Dynamics and statics of flexible axially symmetric shallow shells

Awrejcewicz, J.; Krysko, V. A.; Kravtsova, I. V.

doi:https://doi.org/10.1155/MPE/2006/35672

Mathematical Problems in Engineering

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 035672 | https://doi.org/10.1155/MPE/2006/35672

Dynamics and statics of flexible axially symmetric shallow shells

J. Awrejcewicz,¹V. A. Krysko,²and I. V. Kravtsova²

Received24 Jun 2005

Accepted12 Jul 2005

Published06 Mar 2006

Abstract

In this work, we propose the method for the investigation of stochastic vibrations of deterministic mechanical systems represented by axially symmetric spherical shells. These structure members are widely used as sensitive elements of pressure measuring devices in various branches of measuring and control industry, machine design, and so forth. The proposed method can be easily extended for the investigation of shallow spherical shells, goffer-type membranes, and so on. The so-called charts of control parameters for a shell subjected to a transversal uniformly distributed and local harmonic loading force and resistance moment are constructed. The scenarios of the transition of vibration of shallow-type system into chaotic state are investigated with the use of the theory of differential equations and the theory of nonlinear dynamics. The method of the control of chaotic vibrations of flexible spherical shells subjected to a transversal harmonic load through a synchronized action of either harmonic resistance moment or force is proposed, illustrated, and discussed.

References

J. Awrejcewicz and V. A. Krysko, “Feigenbaum scenario exhibited by thin plate dynamics,” Nonlinear Dynamics, vol. 24, no. 4, pp. 373–398, 2001.
View at: Google Scholar | Zentralblatt MATH
J. Awrejcewicz and A. V. Krysko, “Analysis of complex parametric vibrations of plates and shells using Bubnov-Galerkin approach,” Archive of Applied Mechanics, vol. 73, no. 7, pp. 495–504, 2003.
View at: Publisher Site | Google Scholar
J. Awrejcewicz and V. A. Krysko, Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov-Galerkin and Finite Difference Numerical Methods, Scientific Computation, Springer, Berlin, 2003.
View at: Zentralblatt MATH | MathSciNet
J. Awrejcewicz and V. A. Krysko, “Nonlinear coupled problems in dynamics of shells,” International Journal of Engineering Science, vol. 41, no. 6, pp. 587–607, 2003.
View at: Google Scholar | MathSciNet
J. Awrejcewicz, V. A. Krysko, and A. V. Krysko, “Spatial-temporal chaos and solitons exhibited by von Kármán model,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 7, pp. 1465–1513, 2002.
View at: Google Scholar
J. Awrejcewicz, V. A. Krysko, and A. V. Krysko, “Complex parametric vibrations of flexible rectangular plates,” Meccanica. International Journal of the Italian Association of Theoretical and Applied Mechanics, vol. 39, no. 3, pp. 221–244, 2004.
View at: Google Scholar | MathSciNet
J. Awrejcewicz, V. A. Krysko, and G. G. Narkaitis, “Bifurcations of a thin plate-strip excited transversally and axially,” Nonlinear Dynamics, vol. 32, pp. 187–209, 2003.
View at: Google Scholar
J. Awrejcewicz, V. A. Krysko, and A. F. Vakakis, Nonlinear Dynamics of Continuous Elastic Systems, Springer, Berlin, 2004.
View at: Zentralblatt MATH | MathSciNet
V. F. Fedos'ev, “Application of the step method to analyse stability of compressed rod,” Prikladnaya Matematika i Mekhanika, vol. 27, no. 5, pp. 833–841, 1963 (Russian).
View at: Google Scholar
V. F. Fedos'ev, “On the method of solutions of stability of deformable bodies,” Prikladnaya Matematika i Mekhanika, vol. 27, no. 2, pp. 265–275, 1963 (Russian).
View at: Google Scholar
M. J. Feigenbaum, “The universal metric properties of nonlinear transformations,” Journal of Statistical Physics, vol. 21, no. 6, pp. 669–706, 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Hübler and E. Lüscher, “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaften, vol. 76, no. 2, pp. 67–69, 1989.
View at: Google Scholar
E. A. Jackson, “The entrainment and migration controls of multiple-attractor systems,” Physics Letters. A, vol. 151, no. 9, pp. 478–484, 1990.
View at: Google Scholar | MathSciNet
E. A. Jackson, “On the control of complex dynamic systems,” Physica D. Nonlinear Phenomena, vol. 50, no. 3, pp. 341–366, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. A. Krysko, J. Awrejcewicz, and V. M. Bruk, “On the solution of a coupled thermo-mechanical problem for non-homogeneous {T}imoshenko-type shells,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 409–416, 2002.
View at: Google Scholar | MathSciNet
V. A. Krysko and I. V. Kravtsova, “Stochastic vibrations of flexible flat axisymmetric shells exposed inhomogeneous loading,” in Proceedings of 7th International Conference on Dynamics of System—Theory and Applications (Łódź, 2003), pp. 189–197, 2003.
View at: Google Scholar
V. A. Krysko and I. V. Kravtsova, “Stochastic vibrations of flexible axially symmetric supported along contour spherical shells,” Izviestia VUZ, Maschinostroyeniye, vol. 1, pp. 3–13, 2004 (Russian).
View at: Google Scholar
V. A. Krysko and T. V. Shchekaturova, “Chaotic vibrations of shallow shells,” Izviesta AN Mekhanika Tviordoga Tela, vol. 4, pp. 140–150, 2004 (Russian).
View at: Google Scholar
L. D. Landau, “On the problem of a turbulence,” Doklady Akademii Nauk, vol. 44, no. 8, pp. 339–342, 1944 (Russian).
View at: Google Scholar
B. B. Mandelbrot, The Fractal Geometry of Nature, Schriftenreihe für den Referenten, W. H. Freeman, California, 1982.
View at: Zentralblatt MATH | MathSciNet
P. Manneville and Y. Pomeau, “Different ways to turbulence in dissipative dynamical systems,” Physica D. Nonlinear Phenomena, vol. 1, no. 2, pp. 219–226, 1980.
View at: Google Scholar | MathSciNet
E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. Petrov, V. Gaspar, J. Massere, and K. Showalter, “Controlling chaos in the Belousov-Zhabotinsky reaction,” Nature, vol. 361, no. 6409, pp. 240–243, 1993.
View at: Google Scholar
D. Ruelle and F. Takens, “On the nature of turbulence,” Communications in Mathematical Physics, vol. 20, pp. 167–192, 1971.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, “Controlling chaos in the brain,” Nature, vol. 370, no. 6491, pp. 615–620, 1994.
View at: Google Scholar
T. Shinbrot, C. Grebogi, J. A. Yorke, and E. Ott, “Using small perturbations to control chaos,” Nature, vol. 363, no. 6428, pp. 411–417, 1993.
View at: Google Scholar
J. Singer, Y.-Z. Wang, and H. H. Bau, “Controlling a chaotic system,” Physical Review Letters, vol. 66, no. 9, pp. 1123–1125, 1991.
View at: Google Scholar
S. Smale, “Dynamical systems and turbulence,” in Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), vol. 615 of Lecture Notes in Math., pp. 48–70, Springer, Berlin, 1977.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
N. V. Valishvili, Methods of Computation of Rotational Shells, Mashinostroyeniye, Moscow, 1976.

Copyright

Copyright © 2006 J. Awrejcewicz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

301

Downloads

922

Citations