Reliability assessment of cutting tools based on zero-failure data

Titanium alloys are widely utilized in aeronautical monolithic components due to their excellent mechanical properties. However, the low machinability of titanium alloys results in an uncertain lifetime of cutting tools. Reliability assessment on cutting tools has become more and more significant to the effectiveness and stability of machining systems. Considering that cutting tools are often replaced well before the end of their useful lifetime to avoid failures and thus few tools failure data can be obtained during the machining process of titanium alloys, a reliability assessment model based on zero-failure data is developed to evaluate the reliability of cutting tools. A Weibull distribution model is chosen to describe the life distribution of cutting tools for reliability evaluation. Matching distribution curve method and weighted least squares method are used to estimate the distribution parameters. A novel approach using hierarchical Bayesian method for estimating the prior distribution of failure probability with zero-failure data is proposed by combining the characteristics of Weibull distribution and Incomplete Beta distribution. Reliability analysis for cutting tools with zero-failure data is performed using the proposed method. By comparing with the experience of experts and adopting stochastic simulation, the results of point estimation show plausibility of the proposed approach.


Introduction
Nowadays titanium alloys are being extensively used in aeronautical monolithic components thanks to their excellent mechanical properties. However, titanium and its alloys have been recognized as difficult-to-machine materials [1]. The productivity in machining titanium alloys is adversely restricted because of the low machinability [2].

reliability evaluation of cutting tools
Worn tools can bring the deterioration of surface integrity and poor dimensional accuracy of parts, leading to low productivity and reliability of the processing system. Tools are often replaced well before the end of their useful lifetime to avoid failures and related consequences. It has been reported that only 50-80% of the expected tool life is typically used [3]. So it is essential to evaluate the cutting tool life in machining titanium. Generally speaking, tool wear is considered as a stochastic process and it is a pretty tricky problem to predict tool wear progression accurately.
Cutting tool life can be worked out by taking into consideration the observed distribution of the cutting times to failure. Therefore, reliability research on cutting tools is of great importance to manufacturing system effectiveness and stability.
Wager and Barash [4] performed more than one hundred tool life tests using a high speed steel turning tool and found that the tool life values followed a statistical distribution, which approached normal distribution. Liu and Makis [5] derived a recursive formula to calculate the cutting tool reliability in variable conditions if the failure time distribution of the cutting tool can be represented using an accelerated failure time model. Klim et al. [6] proposed a reliability model taking into account both the flank and the face wear on the cutting tool. Wang et al. [7] developed a reliability-dependent failure rate model to predict the reliability of cutting tools. Rodriguez and Souza [8] developed a reliability-based method combined with process planning for estimating the optimum cutting tool change time.
Their approach took into account that each cutting tool was used for several different operations and was validated for the case of drilling. Ding and He [9] studied on the cutting tool reliability through a proportional hazards model. Salonitis and Kolios [10] put forward a novel approach for the effective reliability assessment of cutting tool wear based on combinations of stochastic response surface and surrogate modeling methods, coupled with Monte Carlo simulations and first-order reliability methods for the estimation of reliability indices. Wang et al. [11] built a mathematical model of dynamic reliability for machining process under the premise of regarding cutting parameters as random variables. The researchers combined moment estimation using maximum likelihood estimation method with dynamic reliability analysis method.
Besides, the monitored data in the machining process was introduced into reliability analysis to reflect the timevarying characteristics. Chen et al. [12] proposed a novel reliability estimation approach to cutting tools based on logistic regression model by using vibration signals. Li et al. [13] used logistic regression models and acoustic emission signal to evaluate cutting tool reliability. Cutting tools are investigated to determine the best maintenance time based on different conditions estimation. Liu et al [14] applied state space model in cutting tool reliability assessment. The system state is treated as a fuzzy event which is uniquely characterized by membership functions defined over the substitute characteristic variable. The wavelet packet energy extracted from the acoustic emission signal was selected as the feature to estimate the tool state

