Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, H-1111 Budapest, Hungary.
Phys Rev E. 2017 Sep;96(3-1):033005. doi: 10.1103/PhysRevE.96.033005. Epub 2017 Sep 19.
The motion of a rigid, spinning disk on a flat surface ends with a dissipation-induced finite-time singularity. The problem of finding the dominant energy absorption mechanism during the last phase of the motion generated a lively debate during the past two decades. Various candidates including air drag and different types of friction have been considered, nevertheless impacts have not been examined until now. We investigate the effect of impacts caused by geometric imperfections of the disk and of the underlying flat surface, through analyzing the dynamics of polygonal disks with unilateral point contacts. Similarly to earlier works, we determine the rate of energy absorption under the assumption of a regular pattern of motion analogous to precession-free motion of a rolling disk. In addition, we demonstrate that the asymptotic stability of this motion depends on parameters of the impact model. In the case of instability, the emerging irregular motion is investigated numerically. We conclude that there exists a range of model parameters (small radii of gyration or small restitution coefficients) in which absorption by impacts dominates all previously investigated mechanisms during the last phase of motion. Nevertheless the parameter values associated with a homogeneous disk on a hard surface are typically not in this range, hence the effect of impacts is in that case not dominant.
刚体旋转盘在平面上的运动最终会以耗散引起的有限时间奇点结束。在过去的二十年中，寻找运动最后阶段主要能量吸收机制的问题引发了激烈的争论。各种候选机制，包括空气阻力和不同类型的摩擦，都已经被考虑过了，但直到现在才对冲击进行了研究。我们通过分析具有单边点接触的多边形盘的动力学，研究了由盘和下垫面的几何不完整性引起的冲击的影响。与早期的工作类似，我们在类似于无进动滚动盘运动的规则运动模式的假设下，确定了能量吸收的速率。此外，我们证明了这种运动的渐近稳定性取决于冲击模型的参数。在不稳定的情况下，我们通过数值方法研究了新兴的不规则运动。我们得出结论，存在一个参数范围（小转动惯量或小恢复系数），在运动的最后阶段，冲击的吸收会超过之前所有研究过的机制。然而，与硬表面上的均匀圆盘相关的参数值通常不在这个范围内，因此在这种情况下，冲击的影响并不占主导地位。
