Auto-identification of dominant modal parameters from multi-batch signals based on weighted SSA to suppress milling vibration

The modal parameters identified from on-site cutting signals can more truly reflect the dynamics of the machine tool in the operating state. However, due to the spindle rotation and position change of movable parts in the cutting process, modal identification based on on-site cutting vibration signals is interfered with the harmonic frequencies, structural time-varying, artificial analysis, and other uncertain factors. The current modal parameter identification methods cannot realize the auto-identification of machine tool structures simultaneously considering the above factors. Therefore, to realize the auto-identification of structural dominant modal parameters eliminating the interference of harmonic, structural time-varying, and artificial analysis, a new weighted SSA (singular spectrum analysis) method is proposed in this paper. First, multi-batch on-site vibration signals are decomposed to extract the eigenvalue and eigen matrix through singular value decomposition (SVD). Then, based on the variance filtering of principal component analysis, a half principal component analysis is proposed to extract the weighted vector of the eigen matrix. After that, the clustering algorithm is adopted to average the sample set, and the power spectrum curve is modified and reconstructed according to the cluster center. The dominant modal parameters are auto-identified with the reconstructed curve and optimized through the genetic algorithm. Finally, cutting tests are conducted to verify the feasibility and effectiveness of the auto-identification and optimization method.


Introduction
The machining performance of machine tools is closely related to the dynamics of machine tool structure [1].With the development of machine tools toward high speed, high precision, and high reliability, it is essential to identify more accurate machine tool dynamic parameters to improve machining performance.Experimental modal analysis (EMA) is the common means to study the dynamic characteristics of machines under static conditions.However, there is a significant difference between the dynamics of machine tools under the cutting state and the static state [2,3].Therefore, there is an urgent need to extract the more accurate modal parameters from on-site monitoring data during the machining process.
For the extraction of machine tool modal parameters under on-site machining, there are two aspects of issues to address.First, the operational modal analysis (OMA) can be used to identify structural modal parameters based on the on-site response signal, but it requires the excitation signal to be a white noise signal [4].For rotating machinery such as machine tools, the excitation usually contains periodic signals, resulting in the harmonic frequency seriously interfering with the identification of modal parameters [5,6].Second, due to the influence of machine tool structural time-varying [7], the uneven workpiece material during the cutting process [8], and the other uncertain factors such as artificial participation in choosing the pole of the stabilization diagram [9,10], there are differences among the identified results of multiple vibration response signals collected under the same cutting parameters.
For the first issue of the harmonic interference, the approaches generally conducted can be divided into two steps.The first step is to distinguish the harmonic frequency, and then some special filters or improved modal identification algorithms are adopted to remove the pseudo mode.Brinker et al. [11] proposed a criterion based on a probability density function to distinguish the natural frequency and harmonic frequency of the structure.Agneni et al. [12] proposed a method to identify harmonic frequencies by the statistic "Entropy."Chen et al. [13] presented a method using correlation functions of the responses to distinguish harmonic frequencies and removing the harmonic effect.This method is applied to a rotating blade to extract modal parameters in its operational state.Once the harmonic signal is identified, harmonic removal can be carried out by comb filter, cepstrum editing, RDT (random decrement technique) [14,15], etc. Kiss et al. [16] adopted a notch filter to extract the basic harmonic frequency and then used a comb filter to eliminate the harmonic components from the structural response signals in the milling process.Mark et al. [17] used the time-domain synchronous averaging method to detune the vibration signal of a uniformly rotating meshing