A general order full-discretization algorithm for chatter avoidance in milling

Chigbogu Godwin Ozoegwu

doi:10.1177/1687814018773811

Abstract

Based on the full-discretization method, this work presents a generalized monodromy matrix as an exact function of the order of polynomial approximation of the milling state for chatter avoidance algorithm. In other words, the computational process is made smarter since the usual derivation of the monodromy matrices on order-by-order basis – a huge analytical involvement that rapidly gets heavier with a rise in the order of approximation – is bypassed. This is the highest possible level of generalization that seems to be the first of its kind among the time-domain methods as the known generalizations are limited to the interpolating/approximating polynomial of the milling state. It then became convenient in this work to study the stability of milling process up to the tenth order. More reliable methods of the rate of convergence analysis were suggested and utilized in consolidating the known result that the best accuracy of the full-discretization method lies with the third and fourth order. It is seen from numerical convergence analyses that, although accuracy most often decreases with rising order beyond the third-order methods, the trend did not persist with a continued rise in order.

Highlights

Regenerative vibration is a violent and noisy chatter of machine tools that limits material removal rate
A wavelet-based approach has been proposed for the stability analysis of periodic delay differential equations with discrete delay and application of the method demonstrated with the milling process by Ding et al.[61]
The usual derivation of monodromy matrices of various full-discretization method (FDM) on order-by-order basis is bypassed

Summary

Introduction

Regenerative vibration is a violent and noisy chatter of machine tools that limits material removal rate. Fourier series expansion of the periodic coefficients It is called the multi-frequency solution method.[13] Insperger and colleagues[22,23,24] introduced a time-domain approach called the semi-discretization method which was later improved for computational efficiency and accuracy by Henninger and Eberhard.[25] The semidiscretization method basically involves discretizing the delayed term while leaving the undelayed terms undiscretized and approximating the periodic coefficient matrices as piecewise constant functions allowing the formation of ordinary differential equations from which a monodromy matrix is constructed. An improved complete discretization method has been proposed very recently.[59] A generalized Runge–Kutta method based on Volterra integral equation of the second kind has been proposed by Niu et al.[60] A wavelet-based approach has been proposed for the stability analysis of periodic delay differential equations with discrete delay and application of the method demonstrated with the milling process by Ding et al.[61].

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Findings

Conclusion