Which parameters influence the service life?

Probabilistic modeling and prediction of the service life and reliability of a milling tool using Bayesian statistics


This article describes the probabilistic modeling and prediction for the tool life and reliability of a milling tool using Bayesian statistics. A Markov Chain Monte Carlo Simulation (MCMC) is applied to the Taylor tool life model to develop the probabilistic model. The a priori probability distributions are determined from the literature research. A training data set and a test data set are generated from subsequent milling attempts. The tests are carried out in a range of cutting speeds from 300 to 400 m / min. The posterior probability distributions are calculated using the training set. Using the trained a ‑ posteriori probability distributions, the test data sets are then predicted at cutting speeds of 300 to 400 m / min. This is followed by a comparison of the probabilistic service life model with the test data sets. The a ‑ posteriori distributions of the tool life are then used to carry out a reliability analysis using the reliability function as a function of the cutting speed. It is shown that the tool life and reliability function can predict the measured tool life within the uncertainty intervals with a maximum prediction error of 18% with only two training data points.


This paper represents probabilistic modeling and prediction for lifetime and reliability of a milling tool using Bayesian statistics. Markov Chain Monte Carlo (MCMC) simulation is applied to Taylor tool life model to develop probabilistic models. Prior probabilities are established from literature review and posterior probabilities of the Taylor model parameters are obtained using milling experiments. The experiments were conducted to train and test the probabilistic models under a range of cutting speeds, 300-400 m / min. In this regard, the training datasets are used to update the tool life model and the test datasets are used to validate the probabilistic tool life model. Posterior distributions of the tool life are used to perform reliability analysis using reliability functions. The probabilistic models investigate the effect of cutting speeds on tool life probability distributions and the reliability functions. It is shown that the tool life posterior and reliability functions predict the measured tool life values ​​within the uncertainty intervals and maximum prediction error of 18%, using only two training data points.


Tool life is an important manufacturing characteristic for machining performance and has a significant impact on workpiece quality and manufacturing productivity. In this context, several performance models have been developed for predicting tool life and its wear. These models can be classified into analytical, numerical, empirical, artificial intelligence (AI) based and hybrid models [1,2,3,4]. Among the empirical models, the Taylor model was defined to predict tool life. It describes an empirical relationship between tool life and cutting speed using the power law

$$ \ mathrm {T} = \ left (\ frac {\ mathrm {C}} {V_ {c}} \ right) ^ {\ frac {1} {n}}. $$

It is T the tool life in minutes by one wear mark width V.B. of for example V.B. = 0.3 mm to be achieved. V.c is the cutting speed in m / min and C. is an empirically determined constant, also called minute speed. These are the downtimesT at different cutting speeds V.c empirically determined and from this the Taylor exponent n certainly. The constantC. theoretically describes the cutting speed with a tool life T = 1 min. The procedure is described in more detail in [5, 6]. The tool life required to reach a predetermined flank or crater wear limit is defined as the cutting time. Flank wear is the most common and preferred wear criterion due to its predictable and regular wear pattern.

Tool wear varies in real applications, even if the cutting tool, machine and cutting conditions are the same. This may be due to the variation in chemical and physical properties of the commercially identical part and tool-to-tool performance. In general, the tool life is characterized by randomness. An exact prediction is therefore difficult. Deterministic models are therefore limited in their application, since the randomness of the results of the tool life is not assessed [7, 8]. This means that stochastic quantities can be used to take this scatter into account for identical service life tests. Based on this background, the uncertainty quantification and the probabilistic prediction of the tool life with Bayesian statistics were carried out. Niaki et al. [9, 10] developed probabilistic models using the Bayesian method to predict tool wear when milling nickel-based materials. A combined Gibbs-Metropolis algorithm was used to estimate the unknown parameters of a non-linear mechanistic cutting performance model. Using the Metropolis algorithm to predict the model parameters and the Gibbs sampler to update the measurement error deviation, the model parameters were predicted with a maximum error of 18%.

The reliability of a system is generally understood to mean the period in which the system fulfills its function and does not fail [11]. The determination of the tool life under process uncertainties can be calculated through the application of reliability techniques. This means that the probability of failure of cutting tools can also be predicted using the reliability functions. The reliability analysis of the service life and wear of cutting tools has already been investigated using various probability distributions and statistical methods [12, 13]. Lin [14] has investigated the basic reliability analysis of cutting tools in high-speed milling by means of experimental tests. He compared the results of the reliability functions, taking into account different cutting speeds and feeds. Elbestawi et al. [15] presented stochastic models for predicting the failure rate when turning hardened steel with ceramic tools. Various types of tool wear, including uniform and chemical wear and premature failure, were modeled using lognormal and Weibull distributions. These quantify the failure behavior of the tool at different cutting speeds using the reliability and probability density functions (WDF), see Section 3.2. It has been shown that the log normal distribution can be used as the tool wear distribution to predict the tool failure rate under different values ​​of the cutting conditions and the surface finish of the workpiece. The Weibull distribution was rated as the most suitable distribution for modeling the micro-fracture of the tool in particular.

