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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.


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TL;DR: The dynamics of a chain of oscillators coupled by fully nonlinear interaction potentials is investigated, including Newton's cradle with Hertzian contact interactions between neighbors, and a rigorous asymptotic description of small amplitude solutions over large times is given.
Abstract: We investigate the dynamics of a chain of oscillators coupled by fully nonlinear interaction potentials This class of models includes Newton's cradle with Hertzian contact interactions between neighbors By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times The envelope equation leading to approximate solutions is a discrete $p$-Schrodinger equation Our results include the existence of long-lived breather solutions to the original model For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times

14 citations

Journal ArticleDOI
TL;DR: In this paper, a perturbation theory is developed for a system of equations which, when linearized, has a plane wave solution with complex frequency of a small imaginary part, and the governing equation for the amplitude becomes a type of generalized nonlinear Schrodinger equation in three dimensions.
Abstract: Modulation of a nonlinear wave in a dissipative and dispersive medium is considered by the method of multiple scales. The slow variables for the amplitude are determined by the coupling between the nonlinearity of the envelope wave and the dissipative or dispersive effect. A perturbation theory is developed for a system of equations which, when linearized, has a plane wave solution with complex frequency of a small imaginary part. Governing equation for the amplitude becomes a type of generalized nonlinear Schrodinger equation in three dimensions. For spatially periodic case, there may be the case that small dissipation can make the wave grow depending on the initial amplitude. As an illustrative example of the general theory, modulation of the convective mode in a fluid layer heated from below is considered.

14 citations

Journal ArticleDOI
TL;DR: In this paper, the steady-state responses of a pipe conveying fluid with a harmonic component of flow speed superposed on a constant mean value in the supercritical regime were investigated.
Abstract: The work investigates steady-state responses of a pipe conveying fluid with a harmonic component of flow speed superposed on a constant mean value in the supercritical regime. If the flow speed exceeds a critical value, the straight configuration of the pipe becomes unstable and bifurcates into two stable curved configurations. The transverse motion measured from each of the curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation. The Galerkin method is employed to discretize the governing equation into a set of coupled nonlinear ordinary differential equations with gyroscopic terms. For the pipes in the supercritical regime, the method of multiple scales is used to determine the steady-state in the vicinity of two-to-one resonance. In the presence of the internal resonance, the subharmonic, the superharmonic and the summation, and the difference resonances exist due to the pulsating fluid. The amplitude–frequency relationships are established with the emphasis on the effects of the viscosity, the pulsating amplitude, the nonlinearity, and the mean flow speed. Some nonlinear phenomena such as the appearance of the peak or jumps pertaining to modal interaction are demonstrated. The numerical integration results are in agreement with the analytical predictions.

14 citations

Journal ArticleDOI
TL;DR: In this paper, the effects of non-ideal boundary conditions (BCs) on fundamental parametric resonance behavior of fluid conveying clamped microbeams are investigated by considering different system parameters.
Abstract: This study aimed to present the effects of non-ideal boundary conditions (BCs) on fundamental parametric resonance behavior of fluid conveying clamped microbeams. Non-ideal BCs are modelled by using the weighting factor (k). Equations of motion are obtained by using the Hamilton’s Principle. A perturbation technique, method of multiple scales, is applied to solve the non-linear equations of motions. In this study, frequency-response curves of fundamental parametric resonance are plotted and the effects of non-ideal BCs are shown. Besides, instability areas of microbeams under ideal and non-ideal BCs are investigated by considering different system parameters. Numerical results show that instability areas significantly changed by the effect of non-ideal BCs.

14 citations

Journal ArticleDOI
TL;DR: In this article, the authors explored enrichment to the method of multiple scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems, where the enrichment follows from an introduced homotopy parameter, which transitions it from linear to nonlinear behavior as the value varies from zero to one.
Abstract: This article explores enrichment to the method of Multiple Scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems. The enrichment follows from an introduced homotopy parameter in the system governing equation, which transitions it from linear to nonlinear behavior as the value varies from zero to one. This same parameter serves as a perturbation quantity in both the asymptotic expansion and the multiple time scales assumed solution form. Two prototypical nonlinear systems are explored. The first considered is a classical forced Duffing oscillator for which periodic solutions near primary resonance are analyzed, and their stability is assessed, as the strengths of the cubic term, the forcing, and a system scaling factor are increased. The second is a classical forced van der Pol oscillator for which quasiperiodic and subharmonic solutions are analyzed. For both systems, comparisons are made between solutions generated using (a) the enriched Multiple Scales approach, (b) the conventional Multiple Scales approach, and (c) numerical simulations. For the Duffing system, important qualitative and quantitative differences are noted between solutions predicted by the enriched and conventional Multiple Scales. For the van der Pol system, increased solution flexibility is noted with the enriched Multiple Scales approach, including the ability to seek subharmonic (and superharmonic) solutions not necessarily close to the linear natural frequency. In both nonlinear systems, comparisons to numerical simulations show strong agreement with results from the enriched technique, and for the Duffing case in particular, even when the system is strongly nonlinear.

14 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202320
202237
202150
202042
201972
201851