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Performance degradation assessment based on condition monitoring plays an important role in ensuring reliable operation of equipment, reducing production downtime and saving maintenance costs, yet performance degradation has strong fuzziness, and the dynamic information is random and fuzzy, making it a challenge how to assess the fuzzy bearing performance degradation. This study proposes a monotonic degradation assessment index of rolling bearings using fuzzy support vector data description (FSVDD) and running time. FSVDD constructs the fuzzy-monitoring coefficient

Rolling bearings, as important components of rotating machinery, not only support the load but also allow relative motion [

Vibration analysis is a powerful tool for fault diagnosis and degradation assessment [

One challenge is how to structure an intelligent assessment model based on the original features. Several scholars have proposed some comprehensive indexes and obtained impressive results. Qiu

This study proposes a monotonic degradation assessment index for rolling bearings using fuzzy support vector data description (FSVDD) and running time. FSVDD is the combination of fuzzy mathematics theory and SVDD, and deals well with the fuzzy matter in small samples. The performance degradation of bearings belongs to this situation. The performance degradation with the strong fuzziness is the intermediate process between the normal running and the final failure. The initial defect, the final failure and the degrees of damage severity with different moments are hard to identify. Meanwhile, the dynamic information which reflects the change of bearing states is fuzzy and random. Some deviations appear when the fuzzy matter is dealt with by the deterministic mathematic method. Besides, the bearing monitored during its whole life is rare due to the difficulties in engineering. Support vector data description (SVDD) is an excellent method of one-class classification for small samples, with the advantages of robustness and high computation [

This paper is organized as follows: In Section 2, the basic theory of FSVDD is introduced, and Section 3 presents the performance assessment method based on FSVDD and running time. The bearing run-to-failure tests and the related analysis are provided in Section 4. Section 5 provides the conclusions from the above studies.

Support vector data description proposed by Tax and Duin [

Assuming a training set contains _{i}, i_{i}_{i}_{i}_{i}_{i}

Reconstituting

All _{i}_{i}_{k}

For a new object _{i},x_{j}_{i}_{φ}

Then the monitoring coefficient of new object

If

SVDD only simply identifies the normal samples and the fault samples, but is unable to accurately distinguish the samples of different degrees of damage severity. Then FSVDD is generated by introducing the fuzzy mathematics theory into SVDD to describe the development process from the initial defect to final failure [

A fuzzy membership degree _{i}_{i}_{i}_{i}_{i}

The fuzzy non-linear vector function is redefined as

From the comparison of _{i}

SVM, SVDD and FSVDD are excellent classifiers for small samples, and have been widely applied to fault diagnosis [

The bearing damage undergoes sustained development with the running time after the initial defect. The damage development trend should be an increasing function of time. Hence we consider that the running time is used to construct a degradation assessment indicator which could reflect the damage development of the bearing more accurately. On the other hand, the vibration signal of bearings usually contains lots of random information which may cause some deviations between the signal and the true damage. It is able to reduce the influence of the vibration randomness that the running time is introduced into the construction of the assessment indicator.

The performance degradation of the bearing is a continuous process of change. A new bearing is installed into the rotating shaft. After a short running-in, it enters a long-term stable working period. Then the minor fault appears, and the defect gradually increases as the bearing fault develops. Finally the bearing fails with a serious defect. According to the running process, the bearing life is divided into three parts: normal stage, degeneration stage and failure stage. In the beginning of the normal stage, there may be a short run-in period. The degradation assessment model is provided based FSVDD and running time as follows:

The time-domain features of vibration signals are extracted. The stability features, such as RMS, square-root amplitude (SRA) and absolute average values (AAV), and the sensitive feature as Kurtosis factor are selected as the inputs of FSVDD in the meantime.

The function of fuzzy membership degree _{i}_{i}_{i}_{i}, a_{1} is the maximum energy of normal state and _{2}

The normal samples are trained to construct the hypersphere. Then the fuzzy-monitoring coefficient

A monotonic degradation assessment index of rolling bearing, damage severity index (DSI), is described based on FSVDD and running time as follows:
_{i}_{id}_{i}

Two assumptions are necessary for _{id}_{i}_{i}_{i}_{i}_{i}_{i}

In _{i}_{i}_{i}

The bearing run-to-failure test is carried out under constant load conditions on the bearing tester to reflect the defect propagation processes. The test bench shown in

The two test bearings, 30311 tapered roller bearings, are installed on both ends of the shaft, while the two steady bearings, N312 cylindrical roller bearings, are fixed at the middle of the shaft. The axial load _{a}_{r}

The vibration signals are transmitted by the screw which touches the outer-race of bearing. The spring exerts a pre-tightening force to ensure the contact of the screw and the outer-race. The acceleration sensor is fixed on the screw with the insulation spacer that insulates electromagnetic interference. In addition, the temperatures of four bearings are monitored by the thermocouple sensors. The axial load _{a}_{r}_{r}, f_{c}, f_{o}, f_{i}, f_{e}_{p}_{b}_{e}

There are three successful experiments in the present study. In each test, there is only one failure bearing which has huge vibration signals overwhelming those of the other three normal bearings. It may result from the individual factors of each bearing. The test 1 failure bearing has inner-race defects while the tests 2 and 3 failure bearings exhibit rolling element defects.

