A novel online chatter detection method in milling process based on multiscale entropy and gradient tree boosting

Mechanical Systems and Signal Processing 135 (2020) 106385

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

A novel online chatter detection method in milling process based on multiscale entropy and gradient tree boosting Kai Li a, Songping He a,⇑, Bin Li a,b, Hongqi Liu b, Xinyong Mao b, Chengming Shi a a b

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China National NC System Engineering Research Center, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Received 26 July 2019 Accepted 19 September 2019

Keywords: Angular synchronous averaging Multiscale permutation entropy Multiscale power spectral entropy Online chatter detection Gradient tree boosting

a b s t r a c t In the milling process, chatter, which results in poor surface quality, dimensional errors, reduced cutter and machine life, is one of the main limitations on performance. In this paper, a novel method of online chatter detection for milling processes is developed. In this method, first, the spindle revolution period component is obtained via angular synchronous averaging (ASA). Then, the residual part related to chatter information is calculated by subtracting the periodic component. Secondly, the multiscale permutation entropy (MPE) and multiscale power spectral entropy (MPSE) of the residual part are calculated, and the Laplacian score (LS) for feature selection is applied to select the optimal sensitive scale features with generalization. Online chatter detection that is based on selected sensitive scale features by splitting signal up into (overlapping) frames in milling process. Finally, a trained gradient tree boosting (GTB) can be used to intelligent diagnosis of the chatter severity level. The analysis results show that the proposed method can effectively detect the onset of chatter under stable cutting conditions and variable cutting conditions, which is simpler and more robust than the existing methods. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Chatter is a self-excited vibration, which is accompanied by instability, chaotic behavior and abnormal fluctuation. Chatter is a major problem that leads to a poor surface finish, low material removal rate, machine tool failure, increased tool wear, excessive noise and increased cost in machining applications [1–3]. To avoid chatter in the production process, the operator always takes time to select the appropriate cutting parameters, which substantially reduces the production efficiency. Although the analysis method can predict chatter and identify the optimal cutting parameters to avoid chatter, the prediction may not be accurate due to the complex environmental interference, variable cutting conditions and simplified errors in the cutting system model [4–6]. Therefore, timely detection of chatter is of high significance for improving the machining efficiency and the quality of the part. Online chatter detection involves signal preprocessing, sensitive feature extraction, and the development of more precise real-time monitoring models [7]. Whether for acceleration signal [8–10], cutting force [7,11–14], motor current [15–17], sound [18,19], or another type of signal, appropriate signal preprocessing for extracting relevant and sensitive features for chatter is vital to accurately and timely chatter detection. For signal preprocessing, various methods have been used in the time domain [10,14,20,21], the frequency domain [22–24] and the time-frequency domain [7,11,12,25-30], to extract ⇑ Corresponding author at: State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology (HUST), 1037 Luoyu Road, Hongshan District, Wuhan 430074, Hubei Province, China. E-mail address: [email protected] (S. He). https://doi.org/10.1016/j.ymssp.2019.106385 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

chatter-related sensitive features. However, for traditional spectral analysis methods that are based on Fourier transform are typically only suitable for stationary signal analysis, which often results in poor robustness as the chatter signal is often nonlinear and instable in the cutting process. Thus, the time-frequency domain signal processing method has been widely used in chatter detection in recent years to locate the time and frequency. In time-frequency analysis, both short-time Fourier transform and S transform are also based on Fourier transform theory, which are more suitable for quasi-stationary signals. Wavelet analysis is a powerful tool in both the time domain and the frequency domain. An effective chatter monitoring method is to use wavelet analysis to decompose the signal and to extract the index based on the decomposed signal. However, it is difficult to determine the suitable wavelet base functions and decomposition levels for the wavelet transform, which substantially influence the analysis results. Empirical mode decomposition (EMD) [28,29] and its improved methods, such as ensemble empirical mode decomposition (EEMD) [9,30] and, variational mode decomposition (VMD) [7,11,12], which is an adaptive analysis method for non-linear and non-stationary signals, have been used recently in chatter detection. Among them, the VMD method, which is an adaptive and non-recursive signal decomposition method that was proposed in 2014, is outstanding in terms of chatter detection performance [31].VMD is more robust than EMD and EEMD in signal preprocessing for chatter detection. Unfortunately, the severe shortcoming of the VMD method lies in the optimal selection of the number of modes K and the quadratic penalty a. The maximum kurtosis calculation, which is based on signal reconstruction, is often highly time-consuming. In addition, the parameters must be reselected for each set of cutting conditions. These disadvantages render challenging the use of VMD in online chatter detection. Angular sampling techniques enable the characterization of the periodic excitations of rotating machinery, even for systems that operate at variable speeds [32]. Meanwhile, the technique of synchronous averaging (SA) can represent the vibration information as a function of the angle of rotation of the rotating machinery and has been widely used in the fault diagnosis of rotating machinery. The motion of the spindle during the milling process is a typical rotary motion. Thus, ASA can be used to extract the periodic component of the raw signal in the milling process. Then, the residual part, which is related to chatter and noise, is calculated by subtracting the periodic component. Compared with the signal preprocessing methods that are discussed above, the ASA method is simpler and its computing costs are cheaper. In addition, for the selection of chatter indicators, identifying chatter from the amplitude spectrum is neither practical nor effective in actual production scenarios. Entropy is an index of the irregularity and complexity that is related to randomness (or, alternatively, uncertainty) from data series, and has been widely applied in chatter detection; the various types of entropy include energy entropy [11,12], power spectral entropy [9,24], sample entropy and approximate entropy [7,33], and permutation entropy [18]. However, due to the background noise and the complexity of the spindle milling system, the information that is associated with chatter is often distributed among various signal time scales and, correspondingly, the changes in the signal dynamics will occur over various scales. Traditional entropy algorithms are single-scale based and, therefore, fail to consider the multiple scales that are inherent in the milling chatter process. Multiscale permutation entropy (MPE) is a nonlinear dynamic method for measuring the randomness and detecting the nonlinear dynamic changes of time series, which was proposed by Aziz and Arif [34] for measuring the irregularity of non-stationary time series. This method is highly robust even in the presence of observational and dynamical noise. Comparing with multiscale sample entropy (MSE), the calculation of MPE is simple and fast, and MPE is invariant with respect to nonlinear monotonous transformations. Power spectral entropy (PSE) is another nonlinear entropy, which is used to characterize signal complexity and has been widely used in signal processing applications, such as speech endpoint detection, medical brain-electrical signal processing, and analysis of the intrinsic dynamics of stock sequence [35–37]. In this study, a novel online chatter detection method for milling processes was proposed that is based on ASA, MPE, MPSE and GTB. ASA was used for cutting force signal preprocessing in a milling process; then, MPE and MPSE were first applied to characterize the multiscale properties of chatter-related signal in the time domain and the frequency domain, respectively, and the Laplacian score (LS) for feature selection was used to select the optimal sensitive scale features with generalization. Finally, for realizing an intelligent diagnosis, gradient tree boosting (GTB) [38] is used, which is an ensemble learning algorithm for multi-classification that can be applied to automatically recognize the chatter severity level. A flow chart of the proposed online chatter detection scheme is presented in Fig. 1. The proposed method was applied under stable cutting conditions and variable cutting conditions, and the results demonstrate the satisfactory performance of the proposed method. 2. Methodology of online chatter monitoring A chatter signal is non-stationary and non-linear. Hence, signal preprocessing and sensitive feature extraction are the key tasks in chatter online detection. Compared with the time-frequency domain analysis that is discussed above, ASA is a simpler and faster signal preprocessing method for extracting chatter-related information. In addition, both MPE and MPSE are used as novel indicators to measure the irregularity of non-stationary signals in various time scales. 2.1. Angular synchronous averaging A cyclostationary process is a stochastic process fsðtÞgt2R that exhibits hidden periodicities in its statistical structure [39]. If the probability density function p of a stochastic process is periodic in t with period T, it can be defined as cyclostationary in the strict sense, i.e. if

