We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally-symmetric ideally-hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal ideal knife-edge rolling point-contact with the level ground. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a 7-dimensional accessible configuration space and three velocity degrees of freedom parameterized by rates of frame lean, steer angle and rear-wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two non-linear dynamics simulations. In the century-old literature several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also served as a check for general-purpose dynamics programs. For the benchmark bicycles we accurately calculate the eigenvalues (the roots of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-old result that this conservative system can have asymptotic stability
Monday, June 21, 2010
Think Riding a Bike Is Easy? Think Again.
An article in today's Daily Telegraph (here) notes how recent scientific studies have demonstrated the complexity of the seemingly simple act of riding a bicycle. In a 2007 paper (here), J.P. Meijaard and co-authors set out the linearized dynamic equations for stable bicycle riding. Here is the abstract from their paper: