Research on milling chatter identification of thin-walled parts based on incremental learning and multi-signal fusion

Chatter is a kind of self-excited vibration that often occurs in the milling process of thin-walled parts, which has become the main factor restricting production efficiency and quality. Due to the occurrence of chatter, the signal becomes more complex and unstable. In order to realize milling chatter detection of thin-walled parts, the method of multi-sensor signal fusion is used. A chatter detection method based on variational mode decomposition (VMD) and nonlinear dimensionless index is proposed by analyzing the characteristics of signals in time–frequency domain. Firstly, a series of intrinsic mode function (IMF) components are obtained by decomposing force and acceleration signals with VMD. When chatter occurs, the energy is transferred to the chatter frequency band. Each IMF signal’s nonlinear energy entropy (EE) is extracted to construct the feature vector. A support vector machine chatter identification model based on multi-sensor signal fusion is established. To solve the problem of model incremental updating, supervised learning and unsupervised learning are combined to provide a method for chatter detection.


Introduction
With the improvement of manufacturing level, high-efficiency and high-precision machining are possible. However, most of the structures of key aerospace parts are mainly thin-walled. Due to such structural characteristics, chatter easily occurs in the milling process, which seriously affects the surface quality and machining efficiency. The existing research shows there are many different chatter forms according to the different chatter formation mechanisms, mainly including regenerative chatter, friction chatter, and modal coupling chatter. Currently, most of the research is mainly focused on regenerative chatter. It mainly involves chatter prediction, chatter identification, and chatter control. The chatter prediction is mainly based on the stability lobe diagram, and then combined with the cutting parameters to determine whether the chatter occurs. However, due to the characteristics of nonlinearity and time-varying modal parameters, the stability analysis is not accurate, so it is difficult to completely avoid the occurrence of chatter [1,2]. Chatter identification and chatter control have attracted more and more attention from researchers. Chatter recognition is a timely diagnosis of chatter state by collecting signals in the processing process and using signal processing methods. It mainly includes three parts: signal acquisition, feature engineering, and pattern recognition. The collected signals mainly include milling force [3] and acceleration signal [4]. As a key factor in the milling process, the change of milling force can reflect most of the information of the machining process. Tansel et al. [5] identified chatter based on standard deviation of torque signal and Teager-Kaiser algorithm, achieving good results. Hino et al. [6] trained and analyzed the milling force signal with fuzzy neural network, and finally accurately predicted the high-speed milling chatter state. HUANG et al. [7] identified chatter through milling force and not only obtained the frequency of chatter but also obtained the influence of milling force on chatter. Feng et al. [8] used fast Fourier transform and wavelet transform to analyze the milling force signal to detect whether chatter occurred. The accuracy of the method was verified by the surface morphology after processing. Li et al. [9] proposed a simple online chatter detection method, which preprocesses the cutting force signal, extracts the MPE and MPSE features, and establishes the mapping relationship between the features and different states. The online chatter detection is realized by using the trained GTB classifier for prediction. Compared with other index detection, this detection index is better and suitable for variable cutting conditions. From the above, it can be found that there is a great correlation between milling force signal and chatter state, so it is reasonable to monitor chatter. However, the relationship between milling force signal and chatter is complex and nonlinear, so the modeling of the relationship between the two is more complicated.
Compared with the milling force signal, the acceleration signal can directly express the information of chatter vibration and has good practicability. KANG et al. [10] used FFT to extract features from vibration signals, and then used selforganizing feature map (SOM) to preprocess and encode spectrum vectors for chatter recognition. Fu et al. [11] collected the vibration signal of the spindle during the machining process using the acceleration sensor and transmitted it to the computer to analyze the milling status online. Sun et al. [12] proposed an optimal entropy method to study the acceleration signal and calculated the threshold of chatter by using the extreme value statistical method. Cao et al. [13] denoised the collected acceleration signal, and obtained the Hilbert-Huang spectrum by Hilbert-Huang transform. Then, the mean and standard deviation of the Hilbert-Huang spectrum were used as indicators to detect whether chatter occurred. Caliskan et al. [14] proposed a chatter detection method based on energy change, which uses Kalman filter to remove irrelevant components. When the vibration energy increases, chatter can be detected. The experimental results show that this method can detect chatter quickly. Chatter is a kind of self-excited vibration. The acceleration signal can directly reflect the chatter state, so the chatter state can be monitored by measuring the acceleration signal. In summary, the milling force signal and acceleration signal are selected in this paper.
