Tensor-Based Automatic Arbitrary Order Computation of the Full-Discretization Method for Milling Stability Analysis

Abstract

The time-domain approaches of milling stability analysis, which are based on the Floquet theory, rely on finite-dimensional approximation of the infinite dimensional monodromy operators of the delayed models of milling. The needed analytical computation of the finite monodromy matrices is quite challenging especially at higher order polynomial approximation of milling states and, thus, in literature only low order approximations are discussed. This work introduces a solution to this problem by unifying and fully computerizing the Full-discretization Method on the framework of generalized polynomial tensor representations of the current and the delayed milling states. The unification makes a monodromy operator an automatically programmable bivariate function of any combination of arbitrary approximation orders of the current and delayed states. Then, fully computerized stability lobe identification can be implement using a Full-discretization Method of any arbitrary combination of approximation orders. As a result, more accurate Full-discretization Methods (here more than 48 in number) than the most accurate of the currently available Full-discretization Methods from literature were identified. A method based on sixth order approximation of the current state and fifth order approximation of the delayed state turned out to be the most accurate for the stability analysis of a popular benchmark milling process and it is shown that approximations of higher order can be computed correctly but can even decrease the stability lobe precision.