Fault detection and classification of the rotor unbalance based on dynamics features and support vector machine

Rotor unbalance faults are one of the high-frequency faults in rotating machinery. As such, their accurate and timely diagnosis is important. In contrast to traditional methods based on static features, the dynamics features and support vector machines (SVM) are combined for the accurate detection and classification of rotor unbalance faults. First, the dynamical trajectories of the rotor system associated with unbalance faults are accurately identified locally based on the deterministic learning theory, which is more sensitive to abnormal changes in the rotor system. Second, entropy dynamics features, including the sample entropy, fuzzy entropy, and permutation entropy, are extracted based on the obtained dynamical trajectory data. Finally, the dynamics features are used to train the fault classifier based on the SVM with a Gaussian kernel function. Experiments on a rotor unbalance fault test rig demonstrate the effectiveness of the proposed method. The accurate detection and classification of rotor unbalance faults were also achieved compared with the results of employing static time or frequency features.


Introduction
As the key component of rotating machinery, rotor systems are widely used in electric power generation, coal mining, ship, and aerospace equipment. One of the most common faults in rotating machinery is rotor unbalance, which has a significant impact on the efficient and stable operation of rotor systems. 1 Unbalance faults may originate from the rotor manufacturing process, which lead to secondary failures of the rotor system, such as the rub-impact phenomenon and broken rotor bars. 2 Therefore, the accurate detection and classification of rotor unbalance faults is a crucial step in the safe operation of rotating machinery.
Rotor unbalance fault diagnosis has attracted the attention of researchers in recent years. Fault-specific symptom parameters are key elements in rotor fault diagnosis. 1 Several time-frequency feature extraction methods have been proposed for rotor unbalance fault symptom analysis. [3][4][5] Zhou et al. 4 used empirical mode decomposition (EMD) to extract the intrinsic mode to represent the different harmonics. Based on the idea of EMD, Chen et al. 5 proposed the variational mode decomposition method to diagnose the unbalance faults of wind power gearboxes. However, the diagnosis results cannot be obtained based solely on time-domain or frequency-domain features. The quantitative diagnosis results still need to be evaluated by analysts based on their experience or by other methods. 6,7 To resolve this issue, machine learning methods combined with fault features have been proposed, [8][9][10] such as support vector machines (SVMs), 11,12 K-nearest neighbor, 13 and deep learning. 14 The main processes of these methods are feature selection and fault classification. Among these machine learning methods, SVM has the advantage of high classification performance, and has been widely used in rotor fault diagnosis. [15][16][17] However, the performance of SVM methods is dependent on the fault feature extraction process. 1 The existing time or frequency domain features are static features, while the nonlinear features, for example, sample entropy (SE), fuzzy entropy (FE), and permutation entropy (PE), are obtained based on the status data of rotor systems. 18 Compared with the static features, system dynamics indices are more sensitive to system changes, and can provide a more accurate representation of the original system by including internal dynamics information. 19 The key step in dynamics index extraction is dependent on the locally accurate identification of the system dynamical trajectory based on deterministic learning, which was proposed by Wang and Hill. 20 The main advantage of deterministic learning is that the system dynamical trajectory can be approximated by radial basis function (RBF) neural networks, in which the neural weights converge in a stable manner. Therefore, the identified trajectory in a dynamical system can be stored in constant neural networks for further feature extraction. The dynamical system trajectory has been applied in the fault diagnosis and pattern recognition of small abrupt and incipient faults, electrocardiogram signals, and the human gait. [21][22][23] Inspired by the above methods, the deterministic learning theory and SVM method have been used to detect rotor unbalance faults. First, the dynamical trajectory of rotor systems with different unbalance faults are identified by deterministic learning, while the SE, FE, and PE are used to extract the dynamic features based on the obtained dynamical trajectory. Second, the corresponding dynamic features are input into the SVM with a Gaussian kernel (Gaussian-SVM) to train the classification model. Finally, the unbalance faults are detected and classified based on the training SVM model. The results of the rotor unbalance fault test rig demonstrate the advantages of the proposed method. The main contribution of this study is the improved sensitivity diagnosis based on the deterministic learning theory and Gaussian-SVM method.
The remainder of this paper is organized as follows. Section 2 presents the problem statement. Section 3 describes the experiments and methods employed in this study. The data analysis and diagnosis results are provided in Section 4. Section 5 presents some discussions. Section 6 presents the conclusions and directions for future study.

