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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This study proposes a new condition diagnosis method for rotating machinery developed using least squares mapping (LSM) and a fuzzy neural network. The non-dimensional symptom parameters (NSPs) in the time domain are defined to reflect the features of the vibration signals measured in each state. A sensitive evaluation method for selecting good symptom parameters using detection index (DI) is also proposed for detecting and distinguishing faults in rotating machinery. In order to raise the diagnosis sensitivity of the symptom parameters the synthetic symptom parameters (SSPs) are obtained by LSM. Moreover, possibility theory and the Dempster & Shafer theory (DST) are used to process the ambiguous relationship between symptoms and fault types. Finally, a sequential diagnosis method, using sequential inference and a fuzzy neural network realized by the partially-linearized neural network (PLNN), is also proposed, by which the conditions of rotating machinery can be identified sequentially. Practical examples of fault diagnosis for a roller bearing are shown to verify that the method is effective.

In the field of machinery diagnosis, vibration signals are often used for fault detection and state discrimination. Machinery diagnosis depends largely on the feature analysis of vibration signals measured for condition diagnosis, because the signals carry dynamic information about the machine state [

Roller bearings are an important part, widely used in rotating machinery. The failure of a rolling bearing may cause the breakdown of a rotating machine, and furthermore, serious consequences may arise due to the failure. Therefore, fault diagnosis of rolling bearings is extremely important for guaranteeing production efficiency and plant safety. Although fault diagnosis of rolling bearings is often artificially carried out using time or frequency analysis of vibration signals, there is a need for a reliable, fast automated diagnosis method thereof. Neural Networks (NN) have potential applications in automated detection and diagnosis of machine failure [

For the above reasons, this paper proposes a novel condition diagnosis method for rotating machinery developed using LSM and a fuzzy neural network realized by the PLNN. The NSPs in the time domain are defined to reflect the vibration signal features measured in each state. To raise the diagnosis sensitivity of the symptom parameters the SSPs are obtained by LSM. Using statistical theory, a detection index (DI) has also been defined to evaluate the applicability of SSPs. The DI can be used to indicate the fitness of a SSP for the PLNN. A sequential diagnosis approach is also proposed through the PLNN to sequentially identify the types of fault of rotating machinery. Diagnostic knowledge for the PLNN is acquired by possibility theory and the DST for solving the problem of ambiguous fault diagnosis. A practical example of condition diagnosis for a roller bearing verifies that the method is effective. The flowchart of the condition diagnostic procedure proposed in this paper is shown in

In this work an accelerometer (PCB MA352A60) with a bandwidth from 5 Hz to 60 kHz and 10 mV/g output was used to measure the vibration signals of the vertical direction in the normal (N), the outer-race defect (O), the inner-race defect (I), and the roller element defect (R) states, respectively. The vibration signals measured by the accelerometer were transformed into a signal recorder (Scope Coder DL750) after being magnified by a sensor signal conditioner (PCB ICP Model 480C02). The original vibration signals in each state are measured at a constant speed (1,500 rpm), and a 150 kg load is also transported on the rotating shaft by the loading equipment (RCS2-RA13R) while the vibration signals are being measured. A high-pass filter with a 5 kHz cut-off frequency was used to cancel noise in the vibration signals for fault diagnosis. Examples of vibration signals measured in each state after filtering are shown in

When a computer is used for condition diagnosis of plant machinery, symptom parameters (SPs) are required to express the information indicated by a signal measured for diagnosing machinery faults. A good symptom parameter can correctly reflect states and the condition trends of plant machinery [_{i}_{i}_{i},
_{i}_{pi}_{i}, x̄_{p}_{p}_{pi}_{p}_{pi}_{vi}_{i}_{v}_{v}_{vi}_{v}_{vi}

Supposing that _{1} and _{2} are values of a symptom parameter (SP) calculated from the signals measured in state 1 and state 2, respectively, and conforming respectively to the normal distributions N(_{1},_{1}) and N(_{2},_{2}). Here, μ and σ are the average and the standard deviation of the SP. The larger the value of |x_{2}−x_{1}| is, the higher the sensitivity of distinguishing the two states by the SP. Because _{2} − _{1} also conforms to the normal distribution N(_{2} − _{1},_{1} + _{2}), there is the following density function about _{2} ≥ _{1} (the same conclusion can be drawn when _{1} ≥ _{2}). The probability can be calculated with the following formula:
_{0} is called the “Discrimination Rate (DR)”. With the substitution:
_{0} can be obtained by:

