Dragomiretskiy and Zosso (2014) proposed a completely non-recursive signal decomposition method, the VMD algorithm. Compared with signal time-frequency domain analysis methods such as EMD, ensemble empirical mode decomposition (EEMD), and wavelet decomposition, VMD breaks away from the common thinking pattern of progressive recursive decomposition and transforms it into solving constrained variational problems, which can better analyze unbalanced signals and solve modal aliasing and adaptive problems (Liu and Tan, 2022; Miao et al., 2022; Song et al., 2022; Zhang et al., 2022).

Suppose that the original signal a(t) is decomposed into u i ( t ) ,   i = 1,2 , … , K by VMD, where u i ( t ) represents the ith IMF component. The algorithm implementation steps are as follows:

1) According to the Hilbert change formula, the unilateral spectrum of the Hilbert transformation of modal components is expressed as follows:

According to the convex optimization theory, to form the above highly nonlinear and non-convex variational problem, the augmented Lagrange function can be obtained: L ( { u i ( t ) } , { ω i } , λ ( t ) ) = α ∑ i = 1 K ‖ ∂ t [ ( δ ( t ) + j π t ) ∗ u i ( t ) ] e − j ω i t ‖ 2 2 +                                   ‖ a ( t ) − ∑ i = 1 K u i ( t ) ‖ 2 + < λ ( t ) , a ( t ) − ∑ i = 1 K u i ( t ) > (4) where α is the penalty factor and λ ( t ) is the Lagrange factor. The alternating direction method of multipliers was used to solve the variational constraint problem of Eq. 4 to determine the optimal solution { u i ( t ) } , { ω i } , λ ( t ) of the Lagrange function.

Particle swarm optimization (PSO) is a classical meta-heuristic optimization algorithm. In recent years, many studies have shown that the performance of the algorithm is greatly affected by the particle update speed, and even falls into local optimization and fails to converge globally (Luo et al., 2021; Yu et al., 2021). Therefore, van den Bergh and Engelbrecht (2002) integrated the concept of quantum mechanics into the updating process of individual position and proposed a QPSO algorithm. Compared with the PSO algorithm, the QPSO algorithm eliminates the concept of speed and uses Eq. 5 to update its position, which improves the global search ability and efficiency and accuracy of algorithm optimization to a certain extent. { p i j t = φ i j t p i j t + ( 1 + φ i j t ) G j t x i j t + 1 = p i j t ± β | x i j t − C j t | ln ( 1 Q ) (5) where p i j t and β are the individual historical optimum and control coefficient, respectively. Q∈ (0,1) represents a random number; C j t represents the mean value of individual optimal particles; x i j t and x i j t + 1 represent individual positions of particles before and after iteration, respectively; φ i j t ∈ (0,1) represents a random number; and G j t represents the globally optimal individual position iterated to round t.

The decomposition effect of the VMD algorithm is significantly different from that of the number of modes K and the parameter configuration of the penalty coefficient α (Kaur et al., 2020; Dhiman et al., 2021a; Dhiman et al., 2021b). To obtain the best decomposition effect, this paper used the QPSO algorithm to optimize the VMD algorithm parameters. Entropy value is an effective method to measure the randomness and complexity of signals (Liu et al., 2020). The more obvious the periodic law of vibration signals, the lower the complexity, and the smaller the entropy value. Therefore, if the IMF component obtained by the VMD algorithm contains more periodic fault characteristic information, the envelope entropy value will be smaller, and the noise interference of the IMF component will be lower. At this point, the IMF component decomposed by the VMD algorithm better represents the characteristic information of the vibration signal, with better robustness. { E i = − ∑ i = 1 N p i , j lg p i , j p i , j = b i ( j ) / ∑ j = 1 N b i ( j ) (6) Eq. 6 shows the formula used to solve the envelope entropy of the IMF component after Hilbert demodulation, where E _i represents the envelope entropy value; b i ( j ) represents the envelope function of IMF-i; and p i , j represents the normalized value of b i ( j ) . As shown in Eq. 7, we used the mean of the k IMF component envelope entropy as the fitness function of the QPSO algorithm, and optimized the VMD algorithm parameters by minimizing the IMF component envelope entropy. f = 1 k ∑ i = 1 k E i (7)

SVM is a machine learning algorithm with strong classification performance in small sample learning. The essential idea is to use a kernel function to map problems that are not linearly separable to a high-dimensional space and establish a hyperplane in the high-dimensional space to classify the samples (Huang et al., 2020).

SVM can be used for binary classification problems, for example, the sample set D = { ( x i , y i ) } , x i ∈ ℝ d , y i ∈ { − 1,1 } , i = 1 ,   2 , … n , where y_i represents the label of the sample, D represents the dimension of the sample, and n represents the number of samples. The mathematical model expression of SVM is shown in Eq. 8: min ω , b || ω || 2 2 + C ∑ i = 1 m ε i s .t .   y i ( ω T x i + b ) ≥ 1 - ε i ,   i = 1 ,  2 ,  … n . ε i ≥ 0 ,   i = 1 ,  2 ,  ⋯ n .     (8)

To improve the generalization performance of the SVM algorithm, Eq. 8 was converted into a dual problem based on convex optimization theory and the kernel method was introduced to obtain Eq. 9: max α ∑ i = 1 m α i − 1 2 ∑ i = 1 m ∑ j = 1 m α i α j y i y j K ( x i , x j ) s .t .   ∑ i = 1 m α i y i = 0 , 0 ≤ α i ≤ C ,   i = 1 ,  2 ,   ⋯ ,   m .     (9) where α i is the Lagrange multiplier and K ( x i , x j ) is the kernel function.

