The Performance Investigation of Smart Diagnosis for Bearings Using Mixed Chaotic Features with Fractional Order

This article presents a performance investigation of a fault detection approach for bearings using different chaotic features with fractional order, where the five different chaotic features and three combinations are clearly described, and the detection achievement is organized. In the architecture of the method, a fractional order chaotic system is first applied to produce a chaotic map of the original vibration signal in the chaotic domain, where small changes in the signal with different bearing statuses might be present; then, a 3D feature map can be obtained. Second, five different features, combination methods, and corresponding extraction functions are introduced. In the third action, the correlation functions of extension theory used to construct the classical domain and joint fields are applied to further define the ranges belonging to different bearing statuses. Finally, testing data are fed into the detection system to verify the performance. The experimental results show that the proposed different chaotic features perform well in the detection of bearings with 7 and 21 mil diameters, and an average accuracy rate of 94.4% was achieved in all cases.


Introduction
The popularization of precision manufacturing has led to increasingly demanding accuracy requirements in machining and manufacturing processes to ensure the normal operation of machines and to avoid damage to ball bearings without warning, which may cause operational failure of the machine or public security accidents. Therefore, it is necessary to detect whether the ball bearing is in a normal state or damaged. Many problems can be avoided if repairs are carried out in time before the bearing fails. Therefore, the purpose of this research is to develop methods and tools to further detect whether a bearing is damaged or not [1,2].
Several studies have shown that physical signals, such as stator currents [3], acoustic signals [4], and vibration signals [5], can be utilized for data exploration in the diagnosis of bearing failures. Among these signals, vibration signals are the most widely used choice. In real industrial production operations, machine vibration signals are considered to be one of the most effective and abundant physical signal sources for detecting bearing failure. Therefore, analyzing vibration signals has become critically important, with wavelet transform and Fourier analysis being two of the most common techniques for studying vibration signals.
when the initial value is determined, we can predict its behavior. The aforementioned three properties of chaotic systems have the ability to change the characteristics of the original signal. The processed signal then undergoes the feature extraction technique proposed in this paper, which can facilitate signal analysis. Two application fields for this method will be discussed. First, because the system is extremely sensitive to initial conditions and does not converge on a single point, the chaotic system offers good performance in signal enlargement. Second, the properties of aperiodic long-term and deterministic behavior are extended using "Circuit Implementation of Synchronized Chaos with Applications to Communications" [23], edited by Kevin M. Cuomo and Alan V. Oppenheim. The spectrum of the chaotic system is wide enough to be applied in signal encryption [24]. We can use this phenomenon to produce chaotic mapping for the analysis, as the spectra of the signals have no limits.
After feature extraction, we choose extension theory as our identification method. Compared to other identification methods, extension theory does not require tedious parameter setting or a significant amount of calculations. These factors save not only computational resources but also computing time [13]. Compared to classical mathematical sets and fuzzy sets, extension sets have a wider area that can describe the degree of belonging. That is, extension sets can describe the degree of belonging in more detail. Taking fuzzy sets [25,26] and extension sets [27] as examples, the former only define the degree of belonging from 0 to 1, while the extension theory extends this to −∞ to +∞. Expanding the degree of belonging can offer a more detailed description, which has the potential to decrease classification errors. For this reason, we choose extension theory for this study. To summarize, we apply a method that combines a fractional order Chen-Lee chaotic system and extension theory to offer a shorter detection time and greater diagnosis rate. Finally, we present five feature extraction methods that have relatively high correct rates.
After the diagnostic procedure is determined, we will explain how to combine vibration signals with chaotic systems, feature extraction, and extension theory. The original vibration signal is a one-dimensional signal that is delayed and fed into the Chen-Lee chaotic system (x, y, z) in three dimensions as a preprocessing step. Then, the five proposed feature formulas are applied to the x, y, and z vectors, which is known as feature extraction. Finally, the diagnostic value calculated from feature extraction serves as a reference to classify and assess the accuracy of the input signal as either a normal state, ball fault, inner race fault, or outer race fault using extension theory.
The organization of this paper is described as follows. In Section 2, we discuss how to obtain the data source and perform data preprocessing. In Section 3, we present the advantages of using chaotic systems, explain how to process chaotic mapping, and demonstrate a phase diagram under various loading and failure states. In Section 4, inspired by the phase diagram, we propose five different feature extraction methods and apply them using the five formulae. In Section 5, we introduce extension theory and define the classic domain and joint field. In Sections 6 and 7, we present the classification results and summarize the diagnosis conclusions.

