The classical trajectory approach has been used to study the nonthermal desorption of CO from a variety of model surfaces to which it is weakly adsorbed. In addition to three degrees of freedom for the CO adsorbate (bond stretching, physisorption, libration), a significant number of lattice degrees of freedom have been included using the generalized Langevin approximation. Nonthermal amounts of energy have been put into both the CO stretching and librational modes at t=0. We find that for initial values of the stretching quantum number upsilon(str)=0-10, desorption does not take place at all within 12.5 ps unless there is also significant librational excitation. The detailed mechanism by which librational energy causes desorption is discussed. The role of the surface is also explored; we find that the probability of desorption is a nonmonotonic function of the Debye frequency of the solid in the range 28-915 cm(-1), and is larger for lattices with either "high" or "low" Debye frequencies than for lattices with "intermediate" Debye frequencies. This result is partially explained in terms of resonances between low frequency fibration and physisorption modes and high frequency phonon modes.