Failure Probability Modeling of Miniature DC Motors and Its Application in Fault Diagnosis

(is paper proposes a fault diagnosis method for miniature DC motors (MDCMs) in the presence of the uncertainties caused by material and random factors of the production process. In this method, the probability models of fault multiple features are established based on the advantage criterion of the maximum overall average membership to determine the distribution of fault multiple features.(e fault diagnosis algorithm is synthesized to obtain the threshold ranges of fault multiple features according to different confidence levels. Experimental test results are presented and analyzed to validate the efficiency and performance of the proposed fault diagnosis method.


Introduction
Due to their small size, lightweight, and easy control, miniature DC motors (MDCMs) are widely used in the industrial fields of home appliances, office automation, automotive parts, etc. [1]. In these applications, the efficient and reliable operation of MDCMs is increasingly important. However, the factory quality inspection of MDCMs mainly relies on the manual experience, which leads to low production efficiency, heavy workload, and missed inspection [2]. erefore, the effective motor fault detection technology on the production line is crucial for the factory quality of MDCMs.
Generally, there are four main techniques for the motor fault diagnosis, which are classified into the signal-based, model-based, data-based, and indicator-based method classes [3,4]. e signal-based method uses sensor equipment to measure the diagnostic signals of the motor, such as current [5], vibration [6], acoustic [7], and thermal [8], and the fault type is detected by comparing and analyzing the diagnostic signals. Measurements of the acoustic signal, vibration signal, and thermal signal are noninvasive. e motor current signature analysis (MCSA) has been successfully applied to the fault diagnosis of DC motors and induction motors, especially under steady-state conditions and rated load [9]. However, the signal-based method requires a large number of sensing devices. e model-based method relies on the theoretical analysis of the motor, which key is to establish the motor model under the fault condition [3]. e motor fault can be detected by comparing the actual parameters with the estimated parameters of the properly functioning motor. Residual analysis and suitable signal processing are used to define the fault indicators [3]. Winding function approach [10], dynamic mesh reluctance approach [3], and finite element approach [11] were used to model. e available models are nice analytical tools predicting motor behavior, but their parameters are not easily determined. e leastsquares (LS) method [11,12], fully decoupled parity equation [13], and universal adaptive stabilizer [14] were used to estimate the motor parameters. Since the motor faults generally change multiple electromechanical parameters, it is difficult to establish a fault prediction model of the complex fault systems. e data-based method is a powerful tool to improve the effectiveness and reliability of the diagnosis since it does not require any knowledge about motor parameters and models [3,4]. Artificial intelligence techniques are widely used in the data-based method, which need to use the healthy and faulty motor data for feature extraction and classification. Several artificial intelligence techniques, such as neural networks [15,16], support vector machine [15], fuzzy logic [17], expert system networks [18], deep learning [19], and algorithms [20], have been developed to detect the motor faults. e combinations of the above techniques have been reported in the literature. e convolutional neural network (CNN) was used for extracting features, and then the support vector machine [21] and deep transfer learning [22] were used to diagnose faults. e application of artificial intelligence techniques in the motor fault diagnosis can promote the automation of the diagnostic process. However, these methods require a large set of training samples [23], and the generalization performance of artificial intelligence techniques has limitations. e indicator-based method is to compare the measured value with the indicator threshold, which is a widely used fault diagnosis method. An and Li [24] proposed the permutation entropy of the reconstructed signal as the fault feature of piezoelectric ceramics. Jafari et al. [25] established four different indices and used the well-established Otsu thresholding technique to set the index thresholds for fault diagnosis. Ali and Liang [26] used d8 for the fault indicator, and the indicator threshold was determined using the universal threshold technique. Irfan et al. [27] designed an adaptive threshold scheme using the statistical decision theory. An accurate indicator threshold is essential for the indicator-based method. In addition, in the manufacturing process of MDCMs, various uncontrollable factors (material nonlinearity and random factors of the production process) lead to uncertainties of the MDCM output performance. erefore, a fixed indicator threshold is difficult to apply to the fault diagnosis of the MDCM.
Fault diagnosis methods of the MDCM need to consider the uncertainty of the production process. Wang et al. [23] used a probability modeling approach for photovoltaic fault diagnosis to solve the problems of the nonlinearity and uncertainty of the PV array output interval. Liu et al. [28] proposed a failure probability calculation method for power equipment based on multicharacteristic parameters. Li et al. [29] proposed an adaptive dynamic update model of the equipment alarm threshold based on a similar proportion and state probability model. Wen and Gao [30] estimated the probability distribution of health indicator points, and the deterioration of the ball screw is evaluated by the probability distribution. e probability density function of fault indicators can be determined using the error comparison method [23] and kernel density estimation [29,30].
However, the probabilistic fault diagnoses of DC motors have not been considered in the previous studies. is paper proposes a novel solution to apply a fault diagnosis method for the MDCM by incorporating the probability modeling and the advantage criterion of the maximum overall average membership.
e novelties and contributions of this paper are summarized as follows: (1) Proposing the advantage criterion of the maximum overall average membership to determine the optimal fitting distribution, which cannot be handled by the traditional hypothesis test (2) Designing a fault diagnosis algorithm based on probability modeling to deal with the uncertainties caused by material and random factors of the production process, improving the accuracy of MDCM fault diagnosis (3) Designing the diagnostic test platform of MDCMs to validate the performance of the proposed fault diagnosis method e rest of this paper is arranged as follows: in Section 2, the structure of the MDCM and data sources are described in detail. In Section 3, the probability models of fault multiple features are established using the advantage criterion of the maximum overall average membership, and the fault diagnosis method is discussed in detail, which is verified in Section 4. Section 5 summarizes this paper and outlines future research directions. Figure 1 shows the schematic diagram of the MDCM discussed in this paper. e MDCM consists of a rotor and a stator (Figure 1(a)). e stator is equipped with a ring permanent magnet, and the rotor consists of three coil windings which are wounded on three teeth [31]. e motor windings are connected by a triangle connection (Figure 1(b)). e parameters of a healthy MDCM used for testing are listed in Table 1.

