Identifying Multidimensional Damage in a Hierarchical Dynamical System

In this paper, we present a novel method for multidimensional damage identification based on a dynamical systems approach to damage evolution. This approach does not depend on the knowledge of particular damage physics, and is appropriate for systems where damage evolves on much slower time scale than the directly observable dynamics. In an experimental context, the phase space reconstruction and locally linear models are used to quantify small distortions occurring in a dynamical system's phase space due to damage accumulation. These measurements are then related to the drifts in damage variables. A mathematical model of a harmonically driven cantilever beam in a force field of two battery-powered electromagnets is used to demonstrate validity of the method. It is explicitly demonstrated that an affine projection of the described feature vector accurately tracks the two competing damage processes. For practical damage identification purposes, the tracking data is analyzed using the proper orthogonal decomposition (POD) and smooth orthogonal decomposition (SOD) methods. Both methods correctly identify the two dominant damage modes. However, the SOD is more impervious to changes in fast-time dynamics and provides a significantly better signal-to-noise ratio. The damage modes identified using SOD are demonstrated to be within a linear transformation from the actual damage states and can be used to reconstruct the corresponding phase space trajectory.