# Grinding Chatter Detection and Identification Based on BEMD and LSSVM

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## Abstract

Grinding chatter is a self-induced vibration which is unfavorable to precision machining processes. This paper proposes a forecasting method for grinding state identification based on bivarition empirical mode decomposition (BEMD) and least squares support vector machine (LSSVM), which allows the monitoring of grinding chatter over time. BEMD is a promising technique in signal processing research which involves the decomposition of two-dimensional signals into a series of bivarition intrinsic mode functions (BIMFs). BEMD and the extraction criterion of its true BIMFs are investigated by processing a complex-value simulation chatter signal. Then the feature vectors which are employed as an amplification for the chatter premonition are discussed. Furthermore, the methodology is tested and validated by experimental data collected from a CNC guideway grinder KD4020X16 in Hangzhou Hangji Machine Tool Co., Ltd. The results illustrate that the BEMD is a superior method in terms of processing non-stationary and nonlinear signals. Meanwhile, the peak to peak, real-time standard deviation and instantaneous energy are proven to be effective feature vectors which reflect the different grinding states. Finally, a LSSVM model is established for grinding status classification based on feature vectors, giving a prediction accuracy rate of 96%.

## Keywords

Grinding chatter BEMD and LSSVM Complex-value chatter signal Feature vector Grinding status classification## 1 Introduction

Grinding is an abrasive machining process which is widely used in modern manufacturing practice to produce high surface quality and close tolerance [1, 2, 3, 4]. Particularly with the increasing mature of ultra-high speed grinding, its advantages are further improved, that providing convenient conditions for development of aerospace technology, transportation, military and other industries [5, 6]. However, grinding chatter is one of the most unfavorable dynamic phenomena in grinding operations including regenerative chatter, frictional chatter and mode coupling chatter. In practice, grinding chatter has negative impacts on the ultimate geometrical workpiece accuracy, surface quality and productivity of machinery. Moreover, it leads to increased wheel wear and adds time and costs to manufacturing [7, 8]. Many theories have been proposed and experiments carried out to discover exactly what mechanism underlies grinding chatter, with the aim of developing reliable suppression methods subsequently [9].

At present, only a few methods for chatter detection have been successfully and practically applied in industry. It is common for trained machine operators to identify the appearance of chatter through experience or observation, meaning that corresponding measurements are not taken at the time that resulting in irreparable loss for the industry. Signal processing techniques and appropriate feature vectors are very important for chatter detection. In the past few decades, either nonlinear time series modeling [10] or spectral analysis [11, 12] has been applied for chatter detection. Additionally, Tansel et al. [13], adopted s-transformation to extract the damping index, making a very descriptive feature of chatter available for inspection in turning operations. Yao et al. [14], presented a two-dimensional feature vector for chatter detection based on the standard deviation of wavelet transforms in drilling machining which had an advantageous identification time. In another study, Gradisek et al. [15] used the coarse-grained entropy rate as a chatter index in grinding and turning, as its value exhibits a drastic drop at the onset of chatter.

It is important to note that the signal processing methods proposed above were mostly based on the theory of Fourier transformation and that these traditional methods are not applicable to processing grinding signals (which are almost non-stationary and nonlinear). They can only detect chatter if it is already in an almost fully developed stage and easily to extract spurious frequency and error information from chatter signals. In order to highly meet the demand of real-world production, it is necessary to detect the onset of chatter before chatter marks have been made on the workpiece. Given this requirement, Rilling et al. [16] proposed a novel method called the bivarition empirical mode decomposition (BEMD). In the third session of the HHT (Hilbert–Huang Transform) International conference, BEMD was successfully applied to the monitoring of wind turbine conditions and displayed its feasibility as a method to determine weak features and integrate information from non-stationary and nonlinear signals [17].

