ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

# Averaging method

## Non-contact atomic force microscope: Modeling and simulation using van der Pol averaging method

Topic and aim. One of the tools which are extremely useful and valuable for creating a topography of surfaces, measuring forces, and manipulating material with nano-meter-scale features is the Atomic force microscope (AFM). Since it can create the image of the surface object in different mediums at the nano-scale, AFM can be used in a wide variety of applications and industries. This work aimed at creating the mathematical model of the non-contact atomic force microscope. Models and Methods.

## Non-Contact Atomic Force Microscope: Modeling and Simulation using Van der Pol Averaging Method

One of the tools which are extremely useful and valuable for creating a topography of surfaces, measuring forces, and manipulating material with nano-meter-scale features is the Atomic force microscope (AFM). Since it can create the image of the surface object in different mediums at the nano-scale, AFM can be used in a wide variety of applications and industries. The lumped parameter model of the atomic force microscope in the non-contact operation mode is utilized to make the mathematical model of the micro cantilever of the AFM in this article.

## The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others

The main moments of the historical development of one of the basic methods of nonlinear systems investigating (the averaging method) are traced. This method is understood as a transition from the so-called exact equation dx/dt = εX(t, x) (ε is small parameter), to the averaging equation dξ/dt = εX0(ξ) + ε2P2(ξ) + ... + εmPm(ξ) by corresponding variable substitution.Bogolyubov–Krylov’s approach to the problem of justifying the averaging method, based on the invariant measure theorem, is analyzed.

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