Vibration analysis of rotating machinery using time–frequency analysis and wavelet techniques
Introduction
Rotating machines cover a wide range of critical facilities and provide the backbone of numerous industries, from gas turbines used in the production of electricity to turbo-machinery utilized to generate power in the aerospace industry. It is vital that these machines run safely over time and under different operational conditions, to ensure continuous productivity and prevent any catastrophic failure, which would lead to extremely expensive repairs and may also endanger lives of the operating personnel.
Generally, a simple condition monitoring system is approached from a pattern classification perspective. It can be decomposed into three general tasks: (1) data acquisition, (2) feature extraction and (3) condition classification [1]. Several parameters may be used for condition monitoring, including temperature, pressure, oil analysis, noise and vibration. The most common method is based on nondestructive vibration measurements using transducers such as accelerometers, velocity pickups and displacement probes. Vibration measurement provides a very efficient way of monitoring the dynamic conditions of a machine such as unbalance, misalignment, mechanical looseness, structural resonance, soft foundation and shaft bow.
Traditional vibration signal analysis has generally relied upon the spectrum analysis via the Fourier Transform (FT). Fourier analysis transforms a signal f(t) from a time-based domain to a frequency-based one, thus generating the spectrum that includes all of the signal's constituent frequencies (fundamental and its harmonics) and which is defined as [2]
Fuelled by its huge success in processing stationary signals in a wealth of application areas, an FT technique has enjoyed other interesting extensions. One such extension is in the particular area of vibrations and machine-health monitoring, called the fast Fourier transform (FFT)-based order analysis (OA) technique, including its order-tracking capability [3]. The OA technique transforms the revolution domain into an order spectrum and any signal that is periodic in the revolution domain will appear as a peak in the order spectrum. By doing so, the OA technique tries to overcome the effect of frequency change on the FFT and hence allows for a better tracking of speed-driven harmonics in rotating machinery. However, the assumption in this technique is that the frequency change within a single time interval is small, so that the necessity of a stationary signal for frequency transformation is not largely violated. If the frequency changes significantly within this time interval, then the FFT will yield an error in the actual value of the signal [4].
An important deficiency of the FFT is its inability to provide any information about the time dependence of the spectrum of the signal analyzed, as results are averaged over the entire duration of the signal. This feature becomes a problem when analyzing non-stationary signals. In such cases, it is often beneficial to acquire a correlation between the time and frequency contents of the signal. Non-stationary signals could be classified into two groups:
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Evolutionary harmonic or frequency-modulated signals: these signals are generated by some underlying periodic time-varying phenomenon like a change in rotational speed during “start-up or coast-down”.
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Transient signals: these signals have short durations and an unpredictable time behavior, and are therefore viewed as being random in nature. Examples of such signals are impact loading and rubbing.
This important limitation of the FFT has led to the introduction of time–frequency signal processing tools, such as the Short-Time Fourier Transform (STFT), the Wigner-Ville Distribution (WVD) and others. The STFT maps a signal into a two-dimensional (2D) function of time and frequency. The difficulty in using the STFT is that the accuracy of extracting frequency information is limited by the length of the window relative to the duration of the signal. Once the window function is defined, the area (time-bandwidth product) of the window function in the time–frequency plane remains fixed, which means that the time and frequency resolutions cannot be increased simultaneously. Consequently, for an STFT, there is a trade-off between time and frequency resolutions [5]. The WVD has a good energy concentration in the time–frequency plane; but suffers from interference terms which appear in the same plane and tend to mislead the signal analysis [6].
The wavelet transform (WT) is a relatively new and powerful tool in the field of signal processing, which overcomes problems that other techniques face, especially in the processing of non-stationary signals. It allows the use of long time intervals, where more precise low-frequency information is desired and also permits the use of shorter time intervals where accurate high-frequency information is desired. It is also employed for the accurate extraction of narrow-band frequency signals. The main advantage gained by using wavelets is the ability to perform a local analysis of a signal, or to zoom on any interval of time without losing the spectral information contained therein. The wavelet analysis is thus capable of revealing some hidden aspects of the data that other signal analysis techniques fail to detect. This property is particularly important for damage (crack) or fault detection applications. One possible drawback of the WT is that the frequency resolution may be quite poor in the higher frequency region. Hence, the WT still faces difficulties when trying to discriminate signals containing high frequency components, such as impact faults (like rubbing).