reliability assessment with zero-failure data
Because of the variety and complexity of aeronautical structures, there exists a great difference in processing parameters when machining different structural features, which results in the uncertainty of cutting tool life. In addition, the long machining time of aeronautical components and various kinds of cutting tools together lead to the difficulty in understanding tool condition. Considering the large fluctuation of tool life in machining titanium alloys, cutting tools are usually replaced before reaching their failure criterion in actual practice in order to avoid the damage caused by worn tools and ensure the surface quality of components [15]. Consequently, cases of zero-failure data in the actual engineering are usually encountered, which refers to that there is no failure cutter in the specific time.
Taking into consideration that few tool failure data can be extracted from the field data in machining process of aerospace components, it is possible to apply the reliability assessment model based on zero-failure data to evaluate reliability of cutting tools.
The cases of zero-failure data may occur in the products with high reliability and long-lifetime, which is a special situation in the time-truncated tests. Martz and Waller [16] firstly studied the reliability evaluation models with zerofailure data and presented a Bayesian zero-failure reliability demonstration testing procedure. Han [17] put forward a method to analysis the optimal confidence limits of reliability parameters. The confidence limits were oriented to the products of which life is subordinate to Weibull distribution. Mao [18] proposed the matching distribution curve method to cope with the problem of reliability analysis with zero-failure data, which had been widely used for its excellent characteristics and ease of application. The Bayesian method has been extensively applied in the estimation of failure probability, which makes full use of information obtained from zero-failure data. Fan and Chang [19] conducted a Bayesian zero-failure reliability demonstration test to design the minimum sample size and testing length subject to a certain specified reliability criterion. Jiang et al. [20] proposed a modified maximum likelihood estimation method for estimating the distribution parameters based on the zero-failure data. Xia [21] used the grey bootstrap method in the information poor theory for the reliability analysis of zero-failure data under the condition of a known or unknown probability distribution of lifetime. Han [22] developed an E-Bayesian estimate method based on hierarchical Bayesian estimation, which was proved to be feasible for zero-failure data. Then the E-Bayesian estimate method attracted the attention of many scholars. Yin et al. [23] investigated an E-Bayesian estimation as a reliability analysis method for the seekers to deal with the zero-failure life testing data.
However, reliability assessment for cutting tools based on zero-failure data has not been studied in the past.
Within the present paper, Weibull distribution is chosen as the life distribution model of cutting tools for reliability evaluation. Matching distribution curve method and weighted least squares method are applied to the distribution parameters estimate process. A new approach using hierarchical Bayesian method for estimating the prior distribution of failure probability with zero-failure data is proposed by combining the characteristics of Weibull distribution and Incomplete Beta distribution (WIB). The zero-failure data of cutting tool life was extracted from field production data in the machining process of titanium alloy components. Reliability analysis for cutting tools with zero-failure data is performed using WIB method. Point estimation of cutting tool reliability is calculated. By comparing with the experience of experts and using stochastic simulation, the results show that the proposed approach is effective and practicable.

Tool Life Distribution Model
Weibull distribution has been widely used for the analysis of reliability due to its versatility and also shown to be an appropriate life distribution for cutting tools [24]. The probability density function (pdf) of two-parameter Weibull distribution is given by: Where β is the shape parameter (β>0), θ is the scale parameter (θ>0), and t is the lifetime.
The parameters β and θ have different influence on the density function curves [25]. The larger of the shape parameter β, the steeper of the pdf curve. For the special case β=1, Weibull distribution becomes exponential distribution. For shape parameter β=2, it is approximate to Rayleigh distribution. While for the value of β in the range 3≤β≤4, the shape of Weibull distribution is close to normal distribution [26]. In the meanwhile, the scale parameter θ does not affect the shape of Weibull distribution, and only has an impact on the timescale of pdf curves. Increasing the value of θ while holding β constant has the effect of stretching out the pdf.
Weibull reliability function R(t) and Weibull distribution function is given by: Weibull distribution has extensive application in reliability assessment in practical cases, as well as in the cases of reliability analysis based on zero-failure data.

