This article may be too long to read and navigate comfortably. Consider splitting content into sub-articles, condensing it, or adding subheadings. Please discuss this issue on the article's talk page. (January 2024)

Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles and their components, due to the forces acting on them. Dynamics falls under a branch of physics known as classical mechanics. Bike motions of interest include balancing, steering, braking, accelerating, suspension activation, and vibration. The study of these motions began in the late 19th century and continues today.^[1][2][3]

Bicycles and motorcycles are both single-track vehicles 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, tricycles, and quadracycles.^[4] As with unicycles, bikes lack lateral stability when stationary, and under most circumstances can only remain upright when moving forward. Experimentation and mathematical analysis have shown that a bike stays upright when it is steered to keep its center of mass over its wheels. This steering is usually supplied by a rider, or in certain circumstances, by the bike itself. Several factors, including geometry, mass distribution, and gyroscopic effect all contribute in varying degrees to this self-stability, but long-standing hypotheses and claims that any single effect, such as gyroscopic or trail, is solely responsible for the stabilizing force have been discredited.^[1][5][6][7]

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 or smaller the turn radius, the more lean is required. This balances the roll torque about the wheel contact patches generated by centrifugal force due to the turn with that of the gravitational force. This lean is usually produced by a momentary steering in the opposite direction, called countersteering. Countersteering skill is usually acquired by motor learning and executed via procedural memory rather than by conscious thought. Unlike other wheeled vehicles, the primary control input on bikes is steering torque, not position.^[8]

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.^[9]

History

The history of the study of bike dynamics is nearly as old as the bicycle itself. It includes contributions from famous scientists such as Rankine, Appell, and Whipple.^[2] In the early 19th century Karl von Drais, credited with inventing the two-wheeled vehicle variously called the laufmaschine, velocipede, draisine, and dandy horse, showed that a rider could balance his device by steering the front wheel.^[2] In 1869, Rankine published an article in The Engineer repeating von Drais' assertion that balance is maintained by steering in the direction of a lean.^[10] In 1897, the French Academy of Sciences made understanding bicycle dynamics the goal of its Prix Fourneyron competition. Thus, by the end of the 19th century, Carlo Bourlet, Emmanuel Carvallo, and Francis Whipple had showed with rigid-body dynamics that some safety bicycles could actually balance themselves if moving at the right speed.^[2] Bourlet won the Prix Fourneyron, and Whipple won the Cambridge University Smith Prize.^[7] It is not clear to whom should go the credit for tilting the steering axis from the vertical which helps make this possible.^[11]

In 1970, David E. H. Jones published an article in Physics Today showing that gyroscopic effects are not necessary for a person to balance a bicycle.^[6] Since 1971, when he identified and named the wobble, weave and capsize modes,^[12] Robin Sharp has written regularly about the behavior of motorcycles and bicycles.^[13] While at Imperial College, London, he worked with David Limebeer and Simos Evangelou.^[14] In the early 1970s, Cornell Aeronautical Laboratory (CAL, later Calspan Corporation 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 this 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,^[15] wobble,^[16] chatter,^[17] simulators,^[18] vehicle modelling,^[19] tire modelling,^[20][21] handling,^[22][23] and minimum lap time maneuvering.^[24][25]

In 2007, Meijaard, et al., published the canonical linearized equations of motion, in the Proceedings of the Royal Society A, along with verification by two different methods.^[2] 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 showing that neither gyroscopic effects nor so-called caster effects due to trail are necessary for a bike to balance itself.^[1] They designed a two-mass-skate bicycle that the equations of motion predict is self-stable even with negative trail, the front wheel contacts the ground in front of the steering axis, and with counter-rotating wheels to cancel any gyroscopic effects. Then they constructed a physical model to validate that prediction. This may require some of the details provided below about steering geometry or stability to be re-evaluated. Bicycle dynamics was named 26 of Discover's 100 top stories of 2011.^[26] In 2013, Eddy Merckx Cycles was awarded over €150,000 with Ghent University to examine bicycle stability.^[27]

Forces

If the bike and rider are considered to be a single system, the forces that act on that system and its components can be roughly divided into two groups: internal and external. The external forces are due to gravity, inertia, contact with the ground, and contact with the atmosphere. The internal forces are caused by the rider and by interaction between components.

External forces

As with all masses, gravity pulls the rider and all the bike components toward the earth. At each tire contact patch there are ground reaction forces with both horizontal and vertical components. The vertical components mostly counteract the force of gravity, but also vary with braking and accelerating. For details, see the section on longitudinal stability below. The horizontal components, due to friction between the wheels and the ground, including rolling resistance, are in response to propulsive forces, braking forces, and turning forces. Aerodynamic forces due to the atmosphere are mostly in the form of drag, but can also be from crosswinds. At normal bicycling speeds on level ground, aerodynamic drag is the largest force resisting forward motion.^[28]: 188 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 of the bike and rider; or simply as inertia in a stationary, inertial reference frame and not forces at all. Gyroscopic forces acting on rotating parts such as wheels, engine, transmission, etc., are also due to the inertia of those rotating parts. They are discussed further in the section on gyroscopic effects below.

