Fine-grained fault recognition method for shaft orbit of rotary machine based on convolutional neural network

2.1. Modeling of fine-grained shaft orbit with different severity of faults

2.1.1. Theory of fine-grained 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 banana-shaped 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 petal-shaped 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 banana-shaped shaft orbit, the severity degree of shaft misalignment increases. The bending of banana-shaped shaft orbits is in the range of 70° to 160°, and banana-shaped shaft orbits are divided into three according to $Arg$. The banana-shaped 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 petal-shaped 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 petal-shaped shaft orbits are in the range of 0.5 to 2.9, and petal-shaped shaft orbits are divided into three parts according to minimum curvature radius. The petal-shaped 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

2.2. Identification method of fine-grained shaft orbits based on improved LeNet-5

In recent years, CNN has been widely used in object recognition, detection, and scene understanding [22]. CNN makes it possible to design an end-to-end 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. LeNet-5 [23] is a classic CNN structure and successfully applied in hand-written digital character recognition, which is a good reference for the identification of fine-grained shaft orbits. In this section, we proposed a CNN-based approach for identification of fine-grained shaft orbits by optimizing the network structure of LeNet-5.

Table 2Part of the simulation dataset for fine-grained shaft orbits

Shape of shaft orbit	Serious	Medium	Slight
Ellipse
Banana
Inner “8”
Outer “8”
Petal

Excluding the input and output layers, LeNet-5 includes 6 layers: three convolutional layers (namely C1, C3 and C5), two pooling layers (namely S2 and S4), and one fully-connected 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 sub-sampling 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 sub-sampling layers S2 and S4, respectively. The number of nodes in fully-connected 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 fully-connected 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 [25-27]. Therefore, the appropriate activation function in this paper is selected among Softplus, LReLU, PReLU, RReLU and ELU functions.

With the advent of various optimization methods, such as adaptive moment estimation (Adam), modified adaptive gradient (Adadelta), root-mean-square 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 fine-grained shaft orbits are selected by experiments in Section 3.1.

3.1. The structure of optimized LeNet-5 for identification of fine-grained shaft orbits

Since LeNet-5 is specifically designed for hand-written 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) LeNet-5 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 LeNet-5.

(3) The Sigmoid active function applied in LeNet-5 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 LeNet-5.

(4) The number of nodes in full connection layer F6 is replaced with 15 to correspond to 15 kinds different fine-grained 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 LeNet-5.

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 fine-grained 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 fully-connected layer. The numbers of kernels in each convolutional layers are set to be the same. In the CP-FF, CCP-FF, CP-CP-FF and CCP-CP-FF 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 CP-CP-CP-FF structure is set to 3×3. The unspecified parameters of the network are the same as those of the primary improved LeNet-5.

It can be seen from Fig. 8 that the test accuracy of the CCP-FF 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 CCP-FF, 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 fully-connected layer

3.1.5. The optimal structure of CNN for the identification of fine-grained 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 fully-connected 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 fine-grained shaft orbits

Table 7The parameters of each layer in the optimal structure of CNN for the identification of fine-grained 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	Fully-connected layer	12×12	120	1×1
F5	Fully-connected layer	1×1	84	1×1
F6	Fully-connected layer	1×1	30	1×1
Output	Output layer	1×1	15	1×1

3.2. The experiment on the identification of simulated fine-grained shaft orbits

To further highlight the advantages of the improved CNN on fine-grained 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, S-type 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.

It can be seen from the Table 8 that the average recognition rate of the optimized LeNet-5 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 LeNet-5 method is 4.76 % higher than that of the primary improved LeNet-5 network method. In single sample testing time, the optimized LeNet-5 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 LeNet-5 network, the test time of a single sample of optimized LeNet-5 method only needs 0.16 milliseconds, which can still meet the real-time performance of the identification.

Although the training time of the optimized LeNet-5 network is longer than that of the primary improved LeNet-5 method and the traditional methods. In practice, the training process can be completed ahead of time, so the real-time 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 fine-grained 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.

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 fine-grained shaft orbits of different unbalance faults are shown in Fig. 12. Two-thirds of measured shaft orbits replace elliptical shaft orbits in simulated training dataset to train the optimized LeNet-5 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 fine-grained shaft orbits of different unbalance faults: a) serious; b) medium; c) slighter

a)

b)

c)

The optimized LeNet-5 is used to test the measured different fine-grained 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.