64 R. Schultz 5.2 Theory This section describes some of the frequency based substructuring and shaker electro-mechanical modeling which was used in this work. This is not intended to be an in-depth examination into either topic. Instead, the following sections provide some of the concepts, terminology, and techniques used to model shakers and connect shaker models to DUT models in this work. 5.2.1 Frequency Based Substructuring This section describes the basic theory behind the FBS used in this work. Far more depth can be found in many papers, including Allen et al. [1] and the course notes by Avitabile [2]. Consider a two-component system consisting of Component A and Component B, as shown in Fig. 5.1. Component B has an input applied at the XB,i degree of freedom (DOF), the outputs are at the o DOF and there is a single connection at the c DOF. The frequency response functions (FRFs) needed for FBS are the between the connections, inputs, and outputs as shown in Eq. (5.1) where the FRF matrices are defined to have outputs on the row dimension, inputs on the column dimension, and frequency lines on the page (third) dimension. The drive point, or connection-connection, FRFs are also needed. With these various FRFs, the FRFs of the coupled system comprised B A fin xA,o xB,c xA,c xB,o xB,i x3 M3 M4 x4 B A M1 K01 M2 K12 K23 x2 x1 K34 K45 x5 M5 M6 x6 K56 C f1 f3 f5 Fig. 5.1 Example two component (left) and three component (right) systems
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