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Research Papers

A Numerical and Experimental Investigation of Period-n Bifurcations in Milling

[+] Author and Article Information
Andrew Honeycutt

Department of Mechanical Engineering and
Engineering Science,
University of North Carolina at Charlotte,
Charlotte, NC 28223-0001

Tony Schmitz

Department of Mechanical Engineering and
Engineering Science,
University of North Carolina at Charlotte,
Charlotte, NC 28223-0001
e-mail: tony.schmitz@uncc.edu

Manuscript received March 8, 2016; final manuscript received June 30, 2016; published online August 8, 2016. Assoc. Editor: Radu Pavel.

J. Manuf. Sci. Eng 139(1), 011003 (Aug 08, 2016) (11 pages) Paper No: MANU-16-1152; doi: 10.1115/1.4034138 History: Received March 08, 2016; Revised June 30, 2016

Numerical and experimental analyses of milling bifurcations, or instabilities, are detailed. The time-delay equations of motions that describe milling behavior are solved numerically and once-per-tooth period sampling is used to generate Poincaré maps. These maps are subsequently used to study the stability behavior, including period-n bifurcations. Once-per-tooth period sampling is also used to generate bifurcation diagrams and stability maps. The numerical studies are combined with experiments, where milling vibration amplitudes are measured for both stable and unstable conditions. The vibration signals are sampled once-per-tooth period to construct experimental Poincaré maps and bifurcation diagrams. The results are compared to numerical stability predictions. The sensitivity of milling bifurcations to changes in natural frequency and damping is also predicted and observed.

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Figures

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Fig. 2

Milling experimental setup with LV, piezo-accelerometer (PA), LT, and CP

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Fig. 3

Poincaré map for period-2 bifurcation. (Left) simulation and (right) experiment.

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Fig. 4

Poincaré map for period-3 bifurcation. (Left) simulation and (right) experiment. The phase space trajectory is represented by the solid line and the once-per-tooth sampled points are displayed as circles.

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Fig. 5

Poincaré map for period-6 bifurcation. (Left) simulation and (right) experiment.

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Fig. 6

Poincaré map for a second period-6 bifurcation. (Left) simulation and (right) experiment.

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Fig. 7

Poincaré map for period-7 bifurcation. (Left) simulation and (right) experiment.

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Fig. 8

Poincaré map for period-8 bifurcation. (Left) simulation and (right) experiment.

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Fig. 9

Poincaré map for period-15 bifurcation. (Left) simulation and (right) experiment.

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Fig. 10

Bifurcation diagram for 3800 rpm and 5 mm radial depth of cut. (Left) simulation and (right) experiment.

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Fig. 1

Milling simulation geometry. The normal and tangential direction cutting forces, Fn and Ft, are identified. The fixed x and y directions, as well as the rotating normal direction, n, are also shown. The angle ϕ defines the tooth angle. The tool feed is to the right for the clockwise tool rotation and the axial depth is in the z direction.

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Fig. 11

Simulated stability map for period-3 experimental setup from Table 1 (M=1μm contour). The transition from stable to unstable behavior occurs at approximately 2.6 mm for a spindle speed of 3800 rpm. The inset shows the bifurcation diagram progression at 3800 rpm from stable to quasi-periodic instability to period-3 and back to quasi-periodic behavior.

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Fig. 12

Variation in bifurcation behavior with changes in natural frequency. Period-6 bifurcation is observed. (Left) simulation and (right) experiment.

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Fig. 13

Variation in bifurcation behavior with changes in natural frequency. Period-6 bifurcation is observed. (Left) simulation and (right) experiment.

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Fig. 14

Variation in bifurcation behavior with changes in natural frequency. Period-7 bifurcation is observed. (Left) simulation and (right) experiment.

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Fig. 15

Variation in bifurcation behavior with changes in natural frequency. Period-15 bifurcation is observed. (Left) simulation and (right) experiment.

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Fig. 16

Milling experimental setup with variable viscous damping. The setup includes a LV, PA, LT, CP, moving conductor, and PM. The top photograph shows the flexure without the PM; the copper conductor is visible inside the parallelogram leaf-type flexure. The lower photograph shows the PM in place. The magnets are positioned on both sides of the copper conductor and provide the eddy current damping effect.

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Fig. 17

Bifurcation diagram for 1.47% damping (3310 rpm). (Left) simulation and (right) experiment. Stable behavior is observed up to approximately 4 mm, period-2 behavior then occurs up to approximately 8 mm, then stable behavior is again seen.

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Fig. 18

Bifurcation diagram for 1.91% damping (3310 rpm). (Left) simulation and (right) experiment.

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Fig. 19

Bifurcation diagram for 2.34% damping (3310 rpm). (Left) simulation and (right) experiment.

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Fig. 20

Bifurcation diagram for 3.55% damping (3310 rpm). (Left) simulation and (right) experiment.

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Fig. 21

Simulated stability map for 1.47% damping (M=1μm contour). As the axial depth is increased, the transition from stable to period-2 (3.8 mm), period-2 back to stable (8.2 mm), and stable to quasi-periodic behavior (9.2 mm) is observed.

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Fig. 22

Simulated stability map for 1.91% damping (M=1μm contour). As the axial depth is increased, the transition from stable to period-2 (4.2 mm) and period-2 back to stable (7.6 mm) is observed.

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Fig. 23

Simulated stability map for 2.34% damping (M=1μm contour). As the axial depth is increased, the transition from stable to period-2 (4.6 mm) and period-2 back to stable (6.8 mm) is observed.

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Fig. 24

Simulated stability map for 3.55% damping (M=1μm contour). Stable behavior is observed at all axial depths.

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