It is obvious that the larger the value of the DI, the larger the value of the “Discrimination Rate (DR = 1 − _{0})” will be, and therefore, the better the SP will be. Thus, the DI can be used as the index of the quality to evaluate the distinguishing sensitivity of the SP. The number of symptom parameters used for the diagnosis and fault types are

In order to raise the diagnosis sensitivity of the symptom parameter, a method for obtaining the new synthetic symptom parameter is proposed as follows. The least squares mapping (LSM) technique aims to increase class separability and consists of the transformation of pattern vectors around arbitrary pre-selected points in the ^{C} space (where _{k}_{ij}_{ij}

_{ij}_{ij}

The transformation matrix _{ij}_{i}_{i}_{i}_{i}

Figure 5 shows an illustration of the projection by the LSM, where

The error vector is:

When

For diagnosis, the new synthetic symptom parameter can be obtained as follows:
_{1}∼P_{8}).

According to the projected results shown in Figure 5(b), the points in state 1 and state 2 are congregated to vector _{1}_{2}

To explain the efficiency of the LSM method, some examples are given. In the present example, we used two symptom parameters (P_{1} and P_{2}) to distinguish the inner race defect (I) and roller element defect (R) states of the bearing. SSP_{1} and SSP_{2} are the new synthetic parameter obtained by the LSM. _{p1}, μ_{p2}, μ_{ssp1} and μ_{ssp2} are the mean values of P_{1}, P_{2}, SSP_{1} and SSP_{2}, respectively. σ_{p1}, σ_{p2}, σ_{ssp1} andσ_{ssp2} are the standard deviations of P_{1}, P_{2}, SSP_{1} and SSP_{2}, respectively.

In many cases of condition diagnosis, symptom parameters are defined to reflect the features of vibration signals measured in each state in order to diagnose faults. However, it is difficult to find one symptom parameter or a few symptom parameters that can identify all of the faults simultaneously. However, the symptom parameters for identification of two states are easy to identify [

As mentioned in the Section 3.2, the larger the value of the DI, the better the SP will be. Therefore, the two best SSPs that have the high sensitivity at each diagnostic step are selected by the DI. As an example, parts of the DI values of each SSP and the selection results are shown in _{1} and SSP_{5} can distinguish the normal (N) and the abnormal states (O, I and R) more easily than the other SSPs. Because all of DI values of SSP_{1} and SSP_{5} for distinguishing these states are larger than those of the other SSPs. Similarly, the SSPs for other diagnostic steps can also be selected. The other selected results of the SSPs are, SSP_{1} and SSP_{5} for the second step, and SSP_{1} and SSP_{2} for the last step, respectively. All of those DIs are larger than 2.12, and therefore all of the distinction rates approach 98.5%.

In most cases of condition diagnosis for rotating machinery, knowledge of distinguishing faults is ambiguous, because the definite relationships between symptom parameters and fault types, even for a single fault, cannot be easily identified. The values of symptom parameters calculated from vibration signals for fault diagnosis are also ambiguous because of the dispersion in the same state. Therefore, it is necessary to solve the ambiguous problem of fault diagnosis and to express uncertainty about the interpretation of the observable.

Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic [

For fuzzy inference, membership functions of SP are necessary. These can be obtained from probability density functions of the symptom parameters using possibility theory. When the probability density function of symptom parameters conforms to the normal distribution, it can be changed to a possibility function _{i})_{i} and _{k} can be calculated as follows:

_{i}_{1}_{i}_{2}_{i}

If _{t}

Where _{1}, W_{2}_{un}

Fuzzy systems rely on a set of rules. In this study, to correctly and effectively identify the condition and the fault type of rotating machinery, we have obtained the following “if-then” rules for condition diagnosis.

