Fault Diagnosis Analysis of Angle Grinder Based on ACD-DE and SVM Hybrid Algorithm

Abstract

Due to the complex structure of the angle grinder and the existence of multiple rotating parts, the coupling phenomenon of the data results in the complexity and chaos of the data. The market scale of angle grinder is huge. Manual diagnosis and traditional diagnosis are difficult to meet the requirements, so a fault diagnosis method of angle grinder that is based on adaptive parameters and chaos theory of dual-strategy differential evolution algorithm (ACD-DE) and SVM model hybrid algorithm is proposed by combining a chaos-mapping algorithm, dynamic and adaptive scale factor, and crossover factor. The effectiveness and robustness of the algorithm are proven by solving eight test functions. The acceleration signal is decomposed by wavelet packet decomposition and reconstruction, and a variety of sensor signals are processed and constructed as feature vectors. The training set and the test set of the fault diagnosis model are divided. SVM model is used as the fault diagnosis model and optimized by ACD-DE. Based on the fault data of the angle grinder, the hybrid algorithm is compared with other optimization algorithms and other machine learning models; the comparison results show that the performance of the improved differential evolution algorithm is improved, in which the precision rate is 98.81%, the recall rate is 98.74%, and the F1 score is 0.9877. Experiments show that the hybrid algorithm has strong diagnosis accuracy and robustness.

1. Introduction

An angle grinder is a kind of grinding tool, which includes rotating parts such as the gearbox, bearing, and motor. The motor consists of rotor and stator [1]. In recent years, the angle grinder has been widely used in the stone-processing industry and machinery-manufacturing industry because of its simple and compact structure, stable operation, and long service life. However, the running state of the angle grinder is that the motor drives the gear and grinding disc to rotate at a high speed. When people work in hand, it is easy to cause serious injuries. Therefore, it is necessary to detect and comprehensively diagnose the potential faults of its rotating parts to prevent the occurrence of major casualties [2]. However, the existing potential fault diagnosis method of angle grinder is traditional manual detection. First, a batch of angle grinders is selected by sampling for manual simulation test, and then, the angle grinder damaged in the test is disassembled, and finally, the faulty parts are judged and comprehensively analyzed. This method has the problems of low automation, backward technology, and low diagnostic accuracy. Moreover, the structure of the angle grinder is complex, and there are multiple rotating parts, which leads to the coupling phenomenon of the collected angle grinder data. The data are complex and disordered, and the annual output of the angle grinder is huge, with a market scale of about CNY 600 billion. For the angle grinder data, the existing manual diagnosis and traditional diagnosis methods are difficult to meet the demand. Therefore, in order to improve the accuracy of angle grinder diagnosis, reduce labor consumption and product consumption, and realize the rapid diagnosis of the entire batch of angle grinder products, this paper introduces an artificial intelligence algorithm to realize complex data and a huge amount of analysis.

Artificial intelligence has developed rapidly in recent years, and some scholars have applied artificial intelligence algorithms to fault diagnosis of different objects. Salman Khalid et al. proposed a fault diagnosis scheme for SPP system using wavelet packet noise reduction and principal component analysis to achieve feature extraction and then using a variety of artificial intelligence algorithms to classify to verify the effectiveness [3]. Salman Khalid et al. proposed a sensor optimization selection method based on machine learning for boiler and steam turbine fault analysis, which greatly reduced the number of sensors and verified the effectiveness of the algorithm. However, in the above two schemes, the algorithm parameters are set based on experience, which makes it difficult to achieve the optimal classification results [4]. Deepam Goyal et al. analyzed the experimental vibration data of different bearing defects under different loads and operating conditions, used discrete wavelet transform to denoise the signal, and used SVM model to complete fault diagnosis. However, the classification accuracy of the proposed algorithm varies greatly under different loads, and the generalization ability of the model is poor [5]. H. Safaeipour et al. proposed an early fault diagnosis scheme for a three-tank system, but the scheme is based on mathematical model, and the modeling process is difficult, and the calculation is large [6]. Konar and Chatopadhyay combined continuous wavelet transform with SVM model for bearing fault diagnosis of induction motor and improved the performance of the algorithm by inputting the characteristic matrix into SVM for diagnosis [7]. However, parameter setting by using grid search method is easily affected by human subjective factors, which makes the best diagnostic performance of the SVM model difficult to achieve. Armaki and Roshanfekr proposed a method to diagnose the rotor rod fracture fault in induction motor and used SVM model to realize the fault classification of induction motor [8]. However, when setting the parameters of SVM model, it depends on expert experience, which makes it difficult to reach the optimal state of results of the SVM model in diagnosis. Although the SVM model performs well in fault diagnosis of rotating parts, whether parameter setting is reasonable or not seriously affects the diagnostic performance of the algorithm. The parameters set by experience are often accidental, which makes it difficult to give full play to the performance of SVM model in diagnosis.

