The Laws of Rolling

The man-made transport was born when the wheel was invented. Since then the study of rolling has started. In 1781 the problem of rolling was mathematically formulated by Ch. Coulomb who offered Coulomb’s law of rolling, but the science of rolling has been purely empirical still until recently. In this paper the exact laws of rolling are analytically derived in terms of elastic and geometric properties of rolling bodies and foundations. Using the mathematical theory of elasticity and the CH-rule, the rolling resistance coefficient is calculated in the cases of: (i) an elastic cylinder rolling over another elastic cylinder of another material, in particular, over an elastic half-space, and an elastic wheel rolling over the rail of another elastic material; (ii) an elastic ball rolling over another elastic ball of another material, particularly over an elastic half-space; (iii) an elastic torus rolling over an elastic half-space of another material, and (iv) a cylinder, or a ball, or a torus rolling over a tightly stretched membrane or over a thin elastic plate. Empirical results of the measurement of the rolling resistance coefficient gained earlier by the railroad and automobile engineers appeared to be in excellent agreement with the results of this analytical calculation based on the suggested rule of rolling. The effect of adhesion was also studied using the exemplary case of an elastic cylinder rolling over an elastic half-space.