumatic trail or trail of the tire is a trail-like effect generated by compliant tires rolling on a hard surface and subject to side loads, as in a turn...
, 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
Camber angle
thumb|100px|From the front of the car, a right wheel with a negative camber angleCamber angle is the angle made by the wheels of a vehicle; specifically, it is the angle between the vertical axis of the wheels used for steering and the vertical axis of the vehicle when viewed from the front or...
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.
High side
A highsider, highside, 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. This can occur under heavy braking, acceleration, a varying road surface, or suspension activation, especially due to interaction with the drive train. It can take the form of a single slip-then-flip or a series of violent oscillations.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 kinematicKinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....
analysis complicated. 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. Bike designs tend to consist of a trade-off between maneuverability and stability.
Rider control inputs
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
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.
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.
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 skidpadSkidpad
A skidpad or skidpan is a large, circular area of flat pavement used for various tests of a car's handling. The most common skidpad use is testing lateral acceleration, measured in g....
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.
Rating schemes
Several schemes have been devised to rate the handling of bikes, particularly motorcycles.- The roll index is the ratio between steering torque and roll or lean angle.
- The acceleration index is the ratio between steering torque and lateral or centripetal acceleration.
- The steering ratio is the ratio between the theoretical turning radius based on ideal tire behavior and the actual turning radius. Values less than one, where the front wheel side slipSlip angleIn vehicle dynamics, slip angle or sideslip angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing...
is greater than the rear wheel side slip, are described as under-steeringUndersteerUndersteer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...
; 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. 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 motorcycleTouring motorcycleA touring motorcycle is a type of motorcycle designed for touring. Although any motorcycle can be used to tour with, manufacturers have brought specific models designed to address the particular needs of these riders...
s tend to have a high Koch index, sport motorcyclesSport bikeA sport bike, also written as sportbike, is a motorcycle optimized for speed, acceleration, braking, and cornering on paved roads, typically at the expense of comfort and fuel economy in comparison to less specialized motorcycles...
tend to have a medium Koch index, and scootersScooter (motorcycle)A scooter is a motorcycle with step-through frame and a platform for the operator's feet. Elements of scooter design have been present in some of the earliest motorcycles, and motorcycles identifiable as scooters have been made from 1914 or earlier...
tend to have a low Koch index. 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. 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. DampingDamping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
, 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
Nonholonomic system
A nonholonomic system in physics and mathematics is a system whose state depends on the path taken to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the...
because its outcome is path
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the 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
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, 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
Scientific modelling
Scientific modelling is the process of generating abstract, conceptual, graphical and/or mathematical models. Science offers a growing collection of methods, techniques and theory about all kinds of specialized scientific modelling...
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):
- x coordinate of rear wheel contact point
- y coordinate of rear wheel contact point
- orientation angle of rear frame (yaw)
- rotation angle of rear wheel
- rotation angle of front wheel
- lean angle of rear frame (roll)
- 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 motionEquation of motion
Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...
of an idealized bike, consisting of
- a rigid frameBicycle frameA bicycle frame is the main component of a bicycle, on to which wheels and other components are fitted. The modern and most common frame design for an upright bicycle is based on the safety bicycle, and consists of two triangles, a main triangle and a paired rear triangle...
, - a rigid fork,
- two knife-edged, rigid wheelsBicycle wheelA bicycle wheel is a wheel, most commonly a wire wheel, designed for bicycle. A pair is often called a wheelset, especially in the context of ready built "off the shelf" performance-oriented 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
Linearization
In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
or two coupled second-order differential equations, the lean equation
and the steer equation
where
is the lean angle of the rear assembly,
is the steer angle of the front assembly relative to the rear assembly and
and 
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
where
is the symmetrical mass matrix which contains terms that include only the mass and geometry of the bike,
is the so-called damping matrix, even though an idealized bike has no dissipation, which contains terms that include the forward speed 
and is asymmetric,
is the so-called stiffness matrix which contains terms that include the gravitational constant 
and 
and is symmetric in 
and asymmetric in 
,
is a vector of lean angle and steer angle, and
is a vector of external forces, the moments mentioned above.