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

3

Sampling signals(e.g.force) in milling process

ASA to remove the rotation frequency and tooth passing frequency in sampled signals

Calculate the MPE of the residual part

Calculate the MSPE of the residual part

Laplacian score for sensitive scale features selection

Online chatter detection by splitting signal up into(overlapping) frames

Identify the severity level of chatter by trained GTB

Fig. 1. Flow chart of the proposed method.

ps ðs1 ; . . . ; sn ; t1 ; . . . t n Þ ¼ ps ðs1 ; . . . ; sn ; t1 þ T; . . . ; t n þ T Þ

ð1Þ

where t could represent a generic variable which is not necessarily time. The involved signals can exhibit different kinds of cyclostationarity according to practical application. In our work, the main focus of chatter detection in milling process is at the first-order cyclostationary (CS1), which is the most basic cyclostationary signal. If S is first-order cyclostationary, the first-order moment or expected value ms(t) is periodic with period T:

ms ðtÞ , EfsðtÞg ¼ ms ðt þ TÞ

ð2Þ

where Efg is the ensemble average and ms is the first-order momentum. In general, in rotating machines, the statistics of vibration signals are periodic due to various cyclic mechanical phenomena. In spindle systems, as well as in the majority of the rotating machines, the signals are periodic with respect to the shaft rotation angle h and, consequently, the signals are intrinsically angle-cyclostationary [40]. Angle h can be used to describe the angular coordinates of rotating machinery.

ps ðs1 ; . . . ; sn ; h1 ; . . . hn Þ ¼ ps ðs1 ; . . . ; sn ; h1 þ T; . . . ; hn þ T Þ

ð3Þ

Each milling signal is typically considered to consist of three components: Sp(h) is a periodic component that contains the harmonic of the spindle rotation frequency. Sc(h) is an aperiodic component that is caused by chatter. Sn(h) is an aperiodic component that is due to the environmental and measurement noise. Thus, this model can be can be expressed as

SðhÞ ¼ Sp ðhÞ þ Sc ðhÞ þ Sn ðhÞ

ð4Þ

where Sp(h) and Sc(h) + Sn(h) are assumed to be uncorrelated. Thus, the signal can be divided into two parts: a periodic component Sp(h) and a residual component Sr(h) = Sc(h) + Sn(h), which can be calculated by subtracting the periodic component. Sp(h) can be obtained via ASA from the component signal the component signal S(h) based on the spindle revolution period.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

Thus, the periodic part is the first order moment (expected values) of the signal during one cycle and is defined as follows [41]:

^ x ½h ¼ Sp ðhÞ ¼ l

K1 1X x½h þ K H K K¼0

ð5Þ

where H is the cycle length and, K is the number of consecutive blocks into which the signal is cut. The above ASA method is simple and effective in eliminating the rotation frequency, the tooth passing frequency and their harmonics. The residual part will be used to calculate the multiscale entropy in the next section for online chatter detection. The chatter indicators MPE and MPSE are linked to the contribution of the aperiodic or random part coming from chatter and other noise sources. The noise component is dominant in the signal when cutting in stable conditions and decreases when the machine is operating in unstable conditions. Therefore, PE or PSE is larger when cutting is stable. However, when chatter occurs, chatter component is dominant and PE or PSE will decreases. 2.2. Multiscale permutation entropy algorithm The permutation entropy (PE) algorithm measures the randomness and detects the dynamic changes of time series, which was proposed in [42]. The PE algorithm is a regularity statistic that relies on the order relations between neighboring values of a time series and is applicable to any real-world data. The PE algorithm provides quantitative information regarding the complexity of a time series. With the onset of chatter, strongly synchronized vibrations occur and these chatter vibrations are manifested in the dynamics as a lowering of the dimensionality of the system and hence, an increase in the predictability of the system dynamics. The main strategy of the PE algorithm is as follows [18]: The series {x(i),i = 1,2,. . .N}, can be reconstructed in phase space as:

8 Xð1Þ ¼ fxð1Þ; xð1 þ kÞ;    ; xð1 þ ðm  1ÞkÞg > > > > > .. > > > <.

XðiÞ ¼ fxðiÞ; xði þ kÞ;    ; xði þ ðm  1ÞkÞg > > > . > > > .. > > : XðN  ðm  1ÞkÞ ¼ fxðN  ðm  1ÞkÞ; xðN  ðm  2ÞkÞ;    ; xðNÞg

ð6Þ

where m is the embedding dimension and k is the time delay For a specified but arbitrary value of i, X(i) in Eq. (6) can be rearranged in increasing order as

XðiÞ ¼ fxði þ ðj1  1ÞkÞ  xði þ ðj2  1ÞkÞ      xði þ ðjm  1ÞkÞg

ð7Þ

Accordingly, any vector XðiÞ can be mapped onto a group of symbols as

XðsÞ ¼ ½j1 ; j2    ; jm 

ð8Þ

where s ¼ 1; 2;    ; k, k  m!. m! is the largest number of distinct symbols and XðsÞ is one of the m! permutations of distinct symbols, which is mapped onto the m number symbols ðj1 ; j2    ; jm Þ in m-dimensional embedding space. The probabilities p1 ; p2    pk of the symbol sequences are calculated. The permutation entropy of k symbol sequences of time series fxðiÞ; i ¼ 1; 2;    ; N g can be defined in the form of Shannon entropy as

