Abstract
In the fault diagnosis of the shaft orbit of rotating machinery, there are few prejudgments about the severity of the faults, which is very important for fault repair. Therefore, a finegrained recognition method is proposed to detect different severity faults by shaft orbit. Since different shaft orbits represent different type and different severity of faults, the convolutional neural network (CNN) is applied for identifying the shaft orbits to recognize the type and severity of the fault. The recognition rate of proposed finegrained fault identification method is 97.96 % on the simulated shaft orbit database, and it takes only 0.31 milliseconds for the recognition of single sample. Experimental result indicated that the classification performance of the proposed method are better than the traditional machine learning models. Moreover, the method is applied for the identification of the measured shaft orbits of rotor with different degree of imbalance faults, and the testing accuracy of the experiments in measured shaft orbits is 97.14 %, which has verified the effectiveness of the proposed finegrained fault recognition method.
Highlights
 A finegrained recognition method is proposed to detect different severity faults by shaft orbit.
 CNN is applied for identifying the shaft orbits to recognize the type and severity of the fault.
 The proposed method achieve good recognition and has a guiding role in the actual fault diagnosis of rotating machinery.
1. Introduction
With the continual development of industry, the structure of rotating machinery is getting more and more complicated. Due to the severe working environment, the rotating machinery is inevitably subject to various faults, which will lead to serious accidents and casualties [1]. Therefore, condition monitoring and fault diagnosis for these key rotating machinery are important for increasing production efficiency, decreasing maintenance charges, and extending service life of the equipment [2, 3].
Faults of rotating machinery are usually characterized as shaft abnormal vibrations [4]. Monitoring and diagnosis of shaft vibration signals are still the main means for the maintenance of rotating machinery, and shaft orbit synthesized by these vibration signals is an important tools for fault diagnosis [2]. Shaft orbit contains abundant information about the shaft working state, and the shaft orbit shape could accurately reflect the running state and fault condition of rotating machinery, such as outer “8” and banana shapes corresponding to shaft misalignment, ellipse corresponding to shaft unbalance [5].
However, there are no clear boundaries between different faults. For example, shaft orbit may change from ellipse to outer “8” shape when the severity of the shaft misalignment increases [6]. Therefore, it is not good enough to identify the graphic merely, and it is necessary to identify the severity of faults reflected by shaft orbits. The ellipse orbit shape with different lengthwidth ratios corresponds to the different fault information, and the difference in the size of the two rings of the outer “8” shape also represents different severity of shaft misalignment [6]. Therefore, there is an urgent need for a method to finely recognize the shape and degree of the shaft orbits.
Traditional identification methods of shaft orbits include three steps: preprocessing, feature extraction, and classification. However, it is always a difficult problem to choose the appropriate feature descriptors and the corresponding classifier. The common feature descriptors used in the identification of shaft orbit are Fourier descriptors (FD) [7], chain code [8], Walsh descriptor (WD) [9], Hu invariant moment [10], histogram of oriented gradients (HOG) [11], comprehensive geometric characteristic (CGC) [2] and accurate Fourier height functions (AFHF) [12]. The commonly used classifiers are BP neural network [10] and support vector machine (SVM) [12]. These traditional pattern recognition methods can only distinguish different kinds of shaft orbits. The features extracted by these methods are not suitable for the identification of finegrained shaft orbits.
The classification performance of traditional methods depends on the feature descriptors to a large extend. Unlike the traditional methods, convolutional neural network (CNN) can automatically extracts features from data. It has higher recognition accuracy and stronger antiinterference ability. At present, CNN has been successfully applied in fault diagnose of rotating machinery. Feature maps reconstructed by the raw vibration signals are usually used as the input of CNN, the reconstruction methods mainly include timefrequency analysis [1317], time series permutation [18] and division of vibration signal map [1921]. However, those feature maps have more complex physical meanings than shaft orbits, which can directly reflect the change of shaft position and the severity of the fault, and the CNN was rarely used in shaft orbit recognition. Therefore, a finegrained fault recognition method for shaft orbit of rotary machine based on CNN is proposed in this paper. First, the theory of finegrained shaft orbit is proposed to divide different shaft orbits into several subclasses which reflect the different severity of the fault. Then, the appropriate network structure of CNN is designed to identify the finegrained shaft orbits.
