Asymptotic analysis of a noncontact AFM microcantilever sensor with external feedback control

Abstract

An external feedback control is inserted in a nonlinear continuum formulation of a noncontact AFM model. The aim of the feedback is to keep the system response to an operationally suitable one, thus allowing reliable measurement of the sample surface by avoiding possible unstable microcantilever sensor motions.\ud The study of the weakly nonlinear system dynamics about the desired fixed point close to primary resonance is carried out via multiple-scale asymptotics, whose outcomes are validated via numerical simulations of the original system equations of motion. The latter include controllable periodic dynamics and additional\ud periodic and distinct quasiperiodic solutions that appear beyond the asymptotic stability thresholds. The results highlight the effectiveness of the applied feedback control technique and also enable the derivation of a comprehensive system bifurcation structure highlighting the stability thresholds for robust controllable\ud AFM dynamics

Influence of a locally-tailored external feedback control on the overall dynamics of a non-contact AFM model

“…The presented outcomes confirm that from a practical viewpoint the acceptable values of the feedback control parameter k g are very low; in particular, far from the resonance frequencies they coincide with the limiting value k g ¼0.0023 which corresponds to the loss of stability of the equilibrium of the relevant Hamiltonian system (i.e., U¼ 0), analyzed in [26]. In the resonance regions, which are shown to be the areas most sensitive to the parameters variation, the operational settings have to be taken under strict control, and must be further reduced in order to avoid errors during the scan operation.…”

“…A Galerkin approximation (the basis function being that of a clamped-spring beam) with a single (first) mode assumption is used for both the controlled vertical displacement of the cantilever base and the reference vertical displacement of the tip, since for the uncontrolled system it has been shown to be sufficient to detect the main non-linear aspects of the response [29]. The IBVP is thus reduced to a system of two ordinary differential equations with one and a half degree-of-freedom, which has the following nondimensional form [26]:…”

“…Note that the control works when the response settles onto the latter, i.e., when z is equal to the expected value z s . The passage from the continuous system with control to the reduced order model is presented in [26], with the underlying formulation being extensively reported in [30], which are referred to for all analytical details. The control is seen to succeed in keeping the system response to the reference one.…”

“…A fortiori, therefore, it is of great interest to study the possible positive or negative effects that control techniques can have on the dynamical response of the system. In this work, reference is made to the model of non-contact AFM with external feedback control proposed by the authors in a companion paper [26]. Therein, a feedback control has been inserted in a non-linear continuum formulation of a non-contact AFM model, with the aim to keep the system response to an operationally suitable one, thus allowing reliable measurement of the sample surface by avoiding possible unstable microcantilever motions.…”

Young’s modulus of multi-layer microcantilevers

“…[10][11][12][13][14][15][16][17] As the critical issues of mechanical performance of microcantilevers, Young's modulus [18][19][20][21][22][23] and resonant frequency [24][25][26][27][28][29][30][31] have been studied by various characterizations. In this research, atomic force microscopy (AFM) measurement is conducted to characterize resonant frequency of multi-layer microcantilevers, and a theoretical model is proposed and discussed including the impact of coating on both Young's modulus and resonant frequency.…”