The reliability analysis performed in the above research is based on data-driven analysis. These were determined with various cutting conditions and tool geometries on the basis of extensive experiments. An alternative method for determining the probability distribution and the reliability function of the tool life is Bayesian statistics, in which an initial “opinion” (prior knowledge) about model parameters of the tool life models can be integrated into the current and future analysis. The prior knowledge (a priori probability) about the model parameters can come from previous experiments or a literature search. As a result, the model parameters can be determined with significantly less experimental effort.

In the following, the Bayesian MCMC method is modeled and applied for the probabilistic prediction of tool life and reliability during milling. The normal distribution is used as a probability distribution function and a reliability function. In addition, the influence of the cutting speed on the tool life and reliability of the milling tool is examined using the probabilistic models.

Experimental setup and empirical data acquisition

To update and finally validate the probabilistic modeling and prediction of the service life and reliability of a milling tool using Bayesian statistics, application-specific test data must first be determined. For this purpose, milling tests were carried out on a 5-axis milling machine Hermle C40; see Fig. 1. For the 2.5-axis tests, a tool holder with three uncoated indexable inserts with the commercial name SPGW09T308 and edge radii of re = 20 µm used with a tool diameter of 32 mm, see Fig. 2. The indexable inserts were designed and manufactured by Zermet Zerspanung GmbH with the ISO cutting material P25. The cubic tool steel AISI 1045 with the dimensions 100 × 100 × 100 mm was used as the test material. The depth of cut was b = 1.5 mm at a feed rate of fz = 0.05 mm / tooth and four cutting speeds of V.c = {300; 325; 350; 400} m / min.

In order to avoid removing the tool from the milling spindle for measuring purposes, a portable microscope 5 Mpx Dino-Lite (AM7915MZT) with the maximum magnification of 220 was used. Recordings of the tool open area were taken at regular intervals. The tool flank wear mark width V.B. the tool wear criterion was measured for three indexable inserts after each interval of 200 mm cutting length and the respective mean value and the standard deviation of the worn indexable inserts until the flank wear criterion of V.B. = 0.3 mm determined.

Fig. 3 shows an example of the recordings of the tool wear growth at a cutting speed of V.c = 350 m / min, which were recorded in 5 steps. To determine the tool life, a linear interpolation was carried out between neighboring data points until the wear limit of 0.3 mm was reached. Figure 4 shows the results of the tool wear tests, with each data point being the mean of the three tests together with a standard deviation error of ± 1σ is specified. The flank wear increases with the machining time. The tool life is reduced when the cutting speed is increased from 300 to 400 m / min. The dashed line represents the tool wear criterion (flank wear) V.B. = 0.3 mm. The measured mean value and the standard deviation of 1σ The tool life for the respective cutting speeds when the wear criterion is reached is shown in Tab. 1.


The probabilistic modeling of tool life is performed using Bayesian inference on the Taylor model. First, the a priori probabilities of the model parameters are determined. The model parameters are then updated using the results of the tests carried out by means of the likelihood function in order to calculate the posteriori probabilities of the parameters of the Taylor model. The Markov Chain Monte Carlo approach (MCMC approach) is then used for the simulation and the parameters of the posterior probability distributions are approximated. The a ‑ posteriori probability distributions finally serve as starting values ​​for the calculation of the reliability functions.

Bayesian inference of the Taylor tool life model

Bayesian inference allows an a priori value (an initial assumption of a parameter) to be adjusted by incorporating new experimental results. According to Eq. (2) the a ‑ posteriori probability \ (p \ left (x | y \ right) \) is the product of the a ‑ priori distribution p(x) with the likelihood function \ (p \ left (y | x \ right) \), divided by the normalization function. Using the Bayesian approach, the a ‑ posteriori distribution of one study can be used as the a ‑ priori distribution of a second study [16, 17].

$$ p \ left (x | y \ right) = \ frac {p \ left (x \ right) p \ left (y | x \ right)} {\ int p \ left (x \ right) p \ left ( y | x \ right) dx} $$

The integral in Eq. (2) is often referred to as marginal probability and generally does not have a closed solution. Therefore, calculation approaches, inter alia. the Markov Chain Monte Carlo (MCMC) simulation, developed to supplement or replace analytical integration and to approximate the posterior distribution [18].

