66 M. Dorosti et al. stiffness/damping matrix of the (sub)structure as generic parameters. In this paper, IPZ model updating is extended to incorporate multiple FRFs. Using multiple FRFs from different actuator/sensor configurations helps to better improve the overall accuracy of the reduced FE model toward the real system since more information on the eigenmodes is used. The remainder of the paper is organized as follows. IPZ model updating using multiple FRFs is discussed in Sect. 8.2. In Sect. 8.3, as a case study, a simulated pinned-sliding beam structure is introduced on which the IPZ model updating technique using multiple FRFs is verified. Finally, some conclusions are drawn in Sect. 8.4. 8.2 IPZ Model Updating Using Multiple FRFs Updating a full FE model is computationally expensive. In addition, the physical parameters causing the difference between the model and the real system may not be known. In this case, updating a reduced-order FE model using generic parameters becomes useful. Using a model reduction technique based on eigenmodes and residual flexibility modes, consider the following reduced-order dynamic equation of an FE model: Mr Rqr CBr Pqr CKrqr Dfr; (8.1) where qr is a set of desired DOFs including actuators, sensors, and unmeasured performance variables DOFs. Mr; Br; Kr 2 Rr r are the reduced order mass, damping, and stiffness matrices and f r 2Rr 1 is the reduced external load column. Using the reduced-order dynamic equation in (8.1), the FRF corresponding to a sensor at DOF i and an actuator at DOF j can be described as Gij.!/ D det. !2Ms Cj!Bs CKs/ det. !2Mr Cj!Br CKr/ ; (8.2) where Ms; Bs; Ks are the so-called substructure matrices which are constructed from the reduced-order matrices Mr; Br; Kr respectively, by eliminating the ith row and the jth column corresponding to the sensor and actuator DOFs. Therefore, Kr can be written as Kr DKr;s C Kr; (8.3) where Kr;s is partitioned such that Kr;s D2 4 Ks;1 0 Ks;2 0 0 0 Ks;3 0 Ks;4 3 5 ; (8.4) and Kr D2 4 0 kr;1 0 kr;2 kr;3 kr;4 0 kr;5 0 3 5 ; (8.5) Ks is derived from (8.4), i.e. Ks D" Ks;1 Ks;2 Ks;3 Ks;4 #: (8.6) In IPZ model updating, a selected number of the eigenvalues of the stiffness matrix Kr and the substructure stiffness matrix Ks are introduced to represent the design parameters p and z for updating poles and zeros, respectively. The poles of the system ( p) are the roots of the denominator in (8.2), while the zeros of FRFGij ( z) are the roots of the numerator.
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