The structure of this paper is arranged as following: the corresponding relationship between the severity of faults and finegrained shaft orbits is analyzed in Section 2.1. The classification method of finegrained shaft orbits based on improved LeNet5 is described in Section 2.2. The experiments about the optimal structure of proposed CNN are presented in Section 3.1. The experiment on the identification of finegrained shaft orbits is introduced in Section 3.2 and 3.3. Finally, conclusions of this paper are summarized in Section 4.
Fig. 1Flow chart of proposed identification method of finegrained shaft orbits based on improved LeNet5
2. Identification method of finegrained shaft orbits based on improved LeNet5
As discussed above, the shape of shaft orbits can reflect the type and the severity of faults. A new theory of finegrained shaft orbits is proposed in this section to divide different shaft orbits into several subclasses that can reflect the severity of the fault. In addition, a CNN based on improved LeNet5 are introduced to identify finegrained shaft orbits. Fig. 1 illustrates the procedure of the proposed method, which consists of two parts: the theory of finegrained shaft orbits and the optimized CNN classification method. Details of each part of the proposed method are described below.
2.1. Modeling of finegrained shaft orbit with different severity of faults
2.1.1. Theory of finegrained shaft orbit
According to the Ref. [6], the shape of shaft orbit changes with different severity of fault. Fig. 2 shows shaft orbits in different degree of misalignment faults. In the condition of Fig. 2(a), the shaft orbit is an elliptical shape close to a circle, which corresponds to light fault, and there is no preload. In the condition of Fig. 2(b), the shaft orbit is a flat ellipse, which corresponds to medium fault, and there is some preload. In the condition of Fig. 2(c), the shaft orbit is outer ‘8’ shape, which corresponds to serious fault, and there is large preload.
Fig. 2Shaft orbits in different degree of misalignment faults
a)
b)
c)
From the perspective of the fault degree, Fig. 2(b) are more serious than Fig. 2(a). From the perspective of the change in curvature, Fig. 2(b) is greater than that of Fig. 2(a). The change in curvature of the shaft orbit reflects the intensity of position change of the shaft and the severity of the fault. Therefore, five indicators that reflect the change in curvature are proposed in this paper to subdivide the shaft orbits shape in Table 1, and the indicators are the slenderness $C$, which corresponds to the elliptical shaft orbit, the bending $Arg$, the ring width ratio $W$, the angle span ratio $F$, and the minimum radius of curvature ${\rho}_{min}$. The slenderness is corresponding to the elliptical shaft orbit, and the bending is corresponding to the bananashaped shaft orbit, and the ring width ratio is corresponding to the outer “8” shaft orbit, and the angle span ratio is corresponding to the inner “8” shaft orbit, and the minimum radius of curvature is corresponding to the petalshaped shaft orbit.
Table 1 shows the corresponding relationship between shaft orbit shapes and the fault types [5]. The shaft orbits are simulated according to [12], and a series of variables are introduced to refine the classification of different shaft orbits.
Table 1The corresponding relationship between the shaft orbits and the fault types
Fault types  Shapes of shaft orbits 
Unbalance  Ellipse 
Misalignment  Banana 
Misalignment  Outer ‘8’ 
Oil whip  Inner ‘8’ 
Oil whirl  Petal 
(1) The slenderness $C$ is defined as $C=l/L$, where $l$ and $L$ represent the short and long axes of the graph, respectively. When the graph is an ellipse, the smaller $C$ is, the slimmer the ellipse is. When the graph is circular, $C=$1. The smaller the slenderness is, the more serious the shaft vibration is, so the more serious the shaft unbalance is. Fig. 3 shows some shaft orbits with different slenderness. The slenderness of elliptical shaft orbits is in the range of 0.1 to 1, and elliptical shaft orbits are divided into three parts according to slenderness $C$. The elliptical shaft orbit represents serious fault when the slenderness is in the range of 0.1 to 0.4, it represents medium fault when the slenderness is in the range of 0.4 to 0.7, and it represents slight fault when the slenderness is in the range of 0.7 to 1.
Fig. 3Part shaft orbits with different slenderness
(2) The bending $Arg$ is defined as the angle between two lines connecting the centroid $P$ of the graph and two “vertices”. The smaller the $Arg$ is, the greater the bending is and the more serious the fault is. As shown in Fig. 4, with the decrease of $Arg$ of the bananashaped shaft orbit, the severity degree of shaft misalignment increases. The bending of bananashaped shaft orbits is in the range of 70° to 160°, and bananashaped shaft orbits are divided into three according to $Arg$. The bananashaped shaft orbit represents serious fault when the bending is in the range of 70° to 100°, it represents medium fault when the bending is in the range of 100° to 130°, and it represents slight fault when the bending is in the range of 130° to 160°.
Fig. 4Part shaft orbits with different bending
(3) The ring width ratio $W$ is defined as $W={w}_{1}/{w}_{2}$, where ${w}_{1}$ and ${w}_{2}$ are the widths of small and large rings, respectively. The characteristic parameters are mainly applicable to the outer “8” shapes. The smaller the $W$ is, the larger the gap between the two rings is, which means that the more serious the changes in the phase of the small ring is, the more serious the fault is. Fig. 5 shows some shaft orbits with different ring width ratio $W$. $W$ increases from left to right, and the severity of fault decreases instead. The ring width ratio of outer “8” shaft orbits are in the range of 0.1 to 1, and outer “8” shaft orbits are divided into three parts according to $W$. The outer “8” shaft orbit represents serious fault when the ring width ratio is in the range of 0.1 to 0.4, it represents medium fault when the ring width ratio is in the range of 0.4 to 0.7, and it represents slight fault when the ring width ratio is in the range of 0.7 to 1.
Fig. 5Part shaft orbits with different ring width ratio
(4) The angle span ratio $F$ is defined as $F={\alpha}_{1}/{\alpha}_{2}$, where ${\alpha}_{1}$ and ${\alpha}_{2}$ are small and big angle at the intersection of the graph, respectively. The smaller the $F$ is, the steeper the phase change is at the intersection of inner “8” shaft orbit, and the more serious the fault is. Fig. 6 shows some shaft orbits with different angle span ratio $F$. $F$ increases from left to right, and the severity degree of fault decreases. The angle span ratio of inner “8” shaft orbits are in the range of 0.1 to 1, and inner “8” shaft orbits are divided into three parts according to the angle span ratio. The inner “8” shaft orbit represents serious fault when the angle span ratio is in the range of 0.1 to 0.4, it represents medium fault when the angle span ratio is in the range of 0.4 to 0.7, and it represents slight fault when the angle span ratio is in the range of 0.7 to 1.
Fig. 6Part shaft orbits with different angle span ratio
(5) The minimum curvature radius ${\rho}_{min}$ is used as a parameter to describe the degree of fault corresponding to petalshaped shaft orbit. The point with the smallest curvature radius on the shaft orbit represents the most violent phase of the shaft. The smaller the value of ${\rho}_{min}$ is, the more serious the failure is. Fig. 7 shows some shaft orbits with different minimum radius of curvature_{, }and the point $M$ in the Fig. 7 is the point with the smallest curvature radius. ${\rho}_{min}$ increases sequentially from left to right, and the severity of oil whip decreases. The minimum radius of curvature of petalshaped shaft orbits are in the range of 0.5 to 2.9, and petalshaped shaft orbits are divided into three parts according to minimum curvature radius. The petalshaped shaft orbit represents serious fault when the minimum curvature radius is in the range of 0.5 to 1.3, it represents medium fault when the minimum curvature radius is in the range of 1.3 to 2.1, and it represents slight fault when the minimum curvature radius is in the range of 2.1 to 2.9.
