Symbolic Stability and Bifurcation Analysis of Time-Periodic Differential-Delay Equations: Applications to High-Speed Machining Models

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PIs: Eric A. Butcher and Edward Bueler University of Alaska, FairbanksProposal # 0114500Symbolic Stability and Bifurcation Analysis of Time-Periodic Differential-Delay Equations: Applications to High-Speed Machining ModelsProject Abstract This project concerns the development and application of a unique method for the symbolic computation of linear stability boundaries in time-periodic differential-delay equations (DDEs). Such equations arise in several linearized models of chatter instability in high-speed machining, including milling with arbitrary immersion level and turning with modulated speed or impedance, as well as in other important engineering and scientific applications. In addition, the local nonlinear bifurcation analysis for the full nonlinear models is also performed. These objectives are accomplished by combining some important topics and recent methodologies in high-speed machining, symbolic computation, and nonlinear dynamics into a single research thrust. By incorporating time-delay into an existing symbolic algorithm for stability analysis of time-periodic systems, the stability boundaries (for example, in the two-parameter plane of spindle speed and depth of cut) which predict chatter in machining operations are obtained analytically. This task constitutes the first phase of the work plan and represents a significant design tool since, given a value of one parameter (say spindle speed), the critical value of the other (cutting depth) is easily obtained. Consequently, this approach provides an attractive alternative to designers who often try to operate in a narrow spindle speed range on the stability chart while retaining as high a cutting speed as possible. Since recent research on chatter instability in turning indicates that linear stability analysis alone is inadequate and should be accompanied by a full nonlinear analysis, the second phase of the work plan, namely the analytical bifurcation analysis for the time-periodic milling models, is accomplished by combining existing computational tools for time-periodic systems with the Hopf bifurcation algorithm for DDEs. The results thus obtained allows the determination of the critical parameter set for loss of global stability and the domain of attraction just prior to loss of local stability. Experimental validation of the theoretical results is made by collaborators at NIST. This project fosters cross-disciplinary education in engineering and mathematics through training graduate students and bringing current research and practice (including symbolic computation) into undergraduate classrooms.