![Tuftscriterium](https://upload.wikimedia.org/wikipedia/commons/8/88/Tuftscriterium.jpg)
Bicycle and motorcycle dynamics is the
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
of the
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
of
bicycle
A bicycle, also called a pedal cycle, bike or cycle, is a human-powered or motor-powered assisted, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A is called a cyclist, or bicyclist.
Bic ...
s and
motorcycle
A motorcycle (motorbike, bike, or trike (if three-wheeled)) is a two or three-wheeled motor vehicle steered by a handlebar. Motorcycle design varies greatly to suit a range of different purposes: long-distance travel, commuting, cruising ...
s and their components, due to the
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s acting on them.
Dynamics falls under a branch of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
known as
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
. Bike motions of interest include
balancing,
steering
Steering is a system of components, linkages, and other parts that allows a driver to control the direction of the vehicle.
Introduction
The most conventional steering arrangement allows a driver to turn the front wheels of a vehicle using ...
,
braking
A brake is a mechanical device that inhibits motion by absorbing energy from a moving system. It is used for slowing or stopping a moving vehicle, wheel, axle, or to prevent its motion, most often accomplished by means of friction.
Backgroun ...
,
accelerating
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
,
suspension
Suspension or suspended may refer to:
Science and engineering
* Suspension (topology), in mathematics
* Suspension (dynamical systems), in mathematics
* Suspension of a ring, in mathematics
* Suspension (chemistry), small solid particles suspend ...
activation, and
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
. The study of these motions began in the late 19th century and continues today.
Bicycles and motorcycles are both
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. In the case of wheeled v ...
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,
tricycle
A tricycle, sometimes abbreviated to trike, is a human-powered (or gasoline or electric motor powered or assisted, or gravity powered) three-wheeled vehicle.
Some tricycles, such as cycle rickshaws (for passenger transport) and freight trikes, ...
s, and
quadracycle
A quadracycle is a four-wheeled human-powered land vehicle. It is also referred to as a quadricycle, quadcycle, pedal car or four-wheeled bicycle amongst other terms.
Quadracycles have been in use since 1853 and have grown into several fami ...
s. As with
unicycle
A unicycle is a vehicle that touches the ground with only one wheel. The most common variation has a bicycle frame, frame with a bicycle saddle, saddle, and has a human-powered vehicle, pedal-driven direct-drive mechanism, direct-drive. A two spee ...
s, bikes lack lateral stability when stationary, and under most circumstances can only remain upright when moving forward.
Experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
ation and
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
have shown that a bike stays upright when it is steered to keep its
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
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
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rotat ...
or
trail
A trail, also known as a path or track, is an unpaved lane or small road usually passing through a natural area. In the United Kingdom and the Republic of Ireland, a path or footpath is the preferred term for a pedestrian or hiking trail. Th ...
, 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
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
or smaller the turn
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
, the more lean is required. This balances the roll torque about the wheel contact patches generated by
centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
due to the turn with that of the
gravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
. This lean is usually produced by a momentary steering in the opposite direction, called
countersteering
Countersteering is 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 ("steer left to turn right"). To negotiate a turn s ...
. Countersteering skill is usually acquired by
motor learning
Motor learning refers broadly to changes in an organism's movements that reflect changes in the structure and function of the nervous system. Motor learning occurs over varying timescales and degrees of complexity: humans learn to walk or talk over ...
and executed via
procedural memory
Procedural memory is a type of implicit memory (unconscious, long-term memory) which aids the performance of particular types of tasks without conscious awareness of these previous experiences.
Procedural memory guides the processes we perform, ...
rather than by conscious thought. Unlike other wheeled vehicles, the primary
control
Control may refer to:
Basic meanings Economics and business
* Control (management), an element of management
* Control, an element of management accounting
* Comptroller (or controller), a senior financial officer in an organization
* Controllin ...
input on bikes is steering
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
, 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, and if the front brake is applied hard enough, bikes can either: skid the front wheel which may or not result in a crash; 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
![Ravensburg Rutenfest 2005 Festzug Draisine](https://upload.wikimedia.org/wikipedia/commons/7/72/Ravensburg_Rutenfest_2005_Festzug_Draisine.jpg)
The history of the study of bike dynamics is nearly as old as the bicycle itself. It includes contributions from famous scientists such as
Rankine Rankine is a surname. Notable people with the surname include:
* William Rankine (1820–1872), Scottish engineer and physicist
** Rankine body an elliptical shape of significance in fluid dynamics, named for Rankine
** Rankine scale, an absolute-te ...
,
Appell Appell is a surname. Notable people with the surname include:
*Dave Appell (1922–2014), American arranger, producer, and musician
* Olga Appell (born 1963), Mexican-American long-distance runner
*Paul Émile Appell or M. P. Appell (1855–1930), ...
, and
Whipple.
In the early 19th century
Karl von Drais
Karl Freiherr von Drais (full name: Karl Friedrich Christian Ludwig Freiherr Drais von Sauerbronn) (29 April 1785 – 10 December 1851) was a noble German forest official and significant inventor in the Biedermeier period. He was born and died ...
, credited with inventing the two-wheeled vehicle variously called the
laufmaschine
The dandy horse, a derogatory term for what was first called a Laufmaschine (in German), then a vélocipède or draisienne (in French and then English), and then a pedestrian curricle or hobby-horse, or swiftwalker, is a human-powered vehicle th ...
,
velocipede
A velocipede () is a human-powered land vehicle with one or more wheels. The most common type of velocipede today is the bicycle.
The term was probably first coined by Karl von Drais in French as ''vélocipède'' for the French translation o ...
,
draisine
A draisine () is a light auxiliary rail vehicle, driven by service personnel, equipped to transport crew and material necessary for the maintenance of railway infrastructure.
The eponymous term is derived from the German inventor Baron Karl Dr ...
, and
dandy horse
The dandy horse, a derogatory term for what was first called a Laufmaschine (in German), then a vélocipède or draisienne (in French and then English), and then a pedestrian curricle or hobby-horse, or swiftwalker, is a human-powered vehicle th ...
, showed that a rider could balance his device by steering the front wheel.
In 1869, Rankine published an article in ''The Engineer'' repeating von Drais's assertion that balance is maintained by steering in the direction of a lean.
In 1897, the
French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
made understanding bicycle dynamics the goal of its Prix Fourneyron competition. Thus, by the end of the 19th century,
Carlo Bourlet,
Emmanuel Carvallo
Emmanuel Carvallo was a French mathematician. He is notable for showing in 1897 that bicycles could be self-stable, for opposing wave models of X-rays in 1900, and for claiming in 1912 that Einstein's Theory of Relativity
The theory of rel ...
, and Francis Whipple had showed with
rigid-body dynamics
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action ...
that some
safety bicycle
A safety bicycle (or simply a safety) is a type of bicycle that became very popular beginning in the late 1880s as an alternative to the penny-farthing ("ordinary") and is now the most common type of bicycle. Early bicycles of this style were kno ...
s could actually balance themselves if moving at the right speed.
Bourlet won the Prix Fourneyron, and Whipple won the
Cambridge University
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
Smith Prize
The Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics and theoretical physics at the University of Cambridge from 1769. Following the reorganization in 1998, they are now awarded under the n ...
.
[ 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 Edward Hugh Jones (20 April 1938 – 19 July 2017) was a British chemist and author, who under the pen name Daedalus was the fictional inventor for DREADCO. Jones' columns as Daedalus were published for 38 years, starting weekly in 1964 in ...
published an article in ''Physics Today
''Physics Today'' is the membership magazine of the American Institute of Physics. First published in May 1948, it is issued on a monthly schedule, and is provided to the members of ten physics societies, including the American Physical Society. I ...
