Improving the self-stability of a bicycle via spectral abscissa minimisation #
10:50 Tuesday in 2Q42.
Part of the Mathematical modelling in sport session.
The bicycle is easy to ride, but surprisingly difficult to model. Refinement of the mathematical model of a bicycle has continued over the last 150 years with contributions from Rankine, Boussinesq, Whipple, Klein, Sommerfeld, Appel, Synge and many others. The canonical, nowadays commonly-accepted model goes back to the 1899 work by Whipple . The Whipple bike is a system consisting of four rigid bodies with knife-edge wheels making it non-holonomic, i.e., requiring for its description more configuration coordinates than the number of its admissible velocities. A fundamental empirical property of real bicycles is their self-stability without any control at a sufficiently high speed . It is believed that deeper understanding of the passive stabilization can provide new principles for the design of more safe and rideable bicycles, including compact and foldable models. However, the theoretical explanation of the self-stability has been highly debated throughout the history of bicycle dynamics to such an extent that a news feature article in Nature described this as “the bicycle problem that nearly broke mathematics” . The reason as to why “simple questions about self-stabilization of bicycles do not have straightforward answers”  lies in the symbolical complexity of the Whipple model that contains seven degrees of freedom and depends on 25 physical and design parameters. In recent numerical simulations, self-stabilization has been observed for some benchmark designs of the Whipple bike . These results suggested further simplification of the model yielding a reduced model of a bicycle with vanishing radii of the wheels, known as the two-mass-skate (TMS) bicycle . Despite the self-stable TMS bike having been successfully realized in recent laboratory experiments, its self-stability still awaits a theoretical explanation.
In this work we find new scaling laws for the two-mass-skate (TMS) bicycle that lead to the design of self-stable machines. These scaling laws optimize the stability of the bicycle by several different criteria simultaneously. The matching of the theoretical scaling laws to the parameters of the TMS bike’s realization demonstrates that the trial-and-error engineering of the bikes selects the most robustly stable species and thus empirically optimizes the bike stability. We have found the optimal solutions directly from the analysis of the sets of exceptional points of the TMS bike model with the help of a general result on the global minimization of the spectral abscissa at an exceptional point of the highest possible order [6, 7]. We stress that all previous results on the self-stability of bicycles even in the linear case have been obtained numerically.
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