Uncoupling Techniques for the Dynamic Characterization of Sub-structures Batista, F. C.1, Maia, N. M. M.2 1 Polytechnic Institute of Leiria, School of Technology and Management, Morro do Lena, 2401-951 Leiria, Portugal 2 IDMEC/IST, Tech. Univ. of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal ABSTRACT The characterization of the dynamic behavior of sub-structures may be of great importance in certain complex structures. A possible approach is through the use of uncoupling techniques. This is the subject of the present paper, where various issues are raised and different possibilities to circumvent the encountered problems are discussed. These problems are mostly due to numerical instabilities associated to the calculation procedures, namely to the ill-conditioning that often appears and causes complications in the inversion of the involved matrices. The location of the possible points of measurement may also influence the results. Several variations to the main algorithms are tried in order to reduce the numerical problems. This study will be based on numerical simulations to compare the various approaches and to draw some final conclusions. 1 INTRODUCTION Recently the subject of structural uncoupling (also referred to by some authors as decoupling) has regained interest among the scientific community [1, 2]. Seeking for the dynamic characterization of sub-structures, there are several techniques combining both experimental and analytical/numerical methods. They can be classified in three different groups, according to the working domain, either modal, mechanical impedance or frequency domain. In the modal domain one has methods that use modal properties, like the classic Craig-Bampton [3], known as ComponentMode Synthesis (CMS). Meanwhile, the Dual Craig–Bampton method has been presented by Rixen [2]. In what mechanical impedance is concerned, one has the system mass, stiffness and damping matrices, as well as external forces and connection forces. Ahmadian et al. and Jalali et al. [4, 5] developed a method that allows the identification of a bolted joint using the physical matrices of the system. In the frequency domain there are a vast number of methods based on the classic one developed by Jetmundesn [6], known as “FRF-based sub-structuring method” (FBS) as classified by de Klerk et al. [7]. D’Ambrogio [1] developed two methods based on system impedance and mobility using only the connection co-ordinates. As it will be explained in this work, one of the methods presented here will have the same result. In Jetmundesn’s method [6] the inherent numerical problems associated to ill-conditioned matrices are well known; moreover, Frequency Response Function (FRF) measurements have inherent associated noise that make the obtaining of acceptable results a difficult job; several strategies have been appointed to overcome the problems, see for instance Ren and Beard [8]; Maia et al. [9] proposed a solution avoiding the connection coordinates. In this work several approaches for the classic uncoupling Jetmundesn’s method are developed, analyzed and compared, through numerical examples using beams. Adding local inertial elements, a possible way of circumventing the illconditioning problems is proposed. 2 THEORETICAL FORMULATION 2.1 FRF coupling Coupling techniques are well known (e.g., see [10]); they can be divided into spatial coupling methods, based upon the system matrices (M, K e C), modal coupling methods, using modal properties (natural frequencies, damping ratios and mode shapes) and frequency response function coupling (FRF coupling). Let us concentrate in this latter one. Let us consider two sub-structures A and B, connected though some co-ordinates as represented in figure 1. Together, they form structure C. Let i represent the co-ordinates exclusively belonging to A, k the ones exclusively belonging to B and j the connection co-ordinates. T. Proulx (ed.), Linking Models and Experiments, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 5, 383 DOI 10.1007/978-1-4419-9305-2_28, © The Society for Experimental Mechanics, Inc. 2011
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