gear pair and diagnose gear defects.Randall et al. [18] applied periodic excitation to a gearbox containing 22 sets of simple harmonic signals with a frequency change amplitude of less than 15% through the exciter.The vibration signal is processed by the cepstral editing method.Randall et al. [19] also compared the harmonic removal methods of cepstral editing and time-domain synchronous averaging (TSA).Regarding the modification of the modal identification algorithm, Mohanty and Rixen improved the LSCE algorithm [20], ITD algorithm [21], and ERA algorithm [22] respectively to cope with the periodic component in the excitation for the known harmonic frequency and verified the correctness of the method.However, these above methods have high requirements for the recognition accuracy of the harmonic frequency and pseudomode.To avoid the step of harmonic recognition, Devriendt et al. [23] proposed the method of TOMA (transmissibility-based operational modal analysis) based on the conduction function.Based on this study, Devriendt et al. [24] further proposed a multi-reference transfer function operational modal analysis method (pTOMA) to reduce the number of required excitation loadings.However, the TOMA method is usually used for offline identification instead of online automatic identification.
For the second issue of the influence of time-varying of machine tool structures, artificial or uncertain factors during the modal analysis, some scholars have studied the combination of machine learning algorithms with classical dynamic analysis methods to realize the auto-identification of modal parameters of structures.Reynders et al. [25] applied a clustering algorithm to divide the response data of multiple measurement points and identify modal parameters.In this method, the poles of stable points are filtered by setting the threshold, and the center value of pole clustering is used as the feature and modal order for parameter identification.Cremona et al. [26] converted the data type of the vibration signal and extracted the parameters by clustering on the premise of retaining the modal information.Mugnaini et al. [27] combined the SSI (stochastic subspace identification) method with the clustering algorithm to realize the automatic operational modal analysis of an airbus H135 helicopter blade.Liu et al. [28] designed a deep neural network (DNN) to optimize the process of identifying modal parameters.The proposed method is verified through a numerical simulation and modal analysis of two actual bridges.Lu et al. [29] optimized the domain adaptation of the deep neural network for structural fault diagnosis, and it was verified that the method has higher classification accuracy and better decision-making by using actual measurement data.Trendafilova et al. [30] introduced the application of singular spectrum analysis (SSA) for vibration analysis of a nonlinear spring-mass system and verified that SSA can be used for vibration analysis especially for structures with more complex or nonlinear structures.According to the above review, there are few reports focused on the auto-identification of machine tools based on on-site cutting vibration signals while simultaneously taking into account the influence of harmonic frequency, structural time-varying, and manual intervention in modal parameter identification.
Therefore, aiming at the two major issues in the identification of modal parameters based on on-site cutting signals, this paper proposes an automatic modal parameter identification method combining weighted SSA and a clustering algorithm.First, SSA is used to decompose the components and extract features of the processed signal set.Then, principal component analysis (PCA) is used to filter the variance of the feature matrix to extract the weighted vector.Third, the sample component set is clustered based on the projection of the weighted vector.After the average of the sample set through the cluster algorithm, the power spectrum curve is modified and reconstructed according to the cluster center.In the reconstructed signal, the influence of the harmonic frequency and time-varying factors are eliminated, and then the modal parameters of the dominant mode of the structure are identified.
The rest of this paper is organized as follows.In Section 2, the identification process based on the presented weighted SSA method is introduced.In Section 3, experimental verification of the weighted SSA method is conducted.In Section 4, the genetic algorithm is adopted to optimize the tool parameters of the spindle tool model.Section 5 gives the conclusion to this paper.

