46 New FRF Based Methods for Substructure Decoupling 469 Exact Eq.(24) Log( I HU 22 [m/N] I ) 10-6 10-4 10-2 100 Frequency [Hz] 1 2 3 4 5 6 7 8 9 FRAC = 0.99791 Fig. 46.5 Driving point FRF at the 2nd DoF of the unknown subsystem: true (solid black line), predicted using Eq. (46.24) (blue asterisks) Table 46.2 List of most recent decoupling methods Ref. Final equation Needs (residual) Needs (coupled) Equation [18] HUjj DHRjkHR kj HRjk HR kk-Hkk HR kj 1 HRjkHR kj-HRjj HRjj HRjk HR kj HR kk ! Hkk (46.27) [18] HUjj D HRjj HRjj -Hjj 1 Ijj HRjj HRjj Hjj (46.28) [18] HUjj DHRjj HRjj HRjk HR kj-Hkj HRjj 1 HRjkHR kj-HRjj HRjj HRjk HR kj HR kk ! Hkj (46.29) [5] HUjj DHRjk HR kk-Hkk 1HR kj-HRjj HRjj HRjk HR kj HR kk ! Hkk (46.30) [2] HUjj D I-Hjj hHRjj i 1 1 Hjj HRjj Hjj (46.31) [11] HUjj D I-Hjj ZRjj Hjk ZR kj Hkj ZRjj Hkk ZR kj ! C Hjj Hkj ! ZRjj ZR kj ! Hjj Hjk Hkj Hkk ! (46.32) [11] HUjj D Hjj Hjk ! T0 @ Ijj 0jk HRjj HR kj ! C Hjj Hjk HRjk Hkj Hkk HR kk !1 A C HRjj HRjk HR kj HR kk ! Hjj Hjk Hkj Hkk ! (46.33) [15] HU D H 0 0 HR! H 0 0 HR! BT BR T !: : : B BR H 0 0 HR! BT BR T !! 1 B BR H 0 0 HR! HRjj HRjk HR kj HR kk ! Hjj Hjk Hkj Hkk ! (46.34) 46.3.2 A Comparison of the Approaches with Well-Known Existing Methods In this section, the performances of proposed methods are compared with those of well-known recent methods. The final equations for these methods, their references and the input data required for each of them are summarized in Table 46.2. Among the first three formulations given in the table, Eq. (46.28) was shown to be the one that produces the smallest error throughout the frequency range [18]. Moreover, among the last three formulations given in the table, Eq. (46.34) was shown to give better results [15]. Note also that Eq. (46.31) is a special case of the Eq. (46.32) as also mentioned in reference [11]. Consequently, it will be more to the point to compare the proposed formulations with Eqs. (46.28), (46.30) and (46.34) in Table 46.2. So, the problem given in Sect. 46.3.1 is also solved by using Eqs. (46.28), (46.30) and (46.34) in addition to employing the proposed formulations, i.e., Eqs. (46.23) and (46.24). Moreover in order to see the effect of increasing noise level in measured FRFs on the performances of different methods, coupled system FRFs are polluted by five different sets of random variables, pij,k and qij,k in Eq. (46.26), with Gaussian distribution, zero mean and standard deviations ranging from 5 10 5 to25 10 5 m/N. Results obtained for the standard deviation of 15 10 5 m/N are given in Fig. 46.6. It is observed that using different pollution sets with the same standard deviation may give slightly different results. Therefore, in order to compare the performances of different formulations and to study the effect of increasing measurement errors (increasing standard deviation of pollution), calculations with each method are repeated 100 times for each standard deviation of pollution, and the averages of the FRAC values are compared in Table 46.3.
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