In terms of signal processing and feature extraction, at present, the chatter feature extraction methods include the time domain method, frequency-domain method, and time-frequency domain method. Time-frequency domain can display the relationship between time and frequency simultaneously. At present, the commonly used time-frequency analysis methods include wavelet packet decomposition (WPT), empirical mode decomposition (EMD), and variational mode decomposition (VMD). Yao et al. [15] found that the energy of each frequency band changed regularly during the chatter development process, and the chatter can be identified by the proportion of energy in the chatter frequency band. Yesilli et al. [16] found that the wavelet packet with the highest proportion of energy is usually selected as the representative feature of chatter, which does not always include the frequency of chatter, thus reducing the accuracy of classification. Ji et al. [17] used improved EMD to decompose the acceleration signal, extract the subsignal of the chatter frequency band to reconstruction, and extract the three-dimensional feature vector of the reconstructed signal for chatter recognition. The experimental results show that this method can effectively identify chatter. For nonlinear and non-stationary signals, variational mode decomposition has been widely used in chatter detection [18]. Yang et al. [19] used optimized VMD to decompose cutting force and extracted approximate entropy and sample entropy as indexes. The experimental results show that the proposed method has higher sensitivity and stability than EMD. Liu et al. [20] used variational mode decomposition and multi-scale permutation entropy to extract chatter features. The experimental results show that time-frequency analysis method can effectively process signals, and when the scale factor is 4, it is more conducive to chatter detection. Mourad et al. [21] fused the dimensional features with the non-dimensional features, and used relative entropy to rank features. Entropy is a physical quantity that represents the degree of system chaos. Based on the concept of entropy, energy entropy can describe the change in energy distribution and is an important index [22]. Zhang et al. [23] proposed a chatter detection method, which uses the energy entropy value to judge the cutting state. When the cutting is stable, the entropy value is small. When the chatter occurs, the energy starts to gather, and the energy entropy will increase.
The identification of chatter is to establish a relationship mapping model between signal features and state labels through machine learning. KUMAR et al. [24] used the mean square error of the input signal as an exponential parameter to measure the chatter. Through these input and output parameters, the adaptive neuro-fuzzy inference system was used to train the chatter model. The effectiveness of the method was verified by a large number of experiments. Chen et al. [25] established a prediction model by using least squares support vector machine, and proved the feasibility of the model for chatter detection and recognition through simulation experiments. Wan et al. [26] integrated multiple weak SVM classifiers through the Adaboost algorithm to form a stronger classifier Adaboost-SVM with better performance, avoiding the problem of chatter classification accuracy degradation caused by sample label errors. Wang et al. [27] extracted two chatter indexes, intersection distance and cross area, from the kurtosis probability density function (KPDF). Combining these two indexes with K-means clustering method, the clustering accuracy is obviously improved. Wang et al. [28] used the structural function method to extract the fractal features of the signal, which solved the problem that the features were sensitive to the process parameters. K-means clustering was used to identify the chatter state. The experimental results show that the recognition accuracy of the single feature used in this method can reach 94.4%. Dun et al. [29] compress the measured signals based on self-encoding, and then cluster the compressed signals by using the mixed clustering method measured by density and distance. The detection accuracy of the experimental measured signals by this method reaches 95.6033%. Li et al. [30] adopted the multi-class support vector machine (MC-SVM) model to make up for the poor classification accuracy of LSVM. The penalty factor c, kernel parameter g, and kernel function of SVM play very important roles in SVM. Therefore, it is necessary to find the best parameters of support vector machine.
In general, the research on chatter monitoring mainly focuses on the feature selection of signals, which must be very sensitive to the change in processing state. The machining of thin-walled workpieces is prone to chatter. The occurrence of chatter means the redistribution of energy. VMD can separate the chatter frequency band. The energy entropy index can consider the energy agglomeration characteristics when chatter occurs. Therefore, the combination of VMD and energy entropy is suitable for milling chatter detection of thin-walled parts. Based on the energy concentration characteristics when chatter occurs, this paper studies the milling chatter monitoring method of thin-walled parts. Aiming at the problem that chatter is difficult to detect, a chatter feature extraction method based on optimized VMD and energy entropy is proposed. Firstly, the maximum envelope kurtosis (MEK) is used as the fitness function of the Bayesian optimization algorithm. The Bayesian optimization algorithm is an intelligent algorithm used to optimize the parameters of VMD. Secondly, the signal is decomposed by VMD using the obtained optimal parameters, calculates the energy ratio of the decomposed signal, and calculates the energy entropy. The energy entropy of force signal and acceleration signal is fused, and the nonlinear support vector machine is used to compare the chatter state prediction accuracy of single signal and fusion signal, which proves the advantage of multi-signal fusion. Finally, in order to identify the chatter state more accurately, an incremental recognition model combining supervised and unsupervised is introduced based on multi-signal fusion. After reaching the update threshold, the samples with consistent recognition are selected for incremental learning, which reduces the error caused by error labels on incremental learning, thus improving the recognition accuracy of the model. The structure of this paper is as follows. The second section introduces the proposed chatter detection method and the mathematical models of VMD and energy entropy. The third section introduces the chatter feature extraction method which optimizes VMD and energy entropy. The fourth section draws and verifies the stability lobe diagram (SLD), and selects the experimental processing parameters according to the SLD. In the fifth section, the experimental data are analyzed. The chatter state is predicted and classified by a nonlinear support vector machine and the combination of supervised learning and unsupervised learning. Finally, the sixth section summarizes the full text.

