An analytical model is presented to predict the nonlinear response of a double-wall sandwich cylindrical shell system subjected to random excitation. Nonlinear spring-dashpot models are integrated into the system to characterize the behavior of the soft core. Donnell's thin shell theory is used to develop the governing nonlinear equations of motion. A Monte Carlo simulation of stationary random processes, multimode Galerkin-type approach, and numerical integration procedures are used to develop linear and nonlinear response solutions of simply supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and response spectral densities. Parametric studies are performed to investigate the effects of nonlinearity, shell thickness, core stiffness, and thickness.