Measurement of structural flexibility matrices for experiments with incomplete reciprocity

This research examines the measurement of accurate structural flexibility matrices from modal vibration experiments using the measured mode shapes, modal frequencies and residual flexibility. The specific research issue presented is the use of measured residual flexibility (the contribution of the unmeasured dynamic modes) in the computation of the flexibility matrix when the experiment has incomplete reciprocity. Incomplete reciprocity exists when the excitation and response sensors are not fully collocated, which is typically the case in modal vibration experiments due to practical constraints. In this situation, the full structural reciprocity is not captured, and thus one partition of the residual flexibility matrix is not measurable.;Two methods are presented for computing the flexibility under these circumstances. The first method estimates the unmeasured partition of the residual flexibility such that it satisfies the constraint of modal orthogonality, but this estimate is generally rank-deficient due to the relatively small number of modal excitations. This method is easily implemented because it requires only the identified modal model, with no assumptions about the underlying second-order behavior of the structure. The second method estimates a full-rank flexibility matrix which is based on an assumed underlying structural connectivity and strain energy distribution. This solution is obtained by parameterizing and scaling the assumed flexibility matrix iteratively to match the known partitions in the measured flexibility matrix. A full-rank flexibility subspace is determined by the parameterization, which is a best fit to both the assumed model form and the measured flexibility. This method generally achieves more accurate results than the rank-deficient solution, provided that the assumed connectivity is consistent with the underlying second-order behavior of the structure. It also allows interpolation of nodal rotations, which are generally difficult to measure. The effects of the locations of modal test excitations on the residual flexibility estimate are also discussed. Experimental applications are presented, including identification of beam bending stiffness, identification of frame joint stiffness, and damage location in a section of aircraft skin and in a highway bridge.