scholarly journals Wavenumber-space band clipping in nonlinear periodic structures

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210052
Author(s):  
Weijian Jiao ◽  
Stefano Gonella
Keyword(s):  
Dispersion Relation ◽  
Multiple Scales ◽  
Periodic Structures ◽  
Dispersion Correction ◽  
Cubic Nonlinearity ◽  
Frequency Shifts ◽  
Weakly Nonlinear ◽  
Practical Applications ◽  
Correction Effect ◽  
Main Effect

In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as an initial condition or as a boundary condition, respectively. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. We then extend the framework to locally resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.

Higher-Order Multiple Scales Analysis of Weakly Nonlinear Lattices With Implications for Directional Stability

2018 ◽  
Author(s):  
Matthew D. Fronk ◽  
Michael J. Leamy
Keyword(s):  
Acoustic Waves ◽  
Local Stability ◽  
Multiple Scales ◽  
Periodic Structures ◽  
Plane Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Springs ◽  
Nonlinear Lattices ◽  
Multiple Scales Analysis ◽  
The Stability

Recent focus has been given to nonlinear periodic structures for their ability to filter, guide, and block elastic and acoustic waves as a function of their amplitude. In particular, two-dimensional (2-D) nonlinear structures possess amplitude-dependent directional bandgaps. However, little attention has been given to the stability of plane waves along different directions in these structures. This study analyzes a 2-D monoatomic shear lattice composed of discrete masses, linear springs, quadratic and cubic nonlinear springs, and linear viscous dampers. A local stability analysis informed by perturbation results retained through the second order suggests that different directions become unstable at different amplitudes in an otherwise symmetrical lattice. Simulations of the lattice’s equation of motion subjected to both line and point forcing are consistent with the local stability results: waves with large amplitudes have spectral growth that differs appreciably at different angles. The results of this analysis could have implications for encryption strategies and damage detection.

Transcritical two-layer flow over topography

1987 ◽  
Vol 178 ◽  
pp. 31-52 ◽  
Author(s):  
W. K. Melville ◽  
Karl R. Helfrich
Keyword(s):  
Steady Flow ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Single Layer ◽  
Cubic Nonlinearity ◽  
Arbitrary Length ◽  
Weakly Nonlinear ◽  
Flow Over Topography ◽  
Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

scholarly journals Heat transfer in the seabed boundary layer

2021 ◽  
Vol 928 ◽  
Author(s):  
S. Michele ◽  
R. Stuhlmeier ◽  
A.G.L. Borthwick
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Free Surface ◽  
Surface Waves ◽  
Multiple Scales ◽  
Perturbation Expansion ◽  
Phase Speed ◽  
Maximum Heat ◽  
Practical Applications ◽  
Free Surface Waves

We present a theoretical model of the temperature distribution in the boundary layer region close to the seabed. Using a perturbation expansion, multiple scales and similarity variables, we show how free-surface waves enhance heat transfer between seawater and a seabed with a solid, horizontal, smooth surface. Maximum heat exchange occurs at a fixed frequency depending on ocean depth, and does not increase monotonically with the length and phase speed of propagating free-surface waves. Close agreement is found between predictions by the analytical model and a finite-difference scheme. It is found that free-surface waves can substantially affect the spatial evolution of temperature in the seabed boundary layer. This suggests a need to extend existing models that neglect the effects of a wave field, especially in view of practical applications in engineering and oceanography.

Method of Complex Amplitudes: Harmonically Excited Oscillator With Strong Cubic Nonlinearity

2003 ◽  
Author(s):  
O. V. Gendelman ◽  
L. I. Manevitch
Keyword(s):  
Multiple Scales ◽  
Duffing Oscillator ◽  
Necessary Condition ◽  
Cubic Nonlinearity ◽  
Slow Time ◽  
Oscillatory Systems ◽  
Exact Procedure ◽  
Excited Oscillator ◽  
Secular Terms ◽  
Approximation Criterion

Application of method of complex amplitudes for investigation of dynamics of nonlinear oscillatory systems is considered. The method is analyzed in details for harmonically forced Duffing oscillator without damping for the case of exact 1:1 resonance. It is demonstrated that the method of complex amplitudes may be formalized as a generalization of standard multiple–scales procedure. Two alternatives for elimination of secular terms in high-order approximations are proposed and compared for various values of small parameter. It is demonstrated that regardless the exact procedure chosen the necessary condition to avoid secularities is the appropriate choice of functions determining the correction of the slow time scale in any approximation. Criterion for use of the method of complex amplitudes is formulated. All analytic results are compared with data of numerical simulation and considerable agreement is observed.

