Kevin D. Murphy’s research while affiliated with University of Louisville and other places

Models are how one understands and predicts observed phenomena, which is the primary objective of all scientific disciplines. Many models have parameters and the values of those parameters are often unknown a priori. Even if the values are known to within some interval, those values carry with them some uncertainty as to which value to use in the model predictions. The parameter value one eventually uses (our estimate) will be different depending on how one defines best value. This chapter focuses on setting up the type of estimation problems frequently encountered in damage identification applications and providing solutions. It discusses the basics of maximum likelihood estimates (MLEs), focusing on why they are desirable as point estimators of parameters. The Bayesian approach is fundamentally different in that it treats the unknown parameters as random variables with joint distribution pΘ(θ).

This chapter focuses on the probabilistic modeling of sequences of observations. It discusses different properties used to describe a random process. The chapter provides an overview of what is to come regarding estimation (our eventual goal). To develop accurate techniques for parameter estimation, a description (model) of the observed structural response data is required. In structural dynamics, random process models are typically comprised of two parts: a deterministic component that is the result of physics-based model and a stochastic component describing the uncertainty present in experimental measurements. The assumption of stationarity is a common one, and it means that the joint probability distribution used to model random process is time-invariant. Spectral analysis allows us to model a given signal as a superposition of some (possibly infinite) number of sine and cosine functions whose arguments are a product of frequency and time.

Statistics are descriptive rather than predictive. In the same way that engineers use stiffness as a parameter in many deterministic structural models, statistics can be thought of as parameters that describe a probabilistic model. Statistics are extremely important in describing the probabilistic model and plays an important role in identifying properties of the structural response data that are indicative of structural damage. This chapter introduces probability, provides the necessary mathematical framework for assigning probability, and clarifies the interpretation of probabilistic statements. Scientists and engineers are frequently interested in using probability to predict the outcomes of an experiment. The chapter discusses a number of important statistics associated with common probability models. It also provides an overview of basic tools needed to construct probabilistic models for entire sequences of observations, such as those produced by one or more sensors located on a structure.

This chapter focuses on complete picture of model-based damage identification. The approach followed in each of the examples has the same basic construct that includes formulating a damage model, estimating the parameters of that model, and using those estimates to detect and identify the damage. In formulating the models, both the deterministic and stochastic modeling tools are used. Several different damage mechanisms are considered including imperfections in shell structures, cracks in plates, and corrosion damage. The chapter discusses the parameters associated with the delamination (location, length, and depth). It then focuses on the identification of the parameters that describe a crack in an aluminum plate. The chapter also quantifies the extent of corrosion damage (presumed uniform) in a homogeneous, isotropic metal plate exposed to a corrosive salt water environment. For corrosion damage, the key damage parameter captured by the model will be the average plate thickness.

This chapter begins with very basic assumptions about the physics of structural damage and move towards the more specific. It notes that in many cases the vibrations of an undamaged, pristine structure is accurately described by a linear model. The bispectrum or bicoherence functions are a popular choice for capturing nonlinearity and are elaborated on in several case studies in the chapter. The chapter considers the strain response of bolted joint structure to a stationary, jointly Gaussian input signal and uses the hypothesis to form a damage detection procedure. It provides a detailed description of how one can use the higher order spectra (HOS) to detect the presence of damage-induced nonlinearity in structures. The chapter considers the more realistic scenario of a multi-degree-of-freedom (MDOF) structure subject to (possibly) a non-Gaussian input. It addresses two numerical surrogate examples include detection of a bilinear stiffness and detection of a cubic stiffness.

This chapter presents an overview of the development of deterministic models in structural dynamics, whose output describes the behavior of the physical structure under consideration. As physical laws tend to be differential in nature, the resulting model equations are often ordinary differential equations (ODEs) or partial differential equations (PDEs). Building a successful structural model is challenging because each problem contains its own unique twists that complicate the process. The modeler has to research related problems, borrowing liberally from the literature, and then develop a model to accommodate the particular problem at hand. The chapter lays out some specifics on the fundamental laws and their applications in modeling. It discusses a few of the more common problems encountered and how they can be addressed. Finally, the chapter focuses on the Volterra series model which is used to model the time-domain response for a causal, weakly nonlinear system with finite memory.

This chapter takes the mechanics of modeling structural systems particularly, structural damage and begins applying it to specific example systems, both numerical and experimental. Plate structures are one of the most common types of engineering structures encountered. Composite materials have become an essential part of many structures and are primarily valued for their high strength-to-weight ratio. One of the primary failure mechanisms is delamination, whereby two or more of the composite layers separate. The chapter describes quasi-static approach in the sense that the analysis linearized about local regions of a globally nonlinear stiffness function located at the joint. The change in spectral properties is a frequently used identifier of damage-induced nonlinearity. The most common beam-foundation model allows for both compressive and tensile stresses to exist across the interface between the beam and the foundation. Axially moving systems are common in engineering applications.