Although some scholars have studied the parameter optimization of SVM—for example, HuYuxia and ZhangHongtao proposed an improved chaotic algorithm and optimized the SVM regression model, which has been verified in Lorenz system, proving that the proposed MSCOA algorithm improves the prediction accuracy of SVM [9]—the algorithm is aimed at the difference between the regression model and the classification model in this paper, and it is difficult to ensure that the algorithm has a better improvement effect in the classification model. Zhou Junbo et al. proposed a rolling bearing fault diagnosis method based on the whale grey wolf optimization algorithm and SVM model. The convergence performance of the algorithm was verified by the bearing data set of Case Western Reserve University [10]. However, although the accuracy of diagnosis can be improved by combining a variety of optimization algorithms, the model is complex, and the operation time is long, which makes it difficult to meet the needs.

The differential evolution algorithm (DE) is widely used in fault diagnosis and other fields because of its fast optimization speed, less controlled parameters, and strong robustness. Some scholars have applied the DE algorithm to fault diagnosis based on the machine SVM model. Cao Longhan et al. proposed a method of optimizing the SVM based on the DE algorithm and applied it to the fault diagnosis of diesel engine valve and established a diesel engine valve clearance fault diagnosis model based on DE algorithm optimized SVM model [11]. Tapas Bhadra et al. proposed an SVM parameter optimization algorithm based on DE algorithm. Experiments show that the performance of the DE algorithm to optimize the SVM model is better than that of a single classifier [12]. The dynamic parameters of SVM model are set by the DE algorithm, which improves the performance of the SVM model. As an optimization algorithm, the search breadth and depth of the DE algorithm have a great impact on the performance of the SVM model. However, with the iteration, the mutation operation in the DE algorithm will rapidly reduce the search breadth and search depth of SVM model, making the algorithm fall into a local optimal solution.

To solve the above problems, a fault diagnosis method of angle grinders that is based on adaptive parameters and chaos theory of dual strategy differential evolution algorithm (ACD-DE) and an SVM model hybrid algorithm is proposed. In view of the complex and huge data of the angle grinder, the ACE-DE-SVM hybrid algorithm improves the depth and breadth of the search and the robustness of the algorithm, reduces the consumption and cost of the calculation time to a certain extent, and improves the efficiency of the actual diagnosis. The research mainly includes three main parts. In the first part, a chaos algorithm is introduced in the initial population stage of DE algorithm to ensure the generation of uniform population, and adaptive parameters are introduced in the mutation to improve the optimization ability of the algorithm. In the second part, the effectiveness of ACD-DE algorithm is verified by eight test functions. In the third part, the actual angle grinder fault data are taken as an example to compare with other algorithms. The convergence of the SVM model optimized by the ACD-DE algorithm is verified.

2. Materials and Methods

2.1. Fault Diagnosis of Angle Grinder Based on SVM Model

The SVM model can solve high-dimensional and small sample problems without relying on the whole data set and has good generalization ability. It has been widely used in the field of fault diagnosis and has always been one of the classical classification algorithms in the field of fault diagnosis. The SVM model finds a partition hyperplane in the angle grinder sample space, classifies the angle grinder sample data of different categories, and ensures that the minimum distance from the angle grinder sample data of different categories to this hyperplane is the largest.

The sample of the angle grinder is set to $\mathrm{S}=\left\{\left({\mathit{x}}_1,y_1\right),\left({\mathit{x}}_2,y_2\right),\left({\mathit{x}}_3,y_3\right),\cdot \cdot \cdot \left({\mathit{x}}_n,y_n\right)\right\}$, each data point ${\mathit{x}}_i=\left\{x_{{}_i}^1,x_{{}_i}^2,x_{{}_i}^3,\cdot \cdot \cdot ,x_{{}_i}^p\right\}$ is a row vector of $p$ features, and each class label is a value $y_i$, and the divided hyperplane can be represented by the following Equation (1) [13].

$${\mathit{w}}^T\mathit{x}+b=0$$

(1)

where $\mathit{w}$ is the normal vector of the hyperplane, which determines the direction of the hyperplane, and $b$ is a bias, which determines the distance between the hyperplane and the origin.

