A Bayesian approach to identifying structural nonlinearity using free-decay response: Application to damage detection in composites

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Abstract

This work discusses a Bayesian approach to approximating the distribution of parameters governing nonlinear structural systems. Specifically, we use a Markov Chain Monte Carlo method for sampling the posterior parameter distributions thus producing both point and interval estimates for parameters. The method is first used to identify both linear and nonlinear parameters in a multiple degree-of-freedom structural systems using free-decay vibrations. The approach is then applied to the problem of identifying the location, size, and depth of delamination in a model composite beam. The influence of additive Gaussian noise on the response data is explored with respect to the quality of the resulting parameter estimates.

Introduction

System identification can be loosely defined as the process of estimating parameters associated with a specified model (or models) from acquired data. There are two main schools of thought in estimation problems: the frequentist approach, often based on the method of maximum likelihood (ML), and the Bayesian approach. Both methods seek to provide the best possible parameter estimates in the face of the inevitable uncertainty (e.g. measurement error) present in the observed data. In fact the likelihood function, describing the joint distribution of the data given the model and model parameters, is the central ingredient in both approaches to estimation. However, the two approaches are fundamentally different in how they treat model parameters resulting in different approaches to inference.

Historically, researchers working on nonlinear system identification problems have tended to focus on ML methods. In a recent review paper Kerschen et al. [1, see Sections 6 and 7 and the many references contained therein], cover numerous available techniques for nonlinear parameter estimation. These approaches are based on time domain (e.g. [2], [3]), frequency domain (e.g. [4], [5], [6]) or higher-order spectral analysis [7], [8]. Each of these techniques takes the best estimate to be the one that minimizes the mean square error between data and model. Although not explicitly stated, this choice of cost function yields ML estimates for the parameters, provided that the uncertainty in the model is taken as additive, iid, jointly Gaussian noise. The specific pros and cons associated with these methods are highlighted in [1] and are therefore not discussed here. Rather, this work is focused on the conceptually different Bayesian alternative to nonlinear parameter estimation problems.

Perhaps the chief benefit of the Bayesian worldview is that it treats the parameters we seek to estimate, denoted by the vector θ, as random variables with joint distribution p(θ). This allows us to make probabilistic statements about our estimated parameters. For example, let us say our goal is to collect data from a structure and estimate the crack length, denoted θ1. The Bayesian framework allows us to make statements such as the following:

There is a 95 percent chance that the crack length θ1 lies in the interval 0.1θ10.5[cm].

There is no way to make an analogous statement using the frequentist view of the world because the crack length would not be treated as a random variable. Instead we would have to make statements about the repeatability of our estimation procedure i.e. the statistical machinery that produced the interval estimate. Thus we might be able to make the statement “The methods we have used produce random intervals which may or may not include the unknown parameter; they succeed in 95 percent of similar circumstances, but we can’t say anything with certainty about [0.1,0.5]” Although in some cases the two methods will produce the same estimate and interval the interpretation of the interval is different, the Bayesian interval being a probability statement about the parameter. Thus, our main reason for gravitating toward the Bayesian viewpoint is that it provides a direct estimate of the information we are after as opposed to related (but different) information. Additional reasons for the Bayesian approach are highlighted in Chapter 1 of [9]. First, the approach is more amenable to more complex data and/or models than is the frequentist (ML) approach (including cases where a frequentist approach does not exist). Secondly, the approach described here does not require asymptotics in approximating the confidence intervals (as does the frequentist approach of using the Fisher Information Matrix to bound the variance of a parameter estimate, see e.g., [10]).

This is certainly not the first paper to explore Bayesian methods in structural dynamics. A general Bayesian approach to structural dynamics problems was put forth in the ’90s with works by Beck and Katafygiotis [11], [12]. These, and subsequent works (see e.g., [13]), relied on asymptotics in order to solve for the marginal parameter distributions which, as will be shown, often take the form of high-dimensional integrals. At around the same time Sohn and Law [14] also developed a Bayesian approach to the structural identification problem. Their approach, also described in subsequent papers [15], [16], circumvented the issue of solving for the desired marginal parameter distributions by using a so-called “branch-and-bound” strategy to instead solve for the most likely damage hypothesis in a damage detection application.