Identification of dominant modal parameters based on weighted SSA
In the actual machining process of the machine tool, due to the rotating movement of the spindle, the existence of a power frequency signal, and other factors, the response includes the influence of periodic forced vibration and harmonics generated by power frequency.The self-power spectrum of the structural vibration response is generated by power frequency, periodic excitation, random excitation, and noise.The periodic excitation will be changed by the transformation of the system structure, such as spindle eccentric force, tooth frequency, gear meshing, ball screw feed, and other factors.Therefore, harmonics occupy more and time-varying components in the system response.The relationship between input components, output components and frequency response of the system is shown in formula (1).
where H(ω) is the frequency response function of machine tool structure.G � ff ( ) is the self-power spectrum of random excitation, and G � TT ( ) represents self-power spectrum of periodic input excitation.G noise (ω) represents the noise input.When the input excitation contains periodic signals, the operational modal analysis cannot identify the modal parameters because it cannot effectively distinguish the random and periodic vibration response of the structure.
Therefore, to extract the modal parameters based on the output response signal, the signal must be reconstructed to extract the modal features.In this paper, the Hankel matrix is constructed for the response signal during the operation of the machine tool, and the characteristic set of the signal is constructed by the singular value method.The identification process of the dominant mode is shown in Fig. 1.

Component decomposition and feature extraction based on the SSA algorithm
The component decomposition and feature extraction based on the SSA algorithm can be divided into three steps.First, the Hankel matrix is constructed.Second, the original signal is decomposed into independent component combinations.Third, the number of features and corresponding signal components are selected based on (1) the proportion of singular values to construct the feature matrix and decomposition component matrix.
Step 1 (construct Hankel matrix): Transform the signal sequence into a Hankel matrix for matrix decomposition according to its sampling points.The original signal sequence can be expressed as: where k is the length of the sampling signal, m is the sliding step, and q = k − m + 1.
Step 2 (decomposition by SVD): Decompose the Hankel matrix to obtain the left matrix and right matrix of X h .Hence, X h is decomposed as: where Λ 1 and Λ 2 are diagonal matrixes, and diagonals are σ 1 , σ 2 , ⋯, σ l .The original signal X h can be expressed as: where σ i is arranged from large to small.X i is sorted according to the contribution to the original signal of each component.
Step 3 (construct decomposition component and eigenvector): According to the contribution of each singular value, the first n singular values are selected as sample features for subsequent feature screening, and the signal is reconstructed according to its corresponding signal decomposition components.
The obtained eigenvector and the corresponding component are: (2)

Extraction of the weighted vector of eigen matrix
The half-PCA (principal component analysis) method is adopted to extract the weighted vector of the eigen matrix obtained in Section 2.1.The principal component analysis method uses the variance of the same feature among samples to filter the feature.Variance filtering is used to compare and filter the variance values of the same latitude characteristics of each sample.If the variance of feature 1 of n samples is very small, the samples have basically no difference in this feature.Therefore, the information entropy of such features is very low and can be filtered.
According to the analysis of the characteristics of the machine tool operation signal, there is considerable harmonic interference in the signal, so each sample should have the same type of harmonic information, and the variance of such characteristics should be the lowest.At the same time, for the differences between different samples, the covariance analysis based on principal component analysis can effectively identify the maximum differences of various samples under the projection of specific dimensions.According to the extraction of weighted vectors, the differences of each sample were comprehensively analyzed to construct a correction curve.The process is shown in Fig. 2.
According to the decomposition results, the most important eigenvector is selected as a weighting vector.The feature vector indicates that the projection of the information contained in the sample feature in this dimension has the maximum difference.Therefore, the sample signal is reconstructed according to the weighted vector and the modified signal vector is obtained: where The original decomposition component matrix is reconstructed as follows: where

Modal correction and identification based on weighted vector
Based on the reconstructed matrix [X f ] add and modified signal vector [X] revise , the K-means algorithm is adopted to average the multisample data.The correction curve and reconstruction curve are fit with the sample center point.Then, the dynamic curve that contains all modal information is extracted and used for parameter identification.K-means is an unsupervised learning algorithm that can divide the eigen matrix of a group of M samples into K clusters without intersection.The clustering objects of the algorithm in this paper are the modified amplitude matrix of [X f ] add and [X] revise .[S] add is the reconstructed signal that contains the decomposed components of machine tool operation response signals from different structures.[S] revise is the revised signal of the corresponding [S] add .The hyperparameter K is the number of sensors arranged in the experiment.
The process can be summarized as shown in Fig. 3. y mean and y revise are obtained through the K-means algorithm.Based on the signal reconstruction principle of singular spectrum decomposition, y mean and y revise are superimposed to obtain the modified dynamic characteristic curve of machine tool operation, which filters harmonic frequency and noise and contains all sample (9) decomposition feature matrix covariance matrix weighting vector of feature matrix

Identification of the dominant modal parameter under idle operation
The proposed weighed SSA is adopted to comprehensively analyze the multisample data set from all measuring points and different batches during idle operating of the machine tool.According to the steps introduced in Section 2, first, the vibration response signal is decomposed based on SSA as shown in Fig. 5.The proportion of each eigenvalue from decomposed components of vibration signals is compared as shown in Fig. 5a.According to the proportion of eigenvalues in Fig. 5a, 10 decomposition components with a total eigenvalue proportion of 90% are selected to construct the decomposition component vector and corresponding feature vector.Then, the 200 × 10 component matrix and corresponding eigen matrix are constructed based on the self-power spectrum of each response signal.The decomposition component matrix is expressed as[X f ].The eigen matrix is expressed as [Σ f ] and the reconstructed signal vector is [X f ] add as shown in Fig. 6a.The half-PCA algorithm is adopted to calculate the weighting vector of the sample feature matrix, which will be linearly added with [X f ] to modify the curve vector linearly superimposed to generate the corrected self-power spectrum signal.The modified self-power spectrum based on the proposed method is shown in Fig. 6b.Compared with the unmodified self-power spectrum as shown in Fig. 6a, the peak trend corresponding to the natural frequency in the frequency domain is more obvious, and the oscillation amplitude within 0-50 Hz is reduced, which effectively avoids the generation of calculation pseudo modes.
The modal parameters identified through the self-power spectrum with two different methods are listed in Table 1.They are also compared with the identified modal parameters through the FRF of impact tests.It shows that using multiple data to carry out weighted analysis is helpful to identify the dynamic parameters of machine tools more accurately and eliminate the influence of various errors.The identification results are quite different from the response data of a single measuring point, which indicates that the data measured by the system measuring point in a single experiment may not contain more complete modal information due to the influence of operating conditions.Through the comparison of the identification results, the method proposed in this paper to modify the modal parameter curve by comprehensive analysis of multiple data is effective and accurate, and making full use of the data of multiple measurement points for comprehensive analysis is helpful to accurately identify the structural characteristics.

