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Given the problems in intelligent gearbox diagnosis methods, it is difficult to obtain the desired information and a large enough sample size to study; therefore, we propose the application of various methods for gearbox fault diagnosis, including wavelet lifting, a support vector machine (SVM) and rule-based reasoning (RBR). In a complex field environment, it is less likely for machines to have the same fault; moreover, the fault features can also vary. Therefore, a SVM could be used for the initial diagnosis. First, gearbox vibration signals were processed with wavelet packet decomposition, and the signal energy coefficients of each frequency band were extracted and used as input feature vectors in SVM for normal and faulty pattern recognition. Second, precision analysis using wavelet lifting could successfully filter out the noisy signals while maintaining the impulse characteristics of the fault; thus effectively extracting the fault frequency of the machine. Lastly, the knowledge base was built based on the field rules summarized by experts to identify the detailed fault type. Results have shown that SVM is a powerful tool to accomplish gearbox fault pattern recognition when the sample size is small, whereas the wavelet lifting scheme can effectively extract fault features, and rule-based reasoning can be used to identify the detailed fault type. Therefore, a method that combines SVM, wavelet lifting and rule-based reasoning ensures effective gearbox fault diagnosis.

With the continuous development of modern industrial large-scale manufacturing and progress in the sciences and technology, machinery, as the major production tool, tends to be large, complex, speedy, continuous and automatic to maximally improve production efficiency and product quality. Machine production efficiency is increasing, and their mechanical structures are becoming more complicated. Once a machine breaks down, the whole production process must stop, which can lead to enormous economic losses and serious personnel injuries. Therefore, reliable and safe equipment operation is required. It has been proved that constantly monitoring equipment conditions and effectively implementing fault diagnosis techniques are the major preventive measures that guarantee safe equipment operation by detecting faults at an early stage to avoid major and fatal accidents.

An intelligent machine fault diagnosis system has been developed rapidly in the past decades by successfully applying new theories. Meanwhile, the large scale and complexity of modern machines, together with the urgent needs of real-time and automatic machine fault diagnosis, have driven the transformation of fault diagnosis technology from artificial diagnosis to intelligent diagnosis.

Among all kinds of intelligent diagnosis methods, pattern recognition based on an Artificial Neural Network (ANN) has been widely used because of its power in self- organizing, unsupervised-learning, and nonlinear pattern classification [

It is well known that the bottleneck of fault diagnosis is a lack of fault samples, which provides SVM a bright application future in machine fault diagnosis. Jack has used SVM to detect the rolling bearing condition [

The wavelet transform is a breakthrough in signal processing technology in the past two decades [

Rule-based reasoning (RBR) is a traditional intelligent diagnosis method. Experience and knowledge will be represented in the form of rules which will be saved in knowledge base, and the reasoning mechanisms will be used to get the diagnosis conclusions with the rules. Considering the engine wear process, Peilin Zhang, Bing Li and Shubao Liang

This study presents a method that combines wavelet lifting, an SVM and rule-based reasoning to diagnose gearbox faults. Gearbox vibration signals are initially processed by wavelet packet decomposition. Then, the energy coefficients of each frequency band are calculated and used as input vectors to the SVM to recognize normal and faulty gearbox patterns. Precise analysis from the wavelet lifting scheme was then utilized to obtain the machine fault feature frequency. Finally, based on the fault feature frequency, the existing diagnostic knowledge and rules were used for logical reasoning to establish a knowledge base to identify fault types. The diagnosis scheme based on an SVM, wavelet lifting and rule-based reasoning methods is shown in

A support vector machine is based on minimizing structural risks. Its algorithm was initially designed for two-class classification. In the field of machine faults, an SVM can simply determine whether there is a fault.

The SVM method is developed by determining the optimal separating hyperplane in for linear separability. The optimal separating hyperplane is not only able to classify all training samples, but also maximizes the distance between the separating hyperplane and points in training samples that are closest to the separating plane.

