Chaotic Oscillations of a Kinematically Excited Flat Shell During Geometrically Non-linear Deformation

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DOI https://doi.org/10.15407/pmach2019.03.026
Journal Journal of Mechanical Engineering
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 3, 2019 (September)
Pages 26-35
Cited by J. of Mech. Eng., 2019, vol. 22, no. 3, pp. 26-35

 

Authors

Konstantin V. Avramov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail: kvavramov@gmail.com ORCID: 0000-0002-8740-693X

Kseniya F. Cheshko, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail: cheshko.ks@gmail.com, ORCID: 0000-0001-8662-4209

Oleg F. Polishchuk, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail: PolischukOleg@nas.gov.ua, ORCID: 0000-0003-1266-9847

 

Abstract

We study the forced oscillations of a cantilevered flat shell of constant curvature. These movements are excited by a kinematic periodic embedding motion. To describe geometrically non-linear deformation, the non-linear theory of Donel shells is used. To build a non-linear dynamic system with a finite number of degrees of freedom, the method of specified forms is used. Since the eigen frequencies of longitudinal and torsional oscillations are much higher than bending ones, the inertial forces in the longitudinal and torsional directions are not taken into account. Therefore, the generalized coordinates of longitudinal and torsional oscillations are expressed in terms of bending ones. As a result, a non-linear dynamic system with respect to bending generalized coordinates is obtained. To calculate the eigen forms of linear oscillations, by using which the non-linear dynamic problem decomposes, the Rayleigh-Ritz method is used. Then only kinematic boundary conditions are satisfied. When the solution converges, the force boundary conditions are automatically satisfied. To study the convergence of eigen frequencies, calculations were performed with a different number of basis functions, which are B-splines. A comparison is made with the experimental data on the analysis of eigen frequencies, with the data published in authors’ previous article. To numerically analyze the non-linear periodic oscillations, a two-point boundary value problem is solved for ordinary differential equations by the shooting method. The stability of periodic motions and their bifurcations are estimated using multipliers. To study the bifurcations of periodic oscillations, the parameter continuation method is applied. In the region of the main resonance, saddle-node bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations are found. To study the steady-state almost periodic and chaotic oscillations, Poincaré sections, spectra of Lyapunov characteristic exponents, and spectral densities are calculated, with the stroboscopic phase portrait used as Poincaré sections. The properties of steady-state oscillations are investigated with a quasistatic change in the frequency of the disturbing action.

 

Keywords: non-linear periodic oscillations of a flat shell, stability of oscillations, almost periodic oscillations, chaotic oscillations.

 

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References

  1. Avramov, K. V. & Mikhlin, Yu. V. (2015). Nelineynaya dinamika uprugikh sistem: v 2-kh t. T. 2: Prilozheniya [Non-linear dynamics of elastic systems: in 2 volumes. Vol. 2: Applications]. Moscow: Institute for Computer Research, 700 p. (in Russian).
  2. Amabili, M. & Paidoussis, M. P. (2003). Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid structure interaction. Applied Mechanics Reviews, vol. 56, iss. 4, pp. 349–381. https://doi.org/10.1115/1.1565084
  3. Amabili, M. (2008). Nonlinear vibrations and stability of shells and plates. Cambridge: Cambridge University Press, 374 p. https://doi.org/10.1017/CBO9780511619694
  4. Parker, T. S. & Chua, L. O. (1989). Practical numerical algorithms for chaotic systems. New York: Springer, 348 p. https://doi.org/10.1007/978-1-4612-3486-9
  5. Meirovitch, L. (1986). Elements of vibration analysis. New York: McGraw-Hill Publishing Company, 495 p.
  6. Awrejcewicz, J., Kurpa, L., & Osetrov, A. (2001). Investigation of the stress-strain state of the laminated shallow shells by R-functions method combined with spline-approximation. Journal of Applied Mathematics and Mechanics, vol. 91, iss. 6, pp. 458–467. https://doi.org/10.1002/zamm.201000164
  7. Hollig, K., Reif, U., & Wipper, J. (2001). Weighted extended B-spline approximation of Dirichlet problems. SIAM Journal on Numerical Analysis, vol. 39, iss. 2, pp. 442–462. https://doi.org/10.1137/S0036142900373208
  8. Cheshko, K. F., Polishchuk, O. F., & Avramov K. V. (2017). Eksperimentalnyy i chislennyy analiz svobodnykh kolebaniy pologoy obolochki [Experimental and numerical analysis of free shallow shell oscillations]. Vestn. NTU «KhPI». Ser. Dinamika i prochnost mashin – Bulletin of NTU “KhPI”. Series: Dynamics and Strength of Machines, iss. 40 (1262), pp. 81–85 (in Russian). https://doi.org/10.20998/2078-9130.2017.40.119720
  9. Avramov, K. V. & Mikhlin, Yu. V. (2015). Nelineynaya dinamika uprugikh sistem: v 2-kh t. T. 1: Modeli, metody, yavleniya [[Non-linear dynamics of elastic systems: in 2 volumes. Vol. 1: Models, methods, phenomena]. Moscow: Institute for Computer Research, 716 p. (in Russian).

 

Received 14 March 2019