The size of the bounded domain plays an important role during occurrence of qualitative distinct phenomena in nonlinear dynamical system. The multimode dispersive waves were generated by spatio-temporal oscillations of solitons in multimode fibre. Roman Starosta is Professor at the Institute of Applied Mechanics, Poznan University of Technology , Poland, where he is the head of the Department of Technical Mechanics. His area of research includes dynamics of structures, fluid mechanics, asymptotic methods, and computational systems of algebra. Professor Starosta is a member of the main board of the Polish Society of Theoretical and Applied Mechanics, and chairman of several editions of the conference on Vibrations in Physical Systems.

Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, as described next. This term is O and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution. It will be of interest to engineers and professionals in mechanical engineering and structural engineering, alongside those interested in vibrations and dynamics. Furthermore, for strong nonlinearities with strong damping effect, the absolute relative error found in this article is only 0.02%, whereas the relative error obtained by MSLP method is 24.18%.

An introduction and tutorial on multiple-scale analysis in solids

Subsequent slip relaxation due to the uncoiling of entanglements will lead to significant strain softening, giving the DE a higher susceptibility to mechanically and electro-mechanically-induced instabilities. From our analyses, we produced mechanical stability phase plots and electromechanical stability design plots, for elastomers of different sliplink-to-crosslink compositions. Such plots may serve as guides to develop dielectric elastomers multi-scale analysis with desired material properties and performances, and tailor to specific applications. This paper presents a new semi-concurrent multi-scale model to study the behaviour of composite materials in softening regime. The traction over the crack is included as a unknown field in the equations system of the problem, and the jump displacement across the discontinuity is obtained with a cohesive constitutive relation (traction-separation law).

Hence, the multiple-scale expansions were used, and it was found that this resonating frequencies cannot excite the breathing mode if the amplitude is small enough. However, if the external drive amplitude is large enough, the junction can switch to the resistive state. You can use these two equations and the initial conditions to determine the leading order solution, you get a combination of a neither-fast-nor-slow decaying exponential and a fast decaying exponential. We propose a continuum model for the description of the dynamics of isolated macromolecules.

Multiple scale analysis

Further, the RBMs arise only in the presence of free-free boundary conditions , while they do not appear in other boundary conditions , see, for example, Strozzi et al. . Also, RBMs are studied experimentally by resonant Raman spectroscopy and numerically by molecular dynamics simulations, see, for example, Araujo et al. , Batra and Gupta , and Rehman et al. , and Ahmad . In this paper we formulate a Laplace-Transform multi-scale expansion procedure to develop asymptotic solutions of weakly nonlinear partial di erential equations. The method is applied to some general non-linear wave and di usion equations. Multiple-scale analysis is a global perturbation scheme that is useful in systems characterized by disparate time scales, such as weak dissipation in an oscillator.

The projected stress over the normal vector of the macro discontinuity is injected in the localized domain in the RVE, obtaining as a dual variable the jump of the displacement field in the macro structure. In this way, during the stable phase of the behaviour, the scale transition is performed in the classical way injecting the strain tensor and obtaining the stress tensor as a dual variable. This new concept leads to a new multi-scale approach with an hybrid injection.

Asymptotic Multiple Scale Method in Time Domain Summary

Finally, we concluded that the modes of the breathing decay to a constant in both cases. Conventional weak-coupling Rayleigh-Schrödinger perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion and the Schrödinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function.

So far, we have obtained the leading-order behavior of the breathing amplitude from and , which is our main objective in this work. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Also, just for the OP’s benefit, this approach that Felix has outlined is called the WKB method.

Motivation for multiple-scale analysis

The use of the symbol \(E\) is deliberate since the Duffing oscillator is a Hamiltonian system with total energy \(E\) given by .

• Also, RBMs are studied experimentally by resonant Raman spectroscopy and numerically by molecular dynamics simulations, see, for example, Araujo et al. , Batra and Gupta , and Rehman et al. , and Ahmad .
• In the paper they considered the direct driven vase, while ours is the parameter with extra phase shift.
• If you want to connect the characteristic roots to the roots of the unperturbed problem then something similar has to be done.
• The interfacial strength due to fibre pullout predicted by the hybrid atomistic-FE model is compared against experimental and molecular dynamics results available in open literature.
• He has authored 850 journal papers and is Editor-in-Chief of three international journals.
• In this paper, two simplified concurrent multiscale methods, one with handshake region and another without handshake region are used to investigate the nanoindentation process on a single crystal of Al at room temperature.

This dependence was found to be an evidence of existence of the relation between DNA functioning and dynamics. The numerical simulations of Gross–Pitaevskii equations were performed to predict spatially localized and temporarily oscillating nonlinear excitation’s. These results resemble with the solutions of sine-Gordon equation known as breather with a difference of slow decay has been explained in the study by Su et https://wizardsdev.com/ al. . An atomistic multiscale modelling approach is used to simulate the nonlinear pullout behaviour of interlinked single walled carbon nano tubes and single layer graphene sheets embedded in an epoxy polymer. The pullout forces have been computed for various configurations of nanocomposites (SWCNT-SWCNT, SLGS-SLGS and hybrid SLGS-SWCNT), also by evaluating the effect provided by three different interlink compounds.

Title:Multiple-Scale Analysis of Quantum Systems

In mathematics and physics, multiple-scale analysis comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. We use a four-parameter material model for polymers – the Edward–Vilgis model, which models polymeric crosslinks, sliplinks, slippage and inextensibility to analyze dielectric elastomers .