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Research Papers

Parameter Inference Under Uncertainty in End-Milling γ′-Strengthened Difficult-to-Machine Alloy

[+] Author and Article Information
Farbod Akhavan Niaki, Laine Mears

International Center for Automotive Research,
Clemson University,
Greenville, SC 29607

Durul Ulutan

Mechanical Engineering,
Bucknell University,
Lewisburg, PA 17837

Manuscript received August 19, 2015; final manuscript received March 10, 2016; published online April 15, 2016. Assoc. Editor: Radu Pavel.

J. Manuf. Sci. Eng 138(6), 061014 (Apr 15, 2016) (10 pages) Paper No: MANU-15-1429; doi: 10.1115/1.4033041 History: Received August 19, 2015; Revised March 10, 2016

Nickel-based alloys are those of materials that are maintaining their strength at high temperature. This feature makes these alloys a suitable candidate for power generation industry. However, high wear rate and tooling cost are known as the challenges in machining Ni-based alloys. The high wear rate causes a rapid failure of the tool, and therefore, fewer data will be available for model development. In addition, variations in material properties and hardness, residual stress, tool runout, and tolerances are some uncontrollable effects adding uncertainties to the currently developed models. To address these challenges, a probabilistic Bayesian approach using Markov Chain Monte Carlo (MCMC) method has been used in this work. The MCMC method is a powerful tool for parameter inference and quantification of embedded uncertainties of models. It is shown that by adding a prior probability to the observation probability, fewer experiments are required for inference. This is specifically useful in model development for difficult-to-machine alloys where high wear rate lowers the cardinality of the dataset. The combined Gibbs–Metropolis algorithm as a subset of MCMC method has been used in this work to quantify the uncertainty of the unknown parameters in a mechanistic tool wear model for end-milling of a difficult-to-machine Ni-based alloy. Maximum of 18% error and average error of 11% in the results show a good potential of this modeling in prediction of parameters in the presence of uncertainties when limited experiments are available.

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Figures

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Fig. 1

Price comparison of different materials (source: McMaster-Carr)

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Fig. 2

Gibbs sampler algorithm

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Fig. 3

Metropolis algorithm

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Fig. 4

Milling schematic [29]

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Fig. 5

Data acquisition with NI-cRIO9103

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Fig. 6

Cutting power of test 3: Vc = 50 m/min and f = 0.1 mm/rev

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Fig. 7

Measured flank wear for tests 1–4: (a) test 1—Vc = 25 m/min and f = 0.1 mm/rev, (b) test 2—Vc = 25 m/min and f = 0.2 mm/rev, (c) test 3—Vc = 50 m/min and f = 0.1 mm/rev, and (d) test 4—Vc = 50 m/min and f = 0.2 mm/rev

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Fig. 8

Flowchart of combined Gibbs–Metropolis algorithm

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Fig. 9

Pilot run samples: (a) trace plot and (b) samples autocorrelation (diverged chain)

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Fig. 10

Main run samples: (a) trace plot and (b) samples autocorrelation (converged chain)

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Fig. 11

Prior and posterior distributions after main run: (a) prior probability of Ki and Kj, (b) posterior probability of K1 and K2, (c) posterior probability of K3 and K1, and (d) posterior probability of K3 and K2

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Fig. 12

Distribution of parameters for prior belief, pilot run, and main run (the y-axis is not normalized)

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Fig. 13

Gamma distribution of the inverse of measurement error variance

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Fig. 14

Posterior predictive distribution algorithm

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Fig. 15

Posterior predictive distribution, measured spindle power (-o- sign)—validation tests 1–4

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Fig. 16

Evolution of mean (E[Ki]) and variance (Var[Ki]) of parameters after 50 runs

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