key: cord-0333764-qwtilddm
authors: Wang, Chenxi; Zhang, Xingwu; Chen, Xuefeng; Yan, Ruqiang; Wang, Peng
title: Weak chatter detection in milling based on sparse dictionary
date: 2020-12-31
journal: Procedia Manufacturing
DOI: 10.1016/j.promfg.2020.05.121
sha: 83902a509cc15b5a3ac73593ee7630fc8044fc83
doc_id: 333764
cord_uid: qwtilddm

Abstract As one of the most unfavorable factors in milling, chatter will lead to poor surface quality, low production efficiency and tool life reduction. Chatter detection, especially weak chatter detection, is an effective way to ensure stable cutting. In this work, the milling dynamic responses based weak chatter detection method is put forward using sparse dictionary. Based on the milling dynamic equations, chatter frequencies are calculated and verified, which are the bases for chatter detection. With the calculated chatter frequencies, the sparse dictionary matrix can be constructed and the orthogonal matching pursuit (OMP) algorithm is adopted for chatter frequencies reconstruction. The proposed method can extract weak chatter frequencies accurately. Experimental results shows that the proposed method can be successful in weak chatter identification.

In machining process, the self-excited vibration, also called chatter, hinders the improvement of production efficiency seriously as well as decreases the tool life and causes damage to workpiece surface quality. To address this, a number of methods have been put forward, including stability limits prediction [1] [2] [3] [4] [5] , chatter detection [6] [7] [8] and chatter suppression [9] [10] [11] [12] . It has to be pointed out that the active chatter suppression strategies are based on the application of chatter detection [7] . Chatter detection is important for practical machining process to ensure the safety of the machining system and the workpiece [13] , which is the focus in this paper.

In previous researches, different kinds of signals have been adopted for chatter detection, such as sound signals [14, 15] , current signals [16, 17] , cutting force signals [18, 19] , acceleration signals [20, 21] and so on. It is worth mentioning that Kuljanic et al. [7, 22] applied multisensory approaches to the intelligent chatter system in the face milling process, which is suitable for industrial conditions. Singh et al. [23] compared different types of measured signals in high speed micro-milling, including force, displacement signals, machined surface topography, etc. Axinte et al. [24] found that vibration signals are more sensitive to chatter onset than cutting forces and acoustic emission. Therefore, in this work, the vibration acceleration signals are selected for chatter identification.

In addition to sensor techniques, the efficient signals processing algorithm is another important issue for chatter identification. In the early research, time domain methods [25] [26] [27] and frequency domain methods [13, 20] are usually adopted. However, the traditional cannot reveal both time domain and frequency domain information simultaneously, which weren't suitable to the non-stationary signals in chatter milling. According to [6] , the efficient signals process algorithms for non-stationary signals are mainly classified as the following two parts: (1) decompose the measured signals and extract the useful components, including ensemble empirical mode decomposition [28] , variation model decomposition [18, 29] , etc. (2) time-frequency analysis methods, such as Hilbert-Huang transform [30] and synchrosqueezing transform [15] . In addition, the entropy

In machining process, the self-excited vibration, also called chatter, hinders the improvement of production efficiency seriously as well as decreases the tool life and causes damage to workpiece surface quality. To address this, a number of methods have been put forward, including stability limits prediction [1] [2] [3] [4] [5] , chatter detection [6] [7] [8] and chatter suppression [9] [10] [11] [12] . It has to be pointed out that the active chatter suppression strategies are based on the application of chatter detection [7] . Chatter detection is important for practical machining process to ensure the safety of the machining system and the workpiece [13] , which is the focus in this paper.

In previous researches, different kinds of signals have been adopted for chatter detection, such as sound signals [14, 15] , current signals [16, 17] , cutting force signals [18, 19] , acceleration signals [20, 21] and so on. It is worth mentioning that Kuljanic et al. [7, 22] applied multisensory approaches to the intelligent chatter system in the face milling process, which is suitable for industrial conditions. Singh et al. [23] compared different types of measured signals in high speed micro-milling, including force, displacement signals, machined surface topography, etc. Axinte et al. [24] found that vibration signals are more sensitive to chatter onset than cutting forces and acoustic emission. Therefore, in this work, the vibration acceleration signals are selected for chatter identification.

In addition to sensor techniques, the efficient signals processing algorithm is another important issue for chatter identification. In the early research, time domain methods [25] [26] [27] and frequency domain methods [13, 20] are usually adopted. However, the traditional cannot reveal both time domain and frequency domain information simultaneously, which weren't suitable to the non-stationary signals in chatter milling. According to [6] , the efficient signals process algorithms for non-stationary signals are mainly classified as the following two parts: (1) decompose the measured signals and extract the useful components, including ensemble empirical mode decomposition [28] , variation model decomposition [18, 29] , etc. (2) time-frequency analysis methods, such as Hilbert-Huang transform [30] and synchrosqueezing transform [15] . In addition, the entropy

In machining process, the self-excited vibration, also called chatter, hinders the improvement of production efficiency seriously as well as decreases the tool life and causes damage to workpiece surface quality. To address this, a number of methods have been put forward, including stability limits prediction [1] [2] [3] [4] [5] , chatter detection [6] [7] [8] and chatter suppression [9] [10] [11] [12] . It has to be pointed out that the active chatter suppression strategies are based on the application of chatter detection [7] . Chatter detection is important for practical machining process to ensure the safety of the machining system and the workpiece [13] , which is the focus in this paper.

In previous researches, different kinds of signals have been adopted for chatter detection, such as sound signals [14, 15] , current signals [16, 17] , cutting force signals [18, 19] , acceleration signals [20, 21] and so on. It is worth mentioning that Kuljanic et al. [7, 22] applied multisensory approaches to the intelligent chatter system in the face milling process, which is suitable for industrial conditions. Singh et al. [23] compared different types of measured signals in high speed micro-milling, including force, displacement signals, machined surface topography, etc. Axinte et al. [24] found that vibration signals are more sensitive to chatter onset than cutting forces and acoustic emission. Therefore, in this work, the vibration acceleration signals are selected for chatter identification.

In addition to sensor techniques, the efficient signals processing algorithm is another important issue for chatter identification. In the early research, time domain methods [25] [26] [27] and frequency domain methods [13, 20] are usually adopted. However, the traditional cannot reveal both time domain and frequency domain information simultaneously, which weren't suitable to the non-stationary signals in chatter milling. According to [6] , the efficient signals process algorithms for non-stationary signals are mainly classified as the following two parts: (1) decompose the measured signals and extract the useful components, including ensemble empirical mode decomposition [28] , variation model decomposition [18, 29] , etc. (2) time-frequency analysis methods, such as Hilbert-Huang transform [30] and synchrosqueezing transform [15] . In addition, the entropy

In machining process, the self-excited vibration, also called chatter, hinders the improvement of production efficiency seriously as well as decreases the tool life and causes damage to workpiece surface quality. To address this, a number of methods have been put forward, including stability limits prediction [1] [2] [3] [4] [5] , chatter detection [6] [7] [8] and chatter suppression [9] [10] [11] [12] . It has to be pointed out that the active chatter suppression strategies are based on the application of chatter detection [7] . Chatter detection is important for practical machining process to ensure the safety of the machining system and the workpiece [13] , which is the focus in this paper.

In previous researches, different kinds of signals have been adopted for chatter detection, such as sound signals [14, 15] , current signals [16, 17] , cutting force signals [18, 19] , acceleration signals [20, 21] and so on. It is worth mentioning that Kuljanic et al. [7, 22] applied multisensory approaches to the intelligent chatter system in the face milling process, which is suitable for industrial conditions. Singh et al. [23] compared different types of measured signals in high speed micro-milling, including force, displacement signals, machined surface topography, etc. Axinte et al. [24] found that vibration signals are more sensitive to chatter onset than cutting forces and acoustic emission. Therefore, in this work, the vibration acceleration signals are selected for chatter identification.

In addition to sensor techniques, the efficient signals processing algorithm is another important issue for chatter identification. In the early research, time domain methods [25] [26] [27] and frequency domain methods [13, 20] are usually adopted. However, the traditional cannot reveal both time domain and frequency domain information simultaneously, which weren't suitable to the non-stationary signals in chatter milling. According to [6] , the efficient signals process algorithms for non-stationary signals are mainly classified as the following two parts: (1) decompose the measured signals and extract the useful components, including ensemble empirical mode decomposition [28] , variation model decomposition [18, 29] , etc. (2) time-frequency analysis methods, such as Hilbert-Huang transform [30] and synchrosqueezing transform [15] . In addition, the entropy method is often adopted for the irregularity and complexity of chatter signals [19] .

The above methods mainly focus on signals processing and constructing complex indexes to extract chatter features without considering the milling dynamic characteristics. In fact, chatter frequencies are the most important features for chatter identification and subsequent chatter suppression. Weak chatter detection is more meaningful because severe chatter is easy to identify. When severe chatter appears, damage might have been caused to tool and spindle. However, very few methods are found to detect weak chatter and extract the corresponding chatter frequencies.

Based on above analysis, this paper proposed weak chatter detection method based on the milling dynamic responses. Based on the calculated chatter frequencies and vibration phase, the chatter dynamic responses can be applied to sparse dictionary construction. Finally, the problem of chatter frequencies reconstruction can be solved by the OMP algorithm.

At first, Insperger et al. [31] proposed multiple chatter frequencies in milling process, including the secondary Hopf bifurcation and the period doubling bifurcation. Lately, in order to obtain the dominant chatter frequency, Dombovari et al. [32] analysed the eigenvectors of the Floquet transition matrix obtained by the semi-discretization method without increasing the computational cost. The chatter frequency prediction method by Insperger will be introduced briefly and verified by experiments.

