A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies 0 Z r Z 1 Z 2 Z c3 0 Frequency [ω] s(H) Fig. 10: Computing the sign count 3.2 Identification of Eigenmodes Using the natural frequencies of the assembled systemobtained using the sign count method, the corresponding eigenmodes can also be identified. At an eigenfrequency of the assembled system, the following holds: H(ωr)q=0, for r =1,2,...,n (23) Since the sign count approximates the location of the eigenfrequency and is limited by the frequency resolution, (23) will most likely not hold exactly: H(ωr)q= ε, for r =1,2,...,n , (24) where || ε|| <<1. This equation is solved by performing an eigenvalue analysis on H(ωr) and selecting the eigenmode (x1) corresponding to the lowest eigenvalue μ1 as the mode shape φr corresponding to the eigenfrequency ωr: (H(ωr)−μ1I)x1 =0, μ1 < μ2 <...< μn then: φr =x1 (25) However, the mode shapes identified in this way have an arbitrary scaling factor and also include a number of interface force terms λ. In order to synthesize the receptance FRFs of the assembled structure, the interface force entries have to expanded to the subsystemresponses and the mode shapes have to be mass normalized. One way of doing this is to compute the driving point FRF of the structure, by taking the inverse of H(ω) at a limited number of points around the natural frequencies. Then, by using the normal mode model (26), the unknown real scalars ar can be determined in a least squares sense. Yj j(ω)= n ∑r=1 a2 r φj,r φ T j,r −ω2 +ω2 r +2i ζr ωωr (26) Note that this scaling scheme could be very sensitive to the estimates of the modal damping ζr and the frequency resolution. 341
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