On the extension of global vibration modes with Ritz-vectors needed for local effects Karim Sherif, Linz Center of Mechatronics GmbH, Altenbergerstr. 69, 4040 Linz, Austria, karim.sherif@lcm.at Wolfgang Witteveen, University of Applied Science - Wels, Stelzhammerstr. 23, 4600 Wels, Austria, Wolfgang.Witteveen@fh-wels Hans Irschik, Johannes Kepler University - Linz, Altenbergerstr. 69, 4040 Linz, Austria, irschik@mechatronik.uni-linz.ac.at Helmut Holl, Johannes Kepler University - Linz, Altenbergerstr. 69, 4040 Linz, Austria, helmut.holl@jku.at Karl Mayrhofer, Siemens VAI Metals Technologies GmbH, Turmstr. 44, 4031 Linz, Austria, mayrhofer.karl@siemens.com Nomenclature FE M mass matrix of FE model FE K stiffness matrix of FE model FE C damping matrix of FE model u nodal DOF vector of FE model u second time-derivative of u u first time-derivative of u f force vector of FE model bu boundary DOF of FE model iu interior DOF of the FE model bn number of boundary DOF in number of interior DOF bf forces acting on bu ext f external force vector kf force vector due to local effects nΦ full matrix of eigenvectors kΦ cΨ matrix of constrained modes cb Φ Craig-Bampton transformation matrix j Ψ matrix of Joint Interface modes j ˆΨ mass-orthogonal local Ritz-vectors uω upper frequency limit q generalized coordinates of reduced model 1 Abstract In Ritz-vector based model reduction techniques, the problem-oriented combination of different kind of Ritz-vectors may significantly influence the quality of the reduction base. It is common to combine global vibration modes with Ritz-vectors, which are necessary to characterize local effects. Even if the global vibration modes and the local Ritz-vectors may be separately orthogonal with respect to the mass and stiffness matrix, the combined reduced system is usually not decoupled. By using common decoupling strategies the separation of the two mode groups is lost. In this contribution, we will present a transformation procedure in order to obtain a combined and decoupled mode base, which is still separable into global vibration modes and Ritz-vectors due to local effects. Due to the clear separation of the two kinds of modes it is possible to give a frequency limit for the relevance of the inertia effects of the second mode group. In case the inertia effects of the second mode group may be neglected, the dimension of the differential equation of motion can be reduced once more again. At our theoretical considerations, an example is presented for the sake of illustration. T. Proulx (ed.), Linking Models and Experiments, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 5, 37 matrix of retained eigenvectors DOI 10.1007/978-1-4419-9305-2_4, © The Society for Experimental Mechanics, Inc. 2011
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