Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Electrohydrodynamic stability of two cylindrical interfaces under the influence of a tangential periodic electric field

Abstract: The electrohydrodynamic stability of two cylindrical interfaces influenced by a periodic tangential field is studied. The model allows for general forms of deformations of the interfaces. Two simultaneous ordinary differential equations of the Mathieu type are obtained. The coupled equations are solved by the method of multiple scales and stability conditions are discussed. It is found that the constant tangential field has a stabilizing effect while the tangential periodic field has a stabilizing influence except at resonance points. Graphs are drawn to illustrate the resonance regions in a parameter space. It is also found that the thickness of the jet plays a role in the stability criterion. The frequency of the modulated field can be used to control the position of the resonance regions. The special cases of large modulation and small modulation are also examined. It is found that for large modulation the electric field exhibits an enhanced destabilizing influence.

Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions

Abstract: The non-linear response of a base-excited slender beam carrying a lumped mass subjected to principal parametric resonance is investigated. The attached mass and its location are so adjusted that the system exhibits 1:3:5 internal resonances. Method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and stability of the system. The steady-state response thus obtained is compared with those found by single- and two-mode analyses and very significant differences are observed in the bifurcation and stability of the response curves. The effects of external and internal detuning, amplitude of excitation and damping on the non-linear steady state, periodic, quasi-periodic and chaotic responses of the system are investigated. Funnel-shaped chaotic orbits, fractal orbits, cascade of period-doubling, torus doubling and intermittency routes to chaos are observed in this system. A simple illustration is given to control chaos by changing the system parameters.

Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom

Abstract: In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions

Localization of harmonics generated in nonlinear shallow water waves.

Abstract: The propagation of nonlinear shallow water waves over a random seabed is studied. A bathymetry which fluctuates randomly from a constant mean adds multiple scattering to resonant interactions and harmonic generation. By the method of multiple scales, nonlinear evolution equations for the harmonic amplitudes are derived. Effects of multiple scattering are shown to be represented by certain linear damping terms with complex coefficients related to the correlation function of the seabed disorder. For any finite number of harmonics, an equation governing the total wave energy is derived. By numerical solution of the amplitude equations, the effects of spatial attenuation (localization) on harmonic generation are studied.