The author of this paper also has made a comparison between EMD and BEMD in extracting features for grinding chatter signals to show the advanced performance of BEMD, that the paper is accepted by the 2016 11th International Conference on Reliability, Maintainability and Safety (ICRMS’ 2016). Thus will not be repeated in details here and just give out some brief conclusions about the distinctions between EMD and BEMD: (1) EMD is initially applied to a one-dimensional signal and extracts zero-mean oscillating components, whereas BEMD is applied to a bivariate signal and extracts zero-mean rotating components; (2) BEMD has calculation efficiency due to process complex-value signals simultaneously and only compute the upper envelope using the maximum points, while EMD can only decompose signals one-by-one and has to obtain both upper and lower envelopes by connecting the extreme points; (3) The number of IMFs derived from signals by EMD are different, and can’t reveal any synchronous characteristics and phase shifting, nor can EMD extract an information fusion function. While the number of IMFs by BEMD is the same, it can extract an information fusion function well and preserve phase differences; (4) BEMD has facilitates the establishment of purified shaft vibration orbits and fully guarantees the correctness of results, which EMD cannot.

Additionally, there are several smart classifiers essential for grinding state identification, such as artificial neural network (ANN) [18, 19], fuzzy logic and support vector machines (SVM). Li et al. [20] used multilayer perceptron ANN to distinguish the tool breakage and cutting chatter. According to the trend of signal in time domain. Bediaga, et al. [21] established the fuzzy logical rule to analyze stability of cutting system. Moreover, Jiang et al. [22] adopted multi-class SVM to identify and classify cutting states that accuracy rate reached 95%. The ANN usually suffers from the problem of intrinsic defeats such as slow study speed, multiple local minima and over-fitting. Also, the prediction ability of fuzzy logic is inaccurate and its theory is still imperfect. SVM overcomes these deficiencies by using the structural risk minimization principle to enhance extensive ability and it also stresses the study of statistical learning rules with a small sample. In order to further improve the learning speed [23]. Suykens proposed a modified version of SVM, i.e. the least squares SVM (LSSVM). In the LSSVM, the non-sensitive loss function is replaced by a quadratic loss function and the inequality constraints are replaced by equality constraints. Through constructing a loss function, the quadratic programming problem is translated into solving linear equation group problems, which simplifies the complexity of calculation [24, 25].

The advantages of BEMD and LSSVM are combined in this paper for detecting and identifying grinding chatter. Section two gives a brief review of BEMD and LSSVM, as well as the extraction criterion of true BIMFs. Moreover, the peak to peak, real-time standard deviation and instantaneous energy are presented as feature vectors for the grinding chatter. In section three, a simulation chatter signal is constructed and then processed by BEMD. Afterwards, peak to peak, real-time standard deviation and instantaneous energy are extracted from BIMFs. In section four, the benefits of the proposed method are further validated experimentally by processing grinding signals which are derived from the grinder KD4020X16, and then a LSSVM model is established to predict the grinding state. Finally, conclusions are presented in section five, which also gives new directions for future work.

## 2 BEMD and LSSVM

### 2.1 A Brief Review of BEMD

#### 2.1.1 Algorithm of BEMD

- (1)
The number of extrema and zeros must be equal or different at most by one.

- (2)
The mean value of the envelope at any point defined by the local maximum points and the envelope as defined by the local minima must be zero.

- S
_{1} -
Select a bivariate signal \(s(t) = x(t) + iy(t)\) and a set of projection directions: \(\varphi_{k} = 2k\pi /N ,\; 1 \le k \le N\)

- S
_{2} - For \(1 \le k \le N.\)
- S
_{21} - Project the signal \(s(t)\) on directions \(\varphi_{k}:\)$$p_{{\varphi_{k} }} = \text{Re} \left[ {s(t)\exp ( - i\varphi_{k} )} \right].$$(1)
- S
_{22} -
Extract all partial maximum points of \(p_{{\varphi_{k} }} (t):\) \(\{ (t_{i}^{k} ,p_{i}^{k} )\}\), where

