Fault Detection in DC Electro Motors Using the Continuous Wavelet Transform

Two time–frequency methods were used to detect typical faults in DC electro motors: the windowed Fourier transform and the continuous wavelet transform. Four groups containing three electro motors each were manufactured with typical faults and examined. These faults included a bearing fault, an increased unbalance, a fragmented brush and a fragmented collector. The velocity of the vibrations at selected points on the electro motors was measured with a laser probe. The parameters of both transforms were selected in order to make both methods comparable. Because of the poor frequency variance, the windowed Fourier transform was, in this case, proven to be inferior to the continuous wavelet transform. Therefore, the continuous wavelet transform was chosen as the primary tool for fault detection.Three criteria were found that successfully discriminated between the typical faults. These were the highest magnitude level, the frequency of the first and second harmonics and the time period between the magnitude pulses in the third (highest) frequency region. If the maximum magnitude levels versus the period of the pulses in the third frequency range are plotted, four distinct regions corresponding to four different faults are obtained. Since the regions do not overlap, linear classifiers can be used with the presented criteria.     

The norms and variances of the Gabor, Morlet and general harmonic wavelet functions

This paper deals with certain properties of the continuous wavelet transform and wavelet functions. The norms and the spreads in time and frequency of the common Gabor and Morlet wavelet functions are presented. It is shown that the norm of the Morlet wavelet function does not satisfy the normalization condition and that the normalized Morlet wavelet function is identical to the Gabor wavelet function with the parameter σ=1.The general harmonic wavelet function is developed using frequency modulation of the Hanning and Hamming window functions. Several properties of the general harmonic wavelet function are also presented and compared to the Gabor wavelet function. The time and frequency spreads of the general harmonic wavelet function are only slightly higher than the time and frequency spreads of the Gabor wavelet function. However, the general harmonic wavelet function is simpler to use than the Gabor wavelet function. In addition, the general harmonic wavelet function can be constructed in such a way that the zero average condition is truly satisfied. The average value of the Gabor wavelet function can approach a value of zero but it cannot reach it.When calculating the continuous wavelet transform, errors occur at the start- and the end-time indexes. This is called the edge effect and is caused by the fact that the wavelet transform is calculated from a signal of finite length. In this paper, we propose a method that uses signal mirroring to reduce the errors caused by the edge effect. The success of the proposed method is demonstrated by using a simulated signal.