The “purified” trial vector set is linearly independent of the baseline system mode shapes in a manner similar to MacNeal’s residual vectors, as follows: > @ > @> @> @ > @ > @ > @0 '<) <, )) ) < ) )) ){ O OL T O OL OL O OL T O OL T O OL OL OL T O OL T M M M M M M (23) > @ > @> @> @ > @> @ > @0 ' 0 <) <, )) ) <) ) { O OL L O OL T O OL T O OL OL OL T O OL T K M K M K O . While the “purified” trial vector set has the above property, it includes an unnecessarily large number of vectors. An appropriate, substantially smaller set of residual vectors is identified by singular value decomposition of the matrix > @ > @' ' < <O T A M , (24) The singular value decomposition process involves solution of the eigenvalue problem, > @> @ > @> @ U U UM M O A ..... 3 2 1 t t t U U UO O O A suitable cut-off criterion, noted below, that has been employed over the past ten years with good success in defining the suitable reduced trial vector set, is 5 1 10 d U U O ON . (26) The augmented trial vector set (replacing the reduction transformation of equation 5) is > @ > @UM' ) )< OL OL (27) Experience over the past ten years on a variety of structural systems with distributed alterations indicates that the size of the augmented trial vector set is on the order of the number of independent scaling parameters, pi. 4.2 MULTI-PARAMETER SENSITIVITY MODELS The form of the resulting Rayleigh-Ritz, multi-parameter sensitivity model (associated with selected values of the scaling parameters) is > @ > @ > @ > @> @ > @0 1 1 » ¼ º « ¬ ª » ' ¼ º « ¬ ª ' ¦ ¦ M O M N i i i O N i i i O p k m p m k , (28) where > @ > @ O OL T OL O K k ) ) , > @ > @ O OL T OL Om M ) ) (29) > @ > @ i OL T OL i K k' )') , > @ > @ i OL T OL im M ' )') . Recovery of mode shapes in terms of physical degrees-of-freedom is accomplished with > @ > @> @M OL ) ) (30) 379
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