*i*indicates number of individual maxima. - S
_{23} -
Interpolate the set of points \(\left\{ {\left( {t_{i}^{k} ,p_{i}^{k} \exp (i\varphi_{k} )} \right)} \right\}\) by cubic spline interpolation to obtain the partial envelope curve in direction \(\varphi_{k}\), namely, \(e_{{\varphi_{k} }} (t).\)

- S
- S
_{3} - Calculate the mean of all envelop curves:$$\bar{m}(t) = \frac{1}{N}\sum\limits_{k = 1}^{N} {e_{{\varphi_{k} }} } (t),$$(2)
- S
_{4} - Subtract the mean \(\bar{m}(t)\) from \(s(t)\) to obtain \(g(t):\)$$g(t) = s(t) - \bar{m}(t).$$(3)
- S
_{5} - Examine if \(g(t)\) is a BIMF:
- S
_{51} -
If not, replace \(s(t)\) by \(g(t)\) and repeat the procedure from step S

_{2}until \(g(t)\) is a BIMF. - S
_{52} -
If it is, record the obtained BIMF and repeat the procedure from step S

_{2}on the residual signal \(g(t).\)

- S

*m*th complex-valued BIMF and \(r_{n} (t)\) denotes the residue.

#### 2.1.2 Extraction Criteria of True BIMFs

It is worth noting that the above-generated BIMFs basically incorporate two components: true BIMFs and spurious BIMFs. These spurious BIMFs cannot exactly reflect the vibration peculiarities of grinding systems in a physical sense and this seriously interferes with the researchers’ efforts to extract the feature vectors from signals and eliminate the mechanism faults of grinders. In general, the generation of spurious BIMFs are summarized by the following factors: (1) the definition of BIMFs is only based on numerical analysis, without referring to its physical significance; (2) the stopping criterion of the sifting process results in an excessive decomposition phenomenon; (3) end effect which can lead to serious deviation from the actual features of the signal is not fully eliminated; (4) either white noise or pulse interference which is superimposed on the vibration signal may produce high frequency spurious components. Considering that the majority of people rely heavily on their experience to estimate the authenticity of BIMFs, this is not conducive to facilitating the expansion of the BEMD method. It is therefore necessary to use an efficient and reliable method to identify and eliminate the spurious BIMFs, a procedure which is of great importance to the extraction of the actual vibration mode and corresponding features of the time-frequency domain.

Extraction criterion of true BIMFs based on correlation coefficient

If \(\zeta_{m} \ge \eta\), |

Reserve the \(m^{th}\) BIMF \(g_{m}\); |

Else |

Estimate \(m^{th}\) BIMF, and \(r_{n} = r_{n} + g_{m}.\) |

*η*in Table 1 is a fixed threshold that is generally adopted as a ratio of the maximum correlation coefficient, where

*δ*is a ratio coefficient larger than 1:

### 2.2 Brief Review of LSSVM

- (1)
Using a training set of the data points \(D = \{ (x_{i} ,y_{i} )|i = 1,2, \ldots ,n\}\), where \(x_{i} \in R^{n}\) is the

