In this paper we address the mechanical properties of carbon nanotubes anchored to atomic force microscopy (AFM) tips in a detailed analysis of experimental results and exhaustive description of a simple model. We show that volume elastic and surface adhesive forces both contribute to the dynamical AFM experimental signals. Their respective weights depend on the nanotube properties and on an experimental parameter: the oscillation amplitude. To quantify the elastic and adhesive contributions, a simple analytical model is used. It enables analytical expressions of the resonance frequency shift and dissipation that can be measured in the atomic force microscopy dynamical frequency modulation mode. It includes the nanotube adhesive contribution to the frequency shift. Experimental data for single-wall and multi-wall carbon nanotubes compare well to the model predictions for different oscillation amplitudes. Three parameters can be extracted: the distance necessary to unstick the nanotube from the surface and two spring constants corresponding to tube compression and to the elastic force required to overcome the adhesion force.