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Chatter detection in milling process based on the energy entropy of VMD and WPD
Author’s Accepted Manuscript Chatter detection in milling process based on the energy entropy of VMD and WPD Zhao Zhang, Hongguang Li, Guang Meng, Xiaotong Tu, Changming Cheng www.elsevier.com/locate/ijmactool
PII: DOI: Reference:
S0890-6955(16)30062-1 http://dx.doi.org/10.1016/j.ijmachtools.2016.06.002 MTM3167
To appear in: International Journal of Machine Tools and Manufacture Received date: 11 January 2016 Revised date: 4 June 2016 Accepted date: 7 June 2016 Cite this article as: Zhao Zhang, Hongguang Li, Guang Meng, Xiaotong Tu and Changming Cheng, Chatter detection in milling process based on the energy entropy of VMD and WPD, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2016.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Chatter detection in milling process based on the energy entropy of VMD and WPD
Zhao Zhang, Hongguang Li*, Guang Meng, Xiaotong Tu, Changming Cheng
State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
[email protected] * Corresponding author.: Tel: +86 21 34206332-816; fax: +86 021 34206814
Abstract This paper presents a novel approach to detect the milling chatter based on energy entropy. By using variational mode decomposition and wavelet packet decomposition, the cutting force signal is decomposed into two group of sub-signals respectively, and each component has limited bandwidth in spectral domain. Since milling chatter is characterized by the change of frequency and energy distribution. Therefore the energy features extracted from the two group of sub-signals are considered and the energy entropies are obtained, which can be utilized to demonstrate the condition of the milling system synthetically. Several milling tests are conducted and the results show that the proposed method can effectively detect the chatter at an early stage.
Keywords: Chatter detection; Variational mode decomposition; Wavelet packet decomposition; Energy entropy; Cutting force
1 Introduction It is well recognized that milling chatter is one of the biggest obstacles in achieving high performance machining operations due to its detrimental nature. Some profound reviews about the chatter problems, e.g. the chatter prediction, chatter 1
detection and chatter control strategies, have been provided by Ehmann and Kapoor [1], Altintas and Weck [2], Quintana and Ciurana [3], Siddhpura and Paurobally [4]. It is noted that the chatter prediction is difficult for industrial users to carry out, hence chatter detection becomes essential for maintaining efficient machining process and the realization of control strategies for chatter suppression. In order to monitor the cutting status, various sensors and signals have been used, for example, displacement signal [5, 6], acceleration signal [7-11], cutting force [12-15], acoustic emission [16-18], motor current [19, 20]. No matter which signal is chosen, the method of signal processing is much more important. Therefore, appropriate feature extraction should be defined for detection in relation to the signal processing, including the time series modeling [10, 23], the spectral analysis [12, 23-26], and the time-frequency analysis [9, 14, 27-29]. There are other available methods as well. For instance, HR. Cao [11] et al. adopted the ensemble empirical mode decomposition to analyze vibration signal, then the C0 complexity and power spectral entropy are extracted as chatter indicators; L. Vela-Martíneza [28] et al. presented an approach to monitor the evolution of cutter tool dynamics with detrended fluctuation analysis. Several smart algorithms were also introduced so far, such as artificial neural network [29, 30], fuzzy logic [31]. Milling chatter is a nonlinear and non-stationary phenomenon. Since the Fourier transform methods conceals the time-domain information, hence it is blind to state transition in non-stationary signals and ineffective for on-line detection of chatter onset. Time-frequency analysis offers an alternative method to identify fault feature for non-stationary process, e.g. variational mode decomposition (VMD) and wavelet packet decomposition (WPD). VMD can decompose a multi-component signal into a discrete number of sub-signals with the specific sparsity properties of its bandwidth in the spectral domain[32]. WPD is an expansion of classical wavelet decomposition, which splits the signal both the low-pass band and high-pass band at all stages. The VMD and WPD have been widely used for non-stationary signals, such as the rolling bearing fault diagnosis[33, 34], physiological signal denoising [35] and cutter tool monitoring [36]. 2
It is well known that regenerative chatter is caused by the interaction of the workpiece and cutting tool. Chatter is characterized by the change of frequency and energy distribution[37]. In the stable milling process, the energy of the tool is dominated by its rotation frequency and harmonics. When chatter occurs, the energy is absorbed to the chatter frequency gradually. Hence, a method for the detection of milling chatter based on the the change of energy distribution is proposed in this paper. First the cutting force signal is decomposed into two group of sub-signals by VMD and WPD respectively. Then the energy features are extracted from the two group of sub-signals and the energy entropies are obtained to demonstrate the condition of the milling system. The effectiveness of the proposed method is validated by a series of milling tests. The results show that the proposed method provides a new way to extract the feature of weak chatter signals under any cutting condition.
