Identifying Local Reductions to Mass and Stiffness with Incomplete Modal Information, Sparsity, and Nonnegative Constraints

Abstract

The l1-norm regularized inverse methods have been suggested as a means to quantify and localize spatially sparse damage from a set of identified natural frequencies. Thus far, damage has been interpreted as changes in stiffness without any appreciable associated changes in mass. In seeking to generalize l1-norm-based inverse methods to encompass damage that produces significant changes in both mass and stiffness, this paper finds that sparsity is too weak a prior to uniquely solve the associated underdetermined inverse problem. However, when damage is defined by local reductions to stiffness and mass, then the addition of a nonnegative constraint when combined with sparsity can yield physically meaningful solutions. This work proposes a two-step model-updating method to obtain sparse and nonnegative solutions. The findings and proposed method were verified using two numerical models: a shear beam and a four-level plane frame. The proposed methodology was experimentally validated using vibration data taken from a four-level bolted steel frame subjected to multiple damage scenarios.