5 Response DOF Selection for Mapping Experimental Normal Modes-2016 Update 37 where [‰aq] is the rectangular partition of [‰fq] corresponding to measured DOFs. The least-squares inverse transformation (assuming there are more fuag DOFs than fqg DOFs) is finally fqgD ‰T aq‰aq 1 ‰T aq fuagD ‰qa fuag. (5.19) Thus the modified Guyan reduction transformation relating “free” and “instrumented” DOFs is fufgD ‰fq ‰qa fuagDŒ‰fa mfuag. (5.20) Application of the reduction transformation, in a symmetric manner, yields the “modified” Guyan reduction TAM mass matrix ŒMaa mDŒ‰fa T mŒMff Œ‰fa m. (5.21) 5.4.3 General Model Order Reduction The general model order reduction transformations for any assumed set of Ritz vectors (e.g. SEREP and others [9, 10]) are: fufgD ‰fq g f qg!fuagD ‰aq g f qg!fqgD ‰T aq‰aq g 1 ‰T aq g f uagD ‰qa g f uag!fufgD ‰fq g ‰qa g fuagDŒ‰fa g fuag. (5.22) Application of the reduction transformation, in a symmetric manner yields the “general” reduction TAM mass matrix ŒMaa g D ‰fa T g Mff ‰fa g (5.23) 5.5 Reduced Order Model Orthogonality and Residual Kinetic Energy The three above types of model order reduction serve as alternatives for development of a TAM mass matrix to be employed in experimental mode evaluations conforming to U.S. Air Force and NASA standards [1, 2]. For the purposes of modal test planning, the adequacy of a selected instrumentation (accelerometer) array may be evaluated by taking the “instrumented” subset partition, [ˆa], of the predicted “free” modal set, [ˆf], and estimating test mode orthogonality, ŒOR DŒˆa TŒMaa Œˆa . (5.24) The simulated “expanded” modes, calculated as, Œˆfa DŒ‰fa Œˆa , (5.25) are then employed to form a residual error matrix, ŒR DŒˆf Œˆfa DŒˆf Œ‰fa Œˆa . (5.26) The residual modal kinetic energy matrix [5] is therefore, ŒRKE DŒMffR ˝ŒR . (5.27) The summed residual kinetic energy for a particular mode ([RKE] column) ideally has an upper bound of 1.0 (or 100 %). This is the case for “classical” Guyan reduction, which has a direct (non-least squares) relationship between “free” and
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