Modal equations for the nonlinear flexural vibrations of a cylindrical shell

Dowell, Earl H.; Ventres, C. S.

doi:10.1016/0020-7683(68)90017-6

Cited by 143 publications

(62 citation statements)

References 9 publications

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“…Consider equations of the nonlinear shallow-shell theory connecting the normal deflection w(x, y, t) and the stress function Φ(x, y, t) [24][25][26][27]:…”

Section: Stability Of Forced Vibration Modes In Nonlinear Shellsmentioning

confidence: 99%

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Mikhlin

Shmatko

Manucharyan

2004

Computers & Structures

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This paper deals with forced vibrations of two-DOF systems with more than one equilibrium positions. Such systems may be obtained by digitization of elastic post-buckling systems. A vibration mode, which is periodic at small force amplitudes and becomes chaotic as the force amplitudes are slowly increased, is selected. It is possible to formulate and solve the problem of stability of a periodic or chaotic vibration mode in a space with greater dimension using the classical Lyapunov stability definition and some calculating procedures. Instability of phase trajectories is used as a criterion of the chaotic behavior in dynamical systems. Trajectories with very close initial values are compared. Use of the Lyapunov stability definition shows mutual stability/instability of the trajectories. Calculations permit to observe an appearance and enlargement of the chaotic behavior regions. Specific results are obtained for the nonautonomous Duffing equation and pendulum system.

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“…Consider equations of the nonlinear shallow-shell theory connecting the normal deflection w(x, y, t) and the stress function Φ(x, y, t) [24][25][26][27]:…”

Section: Stability Of Forced Vibration Modes In Nonlinear Shellsmentioning

confidence: 99%

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Mikhlin

Shmatko

Manucharyan

2004

Computers & Structures

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“…The axi-symmetric mode in the modal expansion was chosen in such a way that the transverse displacement becomes null at the boundaries. Knowing the importance of axi-symmetric modes, [6,7], both using different modal expansions, but containing axi-symmetric modes, obtained a frequency-amplitude relation with hardening behavior. The error was due to the incorrect representation of axi-symmetric modes on the modal expansion, as showed by [10].…”

Section: Introductionmentioning

confidence: 99%

“…The study of the nonlinear vibrations of cylindrical shells began in the middle of the last century with the works by [1][2][3][4][5][6][7][8], among others. In these works either the Ritz or the Galerkin method are used to discretize the shell.…”

Section: Introductionmentioning

confidence: 99%

The Influence of Internal Resonances on the Dynamics of Fluidfilled Cylindrical Shells

Silva¹,

Gonçalves²,

Prado³

2014

Proceedings of 10th World Congress on Computational Mechanics

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“…Mode expansion method in the nonlinear dynamic analysis of rings and cylindrical shells was introduced in (Evensen, 1963, Evensen, 1964, Evensen, 1967, which strongly supported earlier experimental observations of nonlinear effects of large amplitude on vibration frequencies in finite length circular cylindrical shells. Some classic theoretical and experimental developments on nonlinear dynamics of circular rings and cylindrical shells were also published in (Goodier and Mcivor, 1964, Evensen, 1966, Mcivor, 1966, Dowell, 1967, Dowell and Ventres, 1968, Mcivor and Lovell, 1968, Evensen, 1974, Lindberg, 1974. On one hand, the well-known Donnell shallow shell theory (Donnell, 1938) were employed to study the amplitude-frequency characteristics (AFCs) of finite length cylindrical shells (Evensen, 1966, Dowell, 1967, Evensen, 1967, Dowell and Ventres, 1968, which has continuously attracted attentions from scientists (Dowell, 1998, Amabili et al, 1999, Evensen, 1999.…”

Section: Introductionmentioning

confidence: 99%

“…Some classic theoretical and experimental developments on nonlinear dynamics of circular rings and cylindrical shells were also published in (Goodier and Mcivor, 1964, Evensen, 1966, Mcivor, 1966, Dowell, 1967, Dowell and Ventres, 1968, Mcivor and Lovell, 1968, Evensen, 1974, Lindberg, 1974. On one hand, the well-known Donnell shallow shell theory (Donnell, 1938) were employed to study the amplitude-frequency characteristics (AFCs) of finite length cylindrical shells (Evensen, 1966, Dowell, 1967, Evensen, 1967, Dowell and Ventres, 1968, which has continuously attracted attentions from scientists (Dowell, 1998, Amabili et al, 1999, Evensen, 1999. On the other hand, in McIvor and his colleagues' publications on rings and circular cylindrical shells (Goodier and McIvor, 1964, McIvor, 1966, Lovell and McIvor, 1969, vibration features of rings and circular cylindrical shells under nearly uniform radial impulse were characterized by a parametric excitation of axisymmetric and/or asymmetric modes.…”

Section: Introductionmentioning

confidence: 99%

Instability in the transformation between extensional and flexural modes in thin-walled cylindrical shells

Shi

2011

European Journal of Mechanics - A/Solids

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Abstract:With the inclusion of all terms up to the third-order in the membrane strain to consider the geometric nonlinearity in deformation, a third-order implicit model and a fourth-order explicit model for the vibration transformation between extensional and flexural modes in thin-walled cylindrical shells are established and solved numerically. Numerical instability is observed in numerical solutions based on the explicit model. It is found that such numerical instability does not result from the accumulated numerical errors in the numerical integration process, but from the neglecting of higher order terms in the formulation of the problem. With the inclusion of all terms up to the fourth-order in the strain energy, the explicit model can predict a stable vibration history with periodic mode transformations between the two modes. The same 2:1 internal resonance of the vibration mode transformation is predicted by both models. As flexural stress growth is concerned, the two models matches very well during the first group of peaks but lag in phase gradually appears in later group of stress peaks based on the implicit model. This is understood to be resulting from the limited number of terms included in the series expansions based on the explicit model, which allows the most likely excited flexural mode get a larger share of energy transferred from the principle mode and as a result flexural stress arrives at peaks earlier based on the explicit model.

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Modal equations for the nonlinear flexural vibrations of a cylindrical shell

Cited by 143 publications

References 9 publications

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

The Influence of Internal Resonances on the Dynamics of Fluidfilled Cylindrical Shells

Instability in the transformation between extensional and flexural modes in thin-walled cylindrical shells

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