Chaotic Dynamics of Structural Members Under Regular Periodic and White Noise Excitations

  • J. AwrejcewiczEmail author
  • A. V. Krysko
  • I. V. Papkova
  • N. P. Erofeev
  • V. A. Krysko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)


In this work we study PDEs governing beam dynamics under the Timoshenko hypotheses as well as the initial and boundary conditions which are yielded by Hamilton’s variational principle. The analysed beam is subjected to both uniform transversal harmonic load and additive white Gaussian noise. The PDEs are reduced to ODEs by means of the finite difference method employing the finite differences of the second-order accuracy, and then they are solved using the 4th and 6th order Runge-Kutta methods. The numerical results are validated with the applied nodes of the beam partition. The so-called charts of the beam vibration types are constructed versus the amplitude and frequency of harmonic excitation as well as the white noise intensity.

The analysis of numerical results is carried out based on a theoretical background on non-linear dynamical systems with the help of time series, phase portraits, Poincaré maps, power spectra, Lyapunov exponents as well as using different wavelet-based studies. A few novel non-linear phenomena are detected, illustrated and discussed.

In particular, it has been detected that a transition from regular to chaotic beam vibrations without noise has been realised by the modified Ruelle-Takens-Newhouse scenario. Furthermore, it has been shown that in the studied cases, the additive white noise action has not qualitatively changed the mentioned route to chaotic dynamics.


Non-linear dynamics Timoshenko beam Chaos Bifurcations White Gauss noise 



This work has been supported by the Russian Science Foundation (RSF-16-19-10290).


  1. 1.
    Newhouse, S., Ruelle, D., Takens, F.: Occurrence of strange axiom \(A\) attractors near quasi periodic flows on \(T^m, m>3\). Commun. Math. Phys. 64(1), 35–40 (1978)CrossRefzbMATHGoogle Scholar
  2. 2.
    Awrejcewicz, J., Krysko Jr., V.A., Papkova, I.V., Krylov, E.Y., Krysko, A.V.: Spatio-temporal non-linear dynamics and chaos in plates and shells. Nonlinear Stud. 21(2), 313–327 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Horsthemke, W., Lefever, R.: Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology. Springer, Berlin (1984)zbMATHGoogle Scholar
  4. 4.
    Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. 41(245), 744–746 (1921)CrossRefGoogle Scholar
  5. 5.
    Awrejcewicz, J., Krysko, V.A., Papkova, I.V., Krysko, A.V.: Deterministic Chaos in One-Dimensional Continuous Systems. World Scientific, Singapore (2016)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • J. Awrejcewicz
    • 1
    • 2
    Email author
  • A. V. Krysko
    • 3
    • 4
  • I. V. Papkova
    • 5
  • N. P. Erofeev
    • 5
  • V. A. Krysko
    • 5
  1. 1.Department of Automation, Biomechanics and MechatronicsLodz University of TechnologyLodzPoland
  2. 2.Institute of VehiclesWarsaw University of TechnologyWarsawPoland
  3. 3.Department of Applied Mathematics and Systems AnalysisSaratov State Technical UniversitySaratovRussian Federation
  4. 4.Cybernetic InstituteNational Research Tomsk Polytechnic UniversityTomskRussian Federation
  5. 5.Department of Mathematics and ModelingSaratov State Technical UniversitySaratovRussian Federation

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