Two-mass-skate bicycle
Encyclopedia
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 have gyroscopic effects or positive trail to be self-stable. The two-mass and skates aspects of the model were chosen to eliminate design parameters so that the nine that remain, the locations of the masses and the steering geometry, could be more easily analyzed. Instead of full inertia tensors, the total mass of the bike is reduced to just two point masses, one attached to the rear frame
and one attached to the front fork
. Instead of rotating wheels
, the non-holonomic
ground contacts are provided by small-radius skates
.
in 1897 and Francis Whipple in 1899 both developed equations of motion for a bicycle that showed this self-stability. In 1970, Jones demonstrated in Physics Today
that the gyroscopic effects of rotating wheels is not necessary for bike stability. However, positive trail, also called caster trail, was still presumed to be necessary.
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
, University of Wisconsin-Stout
University of Wisconsin-Stout
The University of Wisconsin–Stout is a member campus of the University of Wisconsin System. The school was founded in 1891 in Menomonie, Wisconsin and enrolls more than 9,300 students...
, and Delft University of Technology
Delft University of Technology
Delft University of Technology , also known as TU Delft, is the largest and oldest Dutch public technical university, located in Delft, Netherlands...
to show that it is neither sufficient nor necessary for a bike to have gyroscopic effects or positive trail to be self-stable. The two-mass and skates aspects of the model were chosen to eliminate design parameters so that the nine that remain, the locations of the masses and the steering geometry, could be more easily analyzed. Instead of full inertia tensors, the total mass of the bike is reduced to just two point masses, one attached to the rear frame
Bicycle frame
A bicycle frame is the main component of a bicycle, on to which wheels and other components are fitted. The modern and most common frame design for an upright bicycle is based on the safety bicycle, and consists of two triangles, a main triangle and a paired rear triangle...
and one attached to the front fork
Bicycle fork
A bicycle fork is the portion of a bicycle that holds the front wheel and allows the rider to steer and balance the bicycle. A fork consists of two fork ends which hold the front wheel axle, two blades which join at a fork crown, and a steerer or steering tube to which the handlebars attach ...
. Instead of rotating wheels
Bicycle wheel
A bicycle wheel is a wheel, most commonly a wire wheel, designed for bicycle. A pair is often called a wheelset, especially in the context of ready built "off the shelf" performance-oriented wheels....
, the non-holonomic
Nonholonomic system
A nonholonomic system in physics and mathematics is a system whose state depends on the path taken to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the...
ground contacts are provided by small-radius skates
Ice skate
Ice skates are boots with blades attached to the bottom, used to propel the bearer across a sheet of ice. They are worn as footwear in many sports, including ice hockey, bandy and figure skating. The first ice skates were made from leg bones of horse, ox or deer, and were attached to feet with...
.
History
The self-stability of bicycles was reported as early as 1876. Emmanuel CarvalloEmmanuel 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 had been proven false....
in 1897 and Francis Whipple in 1899 both developed equations of motion for a bicycle that showed this self-stability. In 1970, Jones demonstrated in Physics Today
Physics Today
Physics Today, created in 1948, is the membership journal of the American Institute of Physics. It is provided to 130,000 members of twelve physics societies, including the American Physical Society...
that the gyroscopic effects of rotating wheels is not necessary for bike stability. However, positive trail, also called caster trail, was still presumed to be necessary.
Implications
The existence of the TMS and the self-stability it exhibits suggests that the design envelope of rideable bikes is larger than previously thought. For example, rear-wheel steering may not be as impossible as has been claimed. In the case of the TMS bike, the authors explain that the front assembly center of mass is lower than the rear frame center of mass, so it falls faster in a lean, and it is in front of the steering axis so it tends to steer the front end in the direction of the lean.Physical implementation
A physical example has been created that closely approximates the TMS model and exhibits self-stability without gyroscopic effects from the wheels and without positive trail. Since point masses do not exist in nature, the model can only approximate them with pieces of dense material (lead), mounted on a frame of light materials (carbon fiber and aluminum). To prevent the bike with just two masses, as pictured above, from tipping forward, a third mass is located just above the rear wheel contact patch where it does not alter the dynamics. In place of skates, small-radius wheels that drive nearly identical counter-rotating wheels are used to provide the non-holonomic ground contact with minimal gyroscopic effect. This physical example closely mimics the self-stability of the theoretical model.External links
- About the TMS bicycle at TUDelft
- About the TMS bicycle at Cornell University
- Discussion on Dave MoultonDave MoultonEdward David Moulton is a retired custom bicycle frame builder.Dave Moulton studied at Luton Technical College in England and learned bicycle frame building from Albert "Pop" Hodges in Luton, beginning in 1957. He opened a frame-building business around 1975 in Worcester, England...
's blog about the TMS bicycle with replies from an author of the original ScienceScience (journal)Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
paper - Discussion on Bicycle QuarterlyBicycle QuarterlyBicycle Quarterly is a magazine that examines the history of bicycles, their design, and evolution; with emphasis on Randonneuring bicycles. Articles evaluate equipment and bicycles for performance and function, and articles include footnotes. It is published and edited by Jan Heine, of Seattle,...
author's blog about the TMS bicycle with replies from an author of the original ScienceScience (journal)Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
paper