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Rattlebacks for the rest of us

机译：我们其他人的倾斜

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摘要

Rattlebacks are semi-ellipsoidal tops that have a preferred direction of spin (i.e., a spin-bias). If spun in one direction, the rattleback will exhibit seemingly stable rotary motion. If spun in the other direction, the rattleback will being to wobble and subsequently reverse its spin direction. This behavior is often counter-intuitive for physics and engineering students when they first encounter a rattleback, because it appears to oppose the laws of conservation of momenta, thus this simple toy can be a motivator for further study. This paper develops an accurate dynamic model of a rattleback, in a manner accessible to undergraduate physics and engineering students, using concepts from introductory dynamics, calculus, and numerical methods classes. Starting with a simpler, 2D planar rocking semi-ellipse example, we discuss all necessary steps in detail, including computing the mass moment of inertia tensor, choice of reference frame, conservation of momenta equations, application of kinematic constraints, and accounting for slip via a Coulomb friction model. Basic numerical techniques like numerical derivatives and time-stepping algorithms are employed to predict the temporal response of the system. We also present a simple and intuitive explanation for the mechanism causing the spin-bias of the rattleback. It requires no equations and only a basic understanding of particle dynamics, and thus can be used to explain the intriguing rattleback behavior to students at any level of expertise.
机译：倾斜是具有优选方向的旋转的半椭圆形顶部（即，旋转偏压）。如果在一个方向上旋转，则倾斜将表现出看似稳定的旋转运动。如果沿另一个方向旋转，则响铃将被摆动，然后扭转其旋转方向。当他们第一次遇到一个倾斜时，这种行为往往是对物理和工程学生的反向直观，因为它似乎反对动态的保护规律，因此这种简单的玩具可以是进一步研究的动机。本文以介绍性动态，微积分和数值方法等级的方式，开发了一种准确的倾斜的动态模型，以便使用来自介绍性动态，微积分和数值方法的概念。从简单，2D平面摇摆半椭圆示例开始，我们详细讨论了所有必要的步骤，包括计算惯性张量的质量矩，参考框架的选择，动态方程的保护，运动限制的应用，以及滑动的算法库仑摩擦模型。使用类似数值衍生物和时间阶梯算法的基本数值技术来预测系统的时间响应。我们还为导致倾斜偏向的机制呈现简单而直观的解释。它不需要方程，只有对粒子动态的基本了解，因此可以用于向学生解释任何专业知识水平的兴趣响铃行为。

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