Analysis of rotational motion of cylinder based on rolling friction torque

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 can be solved by taking account of vertical force. Two main origins of vertical force are described. 1) Both a cylinder and a floor are not perfect circle and not 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 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. Then rolling friction torque is also generated. Both are originated by the forces perpendicular to the floor. The rolling friction torque derives velocity decrease of the cylinder. To solve the simultaneous differential equations of rotation and translation is important. This method is useful for studying rolling systems such as trains and cars