Identification of the Taylor model parameters

The probabilistic modeling of the tool life using the normal distribution begins with the definition of the Taylor model parametersC. andn. The a priori probability of the model parametersC. andn is called the common Gaussian distribution p(C.,n) shown. The mean value and the standard deviation of the parameters were taken from [5, 19] for a low-carbon steel:

  • C. = 340 ± 60 m / min (± 1σ)

  • n = 0.26 ± 0.05 (± 1σ)

The a priori distributions are updated using the results of the processing attempts with the likelihood function. The bivariate likelihood function of the measured tool life, taking into account the parameters of the Taylor model, then results as follows:

$$ p \ left (T_ {m} | C, n \ right) = e ^ {- \ frac {\ left (\ left (\ frac {C} {V_ {c}} \ right) ^ {\ frac { 1} {n}} - T_ {m} \ right) ^ {2}} {2 {\ sigma _ {{T_ {m}}}} ^ {2}}} $$

\ (p \ left (T_ {m} | C, n \ right) \) is the likelihood function of the measured tool life Tm given a priori values ​​of the Taylor model parameters (C.,n). σTm is the standard deviation of the measured tool life. A normalization of the probability does not make sense here, since it is not a probability density function of the model parameters (C.,n), but rather the probability of the measured service life data Tm. This probability function describes how probable the measurement result is at a certain cutting speed with given Taylor model parameters.

The a ‑ posteriori probability is calculated using the likelihood function after the a ‑ priori probability has been updated. The posterior probability distribution of tool life is calculated using the normal distribution and is described as follows:

$$ p \ left (C, n | T_ {m} \ right) = \ frac {p \ left (C, n \ right) \ cdot p \ left (T_ {m} | C, n \ right)} { p \ left (T_ {m} \ right)}, $$

where \ (p \ left (C, n | T_ {m} \ right) \) is the common a ‑ posteriori distribution of the parameters (C.,n), calculated by the product of the common a priori distribution p(C.,n) and the likelihood function \ (p \ left (T_ {m} | C, n \ right) \) divided by the normalization factor p(Tm). The common a ‑ posteriori distribution of the parameters is the target distribution and is approximated using MCMC simulation in order to determine the probability distribution of the tool life for Eq. (1) to be calculated.

Application of MCMC to the Taylor model

The MCMC method is i. A. a probabilistic method of drawing a samplex from a probability distribution by a Markov chain mechanism. The resulting distribution is called the target distribution p(x) approximated [20]. The MCMC simulation is used here to determine the parameters of the Taylor model (C.,n) predict. The Metropolis algorithm of the MCMC method is used to take a sample from the proposed distribution q(C.,n) and the posterior target distribution p(C.,n) to approximate. A common normal distribution of the Taylor model parameters (C.,n).

The target distribution p(C.,n) is the product of the common prior distribution and the likelihood function. According to the Metropolis algorithm, the denominator is not calculated here. This is followed by a sample of candidates taken from the proposal distribution (C.,n)New, which is either accepted or rejected, depending on an acceptance ratio to be specified for the acceptance rater. In each iteration, the Markov chain is adapted to (C.,n)Newif the sample is accepted. Otherwise the chain remains at the current value of (C.,n)i. The algorithm moves along N-1-Iterations continued to order N Obtain samples from the target distribution using the following steps:

  1. 1.

    Make a normal a priori distribution p(C.,n) firmly;

  2. 2.

    Make a proposal distribution q(C.,n) firmly;

  3. 3.

    Initialize a sample as a starting value (C.,n)0;

  4. 4.

    Start iterating from i = 0 to i = N − 1:

    • Choose a sample of candidates (C.,n)New from the suggested distribution \ (q \ left ((C, n) ^ {\ text {new}} | (C, n) ^ i \ right) \);

    • Calculate the posteriori distribution \ (p \ left ((C, n) ^ i | T_m \ right) = \) \ (p \ left ((C, n) ^ i \ right) p \ left (T_m | (C, n) ^ i \ right) \);

    • Calculate the acceptance rate \ (r = \ frac {p \ left (\ left (C, n \ right) ^ {\ mathrm {new}} \ right)} {P \ left (\ left (C, n \ right) ^ {i} \ right)} \);

    • Generate a random numberu from a uniform distribution between 0 and 1

      If u ≤ r:

      Accept the suggestion: (C.,n)i + 1 = (C.,n)New;


      Reject the suggestion: (C.,n)i = (C.,n)New;

  5. 5.

    End of iteration.