The proposed chatter detection method
The chatter detection method proposed in this paper is shown in Fig. 1. In the milling of thin-walled workpieces, the original force and acceleration signals are collected and decomposed by variational mode decomposition to calculate the energy entropy characteristics. It is sent to incremental IL-KM-SVM model for online chatter state detection. By collecting samples in real time, when they reach the update threshold, the incremental IL-KM-SVM model is updated so that it is closely linked to the latest cutting process.
In addition, the labels in the original sample data can be determined by off-line chatter detection. The original sample data can be used to train the off-line chatter state identification model, and the model is continuously updated incrementally during cutting. The SLD is an important means of off-line chatter detection. The drawing of the stability lobe diagram is affected by the dynamic characteristics. Firstly, the modal test is carried out. Through the modal analysis of the workpiece, the stability lobe diagram is drawn by combining the milling force coefficient, and the accuracy of SLD is verified by experiments. Therefore, the cutting state can be determined by the off-line chatter detection, and guide the setting of the test to obtain the signal to form the original sample library.

Variational mode decomposition
Variational mode decomposition is a form of adaptively decomposing the collected complex digital signal into multiple effective AM-FM signals (AM-FM) through frequency domain iteration. It is a completely non-recursive adaptive signal processing method based on Wiener filtering, proposed by Dragomiretskiy [31] in 2014. It has been widely used in fault diagnosis as a new signal processing method. VMD can decompose a given complex time-series signal into a certain number of intrinsic mode functions through the iteration of the variational model. The expression is as follows: where f (t) is the time series of the signal; u k (t) is the kth IMF of signal time series decomposition; A k (t) and k (t) is the instantaneous amplitude and phase of the kth IMF.
The goal of the VMD decomposition algorithm is to minimize the sum of the bandwidth of each IMF component. The constraint is that the sum of each IMF component is approximately equal to the original signal.
In solving the center frequency and bandwidth of the time domain signal, it is assumed that the signal can be decomposed into several IMF components. To minimize the sum of the bandwidth of each IMF component, the following constrained variational model expression needs to be established: is the set of center frequencies corresponding to each IMF component after signal decomposition, expressed as To obtain the optimal solution to Eq. (2), the LaGrange multiplier method is adopted, the Lagrange multiplier and quadratic penalty factor are introduced, and the original optimization equation can be rewritten as follows: A group of IMF and its corresponding center frequency can be obtained by iterative solution with the alternating direction method of multiplication algorithm. The spectrum of each IMF can be solved in the frequency domain. The specific form is as follows: k ( ) , f ( ) , and ̂ ( ) are the mathematical expressions of u n+1 k (t), f (t) , and (t) after Fourier transform respectively; n is the number of iterations.
After the frequency spectrum of each IMF is obtained through iterative calculation of Eq. (4), the center frequency can be iterated through the center of gravity of the power spectrum, as shown below: where n+1 k is the center frequency of each IMF. The VMD algorithm is used to iteratively calculate the sensing signal in the milling experiment of thin-walled parts. When the convergence error e is less than the relative error , the iterative calculation of the whole process will terminate and reach the convergence state. The specific form is as follows:

