# Adaptive Parameter Identification Based on Morlet Wavelet and Application in Gearbox Fault Feature Detection

- Shibin Wang
^{1}, - Z. K. Zhu
^{1}Email author, - Yingping He
^{1}and - Weiguo Huang
^{1}

**2010**:842879

https://doi.org/10.1155/2010/842879

© Shibin Wang et al. 2010

**Received: **26 July 2010

**Accepted: **22 October 2010

**Published: **24 October 2010

## Abstract

Localized defects in rotating mechanical parts tend to result in impulse response in vibration signal, which contain important information about system dynamics being analyzed. Thus, parameter identification of impulse response provides a potential approach for localized fault diagnosis. A method combining the Morlet wavelet and correlation filtering, named Cyclic Morlet Wavelet Correlation Filtering (CMWCF), is proposed for identifying both parameters of impulse response and the cyclic period between adjacent impulses. Simulation study concerning cyclic impulse response signal with different SNR shows that CMWCF is effective in identifying the impulse response parameters and the cyclic period. Applications in parameter identification of gearbox vibration signal for localized fault diagnosis show that CMWCF is effective in identifying the parameters and thus provides a feature detection method for gearbox fault diagnosis.

## Keywords

## 1. Introduction

Rotating machines play an important role in many industrial applications, such as aircraft engines, automotive transmission systems, and wind power generation. Most of the machinery was operated by means of gears and other rotating parts, which may develop faults. The study of fault diagnosis of rotating machine by fault feature detection from vibration signals has thus attracted more and more attention over the past decade.

Gearbox, as an important mechanism for transmitting power or rotation, is widely used in industrial applications. The occurrence of impulse response in gearbox vibration signals usually means that there exist mechanical defects or faults. Most gear faults are due to gear damage, such as tooth wear, cracks, scoring, spalling, chipping, and pitting [1, 2]. With such flaws existing on gears, progressive damage will occur and ultimately result in gear tooth breakage, which may cause significant economic loss. For gearbox fault diagnosis, therefore, it is very important to extract the information of impulse response from vibration signals.

So far, different techniques have been proposed to analyze the vibration signal for fault diagnosis, such as time-frequency/time-scale methods, empirical mode decomposition (EMD), and matching pursuit (MP). Time-frequency distribution is a three-dimensional time, frequency, and amplitude representation of a signal, which is commonly used to diagnose faults in mechanical systems because the time-frequency distribution can accurately extract the desired frequencies from a nonstationary signal [3–5]. The time-frequency distributions are linear or bilinear. The former includes the short time Fourier transform, which provides constant resolution for all frequency since it uses the same window for the analysis of the entire signal. The latter includes the Wigner-Ville distribution, the Choi-Williams distribution, and improved ones. There is no doubt that the Wigner-Ville distribution has good concentration in the time-frequency plane. However, even if support areas of the signal do not overlap each other, interference terms will appear to mislead the signal analysis. Time-scale methods often refer to wavelet transform. In wavelet analysis, a signal is analyzed at different scales or resolution: larger time and smaller scale window is used to look at the approximate stationarity of the signal and smaller time and larger scale window at transients. Reference [6] summarizes the application of the wavelet in machine fault diagnosis, including the fault feature extraction, the denoising and extraction of the weak signals, and the system identification.

EMD is an adaptive decomposition method proposed by Huanget al.[7], which in essence extracts the intrinsic oscillation of the signal being analyzed through their characteristic time scales (i.e., local properties of the signal itself) and decomposes the signal into a number of intrinsic mode functions (IMFs), with each IMF corresponding to a specific range of frequency components contained within the signal. Because it still has some shortcomings when it comes to calculating instantaneous frequency [8] or in some cases it may reveal plausible characteristics due to the mode mixing [9, 10], it is untenable in effective application in impulse detection and analysis.

