YB(ω)= k3 −ω2m3 − 1 ; (18) Using the assembly techniques of section 2, the two subsystems can be assembled directly: H(ω)= ZA(ω) LTBT BL −YB(ω) where: BL= 0 0 1 = LTBT T , (19) with the associated DoF vector: q= u1 u2 uγ λ T (20) Now, applying the same kind of reasoning for computing the sign count as before, a Sturmsequence has to be build by releasing one constraint at a timeandcomputing theminor of thematrix thereby obtained. Fromfigure 9 we see that the first two principle leadingminors are exactly the same as before, as expected. The third shows only one zero crossing, which comes fromthe fact that by “constraining” λ, the leading principle minor of ZA is computed, which has one rigid body mode and one natural frequency at infinity. The fourth principle leading minor shows four zero crossings, which are related to the three natural frequencies (indicated by the black markers) of the structure and to the natural frequency of subsystemB (indicated by the starredmarker). Clearly, the sign count computed in this way is incorrect and this is due to two reasons. First of all, one should 0 Z r Z 1 Z 2 Z 3 Z 4 0 Frequency [ω] s(H) Fig. 9: Computing the sign count only apply and release constraints on u (and not on λ)when computing the Sturmsequence. Thus Z3 should not be within the Sturmsequence to compute the sign count of H(ω). Second of all, the last leading principle minor of H(ω) is equal to [24]: Z4 =det(H(ω))=det(YB(ω))det ZA(ω)+L TBTYB(ω)− 1BL (21) So it can be seen that the zero crossing related to the natural frequency of subsystemBwill always be present in the leading principle minor Z4 and that the second termin equation 21 is in fact a primal assembly of the interface stiffness’ of the two subsystems. In order to avoid the zero crossing due to the natural frequency of subsystemB, an alternative minor is used in the Sturmsequence (Z1,Z2,Zc3): Zc3 =det(H(ω))det(YB(ω))− 1 =det ZA(ω)+L TBTYB(ω)− 1BL (22) If the sign count algorithmis applied to this new Sturmsequence, the correct results are obtained, as can be seen in figure 10. S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 340
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