The fault training sample set is (_{i}, y_{i}^{d}

The solution of this optimal problem is the saddle point of the Lagrange function, and the optimal discriminant function is obtained as:

Nonlinear problems can be converted to high-dimensional linear problems with a nonlinear transformation. In high-dimensional space, only the inner-product computation is needed, which can be obtained by using functions in the original low dimensions. According to the relative principles of functional analysis, if one kernel function K(_{i}

The kernel functions commonly used are the RBF kernel, MLP kernel and Multinomial kernel. However, the RBF described as

Using multiresolution analysis and the wavelet packet technique, signals can be decomposed into different frequency bands. Analyzing signals in these frequency bands is called frequency bandwidth analysis. Usually, based on the frequency range where signals of interest are located, users can decompose signals to a certain scale and obtain information from the corresponding frequency bands. Additionally, signals in different frequency bands can be further subject to statistical analyses to obtain feature vectors that represent signal characteristics. Analyzing the signal energy in different frequency bands is called frequency band energy analysis. It is characterized by wide-frequency-range responses when processing nonstationary, transient signals with higher frequency resolution at low frequency and higher time resolution at high frequency. Compared to the FFT, it contains a great deal of non-stationary and nonlinear diagnostic information.

The theoretical basis for wavelet frequency bandwidth analysis is Parseval’s theorem. The time domain energy of

Thus, vibration signals are decomposed into independent frequency bands of different levels by using a conjugate quadrature filter. Not only are these decomposed signals in quadrature to each other in agreement with the law of conservation of energy, but they also contain a large quantity of non-stationary and nonlinear diagnostic information compared to an FFT. Therefore, the signal energy in every frequency band can be used as a feature vector to represent the operation condition of the machine and is useful for machine fault diagnosis.

The procedure for feature vector extraction using wavelets is the following:

Step 1: process vibration signals for wavelet packet decomposition;

Step 2: reconstruct each wavelet packet coefficient, and extract signals in different frequency ranges;

Step 3: acquire _{j}_{jk}

Step 4: use the percentile ratio of the signal energy _{j}

The wavelet lifting transform includes two stages: decomposition and reconstruction. Decomposition consists of splitting, predicting and updating. As shown in

Split: the data series {_{o}_{e}

Predict: suppose _{e}_{o}

Then, the detail signal series is

Update: assume _{e}

Then, the approximation signal series is

Reconstruction of the wavelet lifting is the reverse process of decomposition, and is composed of recovery prediction, recovery updating and merging:

The reconstruction signal s is obtained by merging the odd and even sample series, as shown in

Because of the differences in machine working conditions, vibration signals can provide significant qualitative information. However, there is no one-to-one correspondence between fault features and conclusions because of the complexity of the machines. Therefore, in the diagnosis system used in this study, the fuzzy reasoning strategy was used to perfect rule-based diagnosis methods. The knowledge base is represented by the production rule. The fundamental ideas of fuzzy reasoning are as follows:

Suppose G is a set of a fuzzy proposition, fuzzy characteristics and a fuzzy relation. For simplicity, the fuzzy proposition, fuzzy characteristics and fuzzy relation are together called the fuzzy assertion. Then, a piece of factual information can be presented by a binary group (

One fault symptom may correspond to multiple causes, while one fault cause may also correspond to multiple fault symptoms. Therefore, the relationship between cause and symptom is complicated.

For proper diagnosis, the membership degree between fault causes and fault symptoms needs to be pre-determined. The value of this membership degree can be obtained based on expert experience or theoretical research. Based on years of experience in our lab in field diagnosis, we summarize the rules and establish the knowledge base.

The fuzzy rules of the knowledge base were used for fault cause reasoning to determine the reason for the faults. Then, according to the typical gearbox fault features, the rules of the knowledge base are constructed as shown in _{r}_{m}_{q}

For gearbox fault diagnosis, a fuzzy matrix was established:

In the fuzzy matrix for gearbox fault diagnosis, rows represent sets of fault causes, columns represent sets of faults symptoms, and the values in the matrix represent the membership degree between fault symptoms and causes.

When implementing fault diagnosis with the fuzzy reasoning approach, the fuzzy matrix

The final diagnosis result includes the vectors with relatively large values upon conclusion of the diagnosis. If there are several relatively large values, the existing fault symptom should be considered for the final conclusion.

In this study, SVM, wavelet lifting and diagnosis rules were used to analyze the vibration acceleration signal, according to a broken cog fault of the Z5 gear (tooth 31) in Shaft II of 22 gear-boxes of a high speed wire rolling mill. The gearbox transmission chain of a high-speed wire rolling mill is shown in

As seen in

As mentioned in Section 2.2, the “db10” wavelet was used to decompose the signal into three layers and the energy of each of the eight decomposed frequency bands _{j}_{j}

As shown in

The wavelet energy from the hourly data obtained in early June is set to class 1, which indicates normal conditions; the wavelet energy from data obtained when the rolling mills experienced a fault is set to class 2. Fifteen sets of data were used as SVM input for training. The test data included data from June and September, and each had 15 sets of data. Because gears have different crack patterns, data from September were identified as fault class 2 by the SVM and were significantly different from data in the normal condition.

