Local and global bifurcations in parametrically excited vibrations of nearly square plates

Yang, Xing; Sethna, P. R.

doi:10.1016/0020-7462(91)90052-u

Cited by 49 publications

(33 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This leads to backbone curves with loops, additional branches and bifurcation points. One-to-one resonances were also studied with parametric excitation, see reference [34] for the case of a nearly square plate with parametric in-plane excitation, or reference [35] for a review of the literature.…”

Section: Analytical Perturbative Solutionmentioning

confidence: 99%

Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory

Touzé¹,

Thomas²,

Chaigne³

2002

Journal of Sound and Vibration

103

100

View full text Add to dashboard Cite

In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von K" a arm" a an equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency o kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to o kn ; which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper. #

show abstract

Section: Analytical Perturbative Solutionmentioning

confidence: 99%

Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory

Touzé¹,

Thomas²,

Chaigne³

2002

Journal of Sound and Vibration

103

100

View full text Add to dashboard Cite

show abstract

“…Via the Smale-Birkhoff homoclinic theorem (Guckenheimer and Holmes [1983]), this situation leads to chaotic dynamics of phase points in the vicinity Orbits homoclinic to resonance bands arise in many physical models, particularly those models that are obtained by transforming coordinates into a rotating reference frame and averaging over a fast phase. Examples of such models are Bishop, Flesch, Forest, McLaughlin, and Overman [1990] in the theory of Josephson's junctions; David, Holm, and Tratnik [1990] in nonlinear fiber optics; Holm and Kovačič [1992] in laser-matter interaction; Holmes [1986], Gu and Sethna [1987], and Feng and Sethna [1989], [1990] in the theory of water waves; and Yang and Sethna [1991] in the theory of vibrating plates. All the resonant periodic motions of such physical models, that is, periodic motions whose frequencies coincide with the frequency of the rotating frame, become circles of equilibria.…”

Section: Introductionmentioning

confidence: 99%

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

Kovačič

1993

J Dyn Diff Equat

View full text Add to dashboard Cite

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.

show abstract

“…non-linear oscillators with Z Z -symmetry and analyzed the global behavior of averaged equations. The results obtained in reference [2] indicated that the heteroclinic loops exist and Smale horse and chaotic motions can occur. Based on the studies in reference [2], Feng and Sethna [3] made use of a global perturbation method developed by Kovacic and Wiggins [4] to study further the global bifurcations and chaotic dynamics of thin plate under parametric excitation, and obtained the conditions in which Silnikov-type homoclinic orbits and chaos can occur.…”

Section: Introductionmentioning

confidence: 94%

“…non-linear system and used center manifolds, theory of normal forms to study the degenerate bifurcations. Yang and Sethna [2] used the averaging method to study the local and global bifurcations in parametrically excited nearly square plate. From van Karman equation, they simpli"ed this system to a parametrically excited two-d.o.f.…”

Section: Introductionmentioning

confidence: 99%

Global and Chaotic Dynamics for a Parametrically Excited Thin Plate

Zhang¹

2001

Journal of Sound and Vibration

View full text Add to dashboard Cite

Local and global bifurcations in parametrically excited vibrations of nearly square plates

Cited by 49 publications

References 11 publications

Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory

Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

Global and Chaotic Dynamics for a Parametrically Excited Thin Plate

Contact Info

Product

Resources

About