3.2 TRUNCATED MODE SET APPROXIMATION The most fundamental Rayleigh-Ritz approximation, commonly used in structural dynamics modification[5], employs a truncated set of low frequency eigenvalues as the reduction transformation described by > @ > @> @M OL ) ) , (5) where the reduced baseline structure stiffness and mass matrices, respectively, are > @ > @ > @ OL O OL T OL O K k O ) ) , > @ > @ > @ OL O OL T OL Om M ) ) , , (6) the reduced stiffness and mass sensitivity matrices, respectively, are > @ > @ OL T OL K k' )') , > @ > @ OL T OL m M ' )') , (7) and the reduced altered structure free vibration equation is > @> @ > @> @> @ > @0 ' , ' M O M O p m p k OL OL . (8) A well-known result of this type of trial vector reduction strategy is that the approximate altered structure eigenvalues are generally higher than results for the exact solution, and the approximate mode shapes do not closely follow the exact shapes when parametric alterations are large. 3.3 RESIDUAL VECTOR AUGMENTATION (FOR LOCAL ALTERATIONS) The static displacements for a baseline structure subjected to unit loads (at physical degrees of freedom where the structure is to be altered) described by the columns of a load array, [*], are the solutions of > @> @ > @ * O S K U , (9) A low frequency modal approximation of static displacements for the above system employs the transformation, > @ > @> @ OL L SL q U ) , (10) resulting in the approximate static displacements, > @ > @> @> @ ) * T L OL Lq 1O , > @ > @> @ ) ) * T OL L OL SL U 1O , (11) The difference between the exact and approximate static solutions defines MacNeal’s[6] quasi-static residual vectors, > @ > @ > @ > @> @ > @> @ < ))*{))* T OH H OH T OL L OL O SL SU U K 1 1 1 O O U , (12) which have been mathematically proven to be the quasi-static displacements associated with all of the high-frequency mode shapes. An orthonormalized set of residual vectors is defined by solution of the residual eigenvalue problem[7], > @ > @U U U < <O T k K , > @ > @ U U U < <O T m M , (13) >@> @> @> @>@ > @0 U U U U U M O M k m , > @ > @> @ U U U M ) < 377
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