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Recent Developments in the Theory of Shells pp 459–478Cite as

Localized Parametric Vibrations of Laminated Cylindrical Shell Under Non-uniform Axial Load Periodically Varying with Time

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Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 110))

Abstract

Based on the equivalent single layer model for laminated shells, parametric vibrations of thin laminated non-circular cylindrical shells under non-uniform axial load periodically varying with time are studied. As the governing equations, the non-linear coupled differential equations written in terms of the displacement and stress functions accounting for transverse shears are used. It is assumed that the effective (reduced) shear modulus for an entire laminated package is much less than the reduced Young’s modulus. Using the asymptotic method of Tovstik in combination with the multiple scales method with respect to time, solutions of the governing equations are constructed in the form of functions which are exponentially decay far from some generatrix and growing with time in the case of parametric resonance. The system of two differential equations with periodic in time coefficients and accounting for shears is derived to determine the amplitude of parametric vibrations. The main regions of parametric instability taking into account transverse shears were found. An example of parametric vibrations of a sandwich cylinder with the magnetorheological core affected by a magnetic field is considered.

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References

  1. Awrejcewicz, J., Kurpa, L., Mazur, O.: Dynamical instability of laminated plates with external cutout. Int. J. Non-Linear Mech. 81, 103–114 (2016)

    Article  Google Scholar 

  2. Baruch, M.: Parametric instability of stiffened cylindrical shells. Isr. J. Tech. 7, 297–301 (1969)

    Google Scholar 

  3. Bert, C.W., Birman, V.: Parametric instability of thick, orthotropic, circular cylindrical shells. Acta Mech. 71, 61–76 (1988)

    Article  Google Scholar 

  4. Birman, V., Bert, C.W.: Dynamic stability of reinforced composite cylindrical shells in thermal field. J. Sound Vib. 142(2), 183–190 (1990)

    Article  Google Scholar 

  5. Bogdanov, Y., Mazanik, S., Syroid, Y.: Course of differential equations. Universitetskae, Minsk (1996). (in Russia)

    Google Scholar 

  6. Bogdanovich, A.E.: Dynamic stability of an elastic orthotropic cylindrical shell with allowance for transverse shears. Polymer Mech. 9(2), 268–274 (1973)

    Article  Google Scholar 

  7. Bogdanovich, A.E.: Dynamic stability of a viscoelastic orthotropic cylindrical shell. Polymer Mech. 9(4), 626–632 (1973)

    Article  Google Scholar 

  8. Bogdanovich, A.E., Feldmane, E.G.: Nonlinear parametric vibrations of viscoelastic orthotropic cylindrical shells. Sov. Appl. Mech. 16(4), 305–309 (1980)

    Article  Google Scholar 

  9. Bolotin, V.V.: Dynamic Stability of Elastic Systems. Gostekhizdat, Moscow. Engl. transl.: Holden-Day, San Francisco 1963, German transi.: Verlag der Wissenshaften, Berlin 1960 (1956). (in Russia)

    Google Scholar 

  10. Bondarenko, A.A., Galaka, P.I.: Parametric instability of glass-plastic cylindrical shells. Sov. Appl. Mech. 13(4), 411–414 (1977)

    Article  Google Scholar 

  11. Caughey, T.K.: Subharmonic oscillations of a pendulum. J. Appl. Mech. Trans. ASME 27(4), 754–755 (1960)

    Google Scholar 

  12. Chelomey, V.N.: Dynamic stability of aviation structures. Aeroflot, Moscow (1939). (in Russia)

    Google Scholar 

  13. Darabi, M., Darvizeh, M., Darvizeh, A.: Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading. Compos. Struct. 83(2), 201–211 (2008)

    Article  Google Scholar 

  14. Darabi, M., Ganesan, R.: Non-linear dynamic instability analysis of laminated composite cylindrical shells subjected to periodic axial loads. Compos. Struct. 147, 168–184 (2016)

    Article  Google Scholar 

  15. Donnell, L.H.: Beams. Plates and Shells, McGraw-Hill Inc, New York (1976)

    Google Scholar 

  16. Grigolyuk, E.I., Kulikov, G.M.: Multilayer Reinforced Shells: Calculation of Pneumatic Tires. Mashinostroenie, Moscow (1988). (in Russian)

    Google Scholar 

  17. Kharlamov, S.A.: An example of hetero-parametric excitation of a pendulum by quasi-periodic oscillations of the suspension. Sov. Phys. Dokl. 9(8), 655–657 (1965). (in Russia)

    Google Scholar 

  18. Kochurov, R., Avramov, K.V.: On effect of initial imperfections on parametric vibrations of cylindrical shells with geometrical non-linearity. Int. J. Solids Struct. 49(3), 537–545 (2012)

    Article  Google Scholar 

  19. Kumar, L.R., Datta, P.K., Prabhakara, D.L.: Tension buckling and parametric instability characteristics of doubly curved panels with circular cutout subjected to nonuniform tensile edge loading. Thin-Walled Struct. 42(7), 947–962 (2004)

    Article  Google Scholar 

  20. Machado, S.P., Cortnez, V.H.: Dynamic stability of thin-walled composite beams under periodic transverse excitation. J. Sound Vib. 321(1), 220–241 (2009)

    Article  Google Scholar 

  21. Machado, S.P., Filipich, C.P., Cortnez, V.H.: Parametric vibration of thin-walled composite beams with shear deformation. J. Sound Vib. 305(4), 563–581 (2007)

    Article  Google Scholar 

  22. Markov, A.N.: Dynamic stability of anisotropic cylindrical shells. J. Appl. Math. Mech. 13(2), 145–150 (1949). (in Russia)

