Analysis of Rotational Motion based on Rolling Friction Torque

TL;DR

This paper investigates the real-world factors affecting the motion of rolling cylinders, emphasizing the role of rolling friction torque caused by surface imperfections and deformation, and proposes a differential equation approach to model these effects.

Contribution

It introduces a comprehensive model incorporating rolling friction torque due to surface micro bumps and deformation, improving the understanding of non-ideal rolling motion.

Findings

Rolling friction torque explains the discrepancy between ideal and real rolling motion.

Surface imperfections and deformation generate significant rolling friction torque.

The differential equations model aligns theoretical predictions with observed behavior.

Abstract

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.