Hp ðmÞ ¼ 

k X

Ps lnðPs Þ

ð9Þ

s¼1

where 0  Hp ðmÞ  lnðm!Þ, if Ps ¼ 1=m!, Hp ðmÞ attains its the maximum value, namely, lnðm!Þ. For convenience, Hp ðmÞ is typically normalized as

Hp ¼ Hp ðmÞ=lnðm!Þ

ð10Þ

After normalization, 0  Hp  1.The smaller Hp is, the more regular the time series is; the larger Hp is, the more random the time series is. Therefore, PE is a highly suitable tool for describing the local order structure and enlarging the dynamic changes of time series. However, for some time series, a larger PE value does not always correspond to a highly random time series, especially in other scales, which will often cause unreasonable results [43]. MPE is defined as the PE value set of time series over various scales, and it is calculated as follows: (1) The ‘‘coarse-graining” process is applied to a time series fxðiÞ; i ¼ 1; 2;    ; N g, in which consecutive coarse-grained time series are constructed by averaging a successively increasing number of data points in non-overlapping windows. Each coarse-grained time series y(j s) is calculated via the following equation

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

ðsÞ

yj ¼ where

js X

1

s i¼ðj1Þsþ1

xi ; j ¼ 1; 2;    N=s

ð11Þ

s represents the scale factor and the length of each coarse-grained time series is the integral part of N/s.

(2) The PE is calculated for each ‘‘coarse-grained” time series under the same parameters, and plotted as a function of the scale factor s.This procedure is called multiscale permutation entropy analysis. However, a signal can be embedded into higher-dimensional space via time-delayed embedding to study its dynamics, which requires knowledge of two parameters in advance: the delay parameter k, and the embedding dimension parameter m. To identify the optimal embedding dimension m and the time delay k for MPE calculation, the false nearest neighbor (FNN) function [44]

Table 1 Material properties of Al6061. Material type

Tensile strength (MPa)

Yield strength (MPa)

Fatigue strength (MPa)

Modulus of elasticity (Gpa)

Percentage elongation (%)

Al6061

124

55.2

62.1

68.9

25

Table 2 Tool parameters. Name

Value

The tool materials Coating Helix angle (degree) Tool diameter (mm) Tool blade length (mm) Number of teeth Tool length L1 (mm) Tool overhang

Tungsten carbide No coating 30 8 25 4 65 25

a

b

workpiece Dewesoft

c

Dynamometer 2mm 1.5mm 0.5mm

Charge amplifier

PC

Fig. 2. Experimental setup: (a) test workpiece; (b) data acquisition system; and (c) PC and charge amplifier.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

and the average mutual information (AMI) function [45] are used to calculate the minimum embedding dimension and the optimal delay time, respectively. Detailed discussion of these two methods is beyond the scope of this paper; readers can refer to [44,45].

Table 3 PCB accelerometer parameters. Accelerometer type

Mass(g)

Sensitivity(mv/g)

Frequency range(Hz)

Bandwidth resolution

PCB356A01

1.0

(+10%)5

(+5%) 2 to 5000

(1 to 10000 Hz) 0.003

Fig. 3. FRF of the system in the X and Y directions: (a) tap test with a PCB 086C03 hammer, plastic tip, and PCB accelerometer and (b) transfer functions: real–imaginary parts, coherence, and magnitude.

5 Experimental selected points

bli m (mm)

4

3

2

1

0 1000

2000

3000

4000

5000

6000 (rpm)

7000

8000

Fig. 4. SLD and the selected experimental cutting parameters.

9000

10000

7

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

2.3. Multiscale power spectrum entropy algorithm The multiscale power spectrum entropy (MPSE) is another type of entropy that is used to evaluate the complexity of a signal and reflects the dynamic changes of time series in the frequency domain [37]. PSE is a spectrum detection algorithm that is based on nonlinear feature analysis. In this paper, MPSE is used to calculate the PSE at various scales. Its calculation steps are as follows

Table 4 Cutting parameters for stable and variable cutting conditions. Group

Test

Radial depth of cut (mm)

Axial depth of cut (mm)

Cutting speed (mm/min)

Spindle speed (r/min)

I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot Slot

0.5 1 2 2.5 0.5 1 2 2.5 0.5 1 2 2.5 0.5 1 2 2.5 0.5 1 2 2.5

384 384 384 384 480 480 480 480 576 576 576 576 720 720 720 720 960 960 960 960

4800 4800 4800 4800 6000 6000 6000 6000 7200 7200 7200 7200 9000 9000 9000 9000 12,000 12,000 12,000 12,000

II

III

IV

V

(a) 400 S4

Amplitude(N)

300 S1

200

S3

S2

100 0

-100 -200

-300 -400 -500

0

1

2

3

4 Time(s)

5

6

7

8

(b) 5

Frequency(kHz)

4

0

3.5

-20

3 -40

2.5 2

-60

1.5

-80

Power/frequency(dB/Hz)

20

4.5

1 -100

0.5 0

1

2

3

4

5

6

7

8

-120

Time (s) Fig. 5. Cutting force signals for group IV: (a) the time-domain and (b) the time-frequency-domain spectrographs, which were obtained via STFT.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

(1) Apply the ‘‘coarse-graining” process as for obtaining the MPE; (2) Calculate the power spectral entropy. (i) For a specified one-dimensional time series fxðiÞ; i ¼ 1; 2;    ; N g. the fast Fourier transform (FFT) is used to obtain the power spectrum of series

1 jYðwi Þj2 2pN=s

^ y ðwi Þ ¼ P

ð12Þ

^ y ðwi Þ represents the calculated power spectra at various scales, and; Y(wi) is the fast Fourier where s is the scale factor, P transform (FFT) of each sample.

(a) 80

200

S1

60

200

S3

0 -20 -40

100

100

Amplitude(N)

Amplitude(N)

Amplitude(N)

40 20

50 0 -50

-100 0

0.05 0.1

(b) 50

0.15 0.2

0.25 0.3 Time(s)

0.35 0.4

0.45 0.5

90

Amplitude(N)

30 25

8f

20

f

15

Tooth Passing Harmonics

500

40 8f

30 20

Tooth Passing Harmonics

5

10

4 3 2

50 40 30 Tooth Passing Harmonics

10 0 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency(Hz)

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency(Hz)

30 X 896 Y 11.67 Slight chatter

25 Amplitude(N)

12

Amplitude(N)

6

0.4 0.45 0.5

60

20

f

0 0

1000 1500 2000 25003000 3500 4000 4500 5000 Frequency(Hz)

0.25 0.3 0.35 Time(s)

70

50

10

0 0

0.15 0.2

80

10 5

0.05 0.1

4f

60

35

Amplitude(N)

0.45 0.5 -2000

70

40

(c)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time(s)

80

45

Amplitude(N)

-150

-150 0

4f

0 -50

Amplitude(N)

-100

50

-100

-60 -80

S4

150

150

8 6

X 896 Y 28.33 Severe chatter

20 15

4

10

2

5

Stable 1 0

X 896 Y 0.677 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency(Hz)

0

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency(Hz)

0 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency(Hz)

Fig. 6. Cutting force signals of S1, S3 and S4: (a) the raw cutting force signal; (b) FFT spectrum of the raw signal; (c) the residual part by ASA.

