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

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=LR, 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.

Copyright © 2007 by American Society of Mechanical Engineers
Topics: Pipes , Vibration , Chaos , Shells
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Figure 1

Computational scheme

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Figure 2

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

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Figure 3

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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Figure 4

Maps of control parameters depending on loading corner φ0

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Figure 5

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

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Figure 6

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

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Figure 8

Relation of critical loading q0+(λ)

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Figure 9

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

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Figure 10

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

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Figure 11

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




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