382 K. Motte et al. Fig. 40.5 PDF calculated for measurement filtered around the frequencies given in the legend. Harmonic at 1.5 Hz 0 1.46 Hz 1.48 Hz 1.50 Hz 1.52 Hz 1.54 Hz Amplitude Probability Fig. 40.6 PDF calculated for measurement filtered around the frequencies given in the legend 0 1.30 Hz 1.32 Hz 1.34 Hz 1.36 Hz 1.38 Hz Amplitude Probability heavily the technique remains usable. The results however require to be interpreted and automation is therefore difficult. A possible method is to first use an identification technique, and later the PDF. The PDF will show which identified poles can be attributed to mathematical modes, harmonics or structural modes. 40.3.2 Kurtosis The use of kurtosis has been proposed as a part of the new enhanced frequency domain decomposition technique (EFDD) [14]. The kurtosis describes how peaked or how flat the PDF of a stochastic variable is. The kurtosis is the fourth central moment of a stochastic variable x, normalised with respect to the standard deviation . The kurtosis is written as: .xj ; / D Eh.x /4i 4 3 (40.1) In Eq. (40.1), represents the mean value of x and E denotes the expected value. The standard definition results in a value of 3 for the kurtosis if the PDF is normally distributed (structural mode). Therefore the modified definition is used [Eq. (40.1)], where 3 is subtracted, such that a structural mode results in a value of 0 and a harmonic in a value of 1:5 (value for a sinusoidal component). The measured signal is band pass filtered in consecutive band over the whole frequency band of interest. The kurtosis is calculated and if a negative peak towards 1:5 is observed, a harmonic is indicated. A laboratory set-up was used to demonstrate the use of kurtosis. A beam was excited with white noise and superposed sinusoidal harmonics, with the first harmonic located close to the first structural mode (Fig. 40.7). When the kurtosis is calculated, all the harmonics are clearly indicated by peaks to 1:5 (Fig. 40.8).
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