110 J. Gross et al. Since the generalized coordinates QqB CBIP are a mix of physical DoF related to the interface and modal DoF related to the free part of the substructure, this representation of Bis similar to a Craig-Bampton representation, i.e. it can be easily connected to other FE models. It is not precisely a CB representation of Bbecause the stiffness matrix given in Eq. (10.22) is not block diagonal. Unlike the CB representation, the modal degrees of freedom can not be truncated due to the stiffness coupling resulting from QK B qc and QK B cq. A disadvantage of the CB-IP method is that due to the retention of the interface DoF indefinite mass and stiffness matrices can be obtained. Indefinite mass and stiffness matrices result in nonphysical eigenvalues. The matrices in Eqs. (10.21) and (10.22) can be written as 2 4 QM B qq QM B qc QM B cq QM B cc 3 5D " IC T T TMFE qc T MFE cq MFE cq C TMFE qc T MFE stat# (10.24) 2 4 QK B qq QK B qc QK B cq QK B cc 3 5D 2 4 Ÿ!2 r Ÿ C T Ÿ O!2 r Ÿ FE T Ÿ O!2 r Ÿ FE T Ÿ O !2 r Ÿ FE T Ÿ O !2 r Ÿ FE KFE stat 3 5 (10.25) with D O˚ FE m ˚ C m and D O˚ FE m FE m . The partitions corresponding to the physical interface DoFs, QM B cc and QK B cc, are negative by design of the CB-IP method. The negative mass and stiffness associated with the interface of the TS FE has no equivalent in the modal space of Cto cancel out with while being subtracted. These partitions make the matrices of the substructure Bindefinite. One possible way to overcome this problem is to project the connection DoFs onto the modal space of the combined structure C, using uFE c D ˚ C c qC. In fact, this way was used to calculate the free normal modes in Sect. 10.4.2. When the experimental based representation of the substructure of interest Bis coupled to another substructure of comparable or larger size than the transmission simulator, the excess mass and stiffness due to the TS at the interface properly cancel out because of the coupling terms in QM B CBIP and QK B CBIP. An advantage of the representation obtained with the CB-IP method is that the fixed-interface normal modes and frequencies of the substructure of interest B can be determined directly from Eqs. (10.24) and (10.25) by crossing out the block row and column corresponding to the interface DoF. Thus the fixed-interface normal modes and frequencies for the substructure of interest B can be obtained by performing a free-free modal analysis of the combined substructure C and subsequently solving the EVP for QM B qq and QK B qq. In the partitions corresponding to the modal DoF, QM B qq and QK B qq, the mass and stiffness associated with the TS fixed-interface normal modes is subtracted. The positive mass and stiffness associated with the modes describing the motion of the interface DoF of the combined structure Cstill remains. 10.3 Modeling of the Wind Turbine Rotor Assembly The application of the TS method to structures that are composed of both numerically and experimentally derived substructures requires very accurate models. The Ampair 600 wind turbine blade was investigated by Johansson et al. [7] in great detail. Material tests performed by the authors revealed the structural layup of the composite material as well as material properties which are crucial information for analysts. On this basis the FE modeling and subsequent model updating with respect to experimental modal analysis results of the wind turbine blade can be performed using the HyperWorks suite from Altair Engineering. The FE model of the blade is designed as a composite material structure consisting of both, 2-D shell to model the skin and 3-D solid elements for the core. The hub is considered as a homogeneous structure for the sake of simplicity and is modeled with 3D-solid elements. An overview of FE model parameters is given in Table 10.1. Table 10.1 FEmodel parameters of the blade Eltype No. elts No. nodes Skin Six nodes quadr. trias 4577 9302 Core Eight nodes quadr. tetras 7998 15,888 Hub Eight nodes quadr. tetras 47,227 81,934
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