非弹性转动引力台球的动力学
Dynamics of an Inelastic Gravitational Billiard with Rotation
摘要：
研究非线性动力学公式的开创性物理模型是台球。引力台球为它们的研究提供了一个实验可及的舞台。我们提出了一个数学模型，捕捉描述一个现实的台球任意边界的运动所需的基本动力学，其中包括旋转效应和其他形式的能量耗散。该模型的模拟被施加到抛物线，楔形和双曲线台球正弦驱动。仿真结果表明，抛物线具有稳定的，周期性的运动，而楔形和双曲线（在高驱动频率）出现混沌。在低驱动频率下，双曲线的行为类似于抛物线;即，有规律的运动。模型的预测和以前发表的实验数据之间进行直接比较。在模型中采用的恢复恢复系数的值与实验数据的所有边界形状的调查结果吻合得很好。它表明，数据可以成功地建模与一组简单的参数，而不需要假设的外来能源依赖。
英文摘要：
The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. Gravitational billiards provide an experimentally accessible arena for their investigation. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola (at high driving frequencies) appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola; i.e., has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The value of the coefficient of restitution employed in the model resulted in good agreement with the experimental data for all boundary shapes investigated. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.
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