Machining condition-based stochastic modeling of cutting tool’s life

Abstract

One of the major problems in the application of machining processes is the cutting tool life estimation. In this regard, different studies with various assumptions have been conducted to analyze tool wear characteristics under various cutting conditions to achieve different objectives. Traditional models for the analysis of tool life are mostly based on deterministic approaches, and the variations in cutting conditions are overlooked, and the tool life is not precisely matched with predicted values by these methods. In recent years, researchers have considered using the stochastic approach in forecasting tool life. Among them, Weibull distribution has special significance. One problem in using these approaches is the accurate estimation of tool’s life distribution functions based on the empirical information. In other words, although many researchers have considered Weibull an appropriate distribution for the cutting tool life modeling, however, the estimation of its parameters has certain inherent complexities. In this research, a hybrid methodology is presented to determine the parameters of the tool life distribution, by using the design of experiment (DOE) based on Box-Behnken design (BBD), total time on the test (TTT) transform, and golden section search (GSS). The estimation method of Weibull distribution parameters in this paper is compared with well-known techniques such as the least square method and maximum likelihood estimation. Finally, the proposed methodology was implemented in a case study, and the results were reported. The values of R² for shape and scale parameters are 92.52% and 96.80%, respectively, which confirm the adequacy of the proposed methodology in the practical applications.

Appendix A

To explain the golden section (Heath et al. [40]), a geometrical interpretation is provided. It is assumed that a straight line of length R is divided into two parts so that the ratio of the longer part to the whole line is equal to the ratio of the shorter part to the longer part. In other words, if it is assumed that line AB is divided by point C, such that the length of AC and CB denoted by r₁ and r₂, respectively, and r₂ < r₁, then

$$ s=\frac{r_1}{r_2}=\frac{r_2}{R} $$

(21)

Given that R = r₁ + r₂ , Eq. 18 is rewritten as follows:

$$ s=\frac{r_2}{R}=\frac{r_2}{r_1+{r}_2} $$

(22)

Dividing the numerator and the denominator of Eq. 22 by r₂ and using Eq. 21, we obtain the following:

$$ s=\frac{\raisebox{1ex}{${r}_2$}\!\left/ \!\raisebox{-1ex}{${r}_2$}\right.}{\raisebox{1ex}{${r}_1$}\!\left/ \!\raisebox{-1ex}{${r}_2$}\right.+\raisebox{1ex}{${r}_2$}\!\left/ \!\raisebox{-1ex}{${r}_2$}\right.}=\frac{1}{s+1} $$

(23)

Equation s² + s − 1 = 0 is used to calculate s so that the roots of this equation are obtained through $s=\left(-1+\sqrt{5}\right)/2$ and $s=\left(-1-\sqrt{5}\right)/2$. The positive root of this equation is called the golden section, which is denoted by γ in this paper. This ratio is essentially used by the GSS to optimize the uni-modal functions. This paper seeks to determine the specific value of α denoted by α^∗, which minimizes the SSE. To this end, interval [α_min, α_max] is initially defined to find the optimized point. Then, α₁ and α₂ are calculated, using Eqs. 24, 25, and 26:

$$ \gamma =\frac{-1+\sqrt{5}}{2} $$

(24)

$$ {\alpha}_1=\gamma .{\alpha}_{\mathrm{min}}+\left(1-\gamma \right).{\alpha}_{\mathrm{max}} $$

(25)

$$ {\alpha}_2=\gamma .{\alpha}_{\mathrm{max}}+\left(1-\gamma \right).{\alpha}_{\mathrm{min}} $$

(26)

Following the calculation of α₁ and α₂SSE for each point is calculated and denoted by SSE(α₁) and SSE(α₂), respectively. If SSE(α₁) < SSE(α₂), α^∗ belongs to interval [α_min, α₂]; otherwise, α^∗ belongs to [α₁, α_max]. Given this clarification, each iteration of the GSS involves two search intervals, which only one of them will be selected for the following searches. The lengths of these intervals need to be equal. The explanation provided above expresses the first GSS iteration. In the next iterations, the search interval will be updated, and Eqs. 25 and 26 will be used to obtain α₁ and α₂.The algorithm will continue likewise until the stop condition is met. The stop condition is defined by the GSS when the length of the search interval is less than ε. Figure 8 shows the first iteration of the GSS algorithm in the interval [α_min, α_max], where, e.g., SSE(α₁) < SSE(α₂).

Fig. 8

The first iteration in the golden section search algorithm

Furthermore, two characteristics of the golden sections are initially described to define the convergence rate of the GSS. The first characteristic is 1 − γ = γ², and the second is φ = 1 + γ = (1/γ), leading to the definition of the golden ratio. In the latter, φ is named the golden ratio, which equals to $\varphi =0.5\times \left(1+\sqrt{5}\right)$. If the number of GSS iterations needed to reach the stop condition equals NOI (number of iterations), the convergence rate of the GSS is φ^NOI [41]. Likewise, Shao et al. [42] showed that the length of the search interval by NOI = 15 could decrease to less than 1% of the length of the primary interval in the GSS algorithm. Figure 9 shows the pseudo-code of the GSS algorithm.

Fig. 9

Pseudo-code of the golden section search algorithm