(b) 100 Percentage of false nearest neighbor

Average Mutual Information

(a) 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

1

2

3

4

5 6 Time lag

7

8

9

10

90

Parameters: maxEmb = 10, tau = 1.

80

Data: # datapoints =5000, # dimensions = 1.

70 60 50 40 30 20 10 0

1

2

3

4 5 6 7 8 Number of embeddings

Fig. 7. Outputs of the AMI function and the FNN function: (a) AMI and (b) the percentage of FNN.

9

10

9

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

(a)

0.84

1

0.82

0.9

Spectral Entropy

Permutation Entropy

0.95

0.85 0.8 0.75 0.7

0.6 0.55

(b)

0

2

4

6

8

10 12 Scale

14

16

18

0.7

20

1

Spectral Entropy

Permutation Entropy

0

2

4

6

8

10 12 Scale

14

16

0.85 0.8 0.75 0.7 Stable Slight chatter Severe chatter

0.55

0.78 0.76

0.7

18 16 20 17 19 8 15 7 9 14 5 6 12 11 3 13 4 10 2 1 Scale

0.82

0.9

0.81 Spectral Entropy

0.83

0.95

0.85 0.8 0.75 Stable Slight chatter Severe chatter

0.7 0

2

4

6

8

10 12 Scale

14

16

18

Stable Slight chatter Severe chatter

0.79 0.78 0.77 0.76

20

0.75 0.83

0.81

0.81

Spectral Entropy

0.82

0.8 0.79 0.78

0

2

4

6

8

10 12 Scale

14

16

18

20

Stable Slight chatter Severe chatter

0.8 0.79 0.78 0.77

0.77

0.75

6 5 7 4 3 2 8 17 1 18 16 20 15 9 19 10 13 14 12 11 Scale

0.8

0.82

0.76

Stable Slight chatter Severe chatter

0.72

1

(d) 0.83

20

0.8

0.74

0.65

18

0.82

0.9

0.6

Permutation Entropy

Stable Slight chatter Severe chatter

0.84

0.65

Permutation Entropy

0.76

0.72

0.95

(c)

0.78

0.74

Stable Slight chatter Severe chatter

0.65

0.8

Stable Slight chatter Severe chatter

5 2 6 3 1 17 4 8 7 16 18 19 15 9 14 20 10 11 12 13 Scale

0.76 0.75

11 20 12 4 19 2 3 5 14 18 13 10 1 6 15 9 7 16 17 8 Scale

Fig. 8. MPE and MPSE of the residual parts that are related to chatter information: (a) MPE and MPSE of group I; (b) the rearranged scale of features by LS; (c) MPE and MPSE of group III; and (d) the rearranged scale of features by LS.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

^ y ðwi Þ to obtain P y ðwi Þ: (ii) Normalize P

P^ y ðwi Þ Py ðwi Þ ¼ PN=s ^ i¼1 P y ðwi Þ

ð13Þ

(iii) Calculate the power spectral entropy

(a) 400 S3

S1

300

S4

Amplitude(N)

200 100 0 -100 -200 -300 -400 -500

0.5

1

1.5

2

2.5 Time(s) 0.84 Spectral Entropy

1 0.95

0.9

2s

0.85

3.4s

0.8 0 0.5

1

1.5

(c) 0.08

2 2.5 3 3.5 Time(s) X 2 Y 0.07605

0.07

4

4

4.5

5

0.82

0.8 0.78 1.6s

0.76

0.05 0.04 0.03

0.02 0.01 0

0 0.5

(d) 3.5 3

1

1.5

2 2.5 Time(s)

3

0.72

4.5

X 3.4 Y 0.06864

0.06

3.5

4 4.5

3s

0

0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

Actual Detection

1

1.5

2 2.5 3 3.5 Time(s) X 1.6 Y 0.04151

0

3

0.5

1

1.5

4 4.5

X 3 Y 0.0339

2 2.5 3 Time(s)

3.5

4 4.5

Actual Detection

2.5 Time(s)

2

2

1.5

1.5

1

1

0.5

0.5

0

0.5

3.5

2.5 Time(s)

3.5

0.74

0.75

Difference value

3

Difference value

Permutation Entropy

(b)

0

Variation 1

0 Variation 2

Variation 1

Variation 2

Fig. 9. Online chatter detection results of the proposed method under stable cutting conditions: (a) the cutting force signal; (b) times at which sudden changes in PE and PSE occur; (c) the differences in PE and PSE between two adjacent frames; and (d) comparison of actual change times and detection times.

11

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385 N

H¼

s 1 X

Py ðwi ÞlogðPy ðwi ÞÞ

ð14Þ

i¼1

(iv) For comparison, the result is normalized by the factor log N

MPSE ¼

PNs 1  i¼1 Py ðwi ÞlogðPy ðwi ÞÞ H ¼ logðNs  1Þ logðNs  1Þ

ð15Þ

The MPSE is a nondimensional indicator that is in the range of [0, 1]. As the chatter severity increases, the frequency components gradually gather at the chatter frequency, thereby resulting in a decrease in the PSE [9]. 3. Experimental setup A milling experiment was conducted on the 5axis CNC to verify the proposed methodology, which was built by the National Engineering Research Center of Numerical Control System of China. The workpiece material was a block of

Table 5 Comparison of the actual time with the detection time of the proposed method. Chatter detection

Actual time(s) Detection time(s)

Variation 1

Variation 2

MPE

MPSE

MPE

MPSE

1.96 2

1.96 1.6

3.42 3.4

3.42 3.0

S4

S3

S1

Stable

Slight chatter

Severe chatter

Fig. 10. Surface quality photographs of the workpiece that were captured by a portable microscope under stable cutting conditions.

Total Examples

Base Classifier ( Decision Tree )

Learning

Base Classifier ( Decision Tree )

Learning

Base Classifier ( Decision Tree )

Learning

Fig. 11. Training process of GTB.

Weight

Weight

Weight

Esemble Model

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

Al6061 aluminum of size 150mm  100mm  50mm, which was clamped on the worktable; additional parameters of the workpiece are listed in Table 1. The cutter that was used in the milling experiment is an 8 mm diameter end mill; detailed geometric parameters are listed in Table 2. During milling process, a dynamometer (Kistler 9129A) was used to measure the milling forces with a sampling rate of 10 kHz. A macro-scale image of the machined surface was captured for each group using a portable microscope. The test workpiece was designed as a stepped profile to gradually change the axial depths during milling. Five groups with various cutting conditions were conducted in CNC machining center. The cutting parameters are listed in Table 4 and the experimental test equipment is shown in Fig. 2.

