252 L.G. Trujillo-Franco et al. In this work we propose a novel system parameter monitoring scheme based on algebraic identification. We perform modal testing to the nominal or undamaged structure in order to have a reference for comparisons, and then, we induce changes directly on the structure, simulating failures (e.g., a loose or broken screw or a mass addition), in order to identify in a fast and online fashion the impact of each situation on the system modal parameters. 31.2 Illustrative Vibrating Mechanical System Consider the n Degrees-Of-Freedom (DOF) vibrating mechanical system consisting of a six story building-like structure as shown in Fig. 31.1, where xi, i D1, 2, : : : , n, are the displacements of 6 masses representing the floors or Degrees-OfFreedom (DOF) of the structure, respectively. We model the columns as flexural springs with equivalent stiffness ki and consider the structural damping ratios ci as Rayleigh damping [8]. The simplified mathematical model of this flexible mechanical system of 6 DOF under harmonic and unknown excitation f is given by: MRx CCPx CKx Df .t/; x; f 2R6 (31.1) where x2R6 is the vector of generalized coordinates (displacements) of each floor respect to the main frame reference, and M, Cand Kare symmetric inertia, damping and stiffness 6 6 matrices, respectively, given by: MD 2 66 66 66 66 64 m1 0 0 : : : 0 0 0 m2 0 : : : 0 0 0 0 m3 : : : 0 0 : : : 0 0 0 : : : 0 m6 3 77 77 77 77 75 ; CD 2 66 66 66 66 64 c1Cc2 c2 0 : : : 0 0 c2 c2Cc3 c3 : : : 0 0 0 c3 c3Cc4 : : : 0 0 : : : c6 0 0 0 : : : c1 c6 3 77 77 77 77 75 ; KD 2 66 66 66 66 64 k1Ck2 k2 0 : : : 0 0 k2 k2Ck3 k3 : : : 0 0 0 k3 k3Ck4 : : : 0 0 : : : k6 0 0 0 : : : k6 k6 3 77 77 77 77 75 (31.2) The modal analysis representation of the mathematical model (31.1) is defined in terms of the modal or principal coordinates qi, where i D1, 2, : : : , 6 as follows (see, e.g., [1]) Rqi C2 i!i Pqi C! 2 i qi D‰ Tf (31.3) Fig. 31.1 Six story Building-like structure. (a) Schematic diagram. (b) Physical plant
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