In this study, the nonlinear forced vibration of composite structural systems such as plates, panels and shells reinforced with advanced materials in the presence of linear viscous damping is investigated. Hamilton principle and von Karman-type nonlinear theory are used to obtain the theoretical model of double-curved shells reinforced by carbon nanotubes (CNTs). The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method. By using the multiscale method, the frequency-amplitude relation and nonlinear forced vibration frequency of structural systems are obtained for the first time. Since double-curved shells can be transformed into other structural systems such as spherical and hyperbolicparaboloid shells, rectangular plate and cylindrical panel in special cases, the expressions for nonlinear frequencies can also be used for them. In additional, the backbone curve and the nonlinear frequency/linear frequency ratio are determined as a function of the amplitude in primary resonance for the first time. The results are verified by comparing the reliability and accuracy of the proposed formulation with those in the literature. Finally, a systematic study is aimed at controlling the influence of nonlinearity and types of distribution of CNTs on the frequencies and their quantitative and qualitative variation in the presence of external excitation and viscous damping.