Symmetry and relative equilibria of a bicycle system moving on a revolution surface

Finding the relative equilibria and analyzing their stabilities are of great significance to revealing the intrinsic properties of mechanical systems and developing effective controller. In this paper, we study the symmetry and relative equilibria of a bicycle system moving on a revolution surface. We note that the symmetry group of the bicycle is a three-dimensional Abelian Lie group, and the rolling condition of the two wheels produces four time-invariant first-order linear constraints to the bicycle system. Therefore, we can classify the bicycle dynamics as a general Voronets system whose Lagrangian and constraint distribution are kept invariant under the action of the symmetry group. Applying the Voronets equations to the bicycle dynamics, we obtain a seven-dimensional reduced dynamic system on the reduced constraint space. This system takes time-reversal and lateral symmetries, and has two kinds of relative equilibria: the static equilibria and the dynamic equilibria. Further theoretical analysis shows that both kinds of relative equilibria form one-parameter solution families, and their Jacobian matrices take some specific properties. We then show that a static equilibrium cannot be stable unless all the eigenvalues of the Jacobian matrix are located at the imaginary axis of the complex plane. The stability of the dynamic equilibria is studied by limiting the reduced dynamic system to an invariant manifold, which is established based on the conservation of energy of the system. We prove in a strict mathematical sense that the dynamic equilibria may be Lyapunov stable, but cannot be asymptotically stable. Finally, we employ symbolic computation to carry out numerical simulations in conjunction with the benchmark parameters of a Whipple bicycle. How the revolution surface affects the relative equilibria and their stabilities is then investigated through our numerical simulations.

The influence of bicycle lean on maximal power output during sprint cycling

Competitive cyclists typically sprint out of the saddle and alternately lean their bikes from side-to-side, away from the downstroke pedal. Yet, there is no direct evidence as to whether leaning the bicycle, or conversely, attempting to minimize lean, affects maximal power output during sprint cycling. Here, we modified a cycling ergometer so that it can lean from side-to-side but can also be locked to prevent lean. This modified ergometer made it possible to compare maximal 1-s crank power during non-seated, sprint cycling under three different conditions: locked (no lean), ad libitum lean, and minimal lean. We found that leaning the ergometer ad libitum did not enhance maximal 1-s crank power compared to a locked condition. However, trying to minimize ergometer lean decreased maximal 1-s crank power by an average of 5% compared to leaning ad libitum. IMU-derived measures of ergometer lean provided evidence that, on average, during the ad-lib condition, subjects leaned the ergometer away from the downstroke pedal as in overground cycling. This suggests that our ergometer provides a suitable emulation of lateral bicycle dynamics. Overall, we find that leaning a cycle ergometer ad libitum does not enhance maximal power output, and conversely, trying to minimize lean impairs maximal power output.

Symmetry and Relative Equilibria of a Bicycle System

In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients of these equations appeared to depend on the lean and steer angles only, and in one such equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would always vanish. These properties indeed arise from the symmetry of the bicycle system. From the point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian and constraint distribution invariant. We analyze the dimension relationship between the space of admissible velocities and the tangent space to the group orbit, and then employ the reduced nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local representative of the reduced dynamic system on the reduced constraint space, whose relative equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several families of one-parameter solutions. Finally, we prove that these relative equilibria are Lyapunov (but not asymptotically) stable under certain conditions. However, an isolated asymptotically stable equilibrium may be achieved by restricting the system to an invariant manifold, which is the level set of the reduced constrained energy.

Bicycling Simulator Calibration: Proposed Framework

Bicycling simulation allows for the low-risk experimental study of human factors within transportation environments. A cyclist pedals on a stationary bike trainer, which is instrumented to detect the speed of the wheel and the steering angle of the bicycle. This paper proposes a speed calibration procedure to increase the validity of the simulator results, by using an independent bicycle computer for comparing the simulator speed. The speed ratio, defined as the simulator speed divided by the bike computer speed, approaches one when the simulator is properly calibrated. The effect of tire pressure was analyzed by examining the speed ratio for various tire pressures. The optimal tire pressure was selected as the one that provided a speed ratio closest to one when all other factors were held constant. In the final calibration, a gain factor was used to modify the simulator speed calculation that was embedded in the simulator’s bicycle dynamics model. Following calibration, the final simulation speed was within 99.5% of the bicycle computer speed, indicating that the physical speed of the wheel was accurately modeled in the simulation environment. The calibration procedure uses general equations and techniques that can be applied to other bicycling simulators to calibrate speed measurements and improve the consistency of experimental data worldwide.

Vertical and Longitudinal Characteristics of a Bicycle Tire

ABSTRACT Electric bicycles have undergone a real boom in recent years and play an important role in the area of sustainable mobility. In addition to assisting the rider while accelerating the bicycle, the available electrical energy also offers the possibility to deploy safety systems to reduce the risk of accidents. For instance, active safety systems could help to avoid two major critical braking situations for single-track vehicles: front wheel lockup and nose-over (i.e., falling over the handlebars). An essential prerequisite for the development of such systems is a thorough understanding of tire effects on bicycle dynamics. To date, there are only very few scientific studies concerning bicycle tire characteristics. Thus, test runs on an inner drum tire test bench have been performed to measure vertical and longitudinal characteristics of a typical trekking bike tire. This article presents the main findings such as vertical stiffness and contact patch geometry depending on wheel load and inflation pressure as well as characteristic curves of the longitudinal force depending on slip with variation in road condition, wheel load, speed, and inflation pressure. Based on these valuable insights, further improvements are proposed, and an outlook on the next research steps is given.