12 A. Urbina et al. Furthermore, because tests cannot be performed for many actual use environments, the model is required to extrapolate beyond the data it was developed from. To establish confidence in an extrapolated model prediction, sources of uncertainty must be identified, quantified and propagated to the response quantity of interest at the system model. Recently, there has been an emphasis on developing models of components using first principles, calibrating them from simple exploratory experiments, validating them relative to a different set of experiments and then using them within a more complex model. For example, one could investigate the behavior of mechanical joint using simple experiments, develop a model that explains some phenomenon of interest, validate its performance on a different environment and use it as part of a larger system. What was described above is defined as a hierarchical approach to building a system level model. It is basically a construction of a complex system model by using a building block approach that incorporates simpler component based models and couples them together. This hierarchical model building approach was described in several published papers [1, 2]. To quantify uncertainty, multiple tests of these components should be available from which an estimate of the variability in the components could be obtained. Adding to the uncertainty is the possibility that the interactions of the various components was never tested, thus no information on the coupling of components will be available. In addition, interactions of components could have been tested at excitation levels that are not comparable to those of the full system, thus giving rise to another source of uncertainty. In this paper, we focus on the quantification of uncertainty due to differences in model prediction and experiments, and present a technique to aggregate and propagate these sources from the component level to the applications space. A numerical example based on a structural dynamics application is used to demonstrate the technique. 2.2 Example Problem Description The example problem has the following features: • It is a multi-component problem which involves a mechanical joint which provides an energy dissipating mechanism. • It is a multi-level problem where the phenomena observed at the lowest level is assumed to be present at subsequent levels, i.e. damping in the joints is assumed similar at all levels. This might turn out to be an incorrect assumption. • Experimental data consists of repeated tests on several, nominally identical hardware systems. These are intended to quantify the variability inherent in a physical system. • Simple finite element models are built and calibrated to simulate a particular behavior of the physical hardware. The model parameters have been calibrated from simple, discovery experiments aimed at isolating the particular physical phenomenon that the model is trying to represent. Parametric uncertainty is explored in this paper. The levels of complexity in this problem are defined as follows: Level 1 • Dumbbell configuration: 45 joint samples tested with an impulse type excitation. Level 2 • Three leg configuration (wavelet input): 27 joint samples tested using a wavelet-type. Application level • Three leg configuration (shock input): 27 joint samples tested using a shock-type input excitation. In all levels, acceleration time histories were recorded and energy dissipation was calculated for each experiment and for each model prediction. The particular details of each level are described in the following sections. 2.2.1 Dumbbell Configuration (Level 1) This configuration has two, 30 lb masses bolted at the ends of a single leg (or joint) creating a “dumbbell” looking hardware. This is shown in Fig. 2.1. This configuration is supported by bungee cords to simulate a free-free environment and it is subjected to an impulse excitation, on one of the end masses, provided by an instrumented hammer. The acceleration response of the dumbbell on the side opposite to the excitation is recorded and also shown in Fig. 2.1. From this response, the free decay time history of the response is obtained and used to estimate the energy dissipation of the system at a particular force
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