'' 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 the early 1970s, Cornell Aeronautical Laboratory (CAL, later Calspan Corporation
Calspan Corporation is a science and technology company founded in 1943 as part of the Research Laboratory of the Curtiss-Wright Airplane Division at Buffalo, New York. Calspan consists of four primary operating units: Flight Research, Transportati ...
in Buffalo, NY USA) was sponsored by the Schwinn Bicycle Company and others to study and simulate bicycle and motorcycle dynamics. Portions of this work have now been released to the public and scans of over 30 detailed reports have been posted at thi
TU Delft Bicycle Dynamics site
Since the 1990s, Cossalter, et al., have been researching motorcycle dynamics at the University of Padova. Their research, both experimental and numerical, has covered weave, wobble, chatter, simulators, vehicle modelling, tire modelling, handling, and minimum lap time maneuvering.
In 2007, Meijaard, et al., published the canonical linearized equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
, in the ''Proceedings of the Royal Society A
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905:
* Series A: for papers in physical sciences and mathematics.
* Series B: for papers in life s ...
'', 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 is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
'' 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
A two-mass-skate bicycle (TMS) 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 hav ...
that the equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
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. Bicycle dynamics was named 26 of ''Discover
Discover may refer to:
Art, entertainment, and media
* ''Discover'' (album), a Cactus Jack album
* ''Discover'' (magazine), an American science magazine
Businesses and brands
* DISCover, the ''Digital Interactive Systems Corporation''
* D ...
''s 100 top stories of 2011.
In 2013, Eddy Merckx Cycles
S.A. Cycles Eddy Merckx Rijwielen N.V., better known as Eddy Merckx, is a Belgian brand of high end road bikes, founded by former professional road cyclist Eddy Merckx in 1980.Friebe, Daniel (2012). ''Eddy Merckx, The Cannibal''. . Ebury Press, U ...
was awarded over €150,000 with Ghent University
Ghent University ( nl, Universiteit Gent, abbreviated as UGent) is a public research university located in Ghent, Belgium.
Established before the state of Belgium itself, the university was founded by the Dutch King William I in 1817, when the ...
to examine bicycle stability.
Forces
If the bike and rider are considered to be a single system, the forces that act on that system and its components can be roughly divided into two groups: internal and external. The external forces are due to gravity, inertia, contact with the ground, and contact with the atmosphere. The internal forces are caused by the rider and by interaction between components.
External forces
As with all masses, gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
pulls the rider and all the bike components toward the earth. At each tire contact patch
Contact patch is the portion of a vehicle's tire that is in actual contact with the road surface. It is commonly used in the discussion of pneumatic (i.e. pressurized) tires, where the term is used strictly to describe the portion of the tire’s ...
there are ground reaction forces with both horizontal and vertical components. The vertical components mostly counteract the force of gravity, but also vary with braking and accelerating. For details, see the section on ''longitudinal stability
In flight dynamics, longitudinal stability is the stability of an aircraft in the longitudinal, or pitching, plane. This characteristic is important in determining whether an aircraft pilot will be able to control the aircraft in the pitching pla ...
'' below. The horizontal components, due to friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
between the wheels and the ground, including rolling resistance
Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body (such as a ball, tire, or wheel) rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy nee ...
, are in response to propulsive forces, braking forces, and turning forces. Aerodynamic
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
forces due to the atmosphere are mostly in the form of drag, but can also be from crosswind
A crosswind is any wind that has a perpendicular component to the line or direction of travel. This affects the aerodynamics of many forms of transport. Moving non-parallel to the wind's direction creates a crosswind component on the object and th ...
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 forces in the accelerating reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
of the bike and rider; or simply as inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
in a stationary, inertial reference frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
and not forces at all. ''Gyroscopic
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rotat ...
'' 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, those between components of the bike and rider system, are mostly caused by the rider or by friction. In addition to pedaling, the rider can apply torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
s between the steering mechanism (front fork, handlebars, front wheel, etc.) and rear frame, and between the rider and the rear frame. Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
exists between any parts that move against each other: in the drive train
A drivetrain (also frequently spelled as drive train or sometimes drive-train) is the group of components that deliver mechanical power from the prime mover to the driven components. In automotive engineering, the drivetrain is the components o ...
, between the steering mechanism and the rear frame, etc. In addition to brakes
A brake is a mechanical device that inhibits motion by absorbing energy from a moving system. It is used for slowing or stopping a moving vehicle, wheel, axle, or to prevent its motion, most often accomplished by means of friction.
Background ...
, which create friction between rotating wheels and non-rotating frame parts, many bikes have front and rear suspensions
In chemistry, a suspension is a heterogeneous mixture of a fluid that contains solid particles sufficiently large for sedimentation. The particles may be visible to the naked eye, usually must be larger than one micrometer, and will eventually ...
. Some motorcycles and bicycles have a steering damper
Originally designed in 1966 by Leonard R Jordan Jr the steering damper, or steering stabiliser is a damping device designed to inhibit an undesirable, uncontrolled movement or oscillation of a vehicle steering mechanism, a phenomenon known in moto ...
to dissipate undesirable kinetic energy, and some bicycles have a spring connecting the front fork to the frame to provide a progressive torque that tends to steer the bicycle straight ahead. On bikes with rear suspensions, feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
between the drive train and the suspension is an issue designers attempt to handle with various linkage
Linkage may refer to:
* ''Linkage'' (album), by J-pop singer Mami Kawada, released in 2010
*Linkage (graph theory), the maximum min-degree of any of its subgraphs
*Linkage (horse), an American Thoroughbred racehorse
* Linkage (hierarchical cluster ...
configurations and dampers.
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 stoppie
The stoppie is a motorcycle and bicycle trick in which the back wheel is lifted by abruptly applying the front brake, then, by carefully reducing the brake pressure, the bike is ridden for a short distance on the front wheel. It is also called an ...
s, wheelie
In vehicle acrobatics, a wheelie, or wheelstand, is a vehicle maneuver in which the front wheel or wheels come off the ground due to sufficient torque being applied to the rear wheel or wheels, or rider motion relative to the vehicle. Whe ...
s, brake diving, and most suspension activation. Motions in these two groups are linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
ly decoupled, that is they do not interact with each other to the 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-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
, multibody dynamic analysis with at least two generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
to analyze. At a minimum, two coupled, second-order differential equations are required to capture the principal motions. Exact solutions are not possible, and numerical methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
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 ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to:
Science and medicine
* Kinetics (physics), the study of motion and its causes
** Rigid body kinetics, the study of the motion of rigid bodies
* Chemical ki ...
and just one coordinate.
Balance
When discussing bike balance, it is necessary to distinguish carefully between "stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
**Asymptotic stability
**Linear stability
**Lyapunov stability
**Orbital stability
**Structural stabilit ...
", " self-stability", and "controllability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observabil ...
". Recent research suggests that "rider-controlled stability of bicycles is indeed related to their self-stability."
A bike remains upright when it is steered so that the ground reaction forces exactly balance all the other internal and external forces it experiences, such as gravitational if leaning, inertial or centrifugal if in a turn, gyroscopic if being steered, and aerodynamic if in a crosswind.
Steering may be supplied by a rider or, under certain circumstances, by the bike itself. This self-stability is generated by a combination of several effects that depend on the geometry, mass distribution, and forward speed of the bike. Tires, suspension, steering damping, and frame flex can also influence it, especially in motorcycles.
Even when staying relatively motionless, a rider can balance a bike by the same principle. While performing a track stand
The track stand or standstill is a technique that bicycle riders can use to maintain balance while their bicycle remains stationary or moves only minimal distances. The technique originated in track cycling and is now used by other types of cycl ...
, 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
A fixed-gear bicycle (or fixed-wheel bicycle, commonly known in some places as a fixie) is a bicycle that has a drivetrain with no freewheel mechanism. The freewheel was developed early in the history of bicycle design but the fixed-gear bicyc ...
. 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. One other way that a bike can be balanced, with or without locked steering, is by applying appropriate torques between the bike and rider similar to the way a gymnast can swing up from hanging straight down on uneven parallel bars, a person can start swinging on a swing from rest by pumping their legs, or a double inverted pendulum
A double inverted pendulum is the combination of the inverted pendulum and the double pendulum. The double inverted pendulum is unstable, meaning that it will fall down unless it is controlled in some way. The two main methods of controlling a dou ...
can be controlled with an actuator only at the elbow.
Forward speed
The rider applies torque to the handlebars in order to turn the front wheel and so to control lean and maintain balance. At high speeds, small steering angles quickly move the ground contact points laterally; at low speeds, larger steering angles are required to achieve the same results in the same amount of time. Because of this, it is usually easier to maintain balance at high speeds. As self-stability typically occurs at speeds above a certain threshold, going faster increases the chances that a bike is contributing to its own stability.
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 farther back (closer to the rear wheel) the center of mass is located, the more front wheel lateral movement or bike forward motion is required to regain balance. This can be noticeable on long-wheelbase recumbents
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 ba ...
, choppers
Chopper may refer to:
Arts and entertainment
* Chopper (comics), ''Chopper'' (comics), a horror comic book mini-series
* Chopper (film), ''Chopper'' (film), a 2000 Australian film by and about Mark "Chopper" Read
* Chopper (Judge Dredd), Chopper ...
, and wheelie bike
A wheelie bike, also called a dragster, muscle bike, high-riser, spyder bike or banana bike, is a type of stylized children's bicycle designed in the 1960s to resemble a chopper motorcycle and characterized by ape hanger handlebars, a banan ...
s. It can also be a challenge for touring bikes that carry 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
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle ...
. Just as a broomstick is more easily balanced in the hand than a pencil, a tall bike (with a high center of mass) can be easier to balance when ridden than a low one because the tall bike's lean rate (rate at which its angle of lean increases as it begins to fall over) 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 lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is di ...
. A small force at the end of the lever, the seat or handlebars at the top of the bike, more easily moves a large mass if the mass is closer to the fulcrum, where the tires touch the ground. This is why touring cyclists are advised to carry loads low on a bike, and panniers
A pannier is a basket, bag, box, or similar container, carried in pairs either slung over the back of a beast of burden, or attached to the sides of a bicycle or motorcycle. The term derives from a Middle English borrowing of the Old French ''p ...
hang down on either side of front and rear racks
Rack or racks may refer to:
Storage and installation
* Amp rack, short for amplifier rack, a piece of furniture in which amplifiers are mounted
* Bicycle rack, a frame for storing bicycles when not in use
* Bustle rack, a type of storage bin m ...
.
Trail
A factor that influences how easy or difficult a bike will be to ride is trail
A trail, also known as a path or track, is an unpaved lane or small road usually passing through a natural area. In the United Kingdom and the Republic of Ireland, a path or footpath is the preferred term for a pedestrian or hiking trail. Th ...
, the distance by which the front wheel ground contact point trails behind the steering axis ground contact point. The steering axis is the axis about which the entire steering mechanism (fork, handlebars, front wheel, etc.) pivots. In traditional bike designs, with a steering axis tilted back from the vertical, 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 the head angle, or decreasing the fork rake.
The more trail a traditional bike has, the more stable it feels, although too much trail can make a bike feel difficult to steer. Bikes with negative trail (where the contact patch is in front of where the steering axis intersects the ground), while still rideable, are reported to feel very unstable. Normally, road racing bicycles have more trail than touring bikes but less than mountain bikes. Mountain bikes are designed with less-vertical head angles than road bikes so as to have greater trail and hence improved stability for descents. 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
In cutlery or kitchenware, a fork (from la, furca 'pitchfork') is a utensil, now usually made of metal, whose long handle terminates in a head that branches into several narrow and often slightly curved tines with which one can spear foods ei ...
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 rideable, 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
A motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a yoke, also known as a triple clamp, which consists of an upper yoke joined to a lower yoke via a steering stem, a shaft that runs through the steering h ...
is used to diminish trail.
A small survey by Whitt and Wilson found:
*touring bicycle
A 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 (for ride comfort and to avoid p ...
s with head angles between 72° and 73° and trail between 43 mm and 60 mm
*racing bicycle
A racing bicycle, also known as a road bike is a bicycle designed for competitive road cycling, a sport governed by and according to the rules of the Union Cycliste Internationale (UCI).
Racing bicycles are designed for maximum performance ...
s with head angles between 73° and 74° and trail between 28 mm and 45 mm
*track bicycle
A 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; thus, it has only a single gear ratio and has neither a freewheel nor brakes. Tires are ...
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 is a bicycle company founded by Greg LeMond, the only American winner of the Tour de France.
LeMond initially offered bicycle frames with a geometry based on the racing bikes he used in competition, with a longer top tube and ...
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 51.5 mm to 69 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 51.5 mm to 61 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. Trail can decrease to zero with sufficiently large lean and steer angles, which can alter how stable a bike feels. 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.
Wheelbase
A factor that influences the directional stability of a bike is wheelbase
In both road and rail vehicles, the wheelbase is the horizontal distance between the centers of the front and rear wheels. For road vehicles with more than two axles (e.g. some trucks), the wheelbase is the distance between the steering (front ...
, the horizontal distance between the ground contact points of the front and rear wheels. For a given displacement of the front wheel, due to some disturbance, the angle of the resulting path from the original is inversely proportional to wheelbase. Also, the radius of curvature for a given steer angle and lean angle is proportional to the wheelbase. Finally, the wheelbase increases when the bike is leaned and steered. In the extreme, when the lean angle is 90°, and the bike is steered in the direction of that lean, the wheelbase is increased by the radius of the front and rear wheels.
Steering mechanism mass distribution
Another factor that can also contribute to the self-stability of traditional bike designs is the distribution of mass in the steering mechanism, which includes the front wheel, the fork, and the handlebar. If the center of mass for the steering mechanism is in front of the steering axis, then the pull of gravity will also cause the front wheel to steer in the direction of a lean. This can be seen by leaning a stationary bike to one side. The front wheel will usually also steer to that side independent of any interaction with the ground. Additional parameters, such as the fore-to-aft position of the center of mass and the elevation of the center of mass also contribute to the dynamic behavior of a bike.
Gyroscopic effects
The role of the gyroscopic effect in most bike designs is to help steer the front wheel into the direction of a lean. This phenomenon is called ''precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
'', and the rate at which an object precesses is inversely proportional to its rate of spin. The slower a front wheel spins, the faster it will precess when the bike leans, and vice versa.
The rear wheel is prevented from precessing as the front wheel does by friction of the tires on the ground, and so continues to lean as though it were not spinning at all. Hence gyroscopic forces do not provide any resistance to tipping.