Data Resource
The data used for simulation were obtained from Case Western Reserve University Bearing Data Center of the United States [28], which provides ball bearing test data for normal and faulty bearings. The test stand, depicted in Figure 1, is composed of a 2 hp motor on the left, a torque transducer/encoder in the center, and a dynamometer on the right. The motor shaft is supported by the test bearings, each subjected to a single point of failure (SPOF) using electro-discharge machining (EDM) with diameters of 7, 14, 21, 28, and 40 mils, as specified in the fault specifications, which also indicate fault depths of 0.011 inches. The defective bearings were reinstalled into the test motor. Then, we recorded vibration data for motor loads with different levels of horsepower. Finally, data were collected for normal bearings, single-point drive ends, and fan end defects. Then, we saved the results in MATLAB format. The test specifications are shown in Table 1. The fan end was the last section of the data resource and had a lower sampling rate; only the normal bearing and the SPOF drive end were used. Four states were provided on the inner raceway, rolling element (i.e., ball), outer raceway, and normal types. of failure (SPOF) using electro-discharge machining (EDM) with diameters of 7, 14, 21, 28, and 40 mils, as specified in the fault specifications, which also indicate fault depths of 0.011 inches. The defective bearings were reinstalled into the test motor. Then, we recorded vibration data for motor loads with different levels of horsepower. Finally, data were collected for normal bearings, single-point drive ends, and fan end defects. Then, we saved the results in MATLAB format. The test specifications are shown in Table 1. The fan end was the last section of the data resource and had a lower sampling rate; only the normal bearing and the SPOF drive end were used. Four states were provided on the inner raceway, rolling element (i.e., ball), outer raceway, and normal types.

Data Processing
Data were collected at 12,000 samples/second and 48,000 samples/second for the drive-end bearing experiments. As a result, the lengths of the data may be different.
For the simulation, we chose a sampling frequency of 48 k for the drive-end bearing experiments. However, the data length of 0 HP was different from that of other HPs. The length of the former was 240,000 samples and that of the latter was 480,000 samples. The 0 to 48,000th samples were not taken into consideration because the motor start-up caused one second transient state.
Additionally, data for SPOF with 0.014 inch on the inner race for 0 HP and 1 HP needed to be extended in order to obtain the same data length as other types of faults. After processing, the data were separated into two parts. One part was used for data training, and the other was used to verify the results shown in Table 2.

Data Processing
Data were collected at 12,000 samples/second and 48,000 samples/second for the drive-end bearing experiments. As a result, the lengths of the data may be different.
For the simulation, we chose a sampling frequency of 48 k for the drive-end bearing experiments. However, the data length of 0 HP was different from that of other HPs. The length of the former was 240,000 samples and that of the latter was 480,000 samples. The 0 to 48,000th samples were not taken into consideration because the motor start-up caused one second transient state.
Additionally, data for SPOF with 0.014 inch on the inner race for 0 HP and 1 HP needed to be extended in order to obtain the same data length as other types of faults. After processing, the data were separated into two parts. One part was used for data training, and the other was used to verify the results shown in Table 2.

Chaos Theory
The chaotic phenomenon results from the fact that objects continue replicating the state of motion in previous stages with some certain rules, which leads to unpredictable random effects. Therefore, a chaotic system is often used to discuss the behaviors that cannot be explained in dynamic systems with a single data relationship but can be predicted by a comprehensive data relationship. The chaotic system has some notable properties. First, the motion trajectory of the system is extremely sensitive to the initial value. Additionally, the chaotic system is a nonlinear system with randomness, indicating that the system has some potential principles that govern the system's evolution, which can be predicted in a certain category and regarded as an important factor affecting the operation of the system. In this study, the above properties are used to identify the ball bearings in different states.

