*3.2. Results*

Acquisitions corresponding to operating condition c (rotor blades motionless, no action of the wind) were used to obtain the mean value of the horizontal distance from TLS. For a point P on the wind tower located at H = 65 m height from the tower base, the mean ground distance of D = 47.293 m was

obtained. The procedure described in the Section 2.3.2 was applied. The comparison between raw data and the values obtained by the procedure can be observed in Figure 8.

**Figure 8.** (**a**) Distance DP,TLS of point P on the wind tower located at H = 65 m during acquisition in operating condition c (see Table 3); (**b**) result after applying the proposed methodology; (**c**) frequency distribution of differences between modelled DP,TLS distances (blue dots) and moving average (red line). The standard difference between measured and model DP,TLS distances is 0.6 mm.

The time series of TLS distances of point P, located at the top of the wind tower, is plotted in panel 8(a). Panel 8(b) displays the results obtained by applying the procedure in Section 2.3.2 to the detrended measurements of Panel 8(a). Results provided by the proposed methodology and the mean curve obtained by applying a moving average operator are plotted in Figure 8b, while the frequency distribution of their differences is plotted in Figure 8c. The standard deviation of this frequency distribution provides the precision of TLS estimates of distances equal to 0.6 mm.

Figure 9 shows the plot of TLS ground distances vs. timestamps of four points P on the wind tower located at different heights above the base, obtained by the proposed procedure. Data were collected from 13.41.57 till 13.42.11 CET. These results were obtained in condition a (see Table 3) with a fully operational wind turbine. The peak-to-peak amplitude of oscillations decreased from 15 mm to 2 mm when moving from H = 60 m to H = 15 m on the wind tower.

**Figure 9.** Ground distances from TLS of points at a height of 60 m (d60 line), 40 m (d40 line), 25 m (d25 line) and 15 m (d15 line) on the tower base. Abscissae are the timestamps in sec. The wind turbine is fully operational.

Figure 10 shows the plot of TLS ground distances vs. timestamps of a point P on the wind tower located at H = 60 m above the base. Data were collected from 13.43.39 till 13.44.59 CET. These results were obtained in condition b (see Table 3) with the wind turbine in deactivation mode. Damped oscillations had an amplitude decreasing from 90 mm to a few mm. The average ground distance from the TLS increased and tended to stabilize around a value of 47.293 m.

The corresponding spectrum of vibration frequencies are shown in Figure 11a,b, along with the spectrum obtained for data collected during the stop of the wind turbine in Figure 11c,d. In both cases, two peaks are observed at 0.4 Hz and 40 Hz.

**Figure 10.** Ground distance D60 line from TLS of a point 60 m high on the tower base during the deactivation of the turbine.

**Figure 11.** Spectra of vibration frequencies of point P on the wind tower located at H = 60 m. Spectrum for the data collected during the deactivation of the wind turbine (**a**). Enlargement of the low frequencies (**b**). Spectrum for the data collected during the stop of the wind turbine (**c**). Enlargement of the low frequencies (**d**).

Figure 12 shows the spectrum of vibration frequencies for a point P on the wind tower located at H = 25 m above the base, obtained for data collected during the deactivation of the wind turbine. Additionally, in this case, two peaks can be observed at 0.4 Hz and 40 Hz.

**Figure 12.** Spectrum of vibration frequencies of point P on the wind tower located at H = 25 m. Data have been collected during the deactivation of the turbine (**a**). Enlargement of the low frequencies (**b**).

The spectrum of vibration frequencies obtained by processing GB-RAR data acquired during the stop of the wind turbine is reported in Figure 13. The spectrum refers to a point located at a height H = 25 m above the tower base, corresponding to a slant distance R = 54 m from the radar. Two peaks are observed at 0.4 Hz and 3.1 Hz. It worth noting that besides the common peak at frequency f = 0.4 Hz, the two spectra had di fferent characteristics. In particular, it should be noted that the peak at f = 0.4 Hz measured by GB-RAR in operating condition c was measured by TLS at H = 60 m but not at H = 25 m. Probably, this was due to the di fferent amplitude of oscillations at the top and bottom of the tower.

