Study of Euler’s Belt Formula

Abstract

Euler’s belt formula for a fixed circular drum is well known and is an important formula for mechanical engineers. However, there is little book in which belt equation for a drum of any contour is discussed precisely. This study was carried out to confirm the validity of the Euler’s belt formula applied to a drum of any contour. The calculated result was verified to ascertain whether it satisfied the static equilibrium equations of the force and moment. The results perfectly satisfied the static equilibrium equations of the force. As for an equilibrium of the moment, it was found in the case of a belt wrapped around an elliptical drum that the moment from the belt tension was equal to the integrated moment caused by frictional force and normal force along wrapped length. Normal force distribution under a belt on a cylindrical surface was evaluated experimentally by a pressure sensitive film which was found to correspond qualitatively to the theory. As an example, an equation to calculate the normal pressure and tension of a belt wrapped on an elliptical drum was derived.

Keywords

• belt friction
• curvature
• contact pressure
• statics
• contour integral