Bechhoefer [ 60 ] complied this dataset from a set of bearing experiments. The dataset contains labeled signatures of faulty inner race and outer race of ball bearings. The signatures of faulty inner and outer races were generated at a constant shaft rotational speed of 25 rps rpm and at seven different load conditions: Each vibration signal is recorded for 3 seconds at 48, Hz frequency; it resulted in a signal of length , data points. Fig 1 shows a sample of vibration signals collected from the faulty inner race and outer race of the ball bearing.
These sample signals were collected at The vibration signals for both inner and outer race faults are periodic. The fault characteristic frequencies for inner race fault and outer race fault are outlined by Zhang et al. For both the inner and outer race faults, we obtain a p value of 0.
The raw data includes inner and outer race fault signals of , data points each. For CECP computation, we segment each signal of , data points into 35 sub-signals of 4, data points each.
- LAN Switch Security: What Hackers Know About Your Switches (Networking Technology: Security).
- Quantum Entropies.
- Precalculus: Graphs and Models (5th Edition)!
- Lipids in Health and Disease (Subcellular Biochemistry).
- Quantum Entropies | Ebook | Ellibs Ebookstore.
Fig 2 a plots values of 35 outer race and 35 inner race sub-signals. The overlap of for inner race and outer race signals indicates that is not an effective parameter to distinguish inner race faults from outer race faults.
Mobile main navigation
Fig 2 b presents a CECP map i. It is clear from the figure that the CECP representation is able to separate inner race faults from outer race faults in the plot. The dashed lines represent the lower and upper limit curves. Inset figure is a scaled version of subplot b. Fig 3 presents CECP maps for four different signal lengths. It is evident from the figure that the separation distance between fault classes widens and the variance of of each class decreases with increasing signal length.
This leads to improvement in Dunn index of cluster formation as the signal length increases see Fig 4. Dunn index, which is the ratio of minimum inter-cluster distance to maximum intra-cluster distance [ 62 ], measures the quality of cluster formations: the higher the Dunn index the better the cluster quality. Dunn index for m clusters is defined as 7 where d C i , C j is dissimilarity inter-cluster between clusters C i and C j.
It is defined as , where E a , b is the Euclidean distance between points a and b.
Entropy production and non-Markovian dynamical maps
The stars represent the inner race fault and the circles represent outer race fault. Some figures may not show the limit curves due to axis scale effects. In general, CECP analysis is employed for characterizing time series, which is sensitive to parameter choices. However, in the current work we are more interested in using CECP representation for class separation or classification accuracy with a correct set of application specific parameters. From the sensitivity analysis we found that for load conditions Similarly, for load conditions To verify beyond visual observation of separability, we used a SVM to see how the classification accuracy improves with increasing signal length.
Pier A. Mello and Narendra Kumar
We employed a linear SVM model with 5-fold cross validation. We observed that for all load conditions, the SVM classifier performance improves with increasing signal length, which is consistent with the results of the earlier analysis using Dunn Index. Case Western Reserve University CWRU bearing dataset [ 63 ] includes high quality signals collected at normal and faulty conditions of bearings. Fig 5 shows the setup of the experiment. The setup has test bearings located at the drive-end and the fan-end of the motor.
The faults were introduced in the inner race and outer race and on a ball for both drive-end bearings and the fan-end bearings using electro-discharge machining. The accelerometers attached to the motor housing using magnetic bases were used to measure vibration signals from the setup. One accelerometer was attached on the drive-end of the motor and another on the fan-end of the motor.
Videos from Prospects in Theoretical Physics | Institute for Advanced Study
For some experiments, an additional accelerometer was attached to the base plate supporting the motor. Vibration signals were collected using a channel DAT recorder. Sensor signals were collected at a frequency of 12, Hz. The length of the baseline signals i. Fig 6 shows the sample vibration signals representing the baseline condition, ball fault and inner race fault.
The fault characteristics of the CWRU dataset are exhaustively studied in time and frequency domains by Smith et al. For signals in all the three cases baseline, inner race fault and ball fault we obtain a p value of 0. In this case, the fault diameter for both the ball and inner race is 0.
The bearing considered is drive-end bearing. In this case we analyzed only baseline signals, inner race fault signals, and ball fault signals. The experimental parameters are outlined in Table 2.
All Submission Categories
For all the parameter variations, the fault depth was maintained constant at 0. The results of the sensitivity analysis for a selected set of operating conditions are given in Figs SS23 in S1 File. For both drive-end and fan-end bearings the baseline condition exhibit higher complexity and lower permutation entropy compared to inner race and ball fault conditions.
We observed that the fan-end bearing exhibits bigger class separation between baseline and faulty conditions than the drive-end bearing. To confirm the visual observation of class separability, we used a SVM to see how the classification accuracy improves with increasing signal length.
- Donate to arXiv?
- Bibliographic Information?
- Research Papers.
- The Greek Historians.
- Quantum entropies - Scholarpedia.
- Table of contents.
- Subscriber Login.
- Computer Aided Verification: 9th International Conference, CAV97 Haifa, Israel, June 22–25, 1997 Proceedings.
- Introduction into Capital Theory: A Neo-Austrian Perspective;
We used a linear SVM model with 5-fold cross validation. Similar to the results of the MFPT experiment, we observed that for almost all operating conditions the SVM classifier performance improves with increasing signal length. We considered a dataset provided by the PHM society that contains labeled data on different types of gear degradation [ 65 ]. The experiments were conducted using spur gears and helical gears. For CECP application, we considered experiments with helical gears. Fig 7 shows the experimental setup.
The setup is common for both spur gears and helical gears. Fig 7 shows the details of gear teeth for both spur and helical gears. For our analysis we considered only the helical gears. The setup has an input shaft, an idler shaft and an output shaft on which the gears are mounted.
The input side is on the left and the output side is on the right. Two accelerometers are mounted, one on the input side and the other on the output side. The left helical gear on the input shaft has 16 teeth and the left helical gear on the idler shaft has 48 teeth. The right helical gear on idler shaft has 24 teeth and the right helical gear on the output shaft has 40 teeth.
Experiments with helical gears were performed six times. All in all, there are two fault categories chipped tooth and broken tooth and one baseline category no known faults. From the PHM dataset we picked the case labled helical 1 which has no known gear defects as the baseline case. We selected helical 2 which has a chipped tooth in helical gear with 24 teeth as a chipped tooth gear case and helical 5 which has a broken tooth in helical gear with 24 teeth as a broken tooth gear case. In all the three cases, we used the vibration signals recorded from accelerometer 2 placed on the output side.
The signals were recorded under two different load conditions labeled as low and high and five different rotational speeds, i. For each of these settings, two signals were recorded for four seconds each. Thus, for one fault signal, , data points were generated in eight-second recording at a sampling rate of 66, samples per second.
Fig 8 shows sample signals of length data points each representing the baseline and the fault conditions. For all the three cases baseline, chipped tooth and broken tooth we obtain a p value of 0. The sample signals shown here are taken from experimental condition of high load and 50 rps rpm rotational speed. The results of the sensitivity analysis for a selected set of operating conditions are given in Figs SS41 in S1 File.
In addition to visual observation of class separability, we used a SVM to see how the classification accuracy improves with respect to the increasing signal length. The same pattern is observed when using the Dunn Index values across all operating conditions see Fig 9.