Intelligent real-time monitoring of Computer Numerical Control tool wear based on a fractional-order chaotic self-synchronization system

Abstract

Tool wear and damage are unavoidable in machines which operate for long periods. The main objective of this study was the prediction and understanding of tool wear ahead of time. To achieve this a real-time monitoring system was needed which incidentally also improves product yield and efficiency. Traditionally, tool wear and working life have been estimated by past experience. To monitor the state of the tool, we used a method employing fractional-order chaotic self-synchronization system matched with extension theory to analyze, extract, and measure tool vibration signals. The fractional-order Chen–Lee chaotic system was used to detect differences in micro vibrations, resulting from different degrees of tool wear, and these were introduced into the master and slave systems. The extreme changes produced by micro differences in the chaotic system, and matter element module design based on the extension theory, were used to accurately determine tool status. Chaos theory, wavelet packet analysis, and Fast Fourier transform were then used to compare the results. This method can reduce the number of sensors and the time required for real-time monitoring of expected life compared to previous determination methods where temperature and current diagnostics must be included.

Keywords

Fractional-order chaotic self-synchronization system Chen–Lee system Chua’s circuit extension theory Computer Numerical Control tool wear

Introduction

As we move into Industry 4.0 the need for more efficient and cost-effective machine tools is expanding rapidly. Wear is focused on the cutting tools, in addition to every moving part of the machines themselves, which in turn affect tool wear. Excessively worn tools produce defective products and reduce both yield and quality. To improve this situation, and to increase product yield and efficiency, real-time monitoring of tool condition is a major concern. The objective of this study was the monitoring of the actual state of the tool in real time, which can lead to the establishment of methods that can lead to improved machine efficiency.

The actual condition of a cutting tool can be described as normal, worn, or damaged.^1,2 Normal processing wears a tool, and to clearly describe its state, we used the following descriptions in this study: normal, slight wear, medium wear, and severe wear, which more accurately indicates tool state.

Many recent studies have focused on the diagnosis of mechanical systems, such as bearings,^3–5 AC generators,⁶ gearboxes.^7,8 Several different methods have been used for the monitoring and prediction of tool life and wear.^9–12 Different diagnostic methods have also been used to determine error state, but most of them have given no consideration to the most common cause of defects, the actual tool. In this study, tungsten carbide steel cutters (specification of 10×10×30×75×4T) were used to cut aluminum workpieces, the goal being to determine the rate of tool wear over long periods of machining. The usual methods used for the determination of tool wear involve the use of sensors to monitor temperature, rotor current, and vibration signals.¹³ In practice these sensors can interfere with each other and there is a high probability that readings will be in error. The use of only one sensor in this study eliminates this problem entirely and also reduces cost. In traditional methods, frequency number, usage time, and cutting distance¹⁴ were used to determine wear but this approach is subjective and has no clear differences. Such traditional methods are unsuitable for the determination of tool life for a used tool and are not useful for a real-time monitoring system. The method used in this study requires only one operation cycle for the analysis of tool wear status. An artificial neural network^15–17 is used to analyze the signals that indicate normal and defective operation. Extracted data are processed by Fourier transform to determine eigenvalue and differentiate status. However, an artificial neural network requires many samples for the establishment of a useable database and this makes it unsuitable for real-time monitoring. Furthermore, extracting tool use frequency, tool use time, and tool use distance is cumbersome and cannot be used to determine the life of a tool that has been used. Tool life determination using this method requires brand new tools. The proposed differential algorithm¹⁸ allows the machine to learn the limits and self-adjust the system parameters. However, self-adjustment by the machine can be unreliable and there is a high probability of error. The four tool wear categories are determined by processed signals. Different damage levels have different eigenvalues which can be used to establish a dynamic error matter element model for different wear categories. However, if an optical microscope^19,20 is used to determine wear level, it is time consuming to pull the tool after each processing cycle to check its status. Although this method is accurate and viable, the cost is too high and we did not take it into consideration. Finally, we explored the correlation between the material that is being cut and the resulting tool wear. For this, we referenced previous work,^21–23 where Ni-Hard 4, DIN 1.4301, and TA15 had been tested. These investigations clearly showed that the condition of the worked material correlated with the speed of tool wear.

