Locating the angular position of measured milling forces to determine dual-mechanism global cutting constants

Abstract

Based on the spectrum analysis of the cutting forces, a simple and practical method to determine phase shift between the angular position of measured cutting forces and coordinate origin of the force model is presented. Comparing with the traditional method, the proposed method of determining phase shift does not need to select a specific reference point on the measured cutting force profiles, nor does it need to restrict the cutting conditions to single tooth engaging. Once the phase shift is determined, FFT (fast Fourier transform) is used to decompose the measured cutting force into the nominal cutting forces and the disturbing forces caused by the cutter runout geometry. Dual-mechanism global cutting constants can be extracted from the nominal cutting forces in one cutting test. By using the extracted cutting constants, online cutter runout can be identified from the disturbing force components at spindle frequency. To validate the proposed method, numerical simulations and milling experiments are carried out in this paper.

Appendices

Appendix 1

(CP_ar, CP_br,CP_cr,CP_dr) and (CP_ai, CP_bi,CP_ci,CP_di) are denoted as the real and imaginary parts of (CP_a, CP_b,CP_c,CP_d), and (CP_a, CP_b,CP_c,CP_d) can be expressed as

$$ \left[\begin{array}{c}{CP}_a\\ {}\begin{array}{c}{CP}_b\\ {}\begin{array}{c}{CP}_c\\ {}{CP}_d\end{array}\end{array}\end{array}\right]= CWD(N){\left.\left[\begin{array}{c}\frac{e^{- jN\theta}}{2\left(4-{N}^2\right)}\left(- jN\sin 2\theta -2\cos 2\theta \right)\\ {}\frac{e^{- jN\theta}}{2\left(4-{N}^2\right)}\left( jN\cos 2\theta -2\sin 2\theta \right)-\frac{e^{- jN\theta}}{2 jN}\\ {}\frac{e^{- jN\theta}}{1-{N}^2}\left( jN\cos \theta -\sin \theta \right)\\ {}\frac{e^{- jN\theta}}{1-{N}^2}\left( jN\sin \theta +\cos \theta \right)\end{array}\right]\right|}_{\theta_1}^{\theta_2} $$

(25)

where

$$ CWD(N)=\frac{2R\sin\ \frac{N{d}_a\tan\ \alpha }{2R}}{N\tan\ \alpha }{e}^{-j\frac{N{d}_a\tan\ \alpha }{2R}} $$

(26)

Appendix 2

According to the formulations presented in [16], the cutter runout geometry can be determined with

$$ \rho =\frac{\left|{A}_{ox}(1)\right|}{\sin \left(\frac{\pi }{N}\right)\frac{N}{2\pi }{d}_a{k}_{ts}\left|{P}_3(1)+{k}_{rs}{P}_4(1)\right|} $$

(27)

$$ \lambda =\frac{\pi }{2}-\frac{\pi }{N}-\angle {A}_{ox}(1)+\angle \left[{P}_3(1)+{k}_{rs}{P}_4(1)\right] $$

(28)

or

$$ \rho =\frac{\left|{A}_{oy}(1)\right|}{\sin \left(\frac{\pi }{N}\right)\frac{\mathrm{N}}{2\pi }{d}_a{k}_{ts}\left|-{k}_{rs}{P}_3(1)+{P}_4(1)\right|} $$

(29)

$$ \lambda =\frac{\pi }{2}-\frac{\pi }{N}-\angle {A}_{ox}(1)+\angle \left[-{k}_{rs}{P}_3(1)+{P}_4(1)\right] $$

(30)

where ρ and λ are the magnitude and angular location of cutter runout respectively. Equations (27) and (28) are used for up milling, while Eqs. (29) and (30) are applied for down milling. A_ox and A_oy are denoted as the Fourier coefficients of the measured milling force at the spindle frequency in the X and Y directions. P₃(1) and P₄(1) are expressed as.

$$ \left[\begin{array}{c}{P}_3(1)\\ {}{P}_4(1)\end{array}\right]={\left.\left[\begin{array}{c}\frac{e^{- jN\theta}}{1-{N}^2}\left( jN\cos \theta -\sin \theta \right)\\ {}\frac{e^{- jN\theta}}{1-{N}^2}\left( jN\sin \theta +\cos \theta \right)\end{array}\right]\right|}_{\theta_1}^{\theta_2} $$

(31)