S.T. Trickey

Structural health monitoring through chaotic interrogation

The field of vibration based structural health monitoring involves extracting a ‘feature’ which robustly quantifies damage induced changes to the structure in the presence of ambient variation, that is, changes in ambient temperature, varying moisture levels, etc. In this paper, we present an attractor-based feature derived from the field of nonlinear time-series analysis. Emphasis is placed on the use of chaos for the purposes of system interrogation. The structure is excited with the output of a chaotic oscillator providing a deterministic (low-dimensional) input. Use is made of the Kaplan–Yorke conjecture in order to ‘tune’ the Lyapunov exponents of the driving signal so that varying degrees of damage in the structure will alter the state space properties of the response attractor. The average local attractor variance ratio (ALAVR) is suggested as one possible means of quantifying the state space changes. Finite element results are presented for a thin aluminum cantilever beam subject to increasing damage, as specified by weld line separation, at the clamped end. Comparisons of the ALAVR to two modal features are evaluated through the use of a performance metric.

Use of data-driven phase space models in assessing the strength of a bolted connection in a composite beam

This work explores the role of empirical dynamical models in deducing the level of preload loss in a bolted connection. Specifically, we examine the functional relationship between data gleaned from locations on either side of the connection using nonlinear predictive models. This relationship, as quantified by a measure of prediction error, changes as a function of bolt loosening, thus allowing both the presence and magnitude of the axial load to be identified. The models are based on a phase space portrayal of the system dynamics and require only that the structures response be low dimensional. The technique is demonstrated experimentally on a composite beam fastened to steel plates with four instrumented bolts. Results are compared to a similar approach using an auto-regressive (AR) modeling technique.

The stability of limit-cycle oscillations in a nonlinear aeroelastic system

The effects of a freeplay structural nonlinearity on an aeroelastic system are studied experimentally. Particular attention is paid to the stability of a periodic nonlinear aeroelastic response, known as limit–cycle oscillations (LCOs). The major thrust of this research lies in the application of relatively recently developed techniques from nonlinear dynamics and signal processing to the realm of experimental aeroelasticity. Innovations from the field of nonlinear dynamics include time–delay embedded coordinates to reconstruct system dynamics, a Poincaré section to assess the periodic nature of a response and to prescribe an operating point about which a linear description of the dynamics can be approximated, stochastic perturbations to assess the stability and robustness of responses, and a basin of attraction measure to assess initial condition dependence. A novel system–identification approach is used to generate a linear approximation of the experimental system dynamics about the LCO. This technique makes use of a rotating slotted cylinder gust generator and incorporates a least–squares fit of the resulting transient dynamics. An extension to this method is then developed based on the outcome of relatively large disturbances to the flow and hence airfoil, to obtain global stability.

Detecting impact damage in experimental composite structures: an information-theoretic approach

This work describes a procedure for detecting the presence of damage-induced nonlinearities in composite structures using only the structures vibrational response. Damage is assumed to change the coupling between different locations on the structure from linear to nonlinear. Utilizing concepts from the field of information theory, we are able to deduce the form of the underlying structural model (linear/nonlinear), and hence detect the presence of the damage. Because information theoretics are model independent they may be used to capture both linear and nonlinear dynamical relationships. We describe two such metrics, the time delayed mutual information and time delayed transfer entropy, and show how they may be computed from time series data. We make use of surrogate data techniques in order to place the question of damage in a hypothesis testing framework. Specifically, we construct surrogate data sets from the original that preserve only the linear relationships among the data. We then compute the mutual information and the transfer entropy on both the original and surrogate data and quantify the discrepancy in the results as a measure of nonlinearity in the structure. Thus, we do not require the explicit measurement of a baseline data set. The approach is demonstrated to be effective in diagnosing the presence of impact damage in a thick composite sandwich plate. We also show how the approach can be used to detect impact damage in a composite UAV wing subject to ambient gust loading.