Our method is an improvement on previous work and uses chaotic self-synchronization to extract the eigenvalue of the tool signal focal point. This information is used to determine the dynamic error of the focal point. Analysis of the chaotic characteristics can identify tool status much faster than an artificial neural network which requires learning time. The single sensor used need only be an accelerometer and the resulting accuracy rate is equivalent to that of the three types of sensor needed for other methods. Finally, we compared three types of analysis using the Chen–Lee chaotic self-synchronization system for analysis. Extension theory was then used to analyze the wear state of tools as represented by the current eigenvalue.

In this study the fractional-order derivative method has also been used to improve the effectiveness and determine the initial value of the chaotic system. This method was proposed by Hu and He²⁴ who also used the Fornberg–Whitham equation for all kinds of physical and engineering problems,²⁴ improvement of the Riemann–Liouville derivative,²⁵ and many other applications.^26,27

This method is different from a normal chaotic system where processing has to be done to make a determination. The time required for detection is less and diagnostic precision was better. Rather than using wavelet packet analysis and Fast Fourier transform, these two methods use frequency domain to analyze the signals, which is not an optimal method for target nonlinear system real-time analysis. To realize Industry 4.0 needs, and to improve the status of existing machines, we propose using a smart machine tool real-time wear monitoring system based on fractional-order chaotic self-synchronization system.

Experimental machining system

Figure 1 shows the vertical machining center (QUASER MV154-C) specifications used in this study.²⁸ To measure tool vibration, a triaxial accelerometer was installed on the CNC clamp. The machine was used to process aluminum workpieces for long periods during which the tool status after the processing, and the differences in vibration signal, were observed to determine the current state of tool wear. Figure 2 shows some unprocessed vibration signals from tools at different stages of wear.

Figure 1.

The MV154-C machine used in this study.

Figure 2.

Vibration signals from tools at different stages of wear.

Chaos-fractional-extension theory

Chaos theory is nonlinear, so the measured signals pass through a chaotic system and are influenced by chaotic attractors. The result is that the signal produces a cycle-less movement track. This movement track will produce extremely large changes in response to the input of minute changes, which makes the method very suitable for use in chaotic system. When matched with a master–slave chaotic system we can compare normal and wear status very easily and use the dynamic error to build a matter element module based on the extension theory.

Chaos theory

The master–slave self-synchronization system

The main objective of the master–slave system is control of the slave status by the master system. The master–slave system function^4,29 is demonstrated by equation (1)

{\begin{cases} \dot{X} = A X + F (X) \\ \dot{Y} = A Y + F (Y) + u \end{cases}

(1)

$X \in R^{N}$ and $Y \in R^{N}$ are the status vectors. To realize signal processing, $N$ is a 3 × 3 system matrix. F(X) and F(Y) is the nonlinear vector. u is the nonlinear control item.

Chen–Lee chaotic system

Using the Chen–Lee chaotic system as an example,^30,31 $X = {[x_{1}, x_{2}, x_{3}]}^{T}$ , $Y = {[y_{1}, y_{2}, y_{3}]}^{T}$ , and $U = {[u_{1}, u_{2}, u_{3}]}^{T}$ . The master system MS and slave system SS are demonstrated by equation (2)

{\begin{cases} M a s t e r s y s t e m (M S) = A X + F (X) = [\begin{matrix} a & - x_{3} & 0 \\ x_{3} & b & 0 \\ \frac{1}{3} x_{2} & 0 & c \end{matrix}] \\ S l a v e s y s t e m (S S) = A Y + F (Y) = [\begin{matrix} a & - y_{3} & 0 \\ y_{3} & b & 0 \\ \frac{1}{3} y_{2} & 0 & c \end{matrix}] \end{cases}

(2)

