Damage localization from the null space of changes in the transfer matrix

A new theorem connecting changes in transfer matrices to the spatial location of stiffness related damage is presented. The theorem states that the span of the null space of the change in the transfer matrix (Delta G) contains vectors that are Laplace transforms of excitations for which the dynamic stress field is identically zero over the portion of the domain where the damage is located. A corollary states that if Delta G proves rank deficient at any s it is guaranteed to be rank deficient throughout the plane. It is shown that a sufficient condition for rank deficiency is that the number of independent measurements be larger than the rank of the change in the stiffness matrix resulting from damage. Implementation of the theorem in the time domain involves evaluation of the null space of Delta G along a discretized Bromwich contour, numerical inversion of the resulting s-functions and evaluation of the dynamic response to the computed signals to identify the regions where the stress field is zero. The theorem is most efficiently implemented, however, in the s-domain, where numerical Laplace inversion is avoided and the number of evaluations of the null of Delta G is sharply reduced. Guidelines for implementation of the theorem in the presence of inevitable errors in Delta G and in the model used to compute the stress fields are presented.