Nonlinear Dynamics

, Volume 71, Issue 1–2, pp 187–199 | Cite as

Complete discretization scheme for milling stability prediction

  • Mingzhen Li
  • Guojun Zhang
  • Yu Huang
Original Paper


This study presents a Complete Discretization Scheme (CDS) for milling stability prediction. When compared with the Semi-Discretization Method (SDM) and Full-Discretization Method (FDM), the highlight of CDS is that it discretizes all parts of Delay Differential Equation (DDE), including delay term, time domain term, parameter matrices, and most of all the differential term, by using the numerical iteration method, such as Euler’s method, to replace the direct integration scheme used in SDM and FDM, which greatly simplifies the complexity of the discretization iteration formula. The present study mainly provides a numerical framework than a method that can be theoretically used by different numerical methods for solving Ordinary Differential Equation (ODE), such as Euler’s method, Runge–Kutta method, Adam’s multistep methods, etc., in this framework for derivation of iteration formula with corresponding construction of coefficient matrix of iteration formula. This study presented CDS with Euler’s method (CDSEM) for solving the one degree-of-freedom problem (one DOF) and two DOF motion equations, which are usually used as benchmark problems. When compared with SDM and FDM, the benchmark results of one DOF and two DOF milling stability prediction show that CDSEM can obtain acceptable precision in most ranges. The computational efficiency of SDM and FDM was also determined, and the results show that CDS with Euler’s method is faster than FDM. Furthermore, large approximation parameters (small time interval) were selected by SDM and CDSEM, and the results show that CDS has high effectiveness, accuracy, and reliability.


Milling stability prediction Floquet theory Complete discretization scheme Delay differential equation Chatter 



This work is supported by National Natural Science Foundation of China (NSFC) under Grant No. 50975110 and Grant No. 51121002. It is also supported by The Leading Talent Project of Guangdong Province.


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.State Key Lab of Digital Manufacturing Equipment and Technology, School of Mechanical Science and EngineeringHuazhong University of Science and TechnologyWuhanChina

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