Predicting the remaining useful life of a cutting tool during turning titanium metal matrix composites

Abstract

A cutting tool’s remaining useful life is what is left for a tool, at a particular working age, in order to reach a pre-specified level of acceptable performance. The prediction of remaining useful life is crucial in order to decrease the scrapped products or the unnecessary interruption of the machining process in order to replace the tool. Consequently, the accuracy of its estimation affects the cost of machining, particularly when the product’s material is very expensive. In this article, the remaining useful lifes of 25 identical tools are estimated during turning titanium metal matrix composites. These composites are extensively used in aerospace and aviation industries. Accurate estimation of the remaining useful life has positive impact on product quality in terms of producing the required specifications. In this article, experimental data are gathered, and the proportional hazard model are used in order to model the tool’s reliability and hazard functions with EXAKT software and then the remaining useful life curves are developed for different machining conditions, namely, the cutting speed and the feed rate. The use of the proportional hazard model is validated using a normalization process and Kolmogorov–Smirnov test. The proportionality assumption is verified using log minus log plot. The final result is the development of the curves that represent the tools’ reliability and the remaining useful life for different machining conditions of the titanium metal matrix composites.

Keywords

Remaining useful life metal matrix composites machining

Introduction

The characteristics of titanium metal matrix composites (Ti-MMCs), such as low weight, high mechanical and physical properties, and high stiffness and strength, have brought them up as useful materials in different industries.¹ Although very expensive, Ti-MMCs are a new generation of materials which have proven to be viable in various fields such as biomedical and aerospace industrial. For these reasons, finding the cutting tool’s remaining useful life (RUL) while machiningTi-MMCs is important in order to decrease the scrapped products or the premature replacement of the tool and consequently the cost of machining. The estimation of the cutting tool’s RUL during the machining of Ti-MMCs is more difficult than other metals because of the high wear resistance which causes high wear rate on the cutting tools and also because the nonhomogeneous composition and abrasive properties of the reinforcing particles; the cutting tool confronts particles and matrix whose responses to machining process could be completely different. The machinability ofTi-MMCs is often considered poor because of the material’s characteristics, such as low thermal conductivity and high strength at high temperatures.²

The RUL is generally defined as the expected useful life left for a tool at a particular time of operation.³ Knowing the RUL has an economic impact, since replacing the tool earlier than necessary will cause the loss of valuable resources and time, while replacing the tool later will result in defective products.⁴ The cutting tool’s RUL estimate captures the output of a stochastic process, which is the wear increase, by providing a probabilistic estimate of the tool life as the cutting conditions change. Consequently, a replacement decision can be made in order to avoid unplanned tool failure.

Many researchers tried to estimate the RUL of some cutting tools. For example, Si et al.³ presented an exclusive review of the statistical data–driven approaches which are used to estimate the RUL. Sikorska et al.⁵ discussed the issues that need to be considered when selecting the suitable approach for estimating RUL. The authors also presented a classification of appropriate prognostic models for predicting the RUL of engineering assets. Banjevic⁶ discussed the importance of calculating the RUL in industrial applications. Aramesh et al.⁷ estimated the RUL of cutting tools during turning Ti-MMCs using the reliability function which is based on a Weibull distribution. In their article, the authors assumed three stages of tool wear in order to find the model that best represents the obtained experimental results. Baruah and Chinnam⁸ used hidden Markov models (HMMs) for carrying out diagnostic and prognostic activities for metal cutting tools during drilling process. HMMs were also used with the sensor signals emanating from the machine in order to identify the degradation state of the cutting tool and estimate the RUL. Wang and Wang⁹ used continuous hidden Markov model (CHMM) in order to calculate the RUL during a milling process. The authors considered the cutting forces as the monitoring signals that indicate the tool wear state. The authors also used a Gaussian regression model in order to predict the milling tool’s RUL. Gokulachandran and Mohandas¹⁰ presented two approaches for the prediction of remaining life of carbide-tipped tools: a regression model and an artificial neural network and then the results of both models were compared.

