Determining the embedded dimension of a singular value decomposition Hankel matrix and selecting the singular values representing the intrinsic information of fault features are challenging tasks. Given these issues, this work presents a singular value decomposition-based automatic fault feature extraction method that uses the probability-frequency density information criterion (PFDIC) and dual beetle antennae search (DBAS). DBAS employs embedded dimension and singular values as dynamic variables and PFDIC as a two-stage objective to optimize the best parameters. The optimization results work for singular value decomposition for bearing fault feature extraction. The extracted fault signals combined with envelope demodulation can efficiently diagnose bearing faults. The superiority and applicability of the proposed method are validated by simulation signals, engineering signals, and comparison experiments. Results demonstrate that the proposed method can sufficiently extract fault features and accurately diagnose faults.

1. Introduction

Bearings are widely applied to rotating machinery equipment, such as motors and pumps. Bearing faults may lead to the destruction of equipment, which in turn results in major downtime, economic losses, and even casualties [1, 2]. Therefore, bearing state requires monitoring and diagnosis.

Bearings usually work in harsh environments, where the background noise is strong and suppresses signal impulses. Meanwhile, the manufacturing errors in the bearing can also generate the background noise [3, 4]. Therefore, one of the most critical tasks prior to bearing fault diagnosis is to extract fault features from original signals.

Feature extraction is one of the important steps to diagnose the fault by extracting the fault information in the signal as a symptom function for diagnosis. A wide range of fault feature extraction methods have been explored for bearing fault diagnosis. They include filtering methods [5], signal decomposition methods [6], statistical analysis methods [7], and stochastic resonance [8]. An effective feature extraction method is signal decomposition. This method decomposes a signal into several different mode components and reconstructs the signal by extracting the effective modal component that contains sufficient fault information to enhance the fault feature. Examples of such methods include empirical mode decomposition (EMD) and its improved algorithm [9], variational mode decomposition (VMD) [10], local mean decomposition [11], and singular value decomposition (SVD). SVD is particularly effective in fault feature extraction and has excellent stability and invariability. In this method, a singular value (SV) can present a signal’s intrinsic characteristics and promote the signal-to-noise ratio (SNR). SVD can extract fault features effectively against strong background noise. The effectiveness of SVD has been proved on the basis of many experiments [12]. However, two problems remain: determining the embedded dimension (ED) of the Hankel matrix (HM) and selecting the SV representing the intrinsic features of a fault signal [13]. ED is used to construct the HM, and it plays a crucial role in fault feature extraction results. At the same time, the SVs which represent noise set to 0 represent that the signal is denoised. In some bearing faults, especially in low-speed bearing faults, the SNR is low such that the signal energy distribution and background noise are unclear; in this case, successfully selecting the SV is difficult. A number of studies have explored these issues [14, 15]. Zhao and Ye [16] introduced a method for SV selection based on difference spectra. However, the method’s performance decreases when the background noise signal is strong. Gao et al. [14] presented a method for selecting an effective SV through the asymptotic relationship between SVs and vectors. The extracted fault feature signal matrix is reconstructed by minimizing the asymptotic loss and performs better than that resulting from the ordinary reduction of SVs by thresholding methods [17]. However, the method has some assumptions, such as the orthogonal invariance of noise signals. This assumption is difficult to satisfy in engineering projects.

The aforementioned issues imply that existing SVD methods are not ideal in the selection of EDs and SVs. Intelligent optimization methods, such as the genetic algorithm (GA) and particle swarm optimization (PSO), have attracted increasing attention [18, 19]. These methods can be used to identify the best EDs and SVs. As these optimization algorithms require multiple individuals in the search for optimization solutions, their computational complexity is relatively high. Hence, the current work presents an automatic fault feature extraction method that is based on SVD. Beetle antennae search (BAS) is an efficient optimization method. It enables individual and environmental recognition, it does not need an understanding of function gradient information, and it only requires one individual to search for solutions [20]. Under the designed dual BAS, the ED and SV combination is an associated two-stage objective for optimization. ED is adopted to reconstruct the HM. Each combination based on the selected ED is used to rebuild the extracted fault feature signals. The contributions of this work are as follows: (1) the designed probability-frequency density information criterion (PFDIC) can effectively extract fault features; (2) the proposed algorithm can efficiently select the best ED and SV. According to the designed PFDIC, the extracted signals and normal signals are used to evaluate their similarities. The SV combination with maximum similarity means that the signal contains background noise information. Therefore, the remaining SV is regarded as representative of fault characteristic information and is used for signal reconstruction. The fault feature signal extraction with envelope demodulation can effectively diagnose bearing faults.

