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Nonlinear numerical analysis and averaging method applied atomic force microscopy with viscoelastic term

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Abstract

We investigated the nonlinear behavior of an atomic force microscopy system with the contribution of the medium viscoelasticity term. Atomic force microscopy is one of the most used techniques for analyzing the surface topology of samples of interest. We propose a mathematical model for the nonlinear dynamics of the probe and we also assign a term that describes the viscoelasticity of the medium in which the probe is inserted. For our numerical analysis of nonlinear dynamics, we used the Lyapunov Exponent, Bifurcation Diagram, phase maps and Poincaré maps. We also determine a solution for a set of parameters using perturbation theory. In this paper, it is also shown that using higher approximations of averaging method, which allows to solve systems of differential equations with slowly changing variables instead systems with fast change of variables, is very effective to study dynamics of an atomic force microscopy.

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Abbreviations

r :

Radius of the tip

c :

Structural damping coefficient of the microcantilever

k :

Linear stiffness

\({k}_{\mathrm{c}}\) :

Nonlinear stiffness

\({c}_{\mathrm{s}}\) :

Structural damping coefficient of the media

x :

Displacement of the tip of the microcantilever

m :

Mass of the microcantilever

\({z}_{0}\) :

Equilibrium position

t :

Time

a 1 :

Hamaker constant (repulsive)

a 2 :

Hamaker constant (attractive)

f :

External force amplitude

\(\omega \) :

Angular frequency

\({x}^{1}\) :

Cantilever displacement dimensionless

\({x}^{2}\) :

Cantilever velocity dimensionless

\(h\) :

Integration step for Runge–Kutta method

\({x}_{0}^{1}\) :

Initial condition for displacement dimensionless

\({x}_{0}^{2}\) :

Initial condition for velocity dimensionless

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Funding

This work was financially supported by the Brazilian agencies CAPES and the Brazilian Council for Scientific and Technological Development, CNPq,

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Authors and Affiliations

  1. Department of Electrical, Universidade Tecnológica Federal do Paraná, Ponta Grossa, PR, Brazil

    Mauricio A. Ribeiro, Angelo M. Tusset & Jose M. Balthazar

  2. Department of Production Engineering, Universidade Tecnológica Federal do Paraná, Ponta Grossa, PR, Brazil

    Angelo M. Tusset & Jose M. Balthazar

  3. Faculty of Mechanical Engineering of Bauru, São Paulo State University, Bauru, SP, Brazil

    Jose M. Balthazar

  4. Voronezh, Federal Research Center, Computer Science and Central of Russian Academy of Science, Moscow, Russia

    Galina A. Kurina

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  1. Mauricio A. Ribeiro
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Correspondence to Mauricio A. Ribeiro.

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Ribeiro, M.A., Kurina, G.A., Tusset, A.M. et al. Nonlinear numerical analysis and averaging method applied atomic force microscopy with viscoelastic term. Arch Appl Mech 92, 3817–3827 (2022). https://doi.org/10.1007/s00419-022-02264-5

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