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RESEARCH PAPERS

# A New Curve Tracking Algorithm for Efficient Computation of Stability Boundaries of Cutting Processes

[+] Author and Article Information
Christoph Henninger

Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germanyhenninger@itm.uni-stuttgart.de

Peter Eberhard

Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germanyeberhard@itm.uni-stuttgart.de

J. Comput. Nonlinear Dynam 2(4), 360-365 (May 16, 2007) (6 pages) doi:10.1115/1.2756077 History: Received January 05, 2007; Revised May 16, 2007

## Abstract

Dynamic stability of cutting processes such as milling and turning is mainly restricted by the phenomenon of the regenerative effect, causing self-excited vibration, which is well known as machine-tool chatter. With the semidiscretization method for periodic delay-differential equations, there exists an appropriate method for determining the stability boundary curve in the domain of technological parameters. The stability boundary is implicitly defined as a level set of a function on the parameter domain, which makes the evaluation computationally expensive when using complete enumeration. In order to reduce computational cost, we first investigate two types of curve tracking algorithms finding them not appropriate for computing stability charts as they may get stuck at cusp points or near-branch zones. We then present a new curve tracking method, which overcomes these difficulties and makes it possible to compute stability boundary curves very efficiently.

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## Figures

Figure 1

Model of a two-dimensional upmilling process with a single cutting blade

Figure 2

Stability chart of the high speed upmilling process for a radial depth of cut ratio a∕D=0.05

Figure 3

Rendering implicitly defined curves using Chandler’s algorithm with correct rendering (left) and failure at cusp point (right). 엯, screen pixels; ●, known curve points; ⊗, next curve point; +, test points for function evaluation.

Figure 9

Extending the search domain to a semicircle for determining the continuation point at strong curvature

Figure 10

Overcoming near-branch zones with help of the extremal point E

Figure 11

Overcoming cusp points with help of the extremal point E on the back-side semicircle

Figure 12

Stability chart of milling process computed by the curve tracking algorithm (black) and by complete enumeration (gray)

Figure 4

Algorithm of Morgado and Gomes with 240deg circular search domain S

Figure 5

Handling of near-branch zones at the algorithm of Morgado and Gomes. Correct detection of continuation point (left) and failure (right).

Figure 6

Determination of continuation point R on circular search domain S with angle α

Figure 7

Basic algorithm for polygonization of an implicitly defined curve with adaptive step size control

Figure 8

Problematic cases at polygonization: cusp point (left) and near-branch zone causing jump-over (right)

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