20 R. Fagiani et al. 2.2 FRF Coupling This section is dedicated to revise a methodology for coupling a flexible structure to another structure (e.g. a source machine) using the FRF matrices, e.g. [4]. The objective is to describe an approach for evaluating the power input from a large machine to a flexible structure when forces and responses data of the separated structures can be measured and only acceleration/velocity responses data of the whole coupled structure in operation are available. Figure 2.1 shows structure 1 (source) connected to structure 2 (receiver) through multiple points together with the model the free body diagram for each substructure. Each connection is modelled using a connector system Ci. These can be a rigid connections or flexible-damped connections. Dynamics of structures 1 and 2 is often studied using a discretised model, e.g. as in standard FE or modal analysis, and generally the DOFs at which the internal forces act, u1 D Œu11; u12; : : : ; u1n TI u 2 D Œu21; u22; : : : ; u2n T, are a subset of the DOFs used to discretise the structure, y 1 D Œy11; y12; : : : ; y1m T and y 2 D Œy21; y22; : : : ; y2q T. The same situation arises in experimental modal analysis when the responses evaluated at the connection points are a subset of the responses used for characterizing the dynamic structure behaviour. This situation is pictured in Fig. 2.1. In this case, the DOFs of structure j corresponding to the connected ones can be expressed in terms of the DOFs of the structure using an influence matrix Wj D ŒWj1 : : : Wjn . Here [Wj1 : : : Wjn] are column vectors of zeros and ones, e.g. Wj1 DŒ0;0;1;0:::0 T, where the ones identifies the position of the DOFs corresponding to the DOFs of structure on which the internal forces FCj act. Thus the influence matrix relates displacements and internal forces of structure 2, structure 1, and connectors Cj as FC1.!/ D W1F1C; u1 DWT 1y1 FC2.!/ D W2F2C; u2 DWT 2y2 (2.1) Dynamic equilibrium of each free body diagram in Fig. 2.1 can be written as 8< : FCFC1 DH1.!/y1 F1C DHc.!/.u1 u2/ FC2 DH2.!/y2 )8< : F W1F1C DH1.!/W1u1 F1C DHC.!/.u1 u2/ FC2 DH2.!/y2 (2.2) Since FC1 D W1F1C, combining the first and the second of Eq. (2.6) the relative displacement becomes .u1 u2/ D ŒH1.!/W1 CW1HC.!/ 1F ŒH1.!/W1 CW1HC.!/ 1H1.!/W1u2. The internal force transmitted to structure 2 is thus Fig. 2.1 Schematic representation of a machine-source (structure 1) connected to a receiving structure (structure 2) through multiple points: (a) coupled system, (b) uncoupled system
RkJQdWJsaXNoZXIy MTMzNzEzMQ==