2 Methodology 2.1 Variational mode decomposition VMD is an an adaptive and non-recursive signal decomposition method proposed by K. Dragomiretskiy and D. Zosso [32] in 2014. In the VMD framework, the signal is decomposed into k discrete number of sub-signals, and each component is considered compact around a corresponding center frequency. The bandwidth of a component can be evaluated with constrained variational optimization problem, and the formulated constrained variational problem can be expressed as follows:
j min t (t ) uk (t ) e jk t u k k k t s.t. uk
2 f 2
(1)
k
where uk is the kth component of the signal, and the ωk denotes center frequency of the kth component of the signal, f is the origin signal, δ is the Dirac distribution, t is time script. By introducing a quadratic penalty and Lagrangian multipliers, the above constrained optimization problem can be expressed as [32]:
3
j Luk , k , : t t uk t e jk t t k
2
2
f t uk t t , f t uk t k
2
2
(2)
k
where α denotes the balancing parameter of the data-fidelity constraint. Then Eq. (2) can be solved by the alternate direction method of multipliers (ADMM) , the estimated components in frequency domain are as follows: n 1
u k
f i k ui 1 2 k
2
2
(3)
n 1
where f , ui , and u k the Fourier transforms of y t , yi t ,
t and ykn1 t . It is noted that Eq. (3) contains the Wiener filter structure. The component in time domain can be obtained from the real part of inverse Fourier transform of the filtered signal 2.2 Wavelet packet decomposition WPD is a wavelet transform where the discrete time signal is passed through more filters than the discrete wavelet transform (DWT). It is an extension of the DWT. Instead of dividing only the approximations of the signal, the details are also divided. Therefore, WPD can further decompose the detail information of the signal in the high frequency region. For j levels of decomposition, the WPD produces 2j different sets of coefficients xmj (t ) for a data vector with n data points, where, m=1, 2, …, 2j, are the number of
sets, and the length of xmj (t ) is n/2j. According to the wavelet packet coefficients of different sets, the signal in different frequency region of [fs(m-1)/2j fs(m-1)/2j] can be reconstructed as pmj (t ) , where fs is the sampling frequency. The wavelet packet coefficients could be an indication of certain features of the analyzed signal. 2.3 Energy entropy In the stable milling process, the energy of the milling system is mostly dominated by its rotation frequency and harmonic frequencies. When chatter occurs,
4
the energy is absorbed to the chatter frequency gradually. Hence the frequency and energy distribution are closely related to the milling conditions, which can be illustrated by energy entropy. Entropy is an index of the irregularity and complexity related to randomness ( or, alternatively, uncertainty) from data series. In other words, entropy is a function of the probability distribution. Energy entropy is the extension of entropy in energy domain, which is linked to the distribution of energy components [38, 39]. By using VMD, the milling signal can be decomposed into a set of sub-signals, u1(t), u2(t), …, uN(t). It is noted that these sub-signals include different frequency bands ranging from low to high. The energy contained in each component can be calculated as:
Ei ui (t ) dt
i 1,2,, N
2
(4)
With the j-level wavelet packet transform, 2j sets of sub-band coefficients with n/2j length are obtained, then these signals are reconstructed. Similarity, the energy contained in each reconstructed signal can be expressed as:
Fi
i 1,2,, M
2
pi (t ) dt
(5)
where M = 2j. Since these sub-signals u1(t), u2(t), …, uN(t) and p1(t), p2(t), …, pM(t) include different frequency components, E={E1, E2, …, EN} and F={F1, F2, …, FM} form the energy distribution of the milling system in the frequency domain respectively. For the sake of convenient, these energy values are normalized by:
