ˆkB kB kAB kBA ZC 2 k A (21) in which ˆkB is the modal stiffness approximation for substructure B. Comparing Eq. (21) with Eq. (12) gives the approximation of the transmission simulator modal stiffness as ˆk A W T Z A 2 W (22) To develop the comparison further, consider a finite element model for the transmission simulator, substructure A, alone. In physical coordinates, the equation of motion can be written as AA A AA A A M x K x F (23) The displacement vector xA can be partitioned into the measured degrees of freedom and their compliment xA xAm xAo ® ° ¯° ½ ¾ ° ¿° (24) The transmission simulator modes can be partitioned in the same manner IA IAm IAo ª ¬ « « º ¼ » » (25) Note that that the modal partition IAm was assumed to be full column rank in the previous derivation in Section 2.1. If this is the case, then the physical mass and stiffness matrices in Eq. (23) can be reduced to the measurement degrees of freedom using any of a number of different reduction techniques. This is usually done to generate a reduced mass representation, or testanalysis model (TAM), that is used in test-analysis correlation and analytical model validation [6]. A popular technique for TAM development is called the System Equivalent Reduction Expansion Process (SEREP) [5]. Using this approach, the complete mode shapes of A are written in terms of the measured modal partition IAm as † † † Am Am A A Am Am A Am Am Ao Am T I I I I I I I I I I ª º « » ¬ ¼ (26) 117
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