The techniques above were used to find sensor placement locations for mode shapes, Ritz vectors, as well as a combination of both basis vectors. The NASA eight-bay truss model was used as the analytical test-bed. The Modal Assurance Criteria (MAC) was utilized to compare the sensor placement results and obtain a representation of how well each technique performed. Section 2 presents a detailed look into the Ritz vector extraction method. Section 3 details the sensor placement techniques used. The numerical studies and results are presented in section 4. Finally, the concluding remarks are given in section 5. 1 Ritz Vector Analysis The dynamic equilibrium equation for a discrete n-degree-of-freedom (DOF) structure can be expressed as ( ) ( ) ( ) ( ) xt+ xt+ xt =fut M C K & && (1) where M, C, and K are the ( ) n n× mass, damping and stiffness matrices, respectively, ( ) x t is the ( )1×n position vector, f is the ( )1×n force influence vector, and ( ) u t is the scalar force input signal. The over-dots represent differentiation with respect to time. The first Ritz vector, representing the deflection of the structure under a unit static load [4], is found from the solution of v = f * 1 K , (2) where * 1v is the non-normalized Ritz vector. Although Wilson [4] mass-normalized the Ritz vectors, it was found that in experimental identifications it is more appropriate that the Ritz vectors be unit-normalized as the mass matrix is unknown [5, 6]. The first Ritz vector is unit-normalized as ( ) * 1 * 1 * 1 1 v v v v T = . (3) The subsequent Ritz vectors are found from the solution of i-1 * i v= v K M . (4) Each Ritz vector is orthogonalized using a Gram-Schmidt orthogonalization process ∑ = i-1 j 1 j j * i ivˆ = v - c v (5) where * j T j j c = v v . (6) After each Ritz vector is orthogonalized, it is unit-normalized i Ti i i vˆ vˆ vˆ v = . (7) 2
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