14 A Coupled Approach for Structural Damage Detection with Incomplete Measurements 145 sets and then solving a least-squares problem involving only the active set. Hence, by allowing additional terms in the expansion process, the user can trade off confidence in the expanded dynamic residual for confidence in the measured mode shape DOFs. 14.2.1.5 Damage Location and Extent: MRPT The estimation of damage location and extent is performed via the Minimum Rank Perturbation Theory (MRPT) [17–20]. The underlying philosophy of the MRPT is that reduced rank perturbations to the structural matrices are the manifestation of damage. In this approach, damage results in a zero–nonzero pattern in the dynamic residual as given by Eq. (14.4). Typically, measurement noise and model order reduction or expansion destroys this pattern. Hence, a significant result of this approach is that the zero–nonzero pattern of the dynamic residual will be controlled by the columns of the connectivity matrix that are used to create it. The estimation of damage extent using a minimum rank formulation utilizes the following calculation (assuming that all damage is manifested only in the stiffness matrix): Œ K DŒB ˆTB 1 BT I (14.19) where [B] is the matrix formed of all column vectors of the expanded dynamic residuals for the modes of interest and [ˆ] is the matrix of the expanded modal vectors. 14.3 Example Application of Original Work The example application chosen for this section closely parallels the examples provided in precursory work to provide a context for interpreting the results [6, 11]. 14.3.1 NASA Eight-Bay Truss For the comparisons in this work the NASA eight-bay truss structure [24] is utilized. The eight-bay truss structure is an experimental test article developed to study a variety of damage identification issues. The cantilevered truss includes 32 nodes, each of which was instrumented with triaxial accelerometers. Fifteen unique damage cases were produced by removing individual truss members. The structure and representations of the damage cases (denoted by alphabetic characters) are provided in Fig. 14.1. This work uses 12 modes from a 96-DOF analytical model of the undamaged structure and the damage cases to explore the developments discussed in previous sections. The DOFs that are assumed to be active or measured for the example provided in this section are denoted by circles in Fig. 14.1. At these nodes, full triaxial measurements are utilized, resulting in a total measurement set of 24 DOFs. Case D will be used for the examples provided in this section. 14.3.2 Spring Disassembly The structural matrices for this model include 9,216.96 96/potential entries. However, there are only 320 unique (excluding symmetric entries) nonzero entries in this matrix. After reordering the matrices into analysis and omitted DOFs, the 224 offdiagonal elements are disassembled into 224 springs between DOFs. The 96 diagonal entries are disassembled as springs to ground. Note that this disassembly process does not disallow negative spring stiffness as these springs are usually a simplified representation of a more complicated structural element. It is also important to note that this connectivity pattern becomes an inherent part of subsequent estimations of full residuals, damage location, expanded mode shapes, and damage extent using the procedure outlined in the preceding section.
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