Multi-modal analysis on the intermittent contact dynamics of atomic force microscope

Abstract

a b s t r a c tA multi-modal analysis on the intermittent contact between an atomic force microscope (AFM) with a soft sample is presented. The intermittent contact induces the participation of the higher modes into the motion and various subharmonic motions are shown. The AFM tip mass enhances the coupling of different modes. The AFM tip mass is modeled by the Dirac delta function and the coupling effects are analyzed via the Galerkin method. The necessity of applying multi-modal analysis to the intermittent contact problem is demonstrated. Unlike the impact oscillator model which assumes the impact/contact time is infinitesimal, the contact time can be a significant fractional portion in each cycle, especially for the soft sample case and thus results in different dynamic behavior from that of an impact oscillator.

The inverse problem of sensing the mass and force induced by an adsorbate on a beam nanomechanical resonator

“…When an adsorbate is on a carbon nanotube (CNT)-based resonator with the length of L , the governing equation of the resonator which is modeled as a beam is given as follows [7,8] […”

“…Where m is the resonator mass per unit length; o M and o x are the mass and position of the adsorbate, which is modeled as a concentrated mass by the Dirac delta function of  [7,8] [1…”

“…M and K are the N N  matrices of mass and stiffness, respectively, which are given as the following by using both the orthonormality property of ( ) j   and the integration property of the Dirac function [7,8]:…”

Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies

“…To compute the eigenfrequencies of equation (2.12), the Galerkin method [24,43] is applied, which assumes the following form for W(ξ , τ ): 14) where H is the mode number and φ 0 j (ξ ) is the jth mode shape of a uniform cantilever beam with the zero axial concentrated load (N = 0) as given in equation (2.10) [44]. Clearly, the mode shape of φ 0 j (ξ ) also satisfies the boundary conditions of equation (2.13).…”

“…Therefore, the shifts of resonant frequencies owing to small damping can be ignored in many dynamic models of microstructure vibration in air or vacuum. However, when a microstructure vibrates in liquid [24] or in contact with a viscous material [43], damping plays an important role and cannot be ignored. The eigenfrequency computation of a damped system using the above Galerkin method is presented by Zhang & Murphy [43].…”