Use of Fiber-optic Strain Sensors and Holder Exponents for Detecting and Localizing Damage in an Experimental Plate Structure

The vibration-based structural health monitoring paradigm is predicated on the practitioners ability to acquire accurate structural response data and then to use that information to infer something about the structures health. Here the authors combine advances in both sensing and signal analysis and demonstrate the ability to detect damage in a simple experimental structure. A distributed network of nine fiber-optic strain sensors is used to acquire time series data from a rectangular steel plate where damage is considered as a cut of varying lengths. Both the sensors and the associated optical hardware are described. A new feature, Holder continuity, is then introduced as a means of identifying the presence and location of the cut length. This particular metric is derived from the field of nonlinear dynamics and is based on a phase space description of a structures dynamic response. Specifically, the authors compute the Holder exponent which quantifies the differentiability of the functional relationship between an ‘undamaged’ and a ‘damaged’ structural response. As damage is incurred, this relationship is expected to degrade. Both univariate and multivariate applications of the method are presented. The metric shows sensitivity to damage comparable to that exhibited by the plates modal frequencies, a traditionally used feature in health monitoring applications.

Using chaotic forcing to detect damage in a structure

In this work we develop a numerical test for Holder continuity and apply it and another test for continuity to the difficult problem of detecting damage in structures. We subject a thin metal plate with incremental damage to the plate changes, its filtering properties, and therefore the phase space trajectories of the response chaotic excitation of various bandwidths. Damage to the plate changes its filtering properties and therefore the phase space of the response. Because the data are multivariate (the plate is instrumented with multiple sensors) we use a singular value decomposition of the set of the output time series to reduce the embedding dimension of the response time series. We use two geometric tests to compare an attractor reconstructed from data from an undamaged structure to that reconstructed from data from a damaged structure. These two tests translate to testing for both generalized and differentiable synchronization between responses. We show loss of synchronization of responses with damage to the structure.

Basins of Attraction in Experimental Nonlinear Oscillators

This paper addresses two relatively simple but fundamental questions in nonlinear oscillations: Given an arbitrary initial condition where will the trajectory go, and how long will it take to get there? These related questions are addressed from an experimental perspective where generating global transient behavior has received relatively little attention, despite the fact that a global view of transient behavior provides a much more complete description of the dynamics of a system than a traditional concentration on steady-state behavior. Three different physical systems are studied, each of which exhibits a specific behavior heavily influenced by transient global effects.

A NOTE ON THE RESPONSE SPECTRUM MAP

A plot of frequency, or spectral, content versus a system parameter was introduced in a recent paper by Billings and Boaghe [2001] as a useful alternative to bifurcation diagrams in nonlinear dynamics. The current contribution illustrates the same approach based on data taken from two experimental mechanical systems in which hysteresis is featured.

Development of higher-order spectra for randomly excited quadratic non linear systems: Volterra functional series approach

Higher order spectral analysis techniques are often used to identify nonlinear interactions in modes of dynamical systems. More specifically, the auto and cross- bispectra have proven to be useful tools in testing for the presence of quadratic nonlinearities based on a systems stationary response. In this paper a class of mechanical system represented by a second-order nonlinear equation of motion subject to random forcing is considered. Analytical expressions for the second-order auto- and cross-spectra are determined using a Volterra functional approach and the presence and extent of nonlinear interactions between frequency components are identified. Numerical simulations accompany the analytical solutions to show how modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation. A closed-form solution of the Bispectrum can be used to help identify the source of non-linearity due to interactions at specific frequencies. Possible applications include structural health monitoring where damage is often modeled as a nonlinearity. Advantages of using higher-order spectra techniques will be revealed and pertinent conclusions will be outlined.

Vibration‐Based Damage Assessment Using Novel Function Statistics with Multiple Time Series

Analysis of data from experiments on dynamical systems often centers on the embedding of time series data to reconstruct an atttractor. In our system, we consider output expressed as multiple time series from a circuit designed to simulate a spring‐mass system in both an undamaged and a damaged state. In order to analyze differences in the reconstructed attractors between the damaged and undamaged states we employ a new version of the Continuity Statistic. This statistic was first introduced by Pecora, Carroll and Heagy [1]. Here we use the statistic in the new setting of an embedding of multiple time series and in order to address noisy data, formulate a new null hypothesis. We show that this new continuity statistic is an appropriate tool for showing differences between reconstructed attractors in the specific case of our simulated spring‐mass system in “damaged” and “undamaged” conditions.