In this paper, an algorithm based on two novel shape descriptors and support vector machine (SVM) is proposed to improve the recognition accuracy and speed of shaft orbits of rotating machines. Firstly, two novel shape descriptors, respectively, named accurate Fourier height functions 1 (AFHF1) and accurate Fourier height functions 2 (AFHF2) are presented based on height function (HF) and Fourier transformation. Both AFHF1 and AFHF2 shape descriptors are constant to similarity transforms and also have intrinsic invariance to the starting point change and are more compacted than HF. Therefore, they perform well on the global or local features of the contours of shaft orbits. Then, the AFHF1 and AFHF2 shape descriptors are utilized to extract features of shaft orbits in the simulated dataset and measured dataset. Taking extracted feature vectors as the input, SVM is adopted in order to classify the fault types according to the shapes of shaft orbits. Finally, a series of descriptors including shape context (SC), inner-distance shape context (IDSC), triangular centroid distances (TCDs), and HF were compared to verify the performance of the proposed AFHF1 and AFHF2 shape descriptors. The average accuracy of our method in simulated dataset and measured dataset are all higher than 99.83%, the average recognition time of each sample is no more than 19 milliseconds. The experiments demonstrate that the proposed method has the best recognition accuracy and real-time and antinoise performance.
Rotating machines play a critical role in industrial production [
The shaft orbit [
Identification methods of shaft orbits mainly include two fundamental procedures: feature extraction and classification. In the first procedure, the quality of the extracted features could directly determine the accuracy of the identification on shaft orbits [
Fourier descriptors [
In recent year, Height Function (HF) [
In Figure
The Height Functions for the sample point
Feature vectors of each sampling point are calculated, and they are composed of the feature matrix of the shape
However, there are three obvious deficiencies of HF shape descriptor according to its definition. (1) In HF descriptor, the reference axis direction of the point
Accurate height values for the sample point
The second key procedure of identification of shaft orbit is classification. The main methods of classification are support vector machine (SVM) and the neural network. SVM performs better in accuracy and real time than the neural network when the samples are few and has been successfully applied in many different fields such as fault diagnosis [
In this paper, two novel descriptors called accurate Fourier height functions (AFHFs) including accurate Fourier height function 1 (AFHF1) and accurate Fourier height function 2 (AFHF2) are proposed based on HF and Fourier transformation to improve the accuracy, starting point invariance, and compactness. Then, we take advantage of AFHFs and SVM, and a novel recognition method for the shaft orbits is proposed in Section
In this section, a new identification method of shaft orbit in rotating machine based on improved HF descriptors is introduced. Firstly, AFHFs shape descriptors are presented in detail. Then, the orbit identification method based on AFHFs shape descriptors and support vector machine (SVM) is proposed.
In order to decrease the error of the HF shape descriptor in contour representation, the HF shape descriptor is corrected by using an accurate height value of the contour point.
As shown in Figure
In our method, two improvement algorithms are compensated for the second and the third deficiencies. (1) Fourier transformation is performed on each row of the AHF shape descriptor that is not smoothed, and the phase information is discarded to get the new shape descriptor Accurate Fourier Height Function 1 (AFHF1). (2) Fourier transformation is performed for each row of the AHF shape descriptor that is smoothed, and the phase information is discarded to get the new shape descriptor Accurate Fourier Height Function 2 (AFHF2). The above specific algorithms will be showed in the following two subsections.
Firstly, let
It can be seen from the definition of AFHF1 that its feature matrix dimension is
AFHF1 removes the smoothing process from the original HF, while AFHF2 reserves the HF smoothing process and directly improves it on the basis of the AHF descriptor. By applying Fourier transforms in each row of the AHF shape descriptor and discarding the phase information, a new descriptor AFHF2 can be obtained. Similarly, the final AFHF2 shape descriptor can be defined as follows:
From the definition of AFHF2 shape descriptor, it is not difficult to know that its matrix dimension is
The matrix dimensions of improved AFHF1 and AFHF2 shape descriptors and related descriptors are shown in Table
Dimension comparison of different descriptors feature matrices.