i
Ei E
i
Fi F
(6)
where E= E1+ E2+ …+EN and F= F1+ F2+ …+FM. The energy entropies of the two sub-signals are defined based on the Shannon’s entropy: N
P1 ( i )ln ( i ) i 1
M
P2 (i )ln (i )
(7)
i 1
3 Experimental verification 5
3.1 Experimental setup Table 1 Cutting conditions Milling Radial depth Axial depth of Cutting speed Spindle speed Group Test type
of cut (mm)
cut (mm)
(mm/min)
(r/min)
I
1
Up
4
2.9
108
3600
II
2
Down
slot
0.4
198
6600
3
Down
slot
0.6
198
6600
4
Down
slot
0.7
198
6600
5
Down
2
0.4
108
3600
6
Down
2
2.0
108
3600
7
Down
2
4.0
108
3600
8
Down
slot
0.3
144
4800
9
Down
slot
0.4
144
4800
10
Down
slot
0.6
144
4800
11
Down
slot
0.7
144
4800
12
Down
5
0.5
216
7200
13
Down
5
0.6
216
7200
14
Down
5
4.0
216
7200
15
Down
3
0.7
234
7800
16
Down
3
0.8
234
7800
17
Down
3
3.0
234
7800
18
Down
3
3.2
234
7800
III
IV
V
VI
Cutting tests are carried out to verify the approach on a CNC milling machine.A two-fluted flat end mill cutting tool with 10mm diameter and 35˚ helix angle is used. Besides, the length of the tool and tool blade are 100mm and 40mm, and the tool overhang is 70mm.
6
Fig.1 Cutting force signals for group I: (a) time domain; (b) time-frequency domain; (c) region s1, stable; (d) region s2, stable to chatter; (e) region s3, sever chatter.
Cutting force signals are employed for the chatter detection. The dynamometer Kistler 9272 and supporting Kistler 5070A charge amplifier are used to measure the cutting forces. The workpiece is the aluminum alloy 6061. The sampling frequency is set as 12000 Hz. All tests are conducted without coolant.
7
Fig.2 Cutting force signals in time domain, frequency domain for group II: (a) Test 2, stable; (b) Test 3, slight chatter; (c) Test 4, severe chatter.
3.2 Results and discussion The spindle speed of the cutting tests are carried out from 3600rpm to 7800rpm and the depth of cut from 0.2mm to 5.0mm. Six group of tests with representative results are chosen and the relevant cutting conditions are listed in Table 1. The cutting force signals in time domain, frequency domain for group I and group II are illustrated in Fig.1 and Fig.2 respectively. In Fig.1 and Fig.2, f denotes their rotation frequency and 2f denotes their tooth passing frequency. It is noted that test 1 shows transfer from stable cutting to chatter in the time domain. The milling process is stable in the beginning, then the milling chatter occurs at 19.25s. The time-frequency domain analysis for test 1 further confirm the occurrence of milling chatter. Three typical moments are selected from test 1, as shown in Fig.1(c), Fig.1(d), and Fig.1(e).
8
Fig.3 signals processed by VMD in time domain and frequency domain: (a) s1; (b) s3
As for the group II, it is seen that the milling systems is stable in test 2. The spectrum of the cutting force is dominated by its rotation frequency and harmonics, in the meantime the amplitudes are relative small. But with the increasing of depth of cut, the slight chatter occur in test 3 and sever chatter in test 4, as show in Fig.2(b) and Fig.2(c). With the development of chatter severity level, the amplitudes of the chatter grow substantially. Besides, it is obvious that the chatter frequencies are modulated by their rotation frequency. Even though the occurrence of the chatter can be detected from the time-frequency analysis, it is not convenient for industrial users to carry out. In order to identify the chatter automatically, the signal is decomposed by VMD and WPD respectively. Eight modes are recovered in VMD from low frequency to high 9
frequency. Besides, Daubechies wavelets (db10) are used in WPD and the signals are decomposed for three levels. Only the results of s1 and s3 processed by VMD are shown. The results of s1 and corresponding Fourier spectra are shown in Fig.3(a), while the results of s3 are displayed in Fig.3(b). Besides, the normalized energy ration of the sub-signals are shown in Fig.4.