Problem formulation
A schematic diagram of the rotor unbalance is shown in Figure 1. The black shaft and rotor disk in the figure represent the normal state of the rotor system (the centroid is O 1 ). The red dotted line shows the unbalance fault states (the centroid is O). The unbalance parameter u reflects the degree of the unbalance fault in the rotor system.
The mathematical model of the rotor has been investigated in numerous studies, which can be described in the following form. 7,24 _ where X = ½x, y represents the vibration signal in the x-and y-directions, u represents the unbalance parameter, and different u values will produce different degrees of unbalance faults (i.e. the unbalance parameter u in Figure 1). f x, y, u ð Þ represents the unknown dynamics, and v(x, y, u) represents the system uncertainties and small disturbances.
The unbalance parameter u changes the centroid of the rotor system, and influences the system dynamics behavior. Various unknown system dynamics behaviors can cause the operating state of the rotor system to become uncontrollable. If the unbalance faults cannot be detected accurately, they could produce secondary faults, such as serious rub-impact or shaft bending faults. Therefore, the main goal of this paper is to propose a new fault detection method for the accurate detection and classification of the different unbalance faults.
Assumption 1: The rotor vibration states x and y can be measured by the sensors, and are still bounded after the unbalance force occurs.

Rotor unbalance experiments
The rotor unbalance test rig is shown in Figure 2. The rotor unbalance device consists of a disk and counterweights. The vibrations of the test rig in the x-and ydirections were sampled by two eddy current sensors. Different counterweights produced different degrees of unbalance faults in the rotor system. Based on the rotor dynamical differential equation function (1), different values of the unbalance parameter u will produced different unbalance faults. Therefore, in the experiment on unbalance faults, the counterweights (0, 2.5, 3.6, and 5.0 g) were taken as the unbalance parameter and installed on the disk to inject different degrees of unbalance faults, respectively. The other main components of the test rig are the rotor motor, signal conditioner, and oscilloscope.
Based on the selected counterweights, the corresponding data samples were divided into four classes of unbalance faults: normal class (0 g counterweight), slight unbalance fault class (2.5 g counterweight), moderate unbalance fault class (3.6 g counterweight), and severe unbalance fault class (5.0 g counterweight). The sample frequency was set as 5.12 kHz, and the sampling time was 2 s. In this study, two unbalance fault datasets (a small dataset #1 and big dataset #2) were constructed as follows. For dataset #1, the sampling speed was set as 1200 rpm. The normal class includes 30 sample data obtained with the benchmark test rig. The slight unbalance fault class also includes 30 sample data associated with the 2.5 g counterweight. The other two classes have 240 sample data (each class contained 120 data associated with the 3.6 and 5.0 g counterweights).
For dataset #2, the unbalance data was collected under three different rotor speeds: 1200, 1600, and 1800 rpm. The total number of data samples is 2100. Detailed descriptions of dataset #1 and dataset #2 are provided in Table 1. To evaluate the proposed method, the training and testing datasets were randomly divided 1:1 for dataset #1 and 6:4 for dataset #2.