Fig. 7Part shaft orbits with different minimum radius of curvature
2.1.2. Constructing a simulation dataset of finegrained shaft orbit
According to theory of finegrained shaft orbits, each type of shaft orbits in Table 1 is subdivided into slight, medium, and serious levels by fault feature parameters. Five types of shaft orbits are further subdivided into 15 subtypes of shaft orbits with different degrees. 500 graphs for each subtype of shaft orbits are simulated, 300 of which are randomly selected for training and the remaining 200 are used for testing. Part of the simulation dataset for finegrained shaft orbits are shown in Table 2.
2.2. Identification method of finegrained shaft orbits based on improved LeNet5
In recent years, CNN has been widely used in object recognition, detection, and scene understanding [22]. CNN makes it possible to design an endtoend deep network for identification of shaft orbits. It automatically abstracts the features suitable for classification from the images of shaft orbits, thus avoiding complex process of the traditional feature extraction. LeNet5 [23] is a classic CNN structure and successfully applied in handwritten digital character recognition, which is a good reference for the identification of finegrained shaft orbits. In this section, we proposed a CNNbased approach for identification of finegrained shaft orbits by optimizing the network structure of LeNet5.
Table 2Part of the simulation dataset for finegrained shaft orbits
Shape of shaft orbit  Serious  Medium  Slight 
Ellipse  
Banana  
Inner “8”  
Outer “8”  
Petal 
Excluding the input and output layers, LeNet5 includes 6 layers: three convolutional layers (namely C1, C3 and C5), two pooling layers (namely S2 and S4), and one fullyconnected layer (namely F6). Its input image size is set to 32 × 32, and it has ten nodes corresponding to ten kinds of numbers in MNIST dataset. The kernel size in convolutional layers is fixed to 5 × 5, and in subsampling layer the size is 2 × 2. There are 6, 16 and 120 kernels in convolutional layers C1, C3 and C5, respectively. And there are 6 and 16 kernels in subsampling layers S2 and S4, respectively. The number of nodes in fullyconnected layer F6 is set as 84.
Convolutional layers are used for feature extraction in CNN. The design of the convolutional layer structure mainly includes four parts: the choice of activation function, the number of convolutional layers, the size and number of convolution kernels. Since the size of input image is only 32×32, the range of the number of convolutional layers, pooling layers and fullyconnected layers is from 1 to 3, and the size of convolutional kernels are selected among 3×3, 4×4 and 5×5. The number of convolutional kernels determines the number of linear combinations of network and the ability to extract features. It usually has a wide range from 12 to 512. As for the activation function, it improves the network’s ability of building nonlinear models, and directly affects the convergence speed and recognition rate of CNN. The performance of different active functions in CNN is discussed in Ref. [24, 25]. It is found that there is a problem of gradient disappearance in saturating nonlinear functions. The unsaturated nonlinear function can not only solve those problems, but also accelerate the convergence speed and improve the performance of CNN [2527]. Therefore, the appropriate activation function in this paper is selected among Softplus, LReLU, PReLU, RReLU and ELU functions.
Table 3Range of tested hyperparameters
Hyperparameters  Range 
Number of convolutional layers  1, 2, 3 
Number of pooling layers  1, 2, 3 
Number of fullyconnected layers  1, 2, 3 
Size of convolutional kernels  3×3, 4×4, 5×5 
Number of convolutional kernels  12, 16, 20, 32, 64, 128, 160, 192, 256, 512 
Number of nodes of fullyconnected layers  30, 84, 120 
Pooling method  Max pooling, Mean pooling 
Active function  ReLU, Softplus, LReLU, PReLU, RReLU, ELU 
Optimizer  SGD, Adam, Adadelta, RMSprop 
With the advent of various optimization methods, such as adaptive moment estimation (Adam), modified adaptive gradient (Adadelta), rootmeansquare propagation (RMSprop), etc., it is possible to adjust parameters (weights and biases) during training adaptively [28, 29]. They can mitigate the influence of two changes of gradient descent: the local minimum trap and choosing an appropriate learning rate [29]. Therefore, the optimizer in this paper are selected among stochastic gradient descent (SGD), Adam, Adadelta, and RMSprop.
The range of tested hyperparamenters are shown in Table 3. The hyperparameters of the optimal CNN for the identification of finegrained shaft orbits are selected by experiments in Section 3.1.
