A gear fault diagnosis using Hilbert spectrum based on MODWPT and a comparison with EMD approach

Abstract

When gear fault occurs, the vibration signals always display non-stationary behavior. Therefore time-frequency analysis has become the well-accepted technique for vibration-based gear fault diagnosis. This paper presents the application of a new time-frequency signal processing technique, the Hilbert spectrum based on the maximal overlap discrete wavelet packet transform (MODWPT), to the analysis of simulation signals and gear fault vibration signals measured by the acceleration sensor fixed on the bearing house. As long as the decomposition scale and disjoint dyadic decomposition are chosen suitably, the original signal could be decomposed into a set of monocomponent signals whose instantaneous amplitude and instantaneous frequency own physical meaning. After the instantaneous amplitude and instantaneous frequency of each monocomponent signal are calculated by using MODWPT, the corresponding Hilbert spectrum could be obtained by assembling the instantaneous amplitude and instantaneous frequency. The simulation and practical application examples show that the Hilbert spectrum base on the MODWPT is superior to another competing method, namely, EMD (empirical mode decomposition)-based method, which has been widely used in the gear fault diagnosis.

Introduction

One of the most common failures in rotating machines is the gear failure. An unexpected fault of the gear may cause significant economic losses. It is, therefore, very important to detect gear failure [1], [2]. Currently, fault diagnosis in gear is often dealt with by the following three methods: acoustic signal analysis, debris monitoring and vibration analysis [3], [4], among which the vibration-based diagnosis has become the well-accepted detection technique because of ease of measurement [5]. In fact, when gear fault occurs, the vibration signals always display non-stationary behavior. Therefore how to extract the fault characteristic information from the non-stationary vibration signals is the crux of the gear fault diagnosis [6]. Wavelet analysis has been the most popular analysis technique for non-stationary signal processing because it is capable of providing both time information and frequency information simultaneously [7]. However, once the wavelet bases and the decomposition scales are determined, the results of wavelet transform would be the signal under a certain scale, whose frequency components related only to the sample frequency not the signal itself. Therefore, wavelet analysis is a non-adaptive data analysis method in nature and it is hard to ensure that instantaneous amplitude and instantaneous frequency obtained by wavelet-based approach has clear physical significance [8]. Furthermore, the frequency resolution of the time-frequency distribution obtained by wavelet transform change with the frequencies of the signal, that is, at low frequencies the frequency resolution is better while at high frequencies the frequency resolution is poorer.

In 1998, a new signal analysis method, namely Hilbert–Huang transform (HHT) developed by Huang, is put forward [9]. Hilbert–Huang transform includes empirical mode decomposition (EMD) and associated Hilbert transform. EMD method is based on the local characteristic time scale of signal and could decompose the complicated signal into a number of intrinsic mode functions (IMFs). By analyzing each resulting IMF component that involves the local characteristic of the signal, the characteristic information of the original signal could be extracted more accurately and effectively. In addition, EMD can be regard as a special filter whose bandwidth and central frequency change with the signal itself, therefore, EMD is a self-adaptive signal processing method that can be applied to non-linear and non-stationary process perfectly [10]. However, different problems still exist in EMD and associated Hilbert transform. For example, the spline fitting that is the essential step in EMD and the basis for the corresponding Hilbert spectrum analysis needs improvements. In addition, drawbacks such as the mode mixing, end effects and so on need future attention [11], [12]. With the further study of these problems, the applications of EMD and associated Hilbert spectrum analysis in practice would be more widely. Although so far, all these problems are still underway, Hilbert–Huang transform provide new sight for non-stationary signal processing, that is, individual component signal whose instantaneous amplitude and instantaneous frequency own well-defined physical significance can be obtained by choosing suitable signal decomposition method, furthermore, it is possible to achieve the full time-frequency distribution. So it is necessary to look for an appropriate signal decomposition method.

To avoid the deficiencies in Hilbert–Huang transform, wavelet-based analysis method comes back to our sight once more. Discrete wavelet transform (DWT) has been widely used in gear fault diagnosis due to its good feature. However, DWT algorithm requires the sample size to be exactly a power of 2 for the full transform because of the downsampling step in the DWT. To overcome this disadvantage, naturally we would like to remove the downsampling step in the DWT. Therefore a new algorithm, namely, maximal overlap discrete wavelet transform (MODWT) is developed. However, both DWT and MODWT have good frequency resolution at low frequencies but very poor frequency resolution at high frequencies, naturally maximal overlap discrete wavelet package transform (MODWPT) is considered to replace MODWT for MODWPT gives better frequency resolution [13]. As long as the decomposition scale and disjoint dyadic decomposition are chosen suitably, the complicated signal could be separated into a number of single component signal whose instantaneous amplitude and instantaneous frequency own physical meaning. Different from ordinary DWT, MODWPT not only ensure the important property of circular shift equivariance, but also has no restriction about sample size. In addition each individual component signal obtained by MODWPT has desirable statistical characteristic, which are desirable properties to deal with non-stationary time series in practice [14]. Furthermore, once instantaneous amplitude and instantaneous frequency are calculated, a completed time-frequency distribution could be got, by which the gear working condition can be determined. In this paper, Maximal overlap discrete wavelet packet transform is introduced into various simulation signals analysis and the analysis results demonstrate the validity of Hilbert spectrum based upon MODWPT. In addition, a comparison with another techniques, EMD, is also made. Furthermore, we apply the methodology to analysis of gear vibration data measured under different operating conditions. We also compare the results to another competing method, the EMD-based approach. The analysis result shows that Hilbert spectrum based upon MODWPT is seen to produce better results.

This paper is organized as follows. A brief summary of the Hilbert spectrum based on MODWPT is given in Section 1. In Section 2 we give various simulation signals analysis showing that the Hilbert spectrum based on MODWPT gives excellent results. The analysis results from gear fault vibration signal with crack fault or broken teeth are given in Section 3, which show that superior results can be obtained by replacing the EMD method by MODWPT approach. Finally, we offer the conclusion in Section 4.