Abstract
In this paper, we investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) forces. The dynamic behavior of the system is analyzed through a nonlinear dimensionless mathematical model. Numerical tools as Poincaré maps, Lyapunov exponents, and bifurcation diagrams are accounted for the analysis of the system. With that, the regions in which the system presents chaotic and periodic behaviors are obtained and investigated. Moreover, the fractional calculus is introduced into the mathematical model, employing the Riemann-Liouville kernel discretization in the viscoelastic term of the system. The 0-1 test is implemented to analyze the new dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional (integer derivative) and fractional models reveal the need for the application of control techniques such as Optimum Linear Feedback Control (OLFC), State-Dependent Riccati Equations (SDRE) by using feedback control, and the Time-Delayed Feedback Control. The results of the control techniques are efficient with and without the fractional-order derivative.
Highlights
Technological advances in the development of electromechanical systems are gaining ground in the most diverse branches of engineering science
A special mechanism has been used for sample surfaces analysis at atomic scale, which is the atomic force microscope characterized as a nanoelectromechanical systems (NEMS), mostly referred to as atomic force microscopy technique. is technique is very well established as it is a very precise superficial analysis and has allowed the increase of the understanding and analysis of very small and soft materials such as polymeric materials [5], ceramics [6], biological cells [7], and surface tribological analyses [8]. e mechanisms are found to be actuated by a piezoelectric material which is used for controlling vibrations [9, 10]
Among the AFM in tapping mode (AFM-TM), contact and noncontact with the sample surface stand out, as they can form a three-dimensional image of such surface [11]
Summary
Technological advances in the development of electromechanical systems are gaining ground in the most diverse branches of engineering science. Shock and Vibration motion of the AFM-TM system to investigate the chaotic behavior of the microcantilever beam of the AFM-TM mechanism under VdW forces, where chaos is found and a control design has to be proposed. E authors discuss when the Casimir force becomes as dominant as VdW forces depending on the distance between the AFM microcantilever and the sample surface, concluding that there is a coexistence of the forces at very short distances. We approach computationally the dynamics and control of the AFM-TM model proposed by [14], accounting for the addition of the Casimir force along with the van der Waals forces for the interaction of the tip with the surface of the sample.
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