Time-Variant Reliability Assessment and Its Sensitivity Analysis of Cutting Tool under Invariant Machining Condition Based on Gamma Process

The time-variant reliability and its sensitivity of cutting tools under both wear deterioration and an invariant machining condition are analyzed. The wear process is modeled by a Gamma process which is a continuous-state and continuous-time stochastic process with the independent and nonnegative increment. The time-variant reliability and its sensitivity of cutting tools under six cases are considered in this paper. For the first two cases, the compensation for the cutting tool wear is not carried out. For the last four cases, the off-line or real-time compensation method is adopted. While the off-line compensation method is used, the machining error of cutting tool is supposed to be stochastic. Whether the detection of the real-time wear is accurate or not is discussed when the real-time compensation method is adopted. The numerical examples are analyzed to demonstrate the idea of how the reliability of cutting tools under the invariant machining condition could be improved according to the methods described in this paper.


Introduction
The cutting tool is one of the most important components of machine tools.During manufacturing process, it slides on the surface of the work-piece with a huge friction.Therefore, cutting tool fails due to wear frequently.It has been reported that the downtime due to the cutting tool failure is more than one third of the total down time which is defined as the non-productive lines idling in the manufacturing system 1-3 .Accurate assessment of the cutting tool reliability could result in an optimal replacement strategy for cutting tool, decrease the production cost, and improve the cutting tool reliability.
The reliability assessment of cutting tool has been investigated by many researchers.Klim et al. 4 proposed a reliability model for the quantitative study of the effect of the feed fate variation on the cutting tool wear and life.A deterministic approach based on conditions.It is still valid even if the breakage failure is not considered in the comparative analysis of the cutting tool reliability.This has been proved by the tests in 4 .The cutting tool deterioration process due to wear is a continuous-time and continuous-state stochastic process.Moreover, it is also a monotone increasing stochastic process.Therefore, the Gamma process is employed to model the cutting tool deterioration process.
Gamma process is a stochastic process with independent, nonnegative increment having a gamma distribution with an identical scale parameter.It is a continuous-time and continuous-state stochastic process.Let {X t , t ≥ 0} be a Gamma process.It is with the following properties 17 : 1 X 0 0 with probability one, , for all τ > t ≥ 0, 3 X t has independent increments, where v t is the shape function which is a non-decreasing, right-continuous, real-valued function for t ≥ 0 with v 0 ≡ 0, u > 0 is the scale parameter, and G • is the Gamma distribution.Let X t denote the loss quantity of the cutting tool dimension due to wear LQCTDW at time t, t ≥ 0. In accordance with the definition of the Gamma process, the probability density function of X t is given by where Γ • is the Gamma function, I A x 1 for x ∈ A and I A x 0 for x / ∈ A. Its expectation and variance are, respectively, expressed as

2.3
Empirical studies show that the expected deterioration at time t is often proportional to the power law 17 : where a > 0 or c > 0 and b > 0.
The non-stationary Gamma process with parameters c, b, and u is employed to model the deterioration process of cutting tool due to wear under the invariant machining condition.Here, the invariant machining condition means that the cutting speed, feed rate, depth of cut, work-piece material, work-piece geometry, contact angle, and so on 2 are constants in the machining process.c, b, and u can be estimated by the introduced method in 17 when the data of LQCTDW are collected under the identical machining condition.The data are composed of inspection times t i , i 0, 1, . . ., n, where 0 t 0 < t 1 < t 2 < • • • < t n , and corresponding LQCTDW x i , i 0, 1, . . ., n, where 0

Reliability and Sensitivity Analysis without Compensation and Machining Error of Cutting Tool
The cutting tool reliability model under the invariant machining condition is discussed in the first case where the compensation for the cutting tool wear is not carried out during the machining process and cutting tool is manufactured accurately in the section.Let the maximum permissible machining error of the machine tool be noted by δ. δ is a constant and obtained by referring to the technical parameters of the considered machine tool.The time-variant limit state function of cutting tool in the first case is given by According to Section 2.1, the cutting tool reliability model is The effect of each parameter in 2.6 on the cutting tool reliability could be found by sensitivity analysis.According to the derivation theorem of integration of variable upper limit, the sensitivity of the cutting tool reliability to the maximum permissible machining error δ is calculated by The sensitivity to b is

2.8
The sensitivity to c can be calculated by 2.9 The sensitivity to u is calculated by

Reliability and Sensitivity Analysis with Machining Error of Cutting Tool and without Compensation
In the second case, where cutting tool has the machining error and the compensation for the cutting tool wear is not carried out, the time-variant limit state function of cutting tool under the invariant machining condition is given by where δ d is the machining error cutting tool and equal to the difference between the actual dimension and the ideal one of cutting tool.It is a stochastic real number and follows the normal distribution with expectation δ d 0 and standard deviation σ δ d .According to 2.11 ,