The main differences between this article and the previous researches, which are presented in Banjevic⁶ and Aramesh et al.,⁷ are as follows:

The developed model considers variable machining conditions, while in Banjevic,⁶ the study is conducted under constant machining conditions;

The developed model is based on the proportional hazard model (PHM) which reflects the effect of the machining conditions on the RUL, while in Banjevic⁶ and Aramesh et al.,⁷ the two articles considered the Weibull distribution only and did not take into consideration the machining conditions;

The obtained results of this article are the curves of the RULs which are estimated as functions of the machining conditions, while in Banjevic,⁶ the outputs are the sojourn time and the transition times between the wear states;

The developed curves are based on modelling a nonhomogeneous Markov model, which represents a stochastic process with transition matrix, while in Baruah and Chinnam⁸ and Wang and Wang,⁹ HMMs are considered.

The objective of this article is thus to develop the RUL curves of cutting tools during turning Ti-MMCs under variable conditions. In this study, Ti-6Al-4V is selected among the different Ti-MMCs alloys. The PHM is used to calculate the tool’s reliability and hazard functions and estimate the RUL. The cutting speed $(v)$ and the feed rate $(f)$ are the models’ covariates.

In section ‘Description of the experiment’, the description of the experimental setting is presented. In section ‘Model development’, the PHM for a tool operating under variable conditions is introduced. In section ‘Model validation’, the validation procedures for the obtained PHM are presented. In section ‘RUL’, the theoretical and mathematical basis for the estimation of the RUL is given. Concluding remarks are given in section ‘Conclusion’.

Description of the experiment

Equipment

At the computer numerical control (CNC) turning centre of Polytechnique Montréal, a six-axis Boehringer NG 200 is used in order to conduct the experiments. The used CNC turning centre has live tooling, programmable steady rest, Y-axis, and C-axis.

Tool material

Seco TH1000–coated carbide grade is used. TH1000 is a TiSiN-TiAlN nanolaminate physical vapour deposition (PVD)-coated grade which is produced by Seco global company. Its coating structure includes a nanolaminate PVD top layer for maximum toughness and high chipping resistance. TH1000 is highly resilient to edge fracture which is the kind of wear that typically occurs when turning hard surfaced materials such as MMCs. The experimental setup is shown in Figure 1.

Workpiece material

A cylindrical bar of Ti-6Al-4V alloy matrix reinforced with 10%–12% volume fraction of TiC ceramic particles is used for turning. Ti-MMC alloys have superior mechanical and physical properties when compared to the conventional materials. Ti-6Al-4V is selected among the Ti-MMC alloys because of its high strength and stiffness, high fracture toughness, and good deformability at high temperature.¹¹ The density of conventional Ti-MMCs is 4040 kg/m³, and the stiffness is 200 GPa.¹²

Design of experiment

Design of experiment (DoE) techniques were implemented to investigate the influence of factors on the predicted output result.¹³ The control conditions such as cutting speed and feed rate were used in DoE in titanium metal cutting research.¹⁴ The full factorial designs with two factors, two levels ( $v = 40 and 80 m / \min$ and $f = 0.15 and 0.35 mm / rev$ ) and one centre point ( $v = 60 m / \min$ and $f = 0.25 mm / rev$ ), are used. The ranges of speed and feed rate were selected based on the recommendation of the tool supplier. All combinations of the factors’ settings and a centre point experiment were performed in order to include the possible effect of curvature. In this experiment, changing the depth of cut is not applicable because of the limited amount of material. Table 1 shows the design of the experiment and all combinations of cutting conditions. There are five runs which were executed randomly. Each run was replicated five times; each replication is executed with a new tool. In total, 25 identical tools are used.

Table 1.

Design of experiment.

Run no.	Cutting speed (m/min)	Feed rate (mm/rev)	Depth of cut (mm)	Number of replications
4	80	0.35	0.2	5
3	40	0.35	0.2	5
1	40	0.15	0.2	5
2	80	0.15	0.2	5
5	60	0.25	0.2	5

Each replication uses a new tool on which sequential inspections are performed in order to measure the wear. An Olympus SZ-X12 microscope is used to measure the flank tool wear at discrete points of time through inspections. This procedure continues until the cutting tool fails and becomes dull and no longer operates within acceptable quality. The predefined threshold (VB_Bmax = 0.2 mm) is set to define the tool failure. For example; the experimental data of run 1 and tool 5 is shown in Table 2. The time to failure ( $TTF$ ) is calculated by interpolating between the two measurements around VB_Bmax = 0.2 mm. For example, from Table 2, and by interpolating between the 13th and the 14th inspections, the TTF is found to be 1560 s. As an example, Figure 2 shows the tool wear measurements for the five tools of run 1 when ( $v = 40 m / \min$ , $f = 0.15 mm / rev$ ). The procedure is repeated for the 25 tools. The TTFs of the 25 tools are shown in Table 3.

