Commenced in January 2007
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Paper Count: 30382
Parametric Vibrations of Periodic Shells

Authors: B. Tomczyk, R. Mania

Abstract:

Thin linear-elastic cylindrical circular shells having a micro-periodic structure along two directions tangent to the shell midsurface (biperiodic shells) are object of considerations. The aim of this paper is twofold. First, we formulate an averaged nonasymptotic model for the analysis of parametric vibrations or dynamical stability of periodic shells under consideration, which has constant coefficients and takes into account the effect of a cell size on the overall shell behavior (a length-scale effect). This model is derived employing the tolerance modeling procedure. Second we apply the obtained model to derivation of frequency equation being a starting point in the analysis of parametric vibrations. The effect of the microstructure length oh this frequency equation is discussed.

Keywords: Mathematical Modeling, Micro-periodic shells, length-scale effect, parametric vibrations

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061731

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References:


[1] A. Bensoussan, J. L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. Amsterdam, North-Holland, 1978.
[2] T. Lewiński and J. J. Telega, Plates, Laminates and Shells. Asymptotic Analysis and Homogenization. Singapore: World Scientific Publishing Company, 2000.
[3] J. Awrejcewicz, I. Andrianov and L. Manevitch, Asymptotical Mechanics of Thin-Walled Structures. Berlin: Springer, 2004.
[4] S. A. Ambartsumyan, Theory of Anisotropic Shells. Moscow: Nauka, 1974.
[5] B. Tomczyk, "On the modeling of thin uniperiodic cylindrical shells," J. Theor. Appl. Mech., vol. 41, pp. 755-774, 2003.
[6] B. Tomczyk, "On stability of thin periodically densely stiffened cylindrical shells," J. Theor. Appl. Mech., vol. 43, pp. 427-455, 2005.
[7] B. Tomczyk, "On dynamics and stability of thin periodic cylindrical shells," Diff. Eqs. Nonlin. Mech., ID 79853, pp. 1-23, 2006.
[8] B. Tomczyk, "A non-asymptotic model for the stability analysis of thin biperiodic cylindrical shells," Thin-Walled Struct., vol. 45, pp. 941-944, 2007.
[9] B. Tomczyk, "Vibrations of thin cylindrical shells with a periodic structure," PAMM, vol. 8, pp. 10349-10350, 2008.
[10] B. Tomczyk, B. "Dynamic stability of micro-periodic cylindrical shells," Mechanics and Mechanical Engineering., vol. 14, pp. 137-150, 2010.
[11] B. Tomczyk, "On the modeling of dynamic problems for biperiodically stiffened cylindrical shells," Civil and Environmental Engineering Reports, vol. 5, pp. 179-204, 2010.
[12] B. Tomczyk, "Thin cylindrical shells," in Thermomechanics of Microheterogeneous Solids and Structures. Tolerance Averaging Approach, Part II: Model Equations, C. Wo┼║niak, B. Michalak and J. J─Ödrysiak, Eds. Lodz: Lodz Technical University Press, 2008, pp. 165-175.
[13] B. Tomczyk, "Thin cylindrical shells," in Thermomechanics of Microheterogeneous Solids and Structures. Tolerance Averaging Approach, Part III: Selected Probmems, C. Wo┼║niak, B. Michalak and J. J─Ödrysiak, Eds. Lodz: Lodz Technical University Press, 2008, pp. 383- 411.
[14] B. Tomczyk, "On micro-dynamics of reinforced cylindrical shells," in Mathematical Modeling and Analysis in Continuum Mechanics of Microstructured Media, C. Wo┼║niak, et al., Eds. Gliwice: Silesian Technical University Press, 2010, pp. 121-135.
[15] B. Tomczyk, "Combined modeling of periodically stiffened cylindrical shells," in Selected Topics in Mechanics of the Inhomogeneous Media, C. Wo┼║niak, et al., Eds. Zielona Gora: Zielona Gora University Press, 2010, pp. 79-97.
[16] B. Tomczyk, "A combined model for problems of dynamics and stability of biperiodic cylindrical shells," in Mathematical Methods in Continuum Mechanics, K. Wilmański, B. Michalak and J. J─Ödrysiak, Eds. Lodz: Lodz Technical University Press, 2011, pp. 331-355.
[17] C. Wo┼║niak and E. Wierzbicki, Averaging Techniques in Thermomechanics of Composite Solids. Cz─Östochowa: Cz─Östochowa University Press, 2000.
[18] C. Wo┼║niak, B. Michalak and J. J─Ödrysiak, (Eds.), Thermomechanics of Microheterogeneous Solids and Structures. Tolerance Averaging Approach. Lodz: Lodz Technical University Press, 2008.
[19] C. Wo┼║niak, et al. (Eds.), Mathematical Modeling and Analysis in Continuum Mechanics of Microstructured Media. Gliwice: Silesian Technical University Press, 2010.
[20] C. Wo┼║niak, et al. (Eds.), Selected Topics in Mechanics of the Inhomogeneous Media. Zielona Gora: Zielona Gora University Press, 2010.
[21] S. Kaliski (Ed.), Vibrations. Warsaw-Amsterdam: PWN-Elsevier, 1992.