2 Introduction Accurate Finite Element (FE) models typically require a large number of degrees of freedom (DOF). Due to this huge number of DOF, the time integration of such systems is very expensive. Component Mode Synthesis (CMS) methods significantly reduce the dimension of structural dynamic models, which leads to reduced simulation time and reduced memory requirements. The term ‘component modes’ characterizes Ritz type trial vectors which are used as base vectors in order to approximate the displacement field of a single component. There are two general types of CMS methods; they are known as the fixed-interface and the freeinterface approaches. The fixed-interface CMS technique presented by Hurty [1, 2] and modified by Craig and Bampton [3] is widely used, since the reduction procedure is straightforward and typically produces highly accurate models with relatively few component modes [4]. The free-interface CMS approaches are more attractive than the fixed-interface methods when the component modes are obtained from modal testing. Highly accurate free-interface CMS methods have been developed e.g. by Craig und Chang [5, 6]. The Craig-Chang method is a modified version of the widely used Rubin method [7] and the MacNeal method [8]. A detailed discussion and a comprehensive review on the different CMS methods are given in [9], [10] and [11]. However, in this contribution only the fixedinterface Craig-Bampton method is considered. The Craig-Bampton CMS method can be effectively and consistently used for the reduction of large linear elastic structures, but it is not applicable for structures with nonlinear local effects in terms of contact problems in its original form. This is due to the fact that the latter reduction method does preserve the nodal DOF at interfaces and each interface DOF thus leads to an additional Ritz-vector in the mode base. For an accurate and local application of contact and friction laws in the contact region (nonlinear local effects), the nodal DOF of a contact surface need to be considered as interface DOF, and this leads to an inefficient number of modes. A reduction method for structures with nonlinear local effects has been published in [12], by Apiwattanalunggarn. In the latter approach the fixed-interface Craig-Bampton method has been extended by nonlinear normal modes. Witteveen and Irschik [13] presented an extension of the CraigBampton method that permits solving contact problems of jointed structures. This is possible by extending the component modes of the Craig-Bampton method with local Ritz-vectors or so called Joint Interface Modes (JIM). The latter modes form together with the component modes of the CraigBampton method a suitable Rayleigh-Ritz coordinate transformation for nonlinear contact problems when the potential contact area is time invariant. Due to the fact that for the computation of the JIM, Newton’s third law (principle of equivalence of forces) across the contact region is explicitly accounted at the time of mode generation, only a small number of JIM has to be considered in order to approximate the solution of the contact problem in a reduced subspace. It is the scope of the present contribution to present a decoupling procedure for a mode base consisting of global vibration modes and Ritz-vectors needed for local effects, where the resulting decoupled mode base is still separable into global vibration modes and Ritz-vectors. The paper is organized as follows. Firstly, the fixed-interface Craig-Bampton CMS method and its extension by the JIM are reviewed. Then modified JIM are introduced in order to achieve a reduced system with a block-diagonal decoupled mass matrix. Finally, a generic 501-mass nonlinear structural system is used as an example to demonstrate the method, and some conclusions are drawn. 3 Component mode synthesis The next two subsections give a short review of the fixed-interface Craig-Bampton CMS method and of the computation of its component modes. 3.1 Fixed-interface normal modes and interface constraint modes The equation of motion of a FE model can be formally written as FE FE M u K u f + = , (1) where FE M and FE K are the symmetric mass and stiffness matrices, f represents the time-varying force vector, and the vector u contains the nodal DOF (usually displacements) of the FE model. The vector u is the second derivative of u with respect to time. The following partitioned form of Eq.(1) will 38
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