## 4 An Experimental Case Study
### Wind Tunnel Experiments  
Experiments were carried out in the atmospheric boundary layer wind tunnel of the Institute of Engineering of the National Autonomous University of Mexico and FiiDEM Alliance. The 2 m high and 3 m wide test section has two rotary tables separated by 14 m. The \(5.5\times 5.5\) m contraction at the inlet nozzle, together with a series of honeycomb and a set of fine-grid screens, allows a remarkably low turbulence level (\(I<1\%\)). The flow is accelerated at a velocity between 0.2 m/s and 30 m/s in a closed-loop configuration. Figure 11 shows a schematic view of the test section. Rotary Table 2 was used for the measurements. In the first instance, a test was carried out considering laminar flow conditions; however, the dynamic behavior of the blade was not observed. Subsequently, a specially designed grid was used to generate a turbulent flow, which is produced by the shedding of vortices downstream of the bars. The grid characteristics are the following; the width of the bars \(b=5\) cm, mesh size \(M=35\) cm (i.e., the distance between the centreline of the bars), and the downstream distance from the grid to the section where measurements were performed \(x=5\) m, see Figure 12. The \(x/M>14\) rate assures a homogeneous and isotropic turbulent flow [34], whose main characteristics are a turbulence intensity of \(I=4.2\%\) and an integral length \(L=0.2\) m.  
In a second instance, in order to the evaluate the proposed algebraic identifier of Equation (21), we exposed the wind turbine blade to turbulent flow conditions with a wind speed of 20 m/s in a time span of 0-18 s. At this point, the wind tunnel fan was turned off to generate a free decay response for the interval of 18-40 s, which is then used to synthesize the on-line algebraic identifier (21). Turbulence was generated by placing a fixed pattern grid in front of the blade, as shown in Figure 12, where the measured point on the blade is the same that we used for impact hammer testing. Figure 11: Atmospheric boundary layer wind tunnel of the II-UNAM and FiiDEM Alliance. The experiments were carried out in rotary table 2; the passive grid was fixed to 5 m upstream of the rotary table 2. Time-domain velocity measurements are shown in Figure 13. After 18 s of running the test, as shown on the right side of Figure 13, the fan of the wind tunnel was powered off to generate the free vibrations condition for the blade; at this moment, the algebraic identifier is engaged to estimate in an online fashion the natural frequencies and damping ratios of the first six modes of the blade. In order to derive a method to determine the moment when the algebraic estimator converges to stable calculations, we propose to use a sentinel variable that has a known value of 1, as shown in Figure 14. This unitary variable comes from the characteristic polynomial (11), where the \(N-th\) power coefficient of the complex variable \(s\) precisely equals 1. Figure 12: Experimental set-up for wind tunnel experiments: passive grid, blade, and laser vibrometer. Figure 13: Free decay response in time domain. Complete experiment and decay with fan turned off. The fast and online estimations of the modal parameters are shown in Figures 15 and 16, where the dotted line is the numerical value obtained by the classic off-line peak picking and RFP techniques. Notice that it takes less than a second for the algebraic identifier to give smooth and stable estimations of the modal parameters obtained by solving the roots of the estimated characteristic polynomial of coefficients \(\hat{a}_{k}\). Results of the online modal parameters estimation are reported in Table 3, in addition, the results obtained by the traditional off-line peak picking and RFP methods are compared to the results obtained with the time domain and online algebraic technique. The performance of the evaluated algebraic identifier implies the effective, online, and time-domain determination of two important modal parameters of the system; natural frequencies and damping ratios.