180 A.R. Brink et al. and only updates the component mass and stiffness matrices, as shown in Eqs. (19.7) and (19.8). Since the old modes are used to map the updated mass and stiffness matrices to the old basis, the technique is call REMAP. This is also short for REduced MAtrixPerturbation. Wupdated D ˆnominal ‰updated 0 I ; (19.5) where Œ‰ updated D ŒKnn 1 updatedŒKna updated: (19.6) The stiffness terms in Eq. (19.6) are based on the perturbed parameter sets that are generated during the UQ, as are the mass and stiffness matrices used in Eqs. (19.7) and (19.8), respectively. h OMCBi updated DŒW T updatedŒM updatedŒW updated (19.7) hOKCBi updated DŒW T updatedŒK updatedŒW updated: (19.8) Since this method requires knowledge of the unreduced structure to generate the mass and stiffness matrices, it is ideal for UQ analysis within the same design agency. If the Craig-Bampton reduced order model is transferred to another design agency for implementation into a larger system model, then this technique is not applicable or requires multiple realizations to be transferred. An analysis on a prototypical structure using this technique is presented in Sect. 19.4. 19.3.2 COMP Technique Where the REMAP technique perturbs the mass and stiffness matrices of the reduced order model, the COMP technique instead perturbs the modal matrix, and leaves the mass and stiffness matrices in their nominal condition. To perturb the modal matrix, it is pre-multiplied by a cross-orthogonal modal matrix. Hence the acronymCross-Orthogonal Modal Perturbation. Consider if the component modal matrices were known for both the nominal state and the state that corresponds to an updated parameter set from UQ. Then, the cross-orthogonality matrix of the two modal matrices is Œcˆ DŒˆ T nominal ŒM nominal Œˆ updated: (19.9) If both modal matrices are identical, then Œcˆ is the identity matrix. The Craig-Bampton transformation matrix used in the COMP technique is then Wupdated D cˆˆnominal ‰nominal 0 I : (19.10) For the example problem shown in Sect. 19.4, the cross-orthogonality matrix is generated with a priori knowledge of how the UQ parameter set changes the modal matrix. If the matrix is to be perturbed without this knowledge, care must be taken by the analyst to ensure realistic mode shapes are generated by the perturbation. In addition to perturbing the modal matrix, it is also possible using this technique to perturb the Eigen vector, although this is not considered in this paper. 19.4 Application of REMAP and COMP Techniques In this section the REMAP1, REMAP2 and COMP techniques are applied to the structure shown in Fig. 19.1. The nominal parameters of the structure are given in Table 19.1. In addition to the linear beams that make up the structure, a point mass/inertia is added to the center of each ‘X’ structure. The nominal mass is 2.863 kg and the nominal inertia is
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