A wavelet ψ(t) is a waveform of effectively limited duration that has an average value of zero over time, as described by the following Eq. (2):
To achieve wide ranges of analyses and applications, and a higher signal-resolving power, the wavelet schemes use basis functions other than sines and cosines, which constitute bases of the Fourier analysis. The wavelet functions are composed of a family of basis functions that are capable of describing a signal in a localized time (or space) and frequency (or scale) domains. Choosing the type of the basis function depends on the application as well as the computation efforts required. The appropriate selection of mother wavelets is very important, as results are heavily dependent on the chosen wavelet shape. Wavelets of various shapes exist, including the first-ever wavelet (Haar wavelet) [7] and other types such as Mexican hat-shaped and Gaussian-shaped [6]. However, for singularity analysis and detection, wavelets should have an impulse-like shape to capture the sudden change in the signal. Morlet and Gaussian wavelets are found to have an excellent representation of a singularity (discontinuity) as the vibration signal has a harmonic oscillation plus a singularity representing the fault. It is desirable to have a linear phase scaling function in order to maintain a constant group delay and the envelope of the original vibration signal, which will avoid problems during the signal reconstruction process. Having a wavelet shape resembling that of the targeted fault greatly helps in the wavelet selection decision.
The wavelet transform and its other three types are further discussed in Section 4, where the details of the proposed hybrid wavelet-based approach are clearly expounded.
The paper is organized as follows: Section 1 reviews the two general approaches to vibration analysis, namely the traditional FFT-based and the modern wavelet-based ones. Section 2 describes the experimental setup used and the data collection procedure, whereas Section 3 reports on, and discusses, the experimental results obtained using the FFT approach. In the key Section 4, the 3 main types of wavelets, Continuous Wavelet Transform (CWT), discrete wavelet transformation (DWT) and wavelet packet transform (WPT) used in vibration analysis are first reviewed and then both the CWT-based maxima modulus technique and WPT are applied to the data from the blade-to-stator rubbing test, and their performances are compared. This section also introduces the proposed novel hybrid approach based on a judicious use of these 2 tested techniques, i.e. the CWT-based maximal modulus technique and WPT. Finally, some conclusions are given in Section 5.
Section snippets
Experimental setup and data collection
The test rig was built for research purposes to study blade and shaft vibrations of turbo-machinery (Fig. 1). It is primarily designed to study blade vibrations using a remote sensing system via a set of strain gages bonded to one of the blades. It is also possible to measure shaft vibrations directly by proximity probes or indirectly by accelerometers through their placement near shaft supports. The test rig consists mainly of the following elements:
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A 3-phase induction motor of 220 Volt and 60
Experimental results using an FFT
Three experiments are carried out: 1) free vibration of blades, 2) blade-to-stator rubbing at different speeds and 3) fast start-up and coast-down. The 1st experiment is run to locate the natural frequencies of blades. The 2nd and 3rd experiments are conducted to simulate the occurrence of non-stationary and transient faults for the comparison of the FFT, time–frequency and WT techniques. The expected faults from these experiments are unbalance and misalignment. The unbalance in rotating
Review of Three Wavelet Transforms: CWT, DWT and WPT
In this section, the three major wavelet transforms are briefly reviewed: namely the CWT, DWT and WPT, with an emphasis on the CWT and WPT.
The WT is classified into three types: the CWT, DWT and WPT. J. Morlet along with an A. Grossmann formulated the CWT and defined it as the sum over all time of the signal multiplied by scaled and shifted versions of the mother wavelet function [6]. Hence, the modulus of the CWT for the signal f(t) is given aswhere t is the time, a
Conclusion
The main theme of this work was to study the application of wavelets, in particular the WPT transform, to fault detection in rotating machinery, an area of paramount importance to various industries. All of the signals used in this study were experimentally obtained from a custom-built rotor kit to simulate on a laboratory scale the main operating conditions of rotating machinery in a wide range of industries involving equipments such as turbines, compressors and fans. Although not reported
Acknowledgements
The authors greatly acknowledge the support of King Fahd University of Petroleum & Minerals for this work.
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