At low forward speeds, the precession of the front wheel is too quick, contributing to an uncontrolled bike's tendency to oversteer, start to lean the other way and eventually oscillate and fall over. At high forward speeds, the precession is usually too slow, contributing to an uncontrolled bike's tendency to understeer and eventually fall over without ever having reached the upright position. This instability is very slow, on the order of seconds, and is easy for most riders to counteract. Thus a fast bike may feel stable even though it is actually not self-stable and would fall over if it were uncontrolled.
Another contribution of gyroscopic effects is a roll moment generated by the front wheel during countersteering. For example, steering left causes a moment to the right. The moment is small compared to the moment generated by the out-tracking front wheel, but begins as soon as the rider applies torque to the handlebars and so can be helpful in motorcycle racing
Motorcycle racing (also called moto racing and motorbike racing) is the motorcycle sport of racing motorcycles. Major varieties include motorcycle road racing and off-road racing, both either on circuits or open courses, and track racing. Ot ...
. For more detail, see the section countersteering
Countersteering is 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 ("steer left to turn right"). To negotiate a turn s ...
, below, and the countersteering
Countersteering is 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 ("steer left to turn right"). To negotiate a turn s ...
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 friction and stiff control cables. Thi
video
shows a riderless bicycle exhibiting self-stability.
Longitudinal acceleration
Longitudinal acceleration has been shown to have a large and complex effect on lateral dynamics. In one study, positive acceleration eliminates self stability, and negative acceleration (deceleration) changes the speeds of self stability.[
]
Turning
In order for a bike to turn, that is, change its direction of forward travel, the front wheel must aim approximately in the desired direction, as with any front-wheel steered vehicle. Friction between the wheels and the ground then generates the centripetal acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
necessary to alter the course from straight ahead as a combination of cornering force
Cornering force or side force is the lateral (i.e., parallel to wheel axis) 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 ...
and 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. Camber thrust is generated when a point on the outer surface ...
. 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
In both road and rail vehicles, the wheelbase is the horizontal distance between the centers of the front and rear wheels. For road vehicles with more than two axles (e.g. some trucks), the wheelbase is the distance between the steering (front ...
, is the steer angle, and is the caster angle
250px, θ is the caster angle, the red line is the pivot line, and the grey area is the tire.
250px, Front suspension of a race car, the caster angle is formed by the line between upper and lower ball joint.
The caster angle or castor angle is t ...
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 motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
:
:
where ''v'' is the forward speed, ''r'' is the radius of the turn and ''g'' is the acceleration of gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. 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.
It can also be seen however that this simple 2-dimensional model, essentially an inverted pendulum on a turntable
A phonograph, in its later forms also called a gramophone (as a trademark since 1887, as a generic name in the UK since 1910) or since the 1940s called a record player, or more recently a turntable, is a device for the mechanical and analogu ...
, predicts that the steady-state turn is unstable. If the bike is displaced slightly downwards from its equilibrium lean angle, the torque of gravity increases, that of centrifugal force decreases and the displacement gets amplified. A more-sophisticated model that allows a wheel to steer, adjust the path, and counter the torque of gravity, is necessary to capture the self-stability observed in real bikes.
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 is the approximate radius, is the wheelbase, is the lean angle, is the steering angle, and 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 2''t'' 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 25° 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.
It has been shown that the couple created by gravity and the ground reaction forces is necessary for a bicycle to turn at all. On a custom built bicycle with spring-loaded outriggers that exactly cancel this couple, so that the bicycle and rider may assume any lean angle when traveling in a straight line, riders find it impossible to make a turn. As soon as the wheels deviate from a straight path, the bicycle and rider begin to lean in the opposite direction, and the only way to right them is to steer back onto the straight path.
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, and this torque causes the bike to lean away from the initially steered direction and toward the direction of the desired 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
'' 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
In vehicle dynamics, slip angle or sideslip angle is the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling (i.e., the angle between the forward velocity vector v_x and the vector su ...
s due to weight distribution, tire properties, etc., bikes can experience understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Oversteer is what occurs when a car turns (steers) by more than the amount commanded by the driver. Conversely, understeer is what occu ...
or oversteer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Oversteer is what occurs when a car turns (steers) by more than the amount commanded by the driver. Conversely, understeer is what occu ...
. 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 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. Camber thrust is generated when a point on the outer surface ...
contributes to the centripetal force
A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
necessary to cause the bike to deviate from a straight path, along with cornering force
Cornering force or side force is the lateral (i.e., parallel to wheel axis) 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 ...
due to the slip angle
In vehicle dynamics, slip angle or sideslip angle is the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling (i.e., the angle between the forward velocity vector v_x and the vector su ...
, 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 leans to the left to conserve angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, and the combined center of mass remains nearly 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. This technique may be complicated by additional factors such as headset friction and stiff control cables.
The combined center of mass does move slightly to the left when the rider leans to the right relative to the bike, and the bike leans to the left in response. The action, in space, would have the tires move right, but this is prevented by friction between the tires and the ground, and thus pushes the combined center of mass left. This is a small effect, however, as evidenced by the difficulty most people have in balancing a bike by this method alone.
Gyroscopic effects
As mentioned above in the section on balance, one effect of turning the front wheel is a roll moment caused by gyroscopic precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
. The magnitude of this moment is proportional to the moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
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 (also called moto racing and motorbike racing) is the motorcycle sport of racing motorcycles. Major varieties include motorcycle road racing and off-road racing, both either on circuits or open courses, and track racing. Ot ...
.
Two-wheel steering
Because of theoretical benefits, such as a tighter turning radius at low speed, attempts have been made to construct motorcycles with two-wheel steering. One working prototype by Ian Drysdale in Australia is reported to "work very well." Issues in the design include whether to provide active control of the rear wheel or let it swing freely. In the case of active control, the control algorithm needs to decide between steering with or in the opposite direction of the front wheel, when, and how much. One implementation of two-wheel steering, the Sideways bike
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 ...
, lets the rider control the steering of both wheels directly. Another, the Swing Bike, had the second steering axis in front of the seat so that it could also be controlled by the handlebars.
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
A curb (North American English), or kerb (Commonwealth English except Canada; see spelling differences), is the edge where a raised sidewalk or road median/central reservation meets a street or other roadway.
History
Although curbs have ...
. "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 drive
Front-wheel drive (FWD) is a form of engine and 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 longit ...
mechanism, attempts have been made to construct a rideable rear-wheel steering bike. The Bendix Company built a rear-wheel steering bicycle, and the U.S. Department of Transportation commissioned the construction of a rear-wheel steering motorcycle: both proved to be unrideable. 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
for bicycles with common geometries and mass distributions shows that when moving in reverse, so as to have rear-wheel steering, they are inherently unstable. This does not mean they are unridable, but that the effort to control them is higher. Other, purpose-built designs have been published, however, that do not suffer this problem.
Center steering
Between the extremes of bicycles with classical front-wheel steering and those with strictly rear-wheel steering is a class of bikes with a pivot point somewhere between the two, referred to as center-steering, and similar to articulated steering. An early implementation of the concept was the Phantom bicycle in the early 1870s promoted as a safer alternative to the penny-farthing
The penny-farthing, also known as a high wheel, high wheeler or ordinary, is an early type of bicycle. It was popular in the 1870s and 1880s, with its large front wheel providing high speeds (owing to its travelling a large distance for every r ...
. This design allows for simple front-wheel drive and current implementations appear to be quite stable, even rideable no-hands, as many photographs illustrate.