To realize slave system self-tracking of master system synchronization dynamic error, we designed a control item $u_{1} = u_{2} = u_{3} = 0$ and defined the error status as $e = {[e_{1}, e_{2}, e_{3}]}^{T}$ . Of which, $e_{1} = x_{1} - y_{1}$ , $e_{2} = x_{2} - y_{2}$ , and $e_{3} = x_{3} - y_{3}$ are expressed as in equation (3) after the error system has been processed

[\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \\ {\dot{e}}_{3} \end{matrix}] = [\begin{matrix} a & - e_{3} & 0 \\ e_{3} & b & 0 \\ \frac{1}{3} e_{2} & 0 & c \end{matrix}] [\begin{matrix} e_{1} \\ e_{2} \\ e_{3} \end{matrix}]

(3)

Chaotic phenomenon must produce chaotic attractors, and equation (4) parameters $[a, b, c]$ must be set by equation (4) conditions

a > 0, b < 0, c < 0, and 0 < a < - (b + c)

(4)

Fractional order

Regular systems have all been simplified as an integer-order mathematical model, but physical systems are not all integer order. Actual system signal analysis is almost all in fractional-order form. The main characteristic of fractional order is that it possesses memory. Previously, fractional-order theory was less mature. However, today’s theory is nearly perfect. Most studies that use the fractional-order definition proposed by Grünwald–Letnikov^32,33 can be defined as in equation (5)

D_{e}^{\pm γ} e^{m} \approx \frac{Γ (m + 1)}{Γ (m + 1 \pm γ)} e^{m \pm γ}

(5)

Of which, $m$ is any real number, $e$ is the dynamic error, and γ can be the selected phenomenon as needed. According to |γ|, there are two following rules:

0.00 < | γ | \leq 0.20 : used for quantitative isometric value and proportion application

0.20 < | γ | \leq 1.00 : used for controlling nonisometric values and categorization application

Because fractional order is used, equation (2), Xs must be organized as equation (6) with fractional-order function

\frac{d}{d t} \frac{d^{- γ}}{d t^{- γ}} [\begin{matrix} e_{1} \\ e_{2} \\ e_{3} \end{matrix}] \approx A (e) \frac{d^{- γ}}{d t^{- γ}} [\begin{matrix} e_{1}^{1} \\ e_{2}^{1} \\ e_{3}^{1} \end{matrix}] + \frac{d^{- γ}}{d t^{- γ}} [\begin{matrix} - e_{2} e_{3} e_{1}^{0} \\ e_{1} e_{3} e_{2}^{0} \\ \frac{1}{3} e_{1} e_{2} e_{3}^{0} \end{matrix}] \Rightarrow [\begin{matrix} D^{q} e_{1} \\ D^{q} e_{2} \\ D^{q} e_{3} \end{matrix}] \approx [\begin{matrix} a^{'} & 0 & 0 \\ 0 & b^{'} & 0 \\ 0 & 0 & c^{'} \end{matrix}] [\begin{matrix} e_{1}^{1 + γ} \\ e_{2}^{1 + γ} \\ e_{3}^{1 + γ} \end{matrix}] + [\begin{matrix} \frac{- Γ (1) e_{2} e_{3} e_{1}^{γ}}{Γ (1 + γ)} \\ \frac{Γ (1) e_{1} e_{3} e_{2}^{γ}}{Γ (1 + γ)} \\ \frac{Γ (1) e_{1} e_{2} e_{3}^{γ}}{Γ (1 + γ)} \end{matrix}]

(6)

Of which, $q = (1 - γ)$ and $0 < q < 1$ . $Γ (•)$ is the Gamma function that makes equation (5) produce chaotic attractors, and converted with equation (7)^34,35

a^{'} = \frac{a Γ (2)}{Γ (2 + α)} > 0, b^{'} = \frac{b Γ (2)}{Γ (2 + α)} < 0, c^{'} = \frac{c Γ (2)}{Γ (2 + α)} < 0, where 0 < a^{'} < - (b^{'} + c^{'})

(7)

