A novel stability analysis of systems with multiple time delays and its application to high speed milling chatter

Date of Completion

January 2007


Mathematics|Engineering, Mechanical




An intriguing perspective is presented in studying the stability robustness of systems with multiple rationally independent and uncertain delays. The problem is known to be notoriously complex due to the infinite dimensionality of the system. The process starts with a holographic class coordinate transformation from the delay space to a new set of coordinates. This mapping reduces the dimension of the problem from infinity to a manageably small number. Furthermore it confines the domain of study within a finite dimensional cube with edges of length 2π in the newly introduced domain, what we call the building block. In essence, the mapping collapses the entire set of potential stability switching points onto a small number of building hypersurfaces (upperbounded by n2, where n is the dimension of the dynamics). We further demonstrate that these building hypersurfaces can be implicitly defined and they are completely isolated within the above mentioned cube. It is also shown that the detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture we seek. This novel perspective serves very well for the preparatory steps of our group's recent paradigm called Cluster Treatment of Characteristic Roots (CTCR). As the "building block" concept is valid for the most general class of linear time-invariant, multiple time-delayed systems (LTI-MTDS), this work contains the first treatment of n-dimensional p-time delay cases in full rigor.^ This unique methodology is then utilized to assess the stability of regenerative chatter of non-uniform pitch milling process. With this method and a complementary numerical routine, we provide a process optimization scheme. The end result is a powerful tool to determine some important selections of the milling operation: (i) geometric selection: the pitch angle selection of the tool and, (ii) operational selection: the optimum cutting conditions (i.e., depth of cut and the spindle speeds), for maximizing the metal removal rate while avoiding the onset of undesired regenerative chatter. A case study is provided which displays the capabilities of the technique, as well as a comparison with other numerical approaches to show the advantages of the Building Block concept.^