In this idealized and linearized model, there are many geometric parameters
Bicycle and motorcycle geometry
Bicycle and motorcycle geometry is the collection of key measurements that define a particular bike configuration. Primary among these are wheelbase, steering axis angle, fork offset, and trail...
(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.
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.
Eigenvalues
State variable
A state variable is one of the set of variables that describe the "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour...
s (lean angle, lean rate, steer angle, and steer rate), from the linearized equations in order to analyze the normal mode
Normal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
s 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
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
parts of all eigenvalues (shown in dark blue) are negative, the bike is self-stable. When the imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
parts of any eigenvalues (shown in cyan) are non-zero, the bike exhibits oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
. 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.
There are three forward speeds that can be identified in the plot to the right at which the motion of the bike changes qualitatively:
- 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 polynomialCharacteristic polynomialIn linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
(two of the four eigenvalues have exactly the same value). Below this speed, the bike simply falls over as an inverted pendulumInverted pendulumAn inverted pendulum is a pendulum which has its mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart and pole...
does. - The forward speed at which oscillations do not increase, where the weave mode eigenvalues switch from positive to negative in a Hopf bifurcationHopf bifurcationIn the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...
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. - The forward speed at which non-oscillatory leaning increases, where the capsize mode eigenvalues switch from negative to positive in a pitchfork bifurcationPitchfork bifurcationIn bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations have two types - supercritical or subcritical....
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.
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. 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
Normal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
: 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. 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, 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.
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.
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
Speed wobble
Wobble, shimmy, tank-slapper, speed wobble, and even death wobble are all words and phrases used to describe a quick oscillation of primarily just the steerable wheel of a vehicle. Initially, the rest of the vehicle remains mostly unaffected, until translated into a vehicle yaw oscillation of...
, 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, such as fork asymmetry,
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
Steering damper
A steering damper, steering stabiliser or sprint damper is a damping device designed to inhibit an undesirable, uncontrolled movement or oscillation of a vehicle steering mechanism, a phenomenon known in motorcycling as wobble, or in extreme cases, a tank-slapper...
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 parameters of a bike have on these modes 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 the equations of motion above. Some general conclusions that have been drawn are described here.
The lateral and torsional stiffness of the rear frame
Motorcycle frame
A motorcycle frame includes the head tube that holds the front fork and allows it to pivot. Some motorcycles include the engine as a load-bearing, stressed member. The rear suspension is an integral component in the design. Traditionally frames have been steel, but titanium, aluminium, magnesium,...
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
Motorcycle fork
A motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a pair of triple clamps. It typically incorporates the front suspension and front brake, and allows the bike to be steered via handlebars attached to the top clamp....
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
Tire
A tire or tyre is a ring-shaped covering that fits around a wheel rim to protect it and enable better vehicle performance by providing a flexible cushion that absorbs shock while keeping the wheel in close contact with the ground...
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.
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
Steering damper
A steering damper, steering stabiliser or sprint damper is a damping device designed to inhibit an undesirable, uncontrolled movement or oscillation of a vehicle steering mechanism, a phenomenon known in motorcycling as wobble, or in extreme cases, a tank-slapper...
and excitation of the weave mode is worse for light riders than for heavy riders.
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:
Longitudinal dynamics
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. 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.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 or weight transfer
Weight transfer
Weight transfer and load transfer are two expressions used somewhat confusingly to describe two distinct effects: the change in load borne by different wheels of even perfectly rigid vehicles during acceleration, and the change in center of mass location relative to the wheels because of...
, 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
Motorcycle fork
A motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a pair of triple clamps. It typically incorporates the front suspension and front brake, and allows the bike to be steered via handlebars attached to the top clamp....
designs.
