  We show how spectral submanifold theory can be used to construct
reduced-order models for harmonically excited mechanical systems with internal
resonances. Efficient calculations of periodic and quasi-periodic responses
with the reduced-order models are discussed in this paper and its companion,
Part II, respectively. The dimension of a reduced-order model is determined by
the number of modes involved in the internal resonance, independently of the
dimension of the full system. The periodic responses of the full system are
obtained as equilibria of the reduced-order model on spectral submanifolds. The
forced response curve of periodic orbits then becomes a manifold of equilibria,
which can be easily extracted using parameter continuation. To demonstrate the
effectiveness and efficiency of the reduction, we compute the forced response
curves of several high-dimensional nonlinear mechanical systems, including the
finite-element models of a von K\'arm\'an beam and a plate.