*i*th input data, and \(y_{i} \in \{ - 1, + 1\}\) is the output class. - (2)The regression function in high-dimensional space is constructed:where \({\varvec{\upomega}}\) is the weight vector, \(\varphi (x)\) is a nonlinear function that maps the input data$$y(x) = {\varvec{\upomega}} \cdot \varphi (x) + b,$$(6)
*x*into a low-dimension space and*b*is the bias parameter. - (3)According to the structural risk minimization principle, the optimal \({\varvec{\upomega}}\) and
*b*can be obtained by minimizing the following function:where$$\begin{aligned} \mathop {\hbox{min} }\limits_{\omega ,b,\varepsilon } J({\varvec{\upomega}},\varepsilon ) = \frac{1}{2}\left\| {\varvec{\upomega}} \right\|^{2} + \frac{C}{2}\sum\limits_{i = 1}^{n} {\varepsilon_{i}^{2} } , \hfill \\ {\text{s}} . {\text{t}} .\;y_{i} = {\varvec{\upomega}} \cdot \varphi (x_{i} ) + b + \varepsilon_{i} , \hfill \\ \end{aligned}$$(7)*C*is the penalty coefficient to balance the structural risk and experience risk and \(\varepsilon_{i}\) is the slack variable. - (4)The Lagrange function can be constructed to solve the optimization problem:where \(\alpha_{i}\) represent Lagrange multipliers that can be either positive or negative values. Eq. (8) can be changed to the following equivalent equations:$$L({\varvec{\upomega}},b,\varepsilon ,\alpha ) = J({\varvec{\upomega}},\varepsilon ) - \sum\limits_{i = 1}^{n} {\alpha_{i} ({\varvec{\upomega}} \cdot \varphi (x_{i} ) + b + \varepsilon_{i} - y_{i} )} ,$$(8)$$\left\{ {\begin{array}{*{20}l} {\frac{{\partial L}}{{\partial \omega }} = 0 \Rightarrow \omega - \sum\limits_{{i = 1}}^{n} {\alpha _{i} \varphi (x_{i} ) = 0,} } \\ {\frac{{\partial L}}{{\partial b}} = 0 \Rightarrow \sum\limits_{{i = 1}}^{n} {\alpha _{i} = 0,} } \\ {\frac{{\partial L}}{{\partial \varepsilon _{i} }} = 0 \Rightarrow C\varepsilon _{i} - \alpha _{i} = 0,} \\ {\frac{{\partial L}}{{\partial \alpha _{i} }} = 0 \Rightarrow \omega \cdot \varphi (x_{i} ) + b + \varepsilon _{i} - y_{i} = 0.} \\ \end{array} } \right.$$(9)
- (5)Eliminating \({\varvec{\upomega}}\) and \(\varepsilon_{i}\) and expressing in matrix form gives:where$$\left[ {\begin{array}{*{20}c} 0 & {{\varvec{e}}^{\text{T}} } \\ {\varvec{e}} & {{\varvec{\Omega}}_{i,j} + \varvec{C}^{ - 1} {\varvec{ I}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} b \\ {\varvec{\upalpha}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ {\varvec{y}} \\ \end{array} } \right],$$(10)$${\varvec{e}} = [1,1, \ldots ,1]_{n}^{\text{T}} ,$$$${\varvec{y}} = [y_{1} ,y_{2} , \ldots ,y_{n} ],$$\({\varvec{\Omega}}_{i,j} = (x_{i} ) \times (x_{j} ) = K(x_{i} ,x_{j} )\) is the kernel function.$${\varvec{\upalpha}} = [\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} ]^{\text{T}} ,$$
The commonly used kernel functions are listed as follows [31, 32].

Polynomial kernel function:$$K(x_{i} ,x_{j} ) = ((x_{i} ,x_{j} ) + \theta )^{d} ,\quad d = 1,2, \ldots$$(11)RBF kernel function:$$K(x_{i} ,x_{j} ) = \exp \left( {\frac{{ - \left\| {x_{i} - x_{j} } \right\|^{2} }}{{\sigma^{2} }}} \right).$$(12)Sigmoid kernel function:$$K(x_{i} ,x_{j} ) = \tanh (\upsilon (x_{i} ,x_{j} ) + c).$$(13) - (6)Lastly, the linear model for function estimation is achieved after the optimization problem is solved:$$y(x) = \sum\limits_{i = 1}^{n} {\alpha_{i} K(x_{i} ,x_{j} )} + b.$$(14)

### 2.3 Chatter Feature Vectors Extraction

Numerous experiments have shown that the amplitude of the vibration signal fluctuates within a certain range when the grinder is in a stable grinding state, while the amplitude substantially increases when in a transition state. It later becomes steady again when the grinder is in a chatter state; Therefore, early grinding chatter can be preliminary detected by comparing changes in the time-domain statistical parameters of the signal. In this paper, the peak to peak (*pp*), real-time standard deviation (*Rsd*) and instantaneous energy (*IE*) are conceived as ideal feature vectors that can detect and identify the chatter.

*N*represents the sampling points.

*Rsd*can be described as:

Using these definitions, the *pp*, *Rsd* and *IE* of each BIMF can be monitored every second, achieving initially detecting grinding chatter in real time.

## 3 Application of BEMD to Simulate Chatter Signal

### 3.1 Construction of a Simulation Chatter Signal

*s*(

*t*) was constructed according to the mechanism of chatter and characteristics of the time-frequency domain:

It is clearly seen that the real and imaginary parts of signal *s*(*t*) are composed of two sine signals and white noise, respectively, and that the frequency components of both are 50 rad/s and 100 rad/s. Additionally, the phase of the imaginary parts of signal is shifted by 0.08 rad and 0.024 rad. The output is a harmonic vibration signal which simulates the stable grinding process when *t *≤ 2.5 s. The output is the harmonic vibration signal multiplied with a slant sign also as to simulate the grinding chatter when 2.5 < *t *≤ 3 s. Moreover, the output is the original harmonic vibration signal multiplied by gain coefficients in order to simulate the stable chatter status when 3 < *t *≤ 5 s.

### 3.2 Application of BEMD

Correlation coefficient of BIMFs of simulation chatter signal

No. | BIMFs | Correlation coefficient |
---|---|---|

1 | BIMF | 0.8025 |

2 | BIMF | 0.6281 |

3 | BIMF | 0.0526 |

4 | BIMF | 0.0176 |

5 | BIMF | 0.0043 |

6 | BIMF | 0.0147 |

_{1}and BIMF

_{2}) and that the portions where components are rotating can be identified by a constant phase shift. The cross-correlation function (CCF) of each true BIMF could therefore be obtained and then the phase parameters could be estimated from the CCF [38], as shown in Figure 4. Moreover, the amplitude and the frequency components of the chatter signal which are initially set in the previous signal also could be revealed by carrying out the Hilbert transformation on the true BIMFs, as shown in Figure 5, where the marginal spectrum of real part of signal is plotted as blue solid line and imaginary part of signal is plotted as red-dashed line. The marginal spectrum expresses the amplitude of each frequency in space and represents accumulated amplitude in a statistical sense.

According to Figure 4, it is clearly seen that the phase shifting and synchronization information about the real and imaginary parts of the true BIMFs are well preserved and easily detected. The phase shifting of BIMF_{1} and BIMF_{2} is 0.025 rad and 0.08 rad, respectively, which is similar to the phase as described in the simulation signal. From Figure 5, the frequency components of the real and imaginary parts of the signal (8 Hz and 16 Hz) are accurately revealed as corresponding with the same frequency components, i.e., 50 rad/s and 100 rad/s, as set in the previous signal.

### 3.3 Extraction of the Chatter Feature Vectors Based on BEMD

*pp*are plotted as blue solid lines,

*Rsd*are plotted as a black dot lines, while

*IE*are plotted as a red-dashed lines.

In Figure 6, it is clearly seen that all feature vectors in various grinding states exhibit different behaviors and that the amplitude of the feature vectors is almost constant in a stable grinding state, while the amplitude drastically increases once the grinder turns into chatter. Moreover, the peak to peak and instantaneous energy fluctuation within a certain range when the grinder is in stable chatter state. The real-time standard deviation continuously increases with time and tends towards stability at the end, making it hard to exactly distinguish the transition state and chatter state. But all in all, it is feasible to clearly find out the onset of grinding chatter of great important to take reliable method to suppress the chatter. In summary, the peak to peak, real-time standard deviation and instantaneous energy are significantly distinct and could be used as a predictor for early grinding chatter detection.

## 4 Grinding Experiments and Application of BEMD and LSSVM

### 4.1 Grinding Experiments

In practice, the grinder is more sensitive to the rotational speed, feeding speed and grinding depth of the grinding wheel, which contributes to the unbalance of the grinding vibration. The experiment was therefore carried out in following steps:

Firstly, keep the feeding speed of the workpiece and grinding depth of the wheel constant, and then gradually increase the rotational speed.

