## Effect of flexible supports on the frequencies of nanobeams with tip mass and axial load for applications in atomic force microscopy (AFM)

##### Abstract

This aim of this investigation is to describe the mechanical performance of a beam (probe) used in dynamic atomic force microscopy (dAFM) which can be utilized in scanning the topographical features of biological samples or "pliable" samples in general. These nanobeams can also be used to modify samples by using high frequency oscillating contact forces to remove material or shape nano structures. A nanobeam with arbitrary boundary conditions is studied to investigate different configurations and the effects of the relevant parameters on the natural frequencies.
The nano structure is modelled using the Euler-Bernoulli theory and Eringen's theory of nonlocal continuum or first order stress-gradient theory is incorporated to simulate the dynamics of the system. This theory is effective at nanoscale because it considers the small-scale effects on the mechanical properties of the material. The theory of Nonlocal continuum is based on the assumption that the stress at a single point in the material is influenced by the strains at all the points in the material. This theory is widely applied to the vibration modelling of carbon nanotubes in several studies.
The system is modelled as a beam with a torsional spring boundary condition that is rigidly restrained in the transverse direction at one end. The torsional boundary condition can be tuned, by changing the torsional spring stiffness, such that the compliance of the system matches that of the sample to prevent mechanical damage of both the probe tip and the sample. When the torsional spring stiffness is zero, the beam is pinned and when the stiffness is infinite, the beam is a cantilever. In the first case, a mass is attached to the tip and a linear transverse spring is attached to the nanobeam. The mass and spring model the probe tip and contact force, respectively.
In the second case, 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 cantilever beam. When tip of the nanobeam vibrates, the system behaves like a hammer and chisel.
The motion of the tip of the beam and tip mass can be investigated to observe the tip frequency response, force, acceleration, velocity and displacement. The combined frequencies of the beam and spring-mass systems contain information about the maximum displacement amplitude and therefore the sample penetration depth.