Energy entropy
In the process of stable machining, the energy of milling system is mainly composed of rotation frequency and its harmonics. When the chatter occurs, the energy is concentrated in the frequency band containing the chatter frequency. The energy entropy is a generalization of the energy domain and is nonlinear. Therefore, it is feasible to use the energy entropy to judge whether the chatter occurs in the milling process.
Entropy is a measure of chaos in the physical sense. In energy theory, entropy is related to the uncertainty of signals or random events. Recently, the vibration signal analysis method based on energy entropy has been widely used. In the vibration analysis and monitoring of machine tool machining process, energy entropy is an effective feature.
By using the VMD decomposition method, the signal is decomposed into m IMF components, and the energy E 1 ⋯ E m of each IMF component can be calculated as follows: The total energy E0 of the signal is obtained by adding E 1 ⋯ E m , and the energy of each IMF is normalized. According to the definition of information entropy, the energy entropy of the signal decomposed by VMD can be defined as:

Incremental learning strategy
K-means clustering algorithm can complete the automatic division of different categories of data, which belongs to unsupervised machine learning. When using a clustering algorithm, the sample database is first established. Due to the continuous increase of real-time data, it is necessary to consider the results of the last clustering when updating the clustering model. Each time the model is updated, the data in the sample library is trimmed to remove the previous data samples, so that the sample data is highly correlated with the current machining process, and the clustering result adds a status label to the real-time data.
The support vector machine classification algorithm is widely used in classification problems, but most of its applications are based on the model trained by offline data. During the machining process, the data is updated quickly, and its characteristics will change. The original model will not be able to classify the new data well, so the new data is added to the model to update the model. When updating the model, it is also necessary to take into account the previous training results of the model. The support vector set in the model fully reflects the characteristics of the entire training data, and the support vector set and the newly added samples are used as the training data when the model is updated. This paper proposes an incremental learning strategy that combines unsupervised clustering with supervised training. The specific workflow is shown in Fig. 2. Firstly, the original model is obtained by training the IL-KM and IL-SVM models with the original sample data. When real-time sample data is involved, the IL-SVM model can identify the chatter state in real-time and store the identification results in the result storage. At the same time, IL-KM can cluster the data online and input the clustering results into the result storage. When the set update threshold is reached, samples with consistent results will be screened for the update of the IL-SVM model. Through the combination of unsupervised and supervised training, the number of incorrectly labeled samples is reduced, and the training accuracy of the model is improved during incremental training.

Feature extraction method based on BO-EK-VMD and energy entropy
In the milling process, due to its intermittent cutting characteristics, the collected signal shows nonlinear and unstable characteristics. The amount of signal data collected in the processing process is significant, with some useless information. In the actual condition monitoring, it is difficult to quickly and accurately determine the current processing state directly from the original data. Therefore, it is necessary to process the collected signals and extract their most sensitive features as the basis for identifying the processing state.
The flow of the feature extraction algorithm proposed in this paper is shown in Fig. 3. Firstly, the parameter selection of the variational modal decomposition. The maximum envelope kurtosis is used as the fitness function of Bayesian optimization (BO), and the Bayesian optimization algorithm is used to determine the decomposition level K and penalty factor in the variational modal decomposition optimization; Secondly, the optimized variational modal decomposition is used to decompose the force and acceleration signals to obtain a series of IMF and calculate its energy. According to the definition of entropy, the energy entropy features decomposed by VMD are calculated, and finally input into the support vector machine (SVM) for training.

Selection method of VMD parameters based on maximum EK
According to the decomposition steps of VMD, the decomposition level K, penalty factor and other parameters are required for VMD decomposition. The decomposition level K and penalty factor have the most significant impact on the performance of VMD decomposition. The K value determines the number of decomposition layers of VMD. If the K value is too large, the signal decomposition is prone to fault, and if the K value is too small, the signal decomposition is incomplete or the signal frequency is mixed. The penalty factor affects the bandwidth of the modes. The smaller the penalty factor is, the larger the bandwidth of the modes is, which is prone to mode aliasing and poor noise reduction effect. The larger the penalty factor is, the smaller the bandwidth of each mode is, which may lead to the loss of some vital information in the original signal. However, up to now, there is no exact method for how to set these two parameters, which mainly depends on empirical selection, which seriously restricts the application of VMD decomposition in chatter detection, and envelope spectrum analysis can effectively extract the periodic pulse components in the signal and improve the signal-to-noise ratio of the signal. EK is defined as the ratio of the fourth-order central moment of the envelope spectrum to the fourth power  Fig. 2 Incremental learning framework of the standard deviation. The larger the EK value, the higher the signal-to-noise ratio.
After the VMD parameter optimization index is determined, an iterative search algorithm is needed to search the optimal VMD parameters globally. The Bayesian optimization algorithm is widely used to solve optimization problems because of its fast search speed and high precision. Therefore, this paper is proposed a VMD parameter optimization algorithm based on BO-EK, which can provide high-quality signals for subsequent feature extraction.
VMD is used to preprocess the collected initial signal. In the process of calculating EK, the envelope of the signal must be obtained first, namely: x o (t) is the absolute value obtained by Hilbert transform, which x(t) is the original signal.
Then EK can be expressed as: 4 is the fourth-order central moment of In order to find the optimal [K, ] when the IMFs are maximized, the Bayesian optimization algorithm [32] is introduced. Because the Bayesian optimization algorithm has strong sample validity, it only needs a few iterations to get a good result. See reference [32] for specific algorithms.

