Handling editor: Hinke Osinga

Taking Delay Equation Dynamics to the Cutting Edge

Brian P. Mann, Duke University

Predictive models of manufacturing processes can be applied to help businesses gain a competitive edge. In this time of expanding global markets, it has become essential for manufacturers to improve process efficiencies, maintain stricter part tolerances, and enhance part quality. Furthermore, the motivation for using analytical tools for process optimization, rather than costly trial and error, has perhaps never been greater. One of the most widespread manufacturing operations is machining where the general goal is to sculpt a shaped part by removing material from the bulk [8]. The importance of machining to the aerospace industry has led to the partnering of Dr. Brian Mann at Duke University with Boeing researchers Ryan Hanks, Amy Helvey, and Dr. Keith Young who are investigating the dynamics of a machining process called milling.

Figure 1:

Illustration of the milling process along with labels on some of the important parameters that can influence the dynamic stability of the process. The diagram on the right shows a prototypical stability chart that separates regions in the parameter space of chatter (labeled Unstable Zone) and chatter-free cutting (labeled Stable Zone).

A primary factor that limits the productivity of high-speed milling operations is the onset of self-excited vibrations known as chatter. Chatter can arise due to coupling between the cutting forces and the tool motions. In an attempt to avoid damage to the cutting tool, the machining center, or the workpiece, it is desirable to apply predictive methods to search for regions of chatter-free cutting. While numerical simulation is an approach that can be applied relatively easily, it becomes prohibitively time consuming to explore relatively large regions of parameter space without an analytical approach.

Aside from avoiding regions of chatter vibration, maintaining strict part tolerances is also a central concern in the aerospace industry. Thus parameter combinations that may not result in chatter vibrations, can still be problematic if the motion amplitude is relatively large.

In past work these relatively large oscillations have been called surface location error to describe the contribution of the surface placement error that arises from dynamic tool deflections. Theoretical investigations of these dynamic phenomena are complicated by the fact that the governing equations contain time delay(s), periodic coefficients, and motion dependent discontinuities.

Figure 2:	Schematic diagram of a milling operation showing the relative phasing of vibrations that can result in chip thicknesses (the shaded region) of chatter and chatter-free processes. Pictures show both the case of chatter (bottom right) and chatter-free machining (bottom left).

The focus of many recent works has been the occurrence of new bifurcation phenomena that can occur during milling operations for small radial depths of cut; see Figure 1. In addition to Neimark-Sacker or secondary Hopf bifurcations, period-doubling behavior has now been analytically and/or experimentally confirmed by several researchers [2-6]. An even more recent discovery is the phenomenon known as an isolated island of chatter vibration. To be more specific, an isolated island is a parameter domain in the stability chart that separates or appears to break away from the other regions of chatter vibration. These regions are entirely surrounded by completely stable or chatter-free parameter regions; see Figure 3. Recent works on this topic have attributed this phenomenon to certain common tool geometries (i.e. the tool helix angle).

In the figures that follow, illustrative examples are provided for the types of dynamic behavior that has been uncovered in the combined experimental and theoretical investigations. Figure 3 shows an example stability chart with markers that represent the locations in the parameter space where experiments were performed. The results of Figure 4 show four experimental time series along with the corresponding Poincaré section. Since only displacements were recorded in the experimental tests, visualization of the qualitative features of each attractor required the application of delayed embedding techniques to reconstruct a topologically equivalent phase space in the coordinates displacement vs. delayed displacement. Following the methods suggested in [1], algorithms were developed to graph the mutual information function for the time series and the shifted time series. The first minimum of the mutual information graph was used as the time shift, or delay, between the original time series and the shifted time series. Using the false nearest neighbors approach of [1], the embedding dimension was found to be equivalent to 2 for each of the presented time series. Poincaré sections were created by periodically sampling the displacement and delayed displacement signals as shown in the right-hand graphs of Figure 4.

Figure 3:	Example region of the stability chart where experimental tests were performed to study the isolated island phenomenon in reference [7]. The letters from this graph match the experimental time series and Poincaré Sections shown in the plots of Figure 4.

The first test (labeled case A) is clearly a stable case since both the once-per-period and Poincaré section data show that the system approaches a single fixed-point value. Quasi-periodic motions are observed in the second case of Figure 4 as the tool motions are incommensurate with the period. The remaining two cases, labeled C and D, provide experimental evidence of isolated islands of chatter vibration. These results confirm that theoretical results that predict period-doubling behavior will occur.

Figure 4:	Experimental times series and Poincaré sections used to illustrate the types of behavior that have been uncovered during the collaborative research. Green areas are the continuously sampled data, and black dots are the 1/period sampled displacement data (left column graphs). The Poincaré sections (right column graphs) are displayed in displacement vs. delayed displacement coordinates.

References

[1]		H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Springer, New York, 1996.
[2]		M. A. Davies, J. R. Pratt, B. Dutterer, and T. J. Burns, "Stability prediction for low radial immersion milling," Journal of Manufacturing Science and Engineering 124, no. 2 (2002): 217–225.
[3]		T. Insperger, B.P. Mann, G. Stépán, and P.V. Bayly, "Stability of up-milling and down-milling, Part 1: Alternative analytical methods," International Journal of Machine Tools and Manufacture, 43 (2003): 25–34.
[4]		B.P. Mann, N.K. Garg, K.A. Young, and A.M. Helvey, "Milling bifurcations from structural asym- metry and nonlinear regeneration," Nonlinear Dynamics, 42, no. 4 (2005): 319–337.
[5]		B.P. Mann, T. Insperger, P.V. Bayly, and G. Stépán, "Stability of up-milling and down-milling, Part 2: Experimental Verification," International Journal of Machine Tools and Manufacture, 43 (2003): 35–40.
[6]		B.P. Mann and K.A. Young, "An empirical approach for delayed oscillator stability and parametric identification," Proceedings of the Royal Society A, 462 (2006): 2145–2160.
[7]		B. Patel, B.P. Mann, K.A. Young, "Uncharted islands of chatter instability in milling," International Journal of Machine Tools and Manufacture 48, no. 1 (2008): 124-134.
[8]		J. Tlusty, Manufacturing Processes and Equipment, Prentice Hall, Upper Saddle River, NJ, 1 ed., 2000.