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Roller bearings are in widespread industrial application. In practical applications, the roller bearing is a critical mechanical component and a defect in such a bearing, unless detected in time, would cause malfunction and could even lead to catastrophic failure of the mechanical system. Therefore, the defect detection of the bearing is important for condition monitoring. There are different methods used for detection and diagnosis of bearing defects. Several techniques [1] are applied in measuring the vibration responses from defect bearings, i.e., vibration measurement in time and frequency domains, the shock pulse method, sound pressure and acoustic emission method.
Some researches in the last two decades study mostly on the detection and diagnosis of bearing defects by vibration methods. The vibration analysis of a bearing system is typical of the signal processing for the vibration signal with amplitude modulation. The high frequency resonance technique [2] is usually applied in analyzing the mechanical vibrations with amplitude modulation to derive an envelope signal. When applying the high frequency resonance technique to detect the envelope signal, the vibration signal is operated through a bandpass filter to derive a single mode vibration among the system resonances, and then taking the enveloping transformation to retrieve an envelope signal. In the range of the filtered system resonance, this technique takes advantage of the absence of low-frequency mechanical noise to demodulate a vibration signal and, therefore, provides a low-frequency envelope signal with a high signal-to-noise ratio. Such a signal processing is called the demodulation of mechanical vibrations.
In conventional signal processing, a single mode vibration signal for the bearing vibration is filtered through a bandpass filter, and then the envelope signal which is corresponding to the filtered vibration mode could be rectified and filtered through a lowpass filter with the cutoff frequency at 1 kHz. However, such an envelope detection method is a hardware-based signal processing. There could be serious distortion in the envelope signal. In addition, the hardware of filter is expensive and less flexible in choosing the cutoff frequency.
On the other hand, a number of software-based signal processing methods, such as the filter-based Hilbert transform and the FFT-based Hilbert transform, have been introduced to apply in the high frequency resonance technique. These kinds of signal processing methods could much reduce the cost of hardware. The filter-based Hilbert transform possesses the capability of real-time signal processing. In general, an increase in filter order would improve the frequency response, but could increase the computing burden and cause a serious phase distortion. The FFT-based Hilbert transform has the advantage of high computing speed. Therefore, the FFT-based Hilbert transform is adaptable to signal analysis on line. But it could not be applied in the real-time signal processing. In addition, leakage error is also encountered in the FFT-based Hilbert transform.
In the recent years, the wavelet-based Hilbert transform is introduced to apply in the envelope detection of vibration signals with amplitude modulation. The wavelet-based Hilbert transform has a feature of time-frequency localization capable of exhibiting the instantaneous frequencies of signal and gives a description of how energy distribution over the changes of frequencies from one instance to another. Liu and Qiu [3] use the Morelet wavelet to constructs a Hilbert transform pair that calls for two wavelet transforms, where one wavelet is the Hilbert transform of the other. However, the Morelet wavelet could only be an orthogonal narrow filter and is only suitable for a single-frequency harmonic signal. Nikolaou and Antoniadis [4] base on the use of a complex shifted Morlet wavelet family to construct a matrix from the absolute value of wavelet transform for a vibration signal. By sorting the maximum values of each column in the matrix, an envelope of the vibration signal could be derived and be capable of detecting the appearance of each impulse in the frequency band. But there are very serious distortions in the amplitude and phase. Thus, how to construct a proper enveloping function or propose an envelope detection method to derive a less distorted envelope signal should be the main issue in the signal processing of a bearing vibration.
Based on the high frequency resonance technique, some signal processing methods in this thesis are proposed to derive a less distorted envelope signal and could be applied to the real-time vibration analysis for the bearing defect diagnosis. They are simply described as follows.
In Chapter two, a wavelet-based enveloping function is proposed to implement the high frequency resonance technique. An algorithm for the linear combination of Morlet wavelets is proposed to construct the wavelet-based enveloping function. Under properly choosing the frequency difference for the linear combination of Morlet wavelets, the enveloping function with three parameters which are the low-cutoff frequency, the high-cutoff frequency and the dilation scale could designate a filtering passband and adjust the passband quality for detecting the vibration signal envelope. The filtering passband can be arbitrarily designated by simply setting the parameters of low-cutoff frequency and high-cutoff frequency. In addition, the filtering quality of the wavelet-based envelope function could be guaranteed by properly designated both the dilation scale and the waveform length of the enveloping function.
In Chapter three, the integration of Morlet wavelets is proposed to construct another wavelet-based enveloping function. In comparison with the enveloping function in Chapter two, the enveloping function in Chapter three is further simplified without the parameter of frequency difference. On the other hand, it is shown [5] that the spectrum for the envelope signal has a pattern of sidebands around the defect frequency and its harmonics. Therefore, the logarithm transformation is proposed to apply to the envelope signal to further enhance the defect frequencies under the running condition of non-uniform loading.
In Chapter four, the wavelet-based enveloping function is further studied to construct a three-dimensional envelope spectrum. The wavelet-based enveloping function has a feature of time-frequency localization capable of exhibiting the instantaneous frequencies of signal and gives a description of how energy distribution over the changes of filtering passband from one instance to another. The three-dimensional spectrum of a bearing vibration could be used to describe how energy distribution between the instantaneous frequency and the filtering passband for a vibration signal. Accordingly, the three-dimensional spectrum would be helpful to give a clear view of both characteristic frequencies and system resonances, and possesses the advantage of minimizing the interventions by the end-user in choosing the bandpass frequency.
In Chapter five, a linear least square analysis algorithm derived from the mathematical model of the bearing vibration with amplitude modulation is proposed to estimate the envelope signal for a single mode vibration which is filtered through a bandpass filter. The algorithm could very much reduce the requirement in the calculation for the envelope detection of the single mode vibration signal. Moreover, another study on the sideband reduction around the defect frequencies is to enhance the defect frequencies under the running condition of non-uniform loading. In comparison with the logarithm transformation in Chapter three, the sideband reduce method shows a better result to enhance the defect frequencies.
Accordingly, it is proved from the experimental studies that these proposed methods in this thesis could be effectively applied in the real-time signal processing for the bearing defect diagnosis.
REFERENCES
Tandon, N. and Choudhury, A. A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings.
McFadden , P.D. and Smith, J.D. Vibration monitoring of rolling element bearings by the high frequency resonance technique a review, Tribology International, 1984, 17, 1-18.
Liu, ChongChun and Qiu, Zhengding A method based Morlet wavelet for extracting vibration signal envelope, Proceedings of the Fifth International Conference on Signal Processing, 2000, 1, 337-340.
Nikolaou, N.G. and Antoniadis, I.A. Demodulation of vibration signals generated by defects in rolling element bearings using complex shifted Morlet wavelets, Mechanical Systems and Signal Processing, 2002, 16(4), 677-694.
Su, Y. T. and Lin, S. J. On initial fault detection of tapered roller bearing: frequency domain analysis, Journal of Sound and Vibration, 1992, 155, 75-84.
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