Secondly, keep the feeding speed and rotational speed constant, and then gradually increase the depth of grinding.

Lastly, keep the rotational speed and depth of grinding constant, and then gradually increase the feeding speed.

Grinding parameters

Parameter | Value |
---|---|

Grinding wheel material | Green silicon carbide |

Size of wheel (mm × mm) | \(\phi 600 \times 150\) |

Work-piece material | Gray cast iron 250 |

Size of workpiece (mm × mm × mm) | 3050 × 500 × 500 |

Rotational speed (r/min) | 700 ~ 1100 |

Feeding speed (m/s) | 0.381, 0.254, 0.210 |

Grinding depth (μm) | 5, 10, 15 |

Position of sensors and corresponding sensitivity

Label of sensors | Sensitivity (mV/g) | Position |
---|---|---|

1 | 9.9 | Column |

2 | 10.6 | Column |

3 | 10.4 | Spindle |

4 | 10.1 | Spindle |

5 | 10.4 | Motor |

6 | 10.5 | Motor |

7 | 10.2 | Motor |

8 | 10.1 | Column |

*X*-direction and

*Z*-direction compared to the

*Y*-direction based upon the practical experience and analysis of a considerable portion of the experimental data. For convenience of presentation, this paper selects parts of the

*X*-direction and

*Z*-direction chatter data to construct a complex-valued signal and then eliminates its noise based on the wavelet transform [41, 42]. The newly constructed complex-valued signal is shown in Figure 9, where the blue solid lines represent the

*Z*-direction part and the red-dashed lines represent the

*X*-direction.

From Figure 9, it is seen that the grinding chatter emerges at about 6‒14 s and the transitional phase remains at almost 8 s. It is clearly seen that the amplitude of the vibration signal rapidly expands when the grinder turns into chatter, then the amplitude becomes steady when the grinder gets into stable chatter. However, the signal vibrates more markedly compared with the stable grinding state.

### 4.2 Application of BEMD to Experimental Chatter Signals

Correlation coefficients of the experimental BIMFs

No. | BIMFs | Correlation coefficient |
---|---|---|

1 | BIMF | 0.9656 |

2 | BIMF | 0.0700 |

3 | BIMF | 0.0336 |

4 | BIMF | 0.0156 |

5 | BIMF | 0.0032 |

6 | BIMF | 0.0016 |

7 | BIMF | 0.0021 |

8 | BIMF | 0.0014 |

Phase and maximum of CCF from each of the experimental BIMFs

Parameter | BIMF | BIMF | BIMF | BIMF |
---|---|---|---|---|

Estimated phase (rad) | 0.0562 | 0.0141 | 0.0350 | 0.0562 |

Maximum of CCF | 6517.1 | 148.55 | 74.647 | 7.1760 |

In Figure 11, the marginal spectrum of both the *Z*-direction and *X*-direction shows the same frequency components, which represent about 300 Hz, 580 Hz, 1200 Hz and 1400 Hz. Yet the *Z*-direction BIMFs have a larger amplitude relative to the *X*-direction, and signal vibrates more significantly in the *Z*-direction according to the practical data.

### 4.3 Chatter Feature Vectors Extraction for the Experimental Signal

*pp*,

*Rsd*and

*IE*of real and imaginary parts of the BIMFs, along with time, are shown in Figure 12. The

*pp*of the BIMFs are plotted as blue solid lines,

*Rsd*are plotted as black dot lines, while

*IE*are plotted as red-dashed lines.

*pp*is plotted as blue solid line,

*Rsd*is plotted as black dot line, while

*IE*is plotted as red-dashed line.

From Figure 13, it is seen that the *pp*, *Rsd* and *IE* of chatter grinding are increased to a different degree compared with stable grinding, that it is considered as the significant characteristic to distinguish the grinding state. Therefore, they are all employed as input data for the LSSVM.

