128 K. Sherif and W. Witteveen 11.3 Extension of the Craig-Bampton Reduction Base with Characteristic Attachment Modes for Rolling Contact Problems 11.3.1 Attachment Modes In order to capture the effect of concentrated contact forces, the Craig-Bampton transformation matrix of Eq. (11.8) is supplemented by so-called attachment modes. An attachment mode is defined as the static displacement resulting from applying a unit force at one coordinate of the set ui (or a unit pressure at a predefined sub-area). As previously mentioned, all coordinates of the potential contact area are compressed in the set ui. Further, for the computation of the attachment modes let the set ui be divided into a set uc, which contains all nodes of the potential contact area, and its complement un, which contains all nodes that are not externally loaded during the dynamic simulation. For the computation of the attachment modes, which supplement the Craig-Bampton transformation matrix, one has to restrain all nodes of the set ub and apply a unit force to one coordinate of the set uc and zero forces on both the remainder of the set uc and the set un. Thus, the matrix of attachment modes is computed as 2 4 Kbb Kbc Kbn Kcb Kcc Kcn Knb Knc Knn 3 5 2 4 0 ‰ca ‰na 3 5D 2 4 Rbc Icc 0 3 5 (11.9) where Icc is the (c c) identity matrix and the matrix Rbc denotes the matrix of reaction forces. From the bottom two rows of Eq. (11.9), one can compute the sub-matrices ‰ca and‰na such that the attachment mode matrix‰a is then given by ‰a D 2 4 0 ‰ca ‰na 3 5 (11.10) 11.3.2 Characteristic Attachment Modes As mentioned above, in case of contact problems the number of DOF of the set uc is often very high. This entails that a huge number of attachment modes is needed for approximating the solution to a given accuracy. However, the number of attachment modes may be reduced by seeking a new set of modes that correspond to more natural physical motion of the DOF of the set uc. The new set can be obtained by solving an eigenanalysis on the component mode synthesis mass and stiffness matrices that correspond to the attachment modes, such that KA ¨2 aMA ˆa D0 (11.11) where KA andMA represent, respectively, the component mode synthesis stiffness and mass matrices that correspond to the attachment modes and are defined as: KA D.‰a/ TK‰a; MA D.‰a/ TM‰a (11.12) The eigenvectors ˆa are truncated and only a selected set of modes is retained. Finally, a selected eigenvector from Eq. (11.11) may be transformed from generalized coordinates in FE coordinates by the following linear combination of attachment modes b ‰a D‰aˆa (11.13) The modes of Eq. (11.13) provide, at least in an approximated sense, the principle modes of deformation in the potential contact area. For contact problems where the potential contact area does not vary with respect to time (e.g. bolted connections, welded or screwed parts) the required number of modes for accurately representing local deformations in the contact area is acceptable and the selection of the modes is easy. Thus, by using CAM one can dramatically reduce the size of the system. However, in case of rolling contact problems the potential contact area changes with respect to the rotation angle
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