4. Results and discussions 4.1. Modal analysis To know roughly the cutting stability under various cutting conditions and can be used as a reference for the following measurement results, a tap test was applied at the tool tip in the X and Y directions to obtain the modal parameters of the GTB

DT

SVC 1

1

Desired outputs Classification outputs

Labels

Labels

Desired outputs Classification outputs

0

Desired outputs Classification outputs

Labels

1

0

0

Misclassification

-1

0

5

10 15 20 Test samples

25

30

-1

5

Misclassification

10 15 20 Test samples

25

30

-1

0

5

10 15 20 Test samples

25

30

Fig. 12. Classification results on 30 sets of test samples of GTB, DT and SVC. Table 6 Configurations of GTB. n_estimators

max_leaf_nodes

learning_rate

max_features

loss function

100

4

0.1

2

Deviance

1

Labels

Stable Slight chatter Severe chatter

Misclassification

0 Misclassification

Misclassification

Misclassification -1

0

0.5

1

1.5

2 2.5 Time(s)

3

3.5

4

Fig. 13. Identification results of online chatter severity level via GTB.

4.5

13

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

spindle (including the tool and the tool holder). A low-mass, wide-bandwidth, three-axis accelerometer was mounted at the tool tip to monitor the vibrations in three directions; the parameters are listed in Table 3. The accelerometer was connected to an acquisition system (LMS SCADAS Mobile SCM05), and the sampling rate of the system was 4096 Hz. For a hammer strike, a plastic tip was selected because do not damage the cutter and offer sufficient excitation bandwidth. The frequency responses in the X and Y directions are plotted in Fig. 3. The stability lobe diagram (SLD), which was calculated using the measured frequency responses in the X and Y-directions via zero-order approximation (ZOA) [4], and the selected cutting conditions are presented in Fig. 4. 4.2. Online chatter detection via the proposed method 4.2.1. Case1 In case 1, relatively stable cutting conditions (constant spindle speed) were selected for stepped cutting experiments. Referring to the SLD results in Fig. 4, group I in Table 4 was selected for experimental verification. The cutting force signals in the time domain and in the time-frequency domain are analysed in Fig. 5. Case 1 shows transfer from stable cutting to severe chatter in the time-frequency spectrograph. The S1 and S2 regions are stable, S3 corresponds to slight chatter, and S4 corresponds to severe chatter. Thus, S1, S3 and S4 can be used to evaluate the performance of the proposed method. The online chatter detection steps are as follows: (i) ASA was used to extract the periodic components of the raw cutting force signals in the milling process. Then, the interference of rotation frequency, the tooth passing frequency and their harmonics can be eliminated by subtracting the periodic component, as shown in Fig. 6, among which Fig. 6a shows the raw cutting force signals and FFT spectrum

Table 7 Cutting parameters under variable cutting conditions. Cutting conditions

Depth of cut (mm)

Spindle speed (r/min)

Cutting speed (mm/min)

S1 S2 S3 S4 S5 S6

0.5 1 2 2.5 2.5 2.5

4800 9000 7200 6000 12,000 7200

384 720 576 480 960 576

(a) 300 S6

Amplitude(N)

200

S1

S2

S5

S4

S3

100 0 -100 -200 -300

0

1

2

3

4

5 Time(s)

6

7

8

9

10

(b) 5

Frequency(kHz)

4

0

3.5 -20

3 2.5

-40

2

-60

1.5 -80

1

-100

0.5 0

Power/frequency(dB/Hz)

20

4.5

1

2

3

4

5 6 Time (s)

7

8

9

10

Fig. 14. Cutting force signals under variable cutting conditions: (a) time domain and (b) the time-frequency spectrogram that was obtained via STFT.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

of the raw signals are shown in Fig. 6b. f denotes their rotation frequency and, 4f and, denote their tooth passing frequencies. The residual parts obtained by eliminating rotation frequency, tooth passing frequency and harmonic frequency by ASA are shown in Fig. 6c. The results show that ASA can eliminate periodic components well. (ii) FNN and AMI were applied to calculate the minimum embedding dimension m and the optimal delay time k of MPE and MPSE, respectively. As shown in Fig. 7a, if the results of the function calculation demonstrate that the percentage of FNN immediately drops to 0, the time series of the system does not require additional embedding. The red horizontal dotted line in Fig. 7b corresponds to the threshold that the time delay (1/e) cannot exceed. Therefore, the optimal embedding dimension is m = 4 and the delay time is k = 1.

(a)

0.88

1

0.86

0.95

2.8s

0.84 Spectral Entropy

Permutation Entropy

5.5s

0.9 0.85 7.8s

5.8s

0.82 0.8 8.4s

0.78 4.2s

7.4s

0.76

0.8

4.6s

2.4s

0.74 8.8s

0

1

2

3

4

(b) 0.12

5 6 Time(s)

X 5.5 Y 0.09881

X 4.6 Y 0.1113

Difference value

0.1

7

8

9

10

0

0.09

X 7.8 Y 0.1102

1

2

0.04

4

X

2.8

5 6 Time(s)

7

8

9

10

2.4

X

7.4

X

Y 0.07295

Y 0.06814

0.07

0.06

3

Y 0.08264

0.08

8.8 X Y 0.08593

0.08

0.72

Difference value

0.75

X

8.4

X

Y 0.06169

5.8 Y 0.05688 X

0.06

4.2

Y 0.05162

0.05 0.04

0.03 0.02

0.02 0.01 0

2

3

4

5 6 Time(s)

7

8

9

0

10

0

1

2

3

4

5 6 Time(s)

7

8

9

10

9

9

8

Time(s)

1

Actual Detection

8

7

7

6

6

Time(s)

(c)

0

5

Actual Detection

5

4

4

3

3

2

2

1

1

0 Variation 1

0 Variation 1 Variation 2 Variation 3 Variation 4 Variation 5

Variation 2

Variation 3

Variation 4

Fig. 15. Online chatter detection results of the proposed method under variable cutting conditions: (a) times at which sudden changes in PE and PSE occur; (b) the differences in PE and PSE between two adjacent frames; and (c) a comparison of actual change times and detection times.

Table 8 Comparison of the actual time with the detection time of the proposed method under variable cutting conditions. Chatter detection

Actual time(s) Detection time(s)

Variation 1

Variation 2

Variation 3

Variation 4

MPE

MPSE

MPE

MPSE

MPE

MPSE

MPE

MPSE

4.51 4.6

4.51 4.2

5.82 5.5

5.82 5.8

7.77 7.8

7.77 7.4

8.76 8.8

8.76 8.4

15

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

Labels

1

Stable Slight chatter Severe chatter

0

Misclassification

-1

0

1

2

3

4

5 Time (s)

6

7

8

9

10

Fig. 16. Identification results of online chatter severity level via GTB under variable cutting conditions.