In order to verify the effectiveness of wavelet lifting on data analysis, the field data obtained 74 days before the machine malfunction were analyzed. Through wavelet lifting, the original signals with the spectrum ranging from 0 to 2,500 Hz were decomposed at two levels, as shown in ^{0} is 0∼1,250 Hz, and that of ^{0} is 1,250∼2,500 Hz. Four different bands can be obtained after carrying out decomposition at level 2, among which the spectrum range of ^{1} is 0∼625 Hz, that of ^{1} is 625∼1,250 Hz, that of ^{2} is 1,250∼1,875 Hz, and that of ^{2}^{1}^{2}, and the detail coefficient of the wavelet decomposition at level 2 ^{2} are shown in

According to the rotational speed of motor, the frequency of all parts in a rolling mill can be calculated, among which _{m}^{1} (625∼1,250 Hz) and ^{2} (1,875∼2,500 Hz) respectively, after decomposition at level one and level two. Thus the wavelet lifting only at level one and two are decomposed without any other more decompositions in this paper.

The spectrum obtained from autoregressive spectrum analysis of signals in _{m}_{m}^{0} is 0∼1,250 Hz) obtained by reconstruction of the approximation signal after wavelet lifting decomposition at level 1 is shown in _{m}

Through wavelet lifting analysis, the gear meshing frequency (_{m}_{r}

Here, _{m}_{r}

In _{m}

The ratio can be obtained through the following calculation. The closer this ratio is to 1, the more consistent is the feature frequency of the monitoring spectrum with the calculating frequency of the fault part, and the more possibility there is of a part with some fault. The process of calculation is shown as follows: 1,139.277/1,140.00 = 0.999, 36.751/37.00 = 0.993.

By analyzing September data in _{m}_{m}_{r}

The calculation of the fault conclusion phasor B is shown below:

As calculated in the final result, the maximum value that the fault conclusion corresponds to is 0.874; thus, the corresponding fault cause can be confirmed to be a broken gear tooth. A broken cog was found in gearbox Z5 when the machine was disassembled in the field, which is consistent with the diagnosis conclusion.

Together with the fuzzy reasoning approach in the above fault, we have proved that, in fault diagnosis, the application of fuzzy logic can effectively present some fuzzy information and construct a fuzzy matrix; furthermore, the fault type can be effectively diagnosed with fuzzy reasoning.

In order to describe the ability of intelligent diagnosis method put forward in this paper, we carried out two cases and made comparative analysis with traditional method of Fourier transform.

At 14:00 on Nov. 30th, 2008, through Fourier Transform and wavelet lifting analysis of the original vibration signals, both of these two methods show that the gear meshing frequency of Z5/Z6 in Shaft III in the sixth rack of rolling mill in some factory was 45.9 Hz.

At 4:00 on Jan. 25th, 2008, through Fourier Transform and wavelet lifting analysis of the original vibration signals at low frequency, it is found that the shaft-frequency of Shaft II in gear-box in the second rack of rolling mill was 2.44 Hz.

Some conclusions can be obtained through comparison of the above two cases, and are summarized in

By using wavelet lifting, together with support vector machines and rule-based reasoning fault diagnosis methods, a real fault example of a broken cog in gearbox was analyzed and the following conclusions were drawn:

SVM is suitable for pattern recognition of problems with small sample sizes. In this study, two-class pattern recognition of actual gearbox faults was accomplished for diagnosis using SVM as the classifier. Based on the second generation wavelet packet feature extraction technology, by taking advantage of the fact that resonance occurs in the high frequency bands in the early stages of a fault, interference from noise signals from other frequency bands is effectively avoided through the decomposition and reconstruction of signals at high frequency bands; thus, fault feature extraction was achieved. According to the features of gearbox faults, a fuzzy production approach was applied to reveal fault rules, and rule-based reasoning was achieved through the fuzzy matrix. As demonstrated with actual data, this approach effectively overcomes the difficulty that some rules are difficult to present precisely.

Integrating different diagnosis technologies has become popular in intelligent diagnosis research. Taking advantage of each method in diagnosis inference such that the methods complement each other and create a hybrid diagnosis system is the goal for designing intelligent diagnosis technology.