Considering that there is abnormal noise in the actual collected angle grinder sample data, which makes the angle grinder sample data not linearly separable, soft interval is introduced to solve this problem. The introduction of soft interval can allow some controversial data in the angle grinder samples to be misclassified and avoid the occurrence of over fitting. The optimization problem that consists of objective function and constraints of the SVM model is shown in Equation (2):

$$min\Phi (\mathit{w},\xi ,b)=\frac{1}{2}{\Vert \mathit{w}\Vert }^2+C{\displaystyle {\sum }_{i=1}^n{\xi }_i}$$

(2)

$$s.t. y_i({\mathit{w}}^T{\mathit{x}}_i+b)\ge 1-{\xi }_i ({\xi }_i>0;i=1,2,\cdot \cdot \cdot ,n)$$

where $C$ is the penalty factor, which determines the penalty degree of the algorithm on the angle grinder sample data of error classification, and ${\xi }_i$ is the slack variable.

The constraint is introduced into Equation (2), and the Lagrange function is introduced to obtain Equation (3):

$$L=\Phi (\mathit{w},\xi ,b)-{\displaystyle {\sum }_{i=1}^n{\alpha }_i(y_i({\mathit{w}}^T{\mathit{x}}_i+b)-1+{\xi }_i)}$$

(3)

where ${\alpha }_i$ are the Lagrange multipliers, the partial derivative of the variable in Equation (3) is calculated and set to equal to 0, the kernel function is introduced, and the constraints of soft interval support vector machine on dual variables are used to convert the equation into Equation (4):

$$max\phi (\alpha )={\displaystyle {\sum }_{i=1}^n{\alpha }_i}-\frac{1}{2}{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{j=1}^n{\alpha }_i{\alpha }_j{y}_i{y}_{j}K({\mathit{x}}_i,{\mathit{x}}_j)}}$$

(4)

$$s.t. {\displaystyle {\sum }_{i=1}^n{\alpha }_i{y}_i=0, 0\le {\alpha }_i\le \mathrm{C}, \mathrm{i}=1,2,\cdot \cdot \cdot ,\mathrm{n}}$$

where $K(x_i,x_j)$ is the kernel function, and $C$ is the penalty factor; the Gaussian radial basis kernel function is selected as the kernel function of SVM, so its expression is shown in Equation (5):

$$K({\mathit{x}}_i,{\mathit{x}}_j)=e^{-g{\Vert x_i-x_j\Vert }^2}$$

(5)

where $g$ is the kernel function parameter, which controls the range and shape of Gaussian kernel, and improper selection of SVM model parameters will lead to under fitting or over fitting and will affect the classification results. Therefore, the parameters of SVM model need to be set reasonably.

2.2. ACD-DE Algorithm for Optimizing SVM Model

2.2.1. Chaos Initialization

The calculation accuracy, the result and convergence speed of DE algorithm are closely related to the initialization mode of population. The traditional differential evolution algorithm initializes the population in a random method. The population particles generated by this method are easy to be unevenly distributed in the required space, which makes the algorithm difficult to converge to the global optimal solution. In order to overcome the shortcomings of initializing population, a method of initializing population based on logistic chaotic mapping algorithm is proposed in this paper. Because the chaotic mapping algorithm has the characteristics of nonlinearity, ergodicity, and randomness, when initializing the population, it can not only ensure that the initialized population has good randomness but also makes use of the ergodicity of the algorithm to make every point in the space be traversed without repetition [14]. Therefore, in order to obtain a uniform population, the logistic chaos algorithm is introduced into the process of initializing the population. The equation of logistic chaotic mapping algorithm is shown in Equation (6):

$$x_{i+1}=x_i(1-x_i)\mu , \mu =[0,4]$$

(6)

where $x_i$ is the chaotic sequence generated by Logistic chaotic mapping algorithm, and when $\mu$ is close to 4, the logistic chaotic algorithm has chaotic properties. Therefore, in order to ensure the chaotic state of the system, this paper sets $\mu =4$.

2.2.2. Dynamic Mutation Strategy

DE algorithm uses the mutation strategy to complete the individual mutation. The traditional DE algorithm uses DE/rand/1 as the mutation strategy. Although this strategy ensures the diversity of the population and the search breadth, due to the strong randomness of this strategy, the algorithm deviates from the optimal solution in the operation process. The mutation strategy DE/best/2 selects the best individual in the contemporary population for mutation, which ensures the evolution direction of the population and search depth, but it will increase the possibility of falling into local optimal solution. Therefore, in order to have better search breadth in the initial stage and better search depth in the later stage, a new dynamic mutation strategy is proposed, as shown in Equation (7):

$$\{\begin{array}{cc}\hfill v_j(g+1)=x_{r1}(g)+F\cdot (x_{r2}(g)-x_{r3}(g))\hfill & , rand<p\hfill \\ \hfill v_j(g+1)=x_{r1}(g)+F\cdot (x_{best}(g)-x_{r1}(g))+F\cdot (x_{r2}(g)-x_{r3}(g))\hfill & , other\hfill \end{array}$$

