Reduced-order models for a general class of non-linear oscillators with viscous damping, quadratic and cubic non-linearities, are derived thanks to a real normal form procedure. A special emphasis is put on the treatment of the damping terms, and its effect on the normal dynamics. In particular, it is demonstrated that the type of non-linearity (hardening/softening behaviour) depends on damping. The methodology is then applied for reducing the non-linear dynamics exhibited by a circular cylindrical shell, hamonically excited in the spectral neighbourhood of the natural frequency of an asymmetric mode. The approximation, which consists in using time-invariant manifold instead of computing an accurate time-dependent one, is discussed. The results show that the method is efficient for producing accurate reduced-order models without a special need for intensive numerical computations. Validity limits of the approximation is then assessed by using the amplitude of the forcing as bifurcation parameter. Finally, the method is compared to the Proper Orthogonal Decomposition method (POD).