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Solution of the Problem of Free Vibrations of a Nonthin Orthotropic Shallow Shell of Variable Thickness in the Refined Statement

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We consider the problem of investigation of the spectrum of natural vibrations of a nonthin orthotropic shallow shell variable in two coordinate directions of thickness in the nonclassical statement. The approach to the solution of the obtained two-dimensional boundary-value problem is based on its reduction (by the method of spline-approximation of the unknown functions along one coordinate direction) to the one-dimensional problem with its subsequent solution. We study different cases of boundary conditions imposed on the contours of the shell. We also perform the comparison and analysis of the natural frequencies and modes of vibrations of orthotropic shells of constant and variable thickness.

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References

  1. S. A. Ambartsumyan, General Theory of Anisotropic Shells [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  2. V. D. Budak, A. Ya. Grigorenko, and S. V. Puzyrev, “Free vibrations of rectangular orthotropic shallow shells with varying thickness,” Prikl. Mekh., 43, No. 6, 102–115 (2007); English translation: Int. Appl. Mech., 43, No. 6, 670–682 (2007).

  3. A. Ya. Grigorenko and T. L. Efimova, “Spline-approximation method applied to solve natural-vibration problems for rectangular plates of varying thickness,” Prikl. Mekh., 41, No. 10, 90–99 (2005); English translation: Int. Appl. Mech., 41, No. 10, 1161–1169 (2005).

  4. A. Ya. Grigorenko, Yu. G. Zolotoi, A. P. Prigoda, I. Yu. Zhuk, V. V. Khorishko, and A. V. Ovcharenko, “Experimental investigation of natural vibrations of a thick-walled cylindrical shell by the method of holographic interferometry,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 3, 93–98 (2012); English translation: J. Math. Sci., 194, No. 3, 239–244 (2013).

  5. A. Ya. Grigorenko and A. Yu. Parkhomenko, “Free vibrations of shallow orthotropic shells with variable thickness and rectangular planform,” Prikl. Mekh., 46, No. 8, 29–41 (2010); English translation: Int. Appl. Mech., 46, No. 8, 877–889 (2010).

  6. A. Ya. Grigorenko, S. V. Puzyrev, and E. A. Volchek, “Investigation of free vibrations of noncircular cylindrical shells by the spline-collocation method,” Mat. Met. Fiz.-Mekh. Polya, 54, No. 3, 60–69 (2011); English translation: J. Math. Sci., 185, No. 6, 824–836 (2012).

  7. A. Ya. Grigorenko, T. L. Efimova, and I. A. Loza, “Solution of an axisymmetric problem of free vibrations of piezoceramic hollow cylinders of finite length by the spline collocation method,” Mat. Met. Fiz.-Mekh. Polya, 51, No. 3, 112–120 (2008); English translation: J. Math. Sci., 165, No. 2, 290–300 (2010).

  8. Ya. M. Grigorenko, E. I. Bespalova, A. B. Kitaigorodskii, and A. I. Shinkar’, Free Vibrations of the Elements of Shell Structures [in Russian], Naukova Dumka, Kiev (1986).

  9. Ya. M. Grigorenko and A. T. Vasilenko, Theory of Shells of Variable Stiffness, in: A. N. Guz’ (editor), Methods for the Numerical Analyses of Shells [in Russian], Vol. 4, Naukova Dumka, Kiev (1981).

  10. T. L. Efimova, “Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders,” Mat. Met. Fiz.-Mekh. Polya, 52, No. 1, 92–100 (2009); English translation: J. Math. Sci., 168, No. 4, 613–623 (2010).

  11. S. G. Lekhnitsky, Theory of Elasticity of an Anisotropic Elastic Body, Holden Day, San Francisco (1963).

  12. B. L. Pelekh, Theory of Shells with Finite Shear Stiffness [in Russian], Naukova Dumka, Kiev (1973).

    Google Scholar 

  13. D. J. Dawe, “Finite strip models for vibration of Mindlin plates,” J. Sound Vibrat., 59, No. 3, 441–452 (1978).

    Article  Google Scholar 

  14. R. D. Mindlin, “Influence of rotatory inertia and shear on flexural motion of isotropic elastic plates,” Trans. ASME. J. Appl. Mech., 18, 31–38 (1951).

    MATH  Google Scholar 

  15. T. Mizusawa and Y. Kondo, “Application of the spline element method to analyze vibration of skew Mindlin plates with varying thickness in one direction,” J. Sound Vibrat., 241, No. 3, 485–501 (2001).

    Article  Google Scholar 

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Authors and Affiliations

  1. Tymoshenko Institute of Mechanics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    O. Ya. Grigorenko

  2. Sukhomlyns’kyi Mykolaiv National University, Mykolaiv, Ukraine

    O. Yu. Parkhomenko, L. Ya. Vasil’eva & М. Yu. Borisenko

Authors
  1. O. Ya. Grigorenko
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  2. O. Yu. Parkhomenko
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  3. L. Ya. Vasil’eva
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  4. М. Yu. Borisenko
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 59, No. 1, pp. 121–131, January–March, 2016.

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Grigorenko, O.Y., Parkhomenko, O.Y., Vasil’eva, L.Y. et al. Solution of the Problem of Free Vibrations of a Nonthin Orthotropic Shallow Shell of Variable Thickness in the Refined Statement. J Math Sci 229, 253–268 (2018). https://doi.org/10.1007/s10958-018-3675-6

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  • DOI: https://doi.org/10.1007/s10958-018-3675-6

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