# Why bicycles are so stable?

For almost 150 years now, mathematicians have tried to understand why a bicycle could be so stable. Now, researchers of the Delft University of Technology (TU Delft), working with colleagues from Cornell University and the University of Nottingham, UK, say they've build a model which unravels how a bicycle works. As said a bicycle maker, when designing a bike, only three parameters are used: the general geometry, the distance between the axles, and the angle at which the fork points downwards. The new mathematical model includes 25 parameters and will permit to build bicycles aimed directly at special target groups. Already, a Dutch bicycle manufacturing company is hoping to design better bikes using this model.

For almost 150 years now, mathematicians have tried to understand why a bicycle could be so stable. Now, researchers of the Delft University of Technology (TU Delft), working with colleagues from Cornell University and the University of Nottingham, UK, say they've build a model which unravels how a bicycle works. As said a bicycle maker, when designing a bike, only three parameters are used: the general geometry, the distance between the axles, and the angle at which the fork points downwards. The new mathematical model includes 25 parameters and will permit to build bicycles aimed directly at special target groups. Already, a Dutch bicycle manufacturing company is hoping to design better bikes using this model.

This research was led by Arend Schwab, an assistant professor in applied mechanics at TU Delft working on Bicycle Dynamics. he's also leading the Delft Bicycle Dynamics Lab where he's working with his former student Jodi Kooijman. You can see above the bicycle built by Kooijman to valid the the 3 degree of freedom model. "He measured the steering angle (potentiometer), lean rate and yaw rate (angular rate gyros) and the angular rotation of the rear wheel (Avocet 50 bicycle computer). During 76 runs the bicycle was laterally perterbed and allowed to coast freely at a number of different speeds mostly between 2.5 m/s and 6 m/s." (Credit: Experimental Validation)

"Bicycle manufacturers have never been able to say precisely how a bicycle works," explains Schwab of the Faculty of Mechanical, Maritime and Materials Engineering (3mE). "They have always had to refine their designs purely through experimentation. In our model, they can enter into the computer all of the various factors that influence the stability and handling of their bicycle. The model then calculates how the bicycle will react at specific speeds."

This research work has been commented in Delft Outlook under the name "Bicycles made to measure" (written by Tomas van Dijk). Here are some comments about the stability of a bicycle. "The bike’s speed must be between fourteen and twenty seven kilometres per hour," Kooijman says. "At those speeds, the bicycle is inherently stable. If it goes faster, it will wobble less, but if you then push it sideways it will lean over to one side until it topples. The data match our model predictions exactly." You can see images and videos confirming this on thias page about Treadmill Measurements showing a bicycle's stability at various speeds.

The above article is also available in PDF format (4 pages, 364 KB). This version shows on the last page why we need to briefly steer on the right to make a left hand turn. "We all know intuitively the main combination of forces that ensure we stay upright when riding a bicycle. They involve leaning over and steering and they explain why, when we wish to turn to the right, we have to first turn the front wheel slightly to the left. The action, known as counter steer, results in a force that causes the bicycle to lean over to the other side, which is the direction in which we wanted to go. This also explains why we fall over if we pass too close to a kerb. We just can't manage to get away from it without hitting it."

This is why this new mathematical model is using as much as 25 parameters. "All of them are relate to the two connected motion equations, one for leaning over and one for steering. 'It remains unclear how exactly all these parameters affect the stability,' Schwab says. 'In the final model these parameters appear in fairly complex forms as coefficients to the motion equations. For practical purposes most researchers used to simplify the equations by disregarding certain parameters, but the results tended to be far from ideal. And scientists who failed to make the connection between leaning and steering certainly were on the wrong track altogether.'"

So will we ride better bicycles? Probably soon according to Rob van Regenmortel, product development manager of bicycle manufacturer Batavus. "Van Regenmortel would like to collaborate with Schwab and Kooijman on a future project that will also look at the riding behaviour of the cyclist. The ultimate goal of the bicycle research effort is to include the cyclist's riding behaviour in the model so as to be able to investigate the combination of the bicycle and its rider. 'We could then actually make a ‘tailor-made’ bicycle for everyone,' Van Regenmortel says. 'People who find it difficult to maintain their balance would no longer have to ride a tricycle.'"

Sources: Delft University of Technology (TU Delft) news release, September 18, 2007; and various websites

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