### 4.4 LSSVM Model Prediction

*C*= 1 and

*ε*= 0.1. The grinding parameters, feature vectors, and corresponding prediction results are shown in Table 7.

Prediction results of the LSSVM model

Test No. | Input | Target | Output | Result | Rotational speed (r/min) | Feeding speed (m/s) | Grinding depth (μm) | ||
---|---|---|---|---|---|---|---|---|---|

| | | |||||||

1 | 01710 | 0.4626 | 1.9618 | 1 | 1 | Correct | 992 | 0.381 | 5 |

2 | 0.0382 | 0.0111 | 0.0353 | − 1 | − 1 | Correct | 763 | 0.381 | 5 |

3 | 0.0830 | 0.2870 | 0.2765 | − 1 | 1 | Wrong | 1034 | 0.381 | 5 |

4 | 0.0336 | 0.0113 | 0.0361 | − 1 | − 1 | Correct | 808 | 0.381 | 10 |

5 | 0.0288 | 0.0115 | 0.0383 | − 1 | − 1 | Correct | 943 | 0.381 | 5 |

6 | 0.4903 | 0.0631 | 0.3030 | 1 | 1 | Correct | 808 | 0.381 | 15 |

7 | 0.0920 | 0.0163 | 0.0636 | − 1 | − 1 | Correct | 853 | 0.381 | 15 |

8 | 0.0522 | 0.0176 | 0.0925 | − 1 | − 1 | Correct | 992 | 0.381 | 15 |

9 | 0.1109 | 0.1937 | 0.5600 | 1 | 1 | Correct | 1074 | 0.254 | 15 |

10 | 0.0327 | 0.0131 | 0.0363 | − 1 | − 1 | Correct | 804 | 0.254 | 10 |

11 | 0.0851 | 0.1218 | 0.3987 | 1 | 1 | Correct | 803 | 0.254 | 15 |

12 | 0.0394 | 0.0147 | 0.0513 | − 1 | − 1 | Correct | 844 | 0.254 | 15 |

13 | 0.1162 | 0.2798 | 0.3863 | 1 | 1 | Correct | 936 | 0.254 | 10 |

14 | 0.0421 | 0.0124 | 0.0570 | − 1 | − 1 | Correct | 989 | 0.254 | 5 |

15 | 0.0470 | 0.0149 | 0.0996 | − 1 | − 1 | Correct | 1035 | 0.254 | 5 |

16 | 0.1037 | 0.1263 | 0.3676 | 1 | 1 | Correct | 1040 | 0.254 | 15 |

17 | 0.1602 | 0.5510 | 2.0361 | 1 | 1 | Correct | 760 | 0.210 | 5 |

18 | 0.0325 | 0.0115 | 0.0245 | − 1 | − 1 | Correct | 760 | 0.210 | 10 |

19 | 0.0431 | 0.0122 | 0.0427 | − 1 | − 1 | Correct | 853 | 0.210 | 5 |

20 | 0.1031 | 0.1823 | 0.6225 | 1 | 1 | Correct | 893 | 0.210 | 10 |

21 | 0.0471 | 0.0133 | 0.0702 | − 1 | − 1 | Correct | 935 | 0.210 | 5 |

22 | 0.1787 | 0.3055 | 1.5177 | 1 | 1 | Correct | 936 | 0.210 | 10 |

23 | 0.0507 | 00218 | 0.0884 | − 1 | − 1 | Correct | 1075 | 0.210 | 5 |

24 | 0.1480 | 0.3681 | 0.9967 | 1 | 1 | Correct | 904 | 0.210 | 15 |

25 | 0.0396 | 0.0139 | 0.0417 | − 1 | − 1 | Correct | 860 | 0.210 | 15 |

It is seen that Figure 14 is the visual expression of the LSSVM prediction model. It is clearly divided into two parts: a stable state area and a chatter state area. Moreover, the feature vector distribution of chatter is more extensive than the stable state. Thus, in view of this grinding machine, the feature vectors could be extracted out from the real-time acquired signals by BEMD method, and then tested by this LSSVM model which has high accuracy and efficiency. If the feature vectors are in stable area, it means that the grinder is operating well. Otherwise, the grinding machine is suffering from the chatter, that effective measures should be taken immediately to reduce the damage of chatter. It is also necessary to collect more sample data as training set to improve identification accuracy of the LSSVM model. Using the diagram makes it more intuitive and convenient to judge which area the feature vector is in and then find out whether the grinder is in a stable grinding or chatter state. Consequently, the chatter detection and identification method based on BEMD and LSSVM in this paper has an excellent use for chatter prediction, which is robust under different grinding conditions.

## 5 Conclusions

- (1)
