Search

The six papers in this special issue that develop solutions to the Structural Dynamics Challenge Problem are summarized herein. The goal is to emphasize different tools and approaches applied to various parts of the structural dynamics problem. Specifically the following issues are considered: (1) Development of a mathematical framework for uncertainty quantification of a substructure. (2) Calibration of a mathematical model for the substructure. (3) Validation of the substructure mathematical model. (4) Use of the mathematical model of the substructure, in conjunction with another structure, to make a prediction of the probability that a regulatory limit is surpassed, and discussion of the uncertainty of the prediction. Different methodologies are presented and specific results vary, however, conclusions regarding satisfaction of the regulatory requirement match.

Probabilistic neural nets have been applied in the detection of structural damage. These networks, which rely upon approximating the multivariate density of the training data, have been shown to be effective in some applications. However, quantification of decision errors, which must ultimately be used to rank their effectiveness, has received little attention. In this paper, a two-dimensional probabilistic pattern classifier (PPC) based on an L 2 detector is studied. Each dimension is modelled as a Gaussian random variable in order to directly study the effects introduced by a commonly used density estimator. The region of acceptance is examined for different parameters of the kernel density estimator. The detector behaviour of the original PPC is compared to theory. It is demonstrated that performance is affected by errors introduced by a kernel constructed using a finite set of data, and also from theoretical limitations inherent in the test statistic. A modification of the test statistic is suggested to improve the sensitivity for structural damage detection.

Practically all mechanical structures behave nonlinearly, to some extent, yet most laboratory models for mechanical structures are linear, and most models used for prediction and design are constructed in the linear nite element (FE) framework. This work rst assesses the nonlinearity of the mechanical structure by comparing the structure response to a best-t limear model response obtained from experimental data. Assessment metrics are developed for determining if the system is strongly or weakly nonlinear during this comparison. These same metrics can then be used to validate the use of a linear model.

Linear mathematical models frequently provide good approximations to the input-output relations for real systems. However, ensembles of systems that are nominally identical cannot, usually, be adequately represented with a single model because real systems are stochastic. The randomness in real systems must be modeled if the randomness bears on critical behaviors of the system. The behavior of linear systems can be represented in parametric or non-parametric form; the latter framework is used, here. Among the frameworks available for characterization of system behavior, we choose the frequency response function (FRF). We choose to work with the FRF because many system attributes can be interpreted by inspection of the FRF, and it can be used directly for control design. This paper improves a previously developed Karhunen-Loeve expansion (KLE) representation for linear system behavior based on FRF data. The improvement yields a compact representation of the uncertainty inherent in an ensemble of systems and avoids the introduction of unwanted features in the system representation. This non-parametric, compact representation of the distribution of linear systems can then be used to characterize the performance and stability of a given feedback control law, as well as for control law design.

Linear systems are good approximations of the input-output relationship for many real systems. In practice, ensembles of nominally identical systems can seldom be adequately represented by a single linear system. Some accommodation needs to be made for representing inherent ensemble randomness. One approach would be to use a parameterized model of the system, with the random nature captured in discrete parameters of the parameterized model. Another approach is to use data measured from each system in the ensemble to represent the random nature of the ensemble in a non-parametric form. This is the approach described in this paper. There are several different types of data that could be collected from each system in the ensemble, each type capable of capturing the input-output behavior of the linear system: input-output time domain data, perhaps with specific excitation sequences [1,2]; Markov parameters [2,3]; and Frequency Response Functions (FRFs) [2] to name a few. We choose to work with FRF data because many system attributes can be easily interpreted by inspection of the FRF [4]. FRF data can also be used directly for control design [5,6,7]. This paper develops a Karhunen-Loeve expansion [8] representation for linear system behavior based on FRF data to develop a compact representation of the uncertainty inherent in an ensemble of systems. This non-parametric, compact, representation of the distribution of linear systems can then be used to characterize the performance and stability of a given feedback control law, as well as for control law design [5,6,7].

Natural, random, dynamic environments are ubiquitous, and humans have observed them for millennia. Some random dynamic environments are earthquake ground motions, winds, ocean waves, rough roads, and acoustic pressure environments arising from thunder. Before people could easily conceptualize harmonic motions, they observed random, oscillatory environments. Today, random vibration can be mathematically modeled as the motion of a mechanical system excited by a random input. The mathematical theory of random vibration is essential to the realistic modeling of structural dynamic systems. This paper summarizes the work of some key contributors to the theory of random vibration from the time of Rayleigh and Einstein to the present. Among many other things, we describe the works of (1) Rayleigh, who, in the late nineteenth century introduced ideas about the representation of random signals as the superposition of random harmonics; (2) Schuster, who, in the late nineteenth and early twentieth centuries described, without taking its limit, the spectral density; (3) Einstein who, in 1905, wrote the first actual paper on the theory of random vibration (Brownian movement); and (4) Wiener who, in 1930, formally defined the spectral density. Several graphic examples are included.

An efficient method for performing first passage analyses and inelastic response analyses is presented. The first passage problem is solved by discretizing the equation of motion for the structure in space and in time to obtain the transition probabilities for the displacement of the structure. The response of elastoplastic structures to stationary excitation is characterized by the probability distributions of cumulative plastic deformation and permanent set. The response process of the structure is discretized in time and in space, and an equivalent vibration process is obtained. The half-cycle information is used to predict response information over a given length of time. Results of numerical examples indicate that the accumulation of plastic displacements differs only slightly between the zero-start case and the stationary-start case for elastoplastic structures.