Bicycle Problem that Broke Math: The Caster Theory

Detail of a road bike with a cyclist pedaling on a road.

Have you ever wondered how you keep a bicycle upright while in motion, riding on tires that are less than an inch in width? Riding a bike, at first glance, seems like a tightrope feat. Yet, we consider bicycle riding to be a simple enough feat that even a six-year-old child (or circus bear) can accomplish.

Why do we tend to steer a bike to the right as we lean and turn to the left? How does a bicycle remain stable even when not propelled by a rider?

How Do We Ride Bikes?

Bicycle Problem that Nearly Broke Mathematics: The Caster Theory

According to Nature Research, a publication of scientific journals and essays, pondering these types of questions often leads to significant developments in other wheel-based situations. Prosthetics, robotics, and applications to improve pedal and electric bikes can result from better understanding the physical forces that affect bicycles.

The subject of the article, Jim Papadopoulos, has spent most of his 62 years studying the physical and mathematical dynamics of bicycle propulsion and has joined with a larger team of scientists in recent years to continue to find answers to this seemingly simple question: How does a riderless bicycle stabilize itself?

Note that when you give a riderless bicycle a push, perhaps downhill, it may wobble at first, but then the bike will stabilize and continue on its way.

For over 100 years, bicycles have looked mostly the same. Two wheels, handlebars, a seat, pedals and a chain to propel the system are the main components. Almost all changes in that time have been cosmetic, comfort or weight driven, and gear systems to make pedaling easier.

As far back as 1910, early scientists who were fascinated by the topic proposed that bike stability results from a “gyroscope effect” in which the spinning wheels will tend to resist tilting as long as the applied force lasts.

Caster-Like Behavior

According to the same article by Nature Research, in 1970 a popular-science writer named David Jones viewed the matter of riderless stability differently by comparing the bike wheels to the front wheels or casters of a shopping cart. The bike’s wheels tend to turn in the direction of the motion at some distance, two to four inches behind the steering axis.

What is the steering axis on a bike? The steering axis is a line along the fork that holds the axle and wheel. If that virtual straight line extends from the fork to the ground, the steering axis or bike is ahead of the point where the wheel touches the ground.

Since the front wheel of a bicycle operates like a caster wheel by touching the ground behind the steering axis, Jones believed, the wheel is actually “following” the bike as it is driven by a propelling force like gravity. The wheel will actually turn in the direction the bike is traveling, since the bike, in effect, is leading the wheel.

The Caster Theory Evolves

Papadopoulos and others found the Jones caster theory to be somewhat incomplete without the mathematics to support the theory. More work was necessary. Over time, Papadopoulos created an elaborate set of equations that explained the mechanics of the caster effect.

Science and mathematics that uncover dynamics such as the Jones’s caster theory can lead to solutions in other fields of physical science and regular human activity. Stability achieved by this phenomenon has led researchers to develop improved designs that imitated the bicycle-caster design configuration.  Wheels that follow the leading parts of a structure have inspired new types of walking aids, wheelchairs, and even bikes.

Content repurposed from Nature Research

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