The net aerodynamic drag forces may be considered to act at a single point, called the center of pressure
Center of pressure
The center of pressure is the point on a body where the total sum of a pressure field acts, causing a force and no moment about that point. The total force vector acting at the center of pressure is the value of the integrated vectorial pressure field. The resultant force and center of pressure...
. 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. Also, depending on the shape of the bike and the shape of any fairing
Motorcycle fairing
A motorcycle fairing is a shell placed over the frame of some motorcycles, especially racing motorcycles and sport bikes, with the primary purpose to reduce air drag. The secondary functions are the protection of the rider from airborne hazards and wind-induced hypothermia and of the engine...
that might be installed, aerodynamic lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...
may be present that either increases or further reduces the load on the front wheel.
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
and a center of mass at height 
and at a distance 
in front of the rear wheel hub, and for simplicity, with both wheels locked, can be expressed as:
for the rear wheel and 
for the front wheel.The frictional (horizontal) forces are simply
for the rear wheel and 
for the front wheel,where
is the coefficient of friction, 
is the total massMass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of the bike and rider, and
is the acceleration of gravity. Therefore, if
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
above the horizontal, 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 bicycle
Tandem bicycle
The tandem bicycle or twin is a form of bicycle designed to be ridden by more than one person. The term tandem refers to the seating arrangement , not the number of riders. A bike with two riders side-by-side is called a sociable.-History:Patents related to tandem bicycles date from the late 19th...
s 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
Wheelie
In vehicle acrobatics, a wheelie is a vehicle maneuver in which the front wheel or wheels come off the ground due to extreme torque being applied to the rear wheel or wheels. Wheelies are usually associated with bicycles and motorcycles, but can be done with other vehicles such as cars, especially...
. 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. 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.
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
Bicycle brake systems
A bicycle brake is used to slow down or stop a bicycle. There have been various types of brake used throughout history, and several are still in use today...
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
Stoppie
The stoppie , is a motorcycle and bicycle trick in which the back wheel is lifted and the bike is ridden on the front wheel by carefully applying brake pressure. It is also sometimes called a front wheelie.-General description:...
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 bicycle
Recumbent bicycle
A recumbent bicycle is a bicycle that places the rider in a laid-back reclining position. Most recumbent riders choose this type of design for ergonomic reasons; the rider's weight is distributed comfortably over a larger area, supported by back and buttocks...
s, the front tire will skid instead, possibly causing a loss of balance.
In the case of a front suspension
Suspension (motorcycle)
A motorcycle's suspension serves a dual purpose: contributing to the vehicle's handling and braking, and providing safety and comfort by keeping the vehicle's passengers comfortably isolated from road noise, bumps and vibrations....
, 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
Trail braking
Trail braking is a motorcycle riding and driving technique where the brakes are used beyond the entrance to a turn and are gradually released up to the point of apex....
.
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 rubberRubberNatural rubber, also called India rubber or caoutchouc, is an elastomer that was originally derived from latex, a milky colloid produced by some plants. The plants would be ‘tapped’, that is, an incision made into the bark of the tree and the sticky, milk colored latex sap collected and refined...
on dry asphaltAsphaltAsphalt or , also known as bitumen, is a sticky, black and highly viscous liquid or semi-solid that is present in most crude petroleums and in some natural deposits, it is a substance classed as a pitch...
, - 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 (23.6 in) back from the front wheel contact patch and 120 cm (47.2 in) above, allowing a maximum deceleration of 0.5 g (5 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.
Rear-wheel braking
The rear brake of an upright bicycle can only produce about 0.1 g (1 m/s2) 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-gearFixed-gear bicycle
A fixed-gear bicycle is a bicycle that has no freewheel, meaning it cannot coast, as the pedals are always in motion when the bicycle is moving....
bikes with no other braking mechanism. There are, however, situations that may warrant rear wheel braking
- 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.