Deterministic learning
To diagnose the unbalance faults, the nonlinear dynamical ordinary differential equation (1) was first considered to identify the dynamical trajectory. Under various conditions, the system uncertainties and unknown dynamics could not be clearly decoupled; therefore, these two factors were considered as a whole term to be identified. 21 The following RBF neural network was constructed to estimate the system dynamics.
where A = diag(a x , a y ) represents the designed constant gain parameter;Ŵ is the weight of the RBF neural networks; and S X, u ð Þ is the RBF. The weight updating law is set as whereX = X ÀX,W =Ŵ À W Ã , W Ã is the ideal weight vector, and G = G T . 0, and s is a small value. Based on the dynamics identifier (2) and the weight updating law (3), the unknown rotor dynamics can be accurately identified locally.
where Dn = diag(Dn x , Dn y ) represents the rotor dynamics, and j 1 is the practical identification error.
To obtain and store the dynamical knowledge in constant RBF networks, the neural weights are chosen as the mean value after the stable convergence process.
is the selected time segment between the stable convergence process. Therefore, the system dynamical trajectory can be obtained and stored as Dn = diag(Dn x , Dn y ) for the next dynamics features extraction.
Remark 1: A recent study 7 presented the detailed identification process of the nonlinear system dynamics, which is omitted here for unbalance faults and presented in the Discussion section.

Dynamics features
Based on the obtained dynamical trajectory data (Dn x , Dn y ), the nonlinear features, including the sample entropy (SE), fuzzy entropy (FE), and permutation entropy (PE) as shown in Li et al., 18 were used to extract the dynamics features of the unbalance fault vibration signals. The SE can measure the regularity of the obtained dynamical trajectory data. The FE and PE can characterize the randomness and complexity of the dynamical trajectory data, respectively. The different SE, FE, and PE values indicate the degree of data irregularity of the corresponding unbalance faults. The greater the data irregularity, the larger the SE, FE, and PE values. The corresponding value of SE, FE, and PE values based on the dynamical trajectory data (Dn x , Dn y ) are defined as SE x , SE y , FE x , FE y , PE x , PE y . The subscripts x and y represent the entropy dynamics features in the x-and y-directions, respectively.
Remark 2: In this study, the dynamics features were extracted based on the dynamical trajectory, which can be locally identified via deterministic learning theory. In contrast, the static features are the features extracted from the system vibration states directly, such as the time or frequency domain features.