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

Lattice Boltzmann Study of Flow and Temperature Structures ofNon-Isothermal Laminar Impinging Streams

2013 ◽  
Vol 13 (3) ◽  
pp. 835-850 ◽  
Author(s):  
Wenhuan Zhang ◽  
Zhenhua Chai ◽  
Zhaoli Guo ◽  
Baochang Shi
Keyword(s):  
Boundary Condition ◽  
Reynolds Number ◽  
Temperature Field ◽  
Prandtl Number ◽  
Lattice Boltzmann ◽  
Periodic Structures ◽  
Temperature Fields ◽  
Buoyancy Effect ◽  
Practical Applications ◽  
Boltzmann Method

AbstractPrevious works on impinging streams mainly focused on the structures of flow field, but paid less attention to the structures of temperature field, which are very important in practical applications. In this paper, the influences of the Reynolds number (Re) and Prandtl number (Pr) on the structures of flow and temperature fields of non-isothermal laminar impinging streams are both studied numerically with the lattice Boltzmann method, and two cases with and without buoyancy effect are considered. Numerical results show that the structures are quite different in these cases. Moreover, in the case with buoyancy effect, some new deflection and periodic structures are found, and their independence on the outlet boundary condition is also verified. These findings may help to understand the flow and temperature structures of non-isothermal impinging streams further.

The second-harmonic resonance for nonlinear hydromagnetic waves

1975 ◽  
Vol 14 (1) ◽  
pp. 53-64 ◽  
Author(s):  
Yasuji Matsumoto ◽  
Nobumasa Sugimoto ◽  
Yoshinori Inoue
Keyword(s):  
Multiple Scales ◽  
Second Harmonic ◽  
Geometric Mean ◽  
Collisionless Plasma ◽  
Periodic Wave ◽  
Dynamical Equations ◽  
Weakly Nonlinear ◽  
Acoustic Modes ◽  
Hydromagnetic Waves ◽  
Harmonic Resonance

We investigate second-harmonic resonance for weakly nonlinear hydromagnetic waves travelling in a cold collisionless plasma by the method of multiple scales. We find that the second-harmonic resonance can occur between the magneto-acoustic modes; but it can occur neither between the magneto-acoustic and the Alfvé n modes, nor between the Alfvé n modes. The resonant frequency of the magneto-acoustic modes is characterized by the geometric mean of the ion and electron Larmor frequencies. We obtain steady-state solutions to the dynamical equations governing the second-harmonic resonance. The result of analysis shows that the envelopes of the two resonant waves are composed of two periodic wave- trains, two solitary pulses or a solitary pulse and a phase jump. We also extend the problem to more general dispersive wave systems.

scholarly journals Robustness Analysis of the Collective Dynamics of Nonlinear Periodic Structures Under Parametric Uncertainty

2016 ◽  
Author(s):  
Khaoula Chikhaoui ◽  
Diala Bitar ◽  
Najib Kacem ◽  
Noureddine Bouhaddi
Keyword(s):  
Multiple Scales ◽  
Periodic Structures ◽  
Uncertainty Propagation ◽  
Time Integration ◽  
Robustness Analysis ◽  
Parametric Uncertainty ◽  
Latin Hypercube Sampling ◽  
Algebraic Equations ◽  
Collective Dynamics ◽  
Model Combining

In order to ensure more realistic design of nonlinear periodic structures, the collective dynamics of a coupled pendulums system is investigated under parametric uncertainties. A generic discrete analytical model combining the multiple scales method, the perturbation theory and a standing-wave decomposition is proposed and adapted to the presence of uncertainties. These uncertainties are taken into account through a probabilistic modeling implying that the stochastic parameters vary according to random variables of chosen probability density functions. The proposed model leads to a set of coupled complex algebraic equations written according to the number and positions of the uncertainties in the structure and numerically solved using the time integration Runge-Kutta method. The uncertainty propagation through the established model is finally ensured using the Latin Hypercube Sampling method. The analysis of the dispersion, in term of variability of the frequency and amplitude intervals of the multistability domain shows the effects of uncertainties on the stability and nonlinearity of a three coupled pendulums structure. The nonlinear aspect is strengthened, the multistability domain is wider, more stable branches are obtained and thus the multimode solutions are enhanced.

The development of wavepackets in unstable flows

1981 ◽  
Vol 373 (1755) ◽  
pp. 457-476 ◽  
Keyword(s):  
Dispersion Relation ◽  
Multiple Scales ◽  
Three Dimensional ◽  
Fourier Integral ◽  
Saddle Point Method ◽  
Alternative Derivation ◽  
Closed Form Solutions ◽  
Linear Evolution ◽  
Multiple Scales Technique ◽  
Complex Dispersion

By using a model form of the complex dispersion relation for unstable flows, the linear evolution of a localized three-dimensional wavepacket is determined. The disturbance is expressed as a double Fourier integral which is evaluated asymptotically by the saddle-point method. On making certain approximations, simple closed-form solutions are obtained, some of which resemble the curved wavepackets observed by Gaster & Grant (1975) and some the ‘ elliptic ’ packet found by Benjamin (1961). The range of validity of theories which lead to an elliptic packet is clarified. An alternative derivation of some of the results is given by using a multiple-scales technique. The relative merits of the two methods are thereby illuminated.

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