System parameter $a^{'}, b^{'}, c^{'}$ is the nonzero constant. Phase plane displays various dynamic error behaviors when fractional order is q. To allow equation (6) to realize self-synchronized tracking, its dynamic error is set as: $e_{1} [i] = x [i] - y [i]$ , $e_{2} [i] = x [i + 1] - y [i + 1]$ , $e_{3} [i] = x [i + 2] - y [i + 2]$ , and $i = 1, 2, 3, \dots, n - 2$ . Then $Φ_{1} [i], Φ_{2} [i],$ Φ₃ [i] can be defined as equation (8)

[\begin{matrix} Φ_{1} [i] \\ Φ_{2} [i] \\ Φ_{3} [i] \end{matrix}] = [\begin{matrix} a^{'} & 0 & 0 \\ 0 & b^{'} & 0 \\ 0 & 0 & c^{'} \end{matrix}] [\begin{matrix} (e_{1} {[i]}^{1 + γ}) \\ (e_{2} {[i]}^{1 + γ}) \\ (e_{3} {[i]}^{1 + γ}) \end{matrix}] + [\begin{matrix} \frac{- Γ (1) e_{2} [i] e_{3} [i] {(e_{1} [i])}^{γ}}{Γ (1 + γ)} \\ \frac{Γ (1) e_{1} [i] e_{3} [i] {(e_{2} [i])}^{γ}}{Γ (1 + γ)} \\ \frac{Γ (1) e_{1} [i] e_{2} [i] {(e_{3} [i])}^{γ}}{Γ (1 + γ)} \end{matrix}]

(8)

Extension theory

Studies of extension theory³⁶ began in 1976, and in 1983 Tsai Wen published the first thesis. The extension theory was used to process contradictory problems. The status difference between two items is observed, and special characteristics are induced according to the regularity of the two items. The matter element model can be used to quantify the state, and correlation is used as the basis for determination. This can describe the indicators of tool state in simple terms and can clearly express the current state of the tool. The scope of the fuzzy set $< 0, 1 >$ can be expanded as required for the study. Figure 3 is an illustration of the fuzzy and extension sets.

Figure 3.

Fuzzy set and extension set.

In this study we used extension theory as the method for determining the level of tool damage. The Chen–Lee chaotic system matched with extension theory was used to realize the real-time detection system. The size of the determined classic domain association function was compared with the set matter element model. This was used to determine the characteristic in closest conformation to the tool wear state, which was used to diagnose tool condition by comparison with the associated function.

Signal processing and simulation

Integer- and fractional-order chaotic self-synchronization of dynamic error

The chaotic system was used to convert the tool vibration signal (using the e₁, e₂ dynamic error). The signal was measured based on the four states (normal, light wear, medium wear, and severe wear). The dynamic error chart was drawn with dynamic error e₁, e₂, which is used to construct the extension theory matter element model, as well as to serve as a basis for determining signal status. The eigenvalue changes are shown in Figure 4. The integer-order chaotic self-synchronization dynamic error track and the fractional-order chaotic self-synchronization of dynamic error are shown in Figures 5 and 6.

Figure 4.

Dynamic error ((a) e₁ and (b) e₂) for every tool state.

Figure 5.

Dynamic error plane for each state.

Figure 6.

Fractional-order (0.9 order) dynamic error plane for each state.

We used counters to calculate and mark the accuracy rate of each order. Evaluation showed integer order and 0.9 order to be the optimal order for this system. A comparison of the advantages and disadvantages of integer order and 0.9 order showed that in terms of accuracy rate, integer order was slightly superior to 0.9 order for real-time monitoring. However, in terms of the amount of calculation needed, integer-order calculations took much longer. Diagnostics done during the same time period showed that the number of diagnostic results for fractional order was higher than that of integer order, and the difference in accuracy was small. Because tool damage diagnosis needs to be done in real time, fractional order 0.9 order was chosen for this system to reduce the possible loss of important signals. The tool diagnostic system flow chart is as shown in Figure 7.

Figure 7.

Tool diagnostic system flow chart.