3. Experiment on the identification of finegrained shaft orbits
Firstly, this section establishes a new CNN that is most suitable for the identification of finegrained shaft orbits through optimizing the follow parameters: the number of convolutional layers and fully connected layers, the number and the size of convolution kernel, the nodes number of each fully connected layers, pooling method and optimizer. Secondly, experiments on simulation dataset and measured dataset is designed to prove that the proposed method is suitable for the identification of finegrained shaft orbits.
3.1. The structure of optimized LeNet5 for identification of finegrained shaft orbits
Since LeNet5 is specifically designed for handwritten digits, it differs from the identification of shaft orbits in this paper. Therefore, some preliminary improvements have been made as follows:
(1) The number of feature maps determines the number of linear combinations for network and the ability of features extraction. To improve that, the numbers of feature maps for the first and second pairs of convolution and pooling layers: C1/S2 and C3/S4, are all changed to 12. After adjustment, the total number of linear combinations of the CNN network is increased from 6 by 16 to 12 by 12.
(2) LeNet5 has a large number of parameters in the fully connected layer, which greatly increases the complexity of the network and the training time. And network with many parameters are easily overfitted when the data is insufficient. Therefore, only one full connection layer is retained in the primary improved LeNet5.
(3) The Sigmoid active function applied in LeNet5 is not universal, and it is found that there is a problem of gradient disappearance in Sigmoid functions. ReLU function can overcome that problem and it has a much faster convergence speed than Sigmoid functions. In Ref. [26, 27], ReLU function are all adopted in convolution layers. Hence the ReLU function is selected as the active function in the primary improved LeNet5.
(4) The number of nodes in full connection layer F6 is replaced with 15 to correspond to 15 kinds different finegrained shaft orbits.
(5) SGD in CNN usually faces two challenges: the choice of an appropriate learning rate and how to avoid local minimum trap [28, 29]. Ref [29] has proved that RMSprop is better than SGD in some respects. Hence RMSprop is selected to be the optimizer in the primary improved LeNet5.
After the primary improvement, the network obtains a recognition rate of 93.20 %, and the training parameters of the network are set as follows: the training period is 3000; the batch size is 64; the initial learning rate is 0.0001. Although there is some improvement on the recognition rate, there is still room for further improvement. In order to find the optimal CNN network for the identification of the finegrained shaft orbits, the hyperparameters of CNN network will be optimized in terms of the number of convolution layers, the number and the size of convolutional kernels, the number of fully connected layers, the number of nodes in each fully connected layer, the active function used in convolutional layers, the pooling method and the optimizer.
3.1.1. Numbers of convolutional layers and pooling layers
The structure of the convolutional layer and the pooling layer directly determine the feature extraction ability of CNN. It mainly includes four part: the numbers of convolutional layers, pooling layers, convolutional kernels, and the size of the convolutional kernel. The optimal numbers of convolutional layers and pooling layers are first obtained by experiments, the result of experiments is shown in Fig. 8, where the C means the convolutional layers, the P means pooping layer, and the F means the fullyconnected layer. The numbers of kernels in each convolutional layers are set to be the same. In the CPFF, CCPFF, CPCPFF and CCPCPFF structure, the size of convolutional kernels is all set to 5×5. Due to the limitation of the network structure, the size of convolutional kernels in CPCPCPFF structure is set to 3×3. The unspecified parameters of the network are the same as those of the primary improved LeNet5.
It can be seen from Fig. 8 that the test accuracy of the CCPFF network changes little and can reach the highest 97.35 % when the number of convolution kernel is set to be 128. Therefore, the CNN structure is set to be CCPFF, which includes two convolutional layers and one pooling layer for the following experiments.
Fig. 8The test accuracy of different CNN structure with different convolutional kernels: the C means the convolutional layers, the P means pooping layer, and the F means the fullyconnected layer
3.1.2. The number and size of kernels in each convolutional layer
The numbers of kernels in convolutional layers are also important for CNN. 9 groups of contrast experiments are conducted to find the optimal parameters, and the results of experiments are shown in Table 4. The test accuracy of the network reaches the highest 97.44 % when the numbers of kernels of the first and second convolutional layers are 96 and 192, respectively. It shows that it has a more comprehensive extraction performance for finegrained shaft orbits. Therefore, the numbers of kernels of the first and second convolutional layers in this paper are set to 96 and 192 for the following experiments.