Rule 1: _{i}_{1i} − 3_{1} _{i}_{2i} − 3_{2} _{1} = 0, _{2} = 0, _{un}

Rule 2: _{i}_{1i} + 3_{1} _{i}_{2i} + 3_{2} _{1} = 0, _{2} = 0, _{un}

Rule 3: _{1i} − 3_{1} ≤ _{i}_{1i} + 3_{1}_{1} ≤ 1, 0 ≤ _{2} ≤ 1, 0 ≤ _{un}

Rule 4: _{2i} − 3_{2} ≤ _{i}_{2i} + 3_{2} _{1} ≤ 1, 0 ≤ _{2} ≤ 1, 0 ≤ _{un}

where _{1i} and _{2i} are mean values of symptom parameter _{i}_{1} and _{2} are standard deviations of symptom parameter _{i}_{1}, _{2} and _{un}

Dempster & Shafer theory (DST) provides a rational inference mechanism for the combination relation in the diagnosis problems with uncertainty [_{i}_{j}

Supposing _{i}_{m}_{i}_{m}; W_{j}_{k}_{j}_{k}_{m}_{k}^{′} is the combination possibility function of SP_{i}_{j}_{m}_{k}^{′}can be obtained by:

As mentioned above, the combination possibility functions of SSPs in each sequential diagnosis step are obtained as follows. In the first step of the sequential diagnosis, the normalized combination possibility functions of the normal state possibility _{i}_{j}_{i}_{j}_{i}_{i}_{i}_{i}_{j}_{j}_{j}_{j}

In the second step of the sequential diagnosis, the normalized combination possibility functions of the outer-race defect possibility _{i}_{j}_{i}_{j}_{i}(O), W_{i}(IR)_{i}(U)_{i}, respectively. _{j}(O), W_{j}(IR)_{j}(U)_{j}

The last step of the sequential diagnosis, the normalized combination possibility function of the inner race defect possibility _{i}_{j}_{i}_{j}_{i}(I), W_{i}(R)_{i}(U)_{i}, respectively. W_{j}(I), W_{j}(R) and W_{j}(U) are possibilities of inner race defect (I), rolling element defect (R) and unknown state (U) obtained by _{j}

The main mathematic symbols used in Section 6 are:

_{m}: the neuron number of the m-th layer of an NN, m

_{1}.

_{M}.

_{M}

_{m}; v_{m+1}.

The fuzzy neural network is applied to diagnose the fault types of a rolling bearing by the sequential diagnosis algorithm, and realized with a developed back propagation neural network called as “the partially-linearized neural network” (PLNN). A back propagation neural network is only used for training the data, and the PLNN is used for testing the learned NN. Here, the basic principle of the PLNN for the fault diagnosis is described as follows.

The neuron number of the _{m}_{1}, _{M} and,
_{M}

Even if the NN converges by learning ^{(1)} and ^{(M)}^{(1)}* and ^{(M)}^{(M)}^{(1)}*, partial linear interpolation of the NN is introduced as shown in

In the NN that has converged with the data ^{(1)} and ^{(M)}

_{m}

_{m}_{m+1}.

If all these values are memorized by the computer, when new values
_{m}

Using the operation above, the sigmoid function is partially linearized, as shown in _{1}_{2}_{1}_{2}_{e}

As shown in _{1}_{2}_{1(min)}, _{2(min)} and _{1(max)}, _{2(max)} are the minimum values and the maximum values of _{1} and _{2}, respectively, which have been learned by the PLNN. Therefore, in this work, the values (_{i}_{j}_{i(min)}, _{j(min)} and _{i(max)}, _{j(max)} are the minimum values and the maximum values of _{i}_{j}

In this study, the diagnosis knowledge for training of the PLNN is acquired by the possibility theory and the Dempster & Shafer theory (DST). The possibility functions of the SSPs used for each diagnostic step, as examples, are shown in

In _{1}(N), W_{1}(B)_{1}(U)_{1}_{5}(N), W_{5}(B)_{5}(U)_{5}

In _{1}(O), W_{1}(IR)_{1}(U)_{1}_{5}(O), W_{5}(IR)_{5}(U)_{5}

In _{1}(I), W_{1}(R)_{1}(U)_{1}_{2}(I), W_{2}(R)_{2}(U)_{2} in the inner-race defect, the rolling element defect and the unknown states can also be obtained, respectively.