Table 2.

Experimental results showing the wear of tool 5, at run 1 when $v = 40 m / \min$ and $f = 0.15 mm / rev$ .

Inspection no.	Inspection time (s)	Flank wear (mm)
1	120	0.0525
2	240	0.06
3	360	0.065
4	480	0.0725
5	600	0.0875
6	720	0.1075
7	840	0.1125
8	960	0.12
9	1050	0.125
10	1170	0.135
11	1290	0.165
12	1410	0.175
13	1530	0.1825
14	1650	0.205

Table 3.

Times to failure (TTFs) in seconds for the 25 tools.

$v = 40 m / \min$ ; $f = 0.15 mm / rev$		$v = 80 m / \min$ ; $f = 0.15 mm / rev$		$v = 40 m / \min$ ; $f = 0.35 mm / rev$		$v = 80 m / \min$ ; $f = 0.35 mm / rev$		$v = 60 m / \min$ ; $f = 0.25 mm / rev$
Tool ID	$TTF$	Tool ID	$TTF$	Tool ID	$TTF$	Tool ID	$TTF$	Tool ID	$TTF$
1	1623.3	6	295	11	1240	16	121.4	21	233.5
2	2087	7	267.5	12	1002	17	87.5	22	192
3	1770	8	281.2	13	1320	18	135	23	265
4	1524	9	225.3	14	1263.3	19	135	24	190
5	1560	10	252.7	15	1230	20	121.7	25	160

Model development

The tool life of a cutting tool is a random variable, and its degradation is a stochastic process that depends not only on the age of the tool but also on the covariates which describe the machining conditions.¹⁵ In the PHM, the failure rate is modelled as the product of a baseline failure rate, which is dependent only on the age of the tool and an exponential expression which is the linear sum of $γ_{i} Z_{i}$ , where $Z_{i}$ are the covariates that represent the machining conditions, and $γ_{i}$ are the coefficients or weights of these covariates. The Weibull model is used as a baseline function in modelling the tool failure in Tail et al.,⁴ Mazzuchi and Soyer,¹⁵ and Makis.¹⁶ The Weibull distribution is extensively used in modelling the tool life due to its flexibility in representing many stochastic physical phenomena. The failure rate, which is the hazard function at time $(t)$ , is expressed as in equation (1)

h (t, Z; β, η, γ) = \frac{β}{η} {(\frac{t}{η})}^{β - 1} \exp {\sum_{1}^{m} γ_{i} Z_{i}}

(1)

$β$ is the shape parameter and $η$ is the scale parameter of the Weibull distribution. This model is sometimes called the Weibull parametric regression model. The covariates are the cutting speed $(v)$ and the feed rate $(f)$ . The hazard function for m = 2 in the PHM is given in equation (2)

h (t, Z; β, η, γ_{1}, γ_{2}) = \frac{β}{η} {(\frac{t}{η})}^{β - 1} e^{γ_{1} v + γ_{2} f}

(2)

The hazard function formulation with the PHM leads to a corresponding survival function formulation as shown in equation (3). This survival function formulation $R (t; Z)$ is the basis for determining the cutting tool survival curves. It shows that the survival function at time t for any tool, with vector Z as predictors or covariates, is given as

\begin{matrix} R (t; Z) = P (T > t | Z) = \exp {- \int_{0}^{t} h (t, Z) dt} \\ = \exp {- {(\frac{t}{η})}^{β} e^{γ_{1} v + γ_{2} f}} \end{matrix}

(3)