2.12
When δ d y, the cutting tool reliability is where y must not be more than δ and less than −δ.If y is more than δ or less than −δ, the cutting tool reliability is 0. Therefore, 2.13 can be rewritten by Then, the cutting tool reliability model in the second case is assessed by The sensitivity of 2.15 to σ δ d can be written by The sensitivity to δ, b, c and u can be, respectively, expressed by 2.17 where

Reliability and Sensitivity Analysis with Compensation for Cutting Tool Wear
Nowadays, there are three methods to compensate the cutting tool wear.The first is the offline compensation method such as 21-25 , where the compensation quantity at time t for LQCTDW is estimated by a compensation function prior to machining.The second is the online compensation method such as 26-29 , where the compensation quantity for LQCTDW is determined according to the actual LQCTDW which is measured by the direct or indirect method during machining.This kind of method could be classified into two types.One is the regular compensation method where the actual LQCTDW is measured and then it is compensated periodically in machining process, such as 27, 30-32 .The other is the realtime compensation method where the actual LQCTDW is estimated and then compensated real-timely and continuously, such as 26, 28, 29 .The third is the combination compensation method where two or more compensation methods are combined to decrease the machining error due to LQCTDW, such as 22, 26, 33 .

Reliability and Sensitivity Analysis Using Off-Line Compensation Method
Let the compensation function be denoted by h t in the off-line compensation method, where h t is a continuous real function and h t ∈ 0, ∞ .Then, the time-variant limit state function of cutting tool in the third case where the dimension of cutting tool before working is stochastic and the off-line compensation method used is given by Therefore, the reliability model of cutting tool in the third case could be written by dx dy.

2.24
Its sensitivity to h t and δ can be written, respectively, by 25

Mathematical Problems in Engineering
When h t < 2δ, 2.23 is transformed into dx dy.

2.27
Its sensitivity to h t and δ can be written by

2.28
The sensitivity of the cutting tool reliability in the third case to σ δ d , b, c, and u can be, respectively, expressed using 2.16 , 2.18 , 2.19 , and 2.20 , where the integral upper limit δ y and the integral under limit 0 are only replaced by δ y h t and max −δ y h t , 0 .
In the fourth case, there are two assumptions.One is cutting tool is manufactured accurately or σ δ d of the machining error is close to zero.The other is the off-line compensation method is used.The reliability model of cutting tool under the invariant machining condition could be written by

2.29
When h t ≥ δ, 2.29 is transformed into Its sensitivity to h t and δ can be calculated by

2.31
When h t < δ, 2.29 is transformed into Its sensitivity to h t and δ can be calculated by The sensitivity of 2.29 to b, c and u can be calculated by 2.8 , 2.9 , and 2.10 , where the integral upper limit δ and the integral under limit 0 are only replaced by δ h t and max −δ h t , 0 .

Reliability and Sensitivity Analysis Using Real-Time Compensation Method
When the real-time method is employed to compensate LQCTDW, the time-variant limit state function of cutting tool still can be expressed by 2.21 but h t is the real-time compensation function.h t is determined by measuring LQCTDW.In the fifth case, where the measurement of LQCTDW is accurate, X t − h t in 2.21 is identically equal to 0 and then the reliability model of cutting tool is dy.

2.34
The cutting tool reliability is determined by only two parameters σ δ d and δ.The sensitivity of 2.34 to them are formulated, respectively, by 35

2.36
In the sixth case where the measurement of LQCTDW is not accurate, r t X t − h t is not identically equal to 0 but a stochastic process which is assumed to follow a normal distribution with expectation r t 0 and standard deviation σ r at any time t.σ r could be estimated by the historical data which are collected by the adopted real-time measuring system.Then, the time-variant limit state function of cutting tool in the sixth case is given by where δ d and r t are independent.g 6 t ≥ 0 is equivalent to where Φ • is the cumulative function of standard normal distribution.The sensitivity of 2.39 to δ, σ r , and σ δ d are expressed, respectively, by 41 2.42