The specific contents of this paper are as follows: Section 2 presents the mechanism of SVD fault feature extraction; Section 3 describes the developed method of this study; Sections 4 and 5 show the superiority and applicability of this method with simulation and engineering experiments, respectively; Section 6 describes the comparison experiment; Section 7 presents the conclusions.

2. SVD-Based Fault Feature Extraction

This section introduces the mechanism of SVD-based fault feature extraction. Let be an acquired raw signal of length in an abnormal state. The signal can construct a matrix ,where , and . Matrix . It is called the attractor trajectory matrix or the HM. It has an orthogonal matrix and .

Equation (2) is the matrix of SVD, and . Here,where , , and is the SV of matrix . All nonzero singular values are arranged in the following descending order: . In some situations, the first th SV mainly reflects the object fault signal, and the smaller SVs reflect the noise signals. The mechanism of SVD-based fault feature extraction is shown in Figure 1.

When SVD performs signal processing, equation (2) can be rewritten aswhere , , and , .

Therefore, the SVD operation in the HM can express the original signal as the simple linear superposition of some subsignals. Only by selecting and directly adding SVs representing fault information can fault feature extraction be realized.

3. Developed Method

3.1. Beetle Antennae Search

BAS is a type of metaheuristic intelligent optimization method that is based on group optimization [20, 21]. The BAS algorithm imitates the function of antennae and the random walking mechanism of beetles in nature. Beetles use their antennae on both sides of their bodies to find food resources, and their position represents an optimal solution. If one antenna is close to food, then the aromas it picks up will be strong. The beetle then climbs to this position. Therefore, BAS is inspired by the discovery and search behavior of long-horned beetles.

BAS uses food odor intensity at a given position as the value of the objective function. To search the initial uncertain position, a beetle should be able to crawl randomly in all orientations. Vectors with random directions can be constructed as representatives and can be standardized in any dimensional space.

Normalizing the random searching direction is as follows:

Here, is a random function, and dim denotes the position dimensions.

When a beetle looks for food, it does not know where the food is. It uses two antennae to explore food aromas and climbs toward the direction with the strongest aromas.

The right and left antenna positions are, respectively, obtained aswhere is the iteration number, is the right position, is the left position, is the beetle position, and is the antennae’s sensing length.

The beetle determines its search behavior on the basis of the detected aromas. Therefore, the beetle’s next position is as follows:where , , , and are the searching step size, sign function, optimized function, and the beetle’s movement direction, respectively. If the maximum value is optimized, then ; otherwise, .

The update of and is

3.2. Probability-Frequency Density Information Criterion

In order to perform fault diagnosis accurately, it is necessary to evaluate the similarity between the normal state signal and the abnormal state signal. The PFDIC is used to evaluate the similarities of two signals. The purpose of the PFDIC is to choose the optimal SVs so as to achieve the greatest similarity between reconstructed signals and normal signals. In the design of such a reliable evaluation standard, the higher the similarity of the probability density distribution between reconstructed signals and normal signals, the better the results. The requirements for the similarity of the frequency density distribution are the same as those for the probability density distribution.

The schematic of the PFDIC is shown in Figure 2.

The presented condition should dominate the similarity of the probability density distribution and frequency density distribution.

Given these requirements, the probability density distribution function is designed as follows:

Meanwhile, the frequency density distribution function is

According to equations (11) and (12), the PFDIC can be given aswhere is the probability density function value at t, and are the probability density distribution functions, where represents the reconstructed signals and represents the normal signals; and are the frequency density distribution functions, where represents the reconstructed signals and represents the normal signals; ranges from 1 to .