Chaotic Mapping
As Figure 2 shows, the ball bearing data are first preprocessed and transformed from one-to three-dimensional forms. Next, the data are entered into the chaotic mapping system, including a drive system x i , y i , z i and a response system x o , y o , z o , where x i , y i , z i are the coordinates of a fixed point and x o , y o , z o are the three-dimensional data of the ball bearing. Both systems are calculated by the Chen-Lee chaotic equations [29], represented as Equations (1) and (2).

Chaos Theory
The chaotic phenomenon results from the fact that objects continue replicating the state of motion in previous stages with some certain rules, which leads to unpredictable random effects. Therefore, a chaotic system is often used to discuss the behaviors that cannot be explained in dynamic systems with a single data relationship but can be predicted by a comprehensive data relationship. The chaotic system has some notable properties. First, the motion trajectory of the system is extremely sensitive to the initial value. Additionally, the chaotic system is a nonlinear system with randomness, indicating that the system has some potential principles that govern the system's evolution, which can be predicted in a certain category and regarded as an important factor affecting the operation of the system. In this study, the above properties are used to identify the ball bearings in different states.

Chaotic Mapping
As Figure 2 shows, the ball bearing data are first preprocessed and transformed from one-to three-dimensional forms. Next, the data are entered into the chaotic mapping system, including a drive system xi, yi, zi and a response system xo, yo, zo, where xi, yi, zi are the coordinates of a fixed point and xo, yo, zo are the three-dimensional data of the ball bearing. Both systems are calculated by the Chen-Lee chaotic equations [29], represented as Equations (1) and (2).
(2)  Here, the response is subtracted from the driving system to form a set of dynamic errors, that is, e 1 , e 2 , e 3 . By using the fractional order derivatives of Grünwald-Letnikov [14], as in (3), an extra variable α can be obtained, and the new dynamic errors Φ 1 , Φ 2 , Φ 3 are entered into the extension theory to identify different states of the ball bearing system. After plotting the 3D dynamic error phase diagram with different values of α, the potential features maps are illustrated in Figures 3-8 with α = 0, 0.3 and 0.6, which is also listed in Table 3. For α = 0, in Figure 3a,b, these are similar in both numerical range and shape, while Figure 3c,d have similar shapes, making them difficult to differentiate. Figure 6 has a significant overlap in the range of all four fault states, and the graphs and distribution densities are also similar, making analysis difficult. Both graph parameters are α = 0, so they are not ideal. For α = 0.6, Figures 5 and 8 also have wide numerical ranges, but the distribution densities are very close to the origin, so they are also not suitable for use with parameters that are α = 0.6. Finally, For α = 0.3, in Figure 4b,c, these have similar shapes and ranges, but there is enough difference in the numerical range of Figure 4a,d to allow for classification. In Figure 7, there is a significant difference in the numerical distribution range, allowing for classification. According to the analysis mentioned above, an α value of 0.3 is decided, which is more suitable for diagnosis of bearing status.
Here, the response is subtracted from the driving system to form a set of dynam errors, that is, e1, e2, e3. By using the fractional order derivatives of Grünwald-Letnik [14], as in (3), an extra variable α can be obtained, and the new dynamic errors Φ1, Φ2, are entered into the extension theory to identify different states of the ball bearing syste ≈ + 1 + 1 − After plotting the 3D dynamic error phase diagram with different values of α, t potential features maps are illustrated in Figures 3-8 with α = 0, 0.3 and 0.6, which is al listed in Table 3. For α = 0, in Figure 3a,b, these are similar in both numerical range a shape, while Figure 3c,d have similar shapes, making them difficult to differentiate. Figu 6 has a significant overlap in the range of all four fault states, and the graphs and dist bution densities are also similar, making analysis difficult. Both graph parameters are α 0, so they are not ideal. For α = 0.6, Figures 5 and 8 also have wide numerical ranges, b the distribution densities are very close to the origin, so they are also not suitable for u with parameters that are α = 0.6. Finally, For α = 0.3, in Figure 4b,c, these have simi shapes and ranges, but there is enough difference in the numerical range of Figure 4a,d allow for classification. In Figure 7, there is a significant difference in the numerical dist bution range, allowing for classification. According to the analysis mentioned above, an value of 0.3 is decided, which is more suitable for diagnosis of bearing status. Here, the response is subtracted from the driving system to form a set of dynami errors, that is, e1, e2, e3. By using the fractional order derivatives of Grünwald-Letnikov [14], as in (3), an extra variable α can be obtained, and the new dynamic errors Φ1, Φ2, Φ are entered into the extension theory to identify different states of the ball bearing system After plotting the 3D dynamic error phase diagram with different values of α, th potential features maps are illustrated in Figures 3-8 with α = 0, 0.3 and 0.6, which is also listed in Table 3. For α = 0, in Figure 3a,b, these are similar in both numerical range and shape, while Figure 3c,d have similar shapes, making them difficult to differentiate. Figur 6 has a significant overlap in the range of all four fault states, and the graphs and distri bution densities are also similar, making analysis difficult. Both graph parameters are α 0, so they are not ideal. For α = 0.6, Figures 5 and 8 also have wide numerical ranges, bu the distribution densities are very close to the origin, so they are also not suitable for us with parameters that are α = 0.6. Finally, For α = 0.3, in Figure 4b,c, these have simila shapes and ranges, but there is enough difference in the numerical range of Figure 4a,d to allow for classification. In Figure 7, there is a significant difference in the numerical distri bution range, allowing for classification. According to the analysis mentioned above, an α value of 0.3 is decided, which is more suitable for diagnosis of bearing status.     (c) (d)    Figure 3 0.007 in α = 0 Figure 4 0.007 in α = 0.3 Figure 5 0.007 in α = 0.6 Figure 6 0.014 in α = 0 Figure 7 0.014 in α = 0.3 Figure 8 0.014 in α = 0.6

