One of the important and efficient tools in system analysis is the analysis of responses to harmonic excitations. For linear systems the information on such responses is contained in the frequency response functions, which can be computed analytically. For nonlinear systems there may be even no periodic response to a periodic excitation. Even if such a periodic response exists and is unique, its computation is, in general, a computationally expensive task. In this paper we present a fast method for computing periodic responses to periodic excitations for a class of nonlinear systems. The method allows one to efficiently compute the responses for harmonic excitations corresponding to a grid of excitation frequencies and amplitudes. The results are illustrated by application to a flexible beam with one-sided stiffness subject to harmonic excitation. © 2008 IEEE.