Signal feature extraction based on energy entropy
In the stable milling process, the energy is mainly concentrated in the tool tooth passing frequency and its frequency multiplication. When chatter occurs, the energy is transferred to the vicinity of the chatter frequency, which means that the cutting state changes, and the frequency spectrum and energy distribution of the signal change. Energy entropy is a nonlinear generalization of the energy domain, so it is feasible to use energy entropy to judge whether chatter occurs in the milling process. Energy entropy is introduced in signal feature extraction. Based on the previous analysis, it can be used to represent the cutting conditions in the milling process according to the entropy characteristics. Therefore, this paper proposes a signal feature extraction method based on energy entropy.

Modal experiment setup and results
The stability lobe diagram (SLD) describes the relationship between the spindle speed and the axial cutting depth. A modal hammering experiment can measure an important dynamic characteristic of the stability lobe diagram. The modal test mainly includes excitation, pick-up, and data processing to obtain the modal parameters of the system. Its structure is shown in Fig. 4. The modal tapping experiment is designed to tap the measuring points on the cutting path. The measuring point distribution is shown in Fig. 5. The transverse measuring point interval is 20 mm, the longitudinal measuring point interval is 7 mm, and the acceleration sensor is fixed on the back of the measuring point 7. In order to ensure the test accuracy of the frequency response function, each measuring point needs to be hammered four times and averaged. Enter the analysis software to establish the model and arrange the measuring points. After all the measuring points are hammered, the collected data are imported into the processing software. In the software, the data of each measuring point are matched with the measuring points in the model and processed to obtain the modal parameters. The equipment connection diagram of the hammering test is shown in Fig. 4.
The modal test was carried out on the Dalian VDL-1000E three-axis machine tool. First, fix the acceleration sensor on the workpiece, and the signal picked up by the sensor will be transmitted to the data acquisition equipment. Then, use the software to analyze the transfer function of the system. The modal test and the frequency response function of the workpiece are shown in Fig. 6. The calculation of the obtained frequency response function and the modal parameters of the workpiece are shown in Table 1.

Stability prediction and experimental verification
The methods for drawing the stability lobe map include an experimental method, analytical method, semi-discrete method, and robust discrete method [33,34]. The full discrete method has high accuracy. Therefore, this paper selects the full discrete method for drawing the stability lobe map. The cutting force coefficient is another important input parameter for drawing the stability lobe diagram by the full discrete method. By changing the feed rate in the slot milling experiment, the experimental parameters are shown in Table 2, and K t = 1561.8 MPa; K r = 499.9 MPa can be obtained.  The stability lobe diagram drawn from the above data is shown in Fig. 7. In order to verify the effectiveness of the stability lobe diagram, two points of the stability and chatter cutting parameters are selected, and FFT analyzes their signals. The detailed process parameters of A and B are shown in Table 3. During the experiment, a cemented carbide end milling cutter was used. The tool diameter was 10 mm, the tool had four teeth, the workpiece was made of titanium alloy, the radial cutting depth during cutting was 0.2 mm, and the milling method was forward milling. The original time-domain signal and FFT spectrum obtained are shown in Fig. 8. According to the spindle speed in the experiment, Eq. (11) can calculate the tooth passing frequency F z : where n s is the spindle speed and Z is the number of tool teeth. It can be seen from the Fig. 8 that the spectrum corresponding to point a is mainly composed of the spindle rotation frequency and its frequency doubling, and there are no other frequency components. Chatter frequency (387 Hz) appears in the spectrum corresponding to point B. The results show that the observed results are consistent with the SLD prediction results. Therefore, the processing parameters can be determined through SLD under different processing states, guiding the following experimental and SVM label settings.

