The augmented trial vector set (replacing the reduction transformation of equation 5) is > @ > @U ) )) OL OL . (14) When structural alterations are localized, relatively few residual vectors adequately describe the content of changed system mode shapes, as demonstrated in an offshore oil platform damage sensitivity study[7]. The above described innovation loses its appeal when structural alterations are well-dispersed requiring utilization of many residual vectors. 4. ROBUST STRATEGY (FOR DISPERSED ALTERATIONS) When structural alterations are well-dispersed, parametric structural changes may affect many physical degrees of freedom and require a description in terms of several independent scaling parameters, “pi”. The expressions for altered stiffness and mass matrices in such a situation are > @ > @ > @ ¦ ' N i i i O K K p K 1 , > @ > @ > @ ¦ ' N i i i O M M p M 1 , (15) The altered system free vibration matrix equation for this situation is > @ > @ > @ > @> @ > @0 1 1 » ) ¼ º « ¬ ª » ) ' ¼ º « ¬ ª ' ¦ ¦ O N i i i O N i i i O p K M p M K , (16) Note that equation 1 describes the baseline system’s free vibration behavior. 4.1 RESIDUAL VECTOR FORMULATION Definition of residual vectors associated with dispersed, independent alterations of a baseline structure, described by equation 1, is accomplished by first computing the lowest frequency mode shapes of the baseline structure (equation 5) as well as the lowest mode shapes associated with each independent alteration of the structure > @> @ > @> @> @ > @0 ' ) ' ) i iL iL i O i iL i OK p K M p M O (for i=1,…,N), (17) The selected value of each independent scaling parameter is sufficiently large to produce a substantial change in mode shapes (with respect to the baseline structure). An initial set of trial vectors that adequately (and perhaps redundantly) encompass all potential (low frequency) altered system mode shapes is > @ > @ NL L L < ) ) ) ... 2 1 (18) This set of trial vectors is expressible as the sum of (a) a linear combination of baseline system mode shapes and (b) trial vectors (that are linearly independent of the baseline system mode shapes) > @ > @> @ > @' < ) < COR OL (19) The cross-orthogonality coefficient matrix is determined based on the following least-squares solution > @ > @> @ > @ > @> @ > @0 ' ) < , ) < ) ) COR M M COR M OL O T OL O OL T OL O T OL , (20) where > @ > @> @> @ ) < 0M COR T OL (21) > @ > @> @ < , )) < O T OL OL OL M ' (22) 378
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