In this paper we formulate and analyze a continuum model for the vibration of a noncontacting atomic force microscope (AFM) microbeam in air that consistently incorporates nonlinear geometric and inertia effects, localized atomic interaction, viscoelastic damping and quadratic drag. We investigate a controlled set of experiments that include both free vibration decay of a large Silicon beam and forced vibration response of an AFM Silicon microbeam mapping a Silicon sample for various initial interaction distances. Nonlinear frequency and damping backbone curves are obtained from free vibration decay data and equivalent damping ratios are deduced from forced vibration frequency response. Estimation of the system linear viscoelastic parameters and nonlinear drag parameters is enabled by comparison of the experimental backbone curves with those of a nonlinear modal dynamical system deduced from the continuum model. The calibration results without sample interaction include both a slight softening effect for small amplitude response due to nonlinear inertia and viscoelastic damping and a hardening effect for large amplitude response governed by nonlinear geometric effects and drag. Validation of the nonlinear model is enabled by comparison with the measured forced vibration AFM frequency response below the dynamic jump-to-contact threshold.