Fig.4 normalized energy ration of the sub-signals: (a) group I by VMD and WPD; (b) group II by VMD and WPD
As mentioned above, in the stable process, the energy of the milling system mostly concentrates on its rotation frequency and low order harmonic frequencies (e.g. second-harmonic, third-harmonic and fourth-harmonic, as shown in Fig.3(a)), which can be reflected by the normalized energy ration of u1 and u2 by VMD, and p1 by WPD. When the chatter occurs, the energy of the milling system is absorbed to the chatter frequency, which can be seen from s2 and test 3. The energy distribution of chatter signal comparatively even, which means the uncertainty of the energy distribution will become greater. Hence, the energy entropy can be utilized to identify the chatter automatically. After the signal is decomposed by VMD and WPD respectively, the sub-signals can be obtained. Eq.(7) is utilized and the energy entropies corresponding to these two group of sub-signals are calculated. The results are expressed in Fig.5.
10
Fig.5 Energy entropy of the tests (a) group I; (b) group II; (c) group III; (d) group IV; (e) group V; (f) group VI
Obviously the energy entropies are increased with the increasing of the chatter level. In the stable milling process, take test 8 and test 9, test 12 and test 13 for example, the two energy entropies remain unchanged basically. However, when the slight chatter occurs as shown in s2, the energy entropies increase dramatically compared with s1. The same conclusion can be drawn from test 2 and test 3, which further confirms that the proposed method is sensitive to the onset of the chatter. In the stable milling process, the energy of the system concentrates on the rotation frequency and its harmonics, leading to a small energy entropy. With the development of chatter, the energy components are absorbed to chatter frequencies, hence the energy entropies surely increase. Summing up, the above results show that the proposed method within the energy entropy framework can provide an accurate detection of chattering phenomenon in machining processes. Therefore the method for chatter identification can be applied to real-time milling process with the help of 11
suitable chatter suppression methods and control mechanism.
4 Conclusions This paper presents a simple method for the detection of milling chatter. The complicated non-stationary cutting force signals can be decomposed into two group of sub-signals based on the VMD and WPD respectively. The energy components contained in each frequency band are different and change with the variation of the milling conditions. Therefore, energy entropy is used in this paper to reflect the milling system states in this paper. In the stable milling process, the energy of the system concentrates on its rotation frequency and low order harmonic frequencies, leading to a small energy entropy. With the increase of chatter severity level, the energy components present a distribution phenomenon in frequency domain, hence the energy entropies surely increase. The proposed approach is validated by a series of tests, and the results show that the energy entropy has an excellent performance for chatter premonition identification, especially at an early stage. But it is noticed that the selections of suitable wavelet base function and decomposition levels still rely on the experience at the moment, which have a significant influence on the analysis results. Therefore, the sensitive of different levels to chatter detection will be our next step work.
Acknowledgment This research was supported by the National Key Basic Research Program of China under Grant No. 2014CB046603.