    Google Scholar 

  23. Mikhasev, G.I.: Free and parametric vibrations of cylindrical shells under static and periodic axial loads. Tech. Mech. 17(3), 209–216 (1997)

    Google Scholar 

  24. Mikhasev, G.I.: Some problems on localized vibrations and waves in thin shells. In: Altenbah, H., Eremeyev, V.A. (eds.) Shell-Like Structures. Advanced Theories and Applications. CISM International Centre for Mechanical Sciences, vol. 572, pp. 149–209. Springer (2017)

    Google Scholar 

  25. Mikhasev, G.I., Botogova, M.G.: Effect of edge shears and diaphragms on buckling of thin laminated medium-length cylindrical shells with low effective shear modulus under external pressure. Acta Mech. 228, 2119–2140 (2017)

    Article  Google Scholar 

  26. Mikhasev, G., Botogova, M., Korobko, E.: Theory of thin adaptive laminated shells based on magnetorheological materials and its application in problems on vibration suppression. In: Altenbach, H., Eremeyev, V. (eds.) Shell-Like Structures. Advanced Structured Materials, vol. 15, pp. 727–740. Springer, Heidelberg (2011)

    Google Scholar 

  27. Mikhasev, G., Eremeyev, V., Maevskaya, S., Wilde, K.: Assessment of dynamic characteristics of thin cylindrical sandwich panels with magnetorheological core. Composite Structures (2018). Submitted

    Google Scholar 

  28. Mikhasev, G., Mlechka, I.: Localized buckling of laminated cylindrical shells with low reduced shear modulus under non-uniform axial compression. ZAMM 98(3), 491–508 (2018)

    Article  Google Scholar 

  29. Mikhasev, G.I., Tovstik, P.E.: Localized vibrations and waves in thin shells. Asymptotic Methods. FIZMATLIT, Moscow (2009). (in Russia)

    Google Scholar 

  30. Mushtari, K., Galimov, K.: Nonlinear theory of thin elastic shells. NSF-NASA, Washington (1961)

    Google Scholar 

  31. Ogibalov, P.M., Gribanov, V.F.: Thermostability of plates and shells. Moskov. Universitet, Moscow (1968). (in Russia)

    Google Scholar 

  32. Oniashvili, O.D.: On dynamic stability of shells. Soobsch. Akad. Nauk Gruz. 11(3), 169–175 (1950). (in Russia)

    Google Scholar 

  33. Pisarenko, G.S., Chemeris, A.N.: Dynamic stability of a cylindrical shell. In: Dissipation of Energy with the Vibrations of Mechanical Systems (in Russian). Izd. Naukova Dumka, Kiev, pp. 107–114 (1968). (in Russia)

    Google Scholar 

  34. Sheng, G.G., Wang, X.: Nonlinear vibrations of FG cylindrical shells subjected to parametric and external excitations. Compos. Struct. 191, 78–88 (2018)

    Article  Google Scholar 

  35. Skalak, R., Yarymovich, M.I.: Sub-harmonic oscillations of a pendulum. J. Appl. Mech. Trans. ASME 27, 159–164 (1960)

    Article  Google Scholar 

  36. Sofiyev, A.H., Kuruoglu, N.: Determination of the excitation frequencies of laminated orthotropic non-homogeneous conical shells. Compos. Part B Eng. 132, 151–160 (2018)

    Article  Google Scholar 

  37. Udar, R.S., Datta, P.K.: Parametric combination resonance instability characteristics of laminated composite curved panels with circular cutout subjected to non-uniform loading with damping. Int. J. Mech. Sci. 49(3), 317–334 (2007)

    Article  Google Scholar 

  38. Vijayaraghavan, A., Evan-Ivanowski, R.M.: Parametric instability of circular cylindrical shells. Trans. ASME E 34(4), 985–990 (1967)

    Article  Google Scholar 

  39. Vol’mir, A.S., Smetanina, L.N.: Dynamic stability of glass-reinforced plastic shells. Polymer Mech. 4(l), 79–82 (1968)

    Article  Google Scholar 

  40. Wenzke, W.: Die dynamische Stabilitat der axialpulsierend belasteten Kreiszylynderschale. Wiss. Z. Techn. Hochschule Otto von Queriscke Magdeburg. Bd. 7(1), 93–124 (1963)

    Google Scholar 

  41. Wlassow, W.S.: Nonlinear Theory of Thin Elastic Shells. Akademie-Verlag, Berlin (1958)

    Google Scholar 

  42. Yao, J.C.: Dynamic stability of cylindrical shells and static and periodic axial and radial loads. AIAA J. 1(6), 1391–1396 (1963)

    Article  Google Scholar 

  43. Yao, J.C.: Nonlinear elastic buckling and parametric excitation of a cylinder under axial loads. Trans. ASME 32(1), 109–115 (1965)

    Article  Google Scholar 

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Authors and Affiliations

  1. Belarusian State University, 4 Nezavisimosti Avenue, 220030, Minsk, Belarus

    Gennadi Mikhasev & Rovshen Atayev

Authors
  1. Gennadi Mikhasev
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  2. Rovshen Atayev
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Corresponding author

Correspondence to Gennadi Mikhasev .

Editor information

Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Jacek Chróścielewski

  3. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Victor A. Eremeyev

  4. Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

    Krzysztof Wiśniewski

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Mikhasev, G., Atayev, R. (2019). Localized Parametric Vibrations of Laminated Cylindrical Shell Under Non-uniform Axial Load Periodically Varying with Time. In: Altenbach, H., Chróścielewski, J., Eremeyev, V., Wiśniewski, K. (eds) Recent Developments in the Theory of Shells . Advanced Structured Materials, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-030-17747-8_24

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