Firstly, the original time-domain features are researched. Three stability features, such as RMS, square-root amplitude (SRA), absolute average values (AAV), and one sensitive feature as Kurtosis factor of three bearings are listed in

Secondly, the vector _{i}

The monitoring coefficient _{i}_{i}_{i}_{i}

The left subgraphs are the time domain waveforms at different moments, and the right subgraghs are their corresponding Hilbert spectra. The moment of 4,000 min is the normal stage for the test 1 failure bearing, and the fault characteristic frequencies are not seen in _{i}

Thirdly, the fuzzy membership degree _{i}_{i}

Finally, a new index, DSI, is calculated as

In the run-to-failure tests, there are only two faults types, the inner-race defect and the rolling element defect. Then the outer-race defect and inner-race defect are simulated by increasing the impulse amplitude of the mathematical model to verify again the effectiveness of DSI.

The rolling bearing transfers the main load through elements in rolling contract than in sliding contract [_{i}_{i}

Simplifying the vibration waveform _{n}_{i}_{0}_{m}

Let _{m}_{i}_{m}_{r}_{i}_{n}

In a word, DSI is an excellent degradation assessment index for rolling bearings and has at least the following advantages: (1) DSI is sensitive to the initial defect and grows stably with the development of faults. The stability features, such as RMS, SRA, AAV, reflect the damage development, but it is hard to find the initial defect. The sensitivity feature as Kurtosis Factor is the opposite. DSI is an excellent indicator which is increasing with the damage development and sensitive to the initial defect. Moreover, the run-to-failure experiment and simulation both verify that DSI determines the initial defect earlier. (2) DSI with the defuzzification ability reflects the accelerating relation between the damage development and running time. This advantage of DSI comes from the fuzzy-monitoring coefficient _{i}

The type and width parameter of the kernel function affect the parameters

This paper presents a study of degradation assessment based on FSVDD and the running time. SVDD constructs the monitoring coefficient _{i}

Although the analysis results in this study are acceptable, more tests are necessary for further analysis. Meanwhile, other mathematical methods, such as Principal Component Analysis (PCA), can be used to construct some new indexes. PCA uses an orthogonal transformation to convert a set of possibly correlated variables into linearly uncorrelated variables. The transformation is defined in such a way that the first principal component has the largest possible variance, and each succeeding component in turn has the highest possible variance under the constraint which is orthogonal to the preceding components. If the statistical features are input into PCA, many separate composite characteristics are obtained. Some of the composite characteristics may be excellent assessment indexes.

In addition, DSI could be applied to the performance degradation assessment of other key machine components, such as gears, shafts and ball screws, because their dynamic performance is similar to that of bearings. The remaining life prediction could be carried out by the combination of DSI and the regression methods such as SVM, ANN

This work is supported by National Natural Science Foundation of China (No. 51035007 and No. 51175401), and Doctoral Fund of Ministry of Education of China (No. 20110201130001), and Fork YING TUNG Education Foundation (No. 121052).

Schematic diagram of support vector data description.

Test bench for bearing run-to-failure.

Load diagram of four bearings.

Location of accelerometer sensors and temperature sensors.

Four time-domain features of test 1 failure bearing: (

Four time-domain features of test 2 failure bearing: (

Four time-domain features of test 3 failure bearing: (

Results of SVDD for tests 1–3 failure bearing: (

The Time-domain waveforms and Hilbert spectrums of test 1 failure bearing: (

The Time-domain waveforms and Hilbert spectrums of test 2 failure bearing: (

The Time-domain waveforms and Hilbert spectrums of test 3 failure bearing: (

Results of FSVDD for tests 1–3 failure bearing: (

Comparisons between

Results of degradation assessment for tests 1–3 failure bearing: (

Original features and comprehensive indexes of outer-race defect: (

Original features and comprehensive indexes of inner-race defect: (

Parameters of test bearings and steady bearings.

e | Y | ||||||
---|---|---|---|---|---|---|---|

30311 | 55 | 120 | 16.25 | 16 | 0.35 | 1.7 | 152 |

N312 | 60 | 130 | 19.1 | 16 | — | — | 212 |

Bearing characteristic frequencies.

_{r} |
_{c} |
_{o} |
_{i} |
_{e} |
---|---|---|---|---|

25 | 10.2393 | 163.8288 | 236.1712 | 65.1061 |