Corrected Solvability Conditions for Non-Linear Asymmetric Vibrations of a Circular Plate

Yeo, M.H.; Lee, W.K.

doi:10.1006/jsvi.2002.5037

Cited by 16 publications

(6 citation statements)

References 6 publications

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“…However, they are of significant importance for shells with small curvature only, since cubic non-linear terms become negligible with respect to quadratic terms for large curvatures (e c ( e q if R is small compared to a, see Table 1). The case of Table 5 has been recently addressed by Lee et al (2003) and cases of Tables 6 and 7 extend earlier results of Sridhar et al, 1978), corrected by Yeo and Lee (2002), in the case of circular plates. Table 5 Rules determining if modal interaction between modes U 1 and U 2 is possible, when internal resonance x 2 = 3x 1 is fulfilled…”

Section: Order-three Internal Resonancesmentioning

confidence: 80%

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

Thomas¹,

Touzé²,

Chaigne³

2005

International Journal of Solids and Structures

116

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This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von KármánÕs theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.

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Section: Order-three Internal Resonancesmentioning

confidence: 80%

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

Thomas¹,

Touzé²,

Chaigne³

2005

International Journal of Solids and Structures

116

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“…We can see that there exists an analogy between this problem and asymmetric vibration of circular plates [24][25][26][27][28][29][30][31]. It is meaningful to note that while the circular plates have geometric nonlinearity only (a 3 = 0), the cantilever has geometric and inertial nonlinearities (a 3 = 0).…”

Section: Equations Of Motionmentioning

confidence: 99%

Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

Lee

Pak

2007

Nonlinear Dyn

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Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming nonplanar motion of the beam, we introduce a nonlinear twodegree-of-freedom model by using Galerkin's method based on the first mode in each direction. The system turns out to have two normal modes. Using Synge's stability concept, we examine the stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion obtained analytically.

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“…For studying asymmetric responses, Sridhar, Mook, and Nayfeh [9] derived a general solvability condition for nonlinear interactions in the vibrations of a clamped circular plate. Yeo and Lee [10] re-examined a primary resonance state studied by Sridhar et al and corrected the modulation equations derived by Sridhar et al The results indicate the steady-state response can have not only the form of a standing wave but also the form of a traveling wave. In this paper, the authors follow the work presented in references [9][10][11] and build on their earlier efforts [5,8], and carry out a nonlinear analysis of the asymmetric vibrations of a pressure sensor diaphragm under initial tension.…”

Section: Introductionmentioning

confidence: 97%

Sensor Diaphragm Under Initial Tension: Nonlinear Interactions During Asymmetric Oscillations

Long

Balachandran

2008

Volume 11: Mechanical Systems and Control

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In this article, investigations into the nonlinear asymmetric vibrations of a pressure sensor diaphragm under initial tension are presented. A comprehensive mechanics model based on a plate with in-plane tension is presented and the effect of cubic nonlinearity is studied on the nonlinear asymmetric response when the excitation frequency is close to the natural frequency of an asymmetric mode of the plate. The obtained results show that in the presence of an internal resonance, depending on the initial tension, the response can have not only the form of a standing wave but also the form of a traveling wave. The results of this work should be relevant to diaphragm-type structures used in micro-scale sensors including pressure sensors.

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Corrected Solvability Conditions for Non-Linear Asymmetric Vibrations of a Circular Plate

Cited by 16 publications

References 6 publications

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

Sensor Diaphragm Under Initial Tension: Nonlinear Interactions During Asymmetric Oscillations

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