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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: A perturbed nonlinear Schrödinger equation is derived that describes the evolution of the envelope of circularly polarized electromagnetic field that leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude.
Abstract: We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schr\"odinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schr\"odinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude.

15 citations

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TL;DR: The extended Hamilton principle is utilized to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model by spectral decomposition in space, and the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs.
Abstract: We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize the extended Hamilton principle to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model. By spectral decomposition in space, the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs. We use direct numerical integration in the decoupled system, in which we observe the anomalous power-law decay rate of amplitude in the linearized model. We further develop a semi-analytical scheme to solve the nonlinear equations, using method of multiple scales as a perturbation technique. We replace the expensive numerical time integration with a cubic algebraic equation to solve for frequency response of the system. We observe the super sensitivity of response amplitude to the fractional model parameters at free vibration, and bifurcation in steady-state amplitude at primary resonance.

15 citations

Journal ArticleDOI
TL;DR: An identification scheme that exploits the vibration response and generated voltage of an energy harvester is proposed to estimate parameters representing nonlinear piezoelectric coefficients in this article, and the method of multiple scales is used to determine the approximate solution of the response to a direct resonant excitation.
Abstract: An identification scheme that exploits the vibration response and generated voltage of an energy harvester is proposed to estimate parameters representing nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and a piezoelectric layer using the generalized Hamilton's principle and by accounting for mechanical energy, virtual work, and electric enthalpy. We then use the method of multiple scales to determine the approximate solution of the response to a direct resonant excitation. We show that the nonlinear behavior captured by the method of multiple scales as approximate solution and amplitude and phase modulation equations can be used to estimate parameters of the nonlinear piezoelectric constitutive relations.An identification scheme that exploits the vibration response and generated voltage of an energy harvester is proposed to estimate parameters representing nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and a piezoelectric layer using the generalized Hamilton's principle and by accounting for mechanical energy, virtual work, and electric enthalpy. We then use the method of multiple scales to determine the approximate solution of the response to a direct resonant excitation. We show that the nonlinear behavior captured by the method of multiple scales as approximate solution and amplitude and phase modulation equations can be used to estimate parameters of the nonlinear piezoelectric constitutive relations.

15 citations

Journal ArticleDOI
TL;DR: In this paper, a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition, where the generalized Galerkin method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion.
Abstract: In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

15 citations

Journal ArticleDOI
TL;DR: In this paper, the effect of curvature on the vibrations of a slightly curved resonant microbeam has been investigated by means of direct application of the method of multiple scales (a perturbation method).
Abstract: An investigation into the dynamic behavior of a slightly curved resonant microbeam having nonideal boundary conditions is presented. The model accounts for midplane stretching, an applied axial load, and a small AC harmonic force. The ends of the curved microbeam are on immovable simple supports and the microbeam is resting on a nonlinear elastic foundation. The forced vibration response of curved microbeam due to the small AC load is obtained analytically by means of direct application of the method of multiple scales (a perturbation method). The effects of the nonlinear elastic foundation as well as the effect of curvature on the vibrations of the microbeam are examined. It is found that the effect of curvature is of softening type. For sufficiently high values of the coefficients, the elastic foundation and the axial load may suppress the softening behavior resulting in hardening behavior of the nonlinearity. The frequencies and mode shapes obtained are compared with the ideal boundary conditions case and the differences between them are contrasted on frequency-response curves. The frequency response and nonlinear frequency curves obtained may provide a reference for the choice of reasonable resonant conditions, design, and industrial applications of such systems. Results may be beneficial for future experimental and theoretical works on MEMS.

15 citations


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