The reduced mass and stiffness matrices for component A at the measurement degrees of freedom are then computed using MAr TA T M AATA (27) KAr TA TK AATA (28) It was previously assumed that ICm can be written as a linear combination of columns in IAm. If this is the case, then ICm can be written as follows, ICm IAm J (29) where J is an nA unC coefficient matrix. Post-multiplying Eq. (26) by J gives the approximation of the mode shapes of C at all of the points on the transmission simulator, ˆICA , using the mode shapes of A as a basis. ˆI CA IA J TA IAm J TA ICm (30) Substituting Eq. (30) into the first of Eqs. (16) gives the approximation for the transmission simulator modal mass ˆmA ˆI CA T M AA ˆI CA ICm T T A T M AATA ICm (31) Substituting for transformation TA from Eq. (26) produces ˆmA ICm T I Am †T I A T M AA IA IAm † I Cm (32) or, assuming modes IA are mass normalized ˆmA ICm T I Am †T I Am † I Cm W T W (33) which agrees with Eq. (18). The corresponding modal stiffness approximation is then ˆk A ˆI CA T K AA ˆI CA ICm T I Am †T I A TK AA IA IAm † I Cm (34) or ˆk A ICm T I Am †T Z A 2 I Am † I Cm W T Z A 2 W (35) 118
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