In order to estimate the parameters $(β, η, γ_{1}, γ_{2})$ based on the experimental results, the survival function $R (t; Z)$ and its derivative $\overset{\cdot}{R} (t; Z) = h (t, Z) R (t; Z)$ are used in order to form the maximum likelihood (ML) function.¹⁷ The PHM parameters are then estimated using EXAKT software.¹⁷ The resulting hazard function is given as follows in equation (4)

\begin{matrix} h (t, Z) = \frac{β}{η} {(\frac{t}{η})}^{β - 1} e^{γ_{1} v + γ_{2} f} \\ = \frac{3.90}{23120} {(\frac{t}{23120})}^{2.90} e^{0.203 v + 11.037 f} \end{matrix}

(4)

It is to be noted that the covariate parameter values γ₁ = 0.203 and γ₂ = 11.037 do not represent the effects of the conditions on the hazard rate, because these values are not normalized and depend on the measurements’ scales. A small value for $γ_{1}$ parameter does not mean that the cutting speed $(v)$ has a small effect on the hazard function, because the covariate parameter is multiplied by the covariate value which can be large.¹⁸ Statistical Wald test is used to distinguish between the statistically significant and nonsignificant effects of the covariates; in Table 4, Wald test shows in column 5 that the effect of the cutting speed on the tool wear is more statistically significant than the effect of the feed rate.

Table 4.

Summary of the estimated parameters (based on ML method).

Parameter	Estimate	Significance	Standard error	Wald
Scale $(η)$	23120	−	6041	−
Shape $(β)$	3.902	Yes	0.6792	18.26
$ν$	0.2036	Yes	0.03707	30.18
f	11.04	Yes	2.908	14.41

Based on equation (3), the reliability curve is plotted for each of the combinations of feed at 0.15, 0.25, and 0.35 mm/rev and speed at 40, 60, and 80 m/min as shown in Figure 3. By comparing between these curves, the effect of the machining conditions on the probability of failure is shown. For example, the higher effect of the cutting speed, in comparison to the effect of the feed rate, is clear when comparing between the reliability curve 1 and the reliability curve 7, where both curves have the same feed rates, $f = 0.15 mm / rev$ , but different speeds, $v = 40 m / \min$ and $v = 80 m / \min$ , respectively. However, the smaller effect of the feed rate, in comparison to the effect of the speed, is clear when comparing between the reliability curve 4 and the reliability curve 6, where both curves have the same speed value of $v = 60 m / \min$ but different feed rates of $f = 0.15 and 0.35 mm / rev$ , respectively. Obviously, the effect of the cutting speed is much higher than the effect of the feed rate since the curves 1 and 7 are more distanced. Equation (5) is derived from equation (4), as described in equation (3), and it is used in producing Figure 3

\begin{matrix} R (t, Z) = \exp {- {(\frac{t}{η})}^{β} \exp {\sum_{1}^{m} γ_{i} Z_{i}}} \\ R (t, Z) = \exp {- {(\frac{t}{η})}^{β} e^{γ_{1} v + γ_{2} f}} \\ = \exp {- {(\frac{t}{23120})}^{3.9} e^{0.203 v + 11.037 f}} \end{matrix}

(5)

Model validation

The objective of this section is to present three methods that are used in this article in order to validate the use of the PHM in modelling the degradation of the cutting tool condition due to wear. The results of this validation prove that indeed the PHM represents adequately the degradation of the tool due to wear. In the following subsection, the three methods are presented.

Normalization

Intuitively, the cutting speed’s effect on the cutting tool life should be greater than the feed rate’s effect. To prove this intuitive assertion, the cutting speed and the feed rate effects are calculated using a simple normalization procedure. The normalization is used to adjust the values measured on different scales to a notionally common scale. Since the cutting speed and the feed are in the range $(40, 80 m / \min)$ and $(0.15, 0.35 mm / rev)$ , respectively, the normalization of the ‘overall’ covariate is given as follows