Feature Extraction-Five Feature Extraction Methods for Performance Investigation
In order to identify the four different conditions of the ball bearing system, we must perform feature extraction. Feature extraction is a process used to simplify the initial data that allows data to be allocated to a more manageable group to facilitate learning and   Figure 3 0.007 in α = 0 Figure 4 0.007 in α = 0.3 Figure 5 0.007 in α = 0.6 Figure 6 0.014 in α = 0 Figure 7 0.014 in α = 0.3 Figure 8 0.014 in α = 0.6

Feature Extraction-Five Feature Extraction Methods for Performance Investigation
In order to identify the four different conditions of the ball bearing system, we must perform feature extraction. Feature extraction is a process used to simplify the initial data that allows data to be allocated to a more manageable group to facilitate learning and

Feature Extraction-Five Feature Extraction Methods for Performance Investigation
In order to identify the four different conditions of the ball bearing system, we must perform feature extraction. Feature extraction is a process used to simplify the initial data that allows data to be allocated to a more manageable group to facilitate learning and Sensors 2023, 23, 3801 9 of 14 maintain data integrity. Sometimes, feature extraction can even offer a more complete interpretation of the initial data. The following are the five feature extraction methods used in this article: After using Euclid's distance between 2D dynamic errors and their maxima, the first method takes the maximum values as the characteristics. From the observations of the 2D phase diagram, we found that the graphs exhibit a ribbon distribution with different sizes. Therefore, the maximum values of x and y are deducted from the test points, and the Euclidean distance is taken to make the result close to the long side of the distribution. Then, we take the maximum value as the eigenvalue.
The second method uses only the x direction of dynamic errors, which is the ratio of maximum and minimum value. There are obvious differences between the maximum and minimum values of the four states in the x and y directions when observing the size of the 2D phase diagram. The difference in the x direction is more obvious, so we take the ratio of the maximum and minimum values in the x direction as one of the methods.
In the third method, we subtract the minimum values from the dynamic errors in the x and y directions and then multiply those values. According to the 2D phase diagram, the area of each graphic is different. It is suitable to use this property to identify the vibration signals. Since the area of the graphic is too difficult to calculate, we use the area of the rectangle, which represents the maximum and minimum area of the graphic.
Next, we use Euclid's distance between the average point and origin of the 2D dynamic errors. Each average point of the vibration signals is separated in the 2D phase diagram, so the distribution of these points is useful for identification. We then calculate the value of Euclid's distance between the average point and origin and use it as one of the characteristics for this study.
The last method is almost identical to the first but uses only the y direction, rather than 2D dynamic errors. The range of distribution for the characteristic values is known as the classical domain, which is included in the joint field. By observing the 2D phase diagram, the value of Φ 2 under different error states is shown to be quite different. Therefore, the feature extraction method in method 5 mainly focuses on Φ 2 . We extract the maximum value of the distance between the Φ 2 value of each point and the average value of Φ 2 as the feature value and apply this method to this study.