When some SVs are selected, the value of is calculated according to equation (13). The smaller the value of is, the higher the similarity will be. At the same time, the selected SV combination represents noise information, and the remaining SVs are considered as the fault information of bearings. The remaining SVs can also extract fault signals.

3.3. Dual Beetle Antennae Search Based on PFDIC

Dual BAS is an efficient nonlinear optimization algorithm. It can be used to optimize the parameter optimization in SVD feature extraction. In obtaining the best ED and SVs as a two-stage objective, the dual BAS is designed to optimize the parameters. The first stage is to optimize the ED. Based on the result, the second stage is to optimize the SV combination. The ED is the variable optimization, and the SV combination is the combinatorial optimization. Algorithm 1 shows the detailed steps of the presented method:

(i)Input: Set the maximum number of iterations , . Define the evaluation function with PFDIC. Randomly set beetle positions . Set , . Set the value of on the basis of the optimization purpose. , , , , (best objective function in the first stage), (best objective function in the second stage).
(ii)Output: (Best ED, optimal SV combination),
(iii)While () or (stop criterion) do
(iv)The first stage of BAS : Generate the direction vector unit ; update the position , . Update the next position of the beetle .
(v)While () or (stop criterion) do
(vi)The second stage of BAS : Set ED is . Update , , and of the beetle .
(vii)Calculate the function value of the beetle. If is better than ; ; ;
(viii)End while
(ix)Output the SV combination. Calculate the function value of the beetle. If is better than : .
(x)Update the sensing length of the antennae ; update the step size of search ;
(xi)End while
(xii)Return, .
3.4. Proposed Fault Signal Extraction Method

Figure 3 shows the flowchart of the method presented in this study. In the first stage of the dual BAS, it generates a random number as the ED and constructs the HM. It decomposes the HM with SVD. In the second stage of dual BAS, BAS is employed to optimize the SV combination.

Taking the PFDIC as the objective function, the BAS can fine-tune the optimal SVs via iterative calculation. The optimal objective value is assigned to the optimal value in the first stage. After the operation, the best ED and SVs can be obtained.

3.5. Bearing Fault Characteristic Frequency and Diagnosis

In the bearing faults of a rolling element, a specific impulse is generated when the rolling element hits the inner race or the outer race; this impulse is called the bearing failure’s characteristic frequency. According to [5], is the number of rollers, is the roller diameter, is the bearing pitch diameter, is the contact angle, and is the rotating frequency.

The inner race’s ball pass frequency is denoted as BPFI, the outer race’s ball pass frequency is denoted as BPFO, and the spin frequency of the roller (ball) is denoted as BSF; they can be, respectively, calculated as follows:

The extracted fault signal combined with envelope demodulation can calculate the envelope spectrum. Through the comparison of the dominant frequency and fault characteristic frequency, the bearing fault can be effectively diagnosed.

4. Simulation Experiments

A local bearing’s inner race fault is simulated as follows to analyze the superiority of this study:where is the amplitude modulation of the impulse signal, with A set to 4 and B set to 0.5; is the rotating frequency set to 20 Hz; is the signal damped exponent set to 1,000; is the resonance frequency set to 5,000 Hz. is the tiny fluctuation of the impulse relative to feature cycle . The random fluctuation complies with the standard normal distribution, and the standard deviation is 0.5% of the rotating frequency. The fault characteristic frequency is 120 Hz. is the heavy side function, and is Gaussian noise with a standard deviation of 2. Herein, the sampling frequency is 12,000 Hz, and the sample length is 4,096. The simulation signal and its envelope spectrum are shown in Figure 4.

Figure 4(a) shows that, in the simulation signal (red line), the impulse of the signal is conspicuous. Moreover, in the simulation noise signal, the impulse was suppressed by the noise and is thus difficult to identify. Figure 4(b) shows the envelope spectrum of the simulation signal. The fault characteristic frequency is conspicuous, and its harmonics can be established quickly. In Figure 4(c), the fault characteristic frequency cannot be established from the envelope spectrum. Therefore, the simulation noise signal requires fault feature extraction.

