1 Introduction
2 Co-clustering Framework of GFD
- Feature extraction sub-frame: as the input of this model, the gear vibration signals are collected using several tri-axial accelerometers installed in monitored mechanical equipment, which may be operated in varying working condition environment. Then, the feature vectors \(\left\{ F \right\}\) are obtained using the short-time Fourier transform (STFT) approach, aiming to gain differentiable time-frequency features for effective co-classifier performance;
- Clustering number estimation sub-frame: the Bayesian information criterion (BIC) strategy is adopted to characterize the distribution character of all feature vectors \(\left\{ F \right\}\) and then estimate their co-clustering numbers k & l in row and column, respectively;
- Co-clustering sub-frame: given co-clustering numbers, the conventional as well as modified NMF-based co-clustering classifiers are put into practice to build the varying working condition GFD models and get the classification results in various tasks;
- Parameter regulation sub-frame: aiming to those adjustable parameters involved in BIC and the NMF algorithm, such as the weight factor \(\lambda\) and the transitional dimension \(d\), the gradient ascent (GA) algorithm is implemented to find the optimal values, which reaches a reliable diagnostic accuracy.
3 STFT-based Feature Extraction
4 BIC-based Clustering Number Estimation
Direction | Sample length | Sample size | Clustering number |
---|---|---|---|
Row | \(n\) | \(m\) | \(k\) |
Column | \(m\) | \(n\) | \(l\) |
5 Modified NMF-based Co-clustering
5.1 NMF Theory
5.2 Classical NMF-based Co-clustering
5.3 Modified NMF-based Co-clustering
6 GA-based Parameter Regulator
Symbol | Descriptions | Range |
---|---|---|
\(N\) | The feature-dimension in STFT | \(N \le 80\) |
\(\lambda\) | The weight factor in BIC | \(0 \le \lambda \le 1\) |
\(d\) | The transitional dimension | \(d \le { \hbox{min} }\left( {m,n} \right)\) |
- The step length \(sl\): it represents the speed along the gradient direction during the iteration. We initialize the step length as 0.02 in the parameter \(\lambda\) & \(d/{ \hbox{min} }\left( {m,n} \right)\), where \(0 \le \lambda \le 1\), \(0 < d/{ \hbox{min} }\left( {m,n} \right) \le 1;\)
- The learning function: it has been designed in NMF-based co-clustering classifier as:where \(\varvec{x}_{i}\) represents extracted STFT feature vector from the ith sample; \(NMF\left( \cdot \right)\) means the NMF-based co-clustering classifier, with three input parameters \(\left\{ {\varvec{x}_{i} ,\lambda , \left[ {d/{ \hbox{min} }\left( {m,n} \right)} \right]} \right\}\); \(L_{i}^{r}\) and \(L_{i}^{c}\) means the clustering results of the ith sample in row and column.$$\left\{ {L_{i}^{r} ,L_{i}^{c} } \right\} = NMF\left\{ {\varvec{x}_{i} ,\lambda , \left[ {d/{ \hbox{min} }\left( {m,n} \right)} \right]} \right\},$$(29)
- The validity function: it is calculated by the sum of correct classifications, and it assesses the effectiveness of classification.$$\begin{aligned} & Ac\left( {\lambda , \left[ {d/\hbox{min} \left( {m,n} \right)} \right]} \right) \\ & = \mathop \sum \limits_{i = 1}^{m} zer\left( {L_{i}^{r} - y_{i}^{r} } \right) + \mathop \sum \limits_{i = 1}^{n} zer\left( {L_{i}^{c} - y_{i}^{c} } \right), \\ \end{aligned}$$(30)where \(y_{i}^{r}\) and \(y_{i}^{c}\) represents the label of the ith sample; \(zer\left( \cdot \right)\) means the zero sign function. Notice that, the validity function can only be obtained in those training samples, whose classification labels are known. The optimal \(\lambda\) and \(d/{ \hbox{min} }\left( {m,n} \right)\) is gained according to the training samples and is used in others, called testing samples.$$zer\left( x \right) = \left\{ {\begin{array}{*{20}c} {0, x \ne 0,} \\ {1,x = 0,} \\ \end{array} } \right.$$(31)
7 Experiments and Performance Analysis
7.1 DDS Experimental System
