Runge–Kutta methods for a semi-analytical prediction of milling stability

On the basis of Runge–Kutta methods, this paper proposes two semi-analytical methods to predict the stability of milling processes taking a regenerative effect into account. The corresponding dynamics model is concluded as a coefficient-varying periodic differential equation with a single time delay. Floquet theory is adopted to predict the stability of machining operations by judging the eigenvalues of the state transition matrix. This paper firstly presents the classical fourth-order Runge–Kutta method (CRKM) to solve the differential equation. Through numerical tests and analysis, the convergence rate and the approximation order of the CRKM is not as high as expected, and only small discrete time step size could ensure high computation accuracy. In order to improve the performance of the CRKM, this paper then presents a generalized form of the Runge–Kutta method (GRKM) based on the Volterra integral equation of the second kind. The GRKM has higher convergence rate and computation accuracy, validated by comparisons with the semi-discretization method, etc. Stability lobes of a single degree of freedom (DOF) milling model and a two DOF milling model with the GRKM are provided in this paper.

[1]  Gábor Stépán,et al.  Updated semi‐discretization method for periodic delay‐differential equations with discrete delay , 2004 .

[2]  Gábor Stépán,et al.  On the higher-order semi-discretizations for periodic delayed systems , 2008 .

[3]  Marian Wiercigroch,et al.  Sources of nonlinearities, chatter generation and suppression in metal cutting , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  Gábor Stépán,et al.  On stability prediction for milling , 2005 .

[5]  B. Balachandran,et al.  Stability of Up-milling and Down-milling Operations with Variable Spindle Speed , 2010 .

[6]  Lloyd N. Trefethen,et al.  Barycentric Lagrange Interpolation , 2004, SIAM Rev..

[7]  Yusuf Altintas,et al.  Multi frequency solution of chatter stability for low immersion milling , 2004 .

[8]  Eric A. Butcher,et al.  Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels , 2009 .

[9]  Yusuf Altintas,et al.  Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design , 2000 .

[10]  Balakumar Balachandran,et al.  Nonlinear dynamics of milling processes , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Balakumar Balachandran,et al.  Stability analysis for milling process , 2007 .

[12]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[13]  Liu Qiang,et al.  Solution and Analysis of Chatter Stability for End Milling in the Time-domain , 2008 .

[14]  Tony L. Schmitz,et al.  Effects of Radial Immersion and Cutting Direction on Chatter Instability in End-Milling , 2002 .

[15]  Han Ding,et al.  A full-discretization method for prediction of milling stability , 2010 .

[16]  Keith A. Young,et al.  Chatter vibration and surface location error prediction for helical end mills , 2008 .

[17]  Manfred Weck,et al.  Chatter Stability of Metal Cutting and Grinding , 2004 .

[18]  Zoltan Dombovari,et al.  Chatter stability of milling in frequency and discrete time domain , 2008 .

[19]  Peter Linz,et al.  Analytical and numerical methods for Volterra equations , 1985, SIAM studies in applied and numerical mathematics.

[20]  L. Trefethen,et al.  Two results on polynomial interpolation in equally spaced points , 1991 .

[21]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[22]  Keith A. Young,et al.  Simultaneous Stability and Surface Location Error Predictions in Milling , 2005 .

[23]  Yusuf Altintas,et al.  Analytical Prediction of Stability Lobes in Milling , 1995 .

[24]  Han Ding,et al.  Numerical Integration Method for Prediction of Milling Stability , 2011 .

[25]  Guojun Zhang,et al.  Complete discretization scheme for milling stability prediction , 2013 .

[26]  Balakumar Balachandran,et al.  Dynamics of milling processes with variable time delays , 2006 .

[27]  Han Ding,et al.  Milling stability analysis using the spectral method , 2011 .