- 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.
- Front brake failure.
Suspension
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...
, damping
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
, sprung and unsprung mass, and tire
Motorcycle tyres
Motorcycle tyres provide the only contact with the ground, via the contact patch under normal conditions, and so have a very large influence over motorcycle handling characteristics. Motorcycle tyres have a round cross section to facilitate the leaning necessary when a motorcycle turns...
characteristics. 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
Shaft effect
The shaft effect, also known as elevator effect or shaft jacking, is a phenomenon occurring in shaft-drive motorcycles. This effect occurs because the acceleration being applied to the rear wheel creates a reactive force on the drive shaft. This in turn lifts the rider and the body of the bike,...
on motorcycles, and
squat and brake dive on both.
Vibration
The study of vibration in bikes includes its causes, such as engine balanceEngine balance
Engine balance is the design, construction and tuning of an engine to run smoothly. Improving engine balance reduces vibration and other stresses and can improve the overall performance, efficiency, cost of ownership and reliability of the engine, as well as reducing the stress on other machinery...
, wheel balance
Tire balance
Tire balance, also referred to as tire unbalance or imbalance, describes the distribution of mass within an automobile tire or the wheel to which it is attached. When the tire rotates, asymmetries of mass cause the wheel to wobble, which can cause ride disturbances, usually vertical and lateral...
, ground surface, and aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
; its transmission and absorption; and its effects on the bike, the rider, and safety. An important factor in any vibration analysis is a comparison of the natural frequencies
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
of the system with the possible driving frequencies of the vibration sources. A close match means mechanical resonance
Mechanical resonance
Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies...
that can result in large amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
s. A challenge in vibration damping is to create compliance in certain directions (vertically) without sacrificing frame rigidity needed for power transmission and handling (torsionally
Torsion (mechanics)
In solid mechanics, torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius....
). Another issue with vibration for the bike is the possibility of failure due to material fatigue Effects of vibration on riders include discomfort, loss of efficiency, Hand-Arm Vibration Syndrome
Vibration white finger
Vibration white finger is a secondary form of Raynaud's syndrome, an industrial injury triggered by continuous use of vibrating hand-held machinery. Use of the term Vibration White Finger has generally been superseded by Hand-Arm Vibration Syndrome or HAVS...
, a secondary form Raynaud's disease, and whole body vibration
Whole body vibration
Whole body vibration , as a therapy, was explored by Russian scientist Vladimir Nazarov, who tested vibration on cosmonauts in an effort to decrease the loss of muscle and bone mass in space. As there is minimal gravitational force in space, muscles and bones are not loaded as they normally are on...
. Vibrating instruments may be inaccurate or difficult to read.
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 suspensionBicycle suspension
A bicycle suspension is the system or systems used to suspend the rider and all or part of the bicycle in order to protect them from the roughness of the terrain over which they travel...
s, a variety of techniques have been developed to damp
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
vibrations before they reach the rider. These include materials, such as carbon fiber
Carbon fiber
Carbon fiber, alternatively graphite fiber, carbon graphite or CF, is a material consisting of fibers about 5–10 μm in diameter and composed mostly of carbon atoms. The carbon atoms are bonded together in crystals that are more or less aligned parallel to the long axis of the fiber...
, either in the whole frame
Bicycle suspension
A bicycle suspension is the system or systems used to suspend the rider and all or part of the bicycle in order to protect them from the roughness of the terrain over which they travel...
or just key components such as the front fork
Bicycle fork
A bicycle fork is the portion of a bicycle that holds the front wheel and allows the rider to steer and balance the bicycle. A fork consists of two fork ends which hold the front wheel axle, two blades which join at a fork crown, and a steerer or steering tube to which the handlebars attach ...
