χ-Theory in Gear Geometry

Abstract

In the design of helical gears for pumping action, hydraulic performance is defined by flow rate multiplied by effective work ratio. This is the most important factor in determining the optimum performance. In previous papers, we considered a mathematical model derived mostly from geometrical intuition; however, analytical approach was not completed. In this paper, the optimum design of gear surfaces is thoroughly investigated. It is shown that the optimization of hydraulic performance can be achieved with a particular choice of the parameter χ(y) in the differential equation of meshing gears with parallel axes, where χ(y) is a function characterizing the tooth profile and χ(y) = dφ/d (x/R), φ is rotational angle of rotors, R is the common radius of pitch circles and x, y are coordinates in the plane transverse to gear axes.