Thursday, November 08, 2007

2 A Mathematical Bicycle Model to End all Models...

I was on a hunt for a certain paper written by a student under Dr. A Schwab, at the Delft University of technology in Netherlands. J. Kooijman was a Masters student in mechanical engineering, and probably still is. The subject of his research interested me. He investigated the dynamics of a bicycle and after numerous tests, wrote his engineering thesis and defended it. The topic, on a mathematical model of the bicycle to "end all models".

I finally got it!

"Experimental Validation of a Model for the motion of an Uncontrolled Bicycle" can now be read at this link. Warning, please proceed to attempt reading only if you're engineering savvy. I plan to read some right away. This will add to my 100 MB folder of bicycle related research papers on my computer. I'm :) sad (?)...but I love it!

Check this webpage for some videos and pictures on his test setups.

Here's a primer article for you folks from the University webpage itself. What this means is that a big part of the future of the bicycle industry may be in custom making, or tailoring it to the rider with the help of the computer models, and not just through the knowhow and practices of the frame maker. With the help of these models, we can stray away from conventional bicycles to modern designs, that'll achieve the same or more depending on what the rider intends to do with the bike. Soon we can gravitate towards the Toyota like way of researching customer needs : surveying what their riding styles are like! I think the future is already taking place.

For almost a century and a half, mathematicians have been racking their brains about the bike. How can a rolling bicycle be so stable of its own accord? Delft researchers now say they have completed the model to end all models. Bicycle manufacturer Batavus intends to use it to make better bikes for the elderly and disabled.

By: Tomas van Dijk

The conveyor belt passes at speed under Ir. Jodi Kooijman and his bicycle. Kooijman, an enthusiastic off-road cyclist, pedals until his speedometer indicates sixteen kilometres per hour. On the sideline, Dr Ir. Arend Schwab of the Faculty of Mechanical Engineering, Maritime Technology, and Material Sciences (3me), at the agreed upon moment, yanks a rope attached to the bike’s luggage carrier. For a brief moment Kooijman veers to the right, but his bicycle regains its balance within a fraction of a second, appearing to automatically retrace ‘the line’.

This video-recorded incident took place on a large conveyor belt at the department of motion sciences of Amsterdam Vrije University. The experiment is just one of those the two 3me researchers have carried out in the past couple of years to test a mathematical model defining all the forces that act on a moving bicycle. A publication about this bicycle model recently appeared in the ‘Proceedings of the Dutch Royal Society’, the Royal Dutch Academy of Science.

Schwab shows another video recording. Here, Kooijman is giving a bicycle a hefty push. The bicycle is laden with measuring equipment and the carrier holds a laptop computer that records the bike’s every movement. The unmanned bicycle rolls on following a straight line in the sports centre of Delft University. Kooijman runs after it and pushes the bicycle sideways. The bike wobbles a bit, the handlebars move from side to side, but the bike soon regains it’s straight course.

“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.”

Balance in motion

Ever since the invention of the pedal-driven bicycle around 1860, researchers have been trying to determine what makes a bike fairly stable of its own accord. They added formula after formula, each one of them derived from the laws of motion as defined by Newton and Euler, but they never managed to develop a completely accurate model for predicting a bike’s riding characteristics.

“Bicycle manufacturers never knew exactly how a bike works either,” Schwab says. “They have always had to resort to experiments to improve their products. Not that there’s anything wrong with that, but now they can use our model to feed into a computer all the factors affecting a bike’s steering properties. The model then calculates how the bicycle will behave at different speeds.”

Together with colleagues at Cornell University in the u.s. and at Nottingham University in the u.k. the Delft researchers perused more than fifty publications written by scientists on the subject since the early days of the bicycle. Many mathematicians claim that the bicycle mainly derives its stability from the fact that it takes effort to change the direction of a rotating mass, the gyroscopic effect.

“The gyroscopic effect certainly plays its part,” Schwab says. To demonstrate this, he produces a wheel weighted with lead around the rim, and gives it a mighty jerk. Only with great difficulty can the wheel be made to change direction. “However, mathematicians who took this principle to heart were wrong,” Schwab continues. “When we disregarded the weight of the wheels in our model we discovered that it was still possible to make the bicycle stable. And there is no truth in the idea that bicycles with small wheels are unstable.”


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.

As for the steering properties, the greater the angle at which the fork of the bicycle points forward, the more stable the bicycle will go in a straight line, but also the more difficult it will be to go round corners. “The distribution of mass is also very important,” Kooijman says. “Moving the centre of gravity of a bicycle forward makes it more stable.”

The Delft scientists included twenty five such parameters in their model. 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.”


A model that indicates whether a design will result in a thoroughbred racing bike or in a stable ride suitable for the elderly, is something the bicycle industry has been eagerly awaiting. Rob van Regenmortel, product development manager of bicycle manufacturer Batavus, is following the Delft research effort with an eagle eye. Van Regenmortel: “Traditionally, when designing a bike, we use three parameters: the general geometry, the distance between the axles, and the angle at which the fork points downwards. Most of these properties were established back in the 1970s. Take the angle of the tube that carries the saddle. On our old-fashioned bikes this tube is mounted almost vertically. On bikes made by Gazelle on the other hand, it is inclined slightly more backwards. These are simple design choices all bicycle manufacturers made at one point and which they then more or less stuck to for the simple reason that their products kept selling. Now that we have Schwab’s model, we hope to be able to start designing bicycles aimed directly at special target groups.”

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.”

Ultimately, the model is intended to improve customer communications. “Perhaps we could label bicycles with numbers to give customers an idea of their riding properties. People looking for a bike to carry lots of luggage on holiday could then be recommended a type two bicycle, say, and someone needing transport to work and back might be wanting a slightly more thoroughbred machine, say type four. It’s just an idea.”

But how do you measure people’s riding behaviour? On the conveyor belt in Amsterdam, Kooijman and Schwab have already collected some manned bicycle data through the simple expedient of riding the test bikes themselves. “Scary is the word,” Kooijman says. “You’re cycling at some speed inside an enclosed space without moving forward. It feels very weird. You’re constantly afraid of hitting the wall. We can’t ask elderly or disabled people to ride a bicycle that way to collect data. In future we will have to conduct our tests on the road, and then copy the cycling behaviour in a robot bicycle.”


  1. Anony-Chan2:42 AM

    Thank you very much.

  2. Anonymous6:25 AM

    Great help! Thanks


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