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Numerical Analysis of Dynamic Stability of an Isotropic Plate by Applying Tools Used in Dynamics

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Dynamical Systems in Theoretical Perspective (DSTA 2017)

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Abstract

The aim of the study is to analyze an isotropic plate in terms of its dynamic stability (or its instability), by applying tools that are mainly used in the vibrations theory of dynamical systems e.g. in the theory of bifurcation and chaos. The results achieved by using tools such as phase portraits, Poincaré maps, FFT analysis, the largest Lyapunov exponents were compared with the results obtained by using the Volmir criterion.

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References

  1. Alijani, F., Amabili, M., Karagiozis, K., Bakhtiari-Nejad, F.: Nonlinear vibrations of functionally graded doubly curved shallow shells. J. Sound Vibr. 330(7), 1432–1454 (2011)

    Article  Google Scholar 

  2. Alijani, F., Bakhtiari-Nejad, F., Amabili, M.: Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dyn. 66(3), 251–270 (2011)

    Article  MathSciNet  Google Scholar 

  3. Ari-Gur, J., Simonetta, S.R.: Dynamic pulse buckling of rectangular composite plates. Compos. Part B: Eng. 28(3), 301–308 (1997)

    Article  Google Scholar 

  4. Awrejcewicz, J., Krysko, V.A.: Feigenbaum scenario exhibited by thin plate dynamics. Nonlinear Dyn. 24(4), 373–398 (2001)

    Article  Google Scholar 

  5. Awrejcewicz, J., Krysko, A.V.: Wavelet-based analysis of parametric vibrations of flexible plates. Int. Appl. Mech. 39(9), 997–1028 (2003)

    Article  MathSciNet  Google Scholar 

  6. Awrejcewicz, J., Krysko, V.A., Narkaitis, G.G.: Bifurcations of a thin plate-strip excited transversally and axially. Nonlinear Dyn. 32(2), 187–209 (2003)

    Article  Google Scholar 

  7. Awrejcewicz, J., Krysko, A.V.: Analysis of complex parametric vibrations of plates and shells using Bubnov-Galerkin approach. Arch. Appl. Mech. 73(7), 495–504 (2003)

    Article  Google Scholar 

  8. Awrejcewicz, J., Krysko, V.A., Krysko, A.V.: Complex parametric vibrations of flexible rectangular plates. Meccanica 39(3), 221–244 (2004)

    Article  MathSciNet  Google Scholar 

  9. Awrejcewicz, J., Krysko, V. A., Moldenkova, T.: Mathematical model of dissipative parametric vibrations of flexible plates with nonhomogeneous boundary conditions. Math. Probl. Eng. 16 pp. https://doi.org/10.1155/MPE/2006/85623(2006)

  10. Awrejcewicz, J., Krysko, A.V., Bochkarev, V.V., Babenkova, T.V., Papkova, I.V., Mrozowski, J.: Chaotic vibrations of two-layered beams and plates with geometric, physical and design nonlinearities. Int. J. Bifurcat. Chaos 21(10), 2837–2851 (2011)

    Article  Google Scholar 

  11. Awrejcewicz, J., Papkova, I.V., Krylova, E.U., Krysko, V.A.: Wavelet-based analysis of the regular and chaotic dynamics of rectangular flexible plates subjected to shear-harmonic loading. Shock Vibr. 19(5), 979–994 (2012)

    Article  Google Scholar 

  12. Bazant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic. Fracture and Damage Theories. World Scientific, London (2010)

    Book  Google Scholar 

  13. Bolotin, V.: Dynamic Stability of Elastic Systems, vol. 1 (1962)

    Google Scholar 

  14. Borkowski, L., Perlikowski, P., Kapitaniak, T., Stefanski, A.: Experimental observation of three-frequency quasiperiodic solution in a ring of unidirectionally coupled oscillators. Phys. Rev. E 91(6), 062906 (2015)

    Article  Google Scholar 

  15. Budiansky, B., Roth, R.S.: Axisymmetric dynamic buckling of clamped shallow spherical shells. 597–606 (1962)

    Google Scholar 

  16. Collatz, L.: The Numerical Treatment of Differential Equations, vol. 60. Springer, Berlin (2012)

    Google Scholar 

  17. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19(90), 297–301 (1965)

    Article  MathSciNet  Google Scholar 

  18. Fortuna, Z., Macukow, B., Wasowski, J.: Metody Numeryczne. WNT (2005). ISBN 83-204-3075-5

    Google Scholar 

  19. Gilat, R., Aboudi, J.: Parametric stability of non-linearly elastic composite plates by Lyapunov exponents. J. Sound Vibr. 235(4), 627–637 (2000)

    Article  Google Scholar 

  20. Hsu, Y.C., Forman, R.G.: Elastic-plastic analysis of an infinite sheet having a circular hole under pressure. J. Appl. Mech. 42(2), 347–352 (1975)

