1.2 Problem Description Over the last decades, research has revealed that many aspects have caused the failing of FBSmethods on complex structures. Some of the classic issues are described below, although it should be noted that this overview is by no means exhaustive: Rotational degrees of freedom A lot has been written on the importance of rotational degrees of freedomin dynamic substructuring [7, 18, 9]. The measurement of rotations and torques is however very difficult in practice, so rotational DoF are often neglected. The influence of omitting the rotational information strongly depends on the component’s interface flexibility [4]. To overcome this issue one can either put efforts inmeasuring the rotational DoF [20] or expand translational data to reconstruct the rotational responses [21, 22]. In particular, if one assumes that the interface has only local rigid motions one can construct its response froma minimumof 6 coupling DoF at three nodes. This approach only yields good results up to frequencies where local deformation starts to take place. Continuity of interface In many practical situation the interface between substructures is in fact a continuous surface or curve. Measurements however can usually only be performed on a limited number of discrete points. In that case one can reconstruct the interface continuous behavior using expansion strategies. The simplest method is to consider a rigid behavior aroundmeasured points, but more advancedmethods such as the SEREPmethod [21] can be used to include static deformations obtained froma local finite element model of the interface. Such a local finite element model can also be used to include local dynamic modes in the expansion of the interface measurement points [6]. Recently, the experimental community successfully started using these kind of multiple point connections [23, 19, 2, 15]. Dynamics of joints Dynamic substructuringmethods are sensitive to the couplingmechanisms taking place at the subsystem interfaces. Usually the coupling between the subsystems is either modeled as exact or with linear flexible joints. Inmanyengineering structures, however, people found nonlinear couplingmechanisms between substructures, originating for instance fromfriction between bolted parts. Efforts are made to develop nonlinear models to account for suchmechanisms [8, 10]. The issues described above need to be taken into account in any FBS application to avoidmaking systematic errors. The type of error that we will focus on in this work however, is more general of nature and currently seems to be the biggest bottleneck in FBS: randomexperimental errors. When performing dynamic substructuring using experimental data, measurement errors affect the response of the coupled system. These errors originate for instance frommeasurement noise, collocation errors, added mass effects, etcetera. Since all these errors have a more or less randomperturbation effect on the measured FRFs, we will here designate all these sources as “experimental error”. If the FRFs are polluted with measurement errors the coupling results will be erroneous. The severity of the error in the assembled FRF matrices strongly depends on which FRFs of the subsystems are affected by the measurement errors. The measured interface FRFmatrix for example needs to be inverted in the usual FBSmethods. Due to the matrix inversion, small measurement errors can be significantly amplified, resulting in large errors in the FRFs of the coupled system. Research showed that this is especially problematic around the eigenfrequencies of the components [16, 28, 26], where small errors are greatly amplified. To improve the robustness of FBS a lot of effort has therefore been spend on filtration techniques, using for instance singular value decomposition tomake the inversion less sensitive to small perturbations on the matrix entries [12, 3, 17, 5]. 1.3 Paper Outline In this work we take a different approach to overcome the problemoutlined above. Instead of developing ways to improve the current FBS methods, we try to develop an “inverse free” FBS method that aims to avoid the sensitive matrix inversions. Froma so-called three-field approach to assembly of substructure models, we will develop a procedure for “mixed” assembly of FRF matrices. That is, a method which allows stiffness and flexibility representations of substructure models to be directly assembled without first having to invert (one of) themodels to obtain a common representation. The underlying idea is that each substructuring model involved in the assembly can be expressed in its “natural” form, so (condensed) dynamic stiffness FRFs for models derived fromfinite element (FE) or analytic models and receptance FRFs for models obtained fromexperiments. To this end, the next section presents a general framework for substructure assembly. The resulting “mixed” assembly approach forms a starting point for the new FBSmethod. The key to an alternative FBS method is not only in the mixed assembly however; the strategy for solving the assembled equations is equally important. This aspect is treated in section 3. In section 4 the method is applied to an academic case study, which includes an analysis of the method’s sensitivity and comparison to existing FBS methods. The paper is ended by some conclusions and recommendations in section 5. S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 330
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