The steadystate response of a system with the nonviscous exponential damping and perodic excitation is studied by the generalized cell mapping method based on shorttime Gaussian approximation. The process of the method is firstly introduced, and the moment equations are derived. The steadystate response probability density functions are then presented, in which the effect of damping coefficient and relaxation parameter on the probability density functions is discussed. The validity of the generalized cell mapping method is demonstrated by the results from direct Monte Carlo simulation.