Stability of Milling Process with Variable Spindle Speed Using Runge–Kutta-Based Complete Method

Shujie Lv,Yang Zhao,Jerzy Baranowski

doi:10.1155/2021/6672513

Abstract

The variable-spindle-speed (VSS) technique is effective in preventing regenerative chatter in milling processes. However, spindle-speed-modulation parameters should be deliberately selected to augment the material removal rate. Stability-prediction algorithms of stability predicting play an important role in this respect, as they allow the prediction of stability for all ranges of a given spindle speed. The increase in calculation time in variable-spindle-speed milling, which is caused by the modulation frequency, hinders its practical use in the workshop. In this paper, a Runge–Kutta-based complete discretization method (RKCDM) is presented to predict the stability of milling with variable spindle speeds, which is described by a set of delay differential equations (DDEs) with time-periodic coefficients and time-varying delay. The convergence and calculation efficiency are compared with those of the semidiscretization method (SDM) under different testing configurations and milling conditions. Results show that RKCDM is more accurate and saves at least 50% of the calculation time of SDM. The effects of modulation parameters on the stability of VSS milling are explored through stability lobe diagrams produced from RKCDM.

Highlights

In the milling process, chatter caused by vibrations between the cutter and workpiece hinders manufacturing production
We present a two-degree-of-freedom dynamic milling model expressed by delay differential equations (DDEs) and a Runge–Kutta-based complete discretization method for efficient analysis of VSS milling
Convergence and Computation Efficiency. e convergence rates of Runge–Kutta-based complete discretization method (RKCDM) and semidiscretization method (SDM) with different conditions are compared in Figure 4. e discrete number of time delays τ (τ(t) for VSS milling) is set as m for constant-speed milling and M for VSS milling and is determined according to (17), and the maximal module of the eigenvalues of the transition matrix Φ is set as λ

Summary

Introduction

Chatter caused by vibrations between the cutter and workpiece hinders manufacturing production. Researchers have proposed numerous methods to predict machining stability. Because regenerative chatter is a result of improper phase differences between undulate surfaces caused by successive cuts by a tool, alternative methods usually vary the time lag between cutter teeth. Altintas et al [14], Olgac and Sipahi [15], Jin et al [16], and Sims et al [17] all demonstrated that variable-pitch cutters can avoid or significantly suppress chatter. Niu et al [29] unified different periodic time-variant spindle-speed-modulation schemes in a unique framework, derived and calculated the timevarying delay based on the fourth-order Runge–Kutta method, and calculated and compared the stability of VSS milling with different modulation schemes using variablestep numerical integration. We discuss the VSS modulation effects on the stability of milling with the aid of maps and contour plots of axial depth of cut (ADOC) calculated by RKCDM

Modeling of VSS Process

Stability of Milling with VSS

Simulation and Discussion

20 Chatter

Findings

Conclusion