These designs, such as the Python Lowracer, a recumbent, 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.
Reverse steering
Bicycles have been constructed, for investigation and demonstration purposes, with the steering reversed so that turning the handlebars to the left causes the front wheel to turn to the right, and vica versa. It is possible to ride such a bicycle, but it has been found that riders experienced with normal bicycles find it very difficult to learn, if they can manage it at all.
Tiller effect
Tiller effect is the expression used to describe how handlebars that extend far behind the steering axis (head tube) act like a tiller
A tiller or till is a lever used to steer a vehicle. The mechanism is primarily used in watercraft, where it is attached to an outboard motor, rudder post or stock to provide leverage in the form of torque 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
A cruiser bicycle, also known as a beach cruiser or (formerly) motobike, is a bicycle that usually combines balloon tires, an upright seating posture, a single-speed drivetrain, and straightforward steel construction with expressive styling. ...
s, some recumbents, and some motorcycles.[ It can be troublesome when it limits the ability to steer because of interference or the limits of arm reach.
]
Tires
Tires
A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineering), t ...
have a large influence over bike handling, especially on motorcycles, but also on bicycles. Tires influence bike dynamics in two distinct ways: finite crown radius and force generation. Increase the crown radius of the front tire has been shown to decrease the size or eliminate self stability. Increasing the crown radius of the rear tire has the opposite effect, but to a lesser degree.[
Tires generate the lateral forces necessary for steering and balance through a combination of ]cornering force
Cornering force or side force is the lateral (i.e., parallel to wheel axis) 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 ...
and 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. Camber thrust is generated when a point on the outer surface ...
. 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
In vehicle dynamics, slip angle or sideslip angle is the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling (i.e., the angle between the forward velocity vector v_x and the vector su ...
s due to weight distribution, tire properties, etc., bikes can experience understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Oversteer is what occurs when a car turns (steers) by more than the amount commanded by the driver. Conversely, understeer is what occu ...
or oversteer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Oversteer is what occurs when a car turns (steers) by more than the amount commanded by the driver. Conversely, understeer is what occu ...
. 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 is the portion of a vehicle's tire that is in actual contact with the road surface. It is commonly used in the discussion of pneumatic (i.e. pressurized) tires, where the term is used strictly to describe the portion of the tire’s ...
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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to the plane of the contact patch.
One torque generated by a tire, called the self aligning torque
Self aligning torque, also known as aligning torque, aligning moment, 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. In the presence of a non-zero slip angle, thi ...
, is caused by asymmetries in the side-slip along the length of the contact patch. The resultant force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
of this side-slip occurs behind the geometric center of the contact patch, a distance described as the 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. More technically, it is the distance that the resultant force of side-slip occurs behind th ...
, 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 is one of the angles made by the wheels of a vehicle; specifically, it is the angle between the vertical axis of a wheel and the vertical axis of the vehicle when viewed from the front or rear. It is used in the design of steering an ...
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 kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
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
The primary control input that the rider can make is to apply a torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
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. In motorcycle racing, leaning the torso, moving the body, and projecting a knee to the inside of the turn relative to the bike can also ''cause an aerodynamic yawing moment that facilitates entering and rounding the turn.''
Differences from automobiles
The need to keep a bike upright to avoid injury to the rider and damage to the vehicle even limits the type of maneuverability testing that is commonly performed. For example, while automobile enthusiast publications often perform and quote skidpad
A skidpad is a 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 meters per second squared (m/s2) or the scaled unit ''g''-force. This usage has s ...
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
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
.
*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 slip is greater than the rear wheel side slip, are described as under-steering; equal to one as neutral steering; and greater than one as over-steering. Values less than zero, in which the front wheel must be turned opposite the direction of the curve due to much greater rear wheel side slip than front wheel have been described as counter-steering. Riders tend to prefer neutral or slight over-steering. 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 motorcycle
A touring motorcycle is a type of motorcycle designed for touring. Although almost any motorcycle can be used for this purpose, manufacturers have developed specific models designed to address the particular needs of these riders. Touring motorc ...
s tend to have a high Koch index, sport motorcycles tend to have a medium Koch index, and scooters tend to have a low Koch index. 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
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
. 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
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
system. Damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
, 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
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...
because its outcome is path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
-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 is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, 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
Countersteering is 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 ("steer left to turn right"). To negotiate a turn s ...
Degrees of freedom
The number of degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of a bike depends on the particular model
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
being used. The simplest model that captures the key dynamic features, called the "Whipple model" after Francis Whipple who first developed the equations for it, has four rigid bodies with knife edge wheels rolling without slip on a flat smooth surface, and 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
Roll or Rolls may refer to:
Movement about the longitudinal axis
* Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis
** Roll (aviation), ...
)
#steering angle between rear frame and front end
Adding complexity to the model, such as rider movement, suspension movement, tire compliance, or frame flex, adds degrees of freedom. While the rear frame does pitch with leaning and steering, the pitch angle is completely constrained by the requirement for both wheels to remain on the ground, and so can be calculated geometrically from the other seven variables. If the location of the bike and the rotation of the wheels are ignored, the first five degrees of freedom can also be ignored, and the bike can be described by just two variables: lean angle and steer angle.
Equations of motion
The equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
of an idealized bike, consisting of
*a rigid frame
A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent.
Frame and FRAME may also refer to:
Physical objects
In building construction
*Framing (con ...
,
*a rigid fork,
*two knife-edged, rigid wheels
A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be ...
,
*all connected with frictionless bearings and rolling without friction or slip on a smooth horizontal surface and
*operating at or near the upright and straight-ahead, unstable equilibrium
can be represented by a single fourth-order linearized ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
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 (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
It is possible to calculate eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
, one for each of the four state variable
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
s (lean angle, lean rate, steer angle, and steer rate), from the linearized equations in order to analyze the normal mode
A normal mode of a dynamical 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 free motion described by the normal modes takes place at fixed 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 may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
parts of all eigenvalues (shown in dark blue) are negative, the bike is self-stable. When the imaginary parts of any eigenvalues (shown in cyan) are non-zero, the bike exhibits oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
. 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
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
to the characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
(two of the four eigenvalues have exactly the same value). Below this speed, the bike simply falls over as an inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle ...
does.
#The forward speed at which oscillations do not increase, where the weave mode eigenvalues switch from positive to negative in a Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses ...
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 bifurcation
In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from one fixed point to three fixed points. Pitchfork bifurcations, like Hopf bifurcations, have two ...
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
Bikes, as complex mechanisms, have a variety of modes
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
: 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. Thi
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
Wobble, shimmy, tank-slapper, speed wobble, and even death wobble are all words and phrases used to describe a quick (4–10 Hz) oscillation of primarily just the steerable wheel(s) of a vehicle. Initially, the rest of the vehicle remain ...
'', 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). Also involved is the yawing of the rear frame which may contribute to the wobble when too flexible. 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.
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
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
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 and the wheel spindle affects wobble-mode damping substantially. Long wheelbase
In both road and rail vehicles, the wheelbase is the horizontal distance between the centers of the front and rear wheels. For road vehicles with more than two axles (e.g. some trucks), the wheelbase is the distance between the steering (front ...
and trail
A trail, also known as a path or track, is an unpaved lane or small road usually passing through a natural area. In the United Kingdom and the Republic of Ireland, a path or footpath is the preferred term for a pedestrian or hiking trail. Th ...
and a flat steering-head angle have been found to increase weave-mode damping. Lateral distortion can be countered by locating the front fork torsional axis as low as possible.
