基于滚动摩擦力矩的旋转运动分析
Analysis of Rotational Motion based on Rolling Friction Torque
摘要：
对于圆柱体在水平地板上的无滑移滚动问题，一般的教科书都把圆柱体和地板当作刚体来处理。当忽略空气阻力时，运动方程有一个常速度解。然而，在现实世界中，永久运动不会发生。仅用水平力不能解决这个困难，因为与平移方向相反的水平力会增加绕中心旋转的角速度。因此，需要审查其他机制。造成这一结果的主要原因有两个。1)圆柱体和地板都不是正圆和正平面，而是具有不平坦的表面。表面上的微凸块在垂直于地板的方向上产生小的碰撞。碰撞产生围绕中心的滚动摩擦扭矩。2)强力作用在变形的接触部分上。高速变形对应力应变关系产生历史效应，因为接触部分的压缩波以声速向外扩散。因此，还产生滚动摩擦扭矩。这两个力矩都是由垂直于地板的力引起的。滚动摩擦力矩通过求解转动和平动的微分方程组，消除了教科书上的计算结果与实际的差异。这种方法对于研究诸如火车和汽车之类的滚动系统是有用的。
英文摘要：
In the problem of cylinder rolling without slipping on a horizontal floor, both the cylinder and floor are generally treated as rigid bodies in normal textbooks. When the air resistance is ignored, the equation of motion has a solution with a constant velocity. However, in the real world, permanent motion does not occur. The difficulty cannot be solved only by the horizontal force, because a horizontal force opposite the translational direction increases the angular velocity of rotation around the center. Therefore other mechanisms need to be examined. There are two main reasons for this result. 1) Both a cylinder and a floor are not perfect circle and perfect plane, but have uneven surfaces. The micro bumps on the surface yield small collisions in the direction perpendicular to the floor. The collisions generate a rolling friction torque around the center. 2) A strong force acts on the contact part which is deformed. The high-speed deformation produces a history effect on the relationship between stress and strain, because the compressed wave in the contact part diffuses to the outside at the speed of sound. Therefore a rolling friction torque is also generated. Both torques are caused by forces perpendicular to the floor. The rolling friction torque eliminates the discrepancy between the textbook results and reality by solving the simultaneous differential equations of rotation and translation. This method is useful for studying rolling systems such as trains and cars.
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