关键词: 刚体力学, 双接触点, 滑动-滚动转换, 数值计算, 非线性

Abstract: A bolt-shaped object released from an inclined plane at rest could exhibit intriguing nonlinear phenomena, wherein the twist or rotation amplitude of a rigid body perpendicular to the inclined plane increases upon surpassing a critical condition. When such a rigid body is in contact with two rigid surfaces at two points in the presence of dry friction, four motion modes occur due to the rolling or slipping state at each contact point. The constraints and forces under each mode differ, leading to differences in the dynamical equations. A MATLAB program dynamically switches to the corresponding mode based on the conditions between slipping and rolling states for numerical calculations. By exploring the initial parameters within a certain range, we identify the critical points that determine the increase or decrease of amplitude during release. These critical points correlate with the inclination angle of the inclined plane, semiapex angle of the bolt, and the twist angle at initial release. We utilize a quadric surface to fit the critical points and derive the empirical function of the critical condition. The experimental results demonstrate good agreement with theoretical calculations. This example involving a bolt serves as a practical case study for teaching the analysis of slipping-rolling transitions of a rigid body under two-point contact constraints on a plane in the presence of dry friction using numerical methods.

Key words: rigid body mechanics, dual contact points, slipping-rolling transitions, numerical calculation, nonlinear