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TECHNICAL PAPERS

Simultaneous Stability and Surface Location Error Predictions in Milling

[+] Author and Article Information
Brian P. Mann1

Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65203mannbr@missouri.edu

Keith A. Young

 Advanced Manufacturing R&D, The Boeing Company, St. Louis, MO 62166

Tony L. Schmitz

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611

David N. Dilley

 D3 Vibrations, Inc., 220 S. Main Street, Royal Oak, MI 48067

1

To whom correspondence should be addressed.

J. Manuf. Sci. Eng 127(3), 446-453 (Jul 08, 2004) (8 pages) doi:10.1115/1.1948394 History: Received January 16, 2004; Revised July 08, 2004

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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Copyright © 2005 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Multiple degree of freedom schematic of the milling process: (a) Spatial representation of machine tool structure at discrete locations along the tool; and (b) Up-milling schematic diagram of tool tip in-plane motion

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Figure 2

Comparison of predicted stability boundaries with Euler Integration (dotted line) and TFEA (solid line). Aluminum cutting coeficients, listed in Section 4, were applied along with the modal parameters for the 12.75(mm) tool, listed in Table 1, to create this diagram.

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Figure 3

A comparison of steady-state displacement predictions between Euler Integration (solid line) and TFEA (dotted line). Each row contains the x- and y-tool displacements, with a 1/tooth mark shown by a ◻, for the following cutting parameters: (c) corresponds to [Ω=14300(rpm), b=0.3(mm)]; and (d) corresponds to [Ω=10,700(rpm), b=3.0(mm)]. Aluminum cutting coefficients, listed in Sec. 4, were applied along with the modal parameters for the 12.75(mm) tool, listed in Table 1, to create this diagram.

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Figure 4

Schematic diagram of surface location error experiments: (e) workpieces were mounted on a single degree of freedom flexure and up-milled in the compliant workpiece and rigid tool tests; and (f) down-milling was used in the compliant tool and rigid workpiece tests.

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Figure 5

Comparisons of TFEA fixed point predictions and measured surface location error (∘ indicates a measurement value): (g) Up-milling surface location error measurements for a single degree of freedom flexure obtained with an eddy current displacement transducer; and (h) Down-milling surface location error experimental results for the 19.05(mm) tool of Table 1.

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Figure 6

Down-milling experimental results vs TFEA stability predictions for the 12.75(mm) tool described in Table 1. The symbols in the above diagram are as follows: (a) ∘ is a clearly stable case; (b) ▿ is an unstable cutting test; and (c) + is a borderline unstable case (i.e., not clearly stable or unstable).

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Figure 7

Experimental down-milling measurement data for cases A and B of Fig. 6. Each row contains a 1/tooth passage displacement plot, a Poincaré section shown in delayed coordinates, and a Power Spectral Density (PSD) plot where ∘ marks the tooth passage frequency. Case A [Ω=14625(rpm), b=1.0(mm)] is an example of an unstable period-doubling phenomenon or a flip bifurcation. Case B [Ω=12675(rpm), b=1.5(mm)] is an unstable Hopf bifurcation.

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Figure 8

Experimental down-milling measurement data for cases (C,D,E,F) of Fig. 6. Each row contains a y-axis 1/tooth displacement plot, a Poincaré section shown in delayed coordinates, and a Power Spectral Density (PSD) plot where ∘ marks the tooth passage frequency. Graphics show growth in the dynamic error between each tooth passage, associated with tool runout, for a fixed spindle speed and an increasing depth of cut C [Ω=10725(rpm), b=0.5(mm)], D [Ω=10725(rpm), b=0.75(mm)], E [Ω=10725(rpm), b=1.0(mm)], F [Ω=10725(rpm), b=1.5(mm)].

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Figure 9

Euler integration results for down-milling cases (C,D,E,F) of Fig. 6. The top graph shows two consecutive tooth passages, marked with ∘, and a continuous time trace with the following legend: (C, dotted line; D, dashed line; E, solid line; F, dashed-dotted line). The four bottom graphs are 1/tooth passage displacement samples showing the effect of runout on tool oscillations at the cutter exit.

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