Extension Theory
Generally, extension theory [30][31][32][33] is a method that helps to determine how related an object is to the feature by using an extension set and correlation function. Figure 9 presents a schematic of an extension set and correlation function. Briefly speaking, Figure 9 illustrates two triangles with different colors, and the horizontal and vertical axes represent the eigenvalues and correlation function, respectively. Additionally, assume that the green and blue triangles represent the correlation functions of the ball fault and inner race fault, respectively. Next, we compare K1 and K2, which represent the values of the correlation function shown in Figure 9, to determine which fault state the test point belongs to. Here, the test point is closer to the inner race fault, so the value of the correlation function is larger, i.e., K1 > K2. Thus, we classify this test point as the inner race fault. Eventually, by applying extension theory, we can compare the magnitude of each correlation function to determine which error state an observation belongs to.

2023, 23, x FOR PEER REVIEW
Here, the test point is closer to the inner race fault, so the value of is larger, i.e., K1 > K2. Thus, we classify this test point as the inne by applying extension theory, we can compare the magnitude of ea to determine which error state an observation belongs to.

Experimental Results
We conducted the multiple steps of this research using MATL steps for feature extraction and testing.

•
Use pre-processed training and testing data to generate dyna Lee chaotic mapping system. In this research, we use five methods for feature extraction which is handled by two methods. The weight of each method in 7 show the classical domain and joint field of the second sets using in SPOF diameters. In the table, the "X" symbol represents the thi symbol represents the fourth method.

Experimental Results
We conducted the multiple steps of this research using MATLAB. There are five main steps for feature extraction and testing.

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Use pre-processed training and testing data to generate dynamic errors in the Chen-Lee chaotic mapping system. Compare the values of the correlation functions to determine the testing data (i.e., the vibration signal of the normal state, the ball fault state, the inner race fault state, and the outer race fault state).
In this research, we use five methods for feature extraction and three sets, each of which is handled by two methods. The weight of each method in the sets is 0.5. Tables 4-7 show the classical domain and joint field of the second sets using different HP and 0.007 in SPOF diameters. In the table, the "X" symbol represents the third method, and the "Y" symbol represents the fourth method.    As shown in Table 12, the highest average accuracy in feature extraction was achieved by the second set. For feature extraction, this set used Euclid's distance of the average area of a rectangle, determined by the maximum and minimum values in the 2D dynamic error phase diagram as characteristics. These two characteristics represent, respectively, the position and area information of the graphics. When a difficult condition, such as the 0.014 in SPOF diameter testing data, needs to be determined, the second set offers better effects because it provides the two different properties of dynamic errors.

Conclusions
This study focused on the effects of different feature extraction methods in the diagnosis of ball bearing vibration signals. After generating dynamic errors with the chaotic system, we assessed five feature extraction methods to calculate the classical domains and joint fields and then tested the accuracy of identification with extension theory under different conditions. According to the results, to obtain the highest accuracy ratio, it is necessary to analyze the properties of the dynamic errors in the phase diagram and to determine the appropriate characteristics.
Five feature extraction methods were proposed in this study, of which two were chosen as the basis for Methods 1 to 3. Method 1 used Equations (4) and (5), Method 2 used Equations (6) and (7), and Method 3 used Equations (7) and (8) as the two features X and Y for extension theory classification. The accuracy obtained from experiments using these three methods is presented in Tables 8-12, with Table 12 displaying the average accuracy of the three methods. The accuracy rates for Method 1, Method 2, and Method 3 were found to be 87.8%, 94.4%, and 84.8%, respectively. After comparing the results, we determined that Method 2 achieved the best and most stable classification results among the three methods. This paper also proposed a preprocessing step that utilizes a chaotic mapping system. Five different feature extraction equations were then used to develop three methods that utilized the two main features of X and Y. Finally, extension theory was employed for classification. Ultimately, our results show that the properties of feature extraction are the key factor needed to find the most accurate method for determining the ball bearing vibration signal. In the future, we will use the architecture in this study to develop improved methods for feature extraction with improved accuracy.