On the modal equations of large amplitude flexural vibration of beams, plates, rings and shells

Pandalai, K.A.V.; Sathyamoorthy, M.

doi:10.1016/0020-7462(73)90044-9

Cited by 8 publications

(9 citation statements)

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“…It is a well-known fact that flat plates display a hardening behaviour, as it has been shown both theoretically and experimentally (see e.g. [4][5][6][7][8][9]). Introducing an initial curvature in the middle surface of the structure creates a quadratic nonlinearity, which, in turn, may change the non-linear behaviour to softening type, depending on the balance of the magnitude of quadratic and cubic terms [6,10,11].…”

Section: Introductionmentioning

confidence: 89%

“…[4][5][6][7][8][9]). Introducing an initial curvature in the middle surface of the structure creates a quadratic nonlinearity, which, in turn, may change the non-linear behaviour to softening type, depending on the balance of the magnitude of quadratic and cubic terms [6,10,11]. It is thus a legitimate question to determine the correct non-linear behaviour of shallow spherical shells, and more precisely, the transition from the hardening (flat plate) behaviour to the softening one, as the curvature increases.…”

Section: Introductionmentioning

confidence: 97%

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Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Touzé

Thomas

2006

International Journal of Non-Linear Mechanics

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International audienceNon-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2 a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R → ∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified. © 2006 Elsevier Ltd. All rights reserved

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 97%

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Touzé

Thomas

2006

International Journal of Non-Linear Mechanics

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“…The method shown previously for predicting accurately the type of nonlinearity, is now applied to the case of spherical-cap thin shallow shells with a varying radius of curvature R. Flat plates are known to exhibit a hardening behaviour,as it has been shown both theoretically and experimentally (see e.g. [48,58,35,44,54,46]), which means that the leading cubic coefficient h p ppp is positive. Introducing a radius of curvature R, going to infinity (perfect plate) to finite values (spherical-cap shells) introduces an asymmetry in the restoring force, due to the loss of symmetry of the neutral plane of the shell.…”

Section: Application To Shellsmentioning

confidence: 99%

Normal form theory and nonlinear normal modes: Theoretical settings and applications

Touzé

2014

CISM International Centre for Mechanical Sciences

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International audienceThese lecture notes are related to the CISM course on ”Modal Analysis of nonlinear Mechanical systems”, held at Udine, Italy, from June 25 to 29, 2012. The key concept at the core of all the lessons given during this week is the notion of Nonlinear Normal Mode (NNM), a theoretical tool allowing one toextend, through some well-chosen assumptions and limitations, the linear modes of vibratory systems, tononlinear regimes. More precisely concerning these notes, they are intended to show the explicit link between Normal Form theory and NNMs, for the specific case of vibratory systems displaying polynomialtype nonlinearities. After a brief introduction reviewing the main concepts for deriving the normal formfor a given dynamical system, the relationship between normal form theory and nonlinear normal modes(NNMs) will be the core of the developments

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“…Using the Galerkin method and assumed modes, Pandalai and Sathyamoorthy [131] analyzed the non-linear #exural vibrations of thin elastic orthotropic oval cylindrical shells. Kvasha [132] studied the forced vibrations of three-layer plates and shells made from physically non-linear materials.…”

Section: Closed and Open Composite Shellsmentioning

confidence: 99%

Non-Linear Vibrations of Shell-Type Structures: A Review With Bibliography

Moussaoui¹,

Benamar²

2002

Journal of Sound and Vibration

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On the modal equations of large amplitude flexural vibration of beams, plates, rings and shells

Cited by 8 publications

References 0 publications

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Normal form theory and nonlinear normal modes: Theoretical settings and applications

Non-Linear Vibrations of Shell-Type Structures: A Review With Bibliography

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