x The strength of the FEM is the high resolution of the domain of interest which leads in general to a satisfying accuracy. The drawback of this method is the huge number of degrees of freedom (DOF) which makes it unfeasible for time integration. x The strength of a Ritz vector (commonly called ‘mode’) based approach is the dynamics (time integration) of linear structures. Various Ritz vector based reduction methods have been presented during the last decades. Reviews have been done, among others, by Craig [3], Noor [4] and Zu [5]. Typically, the nonlinearity due to the contact inside a joint has been either neglected or somehow linearized. While this approach preserves the computational efficiency, it may lead to remarkable errors, see exemplarily [6] and [7]. Another approach was introduced by the author and Prof. Irschik at the IMAC 25th [8], 26th [9] and 27th [10]. An extended version of [8] and [9] can be found in [11]. The main idea of the latter approach is the enrichment of existing, well proven mode bases (e.g. Craig – Bampton [12]), by certain problem oriented ‘contact modes’, which we called joint interface modes (JIM). The JIM represent a generalization of the nodal FE DOF of the involved joint surfaces. It has been demonstrated in [8] and [13] that the convergence is superior to the one of the so called interface modes, where the involved contact surfaces are not related to each other at the time of mode generation via Newton’s 3rd law. However, this approach has several drawbacks, namely: x The JIM are obtained by a generalized eigenvalue problem of statically reduced mass and stiffness matrices. So, in principle, the JIM are computed like vibration modes, which is difficult to interpret in a physical sense. x Consequently, the eigenvalues of the mentioned eigenvalue problem are more a mathematical quantity than a meaningful physical frequency for the estimation of the required number of JIM. So to say, the method does not provide an ‘a – priory’ estimation of the required number of JIM. x For the latter mentioned static reduction Guyans method was suggested in [8]. As an improvement in terms of computational efficiency it has been suggested in [10] to discretize the joint area in a certain number of subareas. An open question up to now was, if the chosen discretization is fine enough to represent the joints mechanical characteristics. In this paper a modified computation of JIM is suggested. Instead of a generalized eigenvalue problem of the statically reduced mass and stiffness matrix a orthogonal decomposition (POD) of the statically stiffness matrix is suggested. In a first section the former introduced JIM computation approach is briefly reviewed. Then a rough review on POD is given followed by it’s application for the computation of JIM. Finally two numerical examples are discussed. It will be demonstrated that the JIM gained by a POD based approach converge as quick as the former method and the Hankel singular values can be used for an a priory estimation of the number of required JIM as well as for the evaluation of the joint area discretization. 3. Short review on the former introduced approach for the computation of JIM A jointed flexible body modeled by the FEM can be represented by the equation of motion in the form of B IJ Mx Kx f f , (1) where xis the (n x 1) vector of nodal DOF and the (n x n) time invariant matrices M and K denote the bodies mass and stiffness. The load is a combination of the (n x 1) vectors of external loads Bf and contact forces IJ f . Figure 1: Arbitrary Finite Element structure with a joint 20
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