Chaotic Vibrations of Two Euler-Bernoulli Beams With a Small Clearance

  • Jan Awrejcewicz
  • Vadim A. Krysko
Part of the Scientific Computation book series (SCIENTCOMP)


In this chapter a methodology to detect true/reliable chaos (in terms of non-linear dynamics) is developed on an example of a structure composed of two beams with a small clearance. The Euler-Bernoulli hypothesis is employed, and the contact interaction between beams follows the Kantor model. The complex non-linearity results from the von Kármán geometric non-linearity as well as the non-linearity 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 methods of different accuracy. 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 the Kantz, Wolf and Rosenstein methods and the method of neural networks. In the latter case a spectrum of the Lyapunov exponents is estimated. It has been illustrated how the number of nodes in the employed FDM influences the numerical results regarding chaotic vibrations. We have also shown how increase of the beams distance implies stronger action of the geometric nonlinearity, and hence influence of convergence of the used numerical algorithm for FDM has been demonstrated. Effect of essential dependence of the initial conditions choice on the numerical results of the studied contact problem is presented and discussed.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Jan Awrejcewicz
    • 1
  • Vadim A. Krysko
    • 2
  1. 1.Łódź University of TechnologyŁódźPoland
  2. 2.Department of Mathematics and ModelingSaratov State Technical UniversitySaratovRussia

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