300 S6

200

S1

S2

S4

S3

S5

100

0 -100 -200 -300 0 S1

1

2

3

4

S2

S5

Stable

6

7

8

9

10

S3

Slight chatter

Stable

Stable S4

5

S6

Slight chatter

Severe chatter

Fig. 17. Surface quality photographs of the workpiece that were captured by a portable microscope under variable cutting conditions.

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K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

(iii) Based on the selected embedding dimension m and time delay k, MPE and MPSE were calculated as shown in Fig. 8. According to Fig. 8a, if a single scale is used, it is difficult to detect chatter using the PE and the PSE is also not the optimal scale feature for chatter detection. Fig. 8b shows that the distinguishing capacity of the rearranged features decreases gradually from left to right by LS. According to Fig. 8b, scales 18, 16, and 20 of MPE and scales 6, 5, and 7 of MPSE can be selected as sensitive features. However, the sensitive scales differed among the cutting conditions, as shown in Fig. 8c and Fig. 8d. An appropriately sensitive scale must satisfy two requirements: (1) It can distinguish cutting stability states to the maximum extent and; (2) the entropy decreases as the chatter severity levels increase under this scale. Therefore, after rearranging the multiscale entropy under various cutting conditions, scales 8, 7, and 5 of the MPE and scales 5, 4, and 3 of the MPSE are selected as the sensitive scale feature.

Tooth passing frequency Chatter frequency (a) 45

30

35 u1 25

20

80Hz

10

15 50

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0 500 10001500 200025003000 350040004500 5000 50 320Hz 40 20 30 u2 0 20 -20 10 -60 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 2000 2500 30003500 4000 4500 5000 40 25 640Hz 20 20 15 u3 0 10 5 -30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00 500 1000 1500 20002500 3000 35004000 45005000

Amplitude(N)

60

30

18 960Hz 14 10 u4 0 6 2 896Hz -300 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 15 7

1280Hz

5

u5 0

3

1150Hz

1214Hz 1 -15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00 500 1000 1500200025003000 3500 40004500 5000 15 6 1726Hz

4

u6 0 2

1600Hz

-15 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 2000 2500 3000 3500 40004500 5000 1.4 5 1920Hz 3 1 u7 0 0.6 -3 0.2 -50 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 2000 2500 3000 350040004500 5000 2.5 3 4000Hz 2 2 1 1.5 u8 0 1 -1 0.5 -2 -30 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0 500 1000 1500 2000 2500 3000 3500 40004500 5000

Time(s)

Frequency(Hz)

Fig. 18. IMFs of the raw signal in the time domain and in the frequency domain: (a) S1, stable; and (b) S4, severe chatter.

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

17

(b) 100

90 320Hz 70 50 u1 0 30 -60 10 0 -100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 150020002500 3000 35004000 4500 5000 30 50 25 640Hz 30 20 15 u2 0 10 -30 5 0 -50 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 20002500 3000 3500 4000 4500 5000 30 100 896Hz 25 40 20 15 u3 0 10 -40 5 -1000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00 500 1000 1500 2000 2500 3000 35004000 4500 5000 8 40 1150Hz 1214Hz 6 20

Amplitude(N)

60

u4

0

4

-20

2

1280Hz

-400 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00 500 1000 1500 2000 2500 3000 350040004500 5000 10 20 1726Hz 8 10 6 u5 0 4 -10 2 0 -20 0.05 0.1 0 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2.50 500 100015002000 25003000 3500 40004500 5000 3 4000Hz 2 2 1 1.5 u6 0 1 -1 0.5 -2 0 -3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 15002000 2500 3000 3500 400045005000

Time(s)

Frequency(Hz) Fig. 18 (continued)

(iv) After the above steps have been completed, online chatter detection can be initiated. First, a sliding window with a length of 5000 (0.5 s) was constructed, which split the signal into overlapping frames in the milling process with an overlap of 4000, namely, each window shift was 1000 (0.1 s). Therefore, the time resolution of chatter detection was 0.1 s. Second, sensitive scale features of MPE and MPSE of each frame signal are calculated. The chatter detection results are shown in Fig. 9. Fig. 9a shows the cutting force signal at various axial cutting depths. Fig. 9b shows the times at which sudden changes in PE and PSE were detected. In this paper, a sudden change can be defined as a PE or PSE difference between the signals of two adjacent frames that exceeds 0.05. For the MPE and MPSE in Fig. 9b, there are many troughs in both PE and PSE; however, the maximum difference between the PE of the trough and that of the previous moment is less than 0.04 and the PSE is less than 0.02; hence, these are normal fluctuations that are caused by environmental noise. Fig. 9c compares the actual time and the detected time. Table 5 lists the chatter detection time that are obtained via the proposed method and the corresponding actual time. The results demonstrate that both MPE and MPSE can accurately detect the onset of chatter, and that the detection time of MPSE is 0.4 s earlier than the actual time, which is exactly the difference between window length and window shift. It means that when the window length just covers the chatter signal, the PSE value will change abruptly, which indicate that it is more sensitive to the variation of stability in the frequency domain scale. Fig. 10 shows the corresponding workpiece surface qualities at three axial depths of cut (S1, S3 and S4), which are captured using a portable microscope (amplified 100). The chatter gradually appears and intensifies. (v) To realize an intelligent diagnosis, gradient tree boosting (GTB) can be applied to automatically determine chatter severity levels. Decision trees are chosen as base classifiers. Training process of GTB is shown in Fig. 11. GTB was trained initially with fusion sensitive scale features of various stable states that were obtained from various groups of experiments. To evaluate the performance of the GTB method, 30 sets of test samples were selected, among which 10 sets of features for stable performance; slight chatter and severe chatter were provided. The classification results were compared with those of

18

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

decision tree (DT) and support vector machine classification (SVC), which is commonly used as classification algorithms, as shown in Fig. 12. Labels  1, 0 and 1 represent stable, slight chatter and severe chatter, respectively. The accuracies of the three methods are 100%, 91.12% and 83.33%, respectively. Therefore, the GTB method outperforms the others; the configurations of the GTB are listed in Table 6. Finally, a trained GTB was used to determine the chatter severity; the classification results are shown in Fig. 13. According to the figure, there were a few misclassifications, but with only intermittent occurrence and subsequent return to normal, and there was no continuous misclassification, which is acceptable for chatter detection under actual cutting conditions. Therefore, the GTB method can determine the onset time and the severity levels of chatter accurately and timely.

