mC ICA T I CB T ª ¬ º ¼ MAA MAB MBA MBB ª ¬ « « º ¼ » » ICA ICB ª ¬ « « º ¼ » » ICA T M AA ICA ICB T M BA ICA ICA T M AB ICB ICB T M BB ICB (14) The mass coupling tends to be small in structural finite element models, so the cross terms are often negligible. (This assumption is not necessary, see the derivation below for stiffness for an alternative.) Neglecting those mass coupling terms, one obtains T T C CA AACA CB BBCB m M M I I I I and if the modes are mass normalized the mass matrix of the C system can be written as mC I mA mB (15) in which mA ICA T M AA ICA mB ICB T M BB ICB (16) are the modal mass representations of substructures A and B in substructure C modal space. Equation (15) can then be written as mB I mA (17) If one had a finite element model for C, the mass matrix for B could be computed using these relationships. Of course, if one had a FEM for C one could simply delete A, but this derivation shows how the models for the subcomponents are related. At this point, one can note a similarity between Eq. (17) and Eq. (11). This reveals that the modal substructuring result produces an estimate of the mass matrix for system A, denoted ˆAm , which is given by ˆmA W T W (18) Performing the same analysis for stiffness gives kC ZC 2 k A kAB kBA kB (19) where kA ICA T K AA ICA kAB kBA T I CA T K AB ICB kB ICB T K BB ICB (20) In contrast with mass, the stiffness coupling terms kAB and kBA are not zero. Rearranging Eq. (19) produces 116
RkJQdWJsaXNoZXIy MTMzNzEzMQ==