PCA is a common dimensionality reduction method for the pre-processing of high-dimension feature data. It achieves the compression of feature data by calculating a covariance matrix and correlation coefficient for orthogonal change and finally obtains mutually independent principal components (Fattoh and Safwat, 2022; Jamal et al., 2022). The characteristic compression mathematical expression of PCA for original data D is shown as follows: Y i = a i 1 d 1 + a i 2 d 2 + … + a i p d p = A i T D (10) where A i = [ a i 1 , a i 1 , … , a i p ] T is the characteristic vector corresponding to the covariance matrix, and Y i is the principal component after orthogonal change. Each principal component compressed by the PCA algorithm corresponds to variance in the original data. The sum of the total variances is equal to the sum of the original variables. The ratio of variance to total variance represents the principal component contribution rate.

Considering the influence of the number of modes K and penalty coefficient α on the decomposition effect of the VMD algorithm, the QPSO algorithm was used to optimize the parameters of the VMD algorithm to minimize the envelope entropy. Then, the PCA algorithm was used to compress the IMF components obtained by VMD decomposition under optimal parameters. Finally, the vibration data preprocessed were input into the SVM model to predict the rotor system state. Figure 1 shows the flow chart of the rotor fault diagnosis model based on the QPSO-VMD-PCA-SVM method in this study. The specific realization steps were as follows:

1) Initialize the maximum iteration time T and population number M of the QPSO algorithm. The maximum number of iterations T in this study was 50. The number of populations was set to 100;

The sample data used in this paper were from the open data set of rotor vibration from the state Key Laboratory of Hydraulic Machinery, Ministry of Education of China (Liu et al., 2019; Li et al., 2021a; Li et al., 2021b; Le et al., 2021; Toyoda and Wu, 2021). The sample size of the rotor vibration data set is shown in Table 1. There were 180 samples in four rotor system operating states. The original data set was divided into training and test sets at a ratio of 8:2. The experimental platform of the data set simulated the rotor unbalance, dislocation, and friction by controlling the mass block distribution at the edge of the rotor mass disc, the relative position of the two shafts on the coupling, and the contact between the friction screw shell and the rotating bearing. During signal collection, the rotor speed was set to 1200 r/min, the sampling frequency was set to 2048 Hz, and the sampling time length was 1 s.

According to the rotor system fault diagnosis flow chart shown in Figure 1, the rotor vibration signal was decomposed by VMD, and then the number of modes K and penalty coefficient α were optimized by the QPSO algorithm. After 50 iterations of the optimization algorithm, the optimal values of the modal number K and penalty coefficient α were 8 and 5016.8, respectively. The VMD algorithm was re-initialized according to the optimal value, and the original vibration signals were decomposed. Figure 2 shows the original signals and curves of each modal component of the vibration signals of the rotor system under different working conditions. The yellow curve in Figure 2 represents the envelope entropy curve of each IMF component after Hilbert demodulation. The curve in Figure 2 shows that the optimized IMF components had a relatively obvious cyclical fluctuation trend in the time domain, and the corresponding envelope entropy curves of each component had good stationarity and uniformity. In addition, the IMF components of the vibration signals of the rotor system under different operating conditions were markedly different. Therefore, the QPSO-VMD algorithm proposed in this paper is feasible for the decomposition of the vibration signals of a rotating subsystem. The IMF component obtained could be used to extract the periodic rules of the vibration signals, and was markedly different under different working conditions, providing greater distinguishing feature information for the subsequent fault diagnosis model.

After the original vibration signal was decomposed by the VMD algorithm, the data dimension of the IMF component obtained was 8 times that of the original signal. To improve the robustness of the fault diagnosis model and reduce the sparsity of data in the high-dimensional space, the PCA algorithm was used to perform feature compression on the decomposed vibration signal. Figure 3 shows the characteristic contribution rates of each dimension under different characteristic compression scales. The curve in Figure 3 shows the feature contribution rate was mainly concentrated in feature dimensions 1 to 5, and when the feature dimension exceeded 5, the feature contribution rate tended to be 0. Therefore, this study used the PCA algorithm to compress high-dimensional vibration signals into 5-dimensional data.

To verify the effectiveness of the proposed method, we compared it with ELM, BPNN, and SVM based on different vibration signal decomposition methods, including the QPSO-VMD algorithm and EMD-PCA. Table 2 shows the fault diagnosis accuracy rate of rotor systems under different fault diagnosis methods. In this study, the accuracy rate of each optimal type is marked in bold. Data in the table show that the proposed method had the highest fault diagnosis accuracy in the normal, dislocation, and friction classes, with accuracy reaching 100%. The overall accuracy of the proposed method was also optimal. Compared with other fault diagnosis methods, the overall accuracy was increased by 5.55%. Compared with various fault diagnosis methods based on the QPSO-VMD algorithm, the proposed method achieved the optimal fault diagnosis performance, which indicates that the proposed PCA algorithm effectively improved the robustness of the fault diagnosis model using high-dimensional IMF component compression. Compared with various fault diagnosis methods based on the EMD-PCA algorithm, the proposed method still achieved optimal results, which indicates that the QPSO-VMD decomposition method had a better performance for the analysis of vibration signals in a rotor system.

Vibration signal feature processing methods directly affect the accuracy of rotor fault diagnosis models, as shown in Figure 4, which is an SVM fault diagnosis confusion matrix based on different signal processing methods. By comparing the confusion matrices under different signal processing methods in Figure 4, we found that the fault diagnosis effect of the QPSO-VMD-PCA processing method was the best with an accuracy rate of 97.22%.