Z^{c} = β_{0} + β_{1} x_{1} + β_{2} x_{2} = 16.043 + 4.06 x_{1} + 1.1037 x_{2}

where $x_{1} = (v - 60) / 20$ ; $x_{2} = (f - 0.25) / 0.1$ ; $x_{1}, x_{2} \in [- 1, 1]$ ; $β_{0}, β_{1}, and β_{2}$ are the regression coefficients of the normalized regression equation;¹⁹ and $x_{1}$ and $x_{2}$ are the normalized speed and feed, respectively. After this normalization, it is obvious that the effect of the cutting speed on the cutting tool’s life is approximately four times more than the effect of the feed rate. The effect of the cutting speed on the cutting tool’s survival is also clear in the curves of Figure 3; the nine curves seem to be grouped into three groups according to the speed value, $v = 40, 60, and 80 m / \min$ . In each group, the curves are distinguished from each other by the effect of the feed rate value. Based on a reliability analysis, and using the Taguchi method, Yang et al.²⁰ indicated that the cutting speed is the most significant parameter in the cutting process. Their finding is conforming to what was found in this article.

K-S test

K-S test is originally used as a test of equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution. This test can also be used to test the goodness of fit. K-S test is used to validate the model by evaluating its fit. The test checks the null hypothesis that the cumulative hazard function, which is $(\int_{0}^{t} h (t, Z) dt)$ in equation (3), is distributed exponentially, the exponential distribution being a special case of the Weibull distribution.²¹ Table 5 shows the summary of the goodness-of-fit test which is automatically produced in EXAKT software. The test shows that the PHM offers a good modelling for the degradation data of the 25 tools.

Table 5.

Summary of goodness-of-fit test results.

Test	Observed value	p value	PHM fits data
Kolmogorov–Smirnov	0.198444	0.248773	Not rejected

PHM: proportional hazard model.

Main PHM assumption

The main assumption underlying the use of the PHM is that the change in the tool wear rate, which is the change in the rate of degradation, is proportional to the change in the covariate values. This can be shown graphically in Figure 4.

Figure 4 gives the logarithmic reliability function plot which shows clearly that indeed the assumption of proportionality is satisfied since the combination values of speed and feed produce parallel lines in the log minus log plot. These lines should be parallel, and the distance between the curves should remain proportional across the time.²² Figure 4 shows the analysis of the logarithmic reliability function (log minus log plot) based on equation (3). The linear equation for each speed–feed combination is as follows

Ln [- Ln (R (t; Z))] = β Ln (t) - β Ln (η) + γ_{1} v + γ_{2} f

(6)

The logarithmic reliability function in equation (6) is linear in $Ln (t)$ , and for each combination of speed and feed, the corresponding functions are parallel.⁴ This means that the PHM’s assumption is satisfied, and the model is used in order to calculate the reliability functions of the cutting tool in the given range of the cutting speed and the feed rate.

RUL

The RUL at time $t_{r}$ , also called the mean residual life, is estimated, and it is given in equation (7)

\begin{matrix} RUL (t_{r}) = E (T - t_{r} | T 〉 t_{r}) = \frac{\int_{t_{r}}^{\infty} t f (t, Z) dt}{R (t_{r})} - t_{r} = \frac{\int_{t_{r}}^{\infty} t h (t, Z) R (t, Z) dt}{R (t_{r})} - t_{r} \\ RUL (t_{r}) = \frac{\int_{t_{r}}^{\infty} t \frac{3.90}{23120} {(\frac{t}{23120})}^{2.9} e^{0.203 v + 11.037 f} \exp {- {(\frac{t}{23120})}^{3.9} e^{0.203 v + 11.037 f}} dt}{R (t_{r})} - t_{r} \end{matrix}

(7)

At time $t_{r}$ , the RULs for the cutting tools are estimated using equation (7) and using Mathematica software.²³ In Figures 5 –7, the RUL is estimated at any specific time, $t_{r}$ , and at specific machining conditions, namely, the cutting speed and the feed rate. At a specific cutting time, the RUL varies with the machining conditions. As such, the effect of machining conditions should not be ignored when calculating the RUL. For example, the effect of the feed rate is clear when the cutting speed is constant as shown in Figure 5. At a cutting time t_r = 1000 s and $f = 0.15 mm / rev$ , the RUL is equal to 786.938 s, while with $f = 0.25 mm / rev$ , the RUL = 431.069 s, and with $f = 0.35 mm / rev$ , the RUL = 209.084 s. Similarly, the effect of the cutting speed is clear when the feed rate is constant. For example, in Figures 5 –7, at the cutting time t_r = 200 s, when $f = 0.15 mm / rev$ and $ν = 40 m / \min$ , the RUL is equal to 1506.74 s; while with $v = 60 m / \min$ , the RUL = 406.63 s, and with $v = 80 m / \min$ , the RUL = 53.87 s. At the cutting time t_r = 0, the calculated RUL is equal to the mean TTF of the cutting tool, assuming that the tool will be used at a constant speed and feed from t_r = 0 until failure. In Figure 7, the cutting time scale is limited to 300 s when $v = 80 m / \min$ , which means that the RUL for carbides tool is low when machining the used Ti-MMC at high speed, the same conclusion was also found in Kannan et al.²⁴ and Tomac et al.²⁵

