Bicycle dynamics and its circular solution on a revolution surface

作     者：Jiaming Xiong Nannan Wang Caishan Liu Jiaming Xiong;Nannan Wang;Caishan Liu

作者机构：State Key Laboratory for Turbulence and Complex SystemsCollege of EngineeringPeking UniversityBeijing100871China 

出 版 物：《Acta Mechanica Sinica》 (力学学报（英文版）)

年 卷 期：2020年第36卷第1期

页      面：220-233页

核心收录：

学科分类：08[工学] 0802[工学-机械工程] 080101[工学-一般力学与力学基础] 0701[理学-数学] 0801[工学-力学（可授工学、理学学位）] 0702[理学-物理学] 

基　　金：National Natural Science Foundation of China(Grants 11932001 and 11702002). 

主　　题：Bicycle dynamics Nonholonomic constraints Stability Nonlinear DAEs Circular motion 

摘      要：In this paper,we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution.We employ symbolic manipulations to derive the contact constraint equations from an ordered process,and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations(DAEs),leaving nine coupled differential equations,six contact equations,two holonomic constraint equations and four nonholonomic constraint equations.We then present a complete description of hands-free circular motions,in which the time-dependent variables are eliminated through a rotation transformation.We find that the circular motions,similar to those of the bicycle moving on a horizontal surface,nominally fall into four solution families,characterized by four curves varying with the angular speed of the front wheel.Then,we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface.Furthermore,we directly linearize the nonlinear DAEs,from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates.The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position.We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families.Finally,based on the numerical simulation of the original nonlinear DAEs system,we show that the stable circular motion is not asymptotically stable.