Analysis of the general tool life function of cutting tools by application of the catastrophe theory

Abstract

Production technology planning requires information on tool life T and its relation to cutting speed v. These technological parameters fundamentally affect the economics of the operation of the entire production system. The aim of the present study is to investigate the v-T relationship under technological conditions where this relationship cannot be described by the methods used so far – typically the Taylor formula. As the Taylor formula often cannot be linearized on an log-log scale, a general tool life function has been developed for describing a v-T function with a convex-concave part. The function was validated by the boring of hardened ball bearing rings with the technological data D = 45, 75 and 100 mm, ap = 0.14 mm, fz = 0.005 mm/rev. The general v-T tool life function has been transformed into a quadratic polynomial known in catastrophe theory as cusp catastrophe and allows the description of abrupt changes in the system under study. Using catastrophe theory, an analogy is established between the general tool life function and the cusp catastrophe, allowing topological mapping of the general v-T function. Results were verified by machinability tests in the turning of C35 and C60 conventional and specially deoxidized C-steels during steelmaking. Cutting tests were performed on specimens with a diameter of Dexp = 76 mm and a length of L = 700 mm at a speed of v = 100, 150, 200 m/min and a carbide tool of category P20. It was found that in the convex-concave section of this function, 2–3 cutting speeds can be selected for a given tool life, which is advantageous for harmonizing tool changes in multi-operation technology.