Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves

Results of an experimental and numerical study of parametrically excited nonlinear cross-waves in the vicinity of the cut-off frequency, are reported. Experiments are performed at three cross-wave modes and in the whole range of existence of cross-waves. Numerical studies are based on the solution of the nonlinear Schrödinger equation with a boundary condition at the wavemaker which corresponds to parametric excitation. The validity of the scaling procedure adopted in the model is verified experimentally. Dissipation is incorporated in the model equation and in the wavemaker boundary condition. The influence of the wave breaking on the range of existence of cross-waves is discussed and the relation between the maximum possible steepness of cross-waves and the limits of their existence is obtained.

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Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Acta Mechanica ◽

10.1007/s00707-004-0176-0 ◽

2005 ◽

Vol 175 (1-4) ◽

pp. 139-179

Author(s):

R. T. Hudspeth ◽

R. B. Guenther ◽

S. Fadel

Keyword(s):

Surface Tension ◽

Parametrically Excited ◽

Cross Waves

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Parametrically excited, standing cross-waves

Journal of Fluid Mechanics ◽

10.1017/s0022112088000060 ◽

1988 ◽

Vol 186 ◽

pp. 119-127 ◽

Cited By ~ 23

Author(s):

John Miles

Keyword(s):

Surface Waves ◽

Phase Plane ◽

Evolution Equations ◽

Plane Waves ◽

Wave Problem ◽

Parametrically Excited ◽

The Cross ◽

The Stability ◽

Cross Waves

Luke's (1967) variational formulation for surface waves is extended to incorporate the motion of a wavemaker and applied to the cross-wave problem. Whitham's average-Lagrangian method then is invoked to obtain the evolution equations for the slowly varying complex amplitude of the parametrically excited cross-wave that is associated with symmetric excitation of standing waves in a rectangular tank of width π/k, length l and depth d for which kl = O(1) and kd [Gt ] 1. These evolution equations are Hamiltonian and isomorphic to those for parametric excitation of surface waves in a cylinder that is subjected to a vertical oscillation, for which phase-plane trajectories, stability criteria and the effects of damping are known (Miles 1984a). The formulation and results differ from those of Garrett (1970) in consequence of his linearization of the boundary condition at the wavemaker and his neglect of self-interaction of the cross-waves in the free-surface conditions (although Garrett does incorporate self-interaction in his calculation of the equilibrium amplitude of the cross-waves). These differences have only a small effect on the criterion for the stability of plane waves, but the self-interaction is crucial for the determination of the stability of the cross-waves.

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MULTI-PULSE HETEROCLINIC ORBITS AND CHAOTIC MOTIONS IN A PARAMETRICALLY EXCITED VISCOELASTIC MOVING BELT

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500016 ◽

2013 ◽

Vol 23 (01) ◽

pp. 1350001 ◽

Cited By ~ 6

Author(s):

MINGHUI YAO ◽

WEI ZHANG ◽

JEAN W. ZU

Keyword(s):

Normal Form ◽

Numerical Simulations ◽

Chaotic Dynamics ◽

Dissipative System ◽

Melnikov Method ◽

Heteroclinic Orbits ◽

Unperturbed System ◽

Chaotic Motions ◽

Parametrically Excited ◽

Extended Melnikov Method

This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for nonlinear, nonplanar oscillations of the parametrically excited viscoelastic moving belts by using an extended Melnikov method in the resonant case. Applying the method of multiple scales, the Galerkin's approach and the theory of normal form, the explicit normal form is obtained for the case of 1:1 internal resonance and primary parametric resonance. Studies are performed for the heteroclinic bifurcations of the unperturbed system and for the characteristics of the hyperbolic dynamics of the dissipative system, respectively. The extended Melnikov method is used to investigate the Shilnikov type multi-pulse bifurcations and chaotic dynamics of the viscoelastic moving belt. Based on the investigation, the geometric structure of the multi-pulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multi-pulse chaotic motions can occur. Furthermore, numerical simulations lead to the discovery of the new shapes of chaotic motion. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

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Multi-Pulse Heteroclinic Orbits With a Melnikov Method and Chaotic Dynamics in Motion of Parametrically Excited Viscoelastic Moving Belt

Volume 9: Mechanical Systems and Control, Parts A, B, and C ◽

10.1115/imece2007-41740 ◽

2007 ◽

Author(s):

Ming-Hui Yao ◽

Wei Zhang ◽

Dong-Xing Cao

Keyword(s):

Normal Form ◽

Chaotic Dynamics ◽

Multiple Scales ◽

Melnikov Method ◽

Equation Of Motion ◽

Constitutive Law ◽

Heteroclinic Orbits ◽

Global Dynamics ◽

Chaotic Motions ◽

Parametrically Excited

The multi-pulse heteroclinic orbits and chaotic dynamics of a parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equation of motion for viscoelastic moving belt with the external damping and parametric excitation are determined. The four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of motion for viscoelastic moving belt. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, an extension of the Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for a parametrically excited viscoelastic moving belt. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse heteroclinic orbits of viscoelastic moving belts are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for a parametrically excited viscoelastic moving belt.

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