This work is supported by National Natural Science Foundation of China (Grant No. 50705001), and partly supported by the Fundamental Research Funds for the Central Universities (Grant No. JD0904).

The principle of intelligent fault diagnosis based on SVM, wavelet lifting and RBR.

Decomposition and reconstruction process of the wavelet lifting. (a) decomposition of the wavelet lifting; (b) reconstruction of the wavelet lifting.

The gearbox transmission chain chart.

Comparison between waveform and spectrum with fault or fault-free. (a) time-domain waveform when the fault occurred; (b) spectrum when the fault occurred; (c) time domain waveform under normal condition; (d) spectrum under normal condition.

Wavelet energy. (a) normal wavelet energy profile; (b) faulty wavelet energy profile.

SVM Test Result.

Decomposition of the original signals (the spectrum range 0–2500Hz) through wavelet lifting.

The time domain signals obtained at 1:00 on September 1st, 2006 through wavelet lifting analysis. Approximation signal and detail signal after decomposition through wavelet lifting at level 2. (a) the approximation coefficient ^{1}; (b) the approximation coefficient ^{2}; (c) the detail coefficient ^{2}.

The time domain signals obtained at 1:00 on September 1st, 2006 through wavelet lifting analysis. (a) the spectrum range of ^{1} is 0∼625 Hz; (b) the spectrum range of ^{1} is 625∼1,250 Hz; (c) the spectrum range of ^{2} is 1,250∼1,875 Hz; (d) the spectrum range of ^{2} is 1,875∼2,500 Hz.

Low frequency spectrum (0∼1,250 Hz) of wavelet lifting level 1 decomposition (with the spectrum of ^{0} ranging from 0∼1,250 Hz).

Comparison on the spectrum of Fourier Transform and wavelet lifting analysis of the original signals. (a) time domain waveform of the original signals; (b) spectrum of the original signals after Fourier Transform; (c) time domain waveform of approximation signal reconstruction after decomposition of wavelet lifting at level three; (d) spectrum analysis of wavelet lifting.

The real case of broken cog fault of Z5 (25-tooth) helical gear.

Comparison on spectrum of Fourier Transform and wavelet lifting analysis of the original signals. (a) time domain waveform of the original signal; (b) spectrum after Fourier transform of the original signal; (c) time domain waveform of approximation signal reconstruction after decomposition to wavelet lifting at level three; (d) spectrum analysis of wavelet lifting.

The real case of a broken cog fault of bevel gear Z2 (tooth 35).

The rules of knowledge.

Shaft imbalance | Shaft misalignment | Shaft bending | Foundation deformation | Gear Tooth profile error | Gear wear | Gear tooth breakage | Damage of bearing | |
---|---|---|---|---|---|---|---|---|

_{r} |
0.8 | 0.4 | 0.7 | 0.4 | 0.2 | 0 | 0.2 | 0.4 |

_{r} |
0 | 0.3 | 0 | 0.2 | 0.1 | 0 | 0.1 | 0.1 |

_{r} |
0 | 0.2 | 0 | 0.2 | 0.1 | 0 | 0 | 0.1 |

_{r} |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 |

_{r} |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 |

_{m} |
0 | 0 | 0 | 0 | 0.2 | 0.4 | 0.2 | 0 |

_{m} |
0 | 0 | 0 | 0 | 0.1 | 0.2 | 0.1 | 0 |

_{m} |
0 | 0 | 0 | 0 | 0.1 | 0.2 | 0.1 | 0 |

_{q} >3 |
0 | 0 | 0 | 0 | 0.1 | 0.1 | 0.2 | 0.2 |

0.2 | 0 | 0.3 | 0 | 0.1 | 0.1 | 0 | 0 | |

0 | 0.1 | 0 | 0.2 | 0 | 0 | 0 | 0 | |

_{m} |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Ratios of different frequency components to the feature frequency.

_{m} |
1,140.00 | 1,139.277 | 0.999 |

_{r} |
37.00 | 36.751 | 0.993 |

Improving SNR and extracting features ability compared between Fourier transform (FT) and wavelet transform (WT).

Case 1 | Broken cog fault of helical cylindrical gear | Gear-box of blooming mill | Non-reduction | Mean-square difference |
General | Good |

Case 2 | Broken cog fault of bevel gear | Gear-box of blooming mill | Non-reduction | Mean-square difference |
General | Good |