(7)

where $v_j(g+1)$ is the intermediate of mutation strategy of differential evolution algorithm; $r1$, $r2$, and $r3$ are random numbers randomly generated in $[1,POP]$ that are not equal to each other; $F$ represents the scale factor, usually between $(0,2]$, and rand is the random number between $(0,1)$; $p$ is the adaptive probability, and the calculation formula is shown in Equation (8):

$$p=\frac{ns1}{ns2+ns1}$$

(8)

where $ns1$ is the number of individuals who successfully completed evolution using mutation strategy 1 in the previous generation iteration, $ns2$ is the number of individuals who successfully completed evolution using mutation strategy 2 in the previous generation iteration. p is initially set to 0.5.

2.2.3. Adaptive Parameters

The DE algorithm mainly consists of three control parameters: population POP, scale factor $F$, and crossover factor $Cr$. Population POP is mainly related to individual D. if the population is too small, the population diversity will be reduced, and the search breadth of the algorithm will be affected; if the population is too large, the running time and running cost of the algorithm are increased. Generally, the size of the population is set to $POP\in [5D,10D]$ [15]. Scale factor $F$ and crossover factor $Cr$ are very important to the convergence speed and accuracy of the algorithm. In the traditional DE algorithm, the scale factor F and crossover factor $Cr$ are set as constants. This method will lead to premature convergence to the local optimal solution or require a large number of iterations to converge. In order to solve the above problems, the individual is set to 7D, and the crossover factor $Cr$ of JaDE algorithm is introduced into the ACD-DE algorithm [16]. The self-adaption crossover factor $Cr$ is shown in Equations (9) and (10) below:

$$Cr=rand({\mu }_{Cr},0.1)$$

(9)

$${\mu }_{Cr}=(1-c)\cdot {\mu }_{Cr}+c\cdot mean(S_{Cr})$$

(10)

where $S_{Cr}$ is the set of all successful crossover factor $Cr$ in this generation, and $c$ is a constant between 0 and 1.

The scale factor $F$ is randomly generated according to the Cauchy distribution. Compared with other distribution forms, Cauchy distribution can ensure the diversity of scale factor $F$ and avoid premature convergence caused by greedy selection.

2.3. Flow Chart of the Fault Diagnosis Method of Angle Grinder Based on Hybrid Algorithm of ACD-DE Algorithm and SVM Model

A fault diagnosis method of the angle grinder, which is based on adaptive parameters and chaos theory of dual strategy differential evolution algorithm (ACD-DE) and an SVM model hybrid algorithm, is proposed. The penalty factor $C$ and kernel function parameter $g$ of the SVM model are optimized by the ADE-DE algorithm to improve the performance of the SVM model. The optimization problem that consists of objective function and constraints of the SVM model is used as fitness function to judge the advantages and disadvantages of the algorithm. The specific flow chart of the ADE-DE algorithm optimizing the SVM model is shown in Figure 1.

2.4. Fault Data Acquisition of Angle Grinder

The angle grinder CT13499-115 developed by Zhejiang Crown Power Tools Manu Co., Ltd. in Zhejiang, China was selected as the experimental object. The experimental object is shown in the Figure 2.

By simulating the load applied on the main shaft of the angle grinder on the eddy current dynamometer, the electrical parameters, temperature, and vibration data of the angle grinder during stable operation are collected. The equipment is the data acquisition test bench (10 Nm eddy current motor test system and temperature and triaxial acceleration sensors). The test bench is shown in Figure 3, under the environment of constant humidity and temperature.

According to historical data and relevant literature [17,18,19,20,21], the common fault types of the angle grinder are mainly divided into four categories, which are (1) bearing defects, (2) rotor dynamic balance failure, (3) bevel gear wear, and (4) bearing rubber sleeve defects. The signals of the angle grinder under four types of faults and normal conditions are collected by using the data acquisition test platform, including the current, speed, torque, output power, vibration, and seven temperature signals. The temperature sensors are installed at the rear bearing of the head housing output shaft, the front bearing of the head housing rotor, the brush holder, the rear bearing of the casing rotor, the stator winding, the stator iron core, and the front bearing of the front cover output shaft. The sampling frequency of dynamometer and temperature sensor is set to 1 Hz, and the sampling frequency of acceleration sensor is set to 5 KHz. The relevant parameters of data acquisition are shown in the

Table 1. Relevant parameters of data acquisition.

Number of Samples	Sampling Frequency of Triaxial Acceleration	Sampling Frequency of Electrical Parameters and Temperature
5000	5000 Hz	1 Hz

.

