In 2007, Meijaard, et al. [1] presented the canonical linearized equations of motion for the Whipple bicycle model along with test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. This paper describes benchmarking three other implementations of bike equations of motion: the linearized equations for bicycles written by Papadopoulos and Schwab [2] in JBike6, the non-linear equations for bicycles outlined by Schwab [3] and implemented in MATLAB as a Cornell University class project, and the non-linear equations for motorcycles implemented in FastBike from the Motorcycle Dynamics Research Group at the University of Padua. [4] Some implementations are easier to benchmark than others. For example, JBike6 is designed to produce eigenvalues and easily exposes the coefficients of its linearized equations of motion. At the other extreme, the class project non-linear equations were not originally intended to generate eigenvalues and are implemented in a single 48×48 matrix. Finally, while FastBike does generate eigenvalues, its equations of motion incorporate tire and frame compliance, which cannot be completely disabled. Instead, the tire stiffness parameters must be increased, but not so much as to cause convergence errors in FastBike. In the end, all three implementations generate eigenvalues that match the published benchmark values to varying degrees. JBike6 comes the closest, with agreement of 12 digits or more. The class project is second, with agreement of 12 digits for most forward speeds, but with a loss of measurable agreement near the capsize speed due to a peak in the eigenvalue condition number. Unfortunately, FastBike is limited at this time to exporting eigenvalues with no more than two decimal places, and so agreement can only be found to ±0.005.
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ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
August 28–31, 2011
Washington, DC, USA
Conference Sponsors:
- Design Engineering Division and Computers and Information in Engineering Division
ISBN:
978-0-7918-5481-5
PROCEEDINGS PAPER
Benchmarking Bicycle and Motorcycle Equations of Motion Available to Purchase
Andrew E. Dressel,
Andrew E. Dressel
University of Wisconsin - Milwaukee, Milwaukee, WI
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Adeeb Rahman
Adeeb Rahman
University of Wisconsin - Milwaukee, Milwaukee, WI
Search for other works by this author on:
Andrew E. Dressel
University of Wisconsin - Milwaukee, Milwaukee, WI
Adeeb Rahman
University of Wisconsin - Milwaukee, Milwaukee, WI
Paper No:
DETC2011-47344, pp. 561-569; 9 pages
Published Online:
June 12, 2012
Citation
Dressel, AE, & Rahman, A. "Benchmarking Bicycle and Motorcycle Equations of Motion." Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A and B. Washington, DC, USA. August 28–31, 2011. pp. 561-569. ASME. https://doi.org/10.1115/DETC2011-47344
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