Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field

V. A. Krysko, O. A. Saltykova, A. V. Krysko, I. V. Papkova

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

In this paper is proposed the methodology for determining the true chaos from the position of nonlinear dynamics for distributed mechanical systems in the form of a beam structure of two beams described by the kinematic firstapproximation hypothesis (Euler-Bernoulli). There is a small gap between the beams. The lower beam (beam 2) will be considered as an elastic base for the upper beam (beam 1). The transverse distributed alternating load acts on beam 1. The contact interaction of the beams was taken into account by the Cantor model. This problem has a great nonlinearity due to the account of the geometric nonlinearity of beams according to T. von Karman and the constructive nonlinearity of the structure due to the contact interaction. Differential equations in partial derivatives reduce to the ODE system by the finite differences method (FDM) of the second order accuracy. The resulting system is solved by Runge-Kutta methods of various accuracy orders. In general case, the problem solution essentially depends on the methods of reducing partial differential equations to ODEs and methods for solving the Cauchy problem, boundary and initial conditions. When solving the problem by the method of finite differences with approximation O(h2), the solution will depend on the number of points of the integration interval partition and the time step in solving the Cauchy problem. The analysis of the obtained results is carried out by the nonlinear dynamics methods and the qualitative theory of differential equations. The results are compared for geometrically linear and nonlinear beams, taking into account the contact interaction. The mechanical structure is considered as a system with an infinite number of freedom degrees. A complete coincidence of solutions is achieved, depending on the partitions number in the spatial coordinate in the chaos. The sign of the first Lyapunov exponent is determined by the Kantz, Wolf and Rosenstein methods.

Original languageEnglish
Title of host publication11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1-6
Number of pages6
Volume2017-November
ISBN (Electronic)9781538618196
DOIs
Publication statusPublished - 22 Dec 2017
Externally publishedYes
Event11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Omsk, Russian Federation
Duration: 14 Nov 201716 Nov 2017

Conference

Conference11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017
CountryRussian Federation
CityOmsk
Period14.11.1716.11.17

Fingerprint

MEMS
Temperature distribution
Chaos theory
Differential equations
Runge Kutta methods
Finite difference method
Partial differential equations
Kinematics
Derivatives

Keywords

  • contact interaction
  • finite différencies method
  • first approximation beam model
  • geometric nonlinearity
  • nonlinear dynamics

ASJC Scopus subject areas

  • Aerospace Engineering
  • Control and Systems Engineering
  • Automotive Engineering
  • Computational Mechanics

Cite this

Krysko, V. A., Saltykova, O. A., Krysko, A. V., & Papkova, I. V. (2017). Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field. In 11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings (Vol. 2017-November, pp. 1-6). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/Dynamics.2017.8239471

Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field. / Krysko, V. A.; Saltykova, O. A.; Krysko, A. V.; Papkova, I. V.

11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings. Vol. 2017-November Institute of Electrical and Electronics Engineers Inc., 2017. p. 1-6.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Krysko, VA, Saltykova, OA, Krysko, AV & Papkova, IV 2017, Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field. in 11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings. vol. 2017-November, Institute of Electrical and Electronics Engineers Inc., pp. 1-6, 11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017, Omsk, Russian Federation, 14.11.17. https://doi.org/10.1109/Dynamics.2017.8239471
Krysko VA, Saltykova OA, Krysko AV, Papkova IV. Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field. In 11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings. Vol. 2017-November. Institute of Electrical and Electronics Engineers Inc. 2017. p. 1-6 https://doi.org/10.1109/Dynamics.2017.8239471
Krysko, V. A. ; Saltykova, O. A. ; Krysko, A. V. ; Papkova, I. V. / Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field. 11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings. Vol. 2017-November Institute of Electrical and Electronics Engineers Inc., 2017. pp. 1-6
@inproceedings{d4411172cab846c2a0ef610766cc497e,
title = "Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field",
abstract = "In this paper is proposed the methodology for determining the true chaos from the position of nonlinear dynamics for distributed mechanical systems in the form of a beam structure of two beams described by the kinematic firstapproximation hypothesis (Euler-Bernoulli). There is a small gap between the beams. The lower beam (beam 2) will be considered as an elastic base for the upper beam (beam 1). The transverse distributed alternating load acts on beam 1. The contact interaction of the beams was taken into account by the Cantor model. This problem has a great nonlinearity due to the account of the geometric nonlinearity of beams according to T. von Karman and the constructive nonlinearity of the structure due to the contact interaction. Differential equations in partial derivatives reduce to the ODE system by the finite differences method (FDM) of the second order accuracy. The resulting system is solved by Runge-Kutta methods of various accuracy orders. In general case, the problem solution essentially depends on the methods of reducing partial differential equations to ODEs and methods for solving the Cauchy problem, boundary and initial conditions. When solving the problem by the method of finite differences with approximation O(h2), the solution will depend on the number of points of the integration interval partition and the time step in solving the Cauchy problem. The analysis of the obtained results is carried out by the nonlinear dynamics methods and the qualitative theory of differential equations. The results are compared for geometrically linear and nonlinear beams, taking into account the contact interaction. The mechanical structure is considered as a system with an infinite number of freedom degrees. A complete coincidence of solutions is achieved, depending on the partitions number in the spatial coordinate in the chaos. The sign of the first Lyapunov exponent is determined by the Kantz, Wolf and Rosenstein methods.",
keywords = "contact interaction, finite diff{\'e}rencies method, first approximation beam model, geometric nonlinearity, nonlinear dynamics",
author = "Krysko, {V. A.} and Saltykova, {O. A.} and Krysko, {A. V.} and Papkova, {I. V.}",
year = "2017",
month = "12",
day = "22",
doi = "10.1109/Dynamics.2017.8239471",
language = "English",
volume = "2017-November",
pages = "1--6",
booktitle = "11th International IEEE Scientific and Technical Conference {"}Dynamics of Systems, Mechanisms and Machines{"}, Dynamics 2017 - Proceedings",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
address = "United States",