First, the angle grinder operates for 5 s under no-load condition and 2 min under load condition so that the working state of the angle grinder tends to be thermally stable. Then, electrical parameters, temperature, and vibration data are collected. The number of samples of each fault type is 5000, and different labels are marked for data according to different fault types.

2.5. Feature Extraction of Vibration Data

In this paper, the time domain feature extraction and the frequency domain feature extraction were carried out for the high-frequency vibration signals in three directions, respectively. Among them, five dimensioned and four dimensionless time domain features were selected [22], and the time domain features are shown in

Table 2. Expression of time domain features.

Dimensioned Time Domain Features	Expression	Dimensionless Time Domain Features	Expression
Average value	$\overline{x}=\frac{1}{N}{\displaystyle {\sum }_{i=1}^{N}x(i)}$	Crest factor	$C=\frac{\max \left\{\left|x(i)\right|\right\}}{x_{RMS}}$
Square amplitude	$x_{smr}={\left(\frac{1}{N}\sqrt{{\displaystyle {\sum }_{i=1}^N\left|x(i)\right|}}\right)}^2$	Pulse factor	$I=\frac{\max \left\{\left|x(i)\right|\right\}}{{\overline{x}}_{abs}}$
Variance	$x_v=\frac{1}{N-1}{{\displaystyle {\sum }_{i=1}^N\left(x(i)-\overline{x}\right)}}^2$	Margin factor	$C_L=\frac{\max \left\{\left|x(i)\right|\right\}}{{\left|\frac{1}{N}{\displaystyle {\sum }_{i=1}^N\sqrt{\left|x(i)\right|}}\right|}^2}$
Root mean square	$x_{RMS}=\sqrt{\frac{1}{N}{{\displaystyle {\sum }_{i=1}^N\left(x(i)\right)}}^2}$	Kurtosis factor	$C_K=\frac{\frac{1}{N}{\displaystyle {\sum }_{i=1}^{N}x{(i)}^4}}{x_{RMS}^4}$
Standard deviation	$S=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^N{(x(i)-\overline{x})}^2}}{N}}$

. The time domain features of vibration signals in the x-axis direction under normal and fault conditions are shown in

Table 3. Time domain features of vibration signal in x-axis direction.

Class	Average Value	Square Amplitude	Variance	Root Mean Square	Standard Deviation	Crest Factor	Pulse Factor	Margin Factor	Kurtosis Factor
Normal	0.00011	0.0603	0.0068	0.0825	0.0825	2.6338	3.1830	3.6640	2.4199
1	0.0026	0.1606	0.0556	0.2358	0.2358	3.7471	4.6719	5.5008	2.8930
2	0.0042	0.2413	0.1190	0.3450	0.3450	3.7823	4.6442	5.4068	2.7312
3	0.0045	0.1696	0.0634	0.2519	0.2520	4.0141	5.0498	5.9623	3.2401
4	0.0056	0.2101	0.1004	0.3169	0.3168	3.6462	4.6243	5.4999	2.8037

.

When there is a problem with the angle grinder, the vibration impact of the gear and rotor will change when they rotate at high speed, resulting in changes in the energy distribution in the signal. Therefore, the energy in different frequency bands of the angle grinder vibration signal can be characterized and used for the angle grinder fault diagnosis. The constructed feature is called the energy spectrum of the vibration signal [23]. In this paper, the db3 wavelet basis is used to transform and reconstruct the vibration signal, in which the number of decomposition layers is three, and the energy spectra of different frequency bands are extracted as frequency domain features. The energy spectrum of the vibration signal in the x-axis direction is shown in Figure 4. The nodes are sorted from low-frequency band to high-frequency band. Eight nodes are selected as the frequency domain features of the angle grinder. The frequency domain features of the vibration signal in the x-axis direction under normal and fault conditions are shown in

Table 4. Frequency domain features of vibration signal in x-axis direction.

Class	Node 0	Node 1	Node 2	Node 3	Node 4	Node 5	Node 6	Node 7
Normal	91.1215	7.7860	0.6387	0.2706	0.0364	0.0547	0.0701	0.0217
1	35.7779	25.6934	19.0561	8.1033	2.9719	3.5094	3.0216	1.8661
2	28.4051	57.2857	7.8721	2.5648	0.9393	0.9830	1.1854	0.7642
3	45.5286	38.4758	9.0473	4.1121	1.5175	1.2082	0.7619	0.3481
4	60.8424	17.7864	3.4025	11.7534	3.3283	1.1685	0.8937	0.8243

.

The data set contains a variety of data types, and the data range between each data is different. Therefore, to ensure the diagnosis accuracy, we must normalize the above segmented data set. The min–max standard normalization method was adopted [24], and the normalization function is shown in Equation (11):

$$X=\frac{x_i-x_{\min }}{x_{\max }-x_{\min }}$$

(11)

where X is the normalized data, and $x_{\min }$ and $x_{\max }$ are the minimum and maximum values of the data set before normalization, respectively.