**Figure 13.** Spectrum of vibration frequencies of point P on the wind tower located at H = 25 m. Data have been collected during the stop of the turbine.

#### **4. Numerical Analysis and Discussion**

This section discusses the results presented in Section 3.

#### *4.1. Evaluation of Displacements Due to the Wind*

During the test, anemometric data were acquired, which showed a constant wind direction. The wind speed data make it possible to obtain the thrust on the tower. Given the geometric and mechanical characteristics of the tower, it was possible to calculate the displacements of the points at various heights. The wind pressure can be obtained by the Bernoulli Equation:

$$p = \frac{1}{2}\rho v^2\tag{5}$$

where:

> p = is the wind pressure [N/m2]

ρ = air density [kg/m3] v = air speed [m/sec]

The air pressure, applied to the swept area of the rotor, must be multiplied to a coe fficient less than 0.6 due to the Betz limit [31]. In our case, given the low wind speed, which implies a non-optimal rotor e fficiency, we can use the value 0.4. It is also necessary to take into account the interference factor b (i.e., the ratio of the downstream speed v2 to the upstream speed v1, which in our case can be considered equal to 0.577 [32]). Given that the speed measured by the anemometers and used for our test (installed on the nacelle and behind the rotor) is the downstream speed, the thrust *F* acting on the top of the tower, considered as a cantilever, can therefore be evaluated as [32]:

$$F \cong 0.4\left(\frac{1}{2}\rho \Im v\_2^2 \pi r^2\right) = 0.60\left(\rho v\_2^2 \pi r^2\right) \tag{6}$$

where r is the radius of the rotor. Geometric and material data on the tower can be found in [33]. The displacement δ at the top of the tower, in correspondence of the connection with the nacelle, is obtained by:

$$\mathcal{S} = F \frac{L^3}{3EI} \tag{7}$$

where:

L = cantilever length

E = Young's modulus

J= moment of inertia

For our calculations, we assumed ρ = 1.2 kg/m3, L = 65 m, E = 2.1 × 10<sup>11</sup> <sup>N</sup>/m2, *J* = 0.323 m4. Actually, the lowering changes with the wind speed, and other e ffects a ffect the final results, so only a rough estimate can be obtained with Equation (7). Table 4 shows the displacements of a point 65 m high on the base of the tower computed for di fferent wind speeds. By adding to the displacements, the mean value of the ground distances measured in the operating condition c, i.e., during the stop of the rotor blades, one can obtain the ground distances from TLS for the tower subject to the wind load.


**Table 4.** Theoretical displacements.

#### *4.2. Numerical Analysis of Natural Frequencies*

This section provides a simple numerical analysis for the computation of natural frequencies of a cantilever subject to free oscillations. An approximated value of the circular natural frequencies, in case of a fixed cross section, is given by [34]:

$$
\omega\_n = \alpha\_n^2 \sqrt{\frac{E l}{m L^4}}\tag{8}
$$

where:

> ω*n* = n-th circular natural frequency α*n* = 1.875, 4.694, 7.885

E = Young's modulus J = moment of inertia m = mass per length unit L = cantilever length

The natural frequency fn can be obtained from the circular frequency using the equation:

$$f\_n = \frac{\omega\_n}{2\pi} \tag{9}$$

To bring back the problem to the case of a massless cantilever with a discrete effective mass applied to the free end, we use the effective mass for the n-th frequency *<sup>m</sup>*eff(n), given by:

$$m\_{eff}^{(n)} = \frac{3Ef}{L^3 \alpha\_n^2} \tag{10}$$

This way we can add a *mend* mass actually positioned at the free end of the cantilever and consider a total mass Mn at the free end:

$$\mathbf{M}\_{\mathbf{n}} = m\_{eff}^{(n)} + m\_{\rm end} \tag{11}$$

The n-th natural frequency will be:

$$
\omega\_n = \alpha\_n^2 \sqrt{\frac{3E}{M\_n L^4}}\tag{12}
$$

In our case, given the characteristics of the tower, its mean section and the wind turbine data [33] we can assume E = 2.1 × 10<sup>11</sup> <sup>N</sup>/m2, J = 0.322 m4, m = 2800 kg/m, L = 65 m. By using Equations (9) and (10), we obtain a discrete effective mass of 40,740 kg for the first frequency and of 1120 kg for the second one. We must add the weight of rotor and nacelle, equal to about 8 × 10<sup>4</sup> kg. By using Equations (11) and (12), we get, for the first three natural frequencies, the values f1 = 0.394 Hz, f2 = 3.10 Hz, f3 = 8.596 Hz, and the relevant periods T1 = 2.54 sec, T2 = 0.32 sec, and T3 = 0.11 sec.