Support vector machine
The SVM has been widely used in the field of diagnosis owing to its good optimization and generalization ability, and short training time. [15][16][17] The basic idea of the SVM is to construct a hyperplane for the optimal separation of different classes of input data. It is suitable for nonlinear and high-dimensional conditions. 11 Based on the contents of the SVM, the hyperplane can be described as , where x is the input data vector, v is a vector, and H represents the Hilbert space. To obtain the optimal separation of the input training data with the maximal margin, the main task of the SVM is to find the optimal hyperplane (i.e. min 1 2 jjvjj 2 + C P N q, where C . 0 is the penalty parameter, q50 is a non-negative slack variable and N is the total number of training samples) under the different classification conditions. 12 For the obtained rotor fault dynamics feature training dataset fSE xi , SE yi , FE xi , FE yi , PE xi , PE yi , z i g, i = 1, Á Á Á , N and z i 2 1, 2, 3, 4 f g (associated with the four classes of unbalance). The construction of a suitable hyperplane is dependent on the linear or nonlinear properties of the input training data. In the rotor fault diagnosis process, because the signals of the rotor system are nonlinear and nonstationary, the SVM should have the ability to deal with the nonlinear data and multiclass classification. To address the nonlinear problems, a kernel function is used to construct the decision function of the optimal hyperplane.
where k x i , x ð Þ is the kernel function, x i = ½SE xi , SE yi , FE xi , FE yi , PE xi , PE yi and a i 50 is the Lagrange multiplier. Among the kernel functions of the SVM in existing literature (e.g. linear, polynomial, sigmoid, and Gaussian), the Gaussian kernel function is the most commonly used for the wide convergence domain of such functions. 17 s is the acceptance threshold parameter of the kernel function.
The Gaussian kernel function was input into the SVM (Gaussian-SVM) for the diagnosis of rotor unbalance faults. Based on the Gaussian function, the selection of Dynamics feature extraction phase ffi Rotor unbalance data acquisition: X = ½x, y. ffl RBF neural network identifiers were constructed as shown in (2). Dynamical trajectories were obtained based on the convergence of neural weights Dn = W T S X, u ð Þ. Ð Dynamics features of Dn were calculated, SE x , SE y , FE x , FE y , PE x , PE y . Diagnosis phase ffi Classifier was constructed based on the obtained dynamics features of Dn. ffl The parameters of classification model were determined by using the grid-quadtree selection method Beltrami and da Silva. 17 Repeating step 1-4 as shown in the dynamics feature extraction phase for the test data. Ð Obtaining the diagnosis results after inputting the dynamics features of test data to the constructed classifier.
parameters is highly dependent on the accuracy of classification because the kernel function is directly related to the optimal hyperplane. In this study, the determination of the upper parameter of the Gaussian kernel function is utilized based on the rid-quadtree model selection method, which was proposed by Beltrami and da Silva. 17 To address the multiclass classification problem, the one-against-all (OAA) method was applied to deal with the diagnosis of the four classes of unbalance faults. The reason is because the OAA-SVM is the least timeconsuming among the multiclass classification methods (e.g. one-against-one SVM). 15 The detailed OAA-SVM process is as follows.
For the four classes of rotor unbalance faults, the decision function of the optimal hyperplane is set as follows.
where p belongs to {1,2,3,4}, the p th class is taken as the positive class, and the other three classes are the negative classes.
The input fault data x p considered to the p th class, where p is the maximum label among the P i a p i z i k x p , x À Á + b p .
Based on the selected kernel function (Gaussian-SVM) and the multiclass classifier (OAA-SVM), the accurate diagnosis of the rotor system with four classes unbalance faults was obtained in the following experimental section.
In this study, the deterministic learning and the Gaussian-SVM were combined to detection the rotor unbalance faults more accurately. The proposed method included two phases, which are the dynamics feature extraction phase and the diagnosis phase. The following Figure 3 presents a flowchart of the proposed method, and the algorithm of the proposed method is shown in Table 2. The obtained experimental unbalance data were input into the dynamical RBF neural networks, and the dynamical trajectories of different unbalance fault classes were stored. The nonlinear features (SE, FE, and PE) were extracted based on the dynamical trajectory. Finally, the dynamics features were used to train the classification model based on the Gaussian-SVM.