4.2.2. Case2 As in the practical machining, the cutting conditions are often time-varying. In case 2, variable cutting conditions are used to evaluate the performance of the proposed method. The cutting parameters are shown in Table 7. The cutting parameters are listed in Table 7. Online chatter detection under variable cutting conditions was conducted according to the sensitive scales of PE and PSE that were determined in case 1 and using the trained GTB. According to the time-frequency domain spectrogram in Fig. 14, the amplitude of each frequency band changed substantially under various cutting conditions and had a more complex frequency structure. Moreover, in the frequency range that is close to 900 Hz, many more components were involved in the spectrogram, thereby rendering the stability assessment infeasible. Hence, the proposed method was applied to determine the time and the severity levels of chatter. Fig. 15 shows the variations of MPE and MPSE over time as the cutting conditions change, and compares them with the actual time. Fig. 15b shows the differences of PE or PSE between two adjacent frames; from the results of the graph, only the PE and PSE values of two adjacent frames differ substantially, which can be considered as sudden changes. When the PE and PSE change suddenly, a warning can be issued regarding the state variation from the previous moment of the milling process. The results demonstrate that the proposed method can accurately detect the change times of various milling stability states. In addition, MPSE can also detect changes in two types of stable milling. At 2.4 s, from the signals of cutting force, the cutting parameters changed and a sudden change occurred in PSE simultaneously. However, after 0.4 s, the PSE value returned to the same level as prior to 2.4 s. Therefore, the cutting stability did not change during the 0–4.2 s time period. Combining these two indicators, it would not issue a false alarm. The chatter detection time by the proposed method and actual time are listed in Table 8.

Table 9 IMFs that correspond to various frequency bands for stable performance and severe chatter. Frequency band

1

2

3

4

5

6

Stable Severe chatter

u1, u2 u1

u3 u2

u4 u3

u5 u4

u6, u7 u5

u8 u6

80

Stable

70

Severe chatter

Percentage(%)

60 50 40

Sensitive frequency band 30 20 10 0 1

2

3

4

5

Frequency band Fig. 19. Energy ratios in various frequency bands for stable performance and chatter.

6

19

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

The severity levels of chatter in the milling process can be identified by a trained GTB, as shown in Fig. 16. From the results, except for a few misjudgements of slight chatter, GTB can accurately judge the severity levels of chatter and the corresponding times. The results demonstrate that regions S1, S2 and S4 were chatter-free, the onset of slight chatter in S3 and S5 was detected, and severe chatter occurred in S6. To further demonstrate the accuracy of the chatter severity level detection, surface quality photographs of the workpiece were captured by a portable microscope, which are shown in Fig. 17. The results demonstrate the reliability of the proposed method. In addition, comparing the SLD of Fig. 4, the actual chatter detection results were not fully consistent with the analysis results, which demonstrate the necessity of online chatter detection. 4.2.3. Comparison with VMD [7,12] VMD is a popular method for signal processing and feature extraction in chatter detection. The main strategy of VMD for signal processing is to determine the number of modes (K) and the quadratic penalty (a). Consider group I as an example. According to the maximum kurtosis method of reconstructed signals [6,11], the number of decomposed components K is 8 and the quadratic penalty term a is 1640 for S1. Using the optimized parameters that are identified via the maximum kurtosis method, the decomposed signal in region S1 can be obtained, as shown in Fig. 18a. Region S4 was similarly processed. The number of decomposed components K was 6 and the quadratic penalty term a was 2000; the decomposed signal in region S4 was obtained, as shown in Fig. 18b. According to the figure, the energy ratios increased substantially in frequency

(a) 400

S3

S1

300

S4

200 Amplitude(N)

100

0 -100 -200

-300 -400 0

0.5

(b) 0.02

1

1.5

2

2.5 Time(s)

Sample Entropy

0.015

0.01

0.005

0 0

X 1.5 Y 0.002713

0.5

1

1.5

2 2.5 Time(s)

X Y

3.5

4

4.5

5

1.4 0.7097

X 3 Y 0.6177

0.65 0.6 0.55 X 3.1 Y 0.5211

0.5 X 1.5 Y 0.4802

0.45

3

4

4.5

0.4

0

0.5

1

1.5

2 2.5 Time(s)

3

3.5

4

4.5

4

4.5

1

0.7

X 1.6 Y 0.6045

0.6

X 3.1 Y 0.05911

0.9 X 1.5 Y 0.7988

0.8 X 1.5 Y 0.3576

0.4 0.3

CIAC

0.5 X 3 Y 0.0372

X 3.1 Y 0.6328

0.6

X 1.6 Y 0.6118

0.5

0.1 0

X 3 Y 0.7725

0.7

0.2

0

3.5

0.7

(c) 0.8

CIER

3

0.75

X 1.4 Y 0.0175

Approximate Entropy

-500

0.5

1

1.5

2 2.5 Time(s)

3

3.5

4

4.5

0.4

0

0.5

1

1.5

2 2.5 Time(s)

3

3.5

Fig. 20. Chatter detection results of four additional indicators under stable cutting conditions: (a) the cutting force signal; (b) the times at which sudden changes occur in the sample entropy and in the approximate entropy; and (c) the times at which sudden changes occur in the auto-correlation coefficient (CIAC) and in the energy ratio (CIER).

20

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385

band 3, of which the corresponding IMF is presented in Table 9. According to the results in Table 9, the corresponding IMFs (u3 and u4) are also inconsistent. However, it is impossible to determine in advance whether each frame signal corresponds to stable performance or chatter in practical machining; therefore, it is difficult to determine the optimal parameter values of K and a for VMD. In addition, online chatter detection requires timeliness; however, VMD is too time-consuming for online chatter detection (Fig. 19).

Table 10 Comparison of the actual time and the detection time of the four indicators under stable cutting conditions. Chatter detection

Variation 1

Actual time(s) Detection time(s)

Variation 2

SampEn

ApEn

CIER

CIAC

SampEn

ApEn

CIER

CIAC

1.96 1.5

1.96 1.5

1.96 1.6

1.96 1.6

3.42 No

3.42 3

3.42 3.1

3.42 3.1

(a) 300

S6

Amplitude(N)

200

S1

S2

S5

S4

S3

100

0 -100 -200

-300 0

1

2

3

4

5 Time(s) 1

(b) 0.3

Approximate Entropy

Sample Entropy

0.25 X 2.3 Y 0.205

0.2

0.15

X 4 Y 0.1121

0.1

0.05 X

0

0

2.4

Y 1.11e-16 1 2

0.9

4

7

8

9

10

X 8.4 Y 0.8631

X 5.4 Y 0.6469

X 7.4 Y 0.644

0.6

0.3

X 8.5 Y 0.5686

X 5.5 Y 0.4926

0.5

10

X 4.2 Y 0.4125

X 7.5 Y 0.4309

2.4 X Y 0.3282

0

1

2

3

4

5 6 Time(s)