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Figure captions Fig.1 Cutting force signals for group I: (a) time domain; (b) time-frequency domain; (c) region s1, stable; (d) region s2, stable to chatter; (e) region s3, sever chatter. (TIFF format, 187.4 mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.2 Cutting force signals in time domain, frequency domain for group II: (a) Test 2, stable; (b) Test 3, slight chatter; (c) Test 4, severe chatter. (TIFF format, 97.5 mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.3 signals processed by VMD in time domain and frequency domain: (a) s1; (b) s3 16
(TIFF format, 144.3mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.4 normalized energy ration of the sub-signals: (a) group I by VMD and WPD; (b) group II by VMD and WPD (TIFF format, 75.9mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.5 Energy entropy of the tests (a) group I; (b) group II; (c) group III; (d) group IV; (e) group V; (f) group VI (TIFF format, 118.9mm * 146 mm, 1200 dpi, double-column fitting image) Highlight:
a new approach is presented to detect milling chatter in peripheral milling
the non-stationary chatter signal is decomposed based on the VMD and WPD
energy entropy are utilized to demonstrate the condition of milling system
the method has an excellent performance for chatter identification
17
PII: DOI: Reference:
S0890-6955(16)30062-1 http://dx.doi.org/10.1016/j.ijmachtools.2016.06.002 MTM3167
To appear in: International Journal of Machine Tools and Manufacture Received date: 11 January 2016 Revised date: 4 June 2016 Accepted date: 7 June 2016 Cite this article as: Zhao Zhang, Hongguang Li, Guang Meng, Xiaotong Tu and Changming Cheng, Chatter detection in milling process based on the energy entropy of VMD and WPD, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2016.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Chatter detection in milling process based on the energy entropy of VMD and WPD
Zhao Zhang, Hongguang Li*, Guang Meng, Xiaotong Tu, Changming Cheng
State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
[email protected] * Corresponding author.: Tel: +86 21 34206332-816; fax: +86 021 34206814
Abstract This paper presents a novel approach to detect the milling chatter based on energy entropy. By using variational mode decomposition and wavelet packet decomposition, the cutting force signal is decomposed into two group of sub-signals respectively, and each component has limited bandwidth in spectral domain. Since milling chatter is characterized by the change of frequency and energy distribution. Therefore the energy features extracted from the two group of sub-signals are considered and the energy entropies are obtained, which can be utilized to demonstrate the condition of the milling system synthetically. Several milling tests are conducted and the results show that the proposed method can effectively detect the chatter at an early stage.
Keywords: Chatter detection; Variational mode decomposition; Wavelet packet decomposition; Energy entropy; Cutting force
1 Introduction It is well recognized that milling chatter is one of the biggest obstacles in achieving high performance machining operations due to its detrimental nature. Some profound reviews about the chatter problems, e.g. the chatter prediction, chatter 1
detection and chatter control strategies, have been provided by Ehmann and Kapoor [1], Altintas and Weck [2], Quintana and Ciurana [3], Siddhpura and Paurobally [4]. It is noted that the chatter prediction is difficult for industrial users to carry out, hence chatter detection becomes essential for maintaining efficient machining process and the realization of control strategies for chatter suppression. In order to monitor the cutting status, various sensors and signals have been used, for example, displacement signal [5, 6], acceleration signal [7-11], cutting force [12-15], acoustic emission [16-18], motor current [19, 20]. No matter which signal is chosen, the method of signal processing is much more important. Therefore, appropriate feature extraction should be defined for detection in relation to the signal processing, including the time series modeling [10, 23], the spectral analysis [12, 23-26], and the time-frequency analysis [9, 14, 27-29]. There are other available methods as well. For instance, HR. Cao [11] et al. adopted the ensemble empirical mode decomposition to analyze vibration signal, then the C0 complexity and power spectral entropy are extracted as chatter indicators; L. Vela-Martíneza [28] et al. presented an approach to monitor the evolution of cutter tool dynamics with detrended fluctuation analysis. Several smart algorithms were also introduced so far, such as artificial neural network [29, 30], fuzzy logic [31]. Milling chatter is a nonlinear and non-stationary phenomenon. Since the Fourier transform methods conceals the time-domain information, hence it is blind to state transition in non-stationary signals and ineffective for on-line detection of chatter onset. Time-frequency analysis offers an alternative method to identify fault feature for non-stationary process, e.g. variational mode decomposition (VMD) and wavelet packet decomposition (WPD). VMD can decompose a multi-component signal into a discrete number of sub-signals with the specific sparsity properties of its bandwidth in the spectral domain[32]. WPD is an expansion of classical wavelet decomposition, which splits the signal both the low-pass band and high-pass band at all stages. The VMD and WPD have been widely used for non-stationary signals, such as the rolling bearing fault diagnosis[33, 34], physiological signal denoising [35] and cutter tool monitoring [36]. 2
It is well known that regenerative chatter is caused by the interaction of the workpiece and cutting tool. Chatter is characterized by the change of frequency and energy distribution[37]. In the stable milling process, the energy of the tool is dominated by its rotation frequency and harmonics. When chatter occurs, the energy is absorbed to the chatter frequency gradually. Hence, a method for the detection of milling chatter based on the the change of energy distribution is proposed in this paper. First the cutting force signal is decomposed into two group of sub-signals by VMD and WPD respectively. Then the energy features are extracted from the two group of sub-signals and the energy entropies are obtained to demonstrate the condition of the milling system. The effectiveness of the proposed method is validated by a series of milling tests. The results show that the proposed method provides a new way to extract the feature of weak chatter signals under any cutting condition.