Shape descriptor | Feature dimension |
---|---|
IDSC |
|
HF |
|
TCDs | ( |
AFHF1 | ( |
AFHF2 |
|
As shown in Table
Compared to the original HF descriptor, the contour starting point has almost no significant influence on AFHF1 and AFHF2 descriptor. In order to determine the influence of the starting point on the descriptors, the shape descriptors extracted from the contours
In order to estimate the influence of the starting point on three descriptors, HF, AFHF1, and AFHF2, 100 sample points are sampled at equal intervals in the clockwise direction on the outline of a petal-shaped shaft orbit, as shown in Figure
Comparison of HF, AFHF1, and AFHF2 descriptor from different starting points. (a) Petal-shaped shaft orbit; (b) shape contour 1 of (a); (c) shape contour 2 of (a); (d) HF shape descriptor of (b); (e) HF shape descriptor of (c); (f) AFHF1 shape descriptor of (b); (g) AFHF1 shape descriptor of (c); (h) AFHF2 shape descriptor of (b); (i) AFHF2 shape descriptor of (c).
In order to describe the influence of the starting point on the above three shape descriptors quantitatively, the similarities of three sets of feature matrices are calculated individually by using Equation (
Since the starting point has almost no influence on AFHF1 and AFHF2 shape descriptors, there is no need to use the DP method, which is used by the original HF shape descriptors to obtain the matching result. The SVM, which is a classification algorithm based on statistical learning theory, is adopted as the classifier in this paper. SVM has unique advantages in solving a small amount of samples: nonlinear and high-dimension classification and recognition.
It is the most important for SVM to choose the proper kernel function and its parameters
Combined with the AFHF1 and AFHF2 descriptors proposed in Section
Process of shaft orbit identification method based on AFHFs and SVM.
Using MATLAB software, the shaft orbits are simulated according to the following equation (
Shaft orbit shapes of different faults.
Fault types | Shapes of shaft orbits |
---|---|
Misalignment | Banana-shaped |
Unbalance | Ellipse |
Oil whip | Inner “8” |
Misalignment | Outer “8” |
Oil whirl | Petal-shaped |
200 images of shaft orbits are simulated by MATLAB for each fault type, of which 100 images are for training and the remaining 100 images are for testing. The typical samples of shaft orbit dataset are shown in Figure
The typical samples of simulated shaft orbit dataset. (a) Banana-shaped; (b) petal-shaped; (c) inner “8”; (d) outer “8”; (e) ellipse.
Firstly, a series of descriptors including SC, IDSC, TCDs, and HF were compared to verify the performance of the proposed AFHF1 and AFHF2 shape descriptors. Secondly, in order to illustrate the superiority of SVM, it is compared with BP neural network.
All algorithms selected 100 feature points as samples. SC, IDSC, TCDs, HF, AFHF1, and AFHF2 shape descriptors are used to extract the feature of shaft orbits, respectively. And the parameter
The experiment flowchart.
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 15,
Experimental results of simulated shaft orbit dataset are shown in Tables
Experimental results on simulated shaft orbit dataset by SVM.
Algorithms | Average accuracy (%) | Average feature extracting time of single sample (ms) | Training time (s) | Average testing time of single sample (ms) |
---|---|---|---|---|
SVM + AFHF1 | 99.93 | 16.214 | 0.246 | 0.143 |
SVM + AFHF2 | 99.88 | 18.172 | 0.045 | 0.031 |
SVM + SC | 98.03 | 23.063 | 3.456 | 7.648 |
SVM + IDSC | 96.46 | 46.426 | 3.592 | 7.631 |
SVM + TCDs | 96.04 | 16.187 | 0.407 | 0.457 |
SVM + HF | 95.81 | 18.016 | 0.563 | 0.778 |
Experimental results on simulated shaft orbit dataset by BP neural network.
Algorithms | Average accuracy (%) | Average feature extracting time of single sample (ms) | Training time (s) | Average testing time of single sample (ms) |
---|---|---|---|---|
BP + AFHF2 | 99.66 | 18.172 | 27.39 | 0.380 |
BP + AFHF1 | 99.57 | 16.214 | 61.40 | 0.869 |
BP + HF | 96.50 | 18.016 | 70.16 | 0.924 |
BP + TCDs | 94.10 | 16.187 | 57.17 | 0.726 |
BP + SC | 87.84 | 23.063 | 368.55 | 3.044 |
BP + IDSC | 83.38 | 46.426 | 364.79 | 3.013 |
Accuracy comparison of different algorithms on simulated shaft orbits.
