It has been known that bicycle stability is closely linked to a pair of ordinary differential equations (ODEs). The linearization technique used to derive these ODEs, nevertheless, has yet to be thoroughly examined. For this purpose, we conduct an analysis of the dynamics of the Whipple bicycle, starting with the contact kinematics, using the Gibbs–Appell method. The effort results in a complete nonlinear model with minimal dimensions, from which equilibrium points during the bicycle’s straight and circular motions can be determined. The model can be linearized around these points via a perturbation analysis under no additional assumptions. Given the non-hyperbolic nature of the equilibria, we apply the center manifold theorem to analyze their stability, offering a rigorous derivation of the (well-know) exponential stability of the bicycle in its leaning and steering motions. Finally, a dimensionless index is defined to characterize the influence of physical parameters on the bicycle stability.

中文翻译：

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Whipple自行车动力学的稳定性分析

众所周知，自行车的稳定性与一对常微分方程（ODE）紧密相关。但是，用于导出这些ODE的线性化技术尚待彻底研究。为此，我们使用吉布斯-阿佩尔（Gibbs-Appell）方法对Whipple自行车的动力学进行了分析，从接触运动学开始。通过这种努力，可以得出具有最小尺寸的完整非线性模型，由此可以确定自行车直线运动和圆周运动时的平衡点。无需其他假设，即可通过扰动分析将模型围绕这些点线性化。鉴于均衡的非双曲性质，我们应用中心流形定理来分析其稳定性，在倾斜和转向运动中提供了自行车（众所周知的）指数稳定性的严格推导。最后，定义了无量纲指标来表征物理参数对自行车稳定性的影响。