Numerical Examples and Discussion
This section will show how the proposed reliability assessment method for cutting tool is applied and how the cutting tool reliability is improved using the proposed reliability model and its sensitivity analysis when cutting tool suffers from the failure due to wear under the invariant machining condition by numerical examples.Let δ 7.5 μm, b 0.8, u 2.1, c 5.0, σ r 0.8, σ δ d 1.5.The off-line compensation function h t is assumed to be equal to the expectation function of the cutting tool wear process ct b /u.
The reliability curves of cutting tool R 1 t , R 2 t , R 3 t , and R 4 t are shown in Figure 1.R 5 t and R 6 t are identically equal to 0.99999942669686 and 0.99998974684970, respectively.From Figure 1, it can be observed that R 3 t or R 4 t is much larger than R 1 t and R 2 t with the increasing of t, and R 4 t is slightly more than R 3 t at any time.This implies that the off-line compensation method could improve the reliability of cutting tool greatly when the compensation function is close to the actual wear of cutting tool and the machining error of cutting tool could decrease the cutting tool reliability when the off-line compensation method is used.According to the calculation results, it can be seen that the reliability of cutting tool always could be kept at very high level within the considered time range and the measurement error of the wear decrease the reliability of cutting tool when the real-time compensation method is adopted.Moreover, it is obvious that the real-time compensation method could improve the reliability of cutting tool more effectively than the off-line compensation method.According to Figure 1, it can be obtained that R 1 t is less than R 2 t when t is more than one certain value.It implies that the machining error of cutting tool could increases the reliability when t is more than one certain value.R 1 t is compared with R 2 t with the different σ δ d in Figure 2. Figure 2 shows that the added value of the reliability is larger and larger but the reliability in the early phase is decreased greatly with the increasing of the machining error standard deviation of cutting tool when t is more than one certain value.
The sensitivity curves of the cutting tool reliability in the first case R 1 t to δ, b, c, and u are shown in Figures 3, 4, 5, and 6, respectively, and that of R 2 t , R 3 t , and R 4 t are similar.Here, the considered parameter one of δ, b, c, and u is the only variable.For example, δ    considered parameter at the time t.R 1 t is more sensitive to b and u than δ and c according to Figure 8.
The sensitivity curves of the cutting tool reliability in the second case R 2 t to σ δ d are shown in Figure 7, where σ δ d is the only variable and that to δ is also similar to Figure 6.The sensitivity curves of R 2 t to σ δ d , δ, b, c, and u are shown in Figure 8 simultaneously where   On the basis of Figure 3 to Figure 8, the sensitivity of R 1 t and R 2 t to δ and u are always more than zero but that to other parameters are less than or close to zero.Therefore, R 1 t and R 2 t could be increased by increasing δ or u or by decreasing b, c, or σ δ d in the numerical example.
The sensitivity curves of the cutting tool reliability in the third case R 3 t to h t is shown in Figure 9 where h t is the only variable and that to δ, σ δ d , b, c, and u are not given because they have the similar law to the sensitivity curves of R 2 t .The sensitivity curves of R 3 t to h t , σ δ d , δ, b, c, and u are shown in Figure 10 simultaneously where t is the only variable, δ 7.5 μm, b 0.8, u 2.1, c 5.0, σ δ d 1.5, and h t ct b /u.On the basis of Figure 9, the sensitivity of R 3 t to h t is more than zero when h t is less than one certain positive number and it is less than zero when h t is more than this positive number.The sensitivity of R 3 t to δ is more than zero and has the maximum when δ is the only variable and changes during its domain from Figure 6.R 3 t is the most sensitive to b among all parameters according to Figure 10.
The sensitivity curves of R 4 t to δ and h t are similar to those shown in Figures 6 and 9, respectively.When t is the only variable, δ 7.5 μm, b 0.8, u 2.1, c 5.0, and h t ct b /u, the sensitivity curves of R 4 t to h t , δ, b, c, and u are similar to those in Figure 10.From Figure 9, it can be obtained that the reliability of cutting tool can be improved by using the off-line compensation method only if h t is assigned properly.
The sensitivity curves of the cutting tool reliability in the fifth case R 5 t to δ is shown in Figure 11 where δ is the only variable.The sensitivity curves of R 5 t to σ δ d and R 6 t to σ r and σ δ d is similar to ∂R 3 t /∂σ δ d in Figure 10.The sensitivity curves of R 6 t to δ are similar to those in Figure 11.According to Figures 10 and 11, it can be observed that R 5 t and R 6 t are more sensitive to δ than other parameters, they can be improved by increasing δ, R 5 t could be decreased when σ δ d increases and R 6 t will be decreased with the increasing of σ r or σ δ d .

Conclusions
The cutting tool reliability assessment and its sensitivity analysis under the invariant machining condition are presented in this paper.Here, cutting tool suffers from the failure due to wear and the wear process is modeled by a Gamma process.The deterioration of cutting tool is assumed to be continuous.Therefore, the reliability assessment method for cutting tool is practical.
The sensitivity analysis of the cutting tool reliability offers the approach to improve the reliability under six cases when the machining condition is invariant.

1 Figure 5 :
Figure 5: Sensitivity curves of R 1 t to c with the different machining time t when δ 7.5 μm, u 2.1, b 0.8.

Figure 6 :Figure 7 :
Figure 6: Sensitivity curves of R 1 t with respect to u with the different machining time t when δ 7.5 μm, c 5.0, b 0.8.

X t :
Loss quantity of the dimension of cutting tool due to wear u:Scale parameter of Gamma processΓ • : Gamma function g • : Limit state function x i : Measurement value of X t δ d :Machining error of cutting tool h t : Compensation function of the cutting tool wear σ r :Standard deviation of r t δ:Maximum permissible machining error of the machine tool G • : Gamma distribution a, c, b: Parameters of shape function v t : Shape function of Gamma process