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

PI：Eric A. Butcher and Edward Bueler University of Alaska，FairbanksProposal #0114500 Symbolic Stability and Bifurcation Analysis of Time-Periodic Differential-Delay Equations：Applications to High-Speed Machining ModelsProject Abstract本项目涉及时间周期微分延迟方程（DDE）线性稳定边界符号计算的独特方法的开发和应用。 这种方程出现在高速加工中颤振不稳定性的几个线性化模型中，包括具有任意浸入水平的铣削和具有调制速度或阻抗的车削，以及其他重要的工程和科学应用。 此外，还对全非线性模型进行了局部非线性分岔分析。 这些目标是通过将高速加工、符号计算和非线性动力学中的一些重要主题和最新方法结合到一个单一的研究推力中来实现的。 通过将时间延迟到现有的符号算法的时间周期系统的稳定性分析，稳定性边界（例如，在主轴转速和切削深度的两个参数平面），预测颤振在加工操作中得到解析。 该任务构成了工作计划的第一阶段，并代表了重要的设计工具，因为给定一个参数（例如主轴速度）的值，可以轻松获得另一个参数（切削深度）的临界值。 因此，这种方法为设计人员提供了一种有吸引力的替代方案，这些设计人员经常试图在稳定性图表上的窄主轴速度范围内操作，同时保持尽可能高的切削速度。由于最近的研究颤振不稳定性车削表明，单独的线性稳定性分析是不够的，应该伴随着一个完整的非线性分析，第二阶段的工作计划，即时间周期铣削模型的分析分岔分析，是通过结合现有的计算工具的时间周期系统的Hopf分岔算法的DDE。 由此得到的结果允许的关键参数集的确定损失的全局稳定性和吸引域之前的损失的局部稳定性。 NIST的合作者对理论结果进行了实验验证。 该项目通过培训研究生并将当前的研究和实践（包括符号计算）带入本科课堂，促进工程和数学的跨学科教育。

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.