Identification of dominant modal parameters under cutting conditions
The test setup is shown in Fig. 7a.A cutter with six teeth was used in the milling experiment.The workpiece is aluminum alloy.The clamping height of the workpiece is consistent with the spindle position of idle operation to keep the machining position of the machine tool consistent with the idle running experiments.The total cutting distance in the Y direction is 50 mm, the cutting depth is 1 mm, and the cutting width is 3 mm.The spindle speed is set at 1500 rpm, and the feed speed is set at 300 mm/min.The self-power spectrum of a single measuring point is shown in Fig. 7b.The proposed weighted SSA method is adopted to separate and eliminate the harmonic and noise components of the self-power spectrum for the multisample cutting signal.First, the self-power spectrum of the response signal is decomposed into 200 components through SSA.Among the 200 components, the first-order component, the second-order component, the third-order component, the fourth-order component, the fifth-order component, and the 150th-order component are shown in Fig. 8a-f, respectively.Comparing the signals of each component in Fig. 8, it can be seen that as the order of the singular value component decreases, the corresponding power spectrum component becomes more oscillating, and the amplitude is  Then, according to formulas ( 4) and ( 5) in Section 2.1, the proportion Ratio i of each decomposed component is calculated with the eigenvalue σ i .The calculated result which includes the ratios of eigenvalues for a total of 200 decomposed components is shown in Fig. 9a.It shows that the ratio of eigenvalues of the self-power spectrum decomposition component of the cutting response signal is dispersed compared with the empty running signal.The component signals corresponding to 90% of eigenvalues are selected as the reserved sample signals.Five hundred measured signals are collected to compose the sample set, and the 500 × 100 eigen matrix of the sample set is constructed.
According to Section 2.2, the eigen matrix is processed by half-PCA to generate a weighted vector, and then according to the principle of linear superposition, the modified signals are reconstructed.To eliminate the effect of pseudo mode, the modified signal set is further averaged through the K-means algorithm.After half PCA and K-means weighted averaging processing, the modified self-power spectrum of the cutting signal is generated as shown in Fig. 9b.From Fig. 9b, it can be concluded that the peak of the self-power spectrum in the cutting state of the machine tool is mainly concentrated in the high-frequency range of 500-2000 Hz.
The identified dominant modal parameters are listed in Table 2. From the identification results in Table 2, it can be concluded that the cutting signal can effectively identify some order modal parameters of the machine tool in the machining process after correction, which is relatively consistent with the identification results of the machine tool impact experiment.The weighted SSA proposed in this paper is based on multisample signals.The weighted modified decomposition analysis method based on multisample signals proposed in this paper can effectively analyze the modal parameters of machine tool operation.

Suppression of the dominant mode based on the genetic algorithm
In this section, the spindle tool system is optimized through a genetic algorithm, where the objective function is designed based on the identified dominant modes.According to the designed algorithm, the optimal tool corresponding to different working conditions is selected to realize the suppression of cutting vibration and improvement of machining performance.

Dynamic model based on the dominant mode
The identified modal parameters are used to characterize the physical characteristics of the machine tool, and the machine model is established for structural optimization in the actual machining process.For n-DOF linear systems: Equation ( 11) can be transformed into: According to the eigenvector normalization method, the modal mass of the system is expressed as: where M is a diagonal matrix with order n, and [m] is the diagonal matrix of modal mass, and m i is the modal mass of the ith degree of freedom.{φ} i is the mode shape of the ith order and Combining the above formulas, the mass matrix and stiffness matrix of the machine tool can be obtained by substituting the normalized modal shape into Eq.( 13).The mass normalized modal shape matrix is listed in Table 3.The stiffness normalized modal shape matrix is listed in Table 4.The mass matrix and stiffness matrix of the machine tool can be calculated through Eq. ( 13), as shown in Table 5 and formula (15).The stiffness matrix K is the diagonal matrix.The diagonal stiffness is the total stiffness of each degree of freedom of the system, and the nondiagonal stiffness is the connection stiffness of each degree of freedom.