, seatpost
Seatpost
A bicycle seatpost, seatpin, saddlepole, saddle pillar, or saddle pin is a tube that extends upwards from the bicycle frame to the saddle. The amount that it extends out of the frame can usually be adjusted, and there is usually a mark that indicates the minimum insertion...
, or handlebars
Bicycle handlebar
Bicycle handlebar or often bicycle handlebars refers to the steering mechanism for bicycles; the equivalent of a steering wheel. Besides steering, handlebars also often support a portion of the rider's weight, depending on their riding position, and provide a convenient mounting place for brake...
; tube shapes, such as curved seat stays; and special inserts, such as Zertz by Specialized
Specialized Bicycle Components
Specialized Bicycle Components, more commonly known simply as Specialized, is a major American brand of bicycles and related products. It was founded in 1974 by Mike Sinyard and is based in Morgan Hill, California-History:...
, and Buzzkills by Bontrager
Trek Bicycle Corporation
Trek Bicycle Corporation is a major bicycle and cycling product manufacturer and distributor under brand names Trek, Gary Fisher, Bontrager, Klein and until recently, LeMond Racing Cycles...
.
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 shaftBalance shaft
In piston engine engineering, a balance shaft is an eccentric weighted shaft which offsets vibrations in engine designs that are not inherently balanced...
s, rubber engine mounts, and tire weights
Tire balance
Tire balance, also referred to as tire unbalance or imbalance, describes the distribution of mass within an automobile tire or the wheel to which it is attached. When the tire rotates, asymmetries of mass cause the wheel to wobble, which can cause ride disturbances, usually vertical and lateral...
. The problems that vibration causes have also spawned an industry of after-market parts and systems designed to reduce it. Add-ons include handlebar
Motorcycle handlebar
Motorcycle handlebar refers to the steering mechanism for motorcycles. Handlebars often support part of the rider's weight, and provide a mounting place for controls such as brake, throttle, clutch, horn, light switch, and rear view mirrors....
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
Windshield
The windshield or windscreen of an aircraft, car, bus, motorbike or tram is the front window. Modern windshields are generally made of laminated safety glass, a type of treated glass, which consists of two curved sheets of glass with a plastic layer laminated between them for safety, and are glued...
.
Experimentation
A variety of experiments have been performed in order to verify or disprove various hypotheses about bike dynamics.- David JonesDavid E. H. JonesDavid E. H. Jones is best known as Daedalus, the fictional inventor for DREADCO. Jones' columns as Daedalus were published weekly in the New Scientist starting in the mid-sixties. He then moved on to the journal Nature, and continued to publish for many years. He published two books with columns...
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.
See also
- Bicycle and motorcycle geometryBicycle and motorcycle geometryBicycle and motorcycle geometry is the collection of key measurements that define a particular bike configuration. Primary among these are wheelbase, steering axis angle, fork offset, and trail...
- Bicycle forkBicycle forkA bicycle fork is the portion of a bicycle that holds the front wheel and allows the rider to steer and balance the bicycle. A fork consists of two fork ends which hold the front wheel axle, two blades which join at a fork crown, and a steerer or steering tube to which the handlebars attach ...
- Bicycle tireBicycle tireA bicycle tire is a tire that fits on the wheel of a bicycle, unicycle, tricycle, quadracycle, bicycle trailer, or trailer bike. They may also be used on wheelchairs and handcycles, especially for racing...
- Camber angleCamber anglethumb|100px|From the front of the car, a right wheel with a negative camber angleCamber angle is the angle made by the wheels of a vehicle; specifically, it is the angle between the vertical axis of the wheels used for steering and the vertical axis of the vehicle when viewed from the front or...
- Camber thrustCamber thrustCamber thrust and camber force are terms used to describe the force generated perpendicular to the direction of travel of a rolling tire due to its camber angle and finite contact patch...
- Caster angleCaster anglethumb|250px|θ is the caster angle, the red line is the pivot line, and the grey area is the tire.Caster angle or castor angle is the angular displacement from the vertical axis of the suspension of a steered wheel in a car, bicycle or other vehicle, measured in the longitudinal direction...