Task I | C1 | C2 | C3 | C4 | C5 |
Fault type | Health | Root | Missing | Chipped | Surface |
Task II | D1 | D2 | D3 | D4 | |
Fault severity | Health | Slight | Medium | Heavy | |
Task III | E1 | E2 | E3 | E4 | E5 |
Speed regulator (Hz) | < 5 | 5‒15 | 15‒25 | 25‒35 | > 35 |
Task IV | F1 | F2 | F3 | F4 | F5 |
Torque regulator (N·m) | < 1.83 | 1.83‒5.49 | 5.49‒9.14 | 9.14‒12.80 | > 12.80 |
7.2 GFD Experiments and Performance Analysis
7.2.1 Experimental Setup
A | Column task (rotating speed) | |||||
---|---|---|---|---|---|---|
E1 | E2 | E3 | E4 | E5 | ||
Row task (fault type) | C1 | 25 | 50 | 70 | 10 | 45 |
C2 | 25 | 50 | 70 | 10 | 45 | |
C3 | 25 | 50 | 70 | 10 | 45 | |
C4 | 25 | 50 | 70 | 10 | 45 | |
C5 | 25 | 50 | 70 | 10 | 45 |
B | Column task (load) | |||||
---|---|---|---|---|---|---|
F1 | F2 | F3 | F4 | F5 | ||
Row task (fault type) | C1 | 40 | 40 | 40 | 40 | 40 |
C2 | 40 | 40 | 40 | 40 | 40 | |
C3 | 40 | 40 | 40 | 40 | 40 | |
C4 | 40 | 40 | 40 | 40 | 40 | |
C5 | 40 | 40 | 40 | 40 | 40 |
7.2.2 Experimental Results and Discussion
Rotating speed | Pr (%) | Re (%) | Load | Pr (%) | Re (%) |
---|---|---|---|---|---|
C1 | 92.0 | 99.5 | C1 | 97.0 | 100 |
C2 | 95.0 | 99.3 | C2 | 94.5 | 100 |
C3 | 96.5 | 93.7 | C3 | 98.0 | 94.2 |
C4 | 98.5 | 97.5 | C4 | 99.5 | 98.0 |
C5 | 99.5 | 93.0 | C5 | 100 | 97.1 |
Total | 96.3 | 96.5 | Total | 97.8 | 97.2 |
Rotating speed | C1 | C2 | C3 | C4 | C5 | Pr (%) | Re (%) |
---|---|---|---|---|---|---|---|
R1 | 184 | 1 | 92.0 | 99.5 | |||
R2 | 1 | 138 | 95.0 | 99.3 | |||
R3 | 52 | 2 | |||||
R4 | 10 | 193 | 3 | 96.5 | 93.7 | ||
R5 | 5 | 197 | 98.5 | 97.5 | |||
R6 | 15 | 199 | 99.5 | 93.0 | |||
Total | 96.3 | 96.5 |
Table 4A: | 0.0801 | 0.0918 | 0.2563 | 0.0965 | 0.3236 |
0.3285 | 3.7987 | 3.8255 | 3.2661 | 3.5314 | |
0.2155 | 6.9619 | 6.5298 | 6.6931 | 7.0714 | |
0.7031 | 6.8263 | 7.2954 | 6.7465 | 7.2176 | |
0.2250 | 0.1191 | 0.0545 | 0.2875 | 0.4563 | |
Table 4B: | 3.7887 | 3.5302 | 4.1173 | 2.1937 | 2.4488 |
3.7157 | 0.1592 | 3.4741 | 1.9078 | 2.2279 | |
1.9611 | 1.3846 | 1.5855 | 3.8276 | 3.2316 | |
3.2774 | 0.2309 | 4.7511 | 3.9760 | 3.5468 | |
0.8559 | 0.4857 | 0.1722 | 0.9344 | 3.7734 |
Experiments | Pr (%) | Re (%) |
---|---|---|
Varying rotating speed | ||
Row task (C1‒C5) | 97.0 | 97.2 |
Column task (E1–E5) | 95.3 | 93.5 |
Varying load | ||
Row task (C1‒C5) | 97.8 | 97.5 |
Column task (F1‒F5) | 100 | 100 |
Model | Varying rotating speed | Varying load | ||
---|---|---|---|---|
Row clustering precision (%) | Time consumption (s) | Row clustering precision (%) | Time consumption (s) | |
X-means | 84.1 | 4.923 + 4.856 | 90.5 | 4.919 + 4.774 |
GMM methods | 87.7 | 6.845 + 6.018 | 92.3 | 6.274 + 5.909 |
NMF | 96.3 | 7.991 | 97.8 | 6.845 |
Modified NMF | 97.0 | 9.362 | 97.8 | 8.647 |
7.3 Parameter Regulation Experiments
7.3.1 STFT Dimension Adjustment Experiments
7.3.2 BIC Algorithm Experiments
Algorithm | Clustering number | Sub-clustering number of row task | Precision of row task (%) | ||||
---|---|---|---|---|---|---|---|
Health | Root | Missing | Chipped | Surface | |||
BIC & co-clustering | k = 3 | 1 | 1 | 1 | 60.0 | ||
k = 4 | 1 | 1 | 1 | 1 | 79.8 | ||
k = 5 | 1 | 1 | 1 | 1 | 1 | 96.9 | |
k = 6 | 1 | 2 | 1 | 1 | 1 | 96.3 | |
k = 7 | 1 | 2 | 1 | 1 | 2 | 95.1 | |
k = 8 | 1 | 2 | 2 | 1 | 2 | 94.0 | |
k = 9 | 1 | 3 | 2 | 1 | 2 | 90.4 | |
X-means | k = 12 | 2 | 3 | 3 | 2 | 2 | 84.1 |
7.3.3 GA Parameter Regulator Experiments
Dataset | Initial (\(\lambda\), \(d\)) | Number of iterations | Final (\(\lambda\), \(d\)) | Precision of row task (%) |
---|---|---|---|---|
Varying rotating speed | (0.5, 500) | 21 | (0.46, 156) | 99.3 |
(0.5, 200) | 6 | (0.46, 158) | 99.3 | |
(0.7, 500) | 8 | (0.80, 406) | 90.5 | |
(0.7, 200) | 14 | (0.80, 411) | 91.4 | |
Varying load | (0.5, 500) | 14 | (0.60, 296) | 97.0 |
(0.5, 200) | 9 | (0.60, 285) | 96.9 | |
(0.7, 500) | 13 | (0.64, 296) | 97.0 | |
(0.7, 200) | 8 | (0.64, 279) | 96.3 |