Milling experiment
The test was carried out on the three-axis machine tool VDL-1000E. The PCB acceleration sensor and kister9171A rotary dynamometer collected the acceleration and milling force signals. The acceleration and force sampling frequencies were set to 1000 Hz and 5000 Hz, respectively. The milling experiment site is shown in Fig. 9, and the parameters are provided by the stability lobe diagram drawn based on the modal test. The test parameters are shown in Table 4.

Signal decomposition by BO-EK-VMD
In order to distinguish each frequency band, the BO-EK-VMD algorithm is proposed to decompose the processing signal. When using this algorithm, you must set the decomposition level K, penalty factor , and step length according to the experience of selecting parameters by the central frequency method, the range of decomposition layers K is set to [4,10], and the step length is 1. The range of penalty factor is set to [1000, 6000] in steps of 50 [35]. The acceleration signals obtained using the experimental parameters in Table 3 are used to optimize the VMD using the BO-EK algorithm. The variation curve of the envelope peak of each group of signals with the number of iterations is shown in Fig. 10. The optimized decomposition parameters and the maximum envelope kurtosis are shown in Table 5. Select the acceleration signal under the second group of experimental parameters in Table 4 and use the optimized parameter decomposition of the VMD algorithm. The decomposed signal and its spectrum are shown in Fig. 11. In the figure, red dots represent chatter frequencies. It can be seen from Fig. 11 that the chatter signal is successfully decomposed into 6 IMF, the boundary between each frequency band is clear, and there is no mode aliasing. At the Cutter same time, the chatter frequency is decomposed into the IMF5 frequency band, and the frequency is 387 Hz. Based on avoiding mode aliasing, the proposed BO-EK optimized VMD can successfully separate the chatter frequency band, proving its effectiveness.

Multi-signal feature extraction and fusion
When the cutting state changes from stable cutting to chatter cutting, the signal energy will transfer from cutting frequency to chatter frequency band. This paper selects the acceleration and force signals in X and Y directions to extract the energy entropy feature. Firstly, the feature of stable signal is extracted, and the acceleration time-domain signals in Y direction of the first and third groups of stable cutting in experimental parameter in Table 4 are selected, as shown in Fig. 12. The upper half of the figure is the acceleration signal in the Y direction, and the lower half is the energy entropy of the signal samples divided according to the number of sampling points. The specific rules for sample division are as follows: each sample contains 1 s of time-domain data. It can be seen from the figure that when the processing process is in a stable state, the energy entropy of the signal is small and fluctuates little. The energy entropy of the signal maintains a relative level, so the energy entropy of the acceleration signal in Y direction can be used as one of the feature vectors. The variation law of energy entropy of milling force signal in X and Y directions is basically consistent with that of acceleration signal in X direction.
Then, the feature extraction of the chatter signal is carried out, and the Y-direction acceleration time domain signals of the second and fourth groups of chatter cutting in the experimental parameter in Table 4 are selected, as shown in Fig. 13. The upper half of the figure is the acceleration signal in the Y direction, and the lower half is the energy entropy of the signal samples divided according to the number of sampling points. The specific rules for sample division are consistent with the above. It can be seen from the figure that when the machining process is in the chatter state, the energy entropy of the signal is large and remains around 1.3. The energy entropy of the signal is basically stable without too much fluctuation, so the energy entropy of the Y-direction acceleration signal can be used as one of the eigenvectors. The variation law of energy entropy of milling force signal in X and Y directions is basically consistent with that of acceleration signal in X direction.
As a nonlinear dimensionless parameter, energy entropy does not depend on cutting parameters and signal types, but only depends on the energy distribution of each frequency band of the signal, so it is very suitable for monitoring the milling chatter state of thin-walled parts. The above analysis selects all experimental data as signal samples according to 1 s. Then, the data are divided into 200 groups of chatter state processing signals and 200 groups of stable processing signals, totaling 400 groups of sample data. After feature extraction of the energy entropy of milling force X and Y directions and acceleration X and Y directions, the input dimension of the final feature vector is four, and the total number of samples is 400.

