Research Papers

Application of Bayesian Inference to Milling Force Modeling

[+] Author and Article Information
Jaydeep M. Karandikar, Tony L. Schmitz

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

Ali E. Abbas

Department of Industrial
and Enterprise Systems Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

The z direction is oriented along the tool axis.

The subscript m denotes measured values from cutting experiments. The measured values were assumed to be statistically independent in this study.

If the proposal distribution was chosen to be uniform, then it is not dependent on the current value of x. In that case, x* will be drawn from U(xmin, xmax), where xmin is the minimum value and xmax is the maximum value of x for the uniform proposal distribution. A uniform proposal distribution is therefore less efficient because the random samples are independent of the current state of the chain. The random samples have an equal probability of taking any value between xmin and xmax which leads to many rejections. Using a normal proposal distribution, where x* is dependent on xi is referred to as random walk Metropolis sampling, while the uniform proposal approach where x* is independent of xi is called independent Metropolis Hastings sampling.

A marginal pdf for any variable in a joint distribution is obtained by integrating the remaining variables over all values.

Contributed by the Manufacturing Engineering of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received July 29, 2011; final manuscript received December 19, 2013; published online January 3, 2014. Assoc. Editor: Suhas Joshi.

J. Manuf. Sci. Eng 136(2), 021017 (Feb 20, 2014) (12 pages) Paper No: MANU-11-1259; doi: 10.1115/1.4026365 History: Received July 29, 2011; Revised December 19, 2013

This paper describes the application of Bayesian inference to the identification of force coefficients in milling. Mechanistic cutting force coefficients have been traditionally determined by performing a linear regression to the mean force values measured over a range of feed per tooth values. This linear regression method, however, yields a deterministic result for each coefficient and requires testing at several feed per tooth values to obtain a high level of confidence in the regression analysis. Bayesian inference, on the other hand, provides a systematic and formal way of updating beliefs when new information is available while incorporating uncertainty. In this work, mean force data is used to update the prior probability distributions (initial beliefs) of force coefficients using the Metropolis-Hastings (MH) algorithm Markov chain Monte Carlo (MCMC) approach. Experiments are performed at different radial depths of cut to determine the corresponding force coefficients using both methods and the results are compared.

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Fig. 1

Milling force geometry (a 50% radial immersion up milling cut using a cutter with two teeth is depicted)

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Fig. 2

Histogram of MCMC samples and target distribution (left) and x values for each iteration (right)

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Fig. 3

Traces of Kt and Kn (left) and Kte and Kne (right)

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Fig. 4

Posterior and prior distributions of Kt (top left), Kn (top right), Kte (bottom left), and Kne (bottom right) using a uniform prior. Note that the area under the histogram was normalized to unity in each case.

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Fig. 5

Traces of Kt, and Kn with 0.5 N force measurement uncertainty (left) and 2 N (right)

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Fig. 6

Posterior and prior distributions of Kt with a force uncertainty of σ = 0.5 N (left) and σ = 2 N (right)

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Fig. 7

Posterior and prior distributions of Kt (top left), Kn (top right), Kte (bottom left), and Kne (bottom right) using a normal prior

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Fig. 8

Experimental setup for milling force measurement

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Fig. 9

Linear regression to the mean forces in x (left) and y (right) direction to determine the force coefficients at 25% radial immersion

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Fig. 10

Posterior and prior distributions of Kt (top left), Kn (top right), Kte (bottom left), and Kne (bottom right). The least squares values are identified by the “x” symbols.

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Fig. 11

Comparison of the experimental and simulated force profiles for Fx (left) and Fy (right). The simulation used the force coefficients determined using the MCMC and least squares methods. Note that the oscillations in the experimental data are due to excitation of the dynamometer dynamics by the cutting force frequency content.

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Fig. 12

Linear regression to the mean forces in the x (left) and y (right) directions used to determine the force coefficients at 50% radial immersion

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Fig. 13

Posterior and prior distributions of Kt (top left), Kn (top right), Kte (bottom left), and Kne (bottom right)

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Fig. 14

Comparison of the experimental and simulated force profiles for Fx (left) and Fy (right)

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Fig. 15

Posterior distributions of force coefficients at 25% RI (left) and 50% RI (right)



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