Research Papers

Bayesian Inference for Milling Stability Using a Random Walk Approach

[+] Author and Article Information
Jaydeep Karandikar

Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: jaydeep.karandikar@me.gatech.edu

Michael Traverso

Stanford University,
Stanford, CA 94305
e-mail: miketrav@stanford.edu

Ali Abbas

University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: aliabbas@illinois.edu

Tony Schmitz

University of North Carolina at Charlotte,
Charlotte, NC 28223
e-mail: tony.schmitz@uncc.edu

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received July 29, 2011; final manuscript received February 17, 2014; published online April 11, 2014. Assoc. Editor: Suhas Joshi.

J. Manuf. Sci. Eng 136(3), 031015 (Apr 11, 2014) (11 pages) Paper No: MANU-11-1258; doi: 10.1115/1.4027226 History: Received July 29, 2011; Revised February 17, 2014

Unstable cutting conditions limit the profitability in milling. While analytical and numerical approaches for estimating the limiting axial depth of cut as a function of spindle speed are available, they are generally deterministic in nature. Because uncertainty inherently exists, a Bayesian approach that uses a random walk strategy for establishing a stability model is implemented in this work. The stability boundary is modeled using random walks. The probability of the random walk being the true stability limit is then updated using experimental results. The stability test points are identified using a value of information method. Bayesian inference offers several advantages including the incorporation of uncertainty in the model using a probability distribution (rather than deterministic value), updating the probability distribution using new experimental results, and selecting the experiments such that the expected value added by performing the experiment is maximized. Validation of the Bayesian approach is presented. The experimental results show a convergence to the optimum machining parameters for milling a pocket without prior knowledge of the system dynamics.

Copyright © 2014 by ASME
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Fig. 1

Twenty (left) and 5000 (right) random walks with a normally distributed position step size described by N(0,0.1)

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Fig. 2

Histograms of x at 5 s (left) and 10 s (right)

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Fig. 3

Many sample paths generated in the spindle speed-axial depth domain

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Fig 4

10,000 sample paths after filtering. The paths that cross 0 or 10 mm in the spindle speed range of 4000 rpm to 10,000 rpm have been removed.

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Fig. 5

Histograms of axial depths at 4000 rpm (left) and 10,000 rpm (right)

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Fig. 6

Prior probability of stability in the spindle speed-axial depth domain (left). The probability of stability is 0 at an axial depth of 10 mm.

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Fig. 7

Sample paths remaining after filtering given a stable test result (left) and an unstable test result (right) at and axial depth of 5 mm and spindle speed of 7000 rpm

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Fig. 8

Updated cdf at 7000 rpm given a stable test result (left) and an unstable test result (right) at a test axial depth of 5 mm

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Fig. 9

Posterior cdf for milling stability given a stable test result (left) and an unstable test result (right) at an axial depth of 5 mm and spindle speed of 7000 rpm

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Fig. 10

Tool path for pocket milling

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Fig. 11

Cost of machining at axial depth–spindle speed combinations given that the resultant cut is stable. Notice the steps in the cost function at integer fractions of the pocket depth.

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Fig. 12

Stability results for the value of information testing. The “o” symbols represent a stable result and the “x” symbols indicate an unstable result.

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Fig. 13

Experimental setup for stability testing

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Fig. 14

Frequency content of the acceleration signal (left) and the machined surface (right) at {10,000 rpm, 6.25 mm}. Content is seen only at the tooth passing frequency (167 Hz) and its harmonics.

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Fig. 15

Frequency content of the acceleration signal (left) and the machined surface (right) at {8294 rpm, 8.34 mm}. This unstable cut exhibits content other than tooth passing frequency and its harmonics (left) and chatter marks are observed (right).

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Fig. 16

Posterior stability cdf after 20 tests

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Fig. 17

FRFs for the flexure in the x (left) and the y (right) directions used in the experiments. Note that the dynamic stiffness is an order of magnitude higher in the y direction.

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Fig. 18

Test point selections compared with the analytical stability lobes

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Fig. 19

Experimental validation of the stability lobes

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Fig. 20

The updated cdf at 6500 rpm given a stable test at {7000 rpm, 5 mm} with standard deviations of 0.5 mm and 0.25 mm for the random walk generation

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Fig. 21

Updated posterior cdf given a stable test result at {7000 rpm, 5 mm} for random walks generated using standard deviations of 0.5 mm (left) and 0.25 mm (right)

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Fig. 22

Updated posterior cdf after 20 tests using random walks generated with standard deviations of 0.5 mm (left) and 0.25 mm (right)

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Fig. 23

Test point selection compared with the analytical stability lobes

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Fig. 24

Posterior cdf of stability after eight tests at a spindle speed range of 4000 rpm to 8000 rpm




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