David Chelidze

A Dynamical Systems Approach to Failure Prognosis

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 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 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. @DOI: 10.1115/1.1456908#

A Dynamical Systems Approach to Damage Evolution Tracking, Part 2: Model-Based Validation and Physical Interpretation

In this paper, the hidden variable damage tracking method developed in Part 1 is analyzed using a physics-based mathematical model of the experimental system: a mechanical oscillator with a nonstationary two-well potential. Numerical experiments conducted using the model are in good agreement with the experimental study presented in Part 1, and explicitly show how the tracking metric is related to the slow hidden variable evolution responsible for drift in the fast system parameters. Using the idea of averaging, the slow flow equation governing the hidden variable evolution is obtained. It is shown that the solution to the slow flow equation corresponds to the hidden variable trajectory obtained with the experimental tracking method. Thus we establish in principle the relationship of our algorithm to any underlying physical process. Based on this result, we discuss the application of the tracking method to systems with evolving material damage using the results of some preliminary experiments.@DOI: 10.1115/1.1456907# In Part 1 of this paper, motivated by the need to track damage evolution in machinery, we have developed a nonlinear method for tracking slowly evolving hidden variables. From this perspective, damage is a hidden process causing nonstationarity in a fast, directly observable dynamical system. The method uses a phase space formulation of the damage tracking problem, and uses a tracking metric developed using the short-time reference model prediction error. The method was successfully applied to an electromechanical experimental system consisting of a vibrating beam with a nonlinear potential perturbed by a battery powered electromagnet. The connection between the tracking metric developed in Part 1 and the hidden drift state variable was demonstrated empirically. It was shown that, as expected from the theoretical derivation of the method, the tracking metric is in a one-to-one relationship with the local time average of the measured voltage signal. In this, Part 2, of our paper, a physics based mathematical model of the experimental system is used to study analytically the direct connection between the tracking metric and the hidden drift process. Numerical experiments performed with the model are used to validate the experimental method. The idea of averaging is then used to show that the output of the tracking method is in fact following the solution to the slow flow equation for the drifting process. This provides a physical interpretation for the output of the tracking algorithm, and shows how, in principle, the experimental method can be related to the physics of the damage process. Based on this physical interpretation, we return to the experimental application of the tracking method, and discuss some preliminary results for a system with a crack growing to failure. In the next section, we develop the mathematical model of the battery discharge experiment using a lumped parameter, Lagrangian formulation of the electromechanical system. In Section 3, the tracking method developed in Part 1 is applied to the mathematical model in numerical experiments. Using the output from numerical simulations of the mathematical model, we are able to validate the tracking algorithm. In Section 4, we discuss the method of averaging as it relates to our problem. Finally, in Section 5, we finish with concluding remarks.

Phase space warping: nonlinear time-series analysis for slowly drifting systems

A new general dynamical systems approach to data analysis is presented that allows one to track slowly evolving variables responsible for non-stationarity in a fast subsystem. The method is based on the idea of phase space warping, which refers to the small distortions in the fast subsystems phase space that results from the slow drift, and uses short-time reference model prediction error as its primary measurement of this phenomenon. The basic theory is presented and the issues associated with its implementation in a practical algorithm are discussed. A vector-tracking version of the procedure, based on smooth orthogonal decomposition analysis, is applied to the study of a nonlinear vibrating beam experiment in which a crack propagates to complete fracture. Our method shows that the damage evolution is governed by a scalar process, and we are able to give real-time estimates of the current damage state and identify the governing damage evolution model. Using a final recursive estimation step based on this model, the time to failure is continuously and accurately predicted well in advance of actual failure.

Multimode damage tracking and failure prognosis in electromechanical systems

In this paper a modification to a general-purpose machinery diagnostic/prognostic algorithm that can handle two or more simultaneously occurring failure processes is described. The method is based on a theory that views damage as occurring in a hierarchical dynamical system where slowly evolving, hidden failure processes are causing nonstationarity in a fast, directly observable system. The damage variable tracking is based on statistics calculated using data-based local linear models constructed in the reconstructed phase space of the fast system. These statistics are designed to measure a local change in the fast systems flow caused by the slow-time failure processes. The method is applied to a mathematical model of an experimental electromechanical system consisting of a beam vibrating in a potential field crated by two electromagnets. Two failure modes are introduced through discharging batteries supplying power to these electromagnets. Open circuit terminal voltage of these batteries is a two-dimensional damage variable. Using computer simulations, it is demonstrated both analytically and experimentally that the proposed method can accurately track both damage variables using only a displacement measurements from the vibrating beam. The accurate estimates of remaining time to failure for each battery are given well ahead of actual breakdowns.