After obtaining the possibilities of the SSPs for each diagnostic step, the combination possibility function of each state

In order to verify the diagnostic capability of the PLNN, we used the data measured in each state had not been learned by the PLNN. When inputting the test data into the learnt PLNNs, they can correctly and quickly diagnose those faults with the possibility grades of the corresponding states. The diagnosis results are shown in

According to the diagnosis results above, the normal (N), the outer-race defect (O), the inner-race defect (I), and the roller element defect (R) states of roller bearing can be automatically and correctly identified using the diagnosis methods proposed in this paper.

In order to solve the problem of ambiguity between the symptom parameters and fault types, effectively diagnose faults and automatically identify the condition of a rotating machine, an intelligent diagnosis method was proposed on the basis of the least squares mapping (LSM) and a fuzzy neural network. The main conclusions can be summarized as follows:

A sequential diagnosis method was proposed through which the fuzzy neural network realized by the partially-linearized neural network (PLNN) could sequentially distinguish fault types.

Knowledge for training the PLNN was acquired by possibility theory and the Dempster & Shafer theory (DST). The method of establishing the membership function by converting the probability distribution function of symptom parameters into a possibility function by the possibility theory was proposed, and the combination possibility functions of several symptom parameters were obtained by the DST.

The eight non-dimensional symptom parameters in the time domain were defined for reflecting the features of vibration signals measured in each state. To raise the diagnosis sensitivity of the symptom parameters, the new synthetic symptom parameters (SSPs) were obtained by the LSM method.

The detection index (DI) on the basis of statistical theory was also defined to evaluate the applicability of the SSPs. The DI can be used to select better SSPs for the PLNN.

The practical examples of faults diagnosis of a roller bearing verified the effectiveness of the proposed method. The diagnosis results showed that the faults were sequentially and automatically diagnosed on the basis of the possibilities of the symptom parameters.

Flowchart of the condition diagnosis.

Experimental setup for rolling bearing fault diagnosis.

Bearing defects. (

Vibration signals of bearings after filtering.

Projected example by the LSM (

Flowchart of sequential condition diagnosis.

Possibility function and the probability density function.

Matching examples of possibility function.

The partial linearization of the sigmoid function.

Interpolation by the PLNN.

Partially-linearized neural network for condition diagnosis.

Possibility functions of (_{1} and (_{5}for first diagnostic step.

Possibility functions of (_{1} and (_{5}for second diagnostic step.

Possibility functions of (_{1} and (_{2} for third diagnostic step.

Bearing information for verification.

Bearing outer diameter | 52 mm |

Bearing inner diameter | 25 mm |

Bearing width | 15 mm |

Bearing roller diameter | 7 mm |

The number of the rollers | 11 |

Contact angle | 0 rad |

Outer-race defect | 0.3 × 0.25 mm (width × depth); Early stage |

Inner-race defect | 0.3 × 0.25 mm (width × depth); Early stage |

Rolling element defect | 0.3 × 0.25 mm (width × depth); Early stage |

Diagnosis sensitivity for condition diagnosis.

<0.85 | <80% | Low |

0.85–1.30 | 80%–90% | Slightly low |

1.30–1.65 | 90%–95% | Middle |

1.65–2.33 | 95%–99% | High |

>2.33 | >99% | Very high |

Values of DR and DI before projection.

_{1} |
_{2} | |||||
---|---|---|---|---|---|---|

| ||||||

State | μ_{p1} |
σ_{p1} |
DI_{P1} (DR_{P1}) |
μ_{p2} |
σ_{p2} |
DI_{P2} (DR_{P2}) |

I | 2.38 | 0.35 | 1.12 (86.9%) | 0.72 | 0.17 | 1.19 (87.3%) |

R | 3.12 | 0.56 | 0.435 | 0.168 |

Values of DR and DI after projection.

