Dynamics of a micro-electro-mechanical system associated with an atomic force microscope considering squeeze film damping

Johan S. Duque; Alexander Gutierrez; Daniel Cortés

doi:10.1364/AO.383485

Applied Optics
Vol. 59,
Issue 13,
pp. D76-D80
(2020)
•https://doi.org/10.1364/AO.383485

Dynamics of a micro-electro-mechanical system associated with an atomic force microscope considering squeeze film damping

Johan S. Duque, Alexander Gutierrez, and Daniel Cortés

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Johan S. Duque, Alexander Gutierrez, and Daniel Cortés, "Dynamics of a micro-electro-mechanical system associated with an atomic force microscope considering squeeze film damping," Appl. Opt. 59, D76-D80 (2020)

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About this Article
History
- Original Manuscript: November 15, 2019
- Revised Manuscript: January 27, 2020
- Manuscript Accepted: January 27, 2020
- Published: March 5, 2020
Virtual Issues
Applied Optics Optics theory and practice in Iberoamerica (2020)

Abstract

In the following paper, we present a nonlinear model of an atomic force microscope considering the potential of Lennard–Jones and the nonlinear friction produced by the squeeze film damping effect, between the cantilever and the sample. Specifically, we study the existence and stability of periodic solutions using the lower and upper solution method in the system without friction. The condition for persistence of the homocline orbit was established by Melnikov method when the model has nonlinear friction. In this sense, the analytic and numerical approach is presented to verify the solutions of the model.

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Notation Used in the Paper
$x$	Cantilever displacement
$\dot{x}$	Velocity of the movement
$\ddot{x}$	Acceleration of the movement
$z_{0}$	Initial distance over surface
$A_{1}, A_{2}$	Hamaker constants
$R$	Tip sphere radius
$μ_{e f f}$	Viscosity coefficient
$D$	Width of cantilever
$L$	Length of cantilever
$k_{l}, k_{n l}$	Spring constants
$g (t)$	External excitation
$m (x)$	Auxiliary function
$b_{1}, b_{2}$	Dimensionless constants
$ϵ$	Perturbation parameter
$y_{l}, y_{r}$	Bifurcation curves
$B$	Amplitude of external excitation
$Ω$	Frequency of external excitation
$α (t)$	Lower solution
$β (t)$	Upper solution
$R_{s}$	Solution set

AFM Parameters, $ϵ = 0$
$b_{2}$	$b_{1}$	$a$	Equilibria solutions
0	+	+	Nonlinear center
+	0	$a > - b_{2}^{*}$	Saddle and center
		$a = - b_{2}^{*}$	Saddle and cusp
		$a < - b_{2}^{*}$	No have solutions
+	$b_{1} \geq b_{1}^{*}$	$a > x_{r}$ ó $a < x_{l}$	Nonlinear center
+	$b_{1} < b_{1}^{*}$	$x_{l} < a < x_{r}$	Two nonlinear centers and a saddle
		$a = x_{r}$ or $a = x_{l}$	A nonlinear center and a cusp
		+	Nonlinear center

Notation Used in the Paper
$x$	Cantilever displacement
$\dot{x}$	Velocity of the movement
$\ddot{x}$	Acceleration of the movement
$z_{0}$	Initial distance over surface
$A_{1}, A_{2}$	Hamaker constants
$R$	Tip sphere radius
$μ_{e f f}$	Viscosity coefficient
$D$	Width of cantilever
$L$	Length of cantilever
$k_{l}, k_{n l}$	Spring constants
$g (t)$	External excitation
$m (x)$	Auxiliary function
$b_{1}, b_{2}$	Dimensionless constants
$ϵ$	Perturbation parameter
$y_{l}, y_{r}$	Bifurcation curves
$B$	Amplitude of external excitation
$Ω$	Frequency of external excitation
$α (t)$	Lower solution
$β (t)$	Upper solution
$R_{s}$	Solution set

AFM Parameters, $ϵ = 0$
$b_{2}$	$b_{1}$	$a$	Equilibria solutions
0	+	+	Nonlinear center
+	0	$a > - b_{2}^{*}$	Saddle and center
		$a = - b_{2}^{*}$	Saddle and cusp
		$a < - b_{2}^{*}$	No have solutions
+	$b_{1} \geq b_{1}^{*}$	$a > x_{r}$ ó $a < x_{l}$	Nonlinear center
+	$b_{1} < b_{1}^{*}$	$x_{l} < a < x_{r}$	Two nonlinear centers and a saddle
		$a = x_{r}$ or $a = x_{l}$	A nonlinear center and a cusp
		+	Nonlinear center

Abstract

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Tables (3)

Equations (23)

Applied Optics