Skip to main content
Log in

Simulation of Nonlinear Oscillations in a Micro-Generator of Clock Frequency

  • Published:
Numerical Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider a mathematical model of a new-type micro- generator. The model is based on excitation by electrostatic forces of oscillations of a mobile electrode in a micro gap. The principle of operation of the micro-generator is similar to the well-known theory of clock with a striking escapement mechanism. The difference is that in the equation of motion, the form of the right-hand side takes into account the electrostatic nature of the pulsed action. The numerical analysis shows that with time the bounded oscillations in the phase plane tend to a stable limit cycle, and thus the emerging oscillations are stable towards external perturbations. Studying periodic oscillations in dependence on model parameters relies on the boundary value problem solution for an equation with a discontinuous right-hand side transformed so that to enable application of the method of solution continuation with respect to a parameter. In this way, the domain in the plane of model parameters in which stable limit cycles exist is defined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kostsov, E.G., Status and Prospects of Micro- and Nanoelectromechanics, Avtometriya, 2009, vol. 45, no. 3, pp. 3–52.

    Google Scholar 

  2. Younis, M.I., MEMS Linear and Nonlinear Statics and Dynamics, Springer, 2011.

  3. Greenberg, Ya. S., Pashkin, Yu. A., and Il’ichev, E.V., Nanomechanical Resonators, Usp. Fiz. Nauk, 2012, vol. 182, no. 4, pp. 407–436.

    Article  Google Scholar 

  4. Kostsov, E.G. and Fadeev, S.I., New Electromechanical Resonators for Gigahertz Frequencies, Avtometriya, 2013, vol. 10, no. 2, pp. 115–122.

    Google Scholar 

  5. Fadeev, S.I. and Pimanov, D.O., Study of Periodic Solutions in Mathematical Models of Micromechanics with Periodic Impulse Action, Vestnik NGU, Mat., Mekh., Inf., 2013, vol. 13, no. 3, pp. 122–40.

    MATH  Google Scholar 

  6. Andronov, A.A., Witt, A.A., and Khaikin, S.E., Teoriya kolebanii (Oscillation Theory), Moscow: Fiz.-Mat. Lit., 1959.

    Google Scholar 

  7. Kogai, V.V. and Fadeev, S.I., Applying Parameter Continuation Based on Multiple Shooting Method for Numerical Treatment of Nonlinear Boundary Value Problems, Sib. Zh. Ind. Mat., 2001, vol. 4, no. 1, pp. 83–101.

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful to E.G. Kostsov for suggesting consideration of a micro-generator model similar to the model of simplest clock with the striking escapement mechanism.

Funding

This work was supported by the SB RAS Comprehensive Fundamental Research Program “Interdisciplinary integration studies” for 2018–2020 (project no. 273).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to S. I. Fadeev or V. V. Kogai.

Additional information

Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 4, pp. 489–501.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fadeev, S.I., Kogai, V.V. Simulation of Nonlinear Oscillations in a Micro-Generator of Clock Frequency. Numer. Analys. Appl. 12, 407–417 (2019). https://doi.org/10.1134/S1995423919040086

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423919040086

Keywords

Navigation