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Journal of Computational and Nonlinear Dynamics | Volume 2 | Issue 1 | Research Paper
RESEARCH PAPERS

Routes to Chaos Exhibited by Closed Flexible Cylindrical Shells

[+] Author and Article Information
J. Awrejcewicz

Department of Automatics and Biomechanics (K-16), Technical University of Łódź, 1/15 Stefanowskiego Street, 90-924 Łódź, Polandawrejcew@p.lodz.pl

V. Krysko

Department of Higher Mathematics, Technical University of Saratov, Politehnicheskaya Street, 77, 410054 Saratov, Russiatak@san.ru

N. Saveleva

Department of Higher Mathematics, Technical University of Saratov, Politehnicheskaya Street, 77, 410054 Saratov, Russiaqarx@mail.ru

J. Comput. Nonlinear Dynam 2(1), 1-9 (Sep 18, 2006) (9 pages) doi:10.1115/1.2402923 History: Received June 15, 2005; Revised September 18, 2006
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Complex vibrations of closed cylindrical shells of circular cross section and finite length subjected to nonuniform sign-changeable external load in the frame of classical nonlinear theory are studied. A transition from partial differential equations to ordinary differential equations (Cauchy problem) is carried out using the higher order Bubnov–Galerkin’s approach and Fourier’s representation. On the other hand, the Cauchy problem is solved using the fourth-order Runge–Kutta method. Results are analyzed owing to the application of nonlinear dynamics and qualitative theory of differential equations. The present work is devoted to the analysis of influence of the system dynamics of the following parameters: length of pressure width φ0, relative linear shell dimension λ=L∕R, and frequency ωp and amplitude q0 of external transversal load. Some new scenarios of vibrations of closed cylindrical shells exhibiting a transition from harmonic to chaotic vibrations are illustrated and studied.
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Copyright © 2007 by American Society of Mechanical Engineers
Topics: Pipes , Vibration , Chaos , Shells
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References

1
Awrejcewicz, J., and Krysko, V. A., 2001, “Feigenbaum Scenario Exhibited by Thin Plate Dynamics,” Nonlinear Dyn., 24 , pp. 373–398.
2
Awrejcewicz, J., Krysko, V., and Krysko, A., 2002, “Spatial-Temporal Chaos and Solitons Exhibited by Von Karman Model,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 12 (7), pp. 1465–1513.
3
Awrejcewicz, J., and Krysko, A. V., 2003, “Analysis of Complex Parametric Vibrations of Plates and Shells using Bubnov-Galerkin Approach,” Archive of Applied Mathematics, 73 , pp. 495–504.
4
Awrejcewicz, J., and Krysko, V., 2003, "Nonclassic Thermoelastic Problem in Nonlinear Dynamics of Shells", Springer, Berlin, Germany.
5
Awrejcewicz, J., Krysko, V. A., and Vakakis, A. F., 2004, "Nonlinear Dynamics of Continuous Elastic Systems", Springer, Berlin, Germany.
6
Volmir, A. S., 1963, "Stability of Elastic Systems", Fizmatgiz, Moscow (in Russian).
7
Krysko, V. A., and Kucemako, A. N., 1999, "Stability and Vibrations of Non-Uniform Shells", Saratov University Press, Russia, Saratov (in Russian).
8
Krysko, V. A., and Saveleva, N. E., 2003, “Vibrations of Closed Cylindrical Shells at Non-Axis-Symmetrical Sign-Variable External Load,” "Works of the International Conference “Nonlinear Fluctuations of Mechanical and Biological Systems,”"Saratov University Press, Russia, Saratov (in Russian).
9
Krysko, V. A., and Saveleva, N. E., 2003, "Stochastic Dynamics of Closed Cylindrical Shells at Non-Axis-Symmetrical External Loading", Saratov University Press, Russia, Saratov (in Russian).
10
Krysko, V. A., and Saveleva, N. E., 2004, “Statics and Dynamics of Closed Cylindrical Shells at Non-Uniform Loading,” "Works of the International Conference “Problems of Durability of Materials and Designs on Transport,”"St.-Petersburg, Russia (in Russian).
11
Landau, D., 1944, "On the Problem of Turbulence, Lectures Academy of Sciences of USSR", Moscow, Russia (in Russian), 44 , pp. 339–342.
12
Ruelle, D., and Takens, F., 1971, “On the Nature of Turbulence,” Comp. Math. Phys, 20 (2), pp. 167–192.
13
Feigenbaum, M. J., 1978, “Quantitative Universally for a Class of Nonlinear Transformation,” J. Stat. Phys.  [CrossRef], 19 (1), pp. 25–52.
14
Manneville, P., and Pomeau, Y., 1980, “Different ways to turbulence in dissipative dynamical systems,” Physica D  [CrossRef], 1 , pp. 219–228.
15
Krysko, V. A., and Saveleva, N. E., 2003, “Ruelle-Takens-Feigenbaum Scenario and Counting of Feigenbaum constant,” Proceedings of the 7th Conference on “Dynamical Systems—Theory and Applications , J.Awrejcewicz, A.Owczarek, J.Mrozowski, Łódź Poland, December 8-11, pp. 179–188.

