8 Characterization of Torsional Vibrations: Torsional-Order Based Modal Analysis 81 Fig. 8.1 Fixed Sampling order tracking processing. (a) Waterfall plot. (b) Colormap plot Each of the FFT is calculated by considering an average rpm value over the selected block. The FFTs appear to be horizontal lines in both the colormap and waterfall plot. The rpm step defines the rpm interval between two consecutive FFTs. Of course, Finally, the order, in terms of amplitude and phase, can be extracted from the FFT spectra.The FFT kernel is given in Eq. (8.7). am D 1 N N X nD1 x.n t/cos.2 k p n t/ bm D 1 N N X nD1 x.n t/sin.2 k p n t/ (8.7) Nis the number of time blocks andn t represents the nth time interval. The extracted orders do not normally fall on a single spectral lines. For this reason, often multiple spectral lines are summed together. The main advantage of this method is its computational efficiency. Several drawbacks could be listed. First of all, the blocksize (time interval) is not related to the rpm of the machine. This cause problems both at low and high rotational speed. In fact, at low speeds, the interval is too short to capture the low orders resulting in power leakage (smearing) between closely-spaced orders. At high speeds, the interval is too long to capture rapid variations and spikes in the signal. For minimizing the smearing problem, a Hanning window is typically applied. 8.2.3.2 Resampling Based Order Tracking (AD) The time domain sampling based FFT order tracking (FS) has several drawbacks. Some of these can be solved by resampling from time to angle domain before applying the FFT to the acceleration signals. Assuming that the blocksize remains constant, the resampling implies that the signal is now sampled at a constant rather than at a constant t. The resampling process requires some computational time and it includes two main steps: oversampling and then interpolating in order to get the required -spaced samples. As the technique is based on the angle information, a very accurate tachometer signal is needed in this case. In fact, the resampling intervals are calculated by integrating the speed signal. After the resampling step, the variable-frequency order components are converted into regular sinusoidal signals. The DFT (or FFT) can then be used to process the resampled data and to obtain the order components as spectral lines since the transformation is based on angle domain data rather than time domain data. The kernels of the FFT are reformulated from Eq. (8.7) and they are reported in Eq. (8.8).
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