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 is a branch of classical mechanics, which in turn is a branch of physics. 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.
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. 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.
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.
Although longitudinally stable when stationary, bikes often have a high enough center of mass and a short enough wheelbase to lift a wheel off the ground under sufficient acceleration or deceleration. When braking, depending on the location of the combined center of mass of the bike and rider with respect to the point where the front wheel contacts the ground, bikes can either skid the front wheel or flip the bike and rider over the front wheel. A similar situation is possible while accelerating, but with respect to the rear wheel
History
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. 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. By the end of the 19th century, Emmanuel Carvallo and Francis Whipple showed with rigid-body dynamics that some safety bicycles could actually balance themselves if moving at the right speed. It is not clear to whom should go the credit for tilting the steering axis from the vertical which helps make this possible.
In 1970, David E. H. Jones published an article in Physics Today showing that gyroscopic effects are not necessary to balance a bicycle. Since 1971, when he identified and named the wobble, weave and capsize modes, Robin Sharp has written regularly about the behavior of motorcycles and bicycles. While at Imperial College, London, he worked with David Limebeer and Simos Evangelou. In 2007, Meijaard, et al., published the canonical linearized equations of motion, in the Proceedings of the Royal Society A, along with verification by two different methods. These equations assumed the tires to roll without slip, that is to say, to go where they point, and the rider to be rigidly attached to the rear frame of the bicycle.
In 2011, Kooijman, et al., published an article in Science 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 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.