Time series analysis
Figure 4.15 shows a typical tower top acceleration response in xdirection (i.e. sensor A05) from a parked modal test. Obviously, the stepwise change in the force applied to the tower top excites a wide range of natural frequencies. The relative high frequencies at the beginning of the response are damped out quickly, leaving eventually the single frequency associated with the lowest system mode of vibration. This mode is in literature often denoted with the misleading term "first tower bending mode". The main reason for the domination of the response by the lowest system mode of vibration is that the total mass of the generator, chassis and rotor is relatively large with respect to the tower mass.
The response in Figure 4.15 also shows that the record length is not long enough to capture the complete response history. When the truncated timedomain response is transformed into the frequency domain using the Fast Fourier Transform (FFT)
this may result in socalled leakage errors. In this case the distortion due to leakage is negligible because the truncated response contains only one mode and this mode is truly periodic in time.
The corresponding periodogram estimate of the power spectral density (PSD) is depicted in Fig. 4.16. It shows how the average power of the tower top acceleration response in xdirection is distributed with frequency. The PSD estimate is computed via the Thomson multitaper method, and has a frequency resolution of 500/32768 = 0.0153 Hz. Prior to the computation of the PSD, the acceleration record was preprocessed by filtering the data without phase distortion with an eighth order lowpass digital Butterworth filter with a cutoff frequency of 125 Hz, and subsequently removing the linear trend. It is clear that noise is not the primary problem since the individual peaks indicating the modes are clearly discernible in the plot.
Integrating the total area under the curve reveals that 99 % of the average power is located below 102.7 Hz (73.6 % of the average power is contained in the first system mode of vibration, 5.2% in the fourth system mode, and 0.7% in the fifth system mode). This average power distribution confirms the domination of the first system mode of vibration in the timedomain response.
PSD of sensor A05
PSD of sensor A05
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Frequency [Hz]
15 20 25 30 35 40
Frequency [Hz]
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Frequency [Hz]
Figure 4.16: Periodogram estimate of the power spectral density (PSD) of the tower top acceleration (sensor A05) from measurement M2. Upper figure: signal in the frequency range of 0 to 40 Hz, and lower figure: zoomed in on the second bending mode and higher.
10 15
Frequency [Hz]
Figure 4.16: Periodogram estimate of the power spectral density (PSD) of the tower top acceleration (sensor A05) from measurement M2. Upper figure: signal in the frequency range of 0 to 40 Hz, and lower figure: zoomed in on the second bending mode and higher.
The periodogram estimate of the PSD of both sensor A03 and A16, depicted in upper and lower part of Fig. 4.17 respectively, show a completely different picture. The top figure reveals that both the first and the fourth system mode in xdirection
(second and third mode are not present in this response) dominate the response of sensor A03 (located about halfway between the tower bottom and top, see Table F.1 on page 251). Integrating the total area under the curve reveals that 22.6% of the average power is contained in the first system mode, 29.7% in the fourth system mode, and 12.3% in the fifth system mode. The bottom figure reveals that the response of sensor A16 is dominated by the fourth system mode of vibration in xdirection (the third system mode is not present in this response). Integrating the total area under the curve reveals that 4.3% of the average power is contained in the first system mode, 7.6% in the second system mode, 0.5% in the third system mode, 66.1% in the fourth system mode, and 27.8% in the fifth system mode.
PSD of sensor A03
PSD of sensor A03
05 
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30 
35 
40 
Frequency [Hz]  
PSD of sensor A16 
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Frequency [Hz]
Figure 4.17: Periodogram estimate of the power spectral density (PSD) of sensor A03 (upper figure) and A16 (lower figure) from measurement M2.
15 20 25
Frequency [Hz]
Figure 4.17: Periodogram estimate of the power spectral density (PSD) of sensor A03 (upper figure) and A16 (lower figure) from measurement M2.
Fig. 4.18 shows the periodogram estimates of the PSD of the responses in xdirection as a function of the sensor position with respect to the ground level. This threedimensional plot reveals how the average power distribution changes with the sensor location. It shows that the modes of the Lagerwey LW50/750 wind turbine in xdirection are not closely spaced, and not heavily damped. This implies that the modes are lightly coupled. Consequently, the turbine behaves at resonance predominantly as a singledegreeoffreedom (SDOF) system: the structural response at a natural frequency is (almost) completely determined by that mode.
Fig. 4.19 shows a twodimensional view of Fig. 4.18. The viewpoint is set along the xaxis thereby disregarding the accelerometer position information. This figure clearly reveals that the dynamic response of the Lagerwey LW50/750 wind turbine in xdirection is dominated by the first five bending modes. It should be stressed that the geometrical complexity of the mode shapes, rather than the number of modes observed, determines the number of degrees of freedom.
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leona7 years ago
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