According to the designed algorithm, in the first stage of BAS optimization, the ED is used as the optimization variable. Herein, the search range of the ED is set to 5–200 (the range given by artificial experience). is set to 2, is set to 0.5, and is set to 0.001. The EDs in the first stage are used as the optimization variables in the second stage of optimization, and the variables are set to 0 or 1, where 1 denotes the combination of SVs and 0 denotes the noncombination of SVs. The parameters are optimized by the PFDIC. The numbers of optimization iterations in the first and second stages are set to 100. The objective function after the optimization is shown in Figure 5.

According to the figure of the BSA objective function, the objective value is convergent at generation 13; the best ED is 28; and the best SV combination is 1, 2, 3, 4, and 6. The restructured signal and its enveloped spectrum are shown in Figures 6 and 7.

Figure 7 shows that the restructured signal’s dominant frequency and fault characteristic frequency are the same. The fault characteristic frequency and harmonics are conspicuous. The rotating frequency Fr, in addition to the fault characteristic frequency, can also be determined.

To compare the effects of the selected SVs intuitively, this study employs the ED optimized by the proposed method in SVD. Each SV is applied to the reconstructed signal, and kurtosis is applied to evaluate the SNR of the signal. Kurtosis can evaluate the impulse of fault feature extraction. The larger the value is, the better the feature enhancement is. The results are shown in Table 1.

As shown in Table 1, the extracted fault feature signal based on dual BAS is effective and advanced.

5. Engineering Experiment

5.1. Experiment Platform

The effectiveness of the method is validated by analyzing three vibration fault signals. The experimental platform is set for the condition diagnosis test, and it contains loading equipment, a servo motor, and a motor control system. The platform is shown in Figure 8.

Four bearing condition signals (normal, outer race fault, inner race fault, and roller fault) are used to analyze the effectiveness of the proposed method. As shown in Figure 8, the bearing is artificially faulted by using electrical discharge machining (fault width: 5.0 mm, depth: 0.6 mm). The sampling frequency is 100 kHz. The corresponding fault characteristic frequencies are calculated by equations (14)–(16). The results are shown in Table 2.

5.2. Outer Race Fault

The vibration signal is acquired at a rotating speed of 100 rpm. The length of the data is 262,144. Figure 9(a) presents the normal signal and diagnosis signal. No obvious impulse is noted in the time domain signal. Hence, the impulse signals are suppressed by noise. In the envelope spectrum, the dominant frequency generated by the fault is completely overwhelmed with noise.

With dual BAS, the original signal is filtered, and the envelope is demodulated. On the basis of the programmed convergence, the smaller the parameter is, the better the generation is. After testing, the range of the ED is 15–300. The generation is set to 100. The BSA fitness, filtered signal, and corresponding envelope spectrum are given in Figures 10 and 11.

Figure 10 shows the BSA fitness. The value is convergent at generation 4; the best ED is 43; and the best SV combination is 2, 3, 4, 9, 10, and 11. Figure 11 shows the extracted fault feature signal and its envelope spectrum.

From the time domain signal, the impulse signal is conspicuous, and the noise in the extracted fault feature signal is considerably low. As shown in Figure 11(b), the extracted fault feature signal’s dominant frequency is 8.0 Hz, and it is equal to the characteristic frequency of the bearing’s outer race fault. At the same time, the second harmonic, third harmonic, and fourth harmonic of the dominant frequency are also equal to the corresponding outer race fault’s characteristic frequency. The result proves the effectiveness of the proposed method.

5.3. Inner Race Fault and Roller Fault

The superiority of this method is further verified by the inner race fault signal and roller fault signal. The vibration signal is acquired at a rotating speed of 100 rpm. The signal length is 262, 144, and the selected SVs are shown in Figure 12.

The fault feature signal is extracted on the basis of the selected SVs. Figures 13 and 14, respectively, show the original signal envelope spectrum and the envelope spectrum of the extracted fault feature signal of the inner race fault and roller fault.

As shown in Figure 13(b), the extraction fault feature signal’s dominant frequency is 12.11 Hz, and it is equal to the characteristic frequency of the bearing’s inner race fault. In Figure 14(b), the extraction fault feature signal’s dominant frequency is 7.82 Hz, and it is equal to the characteristic frequency of the bearing’s roller fault.