Multi-signal fusion verification
After extracting energy entropy features of milling force X, Y direction and acceleration X, Y direction, the cutting state is predicted and identified by support vector machine. Support vector machine transforms the input into high-dimensional space through nonlinear transformation and then solves the   optimal hyperplane. The transformation is realized through the kernel function. The commonly used kernel functions are polynomial kernel function, radial basis kernel function, and Sigmoid kernel function. In order to select the appropriate kernel function for the support vector machine, the classification accuracy of the three kernel functions is calculated according to the features extracted in Section 5.2, where c and g are set to 2 and 0.5, respectively, and other parameters are set to the default values, as shown in Fig. 14. It can be seen from the figure that the classification accuracy of the radial basis kernel function for both states is higher than the other two kernel functions. Therefore, the support vector machine with radial basis function kernel is selected to identify the chatter state.
After the kernel function type is selected, the parameters c and g of the support vector machine also significantly impact the recognition accuracy. In order to improve the recognition accuracy of the support vector machine, the Fig. 11 IMF and its spectrum obtained by optimizing VMD decomposition Bayesian optimization algorithm described above is used to optimize the parameters c and g. The optimized objective function is the maximum prediction accuracy of SVM. Before the parameter optimization, the range of parameters c and g is set. In this paper, the value ranges of c and g are set as [0,200] and [0,1000], respectively. The search steps of parameters c and g are 1 and 0.05, respectively. The iterative process is shown in Fig. 15. It can be seen from the figure that the optimal parameters c and g obtained by using the Bayesian optimization algorithm through continuous iterative updates are 80 and 0.05, respectively.
After the parameter optimization is completed, input the parameters into the SVM; 300 groups of cutting state feature samples obtained in Section 5.2 are used as training data sets to train SVM model. The remaining 100 sets of samples are used as the test set of the model to verify the recognition accuracy of the chatter state. The chatter and steady state of the training set and the test set account for 50%, respectively.  Firstly, the identification results of the chatter state for a single signal are shown in Fig. 16. In the figure, the green part represents the number of correctly classified samples and its proportion in the total samples, the red part represents the number of incorrectly classified samples and its proportion in the total samples, and the dark gray part represents the overall prediction accuracy and error rate of the samples. In Fig. 16a, for example, of the 53 stable state predictions, 86.8% were correct and 13.2% were incorrect. Of the 47 unstable states predicted, 91.5% were correct, and 8.5% were wrong. Among the 50 steady-state samples, 92% were correctly predicted as steady-state, and 8% were predicted as unstable state; Of the 50 unstable samples, 86% were correctly classified as unstable and 14% as stable. It can be seen from the figure that the recognition accuracy of force signal energy entropy and acceleration signal energy entropy for the two cutting states is 89% and 91%, respectively, and the recognition accuracy of the acceleration signal is relatively high, which means that the acceleration signal is more sensitive to the change of machining state. The milling force signal features and acceleration signal features are fused and input into the support vector machine for training. Consistent with the above sample classification method, the chatter and steady state of the training set and the test set account for 50%, respectively. The recognition accuracy is shown in Fig. 17. From the figure, it can be seen that the overall recognition accuracy is 95%. In the 50 groups of stable cutting state, two groups were identified as chatter cutting state, and the recognition accuracy was 96%. In the 50 groups of chatter cutting state, three groups were identified as stable cutting state, and the recognition accuracy was 94%. After fusing the force signal features and acceleration signal features, the recognition of the two cutting states is more accurate. In the recognition of steady state, the recognition accuracy of multi-signal fusion is 4% and 6% higher than that of force signal and acceleration signal respectively. In chatter state recognition, the recognition (a)Accuracy of force signal feature recognition ( b)Accuracy of acceleration signal feature recognition accuracy of multi-signal fusion is 8% and 2% higher than that of force signal and vibration signal. In the overall accuracy, the multi-signal fusion is 6% and 4% higher than the force signal and acceleration signal, respectively. The results show that the multi-signal fusion has high accuracy and reliability after the fusion and complementation.

