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Review Article

Milling Bifurcations: A Review of Literature and Experiment

[+] Author and Article Information
Andrew Honeycutt

Department of Mechanical Engineering and
Engineering Science,
University of North Carolina at Charlotte,
Charlotte, NC 28223
e-mail: ahoney15@uncc.edu

Tony L. Schmitz

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

Manuscript received January 16, 2018; final manuscript received August 27, 2018; published online October 5, 2018. Assoc. Editor: Satish Bukkapatnam.

J. Manuf. Sci. Eng 140(12), 120801 (Oct 05, 2018) (19 pages) Paper No: MANU-18-1035; doi: 10.1115/1.4041325 History: Received January 16, 2018; Revised August 27, 2018

This review paper presents a comprehensive analysis of period-n (i.e., motion that repeats every n tooth periods) bifurcations in milling. Although period-n bifurcations in milling were only first reported experimentally in 1998, multiple researchers have since used both simulation and experiment to study their unique behavior in milling. To complement this work, the authors of this paper completed a three year study to answer the fundamental question “Is all chatter bad?”, where time-domain simulation and experiments were combined to: predict and verify the presence of period-2 to period-15 bifurcations; apply subharmonic (periodic) sampling strategies to the automated identification of bifurcation type; establish the sensitivity of bifurcation behavior to the system dynamics, including natural frequency and damping; and predict and verify surface location error (SLE) and surface roughness under both stable and period-2 bifurcation conditions. These results are summarized. To aid in parameter selection that yields period-n behavior, graphical tools including Poincaré maps, bifurcation diagrams, and stability maps are presented.

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Figures

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Fig. 2

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. 1

Example stability lobe diagram. Stable and unstable zones are separated by the stability boundary (or limit). The control parameters are chip width, or axial depth of cut in milling, and spindle speed, which defines the forcing frequency and time delay between teeth.

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Fig. 3

Milling experimental setup with instrumentation including a LV, piezo-accelerometer, LT, and CP. The setup was located on a Haas TM-1 CNC milling machine.

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Fig. 8

Time domain results for a secondary Hopf bifurcation (2850 rpm)

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Fig. 9

Poincaré map for secondary Hopf bifurcation (2850 rpm). The sampled points are arranged in an elliptical distribution.

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Fig. 4

Time domain results for a stable cut (3400 rpm). (Top) time-dependent displacement with periodic samples (circles); (bottom) time-dependent velocity with periodic samples. (Inset) higher magnification view to observe individual periodic samples of the displacement (top) and velocity (bottom).

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Fig. 5

Poincaré map for stable cut (3400 rpm). The sampled points align at a single location for the forced vibration case.

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Fig. 6

Time domain results for a period-2 bifurcation (3310 rpm)

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Fig. 7

Poincaré map for period-2 bifurcation (3310 rpm). The sampled points align at a two fixed locations for the period-2 bifurcation.

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Fig. 23

Milling experimental setup with variable viscous damping. The setup includes a LV, piezo-accelerometer, 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. The setup was located on a Haas TM-1 CNC milling machine.

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Fig. 19

Variation in bifurcation behavior with changes in natural frequency. The natural frequency changes with time as more material is removed. Period-6 bifurcation is observed. (Left) simulation, (right) experiment.

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Fig. 10

Poincaré map for period-2 bifurcation. (Left) simulation, (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. 11

Poincaré map for period-3 bifurcation. (Left) simulation, (right) experiment.

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Fig. 12

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

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Fig. 13

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

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Fig. 14

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

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Fig. 15

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

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Fig. 16

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

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Fig. 17

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

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Fig. 20

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

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Fig. 21

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

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Fig. 22

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

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Fig. 18

Simulated stability map for period-3 experimental setup from Table 1 (M1=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. 30

(a) Once per tooth sampling (τ sampling period); (b) 2τ sampling period; (c) 3τ sampling period; and (d) 4τ sampling period. The zones that appear to be stable and unstable, depending on the sampling period, are marked.

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Fig. 31

New stability map. Period-2 (circle), period-3 (triangle), period-4 (square), period-5 (+), period-6 (diamond), period-7 (×), and secondary Hopf (dot) bifurcations are individually identified. The box indicates the spindle speed range and axial depth (6.4 mm) for the bifurcation diagram in Fig. 32.

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Fig. 32

Bifurcation diagram for an axial depth of 6.4 mm. Secondary Hopf (Hopf), period-2 (2), stable (Stable), and combination secondary Hopf and period-2 (Hopf-2) behaviors are specified.

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Fig. 34

The workpiece included four ribs that were initially machined to the same dimensions. The {5 mm axial depth, 2 mm radial depth} cuts were then performed on one edge at a different spindle speed for each rib. The SLE was calculated as the difference between the commanded, C, and measured, M, rib widths. The flexible direction for the flexure is identified.

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Fig. 24

Bifurcation diagram for 1.47% damping (3310 rpm). (Left) simulation, (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. 25

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

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Fig. 26

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

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Fig. 27

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

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Fig. 28

Simulated stability maps for four damping levels (M1=1 μm contours). (Top left) 1.47% damping. 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. (Top right) 1.91% damping. 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) occurs. (Bottom left) 2.34% damping. 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. (Bottom right) 3.55% damping. Stable behavior is obtained at all axial depths.

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Fig. 29

Simulation results for a spindle speed of 4070 rpm at an axial depth of 3.6 mm. The workpiece x displacement and velocity are shown. (Top row) Time history (left) and Poincaré map (right) for once per tooth sampling (τ sampling period). (Bottom row) Time history (left) and Poincaré map (right) for subharmonic sampling at 2τ.

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Fig. 37

Predicted (left) and measured (right) Poincaré maps for 3600 rpm. Stable behavior is seen with increased amplitude relative to 3300 rpm (Fig. 36).

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Fig. 38

Surface location error prediction from time domain simulation (line) and experimental results from rib cutting tests (circles). The four period-2 bifurcation tests are identified.

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Fig. 35

Predicted (left) and measured (right) Poincaré maps for 3180 rpm. Period-2 behavior is seen. Note that x indicates the flexible direction for the flexure. The feed direction was y for these experiments.

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Fig. 36

Predicted (left) and measured (right) Poincaré maps for 3300 rpm. Stable behavior is seen.

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Fig. 39

Commanded surface (dashed line), CMM scan (solid line), and simulation result (circles) for 3180 rpm (period-2). These results correspond to Fig. 35.

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Fig. 40

Commanded surface (dashed line), CMM scan (solid line), and simulation result (circles) for 3300 rpm (stable). These results correspond to Fig. 36.

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Fig. 41

Period-n bifurcation predictions for the rightmost stability lobe. The system dynamics are the same as those described in Section 4.7.

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Fig. 33

Surface location error/Ra experimental setup with LV, LT, and CP. The feed direction and the flexible direction for the SDOF flexure are also identified. The setup was located on a Haas TM-1 CNC milling machine.

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