154 R. N. Coppolino [ ] = Φ1L 2L . . . NL . (13.63) 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. The “purified” trial vector set, [ ] is defined in a manner similar to MacNeal’s residual vectors [22]. The “purified” trial vector set includes an unnecessarily large number of vectors. A substantially smaller set of residual vectors is identified by singular value decomposition [8], resulting in the augmented trial vector set, OL = OL ’ . (13.64) The form of the resulting SDM, multi-parameter sensitivity model is, kO+ N i=1 pi [ ki] [ϕ] − mO+ N i=1 pi [ mi] [ϕ][λ] =[0] , (13.65) where, [kO] = T OLKO OL , [mO] = T OLMO OL , [ ki] = T OL Ki 2 , [ mi] = T OL Mi OL . (13.66) Recovery of mode shapes in terms of physical degrees-of-freedom is accomplished with, [ ] = OL [ϕ]. (13.67) 13.9.3 RMA Solution Qualities Since its introduction in 2001, RMA has exhibited the capability to accurately follow modal sensitivity trends over an extremely wide range of parametric variation. The simple cantilevered (planar) beam example, provided in Fig. 13.32 below, demonstrates typical RMA performance (“100%” is baseline). Actual cross-orthogonality checks are also excellent. 13.9.4 Test-Analysis Reconciliation Using Cost Function Optimization Reconciliation of a test article’s finite element model with experimental modal data, if conducted in an objective and systematic manner, requires minimization of a cost function. A variety of modal cost functions are employed by many investigators. The present discussion describes a particular cost function that describes a balanced modal frequency and mode shape “error” relationship. Minimization (or optimization) of the modal cost function’s error norm employing gradient based and Monte Carlo strategies are evaluated. Consider the standard expression for the undamped structural dynamics eigenvalue problem, [K] [ ] −[M] [ ] [λ] =[0] . (13.68) When modal test data is substituted into the above expression, [K] [ t] −[M] [ t] [λt] =[R] , (13.69) there is a residual error, [R], due to (a) differences between the FEM and modal test data and (2) measurement error. Premultiplication by the FEM mode shapes results in, TK t − TM t [λt] = TR . (13.70)
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