Matching distribution curve method
The process of matching distribution curve method can be seen in Fig. 1. First of all, the estimation of failure probability pi at truncated time ti is calculated using a certain method. The lifetime probability distribution curve is computed with the aid of least squares method by using the estimation of failure probability ˆi p at truncated time ti.
And then, the estimation of reliability model and parameters of probability distribution are obtained based on the lifetime probability distribution curve. It is considered as the most critical step to estimate the failure probability in matching distribution curve method.

Estimation of failure probability
The expression of the traditional method for estimating failure probability is [27]: Bayesian method is a common approach for estimating the failure probability pi. Different application processes of Bayesian method are discussed as follows.
Suppose there are si tools for life test, and ri tools fail before time ti, and the failure probability of each sample is pi, and the likelihood estimation expression of failure probability is: There is no failure sample at time ti under zero-failure situation, which means ri=0. Then the likelihood estimate of the failure probability changes into: Based on Bayesian theory, the posterior distribution of the failure probability is given by: The estimated failure probability ˆC Bi p of the classical Bayesian method under the quadratic loss condition is provided by: Beta function B(a,b) is defined as: In order to satisfy the precondition of zero-failure data analysis, the failure probability is p1≤p2≤⋯≤pm when the truncated time is t1<t2<...<tm. The density function of pi must be a monotone decreasing function. Consequently, the range of parameters can be determined as 0<a≤1 and 1<b<c [28], where c is the upper bound of b.
According to [29], considering the robustness of Bayesian estimation, the value of hyper parameter a can be determined as a=1, and the value of c should not be large. The range of c can be determined on 2~9 according to [30].
Uniform distribution is chosen as the prior distribution of b and the density function is π(b)=1/(c-1).
When a=1, the hierarchical prior probability density function of failure probability pi can be expressed as: Suppose that the shape parameter β satisfies β>1, the following relationship can be get from the two equations above: ln ln ln ln ln Equation (13) Where: And there is: Generally, the range of pj can be expressed as: Incomplete Beta distribution B(θ1,θ2,a,b) is used to describe the prior distribution of pi. The density function is The estimated failure probability p1 using the hierarchical Bayesian method is given by: Assuming that * m p =1, the hierarchical prior probability density function of failure probability pi (i>1)is: The estimated failure probability ˆi p using the WIB method under the quadratic loss condition is given by: Where: In WIB method, the estimated failure probability ˆi p is a revised value which is not only related to 1 i p − , but also associated with i t and 1 i t − . Hence, this estimation method fully considers the information of zero-failure data.
In the light of the above analysis, the final expressions for estimating the failure probability of the 4 methods are listed in Table 1.

Estimation of distribution parameters
Estimation of the distribution parameters is carried out by utilizing the least-squares method. In order to take advantage of information get from zero-failure data, the weighted least squares method is applied to the parameters estimation process.
The failure probability of the product is pi; when t=ti, the estimation of it is ˆi p . There will be an equation as follows: 1 exp 1, 2,.. ., Hence, the following result can be obtained by taking logarithm of Equation (26) Then μ and σ can be computed by: Thus, the estimation of the scale parameter and shape parameter is:

Estimation of reliability model
The estimate of reliability ˆ( ) R t expression for the product is: It can be seen from Equation (35) that point estimate of reliability model is obtained through the existing zerofailure data.

Case study
The zero-failure data of cutting tools was extracted from the field production data in titanium alloy milling process. All the milling work was carried out on a CNC Vertical Machining Center (RAMBAUDI RAMMATIC-1201G) under fluid cooling condition. Down milling method was adopted. The axial depth of cut ap was 20mm and the radial depth of cut ae was 1mm. The parameters of milling tool are shown in Table 2. The cutting speed and feed rate are shown in Table 3. The material used for the milling job was Ti-6Al-4V alloy. Diameter(mm) 20 Cut length(mm) 40 Overall length(mm) 125