Chatter identification of IL-KM-SVM fusion model
The previous section shows that multi-sensor fusion has better recognition accuracy than single sensor. The IL-KM-SVM fusion model is verified by the feature samples of multi-sensor fusion. Firstly, 150 samples of two states are selected to compare the clustering efficiency of K-means and IL-KM, and the experimental parameters are set as shown in Table 6. In the comparison process between K-means and IL-KM, K-means clustering algorithm directly clusters 300 samples, while IL-KM clustering algorithm uses the cluster centroid obtained from the first 200 samples as the initial centroid, and then clusters the remaining 100 groups of samples. This process is equivalent to adding 100 samples after 200 samples, and then conducting clustering. This process is an incremental learning process, which fully considers the clustering center of last time and improves the clustering efficiency. K-means and IL-KM were carried out 5 experiments to obtain the consumed time, the average value as the final result. The clustering time and iteration times of K-means and IL-KM are shown in Table 7. The clustering time and iteration times of IL-KM are about 13 ms; that is, the state has been judged when it is about 13 ms. The clustering efficiency of IL-KM is about 13.3% higher than that of K-means. The main reason is that IL-KM considers the last clustering result when clustering, thus reducing the number of iterations and improving the clustering efficiency. After verifying the clustering efficiency, the recognition accuracy of IL-KM-SVM model is verified. The update threshold of the model is set to 20 samples; that is, when the number of identified samples reaches 20, the results are filtered, and the samples with consistent recognition results are selected to update the model by incremental learning. The chatter identification algorithm proposed in this paper is implemented on a computer with a central processor frequency of 2.6 GHz using Matlab (2021b) software. According to the set threshold, the incremental model to identify samples at the same time, their model was updated 5 times. The update time of each model is about 6.39 ms. The time to identify the sample is about 0.23 ms; that is, when the chatter starts, the chatter can be identified at about 0.23 ms to meet the real-time requirements of online monitoring.
Combining supervised learning with unsupervised learning, the samples with the same results are selected as the samples for incremental learning of the model, which reduces the problem of sample label errors and improves the recognition accuracy of the model. The recognition accuracy of the IL-KM-SVM model is shown in Fig. 18. It can be seen from Fig. 17 that the recognition accuracy of the SVM model after multi-signal fusion has reached 95%. However, the model is a static model and cannot be updated incrementally according to the data, so the recognition accuracy is lower than that of the IL-KM-SVM model. Through comparison, it can be seen that the overall recognition accuracy has increased by 2%, with a small increase. The main reason is that the number of samples and increments less, in the case of sufficient samples, the advantages of incremental model will be more obvious. At the same time, the comparison between Figs. 17 and 18 shows that the recognition accuracy of stable state and chatter states has been improved.

Conclusions
In this paper, a method based on multi-signal fusion and energy entropy to detect the milling state of thin-walled parts is proposed. Firstly, aiming at the problem of VMD parameter selection, a BO-EK method is proposed to optimize VMD parameters. Then, energy entropy is extracted from IMF to construct the eigenvalue matrix. In order to provide a training label for the support vector machine, the method based on a stable lobe graph is used to determine the machining state and carry out experimental verification. The Bayesian optimization algorithm is used to optimize the parameters of SVM. Compared with the recognition accuracy of single signal and multi-signal fusion for cutting state, the results show that the recognition accuracy of multi-signal fusion for cutting state is higher. The incremental learning model is proposed by combining supervised learning and unsupervised learning. Compared with the SVM model and IL-KM-SVM model, the IL-KM-SVM model is better than the SVM model. The specific conclusions are as follows: (1) Optimizing VMD based on BO-EK can effectively separate the chatter frequency band in the signal. It can predict the machining state through SLD and effectively characterize the cutting vibration state through the trend analysis of IMF energy entropy. The energy entropy used in this method is the evaluation index of the global energy distribution, which is universal to different machine tools in theory. (2) In titanium alloy milling, the acceleration signal is more sensitive to the change of cutting state than the milling force signal, which provides solid theoretical support for the online monitoring of the cutting state. (3) Compare and analyze the recognition accuracy of single signal and multi-signal feature fusion for chatter state under the Bayesian optimized SVM model. The results show that the overall recognition accuracy of multi-signal fusion for chatter state is 8% and 6% higher than that of force signal and acceleration signal, respectively. The incremental learning model combined with supervised and unsupervised training can avoid the impact of sample error labels on the incremental learning model. The results show that the IL-KM-SVM model is better than the SVM model.

Author contribution
Mingwei Zhao has organized the project, analyzed and arranged data, and wrote the manuscript; Caixu Yue contributed the experiments and collected and analyzed data; Xianli Liu helped perform the analysis with constructive discussions.
Funding This research was supported by the National Natural Science Foundation of China (Grant Number 52175393).

Data availability
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.
Code availability Not applicable.

Declarations
Ethics approval The content studied in this article belongs to the field of metal processing and does not involve humans and animals. This article strictly follows the accepted principles of ethical and professional conduct.
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Conflict of interest
The authors declare no competing interests.