In this two-part paper we present a novel method for tracking a slowly evolving hidden damage process responsible for nonstationarity in a fast dynamical system. The development of the method and its application to an electromechanical experiment is the core of Part 1. In Part 2, a mathematical model of the experimental system is developed and used to validate the experimental results. In addition, an analytical connection is established between the tracking method and the physics of the system based on the idea of averaging and the slow flow equations for the hidden process. The tracking method developed in this study uses a nonlinear, two-time-scale modeling strategy based on the delay reconstruction of a system’s phase space. The method treats damage-induced nonstationarity as evolving in a hierarchical dynamical system containing a fast, directly observable subsystem coupled to a slow, hidden subsystem. The utility of the method is demonstrated by tracking battery discharge in a vibrating beam system with a battery-powered electromagnetic restoring force. Applications to systems with evolving material damage are also discussed.

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In this paper, a previously published damage tracking method is extended to provide failure prognosis, and applied experimentally to an electromechanical system with a failing supply battery. The method is based on a dynamical systems approach to the problem of damage evolution. In this approach, damage processes are viewed as occurring in a hierarchical dynamical system consisting of a “fast,” directly observable subsystem coupled to a “slow,” hidden subsystem describing damage evolution. Damage tracking is achieved using a two-time-scale modeling strategy based on phase space reconstruction. Using the reconstructed phase space of the reference (undamaged) system, short-time predictive models are constructed. Fast-time data from later stages of damage evolution of a given system are collected and used to estimate a tracking function by calculating the short time reference model prediction error. In this paper, the tracking metric based on these estimates is used as an input to a nonlinear recursive filter, the output of which provides continuous refined estimates of the current damage (or, equivalently, health) state. Estimates of remaining useful life (or, equivalently, time to failure) are obtained recursively using the current damage state estimates under the assumption of a particular damage evolution model. The method is experimentally demonstrated using an electromechanical system, in which mechanical vibrations of a cantilever beam are dynamically coupled to electrical oscillations in an electromagnet circuit. Discharge of a battery powering the electromagnet (the “damage” process in this case) is tracked using strain gauge measurements from the beam. The method is shown to accurately estimate both the battery state and the time to failure throughout virtually the entire experiment.

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