Fourier Analysis of Milling Force for General Helical Cutters via Spacetime Convolution, Part 2: Applications for Common Cutters and Model Validation

Zheng Cm,Junz Wang Jj

doi:10.4172/2168-9873.1000195

Abstract

Part 2 of this study illustrates the applications of the frequency domain force model put forth in Part 1 for three common helical cutters: the square, taper, and ball end mills. The respective geometric and boundary functions required for the evaluation of the force spectra are derived by applying differential geometry to these three types of cutters including cutters of constant helix angle and constant helix lead. By virtue of the explict expression of Fourier coefficients of milling force, the differences between cutting forces generated by two cutters with constant helix angle and constant helix lead can be described quantitatively. In slot (or half slot) milling for the taper end mills with a constant helix angle and constant helix lead, the strategy for selecting axial depths of cut to reduce force pulsation is presented respectively. Also derived are the specific expressions for the average forces of these three helical cutters in common cutting configurations. Moreover, as an inverse application, a linear equation is formulated for the identification of six shearing and ploughing cutting constants from the measured average cutting forces for a general helical cutter. The frequency domain force model and the identification of the cutting constants are finally demonstrated and validated through experiments with all three types of milling cutters.

Highlights

The convolution force model and the force spectra characteristics of a general helical milling cutter has been presented and analyzed in Part 1 of this paper
The frequency spectra of the milling forces are shown to be characterized by the geometric functions of the helical cutting edge and by the cutting boundary functions which are determined by the axial and radial cutting depths
The task of establishing the specific milling force models for any type of helical cutter is reduced to finding the analytic expressions for the line geometry of its helical cutting edge, as well as the expressions for the cutting boundaries including the entry and exit angle and the limits of the radial angle immersion

Summary

Introduction

The convolution force model and the force spectra characteristics of a general helical milling cutter has been presented and analyzed in Part 1 of this paper. The required geometric and cutting boundary functions will be derived for the cylindrical, taper and ball end mill to complete the force model for each respective cutter. Given the boundary condition of ψ=0 at β=0, ψ can be shown to be Figure 2: Variation of β with respect to cutter height, h/Ro, for two types of taper end mills; ‘+’: constant helix lead and ‘*’: constant helix angle. (a) For 0 ≤ h ≤ da, this is a slot milling process with the axial depth, da, determined by the ball radius R0 and the side step dr0 with da1 =R0 − Within this cutting region, the entry and exit angles are θ1=0 and θ2=π, and the axial cutting range starts from 0 and ends at β2, which is determined by da through Eq (22) or Eq (27).

Rο ψa 0 cscψ dh dψ

Conclusions