The size of the kernels is another important parameter of the convolutional layer. The experimental results of nine different kernel size are shown in Table 5. It shows that the most suitable sizes of kernels of the first and second convolutional layers are 5×5 and 4×4, respectively.
3.1.3. The structure of fullyconnected layers
The structure of the fully connected layer mainly includes the number of fullyconnected layers and the number of nodes of each layer. Four comparative experiments are conducted, and the experimental result is shown in Table 6. It indicates that three fullyconnected layers are suitable for identification of finegrained shaft orbits, and the numbers of each layer’s nodes are 120, 84 and 30, respectively.
Table 4The identification results of CCPFF network with different numbers of convolutional kernels
Numbers of kernels of the first and second convolutional layers  Train time (s)  Test time (s)  Train accuracy  Test accuracy 
64,128  175.44  1.08  100 %  97.33 % 
64,160  192.56  1.16  100 %  97.24 % 
64,192  211.73  1.21  100 %  97.37 % 
96,128  199.85  1.21  100 %  97.28 % 
96,160  220.15  1.29  100 %  97.18 % 
96,192  243.60  1.39  100 %  97.44 % 
128,128  222.28  1.35  100 %  97.35 % 
128,160  248.88  1.44  100 %  97.30 % 
128,192  275.33  1.48  100 %  97.32 % 
Table 5The identification results of CCPFF network with different size of convolutional kernels
The size of kernels of the first and second convolutional layers  Train time (s)  Test time (s)  Train accuracy  Test accuracy 
5×5, 5×5  243.60  1.39  100 %  97.44 % 
5×5, 4×4  250.65  1.38  100 %  97.49 % 
5×5, 3×3  257.57  1.46  98.63 %  96.97 % 
4×4, 5×5  305.69  1.43  98.44 %  96.82 % 
4×4, 4×4  256.67  1.46  99.48 %  97.45 % 
4×4, 3×3  258.90  1.42  99.22 %  97.15 % 
3×3, 5×5  319.49  1.37  98.44 %  96.13 % 
3×3, 4×4  320.27  1.61  100 %  96.91 % 
3×3, 3×3  269.57  1.50  99.48 %  96.66 % 
Table 6The identification results of CCPFF network with different structure of fullyconnected layers
The number of fullyconnected layers  The number of nodes of each fullyconnected layer  Train time (s)  Test time (s)  Test accuracy 
1  84  226.80  1.40  96.13 % 
1  120  238.50  1.39  96.93 % 
2  120, 84  250.65  1.38  97.49 % 
3  120, 84, 30  249.97  1.45  97.61 % 
3.1.4. Optimizer
SGD, Adam, Adadelta, RMSprop are compare in the flowing experiments to select the optimal optimizer of proposed CNN, and the experimental results are shown in Fig. 9. The best test accuracy of SGD, Adam, Adadelta, and RMSprop are 97.75 %, 97.96 %, 97.92 %, and 97.61 %, respectively. The four methods can all achieve a very high test accuracy, among which the Adam method has the highest test accuracy. At the meantime, as can be seen from Fig. 9, the Adam method has the highest convergence rate. Therefore, Adam method is selected to be the optimizer of CNN in this paper.
3.1.5. The optimal structure of CNN for the identification of finegrained shaft orbits
Several experiments are carried out in order to find the optimal activation function and pooling method. Activation function include ReLU, Softplus, LReLU, PReLU, RReLU, ELU and pooling method include max pooling and mean pooling. The experimental result indicates the most suitable activation function is ReLU function and the most suitable pooling method is max pooling. Therefore, the final optimal structure of CNN proposed in this paper is determined, which is shown in Fig. 10. It includes the input layer, two convolutional layers, one pooling layer, three fullyconnected layers and the output layers. The optimizer of the optimal structure of CNN is Adam method. The parameters of each layer s are shown in Table 7.
