Inspection of Eq. 2 shows that the inverted matrix is the Receptance of the original system evaluated at Oj s , therefore 2 2 1 ( ) I I U 'OI I O O ¦ A A A A A T N j j j j M (3) which, after some simple rearranging, can be written as 2 2 1 ( ) I I ' I O U I O O ¦ A A A A A T N j j j j M (4) Eq.4 describes m equations in the 2N modal constants formulated using the jth polynomial eigenvalue equation of the perturbed system. By combining the equations for each perturbed mode a linear system of the type U Q bis obtained. In practice there are only 2n identified modes so the summation upper limit in Eq.4 has to be taken as 2n and, after evaluating it at each of the identified modes, the coefficient matrix dimensions is (mx2n)x2n. As is evident from this derivation, except from possible approximation resulting from modal truncation, the RBN formulation is exact. Obtaining the Coefficient Matrices The linear system of equations implicit in Eq.4 can be conveniently formulated as follows: let ) be the set of identified modes in the original state, at whatever coordinates are measured, and ) the set in the mass modified state (in both cases including the conjugate pairs). The system of equations can be written as ( ) U ) I Q vec (5) where ^ ` 1 2 . U U U U I n , and 1 2 . ª) F º «) F » « » « » « » ¬) F ¼n Q (6) with 1 2 ( , ,... ) F n j j j j diag a a a and 2 ( ) O I ' I O O j i T j i j i j a M (7) As can be seen from Eq.5, the constants UI are coefficients in a projection of the perturbed modes on the basis of the original system. From this perspective one concludes that if the identified eigenvectors (over all coordinates) have the same span in the original and the perturbed conditions the RBN solution is exact, independently of truncation. The Sensitivity Approach Since we shall use sensitivity results to contrast the RBN solution we derive the formula for the scaling constant in the sensitivity scheme for the case of complex eigenvectors. Taking the mass change as 1 ' E M M , where E is a scalar with units of mass, and M1 is a distribution, one has 2 1 ( ) 0 ª º ¬ E O O ¼\ M M C K (8) Differentiating Eq.8 with respect to E, pre-multiplying by \T and evaluating the result at E=0 one gets, after some simple algebra 394
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