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Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance

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Mathematical Modelling and Numerical Analysis of Size-Dependent Structural Members in Temperature Fields

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 142))

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Abstract

The methodology for detecting true chaos (in terms of nonlinear dynamics) developed on the example of a structure composed of two beams with a small clearance is outlined. Euler-Bernoulli hypothesis is employed, and the contact interaction between beams follows Kantor model. The complex nonlinearity results from von Kármán geometric nonlinearity as well as the nonlinearity implied by the contact interaction. The governing PDEs are reduced to ODEs by the second-order finite difference method (FDM). The obtained system of equations is solved by Runge-Kutta method of different accuracies. To purify the signal from errors introduced by numerical methods, the principal component analysis is employed and the sign of the first Lyapunov exponent is estimated by Kantz, Wolf and Rosenstein methods [1].

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Authors and Affiliations

  1. Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Łódź, Poland

    Jan Awrejcewicz

  2. Applied Mathematics and Systems Analysis, Saratov State Technical University, Saratov, Russia

    Anton V. Krysko

  3. Department of Mathematics and Modeling, Saratov State Technical University, Saratov, Russia

    Maxim V. Zhigalov & Vadim A. Krysko

Authors
  1. Jan Awrejcewicz
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  2. Anton V. Krysko
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  3. Maxim V. Zhigalov
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  4. Vadim A. Krysko
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Correspondence to Jan Awrejcewicz .

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Awrejcewicz, J., Krysko, A.V., Zhigalov, M.V., Krysko, V.A. (2021). Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance. In: Mathematical Modelling and Numerical Analysis of Size-Dependent Structural Members in Temperature Fields. Advanced Structured Materials, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-030-55993-9_4

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  • DOI: https://doi.org/10.1007/978-3-030-55993-9_4

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  • Print ISBN: 978-3-030-55992-2

  • Online ISBN: 978-3-030-55993-9

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