2.6. Evaluation Index Model of Model

The precision rate, recall rate, and harmonic average value F1 score were selected as the evaluation indicators [25], as shown in Equations (12)–(14):

$$P=\frac{T_P}{T_P+F_P}\times 100\%$$

(12)

$$R=\frac{T_P}{T_P+F_N}\times 100\%$$

(13)

$$F1=\frac{2PR}{P+R}$$

(14)

where $T_P$ is the number of positive samples correctly identified as positive samples, $F_P$ is the number of negative samples incorrectly identified as positive samples, and $F_N$ is the number of positive samples incorrectly identified as negative samples.

3. Results

3.1. Verification of Test Function

In order to evaluate the performance of the algorithm in many aspects, eight test functions were selected for experiments. The details of the test functions are shown in

Table 5. Test functions.

Test Function	D	s	$f_{\min }$
$F1={\displaystyle \sum _{i=1}^D{x}_i^2}$	30	$\left|s\right|\le 100$	0
$F2={\displaystyle \sum _{i=1}^{D-1}(100(x_i^2-}{x}_{i+1}^{}{)}^2+{(x_i-1)}^2)$	30	$\left|s\right|\le 100$	0
$F3=\left({\displaystyle \sum _{i=1}^D{\left({\displaystyle \sum _{j=1}^i{x}_i}\right)}^2}\right)(1+0.4\left|N(0,1)\right|)$	30	$\left|s\right|\le 32$	0
$F4=-20\exp \left(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^D{x}_i^2}}\right)-\exp (\frac{1}{D}{\displaystyle \sum _{i=1}^D\cos (2\pi x_i)})+20+e$	30	$\left|s\right|\le 32$	0
$F5={\displaystyle \sum _{i=1}^D\frac{x_i^2}{4000}}$	30	R	0
$F6={\displaystyle \sum _{i=1}^D\left|x_i\right|}+{\displaystyle \prod _{i=1}^D\left|x_i\right|}$	30	$\left|s\right|\le 10$	0
$F7=\max \left\{\left|x_i\right|,1\le i\le D\right\}$	30	$\left|s\right|\le 100$	0
$F8=1+\frac{{\displaystyle \sum _{i=1}^D{x}_i^2}}{4000}-{\displaystyle \prod _{i=1}^D\cos (\frac{x_i}{\sqrt{i+1}})}$	30	$\left|s\right|\le 100$	0

. F1–F3 are the unimodal functions with only one extreme value; F4–F8 are the multimodal functions with multiple local optimal solutions. D represents the characteristic dimension, s represents the interval of the value, and $f_{\min }$ represents the theoretical global optimal value.

Because the optimization problem of SVM model is convex function [26], it shows that the optimization problem is continuous and differentiable in the feasible region, and there are one or more minimum values in the feasible region. Therefore, the test function is used to verify the convergence and robustness of ACD-DE algorithm, which provides support for the subsequent fault diagnosis of angle grinder.

On the basis of eight test functions, the ACD-DE algorithm is compared with the traditional DE algorithm and typical DE algorithm variants. The traditional DE algorithm takes DE/rand/1/bin as the mutation strategy and sets the scale factor F = 0.5 and crossover factor Cr = 0.3, and the DE algorithm variants are jDE [27] and SaDE [28].

Considering the scientificity of algorithm and function verification, the specific parameters of the algorithm selected in this paper are the same as those in ref. [27] and ref. [28]. The common parameters of the intelligent optimization algorithm are set as follows: the number of individuals in the population is the same as the characteristic dimension of the function and is set to 30, the number of iterations of the population is set to 3000, the population size is set to 100, and each algorithm runs 20 times, respectively. The evaluation indexes are the average value and standard deviation after 20 times of operation. The experimental data results are shown in

Table 6. Comparison results of four optimization algorithms.

Test Function	DE		jDE		SaDE		ACD-DE
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	8.87 × 10−43	5.12 × 10−43	0.6067	0.7213	6.97 × 10−72	2.75 × 10−71	0	0
F2	8.66× 10−20	6.12 × 10−20	2.91 × 10−23	3.27 × 10−23	5.11 × 10−62	3.74 × 10−62	0	0
F3	152.69	1292.2	2.9050	3.1636	2.7399	2.7042	0.2124	0.2011
F4	7.01 × 10−15	1.34 × 10−15	5.61 × 10−15	1.81 × 10−15	3.99 × 10−15	1.61 × 10−30	2.01 × 10−15	0
F5	2.24 × 10−46	1.15 × 10−46	7.73 × 10−42	1.15 × 10−41	9.47 × 10−77	4.19 × 10−76	8.0 × 10−125	1 × 10−124
F6	2.01 × 10−25	6.47 × 10−26	2.31 × 10−22	1.62 × 10−22	1.39 × 10−37	5.57 × 10−37	3.34 × 10−63	4 × 10−63
F7	2.73 × 10−4	5.55 × 10−5	5.16 × 10−5	6.93 × 10−5	1.08 × 10−4	3.42 × 10−4	0.0025	0.0064
F8	0	0	0	0	0	0	0	0