Diagnosis results
Firstly, dataset #1 was selected to represent the detailed diagnosis process. The time-domain waveforms of the four classes of unbalance faults are shown in Figure 4. Figure 4(a) to (d) were related to the normal class, the slight unbalance class, the moderate unbalance class and the severe unbalance class.
The corresponding frequency and marginal spectrum are shown in Figures 5 and 6, respectively. The timedomain features (e.g. extremum, mean value, variance, and kurtosis) and frequency features (e.g. amplitudefrequency, phase-frequency, and peak difference) which were extracted from the rotor system state are noted as the static features for the further classification comparison. As seen in Figures 5 and 6, the main frequency components of the normal class are concentrated on the fundamental frequency (1X) and second harmonic frequency (2X). The third harmonic frequency (3X) is included in the main frequency components of the slight unbalance class. The third harmonic frequency (3X) and second frequency (2X) are separated more clearly in the moderate unbalance class. The severe unbalance class had a weak quadruple frequency.
Secondly, the deterministic learning theory was used to identify the rotor system dynamics. The dynamical trajectory was obtained from equation (5). The parameters of RBF neural networks, which are used for deterministic learning, were set as: a x = a y = 0:02, G = 1:98, s = 2, andŴ x 0 ð Þ =Ŵ y 0 ð Þ = 0:0. All the data were normalized for the further detection and classificat ion. Therefore, the neuron centers were evenly spaced on [21.   including SE x , SE y , FE x , FE y , PE x , PE y , were directly extracted. The total number of the extracted features are 150 3 6 = 900 for the training dataset #1.
Finally, the extracted dynamics features were input into the Gaussian-SVM to produce the training classification model. To evaluate the effectiveness of the     proposed method, four SVM classifiers were constructed, and the diagnosis results were obtained for the SVM classifier 1 (associated with time-domain features), the SVM classifier 2 (associated with frequencydomain features), the SVM classifier 3 (associated with dynamics features), and the SVM classifier 4 (associated with the hybrid features which are combined with the static features and dynamics features), respectively. The parameters (penalty parameter C and acceptance threshold parameter s) of the four Gaussian-SVM classifiers were determined based on the grid-quadtree selection method,. 17 The values of the four classifiers are listed in Table 3. The parameter determination process is described in the Discussion section.
For test dataset #1, based on the four training classifiers, the detailed classification results and their corresponding confusion matrices are shown in Figures 11  and 12. The figures show that the normal class, slight unbalance fault class, moderate unbalance fault class, and severe class can be accurately classified using the proposed method. It is also shown that the SVM with dynamics features has a higher classification rate (Acc = 97.33%) than the SVM with time-domain features (Acc = 90.67%) or frequency features (Acc = 91.33%). In particular, the detection of rotor unbalance faults reached 100% using SVM classifier 3, with a false alarm rate (FAR) of zero. The detailed comparison of the detection and classification results is shown in Table 4. The static and dynamics features were combined for training SVM classifier 4. The fault detection accuracy was also 100%, while the classification results reached 98%. The similar diagnosis results of SVM classifiers 3 and 4 indicate that the dynamics features have the greatest impact on the classification results and are more sensitive to abnormal unbalance changes in the rotor system.   Dataset #2 was also used to verify the effectiveness of the proposed method. The diagnosis confusion matrices are shown in Figure 13. The detailed comparison of the detection and classification results is presented in Table 5. It can be seen that the classification results of dataset #2 are consistent with the results of dataset #1. Most importantly, based on the dynamics or hybrid features, the FAR valued (FAR = 2/105 = 1.9%) are still much lower than the FAR of SVM classifier 1 (FAR = 15/105 = 14.29%) and classifier 2 (FAR = 8/ 105 = 7.62%). The results of datasets #1 and #2 show that the diagnostic performance of the proposed method has advantages in the case of small samples, which are more suitable for practical applications.

Discussion
Discussion 1: During the construction process of the training classification model, it should be noted that the penalty parameter C and the acceptance threshold parameter s of the Gaussian kernel function are the main focus of researchers. The final diagnosis results rely heavily on these two parameters because the penalty parameter and the acceptance threshold can determine the good region (in contrast to the underfitting and overfitting region). Many methods are available to deal with the selection of the parameters (C, s). In this study, the grid-quadtree selection method proposed by Beltrami and da Silva 17 was employed for parameter selection. In the grid-quadtree method, a quadtree is used to identify uniform regions in the hyperparameter space. This process can be divided into three steps. After inputting the training dataset, the first step is to determine the initial solution. The second step is to start the quadtree division. The last step is to optimize the parameters (C, s) based on the first and second steps. The details of this process can be found in Beltrami and da Silva. 17 Discussion 2: Compared with the latest convolutional neural network-based deep learning methods, the main advantages of the proposed method are that the  dynamics features are first used to construct the training SVM classifier, and the extracted features have a clear physical meaning. The constructed classification model is more sensitive to the dynamics changes under the unbalance faults condition. Otherwise, the required sample size is not too large, which is suitable for real applications. Under the condition of arranging a largescale neural network, the main limitation of the proposed method is that it requires a certain amount of computing power to deal with the RBF structure.

Conclusion
In this study, dynamics features including SE, FE, and PE of rotor unbalance faults were firstly extracted based on the dynamical trajectory (in which was obtained by using the deterministic learning theory), and the diagnosis results based on the dynamics features and Gaussian-SVM achieved more accurate classification of different degrees of rotor unbalance faults. Based on the experiment results, it is indicated that the extracted rotor dynamics features were more sensitivity to the changes of rotor unbalance faults. For the further investigation, the prognosis of rotor unbalance faults and the diagnosis of other rotor faults (e.g. rub-impact faults) based on the dynamics features can be considered.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by the Henan Province Foundation for University Key Teacher [grant