Wavelet packet analysis and Fast Fourier transform

Most current failure analysis signal processing uses wavelet packet analysis and Fast Fourier transform. Wavelet packet analysis is a frequency domain analysis method where the mother wavelet has to be chosen. However, it cannot be known if the chosen mother wavelet is optimal or not. A Fast Fourier transform samples in the time domain and then converts the samples into a discrete time Fourier transform. However, the objective of this study was the real-time analysis of a nonlinear system, and wavelet packet analysis and Fast Fourier transforms were not used. Figures 8 and 9 show the Fast Fourier and wavelet packet analytical chart, respectively. Fast Fourier analysis uses 60 peak value observations and wavelet packet analysis uses the mean signal value of different states to make a determination.

Figure 8.

Fast Fourier analysis of each tool state.

Figure 9.

Wavelet packet analysis of each tool state.

Finally, wavelet packet analysis and discrete Fourier transform were compared in this study, as shown in Figures 8 and 9. This shows that these two methods were not as good as the fractional-order chaotic self-synchronization system.

Experimental results and real-time monitoring

Experimental results

Integer-order Chen–Lee processing was used to synchronize the chaotic system by the dynamic error. The central point value of each status dynamic error plane e₁ and e₂ was used to produce the extension theory matter element model as shown in Table 1.

Table 1.

Dynamic error matter element model.

State	Normal	Slight wear	Moderate wear	Severe wear
Extension matter element model	x[0, 0.007]y[0, −0.013]	x[0.007, 0.013]y[−0.013, −0.018]	x[0.013, 0.02]y[−0.018, −0.027]	x[0.02, 0.03]y[−0.027, −0.04]

Table 2 shows that simulations of the Chen–Lee chaotic system combined with fractal theory and fractional order (at 0.9 order) and the extension detection method had a very high tool wear state accuracy rate. The fractional-order Chen–Lee chaotic extension detection method only requires the use of dynamic error to determine status with extension theory. However, the integer-order Chen–Lee chaotic system must use fractal theory before it can use extension theory to make a determination. The calculations take much longer than those of the method developed for this study. The integer-order Chen–Lee chaotic system offers no advantages for real-time monitoring and we used the fractional-order Chen–Lee chaotic extension detection method.

Table 2.

Dynamic error matter element model.

Tool wear state	Chen–Lee chaotic system				Wavelet packet analysis (%)	Fast Fourier transform (%)
Tool wear state	Integer order (%)	0.9 order (%)	0.8 order (%)	0.7 order (%)	Wavelet packet analysis (%)	Fast Fourier transform (%)
Normal	93.33	92.21	90.33	89.2	88.32	89.23
Slight	94.6	92.5	91.3	90.2	89.24	90.74
Moderate	94.5	90.5	88.37	87.64	85.36	86.54
Severe	97.4	96.18	93.54	90.72	89.43	88.19

Real-time monitoring system

First, we installed tools in each state of wear and operated each at the same speed and power to extract vibration signals and establish a database for each wear condition. We then drew the chaotic dynamic error phase plane diagram according to the database. This was used to build an extendable matter element model for each state, as shown in Table 1. After the database was established, we used LabVIEW to build the human machine interface for the smart tool status monitoring system. The human–machine interface allows the user to view current tool vibration signals and other statuses. Figure 10 shows that the system can accurately determine the condition of the tool and also shows the state with a colored light. Each state uses the matter element model to calculate the extendable associated function to determine the state of wear of the tool as well as access each extracted tool vibration signal to allow the user to determine the actual stability of the machine.

Figure 10.

Tool condition monitoring system identification signal display: (a) normal, (b) light wear, (c) medium wear, and (d) severe wear.