For convenience, the single DOF milling model is adopted for analysis as follows [2]  

where the detailed description can refer to the literature [2] .

Assume that e    is a characteristic multiplier of Eq.(1), then the chatter response can be written as [31] ( ) ( ) ( ) x t p t e p t e     (2) where  is the characteristic exponent, bar denotes complex conjugates and ( ) (

Substituting Eq.(2) to Eq.(1) and with complex derivation, the chatter frequencies with the Hopf and perioddoubling bifurcations can be expressed as [31] Hz, ..., 1, 0,1,...

Hz, 0,1,...

60

where N is the milling tool number,  is the spindle speed in rpm. ln     is the basic chatter frequency [33] and  is the modulus-maximum characteristic multiplier.

In order to verify the above algorithms and describe the chatter identification algorithm conveniently in the subsequent sections, the milling tests are implemented on a three-axis milling machine (VMC-V5) as seen in Fig. 1 speed 5000 rpm, axial milling depth 20 mm, radial milling depth 0.5 mm, feed per tooth 0.02 mm/tooth and downmilling. In addition, The LMS SCA-DASIII data acquisition system and two accelerometers (IMI 608A11 with sensitivity 100 mV/g) are used to measure the acceleration signals in feed and cross-feed directions for chatter frequency identification. In order to show the milling tests intuitively, the finished workpiece surface is given in Fig. 2 , which shows that the milling process went through six cutting states, including stable, weak chatter and severe chatter. The main reason for this phenomenon is that the modal parameters vary with tool position changing and material removing, which can change the stability limits and cause prediction errors of chatter frequencies. The stage ② (weak chatter) is selected for the subsequent analysis. Fig.1 The experimental set-up of milling The measured acceleration signals in stage ② are shown in Fig. 3 in frequency domain, which shows that the chatter frequencies appear (225.6 Hz, 1466 Hz and 1510 Hz). With the chatter frequency prediction method, the predicted chatter frequencies are presented in Table 1 , which shows that the 2rd, 12th and 13th chatter frequencies (marked with red) are very close to the experimental chatter frequencies. The milling tests verified the effectiveness of the chatter prediction method and will be applied to subsequent analysis. 3. Construction of sparse dictionary based on calculated chatter frequency

In order to obtain the specific chatter frequencies, the most important thing is to extract chatter frequencies accurately from the measured signals. Aslan et al. [16] adopted the comb filters to remove the rotating frequency and its harmonics. The residual frequency components are considered to be the chatter frequencies. Caliskan et al. [34] filtered out the force vibration frequencies except the chatter frequencies using the Kalman and bandpass filter and the proposed method was verified with different types of signals, such as sound, acceleration, force, motor current and spindle signals. However, the previous researches focused on how to filtered the other frequencies out, which is inconvenient and indirect. It's better to extract the chatter frequencies directly from the measured signals based on the calculated chatter frequencies and the critical bandwidth in this section. The sparse theory can extract the wanted frequency components according to the constructed sparse dictionary matrix. According to [35] , the chatter feature extraction can be written as the following mathematical form . In order to show the constructed dictionary detailedly, sixteen representative atoms with sixteen different calculated chatter frequencies in Section 2 are presented in Fig. 4 . With the constructed dictionary D , the accurate weak chatter frequencies can be extracted by OMP algorithm for chatter suppression [38] [39] [40] [41] [42] .. 

With the OMP algorithm, the reconstructed results of chatter frequencies are presented in Fig. 5 , which shows that the OMP algorithm can extract the chatter frequencies (225.6 Hz, 1466 Hz and 1510 Hz) accurately, especially for the weak chatter frequencies (225.6 Hz and 1466 Hz), which are very important to early chatter detection and weak chatter suppression. 

In this paper, weak chatter detection is realized using sparse dictionary considering parameters uncertainty. The main contributions are as follows: (1) The chatter frequencies are calculated as the base for chatter frequencies reconstruction. (2) Based on the calculated chatter frequencies, the sparse dictionary is constructed. (3) The OMP algorithm is adopted for chatter frequencies extraction. The present work focuses on the topic of weak chatter detection. The adopted milling dynamic model is simplified. The complex dynamic milling systems, considering helix angle, runout and process damping, may provide more accurate chatter frequencies.

Besides, there exist some better algorithms than the OMP algorithm for NP-hard problem, which can be considered to be applied to chatter frequencies reconstruction. 

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This work is supported by National Natural Science Foundation of China (Nos. 51775408 & 51805406)，and the National Science and Technology Major Project(2017-Ⅰ-0006-0007).

in the dictionary D is called an atom. In order to extract the needed chatter frequencies, D has to be a redundant dictionary [37] , which contains all the possible chatter dynamic responses.

According to [34] , the milling vibration response (or the solution of Eq.(1)) can be expressed as , ,  is the th i chatter frequencies band and f  is the prediction error band. The row number of dictionary D is equal to the measured signals length. The column number of dictionary