How does a tip tap?

There is a demand for good theoretical understanding of the response of an atomic force microscope cantilever to the extremely nonlinear impacts received while tapping a sample. A model and numerical simulations are presented in this paper which provide a very pleasing comparison with experimental results. The dependence of the cantilever amplitude and phase upon the sample stiffness, adhesion and damping are investigated using these simulations, and it is found that `topographic' tapping images are not independent of sample properties, nor will it be trivial to measure materials' properties from the tapping data. The simulation can be applied to other probe microscope configurations as well.