Figures

Grahic Jump Location
+Relation l(λ)
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Relation l(λ)
Figure 7

Relation l(λ)

Grahic Jump Location
+Relation of critical loading q0+(λ)
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Relation of critical loading q0+(λ)
Figure 8

Relation of critical loading q0+(λ)

Grahic Jump Location
+Computational scheme
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Computational scheme
Figure 1

Computational scheme

Grahic Jump Location
+Convergence of Bubnov–Galerkin’s method in the harmonic vibrations zone
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Convergence of Bubnov–Galerkin’s method in the harmonic vibrations zone
Figure 2

Convergence of Bubnov–Galerkin’s method in the harmonic vibrations zone

Grahic Jump Location
+Convergence of Bubnov–Galerkin’s method (time histories w(t), power spectrum S(ω), Poincaré map wt(wt+T) in chaotic zones)
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Convergence of Bubnov–Galerkin’s method (time histories w(t), power spectrum S(ω), Poincaré map wt(wt+T) in chaotic zones)
Figure 3

Convergence of Bubnov–Galerkin’s method (time histories w(t), power spectrum S(ω), Poincaré map wt(wt+T) in chaotic zones)

Grahic Jump Location
+Maps of control parameters depending on loading corner φ0
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Maps of control parameters depending on loading corner φ0
Figure 4

Maps of control parameters depending on loading corner φ0

Grahic Jump Location
+Relation wmax(q0) and vibrations character scale for λ=1
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Relation wmax(q0) and vibrations character scale for λ=1
Figure 5

Relation wmax(q0) and vibrations character scale for λ=1

Grahic Jump Location
+Relations wmax(q0) and vibrations character scales for some values of λ
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Relations wmax(q0) and vibrations character scales for some values of λ
Figure 6

Relations wmax(q0) and vibrations character scales for some values of λ

Grahic Jump Location
+Vibrations at ω0∕2 frequency: 1—yes; 2—no
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Vibrations at ω0∕2 frequency: 1—yes; 2—no
Figure 9

Vibrations at ω0∕2 frequency: 1—yes; 2—no

Grahic Jump Location
+Vibrations at independent frequencies: 1—yes; 2—no
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Vibrations at independent frequencies: 1—yes; 2—no
Figure 10

Vibrations at independent frequencies: 1—yes; 2—no

Grahic Jump Location
+Vibrations at Andronov–Hopf frequency: 1—yes; 2—no
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Vibrations at Andronov–Hopf frequency: 1—yes; 2—no
Figure 11

Vibrations at Andronov–Hopf frequency: 1—yes; 2—no

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