Matching pursuit algorithm, a greedy algorithm that chooses a waveform that is the most adapted to approximate part of the signal at each iteration, is effective in analyzing impulse response signals; however, the excessive computational cost limits its engineering applications [11]. Correlation filtering, enlightened from matching pursuit, is used based on Laplace wavelet to identify the parameters of impulse response by calculating the maximal correlation value, which is employed by Freudinger et al. to identify the modal parameters of a flutter for aerodynamic and structural testing [12]. Similar efforts were made by Zi et al. for the identification of the natural frequency of a hydrogenerator shaft and the wear fault diagnosis of the intake valve of an internal combustion engine [13]. Qi et al. employed Laplace wavelet correlation filtering together with empirical mode decomposition to identify modal parameters [14]. An integrated approach, consisting of empirical mode decomposition, Laplace wavelet correlation filtering, and wavelet finite element model, proposed by Dong et al. for rotor crack detection, was effective in identifying the position and the depth of different cracks [15].

Laplace wavelet correlation filtering is effective in detecting a single transient impulse response. However, localized defects in rotating mechanical parts tend to result in multiple impulse responses, which are generally cyclic impulse responses. Considering that the waveform of Morlet wavelet is in shape similar to transient vibration caused by gearbox localized defects [16, 17] and cyclostationarity matches the key feature of the gearbox vibration [18, 19], Cyclic Morlet Wavelet Correlation Filtering (CMWCF) is thus proposed, which, based on correlation filtering, constructs the cyclic Morlet wavelet and identify both the impulse response parameters and the cyclic period for diagnosed gear fault.

The remainder of the paper is organized as follows. In Section 2, the basic theoretical background concerning CMWCF is introduced. Section 3 gives a simulation study and analysis to verify the proposed method. Section 4 applies the method in gearbox transient feature detection by parameter identification for fault diagnosis. Finally, conclusions are drawn in Section 5.

## 2. Adaptive Parameter Identification Based on Morlet Wavelet

In this section, a method of adaptive parameter identification of Morlet wavelet based on correlation filtering is presented. Using correlation filtering, the parameters of Morlet wavelet are firstly identified to detect the impulse response. Secondly, cyclic Morlet wavelet is constructed to detect the cyclic period between adjacent impulse responses. The proposed method is suitable for not only identifying the parameters of the impulse response but also detecting the cyclic period.

### 2.1. Morlet Wavelet and Parametric Representation

Morlet wavelet is a cosine signal that decays exponentially on both the left and the right sides. This feature makes it very similar to an impulse. It has been used for impulse isolation and mechanical fault diagnosis [16, 17].

where the parameter vector determines the wavelet properties. These parameters are denoted by frequency , damping ratio and time index , respectively.

*dictionary*shown as follows:

and each item in the dictionary is called an *atom*.

### 2.2. Correlation Filtering (CF)

where is the time of the peak value of in the whole time domain.

### 2.3. Cyclic Morlet Wavelet Correlation Filtering (CMWCF)

where
is the time interval between two adjacent cyclic Morlet wavelet atoms, named *cyclic period*. Then, the parameter vector
determines the cyclic Morlet wavelet properties.

where
, associated with the maximum of
, is *cyclic period*.

- (i)
establish Morlet wavelet dictionary;

- (ii)
find optimal Morlet wavelet using correlation filtering based on maximal correlation coefficient criterion;

- (iii)
construct cyclic Morlet wavelet given by (10) obtained in step 2;

- (iv)
find cyclic period using CMWCF based on maximal correlation coefficient criterion.

## 3. Simulation Signal Test

where the frequency , the damping ratio , the time index , and the cyclic period . Obviously, is a real periodic cyclic impulse responses signal. The signal is white noise weight by , and the sampling frequency is 200 Hz in time range .

Figure 1(a) gives the waveform of the simulation signal without noise and Figure 1(b) with noise. Figure 1(c) represents the correlation value , whose peak value locates at one impulse. Figures 1(d) and 1(e) indicate the modal information of frequency and damping ratio parameters revealed from the peak correlation at each time . We obtained frequency and damping ratio which are exactly equal to simulation values ( , ). Because of the multi-impulse the time index is not equal to shown in Figure 1(c). Using the results obtained by correlation filtering, the cyclic Morlet wavelet is constructed. Then, the correlation value , shown in Figure 1(f), is obtained by CMWCF, and the cyclic period associated with is identified, which is also equal to the simulation value ( ). Figure 1(g) gives the comparison between the reconstructed cyclic Morlet wavelet with the obtained results and the simulation signal. To see more clearly, we parallelly move the curve of the reconstructed impulse response. The consistency between them can be obviously seen, so it can be drawn that the proposed method is effective in identifying the cyclic period between adjacent impulses.