Optimization of cutting tool based on genetic algorithm
The dynamic model based on the identified modal parameters is shown in Fig. 10.The system frequency response function fitted by the system mass and stiffness matrix calculated in the previous section is consistent with the cutting identification modal frequency, so the mass and stiffness matrix can be used to construct the objective function of genetic algorithm.M6 and K6 in Fig. 10 are additional tool-modified systems.That is, the genetic  The upper part of Fig. 11b shows the fitting degree of parameter optimization, and the lower part shows the optimization result of M 6 and K 6 .The main determinant of restraining the dominant mode is the dynamic stiffness of the additional tool.The parameter identification results are shown in Table 6.The results show that the main method to suppress the dominant mode in the current cutting state  is to improve the dynamic stiffness between the tool and the spindle, and the influence of quality factors is small.

Verification of tool dynamic optimization
According to the above genetic algorithm optimization results, the dynamic stiffness between the milling cutter and the spindle can be adjusted by changing the overhang of the milling cutter.Therefore, the same milling cutters with three different overhangs were used for the milling experiments.Overhang #3 is set as the original condition of the milling cutter, which is 150 mm.Overhang #1 is 15 mm shorter than overhang #3, which is 135 mm.Overhang 2 is 30 mm shorter than overhang #3, which is 120 mm.The measured vibration signals are shown in Fig. 12.
Operational deformation analysis (ODS) is carried out for the three vibration signals, and the results are shown in Fig. 13.According to the proportion of the ODS decomposition coefficient under the three states, when the cantilever length is 15 mm shorter, the dominant mode in the cutting process of the machine tool disappears, and the contribution of each mode shape to the vibration tends to be the same.When the overhanging extension is shorter than 30 mm, the dynamic stiffness of the spindle tool is enhanced the most in theory, but the ODS decomposition of the machining vibration shows that the dominant mode is still the second-and fourth-order modes, which is close to the proportion of the coefficients of the original overhanging extension.
The surface quality of the machined workpiece is shown in Fig. 14.From the workpiece surface shown in Fig. 14, it can be concluded that when the dynamic stiffness between the spindle and the tool is reasonably improved, the dominant mode of the machine tool can be effectively suppressed and the processing quality can be improved, but the processing performance will not be significantly improved by increasing the stiffness of each part of the machine tool.This section verifies the effectiveness of the method proposed in this paper through experiments.By accurately identifying the dominant cutting mode of the machine tool and taking it as the fitness function of the genetic algorithm, the tool that most conforms to the dynamic characteristics of the machining can be effectively selected.

Conclusion
In this paper, weighted SSA (singular spectrum analysis) is proposed to identify the dominant modal parameters of the machine tool during cutting.In this method, first, singular value decomposition (SVD) is used to decompose the power spectrum signals and extract the signal features.Then, the variance filtering principle is used to extract the weighting vector from the feature matrix.Based on the weighted vector, the characteristic matrix is modified to reconstruct the power spectrum curve.Principal component analysis and clustering methods are used to screen and distinguish a large amount of data from different batches, and the parameters are identified considering the conditions of each dimension.The influence of time-varying factors of machine tools is eliminated, and the identification result is more accurate.Based on the identified modal parameters of the machine tool in the machining process, the accurate dominant modal identification theory is proposed, and the vibration loss function

Fig. 1
Fig.1The identification process of dominant modal parameters

Fig. 2
Fig.2Process of extracting the weighted vector of eigen matrix

Fig. 3 Fig. 4 Fig. 5 Fig. 6
Fig. 3 The process of modified modal curve generation based on clustering

Fig. 8
Fig. 8 Decomposed components.a The first-order component, b the second-order component, c the third-order component, d the fourth-order component, e the fifth-order component, f the 150th-order component optimizes the target parameters.Experimental setup M 6 ∈ [−5, 5], K 6 ∈ [1000, 10000].The parameter optimization steps are shown in Fig. 11a, and the optimization results are shown in Fig. 11b.

Fig. 9 a
Fig. 9 a Proportion of each eigenvalue.b Reconstructed signal

Fig. 10 Fig. 11 a
Fig. 10 Dynamic model of the machine tool based on the identified dominant mode

Table 1
Identified modal parameters based on different signals and methods Fig. 7 a Cutting test setup.b Self-power spectrum of the measured vibration signal

Table 5
The mass matrix based mass normalized mode shapes