Second-Degree Approximation of Orowan's ω-Function

Abstract

“Orowan's ω-function" in the theory of rolling is defined by a definite integral, and the general method of computing it is a time-consuming numerical integration. An alternative method is “zero-degree approximation" in the gradient of the roll surface, because, according to Orowan, the error in the ω-function is less than 0.01. However, this is inconsistent with the aim of his theory to “avoid all mathematical approximations," and the resulting error in overall rolling analysis has not yet been examined. In the present paper, we elaborate on the effects of the approximation by comparing it with numerical integration and second-degree approximation. The error of the latter in the ω-function is 0.0002 at the largest. Since errors in the roll force, torque, and forward slip are much smaller than the error in the ω-function itself, the error could be regarded as negligible even with zero-degree approximation in practical situations. The two equations of second-degree approximation provide reference values in place of numerical integration from the viewpoint of accuracy, and the computing time is sufficiently short for online use.