_{1} |
_{2} | |||||
---|---|---|---|---|---|---|

| ||||||

State | μ_{ssp1} |
σ_{ssp1} |
DI_{ssp1} (DR_{ssp1}) |
μ_{ssp2} |
σ_{ssp2} |
DI_{ssp2} (DR_{ssp2}) |

I | 3.99 | 0.37 | 2.34 (99.04%) | 1.025 | 0.0022 | 2.25 (98.8%) |

R | 5.13 | 0.32 | 1.032 | 0.0022 |

DI values of SSPs for each sequential diagnosis step.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
_{7} |
_{8} | |
---|---|---|---|---|---|---|---|---|

For first step | ||||||||

N:O | 3.11 | 1.43 | 2.10 | 4.93 | 9.48 | 7.11 | ||

N:I | 2.20 | 1.11 | 2.39 | 2.76 | 2.72 | 2.56 | ||

N:R | 3.37 | 0.77 | 1.06 | 1.23 | 2.27 | 1.06 | ||

For second step | ||||||||

O:I | 0.70 | 0.88 | 2.05 | 2.52 | 3.31 | 2.31 | ||

O:R | 2.41 | 1.56 | 0.80 | 1.04 | 1.00 | 0.80 | ||

For third step | ||||||||

I:R | 1.22 | 1.63 | 1.03 | 0.70 | 1.45 | 1.11 |

Training data for first step of sequential diagnosis.

_{1} |
_{5} |
|||
---|---|---|---|---|

1.245 | 0 | 0 | 0 | 1 |

2.76 | 38.7 | 0.5 | 0.02 | 0.48 |

5.35 | 0.665 | 0.333 | 0.38 | 0.287 |

4.52 | 6.18 | 0 | 1 | 0 |

… | … | … | … | … |

Training data for second step of sequential diagnosis.

_{1} |
_{5} |
|||
---|---|---|---|---|

3.15 | 6.17 | 0 | 0 | 1 |

4.13 | 6.42 | 0.333 | 0.333 | 0.333 |

6.08 | 6.5 | 0.978 | 0 | 0.022 |

5.04 | 15.1 | 0 | 1 | 0 |

… | … | … | … | … |

Training data for third step of sequential diagnosis.

_{1} |
_{2} |
|||
---|---|---|---|---|

2.5 | 1.01 | 0 | 0 | 1 |

3.835 | 1.021 | 0.75 | 0 | 0.25 |

5.332 | 1.021 | 0.333 | 0.333 | 0.333 |

5.66 | 1.032 | 0.057 | 0.943 | 0 |

… | … | … | … | … |

Verification result of first step.

_{1} |
_{5} |
||||
---|---|---|---|---|---|

3.025 | 1.854 | 0.811 | 0.112 | 0.105 | N |

2.882 | 1.615 | 0.796 | 0.157 | 0.138 | N |

4.260 | 26.05 | 0.0002 | 0.8405 | 0.1691 | B |

4.961 | 15.53 | 0.0002 | 0.8561 | 0.1462 | B |

1.579 | 30.56 | 0.036 | 0.0928 | 0.9075 | U |

… | … | … | … | … | … |

Verification result of second step.

_{1} |
_{5} |
||||
---|---|---|---|---|---|

6.10 | 6.33 | 0.8607 | 0.0021 | 0.1511 | O |

6.104 | 6.84 | 0.9105 | 0.0059 | 0.1023 | O |

4.22 | 18.44 | 0.1265 | 0.8365 | 0.0732 | I or R |

5.36 | 9.93 | 0.0671 | 0.8012 | 0.1747 | I or R |

2.01 | 25.5 | 0.1011 | 0.0936 | 0.8228 | U |

… | … | … | … | … | … |

Verification result of third step.

_{1} |
_{2} |
||||
---|---|---|---|---|---|

3.81 | 1.025 | 0.9541 | 0.0035 | 0.1231 | I |

4.09 | 1.029 | 0.9027 | 0.0071 | 0.1096 | I |

5.26 | 1.031 | 0.0082 | 0.8974 | 0.1217 | R |

4.73 | 1.033 | 0.0047 | 0.9127 | 0.1056 | R |

6.69 | 0.83 | 0.0767 | 0.0458 | 0.9279 | U |

… | … | … | … | … | … |