Conclusion

In this study, we utilized the fractional-order chaotic self-synchronization system combined with the extension theory determination method. The results showed that this method is viable. Our method uses only one sensor compared to others which use three; this makes it less expensive, reduces the amount of necessary calculation, and provides excellent accuracy. This method also reaches the Industry 4.0 category. If eigenvalues for detecting defective products are added, machine parameters can be adjusted to improve product yield. We used the extendable detection method in this study. First, we introduced the tool vibration signal into the master system and extracted dynamic error produced by the chaotic system. The dynamic error was then synchronized to obtain the displayed eigenvalue used to build the extendable matter element model which detects the current condition of the cutting tool. Although the integer-order Chen–Lee chaotic system is more accurate than the fractional order, the fractional-order Chen–Lee chaotic system is faster. Tool damage occurs instantaneously and after consideration of many factors, the fractional-order Chen–Lee chaotic system was selected for this study. The amount of data needed for calculation is small and not much processing space is needed. The method proposed here is simple, accurate, fast, and extendable. Real-time application of this system showed that this method can accurately determine the state of tool wear. If this system were added to embedded systems, it could provide both malfunction detection and adjustment and control in real time.

Footnotes

Acknowledgements

Technical support and tool data have been provided by Professor Her-Terng Yau and Mr I-Hung Lin, respectively.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial support of this research by Ministry of Science and Technology of R.O.C. under the projects no. MOST-106-2622-E-167-020-CC3 is greatly appreciated.

References

Bhagat

Nalbalwar

SL.

LabVIEW based tool condition monitoring and control for CNC lathe based on parameter analysis. In: 2016 IEEE international conference on Recent Trends in Electronics, Information & Communication Technology (RTEICT), Bangalore, 2016, pp.1386–1388.

Hyeon Sung

Ji-Hyeong

Su-Young

et al. A tool breakage detection system using load signals of spindle motors in CNC machines. In: 2016 eighth International Conference on Ubiquitous and Future Networks (ICUFN),Vienna, 2016, pp.160–163.

Tahir

Khan

Iqbal

, et al. Enhancing fault classification accuracy of ball bearing using central tendency based time domain features. IEEE Access 2016; 99: 72–83.

Yau

Kuo

Chen

, et al. Ball bearing test-rig research and fault diagnosis investigation. IET Sci Meas Technol 2016; 10: 259–265.

Dong

Luo

Zhong

, et al. Fault diagnosis of bearing based on the kernel principal component analysis and optimized k-nearest neighbour model. J Low Freq Noise Vib Active Control 2017; 36: 354–365.

Abad

MRAA

Moosavian

Khazaee

Wavelet transform and least square support vector machine for mechanical fault detection of an alternator using vibration signal. J Low Freq Noise Vib Active Control 2016; 35: 52–63.

Chacon

JLF

Andicoberry

Kappatos

, et al. An experimental study on the applicability of acoustic emission for wind turbine gearbox health diagnosis. J Low Freq Noise Vib Active Control 2016; 35: 64–76.

Kane

Andhare

AB.

Application of psychoacoustics for gear fault diagnosis using artificial neural network. J Low Freq Noise Vib Active Control 2016; 35: 207–220.

Cui

Zhao

Tian

Tool wear in high-speed face milling of AISI H13 steel. Proc IMechE, Part B: J Engineering Manufacture 2012; 226: 1684–1693.

10.

Zhang

, et al. Research on tool wear prediction based on Archard agglomeration scavenging theory. Adv Mech Eng 2017; 9: 1–7

11.

Yang

Wang

, et al. A novel monitoring method for turning tool wear based on support vector machines. Proc IMechE, Part B: J Engineering Manufacture 2016; 230: 1359–1371.

12.

Madhusudana

Budati

Gangadhar

, et al. Fault diagnosis studies of face milling cutter using machine learning approach. J Low Freq Noise Vib Active Control 2016; 35: 128–138.

13.

Duro

Padget

Bowen

, et al. Multi-sensor data fusion framework for CNC machining monitoring. Mech Syst Signal Process 2016; 66–67: 505–520.

14.

Yueh-Ling

Chih-Chieh

Hung-Sheng

Developing a cloud virtual maintenance system for machine tools management. In: 2015 11th international conference on heterogeneous networking for Quality, Reliability, Security and Robustness (QSHINE), Taipei, 2015, pp.358–364.

15.

Shashidhara

Suneel

Gadre

et al. Accuracy improvement for CNC system using wavelet-neural networks. In: Proceedings of IEEE international conference on industrial technology 2000 (IEEE Cat. No. 00TH8482), Goa, 2000, vol. 1 and 2, pp.341–346.