Rather than abandon the goal of parameter estimation or rely on asymptotics, one can make use of a powerful approach for sampling the marginal distributions without having to perform the integration. Such an approach was first proposed by Metropolis et al. [17] in the early 1950's for solving high-dimensional integrals in particle physics. A later work by Hastings [18] extended this general approach, resulted in what is now known as the Markov Chain Monte Carlo (MCMC) approach to approximating probability distributions. MCMC methods have since become extremely popular in implementing Bayesian estimation in a number of fields ranging from ecology [9] to genetics [19] and have recently seen increased use in engineering applications. Beck et al. appear to have pioneered the use of MCMC methods in structural dynamics in Beck and Au [20] and more recently in Cheung and Beck [21]. Additional work by Glaser et al. [22] illustrated the approach in detecting stiffness reduction in beams using static measurements. This method has also been used to estimate failure probabilities in structural reliability problems as part of the “Subset Simulation” approach of Au et al. [23]. In related works, Zhang and Cho [24] used the MCMC approach to help design an evolutionary algorithm for performing system identification, while Kerschen et al. [25] used the MCMC approach to select among competing models for describing the dynamics of a nonlinear mechanical system.

Our goal in this work is to focus on the use of the MCMC approach to Bayesian parameter estimation in nonlinear systems using free-vibration, time-series data. The approach is certainly more general and could be applied to forced structures as well. However, in practice the forcing function is not always obtainable while for the free-decay problem we simply have to include the initial conditions as random variables to be predicted. A different approach that does not require input data would be to perform the analysis using estimated frequencies as a means of comparing model to data as was done in [13]. However here the practitioner has the additional step of estimating the frequencies from observed data. This is compounded by the task of determining the analytical model frequencies which for nonlinear systems can be extremely challenging. Thus, free-decay response data provide for a direct model-to-data comparison and are easily obtainable in experiment. Another practical advantage of this general approach to system identification is that one does not need to measure time-series data from each of the structural degrees-of-freedom (DOF) in order to estimate the associated parameters. For example, a single time-series response from one of the DOFs can, in some cases, be used to estimate model parameters associated with DOFs not observed.

In this work the approach is first used to estimate the parameters associated with a two degree-of-freedom nonlinear structural system. The relationship between the quality of the resulting estimates and the signal-to-noise ratio is explored. We then turn our attention to the difficult problem of estimating and tracking delamination growth in a composite beam model. This model was recently developed in [26] where it was shown to accurately capture the localized buckling that occurs due to the separation of the laminates. Subsequent work by the authors [27] focused on detection of the delamination using a higher-order spectral analysis. Here, the focus is on identifying the damage parameters using only free vibration data, a task to which the Bayesian approach using MCMC to sample the desired parameter distributions is well-suited.

Section snippets

Bayesian approach to structural parameter estimation

In structural dynamics the uncertainty that gives rise to the likelihood function is typically the result of measurement error on the observed data. Assume that we have measured a structural response at a specific location giving the N-point time-serieszn=yn+ηn,n=1Nwhere yn is the noise-free response and ηn is a sequence of independent, identically distributed (iid) samples drawn from a zero-mean Gaussian distribution with variance σG2. In this case we can write the likelihoodpL(z|θ,σG2)=1(2πσG

Markov-chain Monte-Carlo methods

The term “Monte Carlo methods” is used to describe simulation techniques for investigating probability distributions. These techniques can be highly efficient, especially when independent samples can be generated. Unfortunately, posterior distributions used in Bayesian inference are often complicated, making it difficult to draw independent samples. Nevertheless it is often easy to draw a dependent sequence of samples representing posterior distributions. Over the last 60 years, stochastic

Example 1: 2-DOF nonlinear spring–mass-damper system

In order to illustrate the above described identification procedure, consider the two degree-of-freedom (DOF) system described by the second order, nonlinear, ordinary differential equations[M]y¨t+[C]y˙t+[K]yt+g(y˙t,yt)=0where [M]=m100m2[C]=c1+c2c2c2c2[K]=k1+k2k2k2k2are constant coefficient mass, damping and stiffness matrices respectively. The nonlinear function g(·) provides quadratic coupling between masses. Here we consider a quadratic restoring force between masses 1 and 2 so that g(y˙t

Delamination identification in a composite beam

As a second example, we seek to identify the location, extent, and depth of delamination in a composite beam structure. The dynamic beam model was derived previously in [27] and is shown schematically in Fig. 3. This model is low dimensional (only three independent coordinates need to be specified), yet was shown experimentally to accurately capture the localized buckling that occurs due to the presence of the delamination under static loading [26]. The global beam motion is assumed to be

Conclusions

This work has presented an approach for identifying nonlinear, multi-degree-of-freedom structures using only observed free-decay response data. The approach appears to work well for limited, noise corrupted observations that are easily obtainable in experiment. This makes the approach attractive from a practical standpoint with the main drawback being the computational effort required to build the stationary Markov chains. At each step in building process the practitioner is required to

Acknowledgment

The authors would like to acknowledge the Office of Naval Research under contract No. N00014-09-WX-2-1002 for providing funding for this work.

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