The BEMD decomposes the simulation chatter signal derived from a grinding vibration signal generator and is validated by the experimental data which was collected from the CNC guideway grinder KD4020X16 in Hangzhou Hangji Machine Tool Co., Ltd. The results illustrate the suitability of BEMD in terms of processing non-stationary and nonlinear signals and indicating the phase shifting and synchronization information of signals. Meanwhile, the marginal spectrum accurately revealed the actual peculiarities of the signal.

- (2)
The extraction criterion of the true BIMFs based on the correlation coefficient is a reliable technique which successfully identifies and estimates the spurious components. It reserves the main frequency bands which are of great import to the extraction of the actual vibration mode and corresponding features of the time-frequency domain.

- (3)
The peak to peak, standard deviation, and kurtosis values are demonstrated as appropriate feature vectors for early grinding chatter detection.

- (4)
The prediction model based on BEMD and LSSVM shows its feasibility for chatter detection and identification, where the accuracy of this LSSVM model is 96%.

For future work it should be notes that, although the feature vectors based on BEMD showed good performance, these vectors might not be the optimal choice. How to choose and estimate the feature vector is still a challenge for pattern recognition. Furthermore, the selection of the kernel function and the penalty coefficient is also a problem that needs further investigation. In addition, researching smart algorithms for optimization of the vector would be another interesting work.

## Notes

### Authors’ Contributions

H-GC and C-SH was in charge of the whole trial; H-GC, J-YS and W-HC wrote the manuscript; Y-YY and J-CQ assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

Huan-Guo Chen, born in 1977, is currently a doctor, Professor, Master’s tutor at *Zhejiang Province’s Key Laboratory of Reliability Technology for Mechanical and Electrical Product, Zhejiang Sci-Tech University, China*. She received her doctor degree from *Northwestern Polytechnical University, China*, in 2007. Her research interests include on line damage detection and fault diagnosis of intelligent structures.

Jian-Yang Shen, born in 1992, is currently an engineer at *Zhejiang Jiali Technology Co., Ltd., China*. He received his master degree on mechanical engineering from *Zhejiang Sci-Tech University, China*, in 2017.

Wen-Hua Chen, born in 1963, is currently a professor, Vice President and doctoral supervisor at *Zhejiang Province’s Key Laboratory of Reliability Technology for Mechanical and Electrical Product, Zhejiang Sci-Tech University, China*. He received his doctor degree from *Zhejiang University, China*, in 1997. His research interests include on reliability and mechanism.

Chun-Shao Huang, born in 1976, is currently an engineer at *Hangzhou Hangji Machine Tool Co., Ltd*., *China.*

Yong-Yu Yi, born in 1991, is currently an engineer at *Zhejiang Jiali Technology Co., Ltd., China*. He received his master degree on mechanical engineering from *Zhejiang Sci-Tech University, China*, in 2017.

Jia-Cheng Qian, born in 1993, is currently a master candidate at *Zhejiang Sci-Tech University, China*.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Natural Science Foundation of China (Grant No. 51475432), Zhejiang Provincial National Natural Science Foundation of China (Grant No. LZ13E050003), and State Key Program of National Natural Science of China (Grant Nos. U1234207, U1709210).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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