Fundamental frequencies of a torsional cantilever nano beam for dynamic atomic force microscopy (dAFM) in tapping mode

The aim of this investigation is to study the motion of an elastically restrained beam used in tapping mode atomic force microscopy (TM-AFM), which is to be utilized in manufacturing at nano-scale. TM-AFM uses high frequency oscillations to remove material or shape nano structures. Euler–Bernoulli theory and Eringen’s theory of non-local continuum are used to model the nano machining structure composed of a nanobeam and a single degree of freedom, spring-mass system. The system is modelled as a beam with a torsional spring boundary condition that is rigidly restrained in the transverse direction at one end; and at the free end is a transverse linear spring attached to the tip. The other end of the spring is attached to a mass, resulting in a single degree of freedom spring-mass system. When the linear spring constant is infinite, the free end behaves as a beam with a concentrated tip mass. When the mass is infinite, the boundary condition is that of a linear spring. When the tip mass is zero, the configuration is that of a torsionally restrained-free beam. The motion of the tip of the beam and tip mass can be investigated to observe the tip frequency response, displacement and force. The tip displacement frequency contains information about the maximum displacement amplitude and therefore the sample penetration depth.