Improved Estimation of Frequency Response Covariance

In high-consequence systems, critical decisions are made based on analytically predicted system response. In order to have confidence in the predictions, analytical models must be validated, and the effects of uncertainty on system response must be well understood. In systems with high modal density, modal based model validation is often problematic. It becomes more convenient to validate a model by directly using its response in the frequency domain. This then requires a corresponding frequency response based uncertainty quantification and propagation analysis. Covariance propagation is often used to propagate uncertainty and compute second order response statistics. However, linear covariance propagation applied to frequency response is inaccurate at resonances due to nonlinearity. A method is proposed that provides an improved estimate of the first two moments of frequency response uncertainty in the vicinity of resonances. It assumes that the uncertainty is small enough relative to damping such that an impedance based matrix inverse can be approximated using a finite number of terms. The method then propagates higher order moments of modal impedance uncertainty to approximate this inverse. The proposed approach can fit directly into a straightforward covariance propagation approach, and is applicable to substructured systems.