378 J. Baqersad et al. Frequency (Hz) 30 75 99 100100 97 51 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) 1.5 9.5 23.0 28.4 28.7 32.2 57.3 75.2 80.4 84.0 1 28.4 25 21 18 3 82.1 1 3 11 19 20 26 85 9910099 Fig. 35.12 MAC comparison between the edgewise modes of the blades of the turbine assembled to the tower and the cantilevered single blade (the shape of the cantilevered single blade is replicated at each 120ı orientation and their mode shapes are compared using the MAC) approximately a 5 % change in the natural frequencies of the assembled system. However, the change in MAC values is less than 3 % from the previous comparison and the MAC results are similar to Fig. 35.12. 35.3 Observations on Edgewise Modes and Comparisons of Flapwise and Edgewise Modes The results of the study show that there are two types of edgewise modes for free-free and cantilevered three-bladed turbines. Some observations based on the studies can be made: 1. Differential edgewise modes: These modes represent approximately 2/3 of the edgewise modes and tend to be independent of the rotational stiffness of the support. They occur at the frequencies close to the natural frequencies of edgewise modes of the cantilevered blade. Therefore, they tend to be predictable. 2. Collective edgewise modes: These modes represent approximately 1/3 of edgewise modes and are dependent on the rotational stiffness of the support. The deflection of the blades in the turbine for these modes approaches the edgewise modes of a single cantilevered blade when the rotational stiffness of the support increases significantly. However, for the wind turbine assembly, the tower could not provide very high rotational stiffness for the edgewise modes. Furthermore, because the edgewise stiffness is provided to the blades through the generator, the rotational stiffness of the support may not always remain constant during wind turbine operation. Therefore, the frequencies of these modes may change during operation based on the resisting torque provided from the generator. The results of the current paper are compared to results from Part I of the paper pertaining to flapwise modes [19] to help identify the similarities and differences between the edgewise and flapwise modes. A comparison between the results is shown in Table 35.4. 35.4 Conclusion In the current research, a comprehensive investigation on the edgewise modes of a three-bladed system was performed. The modes of the three-bladed turbine were categorized in groups of three and the effects of boundary conditions on the mode shapes of the turbine were studied. It was shown that the translational boundary conditions slightly affects differential edgewise modes of the turbine. However, collective edgewise modes significantly change by changing rotational stiffness of the support. This fact can be used to help predict some modes of a wind turbine assembly when edgewise modes of an individual blade in a cantilevered boundary condition are known. The mode contribution matrix showed the effects of the components on the modes of the wind turbine assembly. It was shown that a pseudo-fixed three-bladed turbine had a higher correlation to the assembled wind turbine than compared to a free-free three-bladed turbine. The results of this investigation showed that some modes of the wind turbine assembly could be predicted by using the modes of a single cantilevered blade, independent of the tower and the boundary condition it places on the hub. Acknowledgement This material is based upon work supported by the National Science Foundation under Grant Number 1230884 (Achieving a Sustainable Energy Pathway for Wind Turbine Blade Manufacturing). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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