The optimal Mexican hat wavelet filter de-noising method based on cross-validation method
Introduction
In mechanical condition monitoring and fault diagnosis, transient signals always contain important information of the monitored objects. Vibration analysis is the best-known technology applied in mechanical condition monitoring, especially for rotating equipments. Signals under considerations are known to be non-stationary, for which the signal parameters are time-varying [1], [2], [3]. For spectral analysis of such type signals, the time–frequency analysis techniques have been widely used [4], [5]. However, if the signal feature components’ energy is low, the amplitude of the interfering noise will be higher than the signal feature components, and at the same time the frequency spectrum of the noise and signal will be mixed together. In this situation, traditional time–frequency analysis methods cannot separate the useful signal features from the noise jamming effectively. Therefore, it is important to pre-process raw signals before analysis since the raw data contain some redundant information.
Wavelet analysis, which is the most popular one for non-stationary signal analysis, overcomes the drawbacks of other techniques by means of analytical functions that are local in both time and frequency. Wavelet transform (WT) method is widely used in mechanical signal de-noising process [6], [7], [8]. However, traditional wavelet de-noising method has some difficulties in the analysis process, such as the selection of the mother wavelet function, the decomposition level of signal, the order of the mother wavelet function, etc. Some papers used wavelet de-noising methods in mechanical fault de-noising and feature extraction, but had not given some useful theoretical methods to ensure the decomposition level or the order of mother wavelet function [9], [10]. Most researchers used comparison methods to choose some optimal mother wavelet functions and decomposition level. In this situation, it is a waste of time and energy to do a lot of contrastive experiments [11]. Wavelet transform can be regarded as the inner product of a time domain signal with the translated wavelet-base function. Therefore, the WT can be regarded as the filter process to the signal, by which the noise in the signal can be de-noised effectively. The WT resulting coefficients reflect the different features of the noise and the useful signal, which can be separated by the filter process. Therefore, by choosing suitable wavelet function and taking the continue wavelet transform (CWT) process, it is feasible to de-noise the interfering noise in the raw signals and keep the effective feature components.
This paper chooses Mexican hat wavelet, which is in shape similar to the mechanical shocking signal, as the mother wavelet function in CWT process. The parameters of the improved Mexican hat wavelet and the wavelet transform scale factor are optimized by the cross-validation method (CVM). The paper is structured as follows. An introduction is given in Section 1. The principle of the Mexican hat wavelet filter de-noising method is discussed in Section 2. The parameters of the improved Mexican hat wavelet are optimized in Section 3. The application of the proposed de-noising method in the gear fault experiment is presented in Section 4. And some conclusions are drawn in Section 5.
Section snippets
Principle of Mexican hat filter de-noising
The wavelet analysis results are series of wavelet coefficients, which indicate the comparability between the signal and the particular wavelet. In order to extract the fault features of the signal more effectively, an appropriate wavelet base function should be selected [12], [13], [14]. The corresponding wavelet family consists of a series of daughter wavelets, which are generated by dilation and translation operations from the mother wavelet and shown as follows:
Parameters confirmation
In the above analysis, the filter de-noising process by CWT has two main steps: one is to select the mother wavelet function parameters m and n, the other is to ensure the scale factor in the CWT process. For the improved Mexican hat wavelet we chosen, we calculate the parameters m and n by cross-validation method (CVM) [20].
Experimental analyses
Experiments were conducted using a 2 hp reliance electric motor, and the acceleration data were collected at locations near to and remote from the motor bearings. Motor bearings were seeded with faults using electro-discharge machining. Faults ranging from 0.007 in. in diameter to 0.040 in. in diameter were introduced separately at the inner raceway, outer raceway and the rolling ball. Faulted bearings were reinstalled into the test motor and vibration data were recorded for motor loads of 0–3
Conclusions
Aimed at the shortage of the traditional wavelet de-noising methods, an optimal Mexican hat wavelet filter de-noising method is proposed in this paper. The mother wavelet shape parameters are optimized by the cross-validation method to be more similar to the mechanical vibration signals. The scale factor in the CWT process is also optimized using the cross-validation method in circle manner. The experimental analysis and comparison to the other two traditional wavelet de-noising methods proved
Acknowledgments
This research was supported by the Scientific research support project for teachers with doctor′s degree, Jiangsu Normal University, China (Grant no. 11XLR15), the National Science Foundation for Young Scientists of China (Grant no. 51075347).
W.Y. Liu was born on 8 March 1984, in Henan, China. He received the PhD degree in Mechatronic Engineering in 2010 from Chongqing University. He joined the School of Mechanical and Electrical Engineering, Jiangsu Normal University, as a lecturer in 2011. His research interests are in the areas of condition monitoring and fault detection of wind turbines, time–frequency analysis, and modeling of non-stationary signals, fault diagnosis, measurement technology and instrument.
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W.Y. Liu was born on 8 March 1984, in Henan, China. He received the PhD degree in Mechatronic Engineering in 2010 from Chongqing University. He joined the School of Mechanical and Electrical Engineering, Jiangsu Normal University, as a lecturer in 2011. His research interests are in the areas of condition monitoring and fault detection of wind turbines, time–frequency analysis, and modeling of non-stationary signals, fault diagnosis, measurement technology and instrument.
J.G. Han was born on September 1963, in Heilongjiang, China. He received the PhD degree in Mechatronic Engineering in 2001. He joined the School of Mechanical and Electrical Engineering, Jiangsu Normal University, as a professor. His research interests are in the areas of mechanical drive dynamic analysis and CAD.