Uncertainty analysis of computational models is essential to obtain a probabilistic description of the output quantities in the presence of uncertain model parameters or model inadequacy. Uncertainty quantification of the numerical model inputs, propagation of uncertainty to the model output and finally the analysis of response statistics, sensitivity, reliability, are all covered within the topic of uncertainty analysis. This has important applications in engineering design in terms of optimizing design variables under parameter fluctuations, obtaining confidence values associated with a novel design amongst others. However, the stochastic analysis of computational models are quite expensive. The objective of the work, described in this paper, is to develop a computational framework for efficient uncertainty analysis of structural dynamic systems. This has been applied to the design optimization of corrugated compliant skins in order to study its response sensitivity to its material properties and geometrical parameters. The sparse grid collocation technique has been utilized here as an efficient uncertainty propagation method for a multidimensional stochastic input. The sensitivity of the solution to the various sources of input uncertainty is studied using the Sobol's indices for sensitivity measure. Important physical insight into the behavior of the model for various uncertainties in geometrical and parametric properties is provided by this analysis.