Acta Scientiarum Naturalium Universitatis Pekinensis ›› 2016, Vol. 52 ›› Issue (4): 713-716.DOI: 10.13209/j.0479-8023.2016.084

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On Nonholonomic Constraints about the Pure Rolling of Point Contact

ZHAO Zhen1, LIU Caishan2, LU Jiandong3   

  1. 1. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191
    2. School of Engineering, Peking University, Beijing 100871
    3. Beijing Institution of Communication, Beijing 102600
  • Received:2015-11-23 Revised:2016-02-13 Online:2016-07-20 Published:2016-07-20
  • Contact: ZHAO Zhen, E-mail: bhzhaozhen(at)


赵振1, 刘才山2, 鲁建东3   

  1. 1. 北京航空航天大学航空科学与工程学院, 北京 100191
    2. 北京大学工学院, 北京 100871
    3. 北京印刷学院, 北京 102600
  • 通讯作者: 赵振, E-mail: bhzhaozhen(at)
  • 基金资助:


Nonholomonic constraints are involved for 3D point-contact problems. The virtual displacements restricted by the constraints are usually given by Appell-Chetaev’s rule. It has not been very clear of the geometric meaning in configuration space for Appell-Chetaev’s rule of nonholonomic constraints. The authors investigate point contact with pure rolling by two rigid bodies in a multibody system to discover its geometric sense. First, the sufficient and necessary conditions of point contact are given. A ball-plane system is presented to demonstrate the validation of the conditions by deducing the system’s obvious contact constraint originating from them. Two geometric restrictions for pure rolling are obtained by the nonholonomic constraints of pure rolling as well as the contact constraint in velocity level. It proves that the virtual displacements of the two restrictions are same as those of the constraints of point contact with pure rolling obtained by Appell-Chetaev’s rule. So, it is thought that the constraints of pure rolling are constructed by the two geometric restrictions.

Key words: point contact, constraint equation, nonholonomic constraint, Appell-Chetaev’s rule


空间物体间点接触纯滚动的相互作用一般包含非完整约束, 而约束所限制的虚位移通常采用速度水平的 Appell-Chetaev 条件给出, 因此点接触纯滚动约束对应的几何意义并不直观。作者从多体系统中两物体沿其轮廓面做点接触纯滚动的问题出发, 探讨此类非完整约束对应的几何意义。首先, 提出两物体保持点接触的充分必要条件, 并以球–面系统为例推导接触时的约束方程。然后, 由空间物体点接触纯滚动的几何和速度约束, 推导此时满足的两种几何限制条件。结果表明, 采用两种几何条件获得的虚位移与速度约束的Appell-Chetaev 条件相同。因此, 可以认为保持点接触纯滚动的空间两物体在位形空间受到两种几何条件的约束限制。

关键词: 点接触, 约束方程, 非完整约束, Appell-Chetaev 条件

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