David Chelidze Departmewnt of Mechanical Engineering & Applied Mechanics, University of Rhode Island, Kingston, RI 02881 e-mail: chelidze@egr.uri.edu Joseph P. Cusumano Department of Engineering, Science & Mechanics, Pennsylvania State University, University Park, PA 16802 e-mail: jpc@crash.esm.psu.edu Anindya Chatterjee Department of Mechanical Engineering, Indian Institute of Science, Bangalore, 560012, India e-mail: anindya@mecheng.iisc.ernet.in A Dynamical Systems Approach to Damage Evolution Tracking, Part 1: Description and Experimental Application 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 develop- ment 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 reconstruc- tion of a system’s phase space. The method treats damage-induced nonstationarity as evolving in a hierarchical dynamical system containing a fast, directly observable sub- system 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 electro- magnetic restoring force. Applications to systems with evolving material damage are also discussed. @DOI: 10.1115/1.1456908# 1 Introduction In this first part of a two-part paper, we develop and experimen- tally apply a method for tracking ‘‘slow’’hidden variables causing drift in the parameters of a ‘‘fast’’ dynamical system. In part two of the paper, we develop a mathematical model of the experimen- tal system, and make an analytical connection between the experi- mental method and the physics of the problem. The idea of time scale separation, where a hidden process re- sponsible for nonstationarity in a fast dynamical system evolves on a much slower time scale than the directly observable dynam- ics, is fundamental to this approach. A study of systems with this type of time scale separation is precisely what is required for the development of next-generation condition-based maintenance and failure prediction technology. For example, a collection of micro- cracks in a spinning shaft can be expected to grow slowly over many hundreds of thousands of typical vibrational time periods ~e.g., shaft revolutions!. The approach developed in this paper is not limited to material damage applications, but can be used for a general class of dynamical systems possessing the needed time scale separation. Possible applications include: the drifting out of alignment of machinery parts, corrosion processes in structural components, moisture accumulation in composite materials or electrical circuits, as well as cases involving material damage of various types. In most cases, online real-time health monitoring and failure prediction for such systems requires assessment of the current state of damage in the system using only readily available vibra- tion data: typically, a direct ‘‘damage sensor’’ is either not avail- able, or requires the removal from service ~or even destructive testing! of the machine in question. In many cases, even if a direct means of damage detection were available, it is not possible to add the required sensors because of cost or weight considerations. The approach presented here is designed to address these difficul- ties by using only readily available vibration data to track the slowly evolving failure process. At the same time, the abstract formulation of the method means that it can in principle be ap- plied to a wide variety of ‘‘damage’’ processes, to some degree independent of damage physics. In the next section, a literature review is presented and used to frame a discussion of fundamental ideas underlying our approach. The tracking procedure based on phase space reconstruction is developed in Section 2. In Section 3 we apply the method to tracking a battery discharge process in an electromechanical ex- perimental system using only strain gauge vibration data. In Sec- tion 4, we discuss the experimental results and compare our method to those based on attractor invariants. Finally, in Section 5 we present our conclusions. 2 Background The main body of previous work relating to the effort presented here stems from studies relating to the development of condition based maintenance technologies, such as Fault Detection and Identification ~FDI! theory. We should mention that most previous research, excluding studies dealing with fault severity assessment, have concentrated on detection of irregularities in the observable system. While the detection process estimates whether or not the system parameters or ‘‘fault indicators’’ have reached certain pre- set failure values, the tracking process as studied here, estimates in real-time the change in the slow state variables causing nonsta- tionarity, a requirement for true prognostics. The main idea behind FDI is to develop a procedure for obtain- ing feature vectors that are sensitive to the changes or failures in systems. There are several strategies for tackling this type of prob- lem ~see, for example, @1,2#, for comparative studies!. One basic approach is to use available signal processing techniques to de- velop a feature vector. We call these techniques heuristic, since they are not based on specific physics or analytical treatment of the problem. The early research was mainly concentrated on de- tecting changes in systems using time or frequency domain statis- tics ~e.g., @3,4#!. Later research considered techniques that carry both frequency and time domain information, such as wavelet transforms or Wigner-Ville distributions @5–9#. Other methods use such data classification schemes as principal component analysis, Contributed by the Technical Committee on Vibrational and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 2001; Revised Dec. 2001. Associate Editor M. I. Friswell. 250 Õ Vol. 124, APRIL 2002 Copyright © 2002 by ASME Transactions of the ASME