7

8

1 X 2.4 Y 0.6617

X 5.4 Y 0.565 X 4.1 Y 0.5169

0.6

0.5 0.4

X 7.4 Y 0.5599

0.3

0.6 X 5.3 Y 0.2169

0.2 X 2.3 Y 0.06819

0.1 0

1

2

4

5 6 Time(s)

7

X 2.4 Y 0.52

0.5

X 7.3 Y 0.2282

8

9

10

X 8.3 Y 0.9223

X 7.3 Y 0.8683

X 5.4 Y 0.6545

X 7.4 Y 0.664 X 8.4 Y 0.5401

X 4.1 Y 0.5766

0.4

X 8.3 Y 0.1165

3

X 4 Y 0.726

0.7

X 8.4 Y 0.4762

9

X 2.8 Y 0.7599

0.8

X 4 Y 0.3254

X 5.3 Y 0.8832

X 2.3 Y 0.9628

0.9

CIAC

0.7

CIER

9

X 4.1 Y 0.8202

0.7

(c) 0.8

0

8

X 2.3 Y 0.8235

0.4

5 6 Time(s)

7

0.8

X 4.1 Y 0.00101

3

6

10

0.3

0

1

2

3

4

5 6 Time(s)

7

8

9

10

Fig. 21. Chatter detection results of four indicators under variable cutting conditions: (a) the cutting force signal; (b) times at which sudden changes occur in the sample entropy and in the approximate entropy; and (c) times at which sudden changes occur in the auto-correlation coefficient (CIAC) and in the energy ratio (CIER).

21

K. Li et al. / Mechanical Systems and Signal Processing 135 (2020) 106385 Table 11 Comparison of the actual times and the detection times for four indicators under variable cutting conditions. Chatter detection

Variation 1 SampEn

ApEn

CIER

CIAC

SampEn

Variation 2 ApEn

CIER

CIAC

SampEn

Variation 3 ApEn

CIER

CIAC

SampEn

Variation 4 ApEn

CIER

CIAC

Actual time(s) Detection time(s)

4.51 2.4

4.51 4.2

4.51 2.4

4.51 2.4

5.82 4.1

5.82 5.5

5.82 4.1

5.82 4.1

7.77 No

7.77 7.5

7.77 5.4

7.77 7.4

8.76 No

8.76 8.5

8.76 7.4

8.76 8.4

4.2.4. Comparison with other detection indicators To evaluate the performances of MPE and MPSE in chatter detection, four common chatter detection indicators were compared with our results. The four chatter detection indicators were the sample entropy, the approximate entropy [7], the autocorrelation coefficient (CIAC) and the energy ratio (CIER) [19]. The approximate entropy and the sample entropy are used to evaluate the complexity of a signal that reflects the probability of generating a new pattern if the time series dimension changes. The auto-correlation coefficient was used to determine the strength of the periodic component of the signal. If the system is stable, the auto-correlation coefficient is approximately 1, whereas it decreases to 0 if the aperiodic component becomes dominant. The energy ratio chatter indicator, namely, CIER, was obtained by using the ratio of the chatter component energy of the signal and the overall energy of signal. For stable cutting conditions, the same ASA signal preprocessing method was used initially and four chatter detection indicators were applied to obtain the chatter detection results, as shown in Fig. 20. According to the figure, the sample entropy is approximately 0 after 1.6 s and it does not change substantially after that; hence, the chatter occurrence after 1.6 s cannot be detected. The other three chatter detection indicators yield similar results. Table 10 lists the chatter detection time using those four indicators and the actual time. From the results, the onset of chatter can be well detected for all three indicators except sample entropy under stable cutting conditions. Fig. 21 shows the detection results under variable cutting conditions. The chatter detection time that were obtained using those four indicators and the actual time are shown in Table 11. According to the results, the sample entropy cannot detect chatter after 4.1 s; CIER can only detect changes in the cutting parameters and cannot detect the onset of chatter, which may be due to the low signal-to-noise ratio under the cutting conditions. Similar results are obtained for CIAC. Among the four indicators, the approximate entropy performs the best; it is because that we can distinctly observe the difference of approximate entropy levels under different stability states. However, the deviation between the time that is obtained by the proposed method in Table 8 and the actual detection time is smaller. 5. Conclusions A simple and novel method for online chatter detection and chatter severity levels identification was presented in this paper. First, the cutting force signal was preprocessed via ASA to eliminate interference from the rotation frequency, the tooth passing frequency and their harmonics. Then, the MPE and the MPSE were extracted from the residual part and LS for features selection was used to select the optimal sensitive scales with generalization through several groups of experiments. Subsequently, a sliding window with a length of 5000 (0.5 s) was used to divide the signal into (overlapping) frames with an overlap of 4000. Online chatter detection can be realized by calculating the PE and the PSE of the sensitive scales for each frame. Due to the differences in entropy levels among the stability conditions, the change in the sensitive scale entropy was used instead of an absolute threshold to detect the onset of chatter. Finally, to realize an intelligent diagnosis, a trained GTB classifier was applied to automatically determine the chatter severity levels. Analysis of the experimental results demonstrated that the proposed method is applicable under stable cutting conditions and under variable cutting conditions and the proposed method outperformed VMD and other four chatter detection indicators. Acknowledgments The research is supported by the National Natural Science Foundation of China under Grant nos. 51875224 and 51705174, the Postdoctoral Science Foundation of China under Grant nos. 2016M602282. References [1] M. Kayhan, E. Budak, An experimental investigation of chatter effects on tool life, Proc. Inst. Mech. Eng. Part B-J. Eng. Manuf. 223 (11) (2009) 1455– 1463. [2] O. Özsßahin, E. Budak, H.N. Özgüven, In-process tool point FRF identification under operational conditions using inverse stability solution, Int. J. Mach. Tools Manuf. 89 (2015) 64–73. [3] J. Fei, B. Lin, S. Yan, M. Ding, J. Xiao, J. Zhang, Chatter mitigation using moving damper, J. Sound Vibr. 410 (2017) 49–63. [4] Y. Altintasß, E. Budak, Analytical prediction of stability lobes in milling, CIRP Ann- Manuf. Technol. 44 (1) (1995) 357–362. [5] H. Cao, X. Zhang, X. Chen, The concept and progress of intelligent spindles: a review, Int. J. Mach. Tools Manuf. 112 (2017) 21–52. [6] M. Postel, O. Özsahin, Y. Altintas, High speed tooltip FRF predictions of arbitrary tool-holder combinations based on operational spindle identification, Int. J. Mach. Tools Manuf. 129 (2018) 48–60.

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