, and the best result is shown bolded. The logarithm of the median convergence after each algorithm runs 20 times is used to draw the convergence curve of the loss function, which is used to compare the convergence speed between the algorithms. The convergence curve is shown in Figure 5.

It can be seen from Table 6 that F1, F2, and F8 converge to the theoretical global optimal solution when ACD-DE algorithm is used. F3, F4, F5, and F6 do not converge to the global optimal solution but are closer to the theoretical global optimal solution than the traditional DE algorithms and typical DE algorithm variants. However, in function F7, the JDE algorithm converges to a better solution.

Compared with other algorithms, ACD-DE algorithm has significantly improved the convergence speed in F3, F4, F5, and F6. ACD-DE algorithm converged faster before 500 generations and fell into local optimal solution after 500 generations in F7. Although the four algorithms can converge to the theoretical global optimal solution, the ACD-DE algorithm has the fastest convergence speed in F8. Combined with Table 6 and Figure 5, it can be seen that under the chaotic mapping algorithm, the global optimization ability has been greatly improved, and the convergence speed of the algorithm has been accelerated. Under the dynamic mutation strategy and adaptive parameters, the algorithm has better local optimization ability in the later stage. The ACD-DE algorithm shows good convergence performance in most test functions, which proves the convergence and robustness of the algorithm. It can be seen that the ACD-DE algorithm can show good convergence and robustness in SVM model.

3.2. Analysis of Experimental Results of SVM Optimized by Algorithms in Angle Grinder Data

In order to verify the convergence and robustness of the proposed algorithm in angle grinder fault diagnosis, experiments were carried out; four algorithms were used to optimize the parameters of SVM model, and the four optimization algorithms were compared, namely DE, jDE, SaDE, and ACD-DE. Considering the scientificity of the algorithms, the specific parameters of the algorithm selected are the same as those in ref. [27] and ref. [28]. The common parameters of the intelligent optimization algorithm are set as follows: the number of individuals in the population is set to 2, the number of iterations of the population is set to 50, the population size is set to 14. The fitness function is set to the accuracy after 5-fold cross validation to avoid over fitting.

The accuracy after 5-fold cross validation is shown in Figure 6. By comparing the four algorithms, it can be found that the ACD-DE algorithm has greatly improved the convergence speed and accuracy. ACD-DE algorithm has completed the convergence in about 24 generations and converged to 1.194%; before the 6th generation, the fitness function of the traditional DE algorithm has been kept at about 10, which shows that the algorithm has poor global search ability in the application situation and converges to 1.772% until the 34th generation; jDE algorithm began to converge from the third generation and converged to 1.482% at the 26th generation; SaDE algorithm converged from the first generation and converges to 1.502% at the 48th generation. Compared with other algorithms, ACD-DE algorithm combines the advantages of the chaotic mapping algorithm and dynamic adjustment, makes up for the shortcomings of traditional DE algorithm that make it easy to fall into local optimization, and improves the convergence speed and convergence accuracy of the algorithm.

In order to further verify the performance of the ACD-DE-SVM hybrid algorithm, the ACD-DE-SVM hybrid algorithm was compared with the non-optimized SVM model and other machine learning models, such as KNN, BP neural network, random forest, and decision tree. The data set used is the above data, and the 5-fold cross validation was carried out. The precision rate, recall rate, and harmonic average value F1 score were selected as the evaluation indicators.

The comparison results are shown in Figure 7, in which the non-optimized SVM model and other machine learning models are trained using the parameters of literatures [25,26,27,28,29].

Table 7. Running time of different models.

Model	Running Time/S
SVM (optimized)	16.11
SVM (non−optimized) [29]	308.21
KNN [30]	10.64
Random forest [31]	24.36
BP neural network [32]	17.98
Decision tree [33]	4.48

shows the running time of different models.

