3.3 Some Comments During the research that preceded this paper, other approaches towards an inverse-free FBSmethod have also been investigated. For example, we have tried solving the LM FBS equations using an iterative Conjugate Gradient approach in order to avoid direct computation of the inverse of the interface flexibility matrices. Furthermore, we attempted to directly invert the mixed assembled equations in eq. (15). After performing Monte Carlo simulations on these variants (see next section), it turned out that they all showed exactly the same sensitivity to errors as the classic FBS methods. The reason is that essentially the same equations are solved, albeit in a different way, and errors propagate and amplify regardless of the exact method used to solve the equations. Hence, we concluded that a truly different approach to solving the equations was needed, which did not involve computing any inverse, whether explicitly or implicitly. The trick of the proposed method is not in the fact that it completely avoids solving/inverting the systemmatrices, since a partial factorization of the assembled equations is still required for the computation of the sign count and inversion of the mixed assembled matrix is still needed at some frequencies for the mode scaling. The key feature however is that the results of these computations are not used directly in further computations. This is in contrast to normal FBS methods, where the inverted (interface) FRF matrices are used to calculate the total assembled FRF matrix. It is through those computations that the sensitivity to measurement errors is propagated to all the FRFs. We therefore hypothesize that in the current approach, the propagation of errors is limited. Whether this is indeed the case will be investigated in the next section. 4 Case Study In this section the results of a case study are presented. The simple problemused for this study is shown in figure 11 and consists of two lightly damped mass-spring-damper systems (modal damping <1%). SubsystemAhas 7 degrees of freedom, subsystemBpossesses 4 DoF and the systems are coupled at a 2 DoF interface. When combined the systemABis formed. The systemparameters are listed in table 1; Rayleigh damping is added (C= αM+βK)with α=0.01 and β=0. E\^ E\@ E\9 E\e E\v E\m E\G `\^ `\m ` \G `\e `\v `\9 `\@ >\^^ >\^9 >\9 >\e >\@^ >\@9 >\m E4e E4@ E4^ E49 `4e `49 `4^ `4@ >4e >4^ >49 7ZARkR 8 V yt Ps{ 7ZARkR 8 Fig. 11: Simple systemused for case study. Component Ais assumed tobe modeled analytically and hence will be expressed in a dynamic stiffness format.We imagine subsystemBto be modeled experimentally and as a result is expressed in the receptance FRF format. In the remainder of this section we will performseveral analyses on this simplesystemto illustrate themethod laid out in the previous two sections. First, in the next subsection we will take the nominal systemmodels (i.e. without any perturbations due to “measurement errors”) and identify the eigenfrequencies and -modes and synthesize the assembled FRF. Thereafter we will performa sensitivity analysis using the Monte Carlomethod in order to determine whether the proposed alternative FBSmethodology is indeed less sensitive tomeasurement errors than the traditional methods. 4.1 Calculation of Assembled Modes and FRFs Using the sign countingmethod described in the previous section, the first analysis of the mixed formof the assembled system (i.e. eq. (15)) was to identify its eigenfrequencies. Obviously, the accuracy with which one can determine these eigenfrequencies is limited by the frequency resolution of the model/measurement of the subsystems. Therefore, we performed an analysis with S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 342
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