2. NOMENCLATURE Matrices: [A] Generalized mass (equation 24) [ 'K] Stiffness alteration [COR] Cross-orthogonality (equation 3) [ 'M] Mass alteration [I] Identity [)] Mode shapes [K] Stiffness [*] Unit loads (equation 9) [M] Mass [<] Trial vectors [OR] Orthogonality (equation 4) [ 'k] Generalized stiffness alteration (equation 7) [U] Physical displacement [ 'k] Generalized mass alteration (equation 7) [k] Generalized stiffness (equation 6) [I] Generalized mode shape (equation 5) [m] Generalized mass (equation 6) [O] Eigenvalues (equation 1), (2Sx frequency)2 [q] Generalized displacement (equation 33) Variable and subscripts: p Structural alteration parameter (equation 2) H High frequency mode shapes a Boundary partition (equation 31) L Low frequency mode shapes n Interior partition mode shape (equation 33) O Baseline system (eq. 1), interior partition (eq. 31) U Residual (equation 12) S “Static” (equation 9) 3. SENSITIVITY ANALYSIS STRATEGIES The present discussion focuses on Rayleigh-Ritz procedures that address structural sensitivities due to stiffness and mass alterations described by large (as opposed to small) parametric variations. Therefore formulations that address computation of eigenvalue and mode shape derivatives are not considered. 3.1 EXACT MODAL ANALYSIS OF BASELINE AND ALTERED STRUCTURES The matrix equations describing exact free vibration of baseline and altered structures, respectively, are > @> @> @> @> @>@0 ) ) O O O O O K M O (1) and > @> @ > @> @> @ > @0 ' ) ' ) O K p K M p M O O (2) It is implicitly assumed that the stiffness and mass changes scale linearly with respect to the parameter, p. Therefore, changes in “beam” depth may not be directly applied, since the axial stiffness (AE) scales linearly with depth and the flexural stiffness (EI) scales as the cube of depth. The appropriate formulation for equation 2 permits linear sensitivity of “AE” and “EI” separately. The relationship between mode shapes of the baseline and altered structures is expressed as the cross-orthogonality of orthonormal mode shape sets, > @ > @> @> @ ) ) 0 M COR T O (3) where > @ > @> @> @ > @o T O o M OR ) ) , 0 0 , > @ > @> @> @ > @ ) ) , 0 OR M T . (4) 376
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