2 Methodology 2.1 Variational mode decomposition VMD is an an adaptive and non-recursive signal decomposition method proposed by K. Dragomiretskiy and D. Zosso [32] in 2014. In the VMD framework, the signal is decomposed into k discrete number of sub-signals, and each component is considered compact around a corresponding center frequency. The bandwidth of a component can be evaluated with constrained variational optimization problem, and the formulated constrained variational problem can be expressed as follows:
j min t (t ) uk (t ) e jk t u k k k t s.t. uk
2 f 2
(1)
k
where uk is the kth component of the signal, and the ωk denotes center frequency of the kth component of the signal, f is the origin signal, δ is the Dirac distribution, t is time script. By introducing a quadratic penalty and Lagrangian multipliers, the above constrained optimization problem can be expressed as [32]:
3
j Luk , k , : t t uk t e jk t t k
2
2
f t uk t t , f t uk t k
2
2
(2)
k
where α denotes the balancing parameter of the data-fidelity constraint. Then Eq. (2) can be solved by the alternate direction method of multipliers (ADMM) , the estimated components in frequency domain are as follows: n 1
u k
f i k ui 1 2 k
2
2
(3)
n 1
where f , ui , and u k the Fourier transforms of y t , yi t ,
t and ykn1 t . It is noted that Eq. (3) contains the Wiener filter structure. The component in time domain can be obtained from the real part of inverse Fourier transform of the filtered signal 2.2 Wavelet packet decomposition WPD is a wavelet transform where the discrete time signal is passed through more filters than the discrete wavelet transform (DWT). It is an extension of the DWT. Instead of dividing only the approximations of the signal, the details are also divided. Therefore, WPD can further decompose the detail information of the signal in the high frequency region. For j levels of decomposition, the WPD produces 2j different sets of coefficients xmj (t ) for a data vector with n data points, where, m=1, 2, …, 2j, are the number of
sets, and the length of xmj (t ) is n/2j. According to the wavelet packet coefficients of different sets, the signal in different frequency region of [fs(m-1)/2j fs(m-1)/2j] can be reconstructed as pmj (t ) , where fs is the sampling frequency. The wavelet packet coefficients could be an indication of certain features of the analyzed signal. 2.3 Energy entropy In the stable milling process, the energy of the milling system is mostly dominated by its rotation frequency and harmonic frequencies. When chatter occurs,
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the energy is absorbed to the chatter frequency gradually. Hence the frequency and energy distribution are closely related to the milling conditions, which can be illustrated by energy entropy. Entropy is an index of the irregularity and complexity related to randomness ( or, alternatively, uncertainty) from data series. In other words, entropy is a function of the probability distribution. Energy entropy is the extension of entropy in energy domain, which is linked to the distribution of energy components [38, 39]. By using VMD, the milling signal can be decomposed into a set of sub-signals, u1(t), u2(t), …, uN(t). It is noted that these sub-signals include different frequency bands ranging from low to high. The energy contained in each component can be calculated as:
Ei ui (t ) dt
i 1,2,, N
2
(4)
With the j-level wavelet packet transform, 2j sets of sub-band coefficients with n/2j length are obtained, then these signals are reconstructed. Similarity, the energy contained in each reconstructed signal can be expressed as:
Fi
i 1,2,, M
2
pi (t ) dt
(5)
where M = 2j. Since these sub-signals u1(t), u2(t), …, uN(t) and p1(t), p2(t), …, pM(t) include different frequency components, E={E1, E2, …, EN} and F={F1, F2, …, FM} form the energy distribution of the milling system in the frequency domain respectively. For the sake of convenient, these energy values are normalized by:
i
Ei E
i
Fi F
(6)
