This work presents an asymptotic approach to predict the response of random structures to an impact in the time domain. The theory is based on the calculation of an asymptotic impulse response function by knowledge of the modal density of the structures involved in the impact. The approach presented predicts a mean response on an ensemble of structures with uncertainty on properties and boundary conditions. Both numerical and experimental results are presented. The theory has been successfully applied to thin and thick cylinders; in the latter case a modal density has been developed that differs from the classical case available for thin cylinders.