- Cornering forceCornering forceCornering force or side force is the lateral force produced by a vehicle tire during cornering.Cornering force is generated by tire slip and is proportional to slip angle at low slip angles. The rate at which cornering force builds up is described by relaxation length...
- CountersteeringCountersteeringCountersteering is the technique used by single-track vehicle operators, such as cyclists and motorcyclists, to initiate a turn toward a given direction by momentarily steering counter to the desired direction...
- HighsiderHighsiderA highsider or highside is a type of motorcycle accident characterized by sudden & violent rotation around the long axis of the motorcycle in the direction of travel...
- LowsiderLowsiderThe lowsider or lowside is a type of motorcycle crash usually occurring in a turn and caused by a loss of grip between the tires and the road surface. It is most often caused by either locking a wheel due to excessive braking or application of excessive power out of or through the turn...
- Motorcycle forkMotorcycle forkA motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a pair of triple clamps. It typically incorporates the front suspension and front brake, and allows the bike to be steered via handlebars attached to the top clamp....
- Parallel parking problemParallel parking problemThe parallel parking problem is a motion planning problem in control theory and mechanics to determine the path a car must take in order to parallel park into a parking space. The front wheels of a car are permitted to turn, but the rear wheels must stay fixed...
- Slip angleSlip angleIn vehicle dynamics, slip angle or sideslip angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing...
- Speed wobbleSpeed wobbleWobble, shimmy, tank-slapper, speed wobble, and even death wobble are all words and phrases used to describe a quick oscillation of primarily just the steerable wheel of a vehicle. Initially, the rest of the vehicle remains mostly unaffected, until translated into a vehicle yaw oscillation of...
- StoppieStoppieThe stoppie , is a motorcycle and bicycle trick in which the back wheel is lifted and the bike is ridden on the front wheel by carefully applying brake pressure. It is also sometimes called a front wheelie.-General description:...
- Trail brakingTrail brakingTrail braking is a motorcycle riding and driving technique where the brakes are used beyond the entrance to a turn and are gradually released up to the point of apex....
- WheelieWheelieIn vehicle acrobatics, a wheelie is a vehicle maneuver in which the front wheel or wheels come off the ground due to extreme torque being applied to the rear wheel or wheels. Wheelies are usually associated with bicycles and motorcycles, but can be done with other vehicles such as cars, especially...
- Outline of motorcycles and motorcyclingOutline of motorcycles and motorcyclingThe following outline is provided as an overview of motorcycles and motorcycling:Motorcycle – single-track, engine-powered, two-wheeled motor vehicle...
Further reading
- ‘An Introduction to Bicycle Geometry and Handling’, Karl Anderson
- ‘What keeps the bicycle upright?’ by Jobst Brandt
- ‘Report on Stability of the Dahon Bicycle’ by John Forester
- Proceedings of the 2010 Bicycle and Motorcycle Dynamics Symposium by various authors
External links
Videos:- Video of riderless bicycle demonstrating self-stability
- Wobble movie (AVI)
- Weave movie (AVI)
- Wobble Crash (Flash)
Research centers:
- Bicycle Dynamics at Delft University of Technology
- Bicycle Mechanics at Cornell University
- Bicycle Science at the University of Illinois
- Motorcycle Dynamics at Padua University
- Control and Power Research Group at Imperial College
- Bicycle dynamics, control and handling at UC Davis
Conferences:
- Bicycle and Motorcycle Dynamics 2010: Symposium on Dynamics and Control of Single Track Vehicles, Delft University of TechnologyDelft University of TechnologyDelft University of Technology , also known as TU Delft, is the largest and oldest Dutch public technical university, located in Delft, Netherlands...
, Oct 20-22, 2010
Discussion groups:
- Single Track Vehicle Dynamics on Google Groups