Cornering weave tendencies are amplified by degraded damping of the rear suspension
Suspension is the system of tires, tire air, springs, shock absorbers and linkages that connects a vehicle to its wheels and allows relative motion between the two. Suspension systems must support both road holding/ handling and ride q ...
. Cornering, camber stiffnesses and relaxation length of the rear tire
A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineering), t ...
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
Originally designed in 1966 by Leonard R Jordan Jr the steering damper, or steering stabiliser is a damping device designed to inhibit an undesirable, uncontrolled movement or oscillation of a vehicle steering mechanism, a phenomenon known in moto ...
and excitation of the weave mode is worse for light riders than for heavy riders.
Riding on treadmills and rollers
Riding on a treadmill
A treadmill is a device generally used for walking, running, or climbing while staying in the same place. Treadmills were introduced before the development of powered machines to harness the power of animals or humans to do work, often a type of ...
is theoretically identical to riding on stationary pavement, and physical testing has confirmed this. Treadmills have been developed specifically for indoor bicycle training. Riding on rollers is still under investigation.
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
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 number of passengers, the amount of luggage, and the location of passengers and luggage, but also with acceleration and deceleration. This phenomenon is known as load 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
*the change in center of mas ...
or 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
*the change in center of mas ...
, 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
A motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a yoke, also known as a triple clamp, which consists of an upper yoke joined to a lower yoke via a steering stem, a shaft that runs through the steering h ...
designs.
The net aerodynamic drag forces may be considered to act at a single point, called the 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 that might be installed, aerodynamic lift
Lift or LIFT may refer to:
Physical devices
* Elevator, or lift, a device used for raising and lowering people or goods
** Paternoster lift, a type of lift using a continuous chain of cars which do not stop
** Patient lift, or Hoyer lift, mobile ...
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
In both road and rail vehicles, the wheelbase is the horizontal distance between the centers of the front and rear wheels. For road vehicles with more than two axles (e.g. some trucks), the wheelbase is the distance between the steering (front ...
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
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
, is the total mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
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, such as on most tandem bicycle
A tandem bicycle or twin is a form of bicycle (occasionally a tricycle) designed to be ridden by more than one person. The term tandem refers to the seating arrangement (fore to aft, not side by side), not the number of riders. Patents relat ...
s or long-wheel-base recumbent bicycles, as well as car
A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods.
The year 1886 is regarded as ...
s, it is less likely that the front wheel can generate enough braking force to flip the bike. This means they can decelerate up to nearly the limit of adhesion of the tires to the road, which could reach 0.8 g if the coefficient of friction is 0.8, which is 40% more than an upright bicycle under even the best conditions. ''Bicycling Science'' author David Gordon Wilson
David Gordon Wilson (11 February 1928 – 2 May 2019) was a British-born engineer who served as a professor of engineering at the Massachusetts Institute of Technology (MIT) in the United States.
Born in Warwickshire, England, Wilson went to th ...
points out that this puts upright bicyclists at particular risk of causing a rear-end collision if they tailgate cars.[
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
In vehicle acrobatics, a wheelie, or wheelstand, is a vehicle maneuver in which the front wheel or wheels come off the ground due to sufficient torque being applied to the rear wheel or wheels, or rider motion relative to the vehicle. Whe ...
. 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
Hillclimbing, also known as hill climbing, speed hillclimbing, or speed hill climbing, is a branch of motorsport in which drivers compete against the clock to complete an uphill course. It is one of the oldest forms of motorsport, since the fir ...
competitions.
Braking according to ground conditions
When braking, the rider in motion is seeking to change the speed of the combined mass ''m'' of rider plus bike. This is a negative acceleration ''a'' in the line of travel. ''F''=''ma'', the acceleration ''a'' causes an inertial
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
forward force ''F'' on mass ''m''.
The braking ''a'' is from an initial speed ''u'' to a final speed ''v'', over a length of time ''t''. The equation ''u'' - ''v'' = ''at'' implies that the greater the acceleration the shorter the time needed to change speed. The stopping distance ''s'' is also shortest when acceleration ''a'' is at the highest possible value compatible with road conditions: the equation ''s'' = ''ut'' + 1/2 ''at''2 makes ''s'' low when ''a'' is high and ''t'' is low.
How much braking force to apply to each wheel depends both on ground conditions and on the balance of weight on the wheels at each instant in time. The total braking force cannot exceed the gravity force on the rider and bike times the coefficient of friction ''μ'' of the tire on the ground. ''mgμ'' >= ''Ff'' + ''Fr''. A skid occurs if the ratio of either ''Ff'' over ''Nf'' or ''Fr'' over ''Nr'' is greater than ''μ'', with a rear wheel skid having less of a negative impact on lateral stability.
When braking, the inertial force ''ma'' in the line of travel, not being co-linear with ''f'', tends to rotate ''m'' about ''f''. This tendency to rotate, an overturning moment, is resisted by a moment from ''mg''.
Taking moments about the front wheel contact point at an instance in time:
*When there is no braking, mass ''m'' is typically above the bottom bracket, about 2/3 of the way back between the front and rear wheels, with ''Nr'' thus greater than ''Nf''.
*In constant light braking, whether because an emergency stop is not required or because poor ground conditions prevent heavy braking, much weight still rests on the rear wheel, meaning that ''Nr'' is still large and ''Fr'' can contribute towards ''a''.
*As braking ''a'' increases, ''Nr'' and ''Fr'' decrease because the moment ''mah'' increases with ''a''. At maximum constant ''a'', clockwise and anti-clockwise moments are equal, at which point ''Nr'' = 0. Any greater ''Ff'' initiates a stoppie.
Other factors:
*Downhill it is much easier to topple over the front wheel because the incline moves the line of ''mg'' closer to ''f''. To try to reduce this tendency the rider can stand back on the pedals to try to keep ''m'' as far back as possible.
*When braking is increasing the center of mass ''m'' may move forward relative to the front wheel, as the rider moves forward relative to the bike, and, if the bike has suspension on the front wheel, the front forks compress under load, changing the bike geometry. This all puts extra load on the front wheel.
*At the end of a brake maneuver, as the rider comes to a halt, the suspension decompresses and pushes the rider back.
Values for ''μ'' vary greatly depending on a number of factors:
*The material that the ground or road surface is made of.
*Whether the ground is wet or dry.
*The smoothness or roughness of the ground.
*The firmness or looseness of the ground.
*The speed of the vehicle, with friction reducing above 30 mph (50 km/h).
*Whether friction is rolling or sliding, with sliding friction at least 10% below peak rolling friction.
Braking
Most of the braking force of standard upright bikes comes from the front wheel. As the analysis above shows, if the brakes
A brake is a mechanical device that inhibits motion by absorbing energy from a moving system. It is used for slowing or stopping a moving vehicle, wheel, axle, or to prevent its motion, most often accomplished by means of friction.
Background ...
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
The stoppie is a motorcycle and bicycle trick in which the back wheel is lifted by abruptly applying the front brake, then, by carefully reducing the brake pressure, the bike is ridden for a short distance on the front wheel. It is also called an ...
'' 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
A cruiser is a type of warship. Modern cruisers are generally the largest ships in a fleet after aircraft carriers and amphibious assault ships, and can usually perform several roles.
The term "cruiser", which has been in use for several hu ...
and 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 ba ...
s, the front tire will skid instead, possibly causing a loss of balance. Assuming no loss of balance, it is possible to calculate optimum braking performance depending on the bike's geometry, the location of center of gravity of bike and rider, and the maximum coefficient of friction.
