The considered FE structure with n nodal DOF consists of two jointed substructures as outlined in figure 1. Note that for the reasons depicted in [11] the jointed structure is considered as a single structure with 6 rigid body DOF. The external forces Bf are acting on the structure exclusively via the interface B and the according DOF are collected in the (nB x 1) vector Bx . The nodal DOF of Bx are outlined in figure 1 by white dots. The non-linear contact forces, which are represented by IJ f , act on the nodal DOF of the joint which are collected in the (nIJ x 1) vector IJ x and outlined in figure 1 by grey dots. The remaining (n - nB - nIJ) DOF are denoted as Rx . According to that scheme, the vector of DOF can be written as ª º ¬ ¼ T T T T B IJ R x x x x . (2) In a first step, the number of DOF n is reduced via a linear superposition of Ritz vectors in the form of x Xq, (3) where the (n x r) matrix X contains r Ritz vectors in its columns. Please refer to [10] for a more detailed description of the reduction procedure. As mentioned, the reduction is based on a discretization of the entire joint area in r subareas. Based on these reduction the reduced (r x r) mass and stiffness matrixes can be given as red T M X MX and (4) red T K X KX. (5) The computation of the JIM in the space of X is based on the eigenvalue problem ª º ª º ¬ ¼ « » ¬ ¼ 2 red *,red red *,red K ȍ M ĭ 0 (6) where r eigenvalues are stored in the (r x r) diagonal matrix *,red ȍ and r eigenvectors in the (r x r) matrix *,red ĭ . The significant reduction of DOF is obtained by considering just the first k eigenvectors for the further considerations where k << r. The considered eigenvectors a collected in the (r x k) matrix red ĭ . The JIM for the entire structure can be computed by applying the reduction rule (3) in the form of JIM red ĭ Xĭ , (7) where the (n x k) matrix JIM ĭ contains the JIM for the joint contact. In the final Ritz vector based computation the matrix of JIM is suggested as enrichment of established mode bases like the one of Craig-Bampton [12]. An exemplarily final transformation rule can be given as ª º ¬ ¼ Craig-Bampton JIM x ĭ ĭ q. (8) As already mentioned in the introduction three drawbacks of this method can be given, namely: x In equation (6) the JIM are obtained by a generalized eigenvalue problem. In principle the JIM are the vibration modes of an artificial structure which is difficult to interpret. x Consequently, the eigenvalues of (6) are more a mathematical quantity than a physical meaningful frequency. These eigenvalues can not be used as an ‘a – priory’ estimation of the required number of JIM. x The reduction (3) determines the quality of the latter JIM. With an increasing number of subareas k the quality of the JIM as well as the computational effort is increasing. The question, whether a particular selection of k is high enough is an open issue. 4. Short Review: POD In this section, a short introduction in the proper orthogonal decomposition is given based on [14]. Let us assume p independent (m x 1) vectors 1y to py which are collected in the (m x p) matrix Y. Proper orthogonal decomposition (POD) of rank g delivers g orthonormal (m x 1) vectors 1u to gu which approximate the space spanned by Y optimal in a Euclidean sense. Note, that the vectors 1u to gu are named proper orthogonal modes (POM). 21
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