where E= E1+ E2+ …+EN and F= F1+ F2+ …+FM. The energy entropies of the two sub-signals are defined based on the Shannon’s entropy: N
P1 ( i )ln ( i ) i 1
M
P2 (i )ln (i )
(7)
i 1
3 Experimental verification 5
3.1 Experimental setup Table 1 Cutting conditions Milling Radial depth Axial depth of Cutting speed Spindle speed Group Test type
of cut (mm)
cut (mm)
(mm/min)
(r/min)
I
1
Up
4
2.9
108
3600
II
2
Down
slot
0.4
198
6600
3
Down
slot
0.6
198
6600
4
Down
slot
0.7
198
6600
5
Down
2
0.4
108
3600
6
Down
2
2.0
108
3600
7
Down
2
4.0
108
3600
8
Down
slot
0.3
144
4800
9
Down
slot
0.4
144
4800
10
Down
slot
0.6
144
4800
11
Down
slot
0.7
144
4800
12
Down
5
0.5
216
7200
13
Down
5
0.6
216
7200
14
Down
5
4.0
216
7200
15
Down
3
0.7
234
7800
16
Down
3
0.8
234
7800
17
Down
3
3.0
234
7800
18
Down
3
3.2
234
7800
III
IV
V
VI
Cutting tests are carried out to verify the approach on a CNC milling machine.A two-fluted flat end mill cutting tool with 10mm diameter and 35˚ helix angle is used. Besides, the length of the tool and tool blade are 100mm and 40mm, and the tool overhang is 70mm.
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Fig.1 Cutting force signals for group I: (a) time domain; (b) time-frequency domain; (c) region s1, stable; (d) region s2, stable to chatter; (e) region s3, sever chatter.
Cutting force signals are employed for the chatter detection. The dynamometer Kistler 9272 and supporting Kistler 5070A charge amplifier are used to measure the cutting forces. The workpiece is the aluminum alloy 6061. The sampling frequency is set as 12000 Hz. All tests are conducted without coolant.
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Fig.2 Cutting force signals in time domain, frequency domain for group II: (a) Test 2, stable; (b) Test 3, slight chatter; (c) Test 4, severe chatter.
3.2 Results and discussion The spindle speed of the cutting tests are carried out from 3600rpm to 7800rpm and the depth of cut from 0.2mm to 5.0mm. Six group of tests with representative results are chosen and the relevant cutting conditions are listed in Table 1. The cutting force signals in time domain, frequency domain for group I and group II are illustrated in Fig.1 and Fig.2 respectively. In Fig.1 and Fig.2, f denotes their rotation frequency and 2f denotes their tooth passing frequency. It is noted that test 1 shows transfer from stable cutting to chatter in the time domain. The milling process is stable in the beginning, then the milling chatter occurs at 19.25s. The time-frequency domain analysis for test 1 further confirm the occurrence of milling chatter. Three typical moments are selected from test 1, as shown in Fig.1(c), Fig.1(d), and Fig.1(e).
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Fig.3 signals processed by VMD in time domain and frequency domain: (a) s1; (b) s3
As for the group II, it is seen that the milling systems is stable in test 2. The spectrum of the cutting force is dominated by its rotation frequency and harmonics, in the meantime the amplitudes are relative small. But with the increasing of depth of cut, the slight chatter occur in test 3 and sever chatter in test 4, as show in Fig.2(b) and Fig.2(c). With the development of chatter severity level, the amplitudes of the chatter grow substantially. Besides, it is obvious that the chatter frequencies are modulated by their rotation frequency. Even though the occurrence of the chatter can be detected from the time-frequency analysis, it is not convenient for industrial users to carry out. In order to identify the chatter automatically, the signal is decomposed by VMD and WPD respectively. Eight modes are recovered in VMD from low frequency to high 9
frequency. Besides, Daubechies wavelets (db10) are used in WPD and the signals are decomposed for three levels. Only the results of s1 and s3 processed by VMD are shown. The results of s1 and corresponding Fourier spectra are shown in Fig.3(a), while the results of s3 are displayed in Fig.3(b). Besides, the normalized energy ration of the sub-signals are shown in Fig.4.
