Reducing computational requirement of stability analysis of milling by partial averaging

Abstract

The trapezoidal rule and Taylor theorem are used to establish a novel reformulation technique called Partial Averaging (PA). PA is seen to reduce computational time (CT) of computer central processing unit (CPU) with- out lose of accuracy. Two types of PA are proposed, one involves use of trapezoidal rule to give a map called Full Trapezoidal Rule Map with Partial Averaging (FTRMPA) while the other is achieved through exact analysis to give a map called Partial Trapezoidal Rule Map with Partial Averaging (PTRMPA). The latter is demonstrated to be more accurate and more time conserving in stability analysis of milling process (both one and two degree of freedom systems were considered) and delayed damped Mathieu equation. With all other things being equal the results of stability analysis of fully-immersed milling process using FTRMPA and PTRMPA are seen to be identical with those of full-discretization by Ding et al. (1) further validating the presented reformulation. PA is applied on the Second Order Least Squares Approximated Full-discretization method of (18) to illustrate its usefulness for reducing the cumbersome analysis and CT of the full-discretization method when analysis gets more complicated by higher order interpolation/ approximation theory.