In order to reduce the excessive autocorrelation due to the dependency of the samples taken with the Metropolis algorithm, the thinning technique is used [21]. In addition, the suggested distribution of the samples is limited within the specified acceptance rate of 15–50% [20, 22, 23]. Due to the process, the initial iterations initially show a transient behavior until they pass into a stationary range. This is commonly referred to as the burn-in period. In order to prevent this influence, this range of iterations is discarded in order to reduce the effects of the initial errors at the beginning of the chain [24].

The observation of the histograms of the parameters is one possibility to evaluate the convergence to the stationary distribution of the chain [21]. The Geweke method is used for this. The last interval of the chain with a smaller interval at the beginning of the chain, e.g. B. the first 10% and the last 50% after removing the initial iterations. If the mean values ​​of two intervals are within a defined tolerance range, the chain is in the steady state [25]. The results of the application of the algorithm described are discussed in Sect. 4 shown.

Reliability function

The reliability function R (t) is the most commonly used function when analyzing tool life data in engineering. This function indicates the probability that a component will function properly for a certain period of time. To determine the reliability function for a cutting tool, the cumulative distribution function must first be found F.(t) (Cumulative distribution function: CDF) of the tool failure can be calculated. The CDF of tool failure refers to the probability that a tool will occur at the timet is worn out and is calculated as follows:

$$ F \ left (t \ right) = \ int _ {0} ^ {t} f \ left (T \ right) dT. $$

Here is T the time at which a tool failure occurs. As a failure criterion at the end of the service life, the flank wear of V.B. = 0.3 mm fixed. The solution of the integral is not trivial and is solved numerically. The reliability function R.(t) is finally the complement of the CDF:

$$ R \ left (t \ right) = 1-F \ left (t \ right) $$

Results of tool life and reliability prediction

This section presents the results of the Taylor model parameter identification and the prediction of tool life and reliability using Bayesian statistics using MCMC simulation. In this context, the parameters of the model are calculated using two tool life data points Tm = {48; 7.6} min at the corresponding cutting speeds of V.c = {300; 400} m / min updated. The trained model is then used to predict the test data points at cutting speeds of 300 to 400 m / min.

Tool life prediction

The modeled common a priori distribution of the model parameters p (C, n) initially contains a covariance matrix, which is defined as independent, see Fig. 5. About the correlation of the parametersC. and n to determine the correlation coefficient is determined. The correlation coefficient is the measure of the linear relationship between two parameters, which are defined as the covariance of the parameters divided by the product of their standard deviations and are calculated as a result

$$ \ rho \ left (C, n \ right) = \ frac {cov \ left (C, n \ right)} {\ sigma _ {C} \ sigma _ {n}}. $$

A Monte Carlo simulation is then carried out to determine the a priori distribution of the tool lifeT using the common a priori distribution of C. and n to determine. Be for the simulation N = 3000 random C.- and n-Samples from the common pre-distribution for Eq. (1) using the cutting speeds V.c = {325; 350} m / min pulled.

Fig. 6 shows the histogram of tool life (in blue), the corresponding logarithmic normal distribution (red line) and the measured tool life (tool life test data) at the cutting speeds mentioned. The log-normal distribution provides a sufficiently good approximation to the histogram of the service life. The right skewed distribution that occurs is based on the power law form of the Taylor equation. The a priori function can initially calculate the measured tool life due to the poorly informative a priori values ​​for the model parametersC. andn, based on literature research, do not predict.

The MCMC simulation follows to determine the a ‑ posteriori distribution of the parametersC. andn. For the simulation, 7500 samples are taken from the proposal distributions q(C.,n) is drawn and the probability function is updated. Fig. 7 shows the curve diagram with the transition from the transient to the stationary range of the distributions for the first 1000 samples of the parametern. These are discarded from the simulation as a burn-in period in order to ensure a steady distribution of the samples.

Fig. 8 shows the histogram and the probability distribution of the samples taken for the parametern. The normal probability distribution (red color) is adapted to the histogram of the samples taken. The acceptance rate of the samples taken, calculated using the MCMC algorithms, is r = 0.34. Fig. 8 shows the trace diagram of the chain that is in the steady state after discarding the burn-in period. This ensures that the samples are evenly distributed.

After tuning the covariance matrix, K (C, n) for the proposal distribution q (C, n) defined as independent and described as follows:

$$ K _ {\ left (C, n \ right)} = \ left [\ begin {array} {cc} 300 & 0 \ 0 & 7e ^ {- 5} \ end {array} \ right] $$

Fig. 9 shows the common a ‑ posteriori distribution of the parametersC. andn using the two update lifetime dates. The mean values ​​of the parametersC. andn are 539 m / min and 0.153 and the associated standard deviation values ​​are 21.6 m / min and 0.013. A comparison of the common a ‑ priori (Fig. 5) and a ‑ posteriori distributions (Fig. 9) shows a reduction in the uncertainties. In addition, the model parameters correlateC. andn with the correlation coefficient of 0.93. The high value of the correlation coefficient compared to 1 describes the strong relationship between the two parametersC. andn.