Fig.4 normalized energy ration of the sub-signals: (a) group I by VMD and WPD; (b) group II by VMD and WPD
As mentioned above, in the stable process, the energy of the milling system mostly concentrates on its rotation frequency and low order harmonic frequencies (e.g. second-harmonic, third-harmonic and fourth-harmonic, as shown in Fig.3(a)), which can be reflected by the normalized energy ration of u1 and u2 by VMD, and p1 by WPD. When the chatter occurs, the energy of the milling system is absorbed to the chatter frequency, which can be seen from s2 and test 3. The energy distribution of chatter signal comparatively even, which means the uncertainty of the energy distribution will become greater. Hence, the energy entropy can be utilized to identify the chatter automatically. After the signal is decomposed by VMD and WPD respectively, the sub-signals can be obtained. Eq.(7) is utilized and the energy entropies corresponding to these two group of sub-signals are calculated. The results are expressed in Fig.5.
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Fig.5 Energy entropy of the tests (a) group I; (b) group II; (c) group III; (d) group IV; (e) group V; (f) group VI
Obviously the energy entropies are increased with the increasing of the chatter level. In the stable milling process, take test 8 and test 9, test 12 and test 13 for example, the two energy entropies remain unchanged basically. However, when the slight chatter occurs as shown in s2, the energy entropies increase dramatically compared with s1. The same conclusion can be drawn from test 2 and test 3, which further confirms that the proposed method is sensitive to the onset of the chatter. In the stable milling process, the energy of the system concentrates on the rotation frequency and its harmonics, leading to a small energy entropy. With the development of chatter, the energy components are absorbed to chatter frequencies, hence the energy entropies surely increase. Summing up, the above results show that the proposed method within the energy entropy framework can provide an accurate detection of chattering phenomenon in machining processes. Therefore the method for chatter identification can be applied to real-time milling process with the help of 11
suitable chatter suppression methods and control mechanism.
4 Conclusions This paper presents a simple method for the detection of milling chatter. The complicated non-stationary cutting force signals can be decomposed into two group of sub-signals based on the VMD and WPD respectively. The energy components contained in each frequency band are different and change with the variation of the milling conditions. Therefore, energy entropy is used in this paper to reflect the milling system states in this paper. In the stable milling process, the energy of the system concentrates on its rotation frequency and low order harmonic frequencies, leading to a small energy entropy. With the increase of chatter severity level, the energy components present a distribution phenomenon in frequency domain, hence the energy entropies surely increase. The proposed approach is validated by a series of tests, and the results show that the energy entropy has an excellent performance for chatter premonition identification, especially at an early stage. But it is noticed that the selections of suitable wavelet base function and decomposition levels still rely on the experience at the moment, which have a significant influence on the analysis results. Therefore, the sensitive of different levels to chatter detection will be our next step work.
Acknowledgment This research was supported by the National Key Basic Research Program of China under Grant No. 2014CB046603.
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Figure captions Fig.1 Cutting force signals for group I: (a) time domain; (b) time-frequency domain; (c) region s1, stable; (d) region s2, stable to chatter; (e) region s3, sever chatter. (TIFF format, 187.4 mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.2 Cutting force signals in time domain, frequency domain for group II: (a) Test 2, stable; (b) Test 3, slight chatter; (c) Test 4, severe chatter. (TIFF format, 97.5 mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.3 signals processed by VMD in time domain and frequency domain: (a) s1; (b) s3 16
(TIFF format, 144.3mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.4 normalized energy ration of the sub-signals: (a) group I by VMD and WPD; (b) group II by VMD and WPD (TIFF format, 75.9mm * 146 mm, 1200 dpi, double-column fitting image)
Fig.5 Energy entropy of the tests (a) group I; (b) group II; (c) group III; (d) group IV; (e) group V; (f) group VI (TIFF format, 118.9mm * 146 mm, 1200 dpi, double-column fitting image) Highlight:
a new approach is presented to detect milling chatter in peripheral milling
the non-stationary chatter signal is decomposed based on the VMD and WPD
energy entropy are utilized to demonstrate the condition of milling system
the method has an excellent performance for chatter identification
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