Data Sources.
By analyzing and solving the mathematical model of the MDCM, the fault multiple features, namely, DC component (P dc ), frequency point (f m ), amplitude of frequency point (P f ), and spectrum area (A), were proposed to diagnose motor faults in [2]. Figure 2 shows the structure diagram of data acquisition of the MDCM with no load. After preamplification, low-pass filtering, analog-todigital (A/D) converter, and fast Fourier transform (FFT) conversion, the frequency domain data of fault multiple features (P dc , f m , P f , and A) can be obtained by the data statistics [2]. In this paper, the healthy motor (HM), the interturn short fault motor (ISFM), the open winding fault motor (OWFM), the winding desoldering fault motor (WDFM), and the abnormal magnet loop fault motor (AMFM) were studied. 110 healthy motors and 60 motors (Xiamen Dazhen Electric Co., Ltd.) with each fault type were randomly selected as experimental samples.

Maximum Overall Average Membership Advantage
Criterion.
e advantage criterion of the maximum overall average membership was used to determine the distribution of the fault multiple features [32,33].
e key to the advantage criterion of the maximum overall average membership is to construct membership functions.
Step 1. Determine the membership function. Let the domain of the fault multiple features' distribution model be U, and A 1 , A 2 , A 3 , and A 4 denote the normal distribution, exponential distribution, Weibull distribution, and lognormal distribution, respectively. ey are the fuzzy subset of F(U). Y is a sample of U, Y � y 1 , y 2 , . . . , y n T , where n is the number of motor represents the fuzzy membership degree of the y j th index in the sample Y to the fuzzy subset A i . e normalization of μ A i (y j ) is as follows: (1) where where ∅(y j − μ/σ) and F(y j ) are the distribution function of the normal distribution and the empirical cumulative distribution function of the sample data, respectively. Parameters σ and μ are estimated from the sample data. 1 − e − y j /λ is the distribution function of the exponential distribution, and the parameter λ is estimated from the sample data. 1 − e − (y j /η) m is the distribution function of the Weibull distribution, and the parameters η and m are estimated from the sample data. ∅(lny j − μ/σ) is the distribution function of the lognormal distribution, and the parameters σ l and μ l are estimated from the sample data.
Exponential distribution membership function μ A 2 (y j ) is expressed as Weibull distribution membership function μ A 3 (y j ) is expressed as  Logarithmic normal distribution membership function μ A 4 (y j ) is expressed as Step 2. Calculate the local membership advantage. e local membership advantage where μ (i) A k (y j ) represents the new sequence of μ A i (y j ) sorted from the largest to the smallest. k represents the sort of the new sequence, and i represents the sort of the original sequence.
Step 3. Calculate the overall average membership advantage. e overall average membership advantage μ A i (Y) is defined as Step 4. Calculate the maximum overall average membership advantage. e maximum overall average membership advantage Figure 3 shows the process of determining the distribution of fault multiple features. e maximum likelihood estimation method was used to estimate the parameters of the hypothetical distribution of fault multiple features. e advantage criterion of the maximum overall average membership was used to determine the distribution type of fault multiple features. e overall average membership advantage μ A i (Y) of fault multiple features was calculated based on the experimental data. e results of the hypothetical distribution of the fault multiple features are shown in Table 2. Because the resolution of the