Fig. 9Experimental results of different optimizers
Fig. 10The structure of the optimal structure of CNN for the identification of finegrained shaft orbits
Table 7The parameters of each layer in the optimal structure of CNN for the identification of finegrained shaft orbits
Layer number  Layer type  The size of kernel  The number of feature maps  The size of feature maps 
C1  Convolutional layer  5×5  96  28×28 
C2  Convolutional layer  4×4  192  25×25 
P3  Pooling layer  2×2  192  12×12 
F4  Fullyconnected layer  12×12  120  1×1 
F5  Fullyconnected layer  1×1  84  1×1 
F6  Fullyconnected layer  1×1  30  1×1 
Output  Output layer  1×1  15  1×1 
3.2. The experiment on the identification of simulated finegrained shaft orbits
To further highlight the advantages of the improved CNN on finegrained shaft orbits classification, several traditional methods are compared. The height function (HF) [30], shape context (SC) [31] and inner distance shape context (IDSC) [32] are selected as shape descriptors to extract the features of shaft orbits, and BP neural network and SVM [7] are used as classifiers. All shape descriptors selected 60 feature points as samples. The parameters of SVM are set as follows: the “linear kernel function” is selected and the other parameters are the default. The parameters of BP neural network in MATLAB toolbox are set as follows: the period is set to 1000, the target error is set to 0.0001, the node number of hidden layer is set to 100, Stype function “logsig” is selected as the excitation function, and linear function “Purelin” is adopted as the output layer excitation function. The experimental results of are shown in Table 8. The results are the average of 20 trials.
Table 8Results of the recognition on 15 kinds of finegrained shaft orbits using different methods
Methods  The average recognition rate  Training period  Training time (s)  Single sample testing time (ms) 
HF+BP  81.50 %  1000  231.09  19.11 
HF+SVM  81.61 %  –  114.30  31.17 
IDSC+BP  77.20 %  1000  486.71  61.13 
IDSC+SVM  86.08 %  –  319.20  96.71 
SC+BP  71.86 %  1000  361.53  35.49 
SC+SVM  87.67 %  –  208.11  67.49 
Primary improved LeNet5  93.20 %  3000  75.54  0.06 
Optimized LeNet5  97.96 %  3000  249.97  0.16 
It can be seen from the Table 8 that the average recognition rate of the optimized LeNet5 method proposed in this paper is the highest 97.96 %, which is much higher than that of traditional methods. In addition, the recognition rate of optimized LeNet5 method is 4.76 % higher than that of the primary improved LeNet5 network method. In single sample testing time, the optimized LeNet5 method is much better than the traditional methods. Although the recognition speed of the proposed method is slightly lower than that of the primary improved LeNet5 network, the test time of a single sample of optimized LeNet5 method only needs 0.16 milliseconds, which can still meet the realtime performance of the identification.
Although the training time of the optimized LeNet5 network is longer than that of the primary improved LeNet5 method and the traditional methods. In practice, the training process can be completed ahead of time, so the realtime performance of the algorithm is not affected.
Therefore, the method proposed in this paper has the best performance, its average time for identifying a shaft orbit is only 0.16 milliseconds and the average recognition rate can reach 97.96 %.
3.3. Experiments on the identification of measured finegrained shaft orbits
The testing bench of rotor (STS1000 online vibration monitoring and analysis system) is used to further verify the practicability of the algorithm, as showed in Fig. 11. The testing bench includes one bearing rotor, several displacement sensors, one signal acquisition unit, one DC motor controller and testing software, etc. The details of the sensors in the testing bench are shown in Table 9. Different degrees of faults for shaft imbalance are generated by adding different amounts of nuts to the counterweight plate and changing the speed of shaft.
