where ICm and IAm are the measured partitions of the experimental and transmission simulator components, respectively. Equation (3) can be solved for the transmission simulator modal response qA IAm T I Am ª¬ º¼ 1 I Am T I CmqC IAm † I CmqC (4) in which † AmI is the left generalized inverse of the transmission simulator modes at the measurement locations. This solution requires that the transmission simulator modal partition at the measurement degrees of freedom be full column rank, which implies that nm tnA. The modal coordinates cannot be directly measured, so these constraints must be written in terms of the physical coordinates before they can be implemented. Premultiplying (4) by IAm and then transforming back into physical coordinates produces xAm IAm IAm † x Cm PAmxCm ˆxCm (5) where PAm is an orthogonal projector onto the column space of IAm, and ˆxCm is then the orthogonal projection of the response of substructure C at the measurement locations onto this space. Therefore, the modal constraints used in the MCFS approach do not strictly enforce the constraint in Eq. (3), but instead enforce the least-squares fit given in Eq. (5). The best synthesis result will be obtained if ˆCm Cm x x | , in which case the columns in ICm can be written as a linear combination of columns in IAm, or R ICm R IAm . The modal constraints can be enforced and the constrained generalized coordinates eliminated from eq. (1) with the transformation qC qA ® ° ¯° ½ ¾ ° ¿° IC W ª ¬ « « º ¼ » » qC TqC (6) in which W IAm † I Cm . The unforced equation of motion for the reduced or coupled system then has the form 0 r C r C mq k q (7) where mr T T IC 0 0 IA ª ¬ « « º ¼ » » T IC W T W (8) and 114
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