26 K. Napolitano and D. Cloutier as many shakers as there are independent force patterns associated with the interface load cells. The baseline FRFs can be defined as {R}=[H] {F}, (3.1) where {R} is all of the measured responses including accelerations and interface load measurements, [H] is the FRF matrix, and {F} is all the applied forces. To simplify this derivation, assume that the responses are accelerations on the test article {a}, and interface forces {fr}, such that {R}= a fr . (3.2) Also assume that the applied forces {F} can be separated into forces applied directly to the test article {fi}, and forces applied to the boundary {fb}, such that {F}= fi fb . (3.3) The FRF matrix [H] is partitioned to obtain a fr = Hai Hab Hri Hrb fi fb . (3.4) In the case where the number of independent reaction force patterns is less than the number of interface load cell measurements, the number of reaction force DOFs can be reduced using force pattern constraint shapes such that {fr}=[Ψ] {fE}, (3.5) where [ ] is a matrix of independent force patterns and {fE} is the force pattern DOFs. Inserting Eq. (3.5) into Eq. (3.4) yields a fE = Hai Hab HEi HEb fi fb , (3.6) where, since {fE}=[ ]+{fE}, [HEi]=[ ]+[Hri] and [HEb]=[ ]+[Hrb]. Note that the term “+” denotes the pseudo-inverse of a matrix. Assuming that an equal number of forces are applied to the boundary as there are independent force patterns, one can perform a partial inversion of the FRF matrix to move the force patterns to the right-hand side and the boundary forces to the left-hand side to obtain a fb = Hai Hab Hbi HbE fi fE , (3.7) where Hai =Hai −HabHEb−1 HEi, Hab =HabHEb−1, Hbi =−HEb−1 HEi, and HbE =HEb−1. For any linear relationship in the form of {x}=[C]{y}, the matrix element Cij is equal to the value of xidue to a unit input at yj, holding all other elements {y} in equal to zero. Applying this property to Eq. (3.7) means that the submatrix Hai is the FRF matrix associated with all interface force patterns equal to zero—the definition of a free-free boundary condition. Most load cells are based on strain gages that are calibrated to give an estimate of force. However, instead of measuring forces directly, assume one measured a series of strain gages of which a linear combination captured all the interface force patterns such that {fr}=[D] {ε}, (3.8)
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