Optimizing the milling process requires a priori knowledge of many process variables. However, the ability to include both milling stability and accuracy information is limited because current methods do not provide simultaneous milling stability and accuracy predictions. The method described within this paper, called Temporal Finite Element Analysis (TFEA), provides an approach for simultaneous prediction of milling stability and surface location error. This paper details the application of this approach to a multiple mode system in two orthogonal directions. The TFEA method forms an approximate analytical solution by dividing the time in the cut into a finite number of elements. The approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map. The formulated dynamic map is then used to determine stability, steady-state surface location error, and to reconstruct the time series for a stable cutting process. Solution convergence is evaluated by simply increasing the number of elements and through comparisons with numerical integration. Analytical predictions are compared to several different milling experiments. An interesting period two behavior, which was originally believed to be a flip bifurcation, was observed during experiment. However, evidence is presented to show this behavior can be attributed to runout in the cutter teeth.

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Drill chatter degrades hole roundness, hole size, and tool life. This wastes time and money in tools, scrap, and hole rework. Chatter prediction in milling and turning has shown significant benefit to industry; however, researchers have been unable to accurately predict chatter in drilling applications. In the past, the drill, including the chisel edge, was modeled as either a fixed-fixed or fixed-pinned beam (Tekinalp, O., and Ulsoy, A. G., 1989, “Modeling and Finite Element Analysis of Drill Bit Vibrations,” ASME J. Eng. Indust. 111, pp. 148–154), but more recent research (Dilley, D. N., Bayly, P. V., and Schaut, A. J., 2005, “Effects of the Chisel Edge on the Chatter Frequency in Drilling,” J. Sound Vib., 281, pp. 423–428) has shown that a fixed-embedded model using springs improves frequency matching. The effects of the drill margins on dynamics have not been studied. The fixed-fixed or fixed-pinned model will be shown to be inappropriate for modeling the effects of margin engagement, while the spring-end boundary condition can better approximate the frequency increase observed experimentally as the drill margins engage deeper into the hole. In addition, the shifted frequency is well below the frequency found from an analytical fixed-fixed or fixed-pinned beam. Evidence that the margins cause the frequency shift is seen in three-dimensional waterfall plots that show this shift for pilot hole drilling (in which the margins are engaged), but not for tube drilling (in which margins are not engaged).

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