Table 9The details of the sensors in the testing bench
Name  Sensitivity  Frequency range  Measuring range  Measuring conditions 
Magnetelectrical absolute vibration velocity transducer  28 mv/mm/s  10 Hz1000 Hz  010 mm/s (Less than 80 Hz)  No strong electromagnetic interference, –30 ℃  +80 ℃ 
Piezoelectric vibration acceleration transducer  98.5 mv/g  0.5 Hz10000 Hz  ± 50 g  –50 ℃  +120 ℃ 
Eddy current vibration displacement pickup  8 mv/um  0 Hz4000 Hz  1.8 mm  –30 ℃  +120 ℃ 
Using the slenderness $C$ presented in Section 2.1 as the indicator, 900 different measured elliptical shaft orbits of different severity are selected in the collected shaft orbits by the testing bench, which include 300 serious, 300 medium and 300 slight faults, respectively. And part of the measured finegrained shaft orbits of different unbalance faults are shown in Fig. 12. Twothirds of measured shaft orbits replace elliptical shaft orbits in simulated training dataset to train the optimized LeNet5 proposed in this paper. And the remaining one third of measured shaft orbits are used for identification to further verify the practicability of the method proposed in this paper. Similar to section 3.1, the primary improved method and the traditional methods are used to set comparative experiments, in which the trained BP networks or SVM models of each method in Section 3.2 are used to identify 300 measured shaft orbits.
Fig. 11Rotor testing bench: a) testing bench; b) counterweight plate
a)
b)
Fig. 12The measured finegrained shaft orbits of different unbalance faults: a) serious; b) medium; c) slighter
a)
b)
c)
The optimized LeNet5 is used to test the measured different finegrained shaft orbits, and the results are shown in Table 10. The networks or SVM models used in the measured experiments are all trained in section 3.2, so the training results are the same as in Table 8, which are not listed in Table 10.
By comparing experimental results on simulated shaft orbits and actual measured shaft orbits in Table 8 and Table 10, the accuracy of identification on the measured data is lower than the simulation data shaft orbit with the same algorithm. This is because measured shaft orbits are more complex than simulated shaft orbits. However, the reduction in the accuracies of the method proposed in this paper is small, not exceeding 0.82 %. It shows that the method proposed in this paper have a great practical performance.
Similar to the analysis of simulation result in section 3.2, the following conclusions can be drawn:
(1) From the perspective of the average recognition rate, the method proposed in this paper is more suitable for identification on actual measured shaft orbit than the primary improved method and the traditional methods.
(2) From the perspective of the single sample testing time, the performance of the method proposed in this paper are much better than the traditional methods.
Table 10Identification result of the measured finegrained shaft orbits for shaft unbalance faults
Methods  The average recognition rate  Single sample testing time (ms) 
HF+BP  80.75 %  20.52 
HF+SVM  80.83 %  32.94 
IDSC+BP  75.86 %  63.67 
IDSC+SVM  84.91 %  98.11 
SC+BP  69.93 %  36.51 
SC+SVM  85.82 %  69.15 
Primary improved LeNet5  91.53 %  0.07 
optimized LeNet5  97.14 %  0.17 
4. Conclusions
This paper proposes a new deep learningbased fault diagnose method for rotating machinery. The proposed method judges the type and severity of the fault by identification the shape of shaft orbit. A new theory of finegrained shaft orbit is proposed to divide different shaft orbits into several subclasses that can reflect the severity of the fault, and the appropriate network structure of CNN is designed to identify the finegrained shaft orbits. The recognition accuracy and realtime performance of the proposed method are verified by a series of contrast experiments. The experimental results show that the proposed method can reach 97.69 % and 97.14 % in the simulated and measured finedgrained shaft orbit dataset, respectively, and the test time of each sample is less than 0.17 ms. It is demonstrated that the proposed method is far superior to the existing traditional algorithms in effectiveness and accuracy and can provide some guidance and support for fault diagnosis of rotating machinery.
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About this article
This research was funded by the National Natural Science Foundation of China (Grant Nos. 51775177, 51675166) and Natural Science Foundation of Shanghai (Grant No. 19ZR1463800).
Bo Wu designed the methods. Songlin Feng collected relevant literature. Guodong Sun analyzed experimental data. Liang Xu wrote the paper. Chenghan Ai checked the grammar of the paper. All of the authors participated in the project, and they read and approved the final manuscript.