current spectrum is 40 Hz and the data interval of f m is narrow, f m does not meet any hypothetical distribution. erefore, it is not necessary to confirm its distribution using the advantage criterion of the maximum overall average membership. It can be seen from Table 2 that μ A 4 (Y) of P dc is the largest among all of them for HM, and it obeys lognormal distribution according to the advantage criterion of the maximum overall average membership. P f and A obey the Weibull distribution and lognormal distribution, respectively. For ISFM, OWFM, WDFM, and AMFM, the process of determining the distribution of fault multiple features is similar to that of HM.

Probability Density Function of Fault Multiple Features.
e probability density functions of the fault multiple features for HM, ISFM, OWFM, WDFM, and AMFM are shown in Table 3. It can be seen from Table 3 {y j |y j ∈ Y, j = 1, 2, ..., n}    rough the setting threshold range of P dc , it is possible to distinguish between HM and OWFM.

reshold of Fault Multiple Features.
e probability distribution of fault multiple features with different fault types was determined using the probability statistics and the advantage criterion of the maximum overall average membership. e threshold range of fault multiple features with different fault types can be calculated by the following equation: where t l and t u are the lower and upper bounds of the threshold range, f(x) is the probability density function of each fault multiple feature, and (1 − α) is the confidence level. By comparing the threshold range of fault multiple features with different confidence levels, the optimal confidence level of fault multiple features was determined. e confidence levels of fault multiple features were set to 95%, 90%, 85%, and 80%, and the performance of different confidence levels was discussed. As shown in Figure 5, the threshold ranges of fault multiple features with different confidence levels were calculated. As can be seen from Figure 5, the larger the confidence level setting is, the larger the upper bound of the threshold range is and the smaller the lower bound of the threshold range is. It means that, as the confidence level increases, the threshold range becomes larger. Table 4 shows the overbound rates for different confidence levels which can be used as a reference for the threshold range setting of the fault multiple features. It can be seen from Table 4 that the smaller the confidence level is, the larger the overbound rates' data and overbound rates are. In order to avoid a faulty motor being misjudged as a HM, the confidence level of fault diagnosis for the HM was set to 90%, and the confidence level of the faulty motor was set to 95% according to the results in Figure 4 and Table 4. e threshold ranges of the HM with a confidence level of 90% were calculated by equation (10) as shown in Figure 6.
By modeling the probability distribution of fault indicators in different fault conditions, the threshold ranges of fault multiple features with a confidence level of 90% were calculated by equation (10). Figure 7 shows the threshold ranges of the ISFM with a confidence level of 95%. e upper and lower bounds of the threshold ranges for each fault multiple feature are shown in Table 5, in which f m � kcpn/60, where k, c, p, and n are the number of commutators, the coefficient determined by the number of commutators, the number of pole pairs, and the rotational speed, respectively. e speed range of the HM is 5400-6700 r/min, so the threshold range of f m is [540,670]. Similarly, the threshold ranges of f m for other faulty motors are shown in Table 5. It can be seen from Table 5 that the threshold ranges of P DC of the HM and P DC of the OWFM do not overlap, and the HM and the OWFM can be diagnosed.