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 torques between the steering mechanism (front fork, handlebars, front wheel, etc.) and rear frame, and between the rider and the rear frame. Friction exists between any parts that move against each other: in the drive train, between the steering mechanism and the rear frame, etc. In addition to brakes, which create friction between rotating wheels and non-rotating frame parts, many bikes have front and rear suspensions. Some motorcycles and bicycles have a steering damper to dissipate undesirable kinetic energy,^[14][29] 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 between the drive train and the suspension is an issue designers attempt to handle with various linkage configurations and dampers.^[30]

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 stoppies, wheelies, brake diving, and most suspension activation. Motions in these two groups are linearly decoupled, that is they do not interact with each other to the first order.^[2] 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, multibody dynamic analysis with at least two generalized coordinates to analyze. At a minimum, two coupled, second-order differential equations are required to capture the principal motions.^[2] Exact solutions are not possible, and numerical methods must be used instead.^[2] 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 and just one coordinate.

Balance

When discussing bike balance, it is necessary to distinguish carefully between "stability", "self-stability", and "controllability". Recent research suggests that "rider-controlled stability of bicycles is indeed related to their self-stability".^[1]

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.^[28] Steering may be supplied by a rider or, under certain circumstances, by the bike itself.^[31] 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 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. 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.^[32]

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.^[5][6] 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 can be controlled with an actuator only at the elbow.^[33]

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.^[34] 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

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.^[35] 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, choppers, and wheelie bikes.^[36] It can also be a challenge for touring bikes that carry a heavy load of gear over or even behind the rear wheel.^[37] Mass over the rear wheel can be more easily controlled if it is lower than mass over the front wheel.^[11]

A bike is also an example of an inverted pendulum. 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.^[38] However, a rider can have the opposite impression of a bike when it is stationary. A top-heavy bike can require more effort to keep upright, when stopped in traffic for example, than a bike which is just as tall but with a lower center of mass. This is an example of a vertical second-class lever. A small force at the end of the lever, the seat or handlebars at the top of the bike, more easily moves a large mass if the mass is closer to the fulcrum, where the tires touch the ground. This is why touring cyclists are advised to carry loads low on a bike, and panniers hang down on either side of front and rear racks.^[39]

Trail

A factor that influences how easy or difficult a bike will be to ride is trail, 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.^[28] 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.^[1]

Trail is a function of head angle, fork offset or rake, and wheel size. Their relationship can be described by this formula:^[40]

${\displaystyle {\text{Trail}}={\frac {(R_{w}\cos(A_{h})-O_{f})}{\sin(A_{h})}}}$

where ${\displaystyle R_{w}}$ is wheel radius, ${\displaystyle A_{h}}$ is the head angle measured clock-wise from the horizontal and ${\displaystyle O_{f}}$ 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,^[41] 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 rake, often a curve in the fork blades forward of the steering axis, is used to diminish trail.^[42] 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.^[1]

In motorcycles, rake refers to the head angle instead, and offset created by the triple tree is used to diminish trail.^[43]

A small survey by Whitt and Wilson^[28] found:

• touring bicycles with head angles between 72° and 73° and trail between 43 mm and 60 mm
• racing bicycles with head angles between 73° and 74° and trail between 28 mm and 45 mm
• track bicycles 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 offers ^[44] 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 71+1⁄4° 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 72+1⁄2° 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.^[45] Trail can decrease to zero with sufficiently large lean and steer angles, which can alter how stable a bike feels.^[11] 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,^[46] 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, 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.^[9] Also, the radius of curvature for a given steer angle and lean angle is proportional to the wheelbase.^[9] 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.^[11]

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.^[47] 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.^[28][47]

Gyroscopic effects

The role of the gyroscopic effect in most bike designs is to help steer the front wheel into the direction of a lean. This phenomenon is called precession, and the rate at which an object precesses is inversely proportional to its rate of spin. The slower a front wheel spins, the faster it will precess when the bike leans, and vice versa.^[48] The rear wheel is prevented from precessing 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.^[49]

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.^[11] 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.^[9] For more detail, see the section countersteering, below, and the countersteering article.

Self-stability

Between the two unstable regimes mentioned in the previous section, and influenced by all the factors described above that contribute to balance (trail, mass distribution, gyroscopic effects, etc.), there may be a range of forward speeds for a given bike design at which these effects steer an uncontrolled bike upright.^[2] 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.^[1]

However, even without self-stability a bike may be ridden by steering it to keep it over its wheels.^[6] 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.^[28] This 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.^[7]

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 necessary to alter the course from straight ahead as a combination of cornering force and camber thrust. The radius of the turn of an upright (not leaning) bike can be roughly approximated, for small steering angles, by:

${\displaystyle r={\frac {w}{\delta \cos \left(\phi \right)}}}$

where ${\displaystyle r\,\!}$ is the approximate radius, ${\displaystyle w\,\!}$ is the wheelbase, ${\displaystyle \delta \,\!}$ is the steer angle, and ${\displaystyle \phi \,\!}$ is the caster angle of the steering axis.^[9]

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:

$$Zdroj:https://en.wikipedia.org?pojem=Bicycle_and_motorcycle_dynamics
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---