Intro: This work started in the Fall of 2002 when I came to Cornell to have my sabbatical leave with Andy Ruina. His line of research began in 1985 when Jim Papadopoulos came to Cornell to work with him (or visa versa). Take a look at  Andy's website on Bicycle Dynamics..

First take a look at this Cornell  VIDEO: Bicycle Stability Demonstration (MP4) (3 MB) to see what is so special about uncontrolled bicycles. Highly unstable at rest but very stable at moderate forward speed!

We would like to think that this is the definitive review paper on the linearized equations of the motion for a bicycle:

J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, A. L. Schwab, 2007 ``Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review,'' Proceedings of the Royal Society A 463:1955-1982. doi:10.1098/rspa.2007.1857,  or  preprint+ESM pdf(578k).

Media Coverage: This paper has drawn some attention in the media. Here you will find some of the media coverage.

History of bicycle steer and dynamics equations.

Over the past 140 years, scores of other people have studied bicycle dynamics, either for a dissertation, a hobby or sometimes as part of a life’s work on vehicles. This sparse and varied research on the dynamics of bicycles modelled as linked rigid bodies was initially reviewed in Hand (1988). A detailed history of bicycle dynamics studies with a large bibliography, which we update as we go along, can be found here.

    Hand, R. S. 1988 ''Comparisons and stability analysis of linearized equations of motion for a basic bicycle model.'' MSc thesis, Cornell University, Ithaca, NY.

News:

April 16, 2007: Today we put the instrumented bicycle from Jodi' MSc work on the big VU treadmill. We feel more confident about the usage of this treadmill for our bicycle handling experiments if the instrumented bicycle on the treadmill shows the same lateral dynamic behaviour as on the road. It turned out to be rather tricky to launch the bicycle and catch it in the  fall. Jodi got really handy by first launching it from standing aside and then stepping to the rear where he could steer, perturb and catch it. The weave speed  of the bicycle on the road is 4.0 m/s (14.4 km/h) [The bicycle is laterally unstable below the weave speed and stable above]. The preliminary result is that the bicycle shows approximately the same weave speed on the treadmill, as you can see from the three videos below:

12 km/h: Unstable (wmv 5.8 Mb)	16 km/h: Stable (wmv 9.4 Mb)	30 km/h: Very Stable (wmv 6.5 Mb)

April 4, 2007: Today we (Jodi and I) went to the VU University Amsterdam and looked at the big treadmill from the faculty of Human Movement Sciences. My friend Knoek van Soest works there and was so nice to let us do some preliminary tests to see if we could use the treadmill for our bicycle handling quality experiments.  The treadmill is big, 3 by 5 m, and has maximum speed of 30 km/h. The minister of Education, Ronald Plasterk,  on a work visit at the VU, also showed genuine interest in our experiments!

April 25, 2006: Jodi Kooijman succesfully defended his Engineering Dynamics  MSc thesis:

J. D. G. Kooijman, Experimental Validation of a Model for the Motion of an Uncontrolled Bicycle, MSc thesis, Delft University of Technology, April 2006. (pdf 4.9Mb)

May 29, 2005: Salome, 3 and 11/12 years old, can ride a bicycle! Thanks to the Cornell/UIUC method. In the last couple of years Andy Ruina, Jim Papadopoulos and Richard Klein gave me advice on how to teach your kid to ride a bike. With our first child, Samuel, I screwed up. I was still unaware of the method so I used trainer wheels. This was bad. It took him (and me) a long time to learn how to ride a bike.
With the next, Simon, things went better. I was now aware of the method and I threw away the trainer wheels. He learned quicker and now far more enjoys riding his bike.
Then with Salome I started out with a scooter. In this way she mastered balancing by leaning and steering without the real danger of falling over. Scooters are slow low riders. After some time she wanted to go faster. So I took the bicycle, dismounted the pedals and lowered the seat. She now could pedal the bike like a scooter and get used to the somewhat different dynamics of the bike opposed to the scooter. Shortly after this she asked for the pedals. To make her really connect to the bike we first went to shop and bought some nice pink and lilac paint (her choice), painted the bike and mounted the pedals. On Sunday we went out on the street walk and after an initial push of she rode of all by herself!

May 24, 2005:Today we did our first serious measurements with the instrumented bike in the big hall at the TUDelft sports centre. We flipped the front fork 180 degrees to get more trail and by such a larger stable forward speed range. Forward speed is now measured with the help of an old Avocet50 speedometer with a 10 magnet ring and coil pickup (thanks Jim!). The trainer wheels are to catch the bike.

