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The instantaneous displacement, velocity and acceleration of a cantilever tip impacting onto a graphite surface are reconstructed. The total dissipated energy and the dissipated energy per cycle of each excited flexural mode during the tip interaction is retrieved. The tip dynamics evolution is studied by wavelet analysis techniques that have general relevance for multi-mode atomic force microscopy, in a regime where few cantilever oscillation cycles characterize the tip–sample interaction.

This work introduces the concept of time–frequency map of the phase difference between the cantilever response signal and the driving signal, calculated with a wavelet cross-correlation technique. The wavelet cross-correlation quantifies the common power and the relative phase between the response of the cantilever and the exciting driver, yielding “instantaneous” information on the driver-response phase delay as a function of frequency. These concepts are introduced through the calculation of the response of a free cantilever subjected to continuous and impulsive excitation over a frequency band.

Wavelet transform analysis is applied to a thermally excited cantilever to get insights into fundamental thermodynamical properties of its motion. The shortcomings of the widely used Fourier analysis are briefly discussed to put into perspective the wavelet transform analysis, used to describe the temporal evolution of the spectral content of the thermal oscillations of a cantilever with an interacting tip. This analysis allows to retrieve the force gradients, the forces and the Hamaker constant in a measurement time of less than 40 ms.