It is to be noted that the RUL, which is also called the mean residual life, is an estimation of the mean remaining life. As such, its value is based on the conditional probability that the tool will live until time $t_{r}$ and then the remaining life is calculated. This value is different from the unconditional estimation of the remaining life while the tool is at t_r = 0. The PHM reflects this statistical result, which is also a real phenomenon. For example, it is well known that the probability of survival until time of a tool that has already been used until time $t_{r}$ is higher than the same probability for a tool that has just started to be used. The practical explanation of this phenomenon is that the more the tool stays in life, the more its probability to stay more in life. This is a natural phenomenon which has been observed in humans, buildings, and other physical objects, and it is the basic natural observation that led to the calculation of the RUL.

Similar to the RUL charts that are given in Figures 6 and 7, RUL charts were obtained in some articles for other applications. For example, In Elsayed,²⁶ the author presented the charts of the reliability function and the RUL using an estimated Weibull distribution for a furnace tube failure, with the scale parameter depending on the operating temperature. The temperature is a time-independent covariate, which is set at a specific value at the beginning of the tube’s life. Also, In Wang et al.,²⁷ the authors presented the graphs for reliability function and the RUL using PHM and Monte Carlo simulation method. Four degradation states of equipment were considered when the RUL graphs were drawn. In Banjevic and Jardine,²⁸ the authors presented the graphs for reliability function and the RUL using Weibull PHM. The RUL charts were obtained for transmission systems of large transporters from a mining company. The wear metal iron is the significant covariate, and the transmission’s oil analysis history is time dependent.

One of the most useful aspects of the RUL charts is the ease of using them in the machining workshops. These charts can be built for any kind of material and tool. The procedure starts by designing, performing some experiments, and the observation of the events of tools’ failures by sequential inspections. The data concerning the covariate values is collected in order to build the PHM by estimating the parameters. The model is validated by checking the proportionality of the hazards and the K-S goodness-of-fit test. Finally, the RUL charts are obtained under different machining conditions.

The difference in the experimental results and results obtained from the developed model is presented in Figure 8 when $v = 40 m / \min$ and $f = 0.15 mm / rev$ . The average experimental RUL for cutting tools 1–5 coincides with the calculated RUL till $t_{r} = 500 s$ . Afterwards, the calculated RUL begins to converge towards nonfailed tools; near 2000 s, the average curve will simply coincides with the tool 2. It is obvious that the calculated RUL almost represents the experimental ones. For example, at $t_{r} = 1000 s$ , average experimental RUL is equal to 713 s, while the calculated RUL is equal to 786.93

Conclusion

During turning Ti-MMCs under variable machining conditions, the experimentally collected data were used to construct the PHM which takes into consideration the TTFs and the effects of the external conditions. The introduction of the PHM as a viable model to represent the degradation phenomena of a cutting tool and the validation of this model with three different techniques are presented. The estimation of the RUL of a cutting tool as a function of the machining conditions is developed. The charts for the RUL of Ti-MMCs under different machining conditions are constructed. The future work aims at incorporating the sensors’ data about the uncontrollable variables, which are the forces and the torque, in the PHM. It is also planned to create the RUL curves in an online system in order to estimate the cutting tool RUL autonomously. The RUL curves will be generated and updated at each inspection time $t_{r}$ , for a specific material and tool, under different and changing machining conditions such as cutting forces, cutting temperatures, progressive wear, acoustic emissions, and vibration signals.

Footnotes

Appendix 1 Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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