Here is the video file(2.8 Mb) of a typical run, and here is the graph(102 KB) with the measured data (no processing, directly out of LabView). Note that we are actually measuring the stability of a steady large turn.

March 11, 2005, first measurement data from the instrumented bicycle. This work is done by my Delft MSc student Jodi Kooijman. This is the bike:

Note the laptop on the rear rack, the manual launcher is Jodi. The bike is instrumented with two rate gyros, one for the lean rate and one for the yaw rate. The steering angle is measured by a potentiometer. The speed is measured (temporarily) by a dynamo (originally meant for the head and tail light).  Data is collected via a USB data collecting box driven by LabVIEW and stored on the laptop. Here is an video file(2.8Mb) of the first test run, and here is the graph(23Kb) with the first measured data.

The final goal is to create a Robot Bicycle by adding a steering torque controller, to stabilize the bike, and a small elector motor to maintain forward speed.

(Spring 2003)

Results: Here are some of my preliminary results on the analysis of the dynamic behavior of a bicycle. For the analysis we used our software system SPACAR. This software has been developed at Delft University of Technology and is capable of doing dynamic analysis of flexible multibody systems. The method is based upon the Finite Element Method where the expressions for the generalized strains can be used as constraint equations to model partly rigid systems. Pure rolling is be modeled by zero generalized slips. The equations of motion are expressed in terms of independent generalized coordinates this to facilitate the numerical integration of the equations of motion.

The model of the bicycle is an ordinary Dutch city bike, like this one:

In a first analysis we look at the steady motion of the upright bicycle with a rigid rider, hands free, and pure rolling. These are the root-loci from the linearized equations of motion with the forward speed v as a parameter.

In order to get an idea about the stability of this upright motion look at the bottom-left figure, the forward speed v versus the Real part of the eigenvalues l. Now its customary to have the parameter, here the forward speed v, on the abscissa and the Real part of the eigenvalue  l on the ordinate. The stability diagram then looks like this:

We see that at a forward speed v of less then 0.9 m/s the bike simple falls over, 4 real eigenvalues l with 2 positive ones. We call this the capsize mode. At a speed of 0.9 m/s two real eigenvalues become identical and start forming a conjugated pair after which we have an unstable oscillatory motion, the so-called weave motion. This weave motion is an oscillatory motion in which the bicycle sways about the headed direction. At about 4.1 m/s this weave becomes stable. But then at about 5.7 m/s the previously stable capsize becomes marginally unstable. So at high speed, v>5.7 m/s, we have an unstable capsize mode but the timescale is so long, l=0.2 1/s or t=5 s, that in practice you can easily correct this mode. Now look at the bottom-right part of the previous figure, the 3D depiction of the root loci as a function of the forward speed, and identify the different modes at increasing speed v.

In a second full nonlinear analysis we look at the motion of the bike by means of a forward dynamic analysis of the perturbed upright motion. The perturbation is a small lateral velocity of 0.1 m/s for the whole bike to start the unstable motion, if present. The results are visualized by a number of VRML (Virtual Reality Modeling Language) files at different initial forward speeds. You can view these VRML files in the Internet Explorer with the plugin Freeware from Cosmo (does not work under Windows-XP) or with the viewer from  Cortona. For the bike to start moving you must click on the red frame of the bike. If you want to see the path of the rear and front wheel, then you can click on one of the wheels. You can change your viewpoint: look in the Viewpoint List located below left. A very nice one is the one called 'Camera', which is a moving camera with stable horizon (as if you were riding along on the rear passenger seat). 

Ok, so now for the VRML movies:
bike3v000.wrl at v=     0 m/s, unstable capsize.
bike3v175.wrl at v=1.75 m/s, unstable weave.
bike3v350.wrl at v=3.50 m/s, unstable weave.
bike3v368.wrl at v=3.68 m/s, stable weave in a curve! (a nice nonlinear result)
bike3v490.wrl at v=4.90 m/s, a stable weave.
bike3v630.wrl at v=6.30 m/s, an unstable capsize.

Note that obtaining a speed of 36 km/h and above is no problem in Ithaca, although I myself do not dare to go that fast.

Links

• Andy Ruina Bicycle Mechanics at Cornell

• JBike6 software

• Delft Bicycle Research

• Hugh Hunt's demystifying Gyrobike page

• Sheldon Brown

• Fixed Gear

• Charles Dauwe Fietsen en Fysica

• Dick Klein's Lose the training wheels