In the case of a front suspension
Suspension or suspended may refer to:
Science and engineering
* Suspension (topology), in mathematics
* Suspension (dynamical systems), in mathematics
* Suspension of a ring, in mathematics
* Suspension (chemistry), small solid particles suspend ...
, especially telescoping fork tube
A motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a yoke, also known as a triple clamp, which consists of an upper yoke joined to a lower yoke via a steering stem, a shaft that runs through the steering ...
s, 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 is a driving and motorcycle riding technique where the brakes are used beyond the entrance to a turn (turn-in), and then gradually released (trailed off). Depending on a number of factors, the driver fully releases brake pressure at ...
''.
Front wheel braking
The limiting factors on the maximum deceleration in front wheel braking are:
*the maximum, limiting value of static friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
between the tire and the ground, often between 0.5 and 0.8 for rubber
Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
on dry asphalt
Asphalt, also known as bitumen (, ), is a sticky, black, highly viscous liquid or semi-solid form of petroleum. It may be found in natural deposits or may be a refined product, and is classed as a pitch. Before the 20th century, the term a ...
,
*the kinetic friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
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 back from the front wheel contact patch and 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 0.25 ''g'' (~2.5 m/s2) deceleration at best,[ because of the decrease in normal force at the rear wheel as described above. All such bikes with only rear braking are subject to this limitation: for example, bikes with only a ]coaster brake
A bicycle brake reduces the speed of a bicycle or prevents it from moving. The three main types are: #Rim brakes, rim brakes, #Disc brakes, disc brakes, and #Drum brakes, drum brakes.
Most bicycle brake systems consist of three main components: ...
, and fixed-gear
A fixed-gear bicycle (or fixed-wheel bicycle, commonly known in some places as a fixie) is a bicycle that has a drivetrain with no freewheel mechanism. The freewheel was developed early in the history of bicycle design but the fixed-gear bicy ...
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.][
*To deliberately induce a rear wheel skid to induce oversteer and achieve a smaller turn radius on tight turns.
*Front brake failure.][
*Recumbent bicycles. Long-wheelbase recumbents require a good rear brake as the CG is near the rear wheel.
]
Braking technique
Expert opinion varies from "use both levers equally at first"
to "the fastest that you can stop any bike of normal wheelbase is to apply the front brake so hard that the rear wheel is just about to lift off the ground,"[ depending on road conditions, rider skill level, and desired fraction of maximum possible deceleration.
]
Suspension
Bikes may have only front, only rear, full suspension or no suspension that operate primarily in the central plane of symmetry; though with some consideration given to lateral compliance. The goals of a bike suspension are to reduce vibration experienced by the rider, maintain wheel contact with the ground, reduce the loss of momentum when riding over an object, reduce impact forces caused by jumps or drops and maintain vehicle trim. The primary suspension parameters are stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
, damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
, sprung and unsprung mass
The unsprung mass (colloquially unsprung weight) of a vehicle is the mass of the suspension, wheels or tracks (as applicable), and other components directly connected to them. This contrasts with the sprung mass (or weight) supported by the ...
, and tire
A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineering), t ...
characteristics. Besides irregularities in the terrain, brake, acceleration, and drive-train forces can also activate the suspension as described above. Examples include bob
Bob, BOB, or B.O.B. may refer to:
Places
* Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname ...
and pedal feedback on bicycles, the 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 ...
on motorcycles, and
squat and brake dive on both.
Vibration
The study of vibrations in bikes includes its causes, such as engine balance
Engine balance refers to how the forces (resulting from combustion or rotating/reciprocating components) are balanced within an internal combustion engine or steam engine. The most commonly used terms are ''primary balance'' and ''secondary bal ...
, wheel balance, ground surface, and aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
; 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
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In ...
of the system with the possible driving frequencies of the vibration sources. A close match means mechanical resonance
Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its ''resonance frequency'' or ''resonant frequency'') close ...
that can result in large amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
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). Another issue with vibration for the bike is the possibility of failure due to material fatigue
In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of ...
Effects of vibration on riders include discomfort, loss of efficiency, Hand-Arm Vibration Syndrome
Vibration white finger (VWF), also known as hand-arm vibration syndrome (HAVS) or dead 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 ...
, a secondary form Raynaud's disease
Raynaud syndrome, also known as Raynaud's phenomenon, eponymously named after the physician Auguste Gabriel Maurice Raynaud, who first described it in his doctoral thesis in 1862, is a medical condition in which the spasm of small arteries cau ...
, and whole body vibration
Whole body vibration is a generic term used when vibrations (mechanical oscillations) of any frequency are transferred to the human body. Humans are exposed to vibration through a contact surface that is in a mechanical vibrating state. Humans are ...
. 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
A tire (American English) or tyre (British English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineering), t ...
and traditional bicycle suspension
Bicycle suspension is the system, or systems, used to ''suspend'' the rider and bicycle in order to insulate them from the roughness of the terrain. Bicycle suspension is used primarily on mountain bikes, but is also common on hybrid bicycles ...
s, a variety of techniques have been developed to damp vibrations before they reach the rider. These include materials, such as carbon fiber
Carbon fiber-reinforced polymers (American English), carbon-fibre-reinforced polymers (Commonwealth English), carbon-fiber-reinforced plastics, carbon-fiber reinforced-thermoplastic (CFRP, CRP, CFRTP), also known as carbon fiber, carbon compo ...
, either in the whole frame
A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent.
Frame and FRAME may also refer to:
Physical objects
In building construction
*Framing (con ...
or just key components such as the front fork, 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 indic ...
, or handlebars; tube shapes, such as curved seat stays;, gel handlebar grips and saddles and special inserts, such as Zertz by Specialized, and Buzzkills by Bontrager
Keith Bontrager (; born December 18, 1954) is a motorcycle racer who became a pioneer in the development of the modern mountain bike. Between 1980 and 1995, he was president of his own Bontrager Company, which continues to develop components for ...
.
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 shaft
Balance shafts are used in piston engines to reduce vibration by cancelling out unbalanced dynamic forces. The counter balance shafts have eccentric weights and rotate in opposite direction to each other, which generates a net vertical force.
Th ...
s, rubber engine mounts, and tire weights. The problems that vibration causes have also spawned an industry of after-market parts and systems designed to reduce it. Add-ons include handlebar weights, isolated foot pegs, and engine counterweights. At high speeds, motorcycles and their riders may also experience aerodynamic flutter or buffeting
Aeroelasticity is the branch of physics and engineering studying the interactions between the inertial, elastic, and aerodynamic forces occurring while an elastic body is exposed to a fluid flow. The study of aeroelasticity may be broadly classi ...
. This can be abated by changing the air flow over key parts, such as the windshield
The windshield (North American English) or windscreen (Commonwealth English) of an aircraft, car, bus, motorbike, truck, train, boat or streetcar is the front window, which provides visibility while protecting occupants from the elements. Mo ...
.
Experimentation
A variety of experiments have been performed in order to verify or disprove various hypotheses about bike dynamics.
* David Jones built several bikes in a search for an unrideable 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.
* Hubbard and Moore have performed measurements with an instrumented bike.
See also
*Bicycle and motorcycle geometry
Bicycle and motorcycle geometry is the collection of key measurements (lengths and angles) that define a particular bike configuration. Primary among these are wheelbase, steering axis angle, fork offset, and trail. These parameters have a major ...
*Bicycle fork
A bicycle fork is the part of a bicycle that holds the front wheel.