The results of CMWCF when increasing the noise amplitude.

0 | 0.4472 | 5.00 | 0.010 | 5.00 | 1.0000 | 2.000 | |

0.1 | 10.0120 | 0.4388 | 5.00 | 0.010 | 3.00 | 0.9522 | 2.000 |

0.2 | 3.9053 | 0.3947 | 5.00 | 0.010 | 5.00 | 0.8454 | 2.000 |

0.3 | 0.6009 | 0.3463 | 5.00 | 0.010 | 3.00 | 0.7427 | 2.000 |

0.4 | −2.3101 | 0.2976 | 5.00 | 0.010 | 7.00 | 0.6253 | 2.000 |

0.5 | −4.1144 | 0.2883 | 5.01 | 0.008 | 3.00 | 0.5378 | 2.000 |

0.6 | −5.6627 | 0.2356 | 5.10 | 0.012 | 4.95 | 0.4167 | 2.000 |

0.7 | −7.0026 | 0.2031 | 4.93 | 0.015 | 7.05 | 0.3846 | 1.995 |

0.8 | −7.8508 | 0.2010 | 4.93 | 0.014 | 6.95 | 0.3676 | 1.995 |

0.9 | −9.0012 | 0.1812 | 5.15 | 0.012 | 6.95 | 0.2688 | 2.010 |

1.0 | −10.0057 | 0.1718 | 4.86 | 0.012 | 6.95 | 0.2851 | 1.990 |

1.1 | −10.8197 | 0.1421 | 4.99 | 0.015 | 0.95 | 0.2802 | 2.000 |

1.2 | −11.5463 | 0.1448 | 4.99 | 0.016 | 1.05 | 0.2695 | 2.000 |

## 4. Application in Gearbox Fault Feature Detection

Working parameters of the third speed gears.

The third speed gears | Constant meshing gears | |||
---|---|---|---|---|

Driving gear | Driven gear | Driving gear | Driven gear | |

Number of teeth | 25 | 27 | 24 | 32 |

Rotating period (s) | 0.05 | 0.054 | 0.04 | 0.03 |

Rotating frequency (Hz) | 20 | 18.5 | 25 | 33.3 |

Meshing frequency (Hz) | 500 | 640 |

In order to identify the cyclic period, the reconstructed Morlet wavelet is used to construct cyclic Morlet wavelet. The correlation coefficient between constructed cyclic Morlet wavelet and the vibration signal under different parameter period is given in Figure 3(d), in which the maximal correlation value is marked and the corresponding period is s. The comparison between the reconstructed cyclic Morlet wavelet and the original vibration signal is given in Figure 3(e), in which, to see more clearly, the curve of the reconstructed one is also parallelly moved. Obviously, as shown in Table 2, the identified cyclic period is consistent with the rotating period of the third speed driving gear. That is to say, the proposed method is effective in identifying the characteristic parameters.

## 5. Conclusions

The cyclic Morlet wavelet correlation filtering (CMWCF) method proposed represents an attempt in the direction of parameter identification and feature detection for fault diagnosis. Both the parameters of the Morlet wavelet associated with the maximal correlation value and the cyclic period are effective in feature detection of the impulse response.

The simulation study demonstrates that the proposed method is effective in identifying parameters of impulse, including frequency, damping ratio, and the time index, and is especially sensitive to the cyclic period. The gearbox application also demonstrates the fact that the method has the capability of parameter identification.

In conclusion, the other gearbox applications have not yet been provided in the paper; however, it conforms that CMWCF provides a feature detection method for gearbox fault diagnosis. Furthermore, the method has the potential applicability for monitoring other rotating mechanical components such as bearings and rotors.

## Declarations

### Acknowledgments

This research is supported partly by the Natural Science Foundation of China (no. 50905021) and the Natural Science Foundation of Jiangsu Province (no. BK2010225).

## Authors’ Affiliations

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