34 Effects of Boundary Conditions on the Structural Dynamics of Wind Turbine Blades—Part 1: Flapwise Modes 357 whereui anduj are displacement vectors of the modes i and j respectively. MAC values close to 1.0 indicate strong similarity between vectors, where values close to 0.0 indicate minimal or no similarity. The MAC values are typically presented in percent (0–100). The MAC values are typically calculated for the vectors with the same length. In some parts of the current work, however, the correlation between a component and a component of an entire assembly is desirable. For instance, the correlation between the mode shapes of an individual blade, and a single blade in a threebladed wind turbine assembly is desirable. The number of degrees of freedom for the mode vectors of the single blade and the three-bladed turbine are different. In these cases, only the displacements of the nodes that existed in both parts are paired up and used in the MAC calculations. Therefore, in order to compare a single blade to one of the blades in the assembly, the single blade displacement is overlaid to a blade in the assembly and only the displacements of nodes that are paired up contribute in the MAC. 34.2.2 Mode Contribution Identification The mode contribution matrix is used to identify the contributions of component modes in the assembled system modes. Based on the structural dynamic modification theory: ŒU2 D UA 1 UB 1 ŒU12 (34.2) where [UA 1 ] and [UB 1 ] are respectively the mode shapes of component A and B before assembly which are organized into a partitioned matrix, [U2] represents the mode shapes of assembled system. [U12] is mode contribution matrix which can be used to obtain the modes of an assembly system by using modes of unmodified components. On the other hand, the mode contribution matrix ([U12]) indicates which modes of components are more critical to obtain the modes of the assembly. The mode contribution matrix can be determined using the following equation [30]: ŒU12 D UA 1 UB 1 T ŒM2 ŒU2 (34.3) where [M2] is the assembled system mass matrix (the subscript “2” indicates that the mass matrix of the assembly was used in the equation whereas the subscript 1 indicates that it is the mass of just one component and not of the three assembled components). The matrix can also be extracted by applying structural dynamic modification theory which essentially leads to the same results as Eq. (34.3). However, in the current paper, Eq. (34.3) is used to extract [U12]. 34.3 Model Description and Cases Studied In order to study effects of boundary conditions on wind turbine blades, seven cases are examined in the current work. A flowchart of the work that was completed in the paper is shown in Fig. 34.1. In the current work, all finite element models were developed using ABAQUS [31]. However, model modifications, mode shape correlations, and matrix multiplications were performed using MATLAB [32]. 34.3.1 Case 1: Modeling a 3D Wind Turbine Blade Using Beam Elements In order to investigate the dynamic characteristics of a wind turbine, a solid element model of the Skystream 4.7™ from Southwest Windpower was initially developed in ABAQUS using the CAD model provided by the manufacturer. The solid finite element model is shown in Fig. 34.2. An extensive description of the finite element modeling and material properties of the blade along with the correlation of the solid models to the experimental measurements can be found in the references [23, 33]. A solid element model can be used to study deformations or to extract stress and strain in the blade; however, using the solid model with numerous DOFs for the study of boundary conditions would be very computationally expensive. Furthermore, in the solid model, due to complexity of the geometry, modes that are essentially flapwise may show small
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