A fork typically consists of two ''blades'' which are joined at the top by a fork ''crown.'' The crown is often at the front. Most suspension forks have an arch connecting the ...
*Bicycle performance
A bicycle's performance is extraordinarily efficient. In terms of the amount of energy a person must expend to travel a given distance, cycling is calculated to be the most efficient self-powered means of transportation.
In terms of the ratio ...
*Bicycle tire
A bicycle tire is a tire that fits on the wheel of a bicycle or similar vehicle. These tires may also be used on tricycles, wheelchairs, and handcycles, frequently for racing. Bicycle tires provide an important source of suspension, generate ...
*Camber angle
Camber angle is one of the angles made by the wheels of a vehicle; specifically, it is the angle between the vertical axis of a wheel and the vertical axis of the vehicle when viewed from the front or rear. It is used in the design of steering an ...
*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. Camber thrust is generated when a point on the outer surface ...
*Caster angle
250px, θ is the caster angle, the red line is the pivot line, and the grey area is the tire.
250px, Front suspension of a race car, the caster angle is formed by the line between upper and lower ball joint.
The caster angle or castor angle is t ...
*Cornering force
Cornering force or side force is the lateral (i.e., parallel to wheel axis) 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 ...
*Countersteering
Countersteering is 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 ("steer left to turn right"). To negotiate a turn s ...
*Highsider
A highsider or highside is a type of motorcycle accident characterized by sudden and violent rotation of the bike around its long axis. This generally happens when the rear wheel loses traction, skids, and then suddenly regains traction, creating ...
*Lowsider
The lowsider or lowside is a type of motorcycle or bicycle crash usually occurring in a turn. It is caused when either the front or rear wheel slides out as a result of either too much braking into the corner, too much acceleration through or out ...
*Motorcycle fork
A motorcycle fork connects a motorcycle's front wheel and axle to its frame, typically via a yoke, also known as a triple clamp, which consists of an upper yoke joined to a lower yoke via a steering stem, a shaft that runs through the steering h ...
*Parallel parking problem
The parallel parking problem is a motion planning problem in control theory and mechanics to determine the path a car must take to parallel park into a parking space. The front wheels of a car are permitted to turn, but the rear wheels must stay ...
*Slip angle
In vehicle dynamics, slip angle or sideslip angle is the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling (i.e., the angle between the forward velocity vector v_x and the vector su ...
*Speed wobble
Wobble, shimmy, tank-slapper, speed wobble, and even death wobble are all words and phrases used to describe a quick (4–10 Hz) oscillation of primarily just the steerable wheel(s) of a vehicle. Initially, the rest of the vehicle remain ...
*Stoppie
The stoppie is a motorcycle and bicycle trick in which the back wheel is lifted by abruptly applying the front brake, then, by carefully reducing the brake pressure, the bike is ridden for a short distance on the front wheel. It is also called an ...
*Trail braking
Trail braking is a driving and motorcycle riding technique where the brakes are used beyond the entrance to a turn (turn-in), and then gradually released (trailed off). Depending on a number of factors, the driver fully releases brake pressure at ...
*Wheelie
In vehicle acrobatics, a wheelie, or wheelstand, is a vehicle maneuver in which the front wheel or wheels come off the ground due to sufficient torque being applied to the rear wheel or wheels, or rider motion relative to the vehicle. Whe ...
*Outline of motorcycles and motorcycling
The following outline is provided as an overview of motorcycles and motorcycling:
Motorcycle — two-wheeled, single-track motor vehicle. Other names include: motorbike, bike, and cycle.
Motorcycling — act of riding a motorcycle, around wh ...
References
Further reading
‘An Introduction to Bicycle Geometry and Handling’
Karl Anderson
by Jobst Brandt
Jobst Brandt (January 14, 1935 – May 5, 2015) was an American mechanical engineer, inventor, bicycle enthusiast, educator, and author.
Early life
Brandt was born in New York City, where his father, the German-born agricultural economist Karl Br ...
‘Report on Stability of the Dahon Bicycle’
by John Forester
External links
Videos:
Video of riderless bicycle demonstrating self-stability
Why bicycles do not fall: Arend Schwab at TEDx Delft 2012
Wobble movie (AVI)
Weave movie (AVI)
on ''Science Friday
''Science Friday'' (known as ''SciFri'' for short) is a weekly call-in talk show that broadcasts each Friday on public radio stations, distributed by WNYC Studios, and carried on over 400 public radio stations. ''SciFri'' is hosted by award-win ...
''
Research centers:
Bicycle Dynamics
at Delft University of Technology
Delft University of Technology ( nl, Technische Universiteit Delft), also known as TU Delft, is the oldest and largest Dutch public technical university, located in Delft, Netherlands. As of 2022 it is ranked by QS World University Rankings among ...
Bicycle Mechanics
at Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...
Bicycle Science
at the University of Illinois
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the University ...
Motorcycle Dynamics
at the University of Padova
The University of Padua ( it, Università degli Studi di Padova, UNIPD) is an Italian university located in the city of Padua, region of Veneto, northern Italy. The University of Padua was founded in 1222 by a group of students and teachers from B ...
Control and Power Research Group
at Imperial College
Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cu ...
Bicycle dynamics, control and handling
at UC Davis
The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institut ...
Bicycle and Motorcycle Engineering Research Laboratory
at the University of Wisconsin-Milwaukee
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the ...
Conferences:
Bicycle and Motorcycle Dynamics 2010
Symposium on Dynamics and Control of Single Track Vehicles, Delft University of Technology
Delft University of Technology ( nl, Technische Universiteit Delft), also known as TU Delft, is the oldest and largest Dutch public technical university, located in Delft, Netherlands. As of 2022 it is ranked by QS World University Rankings among ...
, Oct 20–22, 2010
Single Track Vehicle Dynamics at DSCC 2012
two sessions at the ASME
The American Society of Mechanical Engineers (ASME) is an American professional association that, in its own words, "promotes the art, science, and practice of multidisciplinary engineering and allied sciences around the globe" via "continuing ...
Dynamic Systems and Control Conference in Fort Lauderdale, Florida, USA, October 17–19, 2012
Bicycle and Motorcycle Dynamics 2013
Symposium on Dynamics and Control of Single Track Vehicles, Nihon University
, abbreviated as , is a private university, private research university in Japan. Its predecessor, Nihon Law School (currently the Department of Law), was founded by Yamada Akiyoshi, the Minister of Justice (Japan), Minister of Justice, in 1889. ...
, Nov 11–13, 2013
Bicycle and Motorcycle Dynamics 2016
Symposium on Dynamics and Control of Single Track Vehicles, University of Wisconsin–Milwaukee
The University of Wisconsin–Milwaukee (UW–Milwaukee, UWM, or Milwaukee) is a public urban research university in Milwaukee, Wisconsin. It is the largest university in the Milwaukee metropolitan area and a member of the University of Wiscons ...
, Sept 21–23, 2016
Bicycle and Motorcycle Dynamics 2019
Symposium on Dynamics and Control of Single Track Vehicles, University of Padova
The University of Padua ( it, Università degli Studi di Padova, UNIPD) is an Italian university located in the city of Padua, region of Veneto, northern Italy. The University of Padua was founded in 1222 by a group of students and teachers from B ...
, Sept, 9–11, 2019
Bicycle and Motorcycle Dynamics Conference
Summary page
{{DEFAULTSORT:Bicycle And Motorcycle Dynamics
Control theory
Cycling
Dynamics (mechanics)