application of frequency-domain system identification techniques in ...

6.2. Source identification 135the latter conditions can significantly differ from the structures real-life operatingconditions. The use of input-output models (FRFs, modal models) for the purposeof identifying operational forces from measured response signals, can therefore (insome cases) lead to inaccurate force estimates.As an alternative to the use of forced excitation testing and EMA, use can bemade of ambient vibration testing and OMA. This technique has the advantagethat no artificial excitation devices are required and that the structure can bemodelled under realistic operating conditions while using all ambient vibration asexcitation. A drawback to the use of this method is that the resulting output-onlydynamic model (e.g., cross power spectra, operational modal model) can not bedirectly used for the purpose of relating measured response signals to the forcesacting on the structure.In this Chapter, the possibility will be examined of using the sensitivity-basednormalization technique (introduced in Chapter 4) in order to correctly normalizethe operational mode shapes. Once this task is successfully accomplished, FRFscan be synthesized by means of expression (6.3).6.2.4 Solving the inverse problemOnce a dynamic model (e.g., FRFs) of the test structure is obtained, equation(6.2) makes it possible (at least in theory) to estimate unknown forces that actupon the structure by measuring the resulting structural responses. This can beachieved by finding a solution F (ω f ) to the inverse problem that minimizes thefollowing cost function:N f∑K LS= (ɛ(ω f )) H ɛ(ω f ) (6.5)f=1withɛ(ω f ) = X(ω f ) − H(ω f )F (ω f ) (6.6)Regardless of the derivation of the dynamic model (EMA or OMA), the reliableestimation of forces from a given set of response data is not a simple task. Due tothe ill conditioned nature of the FRF matrix in (6.2), inversion problems can occurespecially for frequencies close to resonances, since in that case H(ω) is dominatedby a single mode. Consequently, an infinite amount of solutions exist that minimizecost function (6.5). In order to find a unique solution, an additional constraint isoften applied to ill conditioned inverse problems (Friswell and Mottershead, 2001).In practice, this can be achieved by adding additional terms to the cost function.In case of the force identification problem, the following constraint can be chosen: