Second-order full-discretization method for milling stability prediction

This paper proposes a second-order full-discretization method for milling stability prediction based on the direct integration scheme. The model of the milling dynamics taking the regenerative effect into account in the state-space form is firstly represented in the integral form. After the time period being equally discretized into a finite set of intervals, the full-discretization method is developed to handle the integration term of the system. On each small time interval, the second-degree Lagrange polynomial is employed to interpolate the state item, and the linear interpolation is utilized to approximate the time-periodic and time delay items, respectively. Then, a discrete dynamical map is deduced to establish the state transition matrix on one time period to predict the milling stability via Floquet theory. The rate of convergence of the method is discussed, and the benchmark example is utilized to verify the effectiveness of the presented algorithm. The MATLAB code of the algorithm is attached in the Appendix.

[1]  Haitao Ma,et al.  Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials , 2004 .

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

[3]  Gábor Stépán,et al.  Semi‐discretization method for delayed systems , 2002 .

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

[5]  Tamás Insperger,et al.  Full-discretization and semi-discretization for milling stability prediction: Some comments , 2010 .

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

[7]  Wŏn-yŏng Yang,et al.  Applied Numerical Methods Using MATLAB , 2005 .

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

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

[10]  Gábor Stépán,et al.  On Stability and Dynamics of Milling at Small Radial Immersion , 2005 .

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

[12]  Gábor Stépán,et al.  On the chatter frequencies of milling processes with runout , 2008 .

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

[14]  A. Galip Ulsoy,et al.  Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter. , 2007, Mathematical biosciences and engineering : MBE.

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

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

[17]  Min Wan,et al.  A unified stability prediction method for milling process with multiple delays , 2010 .

[18]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[19]  Yusuf Altintas,et al.  Analytical Prediction of Chatter Stability in Milling—Part I: General Formulation , 1998 .

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

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

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

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