#### *4.3. Measured vs. Theoretical Displacements*

In the following, we consider the oscillation amplitudes of a point located at H = 65 m above the base of the tower. The choice of a point near the top of the tower allows us to consider displacements of greater amplitude. This is also useful, given the wind speed, which was not high during the test. Figure 14 shows the wind downstream speed during the test. The wind direction was constant during data acquisition.

**Figure 14.** Wind speed in m/sec (downstream) measured by the wind turbine anemometers during the test.

In Table 5 we can see a comparison between the values of the theoretical ground distances and the values measured by TLS in different acquisitions. The theoretical distances have been obtained by subtracting the computed displacements to the mean ground distance D = 47.293 m, obtained in case of motionless rotor blades (see Figure 8a and condition c in Table 3).


**Table 5.** Theoretical and measured ground distances from TLS of a point 65 m high.

Figure 15 shows the displacements of the considered point during the activity of turbine. In the lower part of the figure, the wind speed is represented. Acquisition times are synchronized (see Table 3); the wind direction was constant.

It can be observed that the increase in wind speed leads to a greater deformation and, therefore, to an approach of the point to the TLS located on the opposite side of the rotor. The deformation values δ obtained are in accordance with those obtained with Equation (7).

A continuous oscillation of the point is observed which has a period of around two seconds. This value is not constant and is affected by the noise of the measurements, as well as by the variability of the wind thrust. The amplitude of the oscillations varies from 10 to 20 mm.

**Figure 15.** Ground distance from TLS of a point P on the tower located at H = 65 m during the activity of the turbine, from 13:37:50 to 13:38:12 (**a**) and from 14:00:00 to 14:01:06 (**b**). Below the graphics of the distances, the wind speed in m/sec. X axis is the time, both for upper and lower figures.

## *4.4. Measured vs. Theoretical Frequencies*

In this section, we discuss the properties of the measured vibration spectra and compare them to the theoretical analysis of the wind tower reported in Section 4.2. The 0.4 Hz frequency measured by both TLS and GB-RAR is in agreemen<sup>t</sup> with the value obtained using the analytical formulas for the first natural frequency. The frequency f = 3.1 Hz of the second oscillation mode is measured by GB-RAR but not by TLS, probably due to lower precision of TLS when compared to GB-RAR.

The frequency peak at f = 40 Hz found in all TLS measurements is an artifact due to the rotating mirror. In fact, the laser beam used for measurements in the TLS VZ1000 is addressed and collected by a rotating three-sided mirror, i.e., by the lateral faces of a triangular-based prism. When a scan speed of 120 lines per second is selected, the three-sided mirror rotates with an angular speed of 40 revolutions per second. Even a slight vibration of the rotation axis can cause oscillations in the results at a frequency of 40 Hz (Figure 16a). To confirm this hypothesis, the results of the measurements performed with a slower scanning speed of 60 lines per second were analyzed.

**Figure 16.** The scheme of rotating mirror of VZ1000 TLS (**a**). Ground distance from TLS of a point 65 m high on the tower base during the acquisition with scan speed 60 Hz (**b**). Power Spectrum obtained by FFT for the ground distance from TLS (**c**). Enlargement of Figure 16c (**d**).

Figure 16b shows the displacements of a point 65 m high on the tower base during the acquisitions. Figure 16c shows the power spectrum for the ground distance from TLS of a point 65 m high on the tower base. Two peaks are present: the first at a low frequency, the second at 20 Hz. In Figure 16d the first peak is enlarged, corresponding to a frequency of 0.4 Hz. The results confirm the above hypothesis, so we can attribute the peak at the highest frequency to the vibration of the rotating mirror of the TLS.
