欧拉圆盘不同能量耗散机理之间的关联  

Relations among different energy dissipations of Euler disk

作　　者：朱攀丞 边庆勇 李晋斌[1] 

机构地区：[1]南京航空航天大学理学院,南京210016 [2]南京航空航天大学航空宇航学院,南京210016

出　　处：《物理学报》2015年第17期192-198,共7页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11104143);大学生创新训练计划项目(批准号:2015CX00808)资助的课题~~

摘　　要：本文研究了欧拉圆盘运动过程中盘厚度以及盘面与水平面夹角α两因素对能量耗散的影响.得出圆盘厚度与直径之比x对能量变化中各项因子的影响:x很小时,质心在竖直方向上的动能变化和重力势能变化是系统能量耗散的主要因素;当x>0.4142时,圆盘绕与之平行的轴的转动动能变化成为主要因素,并给出圆盘厚度可忽略的条件.模拟了滚动摩擦、空气黏滞等不同能量耗散方式与x,α的关系,导出各种耗散方式在圆盘运动的过程中的转变规律,并指出x=0.1733,α>18°时能量耗散形式为纯滚动摩擦,这修正了文献[26]结论.The energy dissipation of a disc spinning on a horizontal plane is studied, as the angleαof the coin made with the horizontal plane decreases, while the angular velocity Ω of the point of contact increases. Effect of the ratio x between the thickness and diameter of an Euler disc and theαon the energy dissipation is studied. We find, by using numerical simulation, that when x is small enough, the lose of the kinetic energy and the gravitational potential energy of the mass center is dominant in energy dissipations; when x〉0.4142, the rotational kinetic energy dissipation of the disc around the axis that is parallel to the disc surface, is the leading factor. The requirements in which thickness can be neglected are also obtained, and they can give some hints to the relevant theories and experiments. Our results show that whenα〉10° and b/a〈0.16 the thickness of the disc can be ignored;this conclusion fits Petrie’s experimental data very well. We also discuss the main energy dissipation distributed among different forms： variation of rolling friction and viscous shear of the air with x andα, also show their transition in the process of the motion. Furthermore, we find that the pure rolling friction is the unique dissipation as x=0.1733 andα〉18°, which improves the results obtained before. We speculate that the dominant dissipation is the gliding friction in the final stage of the motion, because when the disc is motionless, one face of the disc lies absolutely in contact with the horizontal surface just before the disc halts. One can assume that they are in contact completely but the disc does not halt, thus axis 1 and axis Z are almost in the same direction. In this case, the energy dissipation of the Euler disc is due to the gliding friction. To some extent, this accounts for the disc final halt.

关 键 词：欧拉圆盘 能量耗散 圆盘厚度 夹角 

分 类 号：O313[理学—一般力学与力学基础]