}

TY - GEN

T1 - Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field

AU - Krysko, V. A.

AU - Saltykova, O. A.

AU - Krysko, A. V.

AU - Papkova, I. V.

PY - 2017/12/22

Y1 - 2017/12/22

N2 - In this paper is proposed the methodology for determining the true chaos from the position of nonlinear dynamics for distributed mechanical systems in the form of a beam structure of two beams described by the kinematic firstapproximation hypothesis (Euler-Bernoulli). There is a small gap between the beams. The lower beam (beam 2) will be considered as an elastic base for the upper beam (beam 1). The transverse distributed alternating load acts on beam 1. The contact interaction of the beams was taken into account by the Cantor model. This problem has a great nonlinearity due to the account of the geometric nonlinearity of beams according to T. von Karman and the constructive nonlinearity of the structure due to the contact interaction. Differential equations in partial derivatives reduce to the ODE system by the finite differences method (FDM) of the second order accuracy. The resulting system is solved by Runge-Kutta methods of various accuracy orders. In general case, the problem solution essentially depends on the methods of reducing partial differential equations to ODEs and methods for solving the Cauchy problem, boundary and initial conditions. When solving the problem by the method of finite differences with approximation O(h2), the solution will depend on the number of points of the integration interval partition and the time step in solving the Cauchy problem. The analysis of the obtained results is carried out by the nonlinear dynamics methods and the qualitative theory of differential equations. The results are compared for geometrically linear and nonlinear beams, taking into account the contact interaction. The mechanical structure is considered as a system with an infinite number of freedom degrees. A complete coincidence of solutions is achieved, depending on the partitions number in the spatial coordinate in the chaos. The sign of the first Lyapunov exponent is determined by the Kantz, Wolf and Rosenstein methods.

AB - In this paper is proposed the methodology for determining the true chaos from the position of nonlinear dynamics for distributed mechanical systems in the form of a beam structure of two beams described by the kinematic firstapproximation hypothesis (Euler-Bernoulli). There is a small gap between the beams. The lower beam (beam 2) will be considered as an elastic base for the upper beam (beam 1). The transverse distributed alternating load acts on beam 1. The contact interaction of the beams was taken into account by the Cantor model. This problem has a great nonlinearity due to the account of the geometric nonlinearity of beams according to T. von Karman and the constructive nonlinearity of the structure due to the contact interaction. Differential equations in partial derivatives reduce to the ODE system by the finite differences method (FDM) of the second order accuracy. The resulting system is solved by Runge-Kutta methods of various accuracy orders. In general case, the problem solution essentially depends on the methods of reducing partial differential equations to ODEs and methods for solving the Cauchy problem, boundary and initial conditions. When solving the problem by the method of finite differences with approximation O(h2), the solution will depend on the number of points of the integration interval partition and the time step in solving the Cauchy problem. The analysis of the obtained results is carried out by the nonlinear dynamics methods and the qualitative theory of differential equations. The results are compared for geometrically linear and nonlinear beams, taking into account the contact interaction. The mechanical structure is considered as a system with an infinite number of freedom degrees. A complete coincidence of solutions is achieved, depending on the partitions number in the spatial coordinate in the chaos. The sign of the first Lyapunov exponent is determined by the Kantz, Wolf and Rosenstein methods.

KW - contact interaction

KW - finite différencies method

KW - first approximation beam model

KW - geometric nonlinearity

KW - nonlinear dynamics

UR - http://www.scopus.com/inward/record.url?scp=85045977234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045977234&partnerID=8YFLogxK

U2 - 10.1109/Dynamics.2017.8239471

DO - 10.1109/Dynamics.2017.8239471

M3 - Conference contribution

VL - 2017-November

SP - 1

EP - 6

BT - 11th International IEEE Scientific and Technical Conference "Dynamics of Systems, Mechanisms and Machines", Dynamics 2017 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

ER -