These results prove that the proposed method can extract fault features effectively and diagnose faults accurately.

6. Comparison Experiment

Comparison experiments are performed to further prove the effectiveness of the proposed method. The comparison experiments include (1) comparison with other SVD methods, (2) comparison with GA and PSO, and (3) comparison with other decomposition methods.

6.1. Comparison with Other SVD Methods

This section compares the proposed method with published methods. The method presented in [22] automatically selects the effective SVs on the basis of the peak of the difference spectrum. In [23], the number of effective SVs was determined using a difference curvature spectrum of incremental singular entropy. In [24], a truncated linear weighting function was proposed to determine the contribution of SVs to the reconstruction of the denoised signal. The presented method is called ISVD. EDs can be determined with the ISVD method. All EDs in the comparison experiments are the same. The authors of [12, 16, 24] presented algorithm steps in detail. The outer race fault signal is acquired at 100 rpm. The extracted fault feature signals and envelope spectra of the three methods are shown in Figures 15 and 16.

As shown in Figure 16, the proposed method’s SV combination is 2, 3, 4, 9, 10, and 11. The results reveal that all methods can find the fault characteristic frequency. However, the results of the proposed method are the most obvious and have the least noise in the envelope spectrum.

6.2. Comparison with GA and PSO

To prove the effectiveness of the dual BAS, this study designs a comparison experiment with GA and PSO. GA and PSO are relatively mature and intelligent algorithms. They have been used in many fields. Herein, the population numbers of GA and PSO are set to 100, and the other settings are consistent with those in the dual BAS. The convergence result of the objective function is shown in Figure 17.

The convergence speed of dual BAS is better than that of GA but is slightly inferior to that of PSO. However, because BAS involves one population, it takes much less time than GA and PSO. The comparison results are shown in Table 3.

Table 3 shows that the computation times of GA and PSO are similar and that the time used for dual BSA only accounts for about 70% of that used for GA and PSO. This result proves the effectiveness of the proposed method.

6.3. Comparison with Other Decomposition Methods

A comparison experiment is conducted between the proposed method, EMD method, and VMD method. The experiment uses the outer race fault experiment data. The EMD method and VMD (eight IMF components) method use kurtosis to choose the best IMF. Figure 18 shows the extracted fault feature signal and its envelope spectrum.

From the processed signal, the three methods extract signals with strong impulse. However, the envelope spectrum shows that the presented SVD method has the smallest noise, as shown in the black dotted line block diagram. To quantify the effects of the fault feature signals extracted by the three methods, this study uses the SNR in the evaluation of denoising performance. The SNR is computed aswhere is the raw signal, is the extracted signal, and is the length of the signal. The larger the SNR value is, the better the denoising effect of the algorithm will be. The SNR results are shown in Table 4.

Table 4 shows that ISVD has the largest SNR value. Moreover, ISVD has the shortest computation time and is thus superior to the EMD and VMD methods. This result proves that the method presented in this work can effectively realize bearing fault diagnosis.

7. Conclusion

This study proposes an automatic fault feature extraction method based on PFDIC and DBAS to diagnose bearing faults. DBAS employs ED and SVs as dynamic variables while PFDIC, in its two-stage objective, searches the best HM and selects the SVs. The results indicate that bearing signal SVD can automatically extract fault features. The extracted fault signals combined with envelope demodulation can efficiently diagnose bearing faults. Simulation signals, bearing fault signals, and comparison experiments are constructed to verify the effectiveness and advancement of the presented method. The results show that the proposed method is superior to other methods. In our future work, we will further improve the algorithm’s performance and apply it to the fault diagnosis of other rotating machineries.

Data Availability

The raw/processed data required cannot be shared at this time as the data also form part of an ongoing study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This work was supported by the National Natural Science Foundation of China under Grant no. 52071090. The authors are thankful to Dr. Yao Lizhong, School of Electrical Engineering, Chongqing University of Science and Technology, for his generous support and suggestions to the work.