From the perspective of shape descriptors, the accuracies of the algorithms using the AFHF1 and AFHF2 shape descriptors are higher than 99.57%, but the highest accuracy of the algorithms using other shape descriptors is 98.03%.
From the perspective of the classifier, the performance of the algorithms using SVM as classifier are slightly better than using BP neural network as classifier in accuracy and real time when AFHF1 and AFHF2 shape descriptors are used to extract feature in the experiments of simulated shaft orbit identification. Therefore, the proposed method has the best performance, its average time for identifying a shaft orbit is less than 19 milliseconds, and the average accuracy is higher than 99.88%.
In order to verify the performance of the proposed algorithm, a rotor test bench is used to produce different faults, and the same methods which are used in the simulation experiment are adopted for contrast. The measured database of shaft orbits is created on the bearing rotor test bench as shown in Figure
Structure of bearing rotor test bench.
As shown in Figure
Measured shaft orbit. (a) Original shaft orbit. (b) Shaft orbit after preprocessing.
Experimental results on actual measured shaft orbit dataset by SVM.
Algorithms | Average accuracy (%) | Average feature extracting time of single sample (ms) | Average training time (s) | Average testing time of single sample (ms) |
---|---|---|---|---|
SVM + AFHF2 | 99.86 | 18.521 | 0.048 | 0.041 |
SVM + AFHF1 | 99.83 | 16.374 | 0.270 | 0.244 |
SVM + SC | 96.16 | 23.542 | 4.954 | 9.594 |
SVM + TCDs | 95.34 | 16.354 | 0.385 | 0.457 |
SVM + IDSC | 94.21 | 46.592 | 4.823 | 8.907 |
SVM + HF | 92.52 | 18.539 | 0.650 | 1.050 |
Experimental results on actual measured shaft orbit dataset by BP neural network.
Algorithms | Average accuracy (%) | Average feature extracting time of single sample (ms) | Average training time (s) | Average testing time of single sample (ms) |
---|---|---|---|---|
BP + AFHF2 | 99.49 | 18.521 | 26.04 | 0.345 |
BP + AFHF1 | 99.34 | 16.374 | 58.13 | 0.756 |
BP + HF | 93.34 | 18.539 | 68.77 | 0.879 |
BP + TCDs | 92.84 | 16.354 | 57.33 | 0.724 |
BP + SC | 84.59 | 23.542 | 369.91 | 3.044 |
BP + IDSC | 76.30 | 46.592 | 376.39 | 3.051 |
By comparing experimental results on simulated shaft orbits and actual measured shaft orbits in Tables
Similar to the analysis of simulation result in section 3.1.2, the following conclusions can be drawn: From the perspective of shape descriptors, AFHF1 and AFHF2 shape descriptors are more suitable for identification on actual measured shaft orbit than SC, IDSC, TCDs, and HF shape descriptors. From the perspective of the classifier, the performance of the algorithms using SVM as classifier is slightly better than that using BP neural network as classifier in accuracy and real time when AFHF1 and AFHF2 shape descriptors are used to extract feature in the experiments of actual measured shaft orbit identification.
An algorithm based on two novel shape descriptors and SVM is proposed. In the algorithm, two novel shape descriptors, respectively, named AFHF1 and AFHF2 based on HF and Fourier transformation are presented to extract features from shaft orbits. Both AFHF1 and AFHF2 shape descriptors are constant to similarity transforms, also have intrinsic invariance to the starting point change and are more compacted than HF. Therefore, they perform well on the global or local features of the contours of shaft orbits. SVM is adopted as the classifier to efficiently identify the extracted feature. In our experiments, AFHF1 and AFHF2 are compared with SC, IDSC, HF, and TCDs, and BP neural network is compared with SVM. The identification accuracies on the simulated and actual measured shaft orbit datasets and the comparisons with the algorithms mentioned above are obtained. The experiments show that the proposed algorithm can quickly and accurately identify shaft orbits. As for simulated and measured shaft orbits, the average time of identifications is all less than 19 milliseconds and the average of accuracies are all higher than 99.83%. The proposed method has the best recognition accuracy, and real-time and antinoise performance.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study is supported by the National Natural Science Foundation of China (nos. 51775177and 51675166).