Helix angle 48°
Corner radius (mm) 0 Each milling job was stopped at a specific time, and 30 sets of data were collected. None of the milling tools was failure at last.
All of the milling tools are divided into 6 groups, the cutting times are shown in Table 4.
According to the definition of zero-failure data model above, the minimum time in each group is regarded as the truncated time. Then the zero-failure data can be obtained, which is shown in Table 5. According to the experience of the experts, the reliability of the milling tool is no more than 0.3 after 180min and no more than 0.2 after 240min.
It can be seen that only two cutters have been worked for more than 180 minutes, which indicates that the failure probability is high when the cutting time reaches 180 minutes. The experience of the experts is used for evaluating the accuracy of the 4 models proposed above. Assuming that the shape parameter β>1 [32]. The failure probability of milling tools at each truncated time is calculated by using the 4 methods mentioned above, and the results are shown in Table 6. The parameter value of c is set as 5 when estimating ˆH Bi p and ˆW IBi p . Comparing with the experiences of specialists, the failure probability estimation results of the milling tools are more acceptable in engineering applications using the WIB method proposed in this paper. The estimated reliability is about 0.35 at the truncated time t6=148min. It can be found that in the other three cases, the results of reliability estimation are more than 0.8 at t6=148min, which is out of accord with the actual situation. In conclusion, the accuracy of these models have a certain degree of inadequacy when used for reliability estimation of milling tools. Therefore, the WIB method is chosen as the failure probability estimation method in the following.
The parameters of the lifetime probability distribution of the milling tool can be gained using weighted least square method and the results are β =1.37 and  θ =147.31, which satisfy the assumption that β>1.
The reliability R(t) of the milling tool can be expressed as: 1.37 ( ) exp 147.31 According to the experiences of specialists, the reliability of the milling tool is expected no more than 0.3 when the cutting time is more than 180min and no more than 0.2 when the cutting time is more than 240min. The reliability at 180min and 240min can be obtained from the reliability function: From Equation (37), the WIB method proposed in this paper is more suitable for engineering application and consistent with the experts' knowledge.
The expectation of cutting tool life can be obtained through Weibull distribution function, which can be expressed as: Considering the value of c is determined based on experience, the influence of c on the estimation result should be evaluated. By using different values of c, the estimated reliability is shown in Table 7. The estimation of the parameters of lifetime probability distribution as well as the reliability of the milling tool at 180min and 240min are shown in Table 8.
It can be found that with the increase of c, the reliability increases gradually. The change of c has influence on the estimation of reliability. In the meanwhile, the estimation results utilizing different values of c are consistent with expert knowledge yet. So it is concluded that the WIB method has good robustness, which is still applicable even if the value of c is different due to empirical information.  9, which is the same as the grouping condition of zero-failure data of cutting tools. The minimum time in each group is regarded as the truncated time. Subsequently, estimates of distribution parameters were performed based on the 30 pseudorandom numbers. The processes of pseudorandom numbers generation and parameters estimation are repeated ten times and average values are taken as the results. TABLE 10 shows the simulation results using different methods mentioned above. It is worthy noted that WIB method shows the minimum error comparing with other method, which can be chosen for reliability assessment of cutting tools coping with zero-failure data.

Conclusion
(1) In the case of zero-failure data, this paper proposed a novel methodology for estimating the prior distribution of failure probability of cutting tools. The new method is based on the hierarchical Bayesian method by combining the characteristics of Weibull distribution and Incomplete Beta distribution.
(2) Weibull distribution is chosen as cutting tool life distribution model for reliability evaluation. Matching distribution curve method and weighted least squares method are applied to estimate the distribution parameters.
(3) The failure probability of each truncated time can be get from WIB method, which fully considers the information of zero-failure data collected from the workshop. Subsequently, point estimation of cutting tool reliability is obtained. By comparing with the experience information given by specialists, estimation results show good agreement with expert knowledge. Finally, the robustness of this method is discussed. The proposed method is still applicable even if the value of c is different due to empirical information.
(4) Random simulation is adopted to evaluate the performance of different methods. WIB method shows the minimum error comparing with other method, which can be chosen for reliability assessment of cutting tools based on zero-failure data.

Availability of data and materials
There is no data and materials available because all the data mentioned in this paper is listed in the manuscript.