| | Mechanics of Solids A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544 Online ISSN 1934-7936 |
Archive of Issues
Total articles in the database: | | 12804 |
In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
In English (Mech. Solids): | | 4760 |
|
<< Previous article | Volume 39, Issue 5 / 2004 | Next article >> |
V. A. Krys'ko and T. V. Shchekaturova, "Chaotic vibrations of cone shells," Mech. Solids. 39 (5), 124-133 (2004) |
Year |
2004 |
Volume |
39 |
Number |
5 |
Pages |
124-133 |
Title |
Chaotic vibrations of cone shells |
Author(s) |
V. A. Krys'ko (Saratov)
T. V. Shchekaturova (Saratov) |
Abstract |
Chaotic vibrations of deterministic, geometrically nonlinear, shallow, isotropic cone shells of revolution subjected to a cyclic load are studied. The influence of the inertial forces along the directions tangent to the middle surface and the rotational inertia of the normal cross-section are ignored. The Ritz method is utilized. The initial value problem for ordinary differential equations is solved by the Runge-Kutta fourth-order method.
The numerical analysis is based on the nonlinear dynamics and qualitative theory of differential equations. New scenarios for the transition of vibrations of flexible cone shells from harmonic to chaotic ones are discovered. The development of various types of vibration is studied as depending on a number of parameters, including the rise of the shell, the amplitude and frequency of the excitation force, and the number of terms preserved in series expansions of the basic functions. This enables one to consider an important issue of the approximation of a distributed system by a discrete one and to obtain an answer to the question whether the chaos disappears for sufficiently large number of terms preserved in the series expansions of the basic functions. |
References |
1. | J. Awrejcewicz and V. A. Krysko, Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov-Galerkin
and Finite Difference Numerical Methods, Springer, Berlin, 2003. |
2. | J. Awrejcewicz, V. A. Krysko, and A. V. Krysko,
"Regular and chaotic behavior of flexible plates,"
in Europ. Congr. on Computational Methods in Applied Sciences and Engineering (CD ROM), Eccomas Computational Fluid Dynamics Conf.,
Swansea, UK, 2001. |
3. | J. Awrejcewicz, V. A. Krysko, and A. V. Krysko,
"Spatio-temporal chaos and solutions exhibited by von Kàrman
model," Int. J. Bifurcations and Chaos, Vol. 12, No. 7, pp. 1465-1513, 2002. |
4. | V. A. Krysko and J. Awrejcewicz,
"Bifurcations of thin plates transversally and sinusoidally
excited,"
in H. Grundmann and J. I. Schuelee (Editors), Structural Dynamics. EURODYN,
Swets Zeitinger Lisse, pp. 529-534, 2002. |
5. | J. Awrejcewicz and V. A. Krysko,
"Nonlinear coupled problems of dynamics of shells,"
Intern. J. Engng. Sci., Vol. 41, No. 6, pp. 587-607, 2003. |
6. | J. Awrejcewicz and V. A. Krysko,
"Feigenbaum scenario exhibited by thin plate dynamics,"
Nonlinear Dynamics, Vol. 24, No. 4, pp. 373-398, 2001. |
7. | V. A. Krys'ko, A. A. Sopenko, and E. V. Salii,
"Complex vibrations of geometrically and physically nonlinear
shallow shells with a rectangular plane contour,"
Izv. Vuzov. Prikladnaya Nelineinaya Dinamika,
Vol. 10, No. 1-2, pp. 92-103, 2002. |
8. | V. V. Novozhilov, Fundamentals of Nonlinear Elasticity [in Russian], Gostekhizdat, Leningrad, Moscow, 1948. |
9. | A. S. Vol'mir, Stability of Deformable Systems [in Russian], Nauka, Moscow, 1967. |
10. | V. I. Feodos'ev, "One approach for solving stability problems for deformable systems,"
PMM [Applied Mathematics and Mechanics], Vol. 27, No. 2, pp. 265-275, 1963. |
11. | B. Ya. Kantor, Nonlinear Problems in the Theory of Inhomogeneous Shallow Shells [in Russian], Naukova Dumka, Kiev, 1971. |
12. | E. N. Lorenz, "Deterministic nonperiodic flow,"
Atmos. Sci., Vol. 20, No. 1, pp. 130-141, 1962. |
13. | J. H. Curry, J. R. Herring, J. Loncaric, and S. A. Orszag,
"Order disorder in two- and three-dimensional Bénard convection,"
J. Fluid. Mach., Vol. 147, pp. 1-38, 1984. |
14. | Li Tien-Yien and J. A. Yorke, "Period three implies chaos,"
Amer. Math. Monthly, Vol. 82, pp. 985-992, 1975. |
15. | A. N. Sharkovskii, "Coexistence of cycles of a continuous transformation of the straight line into itself," Ukrainskii Matem. Zh., Vol. 16, No. 1, pp. 61-71, 1964. |
16. | L. D. Landau, "On the Problem of Turbulence,"
Doklady AN SSSR, Vol. 44, No. 8, pp. 339-342, 1944. |
17. | M. J. Feigenbaum, "The universal metric properties of nonlinear transformations,"
J. Stat. Phys., Vol. 21, No. 6, pp. 669-706, 1979. |
18. | D. Ruelle and F. Takens, "On the nature of turbulence,"
Commun. Math. Phys., Vol. 20, No. 3, pp. 167-192, 1971. |
19. | P. Manneville and Y. Pomean, "Different ways to turbulence in dissipative dynamical systems,"
Physica, Vol. 1, No. 2, pp. 219-226, 1980. |
20. | V. A. Krys'ko and A. V. Krys'ko,
"Problems of bifurcations and hard instability in the nonlinear theory of plates,"
in Mechanics of Plates and Shells in the 21st Century
[in Russian], pp. 50-67, Izd-vo SGTU, Saratov, 1999. |
21. | P. Collet and J. P. Eckmann,
Iterated Maps on the Interval as Dynamical Systems,
Birkhauser, Boston, 1980. |
22. | L. P. Shil'nikov, "Theory of bifurcations and turbulence,"
in Issues of Nonlinear and Turbulent Processes in Physics. Part 2 [in Russian], pp. 118-124, Naukova Dumka, Kiev, 1985. |
23. | S. Smale, "Dynamical systems and turbulence," in
Lect. Notes. Math., No. 615, pp. 48-70, Springer, Berlin, 1977. |
|
Received |
22 May 2003 |
<< Previous article | Volume 39, Issue 5 / 2004 | Next article >> |
|
If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|