Fault Diagnosis Algorithm.
e fault classification indicator d i (m) was defined to measure the membership degree of the tested motor (m) belonging to the ith motor type (HM, ISFM, OWFM, WDFM, and AMFM), and it can be calculated by the following equation: where m j is the jth fault multiple features of the tested motor m. u ij (m j ) reflects whether m j is in the threshold range of the ith motor type, and it can be calculated by the following equation: where t l ij and t u ij are the lower and upper bound of the threshold range of the jth fault multiple features for the ith motor type (Table 4). According to equation (13), the tested motor was classified.
3.6. Steps of the Fault Diagnosis Method. Based on the above analysis, the MDCM fault diagnosis method is proposed and shown in Figure 8. e steps of the proposed fault diagnosis method can be summarized as follows.
Step 1. Data acquisition. e armature currents are processed by a preamplifier, a low-pass filter, an A/D converter, and FFT transformation, and the values of fault multiple features are obtained.
Step 2. Confirm the probabilistic model. Build the probability distribution model of fault multiple features, calculate the overall average membership advantage of each fault multiple feature, and then determine the probabilistic model of each fault multiple feature based on the advantage criterion of the maximum overall average membership.
Step 3. Calculate threshold ranges. Calculate the threshold ranges of each fault multiple feature using the probability model.   Figure 9, which mainly includes the data collection and processing module, the microprocessing module, and the LCD display module. e function of the signal acquisition and processing module is to amplify the current signal and remove the high-frequency noise generated by the commutation. e STM32F107VCT chip is selected as the processor of the microprocessor module, which mainly plays the role of A/D sampling, FFT transformation, and fault diagnosis. e LCD display module mainly displays the fault multiple feature values and diagnosis results. 100 healthy motors and 30 motors (Xiamen Dazhen Electric Co., Ltd.) with each fault type were selected to test the proposed method.

Diagnostic
Results. e fault diagnosis results of motors with different faults are shown in Table 6. Table 6 shows that one of the 100 HMs is misjudged as other faults, and the 8 Shock and Vibration accuracy rate of HM is 99%. e reason is that the confidence level of HM was set to 90%, and the threshold ranges of fault multiple features were narrowed accordingly. From Table 6, the accuracy rate of ISFM, OWFM, WDFM, and AMFM is, respectively, 96.67%, 96.67%, 100%, and 93.33%, and the faulty motor has not been diagnosed as the healthy motor. e reason is that the confidence levels of ISFM, OWFM, WDFM, and AMFM were all set to 95%.    Step 1 Data acquisition Step 2 Confirm probabilistic model Step 3 Calculate threshold ranges

Shock and Vibration
To further calculate the diagnostic accuracy of the proposed method, 3000 motors were randomly selected from the production line, and the diagnostic accuracy of motors is calculated and shown in Table 7. It can be seen from Table 7 that the accuracy rate of HMs is 99.76%, and the faulty motor has not been diagnosed as the healthy motor. It means that the proposed method can be applied to the motor factory testing.

Conclusions
is paper proposes a probability modeling approach by the threshold ranges of each fault multiple feature. By calculating the overall average membership advantage of the four distribution functions, the probability density of fault multiple features can be determined; then, the threshold ranges of fault multiple features can be obtained. Finally, the validity of the fault diagnosis method is verified. In this paper, the main findings and the shortcomings can be summarized as follows: (1) e membership functions of the normal distribution, exponential distribution, Weibull distribution, and lognormal distribution are established to calculate the maximum overall average membership advantage of the fault multiple features. (2) e maximum overall average membership advantage criterion is used to determine the probability distribution model of fault multiple features, which solves the problem that the classical statistical analysis is difficult to determine the optimal probability distribution model. Data Availability e feature data of motor faults used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.