The posterior distributions of the standing time are calculated using the Monte Carlo simulation by dividing the marginal posterior distributions of C. and n into Eq. (1) can be inserted. Fig. 10 shows the normal a ‑ posteriori distributions of tool life in red with the mean value and the standard deviation of 2σ for the four cutting speeds of V.c = {300; 325; 350; 400} m / min. The a ‑ priori distributions are also shown in the figures with the log normal distributions in blue. This shows that the measured tool life at all cutting speeds is within the uncertainty interval of ± 2σ of the a ‑ posteriori functions lie, but the a ‑ priori functions cannot predict them. It is important to note that the posterior functions were updated with only two training data points.

The measured and predicted tool service lives are given in addition to the percentage of the prediction error in Tab. 2. The predicted values ​​are calculated with the mean and a standard deviation interval of 1σ specified. The percentage error values ​​between the measured and predicted mean tool life are calculated as follows:

$$ \% _ {\ text {error}} = \ left | \ frac {F _ {\ text {measured}} - F _ {\ text {predicted}}} {F _ {\ text {measured}}} \ right | \ cdot 100. $$

The determined errors are between 0% at a cutting speed of 325 m / min and 18% at a cutting speed of 350 m / min. All measured service life values ​​are within the uncertainty intervals of ± 2σ of the predicted distributions.

Reliability prediction

Based on the determined tool life, according to Eqs. (5) and (6) predicted the reliability. Fig. 11 shows the reliability functions R.(T) of the tools at cutting speeds of V.c = {300; 325; 350; 400} m / min together with the measured tool life when reaching the average flank wear mark width of V.B. = 0.3 mm. The calculated mean values ​​of the reliability functions at the reliability value of 0.5 correspond to the predicted tool life at each cutting speed.

The slope of the reliability function increases as the cutting speed increases from 300 to 400 m / min. The higher the cutting speed, the higher the decrease in reliability. At the cutting speed of V.c = 350 mm / min, the a ‑ posteriori reliability function cannot predict the measured data points with sufficient accuracy compared to the other a ‑ posteriori functions. This is due to the higher, previously determined percentage of errors, which was calculated using Eq. (9) is given in Tab. 2.

The predicted reliability values ​​are summarized in Tab. 3. They correspond to the measured tool life data points. The reliability of the milling tool at the cutting speeds of V.c = {300; 325} m / min is closer to the function mean value 0.5 due to the more accurate prediction of the test data points at these cutting speeds. Basically, the reliability of the tool decreases with increasing cutting length. It decreases as the cutting speed increases. Finally, it should be noted that the probability of failure of a cutting tool at a certain cutting time can be determined with the aid of the reliability function.


The investigations presented include the probabilistic prediction of the tool life and the reliability analysis of a milling tool at cutting speeds of 300 to 400 m / min. The Bayesian MCMC method was applied to the Taylor model to determine the probability distributions of tool life. The numerical quantification and minimization of the tool life uncertainty were carried out using the Metropolis algorithm of the MCMC simulation. The a priori distributions were modeled as log normal distributions. The posterior distributions were measured after updating the model parameters using two measured tool lives Tm = {7.6; 48} min with the respective cutting speeds of V.c = {300; 400} m / min determined. Finally, using the reliability function, a reliability prediction for reaching the specified wear mark width was modeled and predicted.

In this research, the uncertainties of the Taylor model parameters were identifiedC. andn, and the tool life distribution is minimized at all cutting speeds. The determined a ‑ posteriori distributions of the tool life could all test data points within the uncertainty intervals with a maximum prediction error of 18% (at V.c = 350 m / min). Furthermore, the reliability functions were able to adjust the test data points to the function mean values ​​(with the exception of the cutting speed of V.c = 350 m / min).

The probabilistic modeling and prediction of the tool life and reliability of a milling tool carried out by means of Bayesian statistics enables the tool life to be predicted and the reliability of a milling tool to be determined with a certain accuracy within a defined uncertainty interval. The model developed is not generally applicable and is only intended for this application. The model must be adapted depending on the tool geometries, workpiece materials and cutting speeds. For example, for each combination of cutting tool and workpiece material as well as for a new cutting speed, the parametersC. andn redefined and the service life and the reliability function recalculated.


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