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Research Papers

Chaotic Vibrations of Sector-Type Spherical Shells

[+] Author and Article Information
A. V. Krysko, I. V. Papkova

Department of Applied Mathematics, Technical University of Saratov, St. Polytechnical 77, 410054 Saratov, Russia

J. Awrejcewicz

Department of Automatics and Biomechanics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Polandawrejcew@p.lodz.pl

J. Comput. Nonlinear Dynam 3(4), 041005 (Aug 21, 2008) (17 pages) doi:10.1115/1.2908134 History: Received November 24, 2005; Revised July 26, 2007; Published August 21, 2008

In this work, chaotic vibrations of shallow sector-type spherical shells are studied. A sector-type shallow shell is understood as a shell defined by a sector with associated boundary conditions and obtained by cutting a spherical shell for a given angle θk, or it is a sector of a shallow spherical cap associated with the mentioned angle. Both static stability and complex nonlinear dynamics of the mentioned mechanical objects subjected to transversal uniformly distributed sign-changeable load are analyzed, and the so-called vibration charts and scales regarding the chosen control parameters are reported. In particular, scenarios of transition from regular to chaotic dynamics of the mentioned shells are investigated. A novel method to control chaotic dynamics of the studied flexible spherical shells driven by transversal sign-changeable load via synchronized action of the sign-changeable antitorque is proposed and applied. All investigations are carried out within the fields of qualitative theory of differential equations and nonlinear dynamics.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Computational scheme

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Figure 2

Relation q(wstable) for θk=0

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Figure 3

Relation q(wstable) for θk=0

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Figure 4

Relation q(wstable) for θk=32π

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Figure 5

Relation q(wstable) for θk=π

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Figure 6

Relation q(wstable) for θk=π∕2

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Figure 20

Relation wmax(q0) and vibration character scales for θk=π∕4 (a); π∕2 (b); π (c); 3π∕2 (d)

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Figure 21

Relation wmax(q0) and vibration character scales for various b: 10 (a), 12 (b), 15 (c), 20 (d)

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Figure 22

Relations wmax(q0) and vibration character scales

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Figure 11

Charts of vibration types {qo,ωp}

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Figure 12

Relations wmax(q0)

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Figure 13

Charts of vibration types

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Figure 14

Relations wmax(q0) and scales of vibration character for θk=π, b=9

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Figure 15

Relations wmax(q0) and scales of vibration character for θk=π∕2, b=10

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Vibration type charts for {q0,ωp} and for sector angle θk=0

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Figure 8

Vibration-type charts for {q0,ωp} and for angle sector θk=0

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Figure 9

Charts of vibrations type {q0,ωp}.

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Figure 10

Curves of functions q0(wmax) and vibration-type scales

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Figure 16

Time histories depending on the partition numbers n=m for q0=0.25

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Figure 17

Time histories depending on the partition numbers n=m for q0=0.76

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Figure 18

Relation wmax(q0) and vibration character scales

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Figure 19

Relation wmax(q0) and scales of vibration character for simply supported movable contour (a) and movable clamping (b)

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