    Article  Google Scholar 

  21. Hutchinson, J.W., Budiansky, B.: Dynamic buckling estimates. AIAA J. 4(3), 525–530 (1966)

    Article  Google Scholar 

  22. Kapitaniak, T., Wojewoda, J.: Bifurkacje i chaos. Wydawnictwo Naukowe PWN, Wydawnictwo Politechniki Lodzkiej (2000)

    Google Scholar 

  23. Kleiber, M., Kotula, W., Saran, M.: Numerical analysis of dynamic quasi-bifurcation. Eng. Comput. 4(1), 48–52 (1987)

    Article  Google Scholar 

  24. Kolakowski, Z.: Some aspects of dynamic interactive buckling of composite columns. Thin-Walled Struct. 45(10), 866–871 (2007)

    Article  Google Scholar 

  25. Kolakowski, Z., Kubiak, T.: Interactive dynamic buckling of orthotropic thin-walled channels subjected to in-plane pulse loading. Compos. Struct. 81(2), 222–232 (2007)

    Article  Google Scholar 

  26. Kowal-Michalska, K.: About some important parameters in dynamic buckling analysis of plated structures subjected to pulse loading. Mech. Mech. Eng. 14(2), 269–279 (2010)

    Google Scholar 

  27. Kubiak, T.: Criteria of dynamic buckling estimation of thin-walled structures. Thin-Walled Struct. 45(10), 888–892 (2007)

    Article  Google Scholar 

  28. Kubiak, T., Kolakowski, Z., Kowal-Michalska, K., Mania, R., Swiniarski, J.: Dynamic response of conical and spherical shell structures subjected to blast pressure. In: Proceedings of SSDS’Rio (2010)

    Google Scholar 

  29. Mania, R., Kowal-Michalska, K.: Behaviour of composite columns of closed cross-section under in-plane compressive pulse loading. Thin-Walled Struct. 45(10), 902–905 (2007)

    Article  Google Scholar 

  30. Michlin, S.G., Smolnicki, C.L.: Approximate Methods for the Solution of Integral and Differential Equations. PWN (1970)

    Google Scholar 

  31. Moorthy, J., Reddy, J.N., Plaut, R.H.: Parametric instability of laminated composite plates with transverse shear deformation. Inte. J. Solids Struct. 26(7), 801–811 (1990)

    Article  Google Scholar 

  32. Petry, D., Fahlbusch, G.: Dynamic buckling of thin isotropic plates subjected to in-plane impact. Thin-Walled Struct. 38(3), 267–283 (2000)

    Article  Google Scholar 

  33. Raftoyiannis, I.G., Kounadis, A.N.: Dynamic buckling of 2-DOF systems with mode interaction under step loading. Int. J. Non-linear Mech. 35(3), 531–542 (2000)

    Article  Google Scholar 

  34. Shariyat, M.: Thermal buckling analysis of rectangular composite plates with temperature-dependent properties based on a layerwise theory. Thin-Walled Struct. 45(4), 439–452 (2007)

    Article  Google Scholar 

  35. Touati, D., Cederbaum, G.: Influence of large deflections on the dynamic stability of nonlinear viscoelastic plates. Acta Mech. 113(1–4), 215–231 (1995)

    Article  Google Scholar 

  36. Volmir, A.S.: Nonlinear Dynamics Plates and Shells. Science, Moscow (1972)

    Google Scholar 

  37. Wang, Y.G., Song, H.F., Li, D., Wang, J.: Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution. Arch. Appl. Mech. 80(7), 815–828 (2010)

    Article  Google Scholar 

  38. Wu, G.Y., Shih, Y.S.: Analysis of dynamic instability for arbitrarily laminated skew plates. J. Sound Vib. 292(1), 315–340 (2006)

    Article  Google Scholar 

  39. Yeh, Y.L., Lai, H.Y.: Chaotic and bifurcation dynamics for a simply supported rectangular plate of thermo-mechanical coupling in large deflection. Chaos, Solitons Fractals 13(7), 1493–1506 (2002)

    Article  MathSciNet  Google Scholar 

  40. Yuda, H., Zhiqiang, Z.: Bifurcation and chaos of thin circular functionally graded plate in thermal environment. Chaos, Solitons Fractals 44(9), 739–750 (2011)

    Article  Google Scholar 

  41. Zhang, T., Liu, T.G., Zhao, Y., Luo, J.Z.: Nonlinear dynamic buckling of stiffened plates under in-plane impact load. J. Zhejiang Univ. Sci. 5(5), 609–617 (2004)

    Article  Google Scholar 

  42. Zizicas, G.A.: Dynamic buckling of thin plates. Trans. ASME 74(7), 1257–1268 (1952)

    Google Scholar 

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Acknowledgements

This work has been supported by the internal grant awarded by the Lodz University of Technology with the Young Scientists Fund.

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Authors and Affiliations

  1. Department of Strength of Materials, Lodz University of Technology, Stefanowskiego 1/15, 90-924, Lodz, Poland

    Lukasz Borkowski

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  1. Lukasz Borkowski
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Correspondence to Lukasz Borkowski .

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Editors and Affiliations

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

    Jan Awrejcewicz

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Borkowski, L. (2018). Numerical Analysis of Dynamic Stability of an Isotropic Plate by Applying Tools Used in Dynamics. In: Awrejcewicz, J. (eds) Dynamical Systems in Theoretical Perspective. DSTA 2017. Springer Proceedings in Mathematics & Statistics, vol 248. Springer, Cham. https://doi.org/10.1007/978-3-319-96598-7_6

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