Reconstructing slow-time dynamics from fast-time measurements

This paper considers a dynamical system subjected to damage evolution in variable operating conditions to illustrate the reconstruction of slow-time (damage) dynamics using fast-time (vibration) measurements. Working in the reconstructed fast-time phase space, phase space warping-based feature vectors are constructed for slow-time damage identification. A subspace of the feature space corresponding to the changes in the operating conditions is identified by applying smooth orthogonal decomposition (SOD) to the initial set of feature vectors. Damage trajectory is then reconstructed by applying SOD to the feature subspace not related to the changes in the operating conditions. The theory is validated experimentally using a vibrating beam, with a variable nonlinear potential field, subjected to fatigue damage. It is shown that the changes in the operating condition (or the potential field) can be successfully separated from the changes caused by damage (or fatigue) accumulation and SOD can identify the slow-time damage trajectory.

Reduced Order Models for Systems with Disparate Spatial and Temporal Scales

Simulations and parametric studies of large-scale models can be facilitated by high-fidelity reduced order models (ROMs). One of the methods to obtain these ROMs, which is of considerable current interest, is using linear subspaces obtained from spatiotemporal decompositions including proper orthogonal decomposition (POD) and smooth orthogonal decomposition (SOD). Previous studies showed that SOD has advantages over POD in obtaining a lower dimensional ROM. However, in many dynamical models, the data matrices used in multivariate analysis can be ill-conditioned. This leads to non-optimal results with POD, since it only aims to find the subspace on which the data projection has the maximal variance. Therefore, POD is likely to overlook small-variance state-variables.

Model Order Reduction of Nonlinear Euler-Bernoulli Beam

Numerical simulations of large-scale models of complex systems are essential to modern research and development. However these simulations are also problematic by requiring excessive computational resources and large data storage. High fidelity reduced order models (ROMs) can be used to overcome these difficulties, but are hard to develop and test. A new framework for identifying subspaces suitable for ROM development has been recently proposed. This framework is based on two new concepts: (1) dynamic consistency which indicates how well does the ROM preserve the dynamical properties of the full-scale model; and (2) subspace robustness which indicates the suitability of ROM for a range of initial conditions, forcing amplitudes, and system parameters. This framework has been tested on relatively low-dimensional systems; however, its feasibility for more complex systems is still unexplored.

Slow-time changes in human EMG muscle fatigue states are fully represented in movement kinematics.

The ability to identify physiologic fatigue and related changes in kinematics can provide an important tool for diagnosing fatigue-related injuries. This study examined an exhaustive cycling task to demonstrate how changes in movement kinematics and variability reflect underlying changes in local muscle states. Motion kinematics data were used to construct fatigue features. Their multivariate analysis, based on smooth orthogonal decomposition, was used to reconstruct physiological fatigue. Two different features composed of (1) standard statistical metrics (SSM), which were a collection of standard long-time measures, and (2) phase space warping (PSW)-based metrics, which characterized short-time variations in the phase space trajectories, were considered. Movement kinematics and surface electromyography (EMG) signals were measured from the lower extremities of seven highly trained cyclists as they cycled to voluntary exhaustion on a stationary bicycle. Mean and median frequencies from the EMG time series were computed to measure the local fatigue dynamics of individual muscles independent of the SSM- and PSW-based features, which were extracted solely from the kinematics data. A nonlinear analysis of kinematic features was shown to be essential for capturing full multidimensional fatigue dynamics. A four-dimensional fatigue manifold identified using a nonlinear PSW-based analysis of kinematics data was shown to adequately predict all EMG-based individual muscle fatigue trends. While SSM-based analyses showed similar dominant global fatigue trends, they failed to capture individual muscle activities in a low-dimensional manifold. Therefore, the nonlinear PSW-based analysis of strictly kinematic time series data directly predicted all of the local muscle fatigue trends in a low-dimensional systemic fatigue trajectory. These results provide the first direct quantitative link between changes in muscle fatigue dynamics and resulting changes in movement kinematics.

Smooth local subspace projection for nonlinear noise reduction

Many nonlinear or chaotic time series exhibit an innate broad spectrum, which makes noise reduction difficult. Local projective noise reduction is one of the most effective tools. It is based on proper orthogonal decomposition (POD) and works for both map-like and continuously sampled time series. However, POD only looks at geometrical or topological properties of data and does not take into account the temporal characteristics of time series. Here, we present a new smooth projective noise reduction method. It uses smooth orthogonal decomposition (SOD) of bundles of reconstructed short-time trajectory strands to identify smooth local subspaces. Restricting trajectories to these subspaces imposes temporal smoothness on the filtered time series. It is shown that SOD-based noise reduction significantly outperforms the POD-based method for continuously sampled noisy time series.