Complex Parametric Vibrations of Flexible Rectangular Plates

Awrejcewicz, Jan; Krysko, Vadim A.; Krysko, Anton V.

doi:10.1023/b:mecc.0000022845.52667.b0

Cited by 20 publications

(12 citation statements)

References 21 publications

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“…It must be pointed out that new and promising methodologies are appearing for the chaotic analysis and simulation of continuous systems [42][43][44][45]. The high-order approximations and the extended Melnikov-type method [47] should be applied for further research in heated plates and shells.…”

Section: Discussionmentioning

confidence: 99%

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Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Wang

Song

et al. 2009

Arch Appl Mech

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The chaotic vibrations of a bimetallic shallow shell of revolution under time-varying temperature excitation are investigated in the present study. The governing equations are established in forms similar to those of classical single-layered shell theory by re-determination of reference surface. The nonlinear differential equation in time-mode is derived by variational method following an assumed spatial-mode. The Melnikov function is established theoretically to estimate regions of the chaos, and the Poincaré map, phase portrait, Lyapunov exponent, and Lyapunov dimension are used to determine if a chaotic motion really appears. Further investigations are developed by means of detailed numerical simulation, and both the bifurcation diagrams and corresponding maximum Lyapunov exponent are illustrated. The influence of static and time-dependent temperature parameters, height parameter of the shell, and damping parameter on the dynamic characteristics is examined. Interesting phenomena such as the onset of chaos, transient chaotic motion, chaos with interior crisis and period window, period-doubling scenario and reversed period-doubling bifurcation leading to chaos, jump phenomena, and chaos suddenly converting to period orbit have been observed from these figures.

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Section: Discussionmentioning

confidence: 99%

“…Some other well-established achievements related to this issue have been exhibited in Refs. [42][43][44][45] by adopting a rather high-order approximation while remaining relatively simple in mathematical relations.…”

Section: Nonlinear Differential Equations For Time Functionmentioning

confidence: 99%

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Wang

Song

et al. 2009

Arch Appl Mech

View full text Add to dashboard Cite

show abstract

“…It must be pointed out that new and promising methodologies are appearing for the chaotic analysis and simulation of continuous systems [37][38][39][40]. The high-order approximations and the extended Melnikov-type method [42] should be applied for further research in heated plates and shells.…”

Section: Discussionmentioning

confidence: 99%

“…To overcome these potential problems, a rather higher order approximation is expected to sufficiently represent the properties of the original plate, while remaining relatively simple in mathematical relations when applying this approach. Some well-established achievements related to this issue have been exhibited in [37][38][39][40].…”

Section: Nonlinear Differential Equations For Time Functionmentioning

confidence: 99%

Bifurcations and chaos of bimetallic circular plates subjected to periodic heat load

Wang

2008

Z Angew Math Mech

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The chaotic vibrations of bimetallic thin circular plates with geometric nonlinearity subjected to periodic time-varying temperature load are investigated. The governing equations are formulated in forms similar to those of classical von Kármán's single-layered plate theory by re-determination of reference surface of coordinate. The nonlinear differential equation in time-mode is derived by variational method following an assumed spatial-mode. The Melnikov function is established theoretically to estimate regions of the chaos, and the Poincaré map, phase portrait, Lyapunov exponent, and Lyapunov dimension are used to evaluate if a chaotic motion really appears. Further investigations are developed by means of detailed numerical simulation, the combined bifurcation diagram and its corresponding maximum Lyapunov exponent are illustrated. The influence of static and time-dependent temperature parameters, and damping parameter on the dynamic characteristics is examined. Interesting phenomena such as the onset of chaos, transient chaos, chaos with periodic windows, period-doubling scenario and reversed period-doubling bifurcation leading to chaos, jump phenomena, and chaos suddenly converts to period orbit have been observed.

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“…"set-up method" proposed firstly by Fedos'ev 12 and widely applied in works. 13,20 It has been shown in Ref. 5 that differentiation with respect to the parameter and the Cauchy problem for ODEs with independent variable t should be solved on the fixed time interval 0 ≤ t ≤ 1, if the initial system behaves as the Ritz system for a certain functional.…”

Section: Historical Introductionmentioning

confidence: 99%

Transitions From Regular to Chaotic Vibrations of Spherical and Conical Axially-Symmetric Shells

Awrejcewicz

Krysko

Shchekaturova

2005

Int. J. Str. Stab. Dyn.

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By the variational principle, the chaotic vibrations of deterministic geometrically nonlinear elastic spherical and conical axially symmetric shells with non-homogeneous thickness subjected to a transversal harmonic load are analyzed. The material of the shells is assumed to be isotropic and of the Hookean type. Inertial forces tangent to the averaged surface and inertia of rotation of the cross-section are neglected. By the Ritz procedure, the original PDEs are transferred to the ODEs (Cauchy problem), which are then solved by the fourth-order Runge-Kutta method. In the numerical studies, scenarios of transitions from harmonic to chaotic states for vibrations of flexible spherical and conical shells are detected. Various vibrational states for different combinations of the following control parameters: shell's deflection arrow, the amplitude and frequency of the exciting force, number of modes considered, boundary conditions, and the thickness and shape of the shell cross-section are studied. By adjusting the above parameters, we can detect the transition of a continuous system to the lumped one, and the transition from the harmonic to chaotic vibrations.Keywords: Ritz method; shells; chaos; ordinary and partial differential equations. Int. J. Str. Stab. Dyn. 2005.05:359-385. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ DAVIS on 02/07/15. For personal use only.

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Complex Parametric Vibrations of Flexible Rectangular Plates

Cited by 20 publications

References 21 publications

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Bifurcations and chaos of bimetallic circular plates subjected to periodic heat load

Transitions From Regular to Chaotic Vibrations of Spherical and Conical Axially-Symmetric Shells

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