Jaydeep Karandikar

Uncertainty in Machining: Workshop Summary and Contributions

A National Science Foundation-sponsored workshop was held from Feb. 24–26, 2010 in Arlington, VA. The purpose of this “Uncertainty in machining” workshop was to address uncertainty and risk in machining and related manufacturing operations. The application of decision theory, which defines how rational decision makers should make decisions in the presence of uncertainty, was discussed. A summary of the meeting outcomes is presented. To aid in the application of decision theory to manufacturing process modeling, an example of Bayesian inference for the well-known mechanistic turning force model is provided. The discrete grid method is presented, and updating is performed using force data from the Assessment of Machining Models study completed by the National Institute of Standards and Technology. [DOI: 10.1115/1.4004923]

Application of Bayesian Inference to Milling Force Modeling

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 MetropolisHastings (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. [DOI: 10.1115/1.4026365]

TOOL LIFE PREDICTION USING RANDOM WALK BAYESIAN UPDATING

According to the Taylor tool life equation, tool life reduces with increasing cutting speed. The influence of additional factors can also be incorporated. However, tool wear is generally considered a stochastic process with uncertainty in the model constants. In this work, Bayesian inference is applied to predict tool life for milling/turning operations using the random walk/surface methods. For milling, Bayesian inference using a random walk approach is applied to the well-known Taylor tool life model. Tool wear tests are performed using an uncoated carbide tool and AISI 1018 steel work material. Test results are used to update the probability distribution of tool life. The updated beliefs are then applied to predict tool life using a probability distribution. For turning, both cutting speed and feed are considered. Bayesian updating is performed using the random surface technique. Turning tests are completed using a coated carbide tool and forged AISI 4137 chrome alloy steel. The test results are applied to update the probability distribution of tool life and the updated beliefs are used to predict tool life. While this work uses the Taylor model, by following the procedures described here, the technique can be applied to other tool life models as well.

Remaining Useful Tool Life Predictions Using Bayesian Inference

Tool wear is an important limitation to machining productivity. In this paper, remaining useful tool life predictions using the random walk method of Bayesian inference is demonstrated. End milling tests were performed on a titanium workpiece and spindle power was recorded. The power root mean square value in the time domain was found to be sensitive to tool wear and was used for tool life predictions. Sample power root mean square growth curves were generated and the probability of each curve being the true growth curve was updated using Bayes’ rule. The updated probabilities were used to determine the remaining useful tool life. Results show good agreement between the predicted tool life and the true remaining life. The proposed method takes into account the uncertainty in tool life and the percentage of nominal power root mean square value at the end of tool life.Copyright

Bayesian Inference for Milling Stability Using a Random Walk Approach

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.

Optimal Machining Parameter Selection in Titanium Milling Using a Decision Analytic Framework

The paper presents a decision analytic framework for optimal machining parameter selection in titanium milling, considering uncertainty in tool life and stability. An influence diagram showing the decision situation, the corresponding uncertainties and the value was developed. The objective was to select optimum machining parameters to minimize machining cost while considering uncertainty in tool life and stability. The uncertainties were characterized using a probability distribution taking into account all available information. The cost associated with tool failure and unstable cutting conditions was incorporated in the cost formulation. A process probability tree showing the uncertainties and the corresponding costs was constructed. The optimal machining parameters were selected which minimize the expected machining cost. Results show a 90% reduction in machining cost as compared to manufacturer recommended parameters. A discussion section regarding inference and experimental design is included.Copyright

Combining Process Dynamics and Tool Wear in the Milling Super Diagram

This paper describes the milling “super diagram” that incorporates limitations to milling productivity and part quality imposed by stability, surface location error (part errors due to forced vibrations), and tool wear. Combinations of axial depth of cut and spindle speed that offer stable cutting conditions with an acceptable, user-defined surface location error level are identified by a gray-scale color coding scheme. The effect of tool wear is included through the force model coefficients (that relate the cutting force to the chip area) used for process dynamics prediction. Because the force model coefficients vary as a function of the volume of material removed, a unique super diagram is constructed for any user-defined volume of material removed with the selected cutter. For example, preferred operating conditions for a new tool can be compared to those for a worn tool. Additionally, user beliefs about data and model accuracy are applied to identify safety margins relative to the deterministic boundaries in the diagrams. Experimental results are provided for an inserted (carbide) cutter used to machine 1018 steel. The wear behavior is characterized as changes in the force model coefficients as a function of the volume of material removed. The flank wear is also measured using an on-machine microscope (to avoid tool removal from the spindle) and correlated to the force model coefficients. Stability diagrams are developed that correspond to the new and worn tool performance and experimental results are provided to verify changes in the process stability due to tool wear.Copyright

A Sequential Greedy Search Algorithm With Bayesian Updating for Testing in High-Speed Milling Operations

This paper describes a probabilistic greedy search optimization algorithm for stability testing in high-speed milling. The test parameters (i.e., the experiment setup decisions) consist of the axial depth of cut and the spindle speed. These parameters are selected to maximize the expected value of profit using a greedy search approach (an approach that maximizes the expected value of each stage one step at a time). After a test is performed, Bayesian updating is applied to determine the posterior distribution of stability. The algorithm is then repeated to identify a new test point. The motivation for this work is that, while deterministic models for milling stability prediction are available, uncertainty in the inputs always exists. In this study, it is assumed that the tool point frequency response function, which is required for stability lobe diagram development, is unknown. Therefore, the probability of stability over the selected axial depth-spindle speed domain must be determined experimentally. The greedy search algorithm identifies the maximum expected value of profit within the selected domain, where profit is determined from the product of the profit function and the stability cumulative distribution function, referred to as the probability of stability. This optimal point is then tested to evaluate stability. Whether stable or unstable, the results are used to update the probability of stability. A stable test updates all axial depths smaller than test depth to be stable at the selected spindle speed, while an unstable test specifies that all axial depths above the test depth are unstable. After updating, a new test point is selected by the greedy search algorithm and the process is repeated. This select/test/update sequence is repeated until a preselected stopping criterion is reached. This paper presents both numerical results and experimental validation that the optimization/updating approach quickly converges to the well-known stability lobe behavior described in the literature. However, in this probabilistic technique the issue of uncertainty is also addressed and results can be obtained even if no information about the dynamic system is available.Copyright