T transformation matrix applied in the reduced technique ĭ, mĭ , sĭ respectively complete eigenvector matrix, master and slave subset cĭ , nĭ respectively constraint and normal modes of Craig-Bampton reduction technique 1 m,gen ĭ generalized master subset eigenvector inverse matrix 2 rȦ modal eigenvalues diagonal matrix INTRODUCTION In the experimental dynamic analysis, the selection of the points where the accelerometers have to be placed is a great problem. In the same way, when a model reduction is performed, it is not easy to define the master nodes. This is due to the deep relation between the nodes selection and the fundamental physical and dynamic properties of the body, in other words, the spatial incompleteness of the described degrees of freedom of a model and the modal model incompleteness of the dynamic behavior that the model represents could deeply change with respect to the nodes used for modal analysis or modal reduction. The places where sensors have to be put are usually selected on the basis of engineer’s experience. It then becomes important to identify a method for choosing the master dofs that allow a good depiction of the dynamic behavior of the reduced model and an improvement of the numerical stability of modal matrices. Therefore, the objective of sensor placement strategy is to select locations that render the corresponding target mode shape partitions as linearly independent as possible and, at the same time, maximize the modal responses within the sensor data. These effective independence criteria (EI) are recommended by Friswell and Mottershead [1] for modal testing and modal updating. Another criterion is based on modal kinetic energy estimations that can be used to help determine a good candidate node set. In [10] the proposal of a connection between these two influencing sensor placement methods has been analyzed. In [12] different optimal sensor placement techniques, based on the maximization of the Fisher information matrix and the EI criteria or based on energetic approaches, such as the eigenvalue vector product (EVP), have been compared and applied on a bridge structure with the aim of maximizing the dynamic data information is depicted. A complementary or alternative approach for master dofs selection in dynamic condensation approaches and optimal sensor placement techniques has to implement the following keypoints: x integration of effective independence criteria and modal kinetic energy approaches in an adaptable and flexible formulation in order to maximize the modal description and the orthogonality properties and minimize the ill-conditioning of mass and stiffness matrices of the reduced model; x selection of master dofs aside from the knowledge of a starting dynamic model and, therefore, with a very low computational cost; the approach has to reduce both analytical and experimental data, starting from the knowledge of the geometrical data of candidate node set and the modal properties; x increment of model dofs and modal order by adding some auxiliary dofs to previously selected master nodes without iterative procedure in order to simplify the modelling process taking into account important nodes or the connection between sub-structures. In order to validate the alternative approach named MoGeSeC (Modal-Geometrical Selection Criterion), in this paper, an extensive application of the proposed criterion is applied on a automotive component that possess rich dynamic properties, such as a crankshaft, and the results are compared with obtained from the EI criterion. SELECTION CRITERIA OF MASTER NODES: MOGESEC AND EI In order to maximize modal properties and eigenvector orthogonality and to reduce the ill-conditioning of the reduced model the Modal-Geometrical Selection Criterion of master dofs is here proposed and compared to the most used criterion called Effective Independence. A brief discussion on the mathematical basis of the two criteria is presented, then a comparison between the obtained results is conducted. 282
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