Compared with the non-optimized SVM model, the ACD-DE-SVM hybrid algorithm has greatly improved the precision rate, recall rate, and F1 score by 7.25%, 7.17%, and 0.0721%, respectively. It can be seen that ACD-DE algorithm greatly improves the classification performance of SVM model. Compared with other machine learning models, the ACD-DE-SVM hybrid algorithm has an improved accuracy by 1.77%, 1.33%, 6.45%, and 2.1%; recall by 1.7%, 1.33%, 3.75%, and 2.73%; and F1 score by 0.0173, 0.0133, 0.0512, and 0.02594, respectively. It can be seen from Table 7 that the running time of the ACD-DE-SVM hybrid algorithm is greatly reduced compared with SVM model. Combined with Figure 7 and Table 7, it can be seen that ACD-DE-SVM hybrid algorithm is better than random forest and neural network in terms of running time and running results. Although the running time of ACD-DE-SVM hybrid algorithm is 5.47 s and 11.63 s longer than the KNN algorithm and decision tree algorithm, respectively, the running results are better than these two algorithms. Therefore, the convergence and robustness of ACD-DE-SVM hybrid algorithm were further verified.

In order to further verify the generalization ability of ACD-DE-SVM hybrid algorithm, it was applied to the actual diagnosis environment of the angle grinder. First, 50 angle grinders that were tested in the company were randomly selected. Whether there was a fault and the specific fault situation were unknown. Secondly, the angle grinder was installed on the data acquisition test bench. After waiting for the angle grinder to reach the thermal stability state, the data were collected, and the collection time is 1 min; that is, the number of samples collected by each machine is 60. Then, the collected data were integrated and processed according to the data set format. Finally, the processed data were input into the ACD-DE-SVM hybrid algorithm to judge the fault type. In order to avoid the contingency of the diagnosis result, the result with the most times was selected as the model diagnosis result of the angle grinder.

By comparing the diagnosis results of the model with the actual disassembly results, the accuracy of the model diagnosis is 96%, and the diagnosis results of two angle grinders was inconsistent with the actual disassembly results. Some diagnosis results are shown in

Table 8. Comparison between diagnosis results and actual disassembly results.

Number of Angle Grinder	Diagnosis Result	Disassembly Result
1	Normal	Normal
2	Bevel gear wear	Bevel gear wear
3	Normal	Normal
4	Normal	Normal
…	…	…
48	Bearing defects	Bevel gear wear
49	Normal	Normal
50	Bevel gear wear	Normal

. The results of diagnosis errors are shown in bold in the Table 8.

Figure 8 shows the disassembly results of the angle grinder No. 2. The rotor and bearing of the angle grinder have no obvious faults. The surface of the rotor commutator is worn smoothly, the bearing operates flexibly, the stator and rotor iron core and coil are normal, and there is no wear and damage, but some tooth surfaces of the bevel gear have serious wear (the range drawn by the red circle in Figure 8b,c).

4. Conclusions

(1)

Aiming at the problem that the model makes it difficult to meet the required accuracy requirements due to the penalty factor C and kernel function parameter g in the SVM model, this paper introduces intelligent optimization algorithm to realize parameter optimization. The traditional DE algorithm uses random initialization population and fixed difference strategy, scale factor, and crossover factor when optimizing the algorithm, which makes the algorithm easily fall into local optimal value. Therefore, an ACD-DE algorithm to optimize the parameters of SVM algorithm is proposed and applied to the fault diagnosis of angle grinder.

(2)

The ACD-DE algorithm was calculated and analyzed in eight test functions. By comparing with the traditional DE algorithm and two variants of the DE algorithm, the convergence and robustness of the algorithm were verified. The optimization problem of SVM algorithm is introduced. The correlation between the optimization problem and the test functions was verified, and the ACD-DE algorithm was used to optimize the SVM model.

(3)

The angle grinders were used to collect temperature, electrical parameters, and vibration signals, and the vibration signals are processed in time and frequency domain. The data set was constructed, and the performances of the ACD-DE-SVM hybrid algorithm, DE-SVM algorithm, iDE-SVM algorithm, and SaDE-SVM algorithm were compared. Then, the obtained penalty factor C and kernel function parameter g were encapsulated in the model and compared with other machine learning models. Finally, in order to verify the generalization ability of the ACD-DE-SVM hybrid algorithm, it was applied to the fault diagnosis of an actual angle grinder. The comparison of various DE algorithms shows that the ACD-DE algorithm has a great improvement in convergence speed and convergence accuracy, which were increased by 0.578%, 0.288%, and 0.308%, respectively. By comparing ACD-DE-SVM hybrid algorithm with other machine learning models, it can be concluded that the ACD-DE-SVM hybrid algorithm effectively improves the accuracy, recall, and F1 value of the algorithm on the basis of reducing the running time of the algorithm. In the actual angle grinder fault diagnosis, comparing the model diagnosis results with the actual disassembly results, the accuracy of the model diagnosis is 96%, which verifies the generalization ability of the algorithm.