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.
[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.
[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.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:54898", author = "B. Tomczyk and R. Mania", title = "Parametric Vibrations of Periodic Shells", 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 = "Micro-periodic shells, mathematical modeling,length-scale effect, parametric vibrations", volume = "5", number = "11", pages = "2285-10", }