16.

Salimi

Özdemir

Safarian

Designing an artificial neural network based model for online prediction of tool life in turning. Int J Adv Des Manuf Technol 2015; 8.

17.

Drouillet

Karandikar

Nath

, et al. Tool life predictions in milling using spindle power with the neural network technique. J Manuf Processes 2016; 22: 161–168.

18.

Yang

Zhou

Tsui

KL.

Differential evolution-based feature selection and parameter optimisation for extreme learning machine in tool wear estimation. Int J Prod Res 2016; 54: 4703–4721.

19.

Aramesh

Attia

Kishawy

, et al. Estimating the remaining useful tool life of worn tools under different cutting parameters: a survival life analysis during turning of titanium metal matrix composites (Ti-MMCs). CIRP J Manuf Sci Technol 2016; 12: 35–43.

20.

Dutta

Pal

Sen

On-machine tool prediction of flank wear from machined surface images using texture analyses and support vector regression. Precis Eng 2016; 43: 34–42.

21.

Kir

Oktem

Col

, et al. Determination of the cutting-tool performance of high-alloyed white cast iron (Ni-Hard 4) using the Taguchi method. Mater Tehnol 2016; 50: 239–246.

22.

Dubovska

Majerik

Cep

, et al. Investigating the influence of cutting speed on the tool life of a cutting insert while cutting DIN 1.4301 steel. Mater Tehnol 2016; 50: 439–445.

23.

Liu

, et al. Tool life and surface integrity in high-speed milling of titanium alloy TA15 with PCD/PCBN tools. Chin J Aeronaut 2012; 25: 784–790.

24.

J-H.

On fractal space-time and fractional calculus. Thermal Science 2016; 20: 773–777.

25.

Khan

Faraz

, et al. A new fractional analytical approach via a modified Riemann–Liouville derivative. Appl Math Lett 2012; 25: 1340–1346.

26.

Sayevand

Pichaghchi

Analysis of nonlinear fractional KdV equation based on He’s fractional derivative.Nonlinear Science Letters A 2016; 3: 77–85.

27.

Wang

Zhang

Y-F

Liu

J-G

et al. A short review on analytical methods for fractional equations with he’s fractional derivative. Thermal Science 2017; 21: 36–36.

28.

I. Quaser Machine Tools, http://www.quaser.com/english/products.php?mode=proDetail&cid=1478590037&pid=1467253710&topcid=1467253473.

29.

Liu

Guo

RW.

Control problems of Chen-Lee system by adaptive control method. Nonlinear Dyn 2017; 87: 503–510.

30.

Wang

Lao

Chen

, et al. Implementation of the fractional-order Chen-Lee system by electronic circuit. Int J Bifurcation Chaos 2013; 23: 10.

31.

Kuo

Hsieh

Yau

, et al. Research and development of a chaotic signal synchronization error dynamics-based ball bearing fault diagnostor. Entropy 2014; 16: 5358–5376.

32.

Chen

Lin

, et al. Phonographic signal with a fractional-order chaotic system: a novel and simple algorithm for analyzing residual arteriovenous access stenosis. Med Biol Eng Comput 2013; 51: 1011–1019.

33.

Rajan

Muniyappan

Park

, et al. Stability of fractional differential equation with boundary conditions. J Comput Anal Appl 2017; 23: 750–757.

34.

C-M

, et al. Synchronizing chaotification with support vector machine and wolf pack search algorithm for estimation of peripheral vascular occlusion in diabetes mellitus. Biomed Signal Process Control 2014; 9: 45–55.

35.

Chen

W-L

Chen

Lin

C-H

, et al. Phonographic signal with a fractional-order chaotic system: a novel and simple algorithm for analyzing residual arteriovenous access stenosis. Med Biol Eng Comput 2013; 51: 1011–1019.

36.

Wang

Huang

Liou

KJ.

Application of extension theory with chaotic signal synchronization on detecting islanding effect of photovoltaic power system. Int J Photoenergy 2015; 2015: 1–10.