Some uncertain parameters were then recognized in the modeling of this lumped-mass model. Notice that the application of this model was defined to be the accurate prediction of the system’s time-history response. From this viewpoint, not only mi and ki, but also damping parameters related to ci will influence this prediction. The damping parameters were thus considered to be uncertain parameters in the calculation. Furthermore, the parameters, Gap and ', were also considered to have uncertain values in the nonlinear system calculation. 3. Response feature for structural model validation 3.1. General about feature selection Response features are quantities that can be used to compare the measured and calculated system response. When used for model validation, the extracted response features should be sensitive to the target uncertain parameters of the numerical model. It should be noted that the intended purpose of the numerical simulation should also be considered when performing feature selection because the parameters that most influence the response may be different depending on in the intended purpose of the analysis. Dimensionality is another important consideration in the feature selection process. The feature dimension is the number of independent scalar quantities that are necessary to describe the feature. Low-dimensional features are preferable to high-dimensional features because this makes it easier to compare values and to statistically analyze their trends. Furthermore, the feature selection should reflect the type of response that is being considered, such as linearity or nonlinearity. Some response features that have been suggested for dynamic response calculations are [4]: - Linear, stationary, Gaussian vibrations: Direct and inverse Fourier transforms, Power spectral density, Input-output transfer functions, Frequency responses, Modal parameter. - Transient dynamics and mechanical shock response: Peak values, Energy content, Decrement and exponential damping, Shock response spectrum, temporal moments. - General-purpose time-series analysis: AR, ARMA, ARX, AR-ARX models, Time-frequency transforms, Wavelet transform, Principal component decomposition. - Unstable, chaotic, multiple-scale dynamics: Holder exponent, State-space maps, Time-frequency and higher-order transforms, Symmetric dot pattern, Fractal analysis. It should be emphasized that there is no one feature that will be applicable to all structural dynamics predictive modeling scenarios. If multiple aspects of the system response are of interest, the model validation process may require different features to be extracted from the data in an effort to validate different aspects of the modeling process. 3.2. Discussion: feature extraction for the structural dynamics model validation In this study, the damping parameters were defined to be the most influential uncertain parameters in the application of dynamic response analysis of the test-bed structure. Table 1 presents some extracted response features expected to be used in the damping parameters validation. Blue results are all from experimental data, and red and green results are from numerical outputs. Notice that the numerical output used in deriving green results was more accurate than red one because assigned damping ratios in the green output were values obtained form an experimental modal analysis. Considering the physical meaning of damping, the parameters should influence on the amplitude behaviour of the response, which is directly related to the energy dissipating behaviour. In the use of a random and stationary response from the linear system; i.e., w/o the bumper-column mechanism, the standard deviation, which was one of basic statistics, showed sensitivity to the accuracies of damping parameters. The feature that had the same physical meaning for a transient response was the energy value of temporal moments, and it also showed almost similar sensitivity as the standard deviation. 156
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