Instantaneous energy density as a feature for gear fault detection

doi:10.1016/j.ymssp.2004.12.001

Mechanical Systems and Signal Processing

Volume 20, Issue 5, July 2006, Pages 1239-1253

https://doi.org/10.1016/j.ymssp.2004.12.001 Get rights and content

Abstract

In this work, energy-based features for gear fault diagnosis and prediction are proposed. The instantaneous energy density is shown to obtain high values when defected teeth are engaged. Three methods are compared in terms of sensitivity, reliability and computation effectiveness. The Wigner–Ville distribution is contrasted to the wavelet transform and the newly proposed empirical mode decomposition scheme. It is shown that all three methods are capable of a reliable prediction. An empirical law, which relates the energy content to the crack magnitude is established.

Introduction

Gear mechanisms are widely used in rotating machinery. For this reason, gear health monitoring has been the subject of intensive investigation and research. Among several other methods, vibration measurement and analysis is considered as the most general basis for fault detection. In the past years, the major effort has been applied to the development of reliable methods that would manifest the existence of faults. In this context, successful results have been mentioned by use of power cepstrum [1], wavelet analysis [2], application of the Wigner–Ville distribution [3], statistical methods [4] and cyclostationary processes theory [5]. However, practical condition monitoring systems fail to provide an early warning before final failure, or even lead to false alarms and unnecessary shut downs of the machines. The selection of features used for fault detection is probably the most critical step in the diagnostic procedure. In a number of studies, pattern recognition techniques are applied [6]. The progression of a fault is usually detected by changes in the pattern of the time–frequency distribution of the vibration signal.

Faults by their nature are transient events, causing a parceling of the energy of the vibration signal. It is thus expected that the energy density will significantly change at the moment that damaged teeth are engaged. Energy may be easily computed as it results from a quadratic form of the signal and subsequently used as a prognostic feature. Such an approach has been adopted by Dejie et al. [7] and Zheng et al. [8]. In [7] the Hilbert energy spectrum is used and in [8] the signal processing tool is the continuous wavelet transform (CWT). In this work, energy density is worked out from two well-known time–frequency analysis methods namely the Wigner–Ville (WVD) distribution and the CWT. In addition, the recently introduced empirical mode decomposition (EMD) scheme is also used to derive energy-based features. In 3 Gear damage prediction based on the Wigner–Ville distribution, 4 Local energy density derived from the scalogram, 5 Energy-based features derived from EMD, the derivation of instantaneous energy density is detailed. A comparative study is presented in Section 6. It is shown that whatever the method used is, energy associated with faults can be used for predicting the evolution of gear damage.

Section snippets

Experimental test rig

The experimental rig consists of two electrical machines, a pair of spur gears, a power supply unit with the necessary speed control electronics and the data acquisition system. Referring to Fig. 1, a DC machine of 1.5 kW rotates the pinion. The load is provided by an AC asynchronous machine, which is configured as a brake. The transmission ratio is $35 / 19 = 1.842$ , which means that an increase in rotational speed is achieved. The characteristics of the spur gear pair are given in Table 1. The

The Wigner–Ville distribution

The WVD is a very important quadratic-form time–frequency distribution with optimised resolution in both time and frequency domains. It was brought to attention by a series of papers from Claasen and Mecklenbrauker [9]. For a classification of time–frequency distributions see [10]. The WVD of a function f is computed by correlating the function with itself, the correlation being a product of the function at a past time with the function at a future time. The analytical form is $W_{xx} (t, f) = \int_{- \infty}^{+ \infty} x^{*} (t -$

The continuous wavelet transform

The CWT of a signal $f (t)$ is defined as the inner product of $f (t)$ with a wavelet function $ψ (t)$ , [11] $Wf (u, s) = 〈 f, ψ_{u, s} 〉 = \frac{1}{\sqrt{s}} \int_{- \infty}^{+ \infty} f (t) ψ^{*} (\frac{t - u}{s}) d t .$

The parameter $u$ is called translation and is normally associated with time. The parameter s is called scale and is proportional to the inverse of frequency. The time–frequency resolution of the transform depends on the properties of the selected wavelet function $ψ_{u, s}$ .

Using Parseval's theorem, the total energy of a signal $f (t)$ can be expressed as $E = \int_{- \infty}^{+ \infty} f^{2} (t) d t =$

Empirical mode decomposition

The EMD method, pioneered by Huang et al. [12], decomposes a time-series into a finite set of oscillatory functions called the intrinsic mode functions (IMF). An IMF is a function that satisfies two conditions: (1) the number of extrema and the number of zero crossings must either equal or differ at most by one; (2) the running mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. The intrinsic modes represent the embedded time scales in

Discussion

In this paragraph, a comparison of the three methods studied in previous paragraphs is presented on the basis of a reliable energy-based feature extraction procedure for gear fault diagnosis and prediction. In addition, a prediction based on the squared envelope of the signal is presented and the elaboration of the time–frequency approach is justified. First, the unsmoothed WVD successfully diagnosed the problem in all cases. Localisation in both time and frequency domains has been very good

Conclusions

It has been shown that the instantaneous energy of vibration resulting from a pair of spur gears can be used for gear condition monitoring. In this sense, instantaneous energy density was proposed as a general diagnostic feature. At least three signal processing tools have been shown to provide useful energy-based features.

The WVD has the main advantages of simplicity and fast computation. The wavelet transform offers the choice of customising the resolution in time or frequency domains.

Acknowledgements

This work was supported by the Greek Secreteriat of Research and Technology Grant 01EΔ330.

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Instantaneous energy density as a feature for gear fault detection

Abstract

Introduction

Section snippets

Experimental test rig

The Wigner–Ville distribution

The continuous wavelet